ELECTRONICS IN NUCLEAR PHYSICS*)

W. C. Elmore

Most of the measuring instruments currently used in nuclear physics contain electronic circuits. As a result of intensive work on the design of measuring instruments, as well as on the development of new and the improvement of old radar circuits and other circuits that found application during the war, the level of electronic instrumentation has risen considerably and has come to include new areas.

If one turns to such problems as monitoring with the aid of electronic instruments, accelerators, and electronic circuits for the study of cosmic radiation, then a sufficiently complete description of all the electronic instruments used here would require at least an entire book.

The present article does not claim to be so complete an exposition. Its aim is to give a brief account of several sections of electronic theory and to acquaint the reader with certain ideas underlying the calculation of special circuits that are very useful for researchers working in the field of nuclear physics.

Since the size of the review is limited, not all typical circuits are considered. In particular, the circuits originally used with Geiger–Müller counters are not discussed, nor are direct-current amplifiers. The main attention is devoted to the theory and calculation of circuits intended for amplifying and recording pulses from an ionization chamber or a proportional counter that collects electrons. It is precisely in this area that a number of substantial advances achieved in recent years by instrumentation connected with nuclear physics belong. The author hopes that the present exposition will prove useful for researchers working in the field of phy-

) W. C. Elmore, “Electronics for the nuclear physicist,” Nucleonics 2*, 16 (1948). Translated by V. A. Troitskaya.

knowledge of the nucleus and who expect to use modern techniques. In addition, the author hopes that this survey will be of interest also to persons working on the design of electronic instruments, in particular instruments for research in nuclear physics.

I. THE LAPLACE TRANSFORM METHOD

For calculating and understanding the operation of circuits containing electronic devices, it is necessary to have some information about transient phenomena in electric circuits. The most convenient method for analyzing transient phenomena is based on the Laplace transform. A complete treatment of this method would require considerable space. Nevertheless, we shall try to present, in concise form, material sufficient for solving most problems concerning transient phenomena that arise in the calculation of electronic circuits. The typical problem that we shall consider is the following: in a circuit consisting of inductances, capacitances, resistances, and other elements whose action is essentially linear (for example, electron tubes operating in class-A mode), up to the instant of time \(t = 0\) there are no disturbances of any kind. At this instant, in some section of the circuit, a source of e.m.f. or current begins to act. Then currents begin to flow in the circuit, voltages appear, and so on. The source of disturbance, in the general case, may be denoted as the “cause” \(c(t)\). We shall assume, therefore, that \(c(t)\) is always equal to zero for \(t < 0\). Apart from this condition, no restrictions are imposed on the form of the function \(c(t)\) (except those determined by very broad mathematical requirements, always satisfied in practical cases).

The action that \(c(t)\) produces in some part of the circuit may be called the “effect” \(e(t)\). This effect includes a component corresponding to the steady-state regime and a component corresponding to the transient regime, gradually dying out with time. The classical method of analyzing transient phenomena usually consists in finding the general solution of the differential equations of the circuit under consideration and determining arbitrary constants on the basis of the conditions existing at the instant of time \(t = 0\) (initial conditions). This method, however, often requires great labor in comparison with the Laplace method, to which we now turn.

The Laplace method is based on the functional transform

\[ F(s)=\int_{0}^{\infty} f(t)e^{-st}\,dt, \tag{1} \]

where the function of time \(f(t)\) is equal to 0 for \(t<0\), and the variable \(s=\sigma+i\omega\) is a complex quantity, the real part of which, \(\sigma\), assumes positive values sufficient to ensure the absolute convergence of the integral (1). For any function of time \(f(t)\), equation (1) uniquely determines a certain function \(F(s)\), called the “image” of the function \(f(t)\). Conversely, it can be shown that to any function \(F(s)\) there corresponds a unique function of time \(f(t)\) satisfying equation (1). The function \(f(t)\) is called the original function for \(F(s)\).

There are exact requirements concerning the behavior of the function \(f(t)\) at infinity, which we shall not give. One of the essential advantages of transformation (1) is that complicated functions with discontinuities, often consisting of several parts, each of which has a different analytic expression, are transformed into simple continuous functions. As a first example, let us consider the so-called unit function (“step”), defined as follows:

\[ u(t)=0 \text{ for } t<0;\quad u(t)=1 \text{ for } t \geq 0. \]

The image of this function has the form

\[ F(s)=\int_0^\infty e^{-st}\,dt=\frac{1}{s}; \]

\[ \sigma \gg c>0. \]

As a second example, consider the function

\[ f(t)=0 \quad \text{for } t<0; \]

\[ f(t)=e^{-at} \quad \text{for } t \geq 0. \]

In this case

\[ F(s)=\int_0^\infty e^{-(a+s)t}\,dt=\frac{1}{s+a}; \]

\[ \sigma \gg c>\operatorname{Re} a. \]

Tables in which the images \(F(s)\) are given for various original functions \(f(t)\) can be found in standard handbooks.

These tables should be used if it is necessary to find \(F(s)\) for a given \(f(t)\), and conversely. Some pairs of the most frequently occurring transforms are given in Table I. We note that part of these pairs has been obtained by carrying out Laplace transformations on operations.

Table 1

Some initial functions and their images obtained by means of the Laplace transform

\(f(t)\)	\(F(s)\)
\(a f_1(t)+b f_2(t)\)	\(aF_1(s)+bF_2(s)\)
\(\dfrac{d f(t)}{dt}\)	\(sF(s)-f(+0)^{*)}\)
\(\displaystyle \int f(t)\,dt\)	\(\dfrac{F(s)}{s}+\dfrac{f(+0)}{s}\)
\(e^{-at}f(t)\)	\(F(s+a)\)
\(f(t-a)\)	\(e^{-as}F(s)\)
\(0\) for \(-\infty<t<a\)	\(e^{-as}F(s)\)
\(u_1(t)\) (\(\delta\)-function for \(t=0\))	\(1\)
\(u_2(t)\) (derivative of the \(\delta\)-function)	\(s\)
\(u(t)\) (unit function)	\(\dfrac{1}{s}\)
\(\dfrac{t^{n-1}}{(n-1)!}\)	\(\dfrac{1}{s^n}\)
\(e^{-at}\)	\(\dfrac{1}{s+a}\)
\(\left(\dfrac{1}{b}\right)\sin bt\)	\(\dfrac{1}{s^2+b^2}\)
\(\cos bt\)	\(\dfrac{s}{s^2+b^2}\)
\(\left(\dfrac{1}{b}\right)e^{-at}\sin bt\)	\(\dfrac{1}{(s+a)^2+b^2}\)

*) \(f(+0)\) denotes the value of \(f(t)\) at the instant \(t=0\). The sign \(+\) indicates that if \(f(t)\) undergoes a discontinuity at the instant \(t=0\), then \(f(+0)\) denotes the value attained by the function as zero is approached from the positive side.

Fig. 1. Series circuit for illustrating the Laplace-transform method.

The method by which the Laplace transform is used to obtain the transient regime of an electrical circuit may be set forth most simply by considering a particular case. The simple series circuit shown in Fig. 1 contains the most frequently encountered elements. It is required to determine the current \(i(t)\) that will flow in the circuit if, at the instant \(t=0\), the key connecting the source of a given voltage \(V(t)\) to the circuit, which until then has been in an undisturbed state, is closed.

The differential equation of the circuit has the form:

\[ L\frac{di}{dt}+Ri+\frac{1}{C}\int i\,dt=V(t). \tag{2} \]

Multiplying each term of the equation by \(e^{-st}\,dt\) and integrating from \(0\) to \(\infty\), we find (with the aid of Table 1) the transform of equation (2):

\[ Ls\,I(s)+I(s)+\frac{I(s)}{Cs}=E(s). \tag{3} \]

In obtaining from Table 1 the transforms of the derivative and the indefinite integral of \(i(t)\), we have used the fact that \(i_0=q_0=0\).

Equation (3) can be solved algebraically, as a result of which we obtain for the transform of the current the expression

\[ I(s)=\frac{E(s)}{Ls+R+\dfrac{1}{Cs}}. \tag{4} \]

The current \(i(t)\) is the original function corresponding to the transform \(I(s)\). This original function can be determined either from tables analogous to Table 1, or by direct methods, which cannot be described here. If \(V(t)\) is given, then \(E(s)\) can be calculated. For example, if \(V(t)\) is a battery with emf equal to \(\mathcal{E}_0\), then \(V(t)=\mathcal{E}_0 u(t)\) and \(E(s)=\dfrac{\mathcal{E}_0}{s}\). In this case

\[ I(s)=\frac{\mathcal{E}_0}{Ls^2+Rs+\dfrac{1}{C}} =\frac{\mathcal{E}_0}{L}\, \frac{1}{\left(s+\dfrac{R}{2L}\right)^2+\dfrac{1}{LC}-\dfrac{R^2}{4L^2}}, \]

and for \(i(t)\) we obtain the familiar result:

\[ i(t)= \frac{\mathcal{E}_0}{L\sqrt{\dfrac{1}{LC}-\dfrac{R^2}{4L^2}}} \exp\left(-\frac{R}{2L}t\right) \sin\left(\sqrt{\frac{1}{LC}-\frac{R^2}{4L^2}}\,t\right). \tag{5} \]

The method used in this example is very general and can be applied to the solution of any differential equation with constant coefficients under given initial conditions. Let us note that the denominator on the right-hand side of equation (4) is simply the complex impedance of the circuit, in which the usual \(i\omega\) has been replaced by the complex quantity \(s\). This circumstance makes the Laplace transform method especially useful in the case of electrical circuits.

From the example we have considered it follows that the transient regime of a circuit can be obtained by considering the steady-state regime under the action of a sinusoidal disturbance. To do this one proceeds as follows.

1. The desired solution (relating the “cause” and the “effect”) is obtained for the case of the steady state by using the expressions for complex impedances. Replacing all \(i\omega\) by \(s\), one obtains the function called the “system function” (for the preceding example the system function

\[ G(s)=\frac{1}{Ls+R+\frac{1}{sC}} \]

can be obtained from the expression for the complex admittance relating the current and voltage in the circuit of Fig. 1 in the steady-state case, under the action of a sinusoidal e.m.f.).

1. Next, the transform of the “cause” \(c(t)\) is obtained

\[ C(s)=\int_{0}^{\infty} c(t)e^{-st}\,dt=L[c(t)]. \]

1. The transform of the “effect” is obtained in the form \(E(s)=G(s)C(s)\).

2. From tables or by other methods the original function for \(E(s)\) is found; the obtained original function represents the solution for the transient regime. Formally this process of finding the original function may be written in the form

\[ e(t)=L^{-1}[E(s)]. \]

Let us now consider several examples which are very often encountered in electronic circuits.

II. TRANSIENT REGIME OF VARIOUS CIRCUITS WHEN A UNIT VOLTAGE STEP IS APPLIED

1. “Differentiation” of a unit step

An input signal in the form of a unit voltage step is applied to a simple circuit consisting of \(R\) and \(C\), shown in Fig. 2,a. It is required to find the form of the signal arising across the resistance \(R\).

Fig. 2. The simplest circuit of \(R\) and \(C\) and its transient regime under the action of a unit voltage step.

Such a circuit is often used to couple two points of a circuit that have different constant potentials. It is also used for pulse shaping in pulse amplifiers. For this circuit the system function takes the form:

$$ G(s)=\frac{R}{R+\frac{1}{Cs}}=\frac{\tau s}{1+\tau s}, \qquad \text{where } \tau=RC. $$

The transform of the output voltage is in this case equal to

$$ E_0(s)=\frac{1}{s}G(s)=\frac{\tau}{1+\tau s}, $$

and the original function corresponding to this transform has the form:

$$ V(t)=\exp\left(-\frac{t}{\tau}\right). $$

The input and output voltages are shown in Fig. 2, б.

2. Double “differentiation” of a unit voltage step

In most amplifiers two or more coupling circuits consisting of \(R\) and \(C\) are necessary. In the case of an amplifier intended to amplify pulses from some electrical radiation detector, one of these circuits can be used for pulse shaping. For the case of two coupling circuits it is necessary to consider the equivalent circuit shown in Fig. 3. For the system function we now have the expression

Fig. 3. Circuit and graph illustrating double differentiation of a unit function.

$$ G(s)=\frac{\tau_1 s}{1+\tau_1 s}\cdot\frac{\tau_2 s}{1+\tau_2 s}. $$

The transform of the output signal corresponding to a unit voltage step applied to the input has the form

$$ E_0(s)=\frac{1}{s}G(s), $$

from which one can obtain that

\[ V(t)=\frac{1}{\tau_{2}-\tau_{1}}\left[\tau_{2}\exp\left(-\frac{t}{\tau_{1}}\right)-\tau_{1}\exp\left(-\frac{t}{\tau_{2}}\right)\right]. \]

Additional differentiating cascades cause “overshoots” in the transient-response curves (Fig. 3). To avoid this phenomenon in circuits used for pulse shaping, one time constant must be considerably smaller than the others (by a factor of one hundred for a 1% relative overshoot).

3. Anode load in a resistance amplifier stage

Replacing the pentode in a pentode amplifier stage by a current generator, we arrive at the consideration of transients in the circuit shown in Fig. 4a, where \(R\) is the anode resistance and \(C\) is the parasitic capacitance shunting this resistance. For this case

Fig. 4. Equivalent anode load for a resistance amplifier stage and its transient response under the action of a unit voltage step.

\[ G(s)=Z(s)=\frac{R}{1+\tau s};\qquad \tau=RC, \]

so that

\[ E_{0}(s)=\frac{I_{0}}{s}\,G(s). \]

The output voltage, which is the initial function for \(E_{0}(s)\), has the following form (see Fig. 4b):

\[ V(t)=I_{0}R\left[1-\exp\left(-\frac{t}{\tau}\right)\right]. \]

The output voltage rises from 10% to 90% of its maximum value in the time \(2.25\tau\). This time, according to the definition given below, is the so-called rise time.

4. Amplifier with an anode load compensated by inductance

For given values of \(R\) and \(C\), a somewhat more rapid rise of the pulse can be obtained if a small inductance is connected in series with \(R\). In this case one says that the anode load of the amplifier is compensated by inductance. The system function for the equivalent circuit shown in Fig. 5a takes the form

Fig. 5. Equivalent anode load for a resistance-amplifier stage, compensated by capacitance, and its transient behavior under the action of a unit voltage step.

\[ G(s)=Z(s)=\frac{1}{\dfrac{1}{R+Ls}+Cs} =\frac{R+Ls}{1+RCs+LCs^{2}}. \]

The subsequent analysis is simplified if we put \(R=1,\ C=1\). Then \(L=aR^{2}C=a\). This choice of scale normalizes the system function, as well as the expression for the output voltage \(V(t)\), arising under the action of a unit current step at the input of the circuit (Fig. 5a), and establishes the time scale in units of \(RC\). For brevity, in what follows we shall call \(V(t)\) the “transient behavior” of the circuit under consideration. The normalized system function takes the form

\[ g(s)=\frac{1+as}{1+s+as^{2}}. \]

To represent the “transient behavior” arising under the action of a unit current step, we obtain

\[ E_{0}(s)=\frac{1}{s}g(s)= \frac{\dfrac{1}{s}\left(s+\dfrac{1}{a}\right)} {(s+s_{1})(s+s_{2})}, \]

where

\[ s_{1},\,s_{2}=-\frac{1}{2a}\pm\sqrt{\frac{1}{4a^{2}}-\frac{1}{a}}. \]

Using the pair of operators (109) given in Gardner’s work

and Barnes\(^1\), we find for the “transient regime” the expression

\[ V_0(t)=1-\frac{\frac{1}{\alpha}-s_1}{s_1(s_2-s_1)}e^{-s_1t} -\frac{\frac{1}{\alpha}-s_2}{s_2(s_1-s_2)}e^{-s_2t}, \]

which, if desired, may also be represented in another form. The “transient regime” will be oscillatory if \(s_1\) and \(s_2\) have an imaginary part. In order that there be no oscillations, it is evidently necessary that \(\alpha \leq \frac{1}{4}\). If \(\alpha=\frac{1}{4}\), then the circuit is said to be critically compensated, and for this case

\[ E_0(s)=\frac{1}{s}\, \frac{\left(1+\frac{1}{4}s\right)} {\left(1+\frac{1}{2}s\right)^2}, \]

whence, using pair (2,138) from the work of Gardner and Barnes, we obtain:

\[ V_0(t)=1-(1+t)e^{-2t}. \]

For the critically compensated case the rise time of the pulse from \(10\%\) to \(90\%\) of its maximum value is equal to \(1.55\tau\). Thus, compensation reduces the rise time by a factor of \(2.25/1.55\).

When using the compensation method in practice, it is convenient to make the inductance variable, with a movable core, which is set in the required position after investigation of the transient regime by means of a cathode oscillograph. In Fig. 5, б several cases are given corresponding to different values of \(\alpha\). It should be noted that, if in the example considered the voltage at the output is to be monotonic (i.e., have no pulsations), then the roots of the denominator \(G(s)\) must not have an imaginary part. This condition, however, is neither sufficient nor necessary in order to ensure a monotonic increase of the voltage in the case of an arbitrary circuit.

5. Transient regime in an amplifier (general case)

Let \(G(s)\) denote the complex gain coefficient of a linear amplifier. If \(E_i(s)=L[e_i(t)]\) is the transform of the input signal, then the signal at the output is determined by the expression

\[ V_0(t)=L^{-1}[G(s)E_i(s)]. \]

In investigating the transient regime of a narrow-band amplifier it is often convenient to decompose \(G(s)\) into two factors: \(G(s)=G_1(s)G_2(s)\), where the first factor is the system function of an equivalent amplifier having a flat frequency characteristic at high frequencies, and the second factor is the system function of an equivalent amplifier having a flat characteristic at low frequencies (down to zero-

frequency). The transient regime for large and small times can then be considered independently. The curves of the dependence of the gain coefficient on frequency, corresponding to \(G_1\) and \(G_2\), are shown in Fig. 6.

Fig. 6. Frequency characteristic of the amplifier: dependence of the gain coefficient on frequency

Fig. 7. Equivalent circuit of an amplifier stage.

6. Amplifier with two \(RC\) circuits limiting the gain in the low- and high-frequency regions

The system function for the amplifier stage shown in Fig. 7 has the form

\[ G(s)=\frac{s\tau_1}{1+\tau_1 s}\cdot \frac{R_2G_m}{1+\tau_2 s}, \]

where \(\tau_1=R_1C_1\) and \(\tau_2=R_2C_2\), and \(G_m\) is the transconductance of the tube.

The lower and upper frequencies on the frequency characteristic corresponding to half power are located at the points

\[ f_1=\frac{1}{2\pi\tau_1} \]

and

\[ f_2=\frac{1}{2\pi\tau_2}, \]

respectively. The normalized system function will evidently have the form

\[ \frac{G(s)}{R_2G_m}\equiv g(s)= \frac{s\tau_1}{(1+\tau_1s)(1+\tau_2s)}. \]

Table II

Relative pulse amplitudes

\(\lambda\)	\(\lambda^{\frac{1}{1-\lambda}}\)
0.5	0.250
1.25	0.410
2.5	0.543
5.0	0.669
12.5	0.803
25	0.815
50	0.923
125	0.962
250	0.978
\(\infty\)	1.000

Knowing the transform of the “transient regime” arising under the action of a unit impulse,

\[ E_0(s)=\frac{RG_m}{s}g(s), \]

we find the “transient regime”

\[ V_0(t)=R_2G_m\frac{\tau_1}{\tau_1-\tau_2} \left[ \exp\left(-\frac{t}{\tau_1}\right) - \exp\left(-\frac{t}{\tau_2}\right) \right]. \]

Setting

\[ \frac{dV_0}{dt}=0, \]

we find that the pulse has amplitude

\[ V_{\max}=R_2G_m\lambda^{\frac{1}{1-\lambda}}, \]

where

\[ \lambda\equiv\frac{\tau_1}{\tau_2}. \]

Some values of the function \(\lambda^{\frac{1}{1-\lambda}}\) are given in Table II.

III. THEORY OF THE TRANSIENT REGIME IN A VOLTAGE AMPLIFIER

1. Equivalent circuits for a triode or pentode operating in the linear regime

We assume that the reader is familiar with the usual characteristics of radio tubes. The values of these curves, with their corresponding numerical data, are invaluable in the practical calculation of circuits. However, for the mathematical analysis of the transient regime it is necessary to construct an equivalent circuit for each electron tube of the amplifier. It is assumed here that, in the signal range used, the tube operates linearly. Let us consider a triode or pentode whose screen voltages are maintained constant with respect to the cathode. In the absence of grid currents,

\[ \left. \begin{aligned} I_g &= 0,\\ I_a &= I_a(U_g, U_a), \end{aligned} \right\} \tag{6} \]

where \(I_g\) is the grid current, \(I_a\) the anode current, and \(U_g\) and \(U_a\) the grid and anode potentials, measured with respect to the cathode. Taking \(i_a=\Delta I_a\), \(V_g=\Delta U_g\), etc., we obtain:

\[ i_a=\left(\frac{\partial I_a}{\partial U_g}\right)_{U_a}V_g+ \left(\frac{\partial I_a}{\partial U_a}\right)_{U_g}V_a =G_mV_g+\frac{V_a}{r_a}. \tag{7} \]

where

\[ G_m=\left(\frac{\partial I_a}{\partial U_g}\right)_{U_a} \quad \text{and} \quad \frac{1}{r_a}=\left(\frac{\partial I_a}{\partial U_a}\right)_{U_g}. \]

\(G_m\) is the transconductance, and \(r_a\) is the internal resistance of the tube. For small changes of current and voltage it is usually assumed that \(G_m\) and \(r_a\) are constant, i.e. that the tube operates linearly.

If a resistance \(R_L\) is now connected in series with the anode, then

\[ V_a=-i_aR_L, \]

and equation (7) takes the form:

\[ +\mu V_g=i_ar_a+i_aR_L, \tag{8} \]

where \(\mu=G_m r_a=\left(\dfrac{\partial U_a}{\partial U_g}\right)_{I_a}\) is the amplification factor of the tube. The two equivalent circuits corresponding to equations (7) and (8) are shown in Fig. 8, \(a\) and \(b\). The parallel equivalent circuit is the most convenient for a pentode, since in most cases of practical interest \(r_a \gg R_L\).

Fig. 8. Equivalent circuits of a triode or pentode. \(a\)—series circuit, \(b\)—parallel circuit.

If the screen voltage of the pentode \(U_s\) may vary (with respect to the cathode), then

\[ I_g=0;\qquad I_a=I_a(U_g, U_s, U_a);\qquad I_s=I_s(U_g, U_s, U_a). \]

Hence six independent tube parameters are obtained. By means of the method described above, it is not difficult to obtain the complete equivalent circuit for this case. This circuit must include the plate and screen resistances \(r_a\) and \(r_s\), and four voltage (or current) sources. In deriving the two simple equivalent circuits we neglected the interelectrode capacitances. Since the capacitance between the plate and the grid of a triode causes a reduction of the input resistance of the tube, this type of tube is usually not used in a wide-band (“fast”) amplifier. The shielding present in a pentode reduces the coupling between grid and plate so much that it may be neglected. Thus, in order that the equivalent circuits given should be sufficiently accurate, it is necessary to specify the input and output capacitances. The equivalent circuit for any amplifier stage can be obtained from consideration of Fig. 9. It is assumed that the screen and grid bias resistances are completely shunted by capacitances. In addition, it is assumed that \(R_g \gg R\) and \(r_a \gg R\).

Fig. 9. \(a\)—typical amplifier stage; \(b\)—equivalent circuit for high frequencies; \(v\)—equivalent circuit for low frequencies.

2. Transient regime in the equivalent circuit of a high-frequency amplifier\(^2\)

Let \(G_2(s)\) denote the gain coefficient of an equivalent high-frequency amplifier which has an ideal characteristic at low frequencies. If \(G_2(0)\) represents the gain of such an idealized amplifier at zero frequency, then

function of the system

\[ g_2(s)=\frac{G_2(s)}{G_2(0)}. \tag{9} \]

we shall call the normalized gain function. The computation of the initial function for the system function (9), under the condition that the “cause” is the unit function \(u(t)\) (for brevity we have agreed to call this initial function the “transient”), is the usual way of investigating the properties of an amplifier in the transient regime for signals possessing high-frequency components (“fast” signals). This initial function (“transient”) is found from the inverse Laplace transform:

\[ V(t)=L^{-1}\left[\frac{1}{s}\,g_2(s)\right]. \tag{10} \]

The time derivative of this initial function is equal to

\[ V'(t)=L^{-1}[g_2(s)]. \tag{11} \]

The transform corresponding to equation (11) will be

\[ g_2(s)=\int_{0}^{\infty} V'(t)e^{-st}\,dt. \tag{12} \]

Several typical “transient” curves are shown in Fig. 10 (these curves depict the initial functions), giving the voltage at the amplifier output that arises under the action of a unit voltage step at its input. Curves 1 and 2 have “overshoots.” In curve 1 these “overshoots” are oscillatory in character; the “overshoot” of curve 2 decays monotonically. Curves 3 and 4 increase monotonically, with curve 4 reaching a certain value in the minimum time, depending on the type and number of tubes, as well as on the total gain of the amplifier. Curves of types 3 and 4 are considered more desirable for an amplifier of transient processes than curves 1 and 2. In the subsequent exposition we shall confine ourselves to considering amplifiers having a monotonic characteristic.

Fig. 10. Some typical transient curves.

In many cases the delay time \((T_D)\) and the rise time \((T_R)\) characterize the amplifier transient regime in the high-frequency region sufficiently completely. The delay time is often defined as the interval of time between the instant at which the input to the amplifier receives

voltage point, and the instant at which the output voltage reaches one half of its final value (Fig. 11, a).

The rise time is then defined as a quantity inversely proportional to the slope of the tangent to the curve of Fig. 11, a at the point where the output voltage has reached one half of its final value.

Another accepted definition is the definition of \(T_R\) as the distance between the abscissas of the curve of Fig. 11, a, for which the output voltage reaches 10% and 90% of its final value.

These definitions are inconvenient for calculations. This prompted the introduction of certain new definitions, which are applicable only

Fig. 11. Graphs illustrating the definition of rise time and delay time.

to a monotonically increasing voltage at the output. The delay time may be defined as

\[ T_D=\int_0^\infty tV'(t)\,dt . \tag{13} \]

Defined in this way, \(T_D\) is the abscissa of the “center of gravity” of the figure formed by the curve \(V'(t)\) (Fig. 11, b). For the rise time we have the following definition:

\[ T_R^2=2\pi\int_0^\infty (t-T_D)^2V'(t)\,dt =2\pi\left[\int_0^\infty t^2V'(t)\,dt-T_D^2\right]. \tag{14} \]

With this definition, the rise time is \(\sqrt{2\pi}\) times greater than the “standard deviation” of the curve \(V'(t)\). The coefficient \(\sqrt{2\pi}\) is chosen so that this value of the rise time coincides with the value determined from the slope of the curve \(V(t)\) in the case when \(V'(t)\) is a Gaussian error curve. Let us note that such a form of the curve \(V'(t)\) is the limiting case which occurs if the number of amplifier stages increases without bound. In practical cases the values of \(T_D\) and \(T_R\), obtained from (13) and (14), will differ only slightly from the “laboratory” definitions of \(T_D\) and \(T_R\) given above. We shall now show the целе-

reasonableness of introducing \(T_D\) and \(T_R\) according to (13) and (14). Expand \(e^{-st}\) in expression (12) into an infinite series. Then we obtain:

\[ g_2(s)=1-s\int_0^\infty tV'(t)\,dt+\frac{s^3}{2}\int_0^\infty t^2V'(t)\,dt-\ldots \]

Consequently, if the normalized amplification function is expanded in powers of \(s\),

\[ g_2(s)=1+As+Bs^2+\ldots, \]

then

\[ T_D=-A \]

and

\[ T_R=\sqrt{2\pi}\,(2B-A^2)^{\frac12}. \]

It is useful to consider an amplification function of the form

\[ g_2(s)=\frac{1+a_1s+a_2s^3+\ldots}{1+b_1s+b_2s^3+\ldots}, \]

since such a function is often encountered in practice. In this case

\[ T_D=b_1-a_1 \quad \text{and} \quad T_R=\sqrt{2\pi}\,[b_1^2-a_1^2+2(a_2-b_2)]^{\frac12}. \tag{15} \]

Let us now consider two examples, namely, an amplifier cascade on resistances and the same cascade compensated by a series-connected inductance. In Section II expressions for the “transient regime” (arising under the action of a unit step) were derived for both of these cases.

The normalized amplification function for a resistance amplifier cascade takes the form

\[ g_2(s)=\frac{1}{1+RCs}. \]

Calculating the delay time and the rise time by formulas (15), we find:

\[ T_D=RC \quad \text{and} \quad T_R=\sqrt{2\pi}RC=2.51RC, \]

i.e. the delay time is equal to the time constant of the anode circuit, while the rise time is approximately two and a half times greater than this time constant. Determining \(T_R\) by the “slope” gives the value \(T_R=2RC\), and determining it “from 10% to 90%” gives the value \(T_R=2.2RC\). The ratio of amplification to rise time, computed from the values \(G_0=G_mR\) and \(T_R=\sqrt{2\pi}RC\), is equal to

\[ \frac{G_0}{T_R}=\frac{G_mR}{\sqrt{2\pi}RC}=\frac{G_m}{\sqrt{2\pi}C}. \tag{16} \]

ELECTRONICS IN NUCLEAR PHYSICS

Obviously, the quantity \(\dfrac{G_m}{\sqrt{2\pi C}}\) characterizes the suitability of a tube for amplifying voltage steps and fast pulses. For 6AC7, \(G_m = 0.009\ \Omega^{-1}\), \(C = 8 + 11 = 19\ \mu\mu\mathrm{F}\), which gives \(\dfrac{G_0}{T_R} = 190\) per microsecond. If one takes into account the distributed capacitance possessed by the wire resistor, then \(C = 23\ \mu\mu\mathrm{F}\) (approximately) and \(\dfrac{G_0}{T_R} = 155\) per microsecond. For 6AK5, \(G_m = 0.005\ \Omega^{-1}\), \(C = 4.3 + 2.1 = 6.4\ \mu\mu\mathrm{F}\), and \(\dfrac{G_0}{T_R} = 312\) per microsecond. Taking into account the distributed capacitance of the resistor increases the value of \(C\) to \(10.4\ \mu\mu\mathrm{F}\) and lowers the value of \(\dfrac{G_0}{T_R}\) to 208 per microsecond.

The normalized gain function for an amplifier stage compensated by a series-connected inductance (see Fig. 5) can be written in the form

\[ g_2(s)=\frac{1+\frac{1}{4}RCs}{1+RCs+\frac{1}{4}R^2C^2s^2}. \]

With the aid of equation (15) we obtain:

\[ T_D=\frac{3}{4}RC, \]

\[ T_R=\sqrt{\frac{7}{16}}\sqrt{2\pi RC}=1.66RC. \]

Proceeding from the definition “from 10% to 90%,” we obtain for \(T_R\) the value \(1.5RC\). The ratio of gain to rise time is now equal to

\[ \frac{G_0}{T_R}=\sqrt{\frac{16}{7}}\frac{G_m}{\sqrt{2\pi C}}. \tag{17} \]

In the general case this ratio has the following form:

\[ \frac{G_0}{T_R}=S\frac{G_m}{\sqrt{2\pi C}}, \tag{18} \]

where \(S\) can be defined as a constant characterizing the rise time for the type of stage under consideration. Obviously, for a resistance stage \(S=1\), and for a resistance stage compensated by a series-connected inductance \(S=\left(\dfrac{16}{7}\right)^{1/2}\). The theoretical limit for \(S\), which can be reached (using two-terminal networks for compensation) under the condition

in the absence of overshoots, is apparently the value \(S=\dfrac{3}{\sqrt{2}}=2.12\).

Let us now consider some theorems applicable to multistage amplifiers, in which the gain function of the entire amplifier is equal to the product of the gain functions of the individual stages (an amplifier without feedback). The proof of these theorems is based chiefly on definitions (13) and (14) and may be found in the paper by Gardner and Barnes\(^1\).

A. In a multistage amplifier the delay times and the squares of the rise times add:

\[ T_D=\sum_{1}^{n} T_{D_i};\qquad T_R^2=\sum_{1}^{n} T_{R_i}^{2}. \tag{19} \]

B. For a given gain, the rise time in the amplifier is least in the case when all \(n\) stages have the same rise time \(T_{R_1}\). If this is so, then

\[ T_R=\sqrt{n}\,T_{R_1}. \tag{20} \]

C. The gain in each stage of an amplifier with rise time \(T_R\) must be equal to [from formulas (18) and (20)]

\[ G_1=\frac{S}{\sqrt{n}}\left(\frac{G_m}{\sqrt{2\pi C}}\right)T_R, \tag{21} \]

which gives for the total gain the quantity

\[ G_t=G_1^n. \tag{22} \]

D. If \(G_t\) is known, then the rise time \(T_R\) depends on \(n\) in the following way [from equations (21) and (22)]:

\[ T_R=\left(\frac{\sqrt{2\pi C}}{G_m}\,\frac{1}{S}\right)\sqrt{n}\,G_t^{\frac{1}{n}}. \tag{23} \]

It follows that the minimum rise time is obtained for \(n=2\ln G_t\). In this case

\[ G_1=e^{\frac{1}{2}}=1.65\ldots \]

For example, if \(G_t=10^5\), then \(n=23\) stages, and the rise time will be greater both for a smaller and for a larger number of stages, provided only that the total gain remains constant. The minimum rise time, obtained when \(G_1=e^{\frac{1}{2}}\), is equal to

\[ T_{\min}=\frac{1}{S}\left(\frac{\sqrt{2\pi C}}{G_m}\right)\sqrt{2\ln G_t}. \tag{24} \]

Let us now calculate the minimum rise time that can be obtained with ordinary tubes in an amplifier compensated by inductances. Let \(S=1.5\), \(C=22\cdot 10^{-12}\ \mathrm{F}\), \(G_m=0.009\ \mathrm{ohm}^{-1}\), \(G_t=10^5\). (A 23-stage amplifier using 6AC7 tubes, compensated by inductances and having a voltage gain equal to \(10^5\).)

Substituting these data into (24), we obtain:

\[ T_{\min}=\frac{1}{1.5}\left(\frac{2.5\cdot 22\cdot 10^{-12}}{0.009}\right)\cdot \sqrt{\frac{2\cdot 2.72\cdot 5}{0.434}}, \]

i.e.

\[ T_{\min}=0.032\ \mu\mathrm{sec}. \]

In such an amplifier the anode-load resistances must be only \(180\ \mathrm{ohm}\), which leads to a narrowing of the region of linear operation of the amplifier to a voltage interval of about \(1\ \mathrm{V}\). A more reasonably designed amplifier may contain 9 stages. In this case the rise time, calculated from (23), proves to be equal to \(0.044\ \mu\mathrm{sec}\). The anode-load resistances will be \(400\ \mathrm{ohm}\), which increases the region of linear operation of the amplifier to a range of several volts. By using 6AG7 and 6V6 tubes in the last two stages and neglecting a certain increase in rise time, one can construct an amplifier, compensated by inductances, which will operate linearly in the region of positive pulses up to values of \(40\ \mathrm{V}\). The rise time in such an amplifier is \(0.06\ \mu\mathrm{sec}\). By employing four-terminal-network circuits as an intermediate stage (not described in this survey), the rise time can be reduced to approximately \(0.035\ \mu\mathrm{sec}\), without impairing the monotonic character of the “transient regime.”

The use of such intermediate stages in a 23-stage amplifier reduces the rise time to \(0.02\ \mu\mathrm{sec}\). An even faster amplifier can be realized if most of the 6AC7 tubes are replaced by 6AK5 tubes.

In concluding the present section on transient phenomena in an equivalent high-frequency amplifier, it is useful to mention the relation between the rise time and the upper frequency corresponding to the half-power value delivered by the amplifier (i.e. the frequency at which the magnitude of the gain of sinusoidal voltage falls to \(\sqrt{2}/2=70\%\), and the magnitude of the power gain to \(50\%\)).

As was noted above, it can be shown that if a unit voltage pulse is applied to the input of an amplifier, then the shape of the curve representing the time derivative of the voltage at the amplifier output approaches, as the number of stages is increased, the form of a Gaussian error curve (on the assumption that the voltage curve

at the output of each cascade stage is monotonic). Similarly, the frequency characteristic of the gain also approaches a Gaussian error curve as \(n\) increases. The frequency characteristic, expressed by a Gaussian curve and providing the rise time \(T_R\), has the form

\[ |g_2(i\omega)|=\exp(-\pi T_R^2 f^2). \]

Equating this expression to \(\dfrac{1}{\sqrt{2}}\), we obtain:

\[ T_R f_2=\sqrt{\frac{\ln 2}{2\pi}}=\frac{1}{3.015}=\frac{1}{3}, \tag{25} \]

where \(f_2\) is the upper frequency corresponding to the half value of the delivered power. It has been found experimentally that this relation remains, with almost exact validity, true for any amplifier used for amplifying pulses from ionization chambers, and thus this relation is useful to remember.

For example, the upper frequency for an amplifier with a rise time of \(0.1\ \mu\text{sec}\) is \(3.3\cdot 10^6\) cycles. Here it is assumed, of course, that by applying a unit voltage impulse to each amplifier stage, we obtain at the output of the stage a monotonically increasing voltage.

3. Transient response in an equivalent low-frequency amplifier

Suppose that the normalized gain function is equal to \(g_1(s)\). The “transient response” corresponding to \(g_1(s)\) (caused by switching on the voltage in the form of a unit step function \(u(t)\)) is equal to one for \(t=0\), and then, as time increases, approaches zero. If we want the amplifier not to distort a pulse of duration \(\Delta t\), it is necessary to require that the “transient-response” curve arising when a unit voltage impulse is applied to the input of the amplifier deviate during the time \(\Delta t\) from unity by less than some specified amount, expressed as a small fraction \(\delta\). Fig. 12 illustrates this requirement. The most convenient method for analyzing the properties of the “transient-response” curve is to expand \(g_1(s)\) in a power series in powers of

Fig. 12. Transient-response curve illustrating the definition of \(\delta\).

\[ g_1(s)=1+\frac{a_1}{s}+\frac{a_2}{s^2}+\cdots . \tag{26} \]

In this expression the first term is equal to unity, since \(g_1(s)\) is normalized. Further, for small values of \(t\) (i.e., for \(t\) of the order of \(\Delta t\)) the “transient regime” will be

\[ V(t)=L^{-1}\left[\frac{1}{s}g_1(s)\right] =L^{-1}\left[\frac{1}{s}+\frac{a_1}{s^2}+\frac{a_2}{s^3}+\cdots\right] = \]

\[ =1+a_1t+a_2\frac{t^2}{2!}+\cdots . \tag{27} \]

This power series converges rapidly. The second term of the series explains the linear fall of the “transient-regime” curve for small \(t\). To illustrate this method of analysis, suppose that the amplifier contains, between stages, \(n\) intermediate couplings, whose circuit is shown in Fig. 13. Then

\[ g_1(s)=\left(\frac{\tau s}{1+\tau s}\right)^n= \]

\[ =1-\frac{n}{\tau s}+\frac{n(n+1)}{2!\tau^2s^2}-\cdots, \]

so that

\[ V(t)=1-n\frac{t}{\tau}+ \]

\[ +\frac{n(n+1)}{2\cdot 2!}\left(\frac{t}{\tau}\right)^2-\cdots \]

Fig. 13. Coupling circuit producing the fall of the transient-regime curve.

It is obvious that at first the “transient-regime” curve falls with the rate \(-\dfrac{n}{\tau}\), and therefore \(\tau\)—the time constant of the coupling circuits—must be chosen in such a way as to satisfy the relation

\[ \delta = n\frac{\Delta t}{\tau}. \tag{28} \]

As an example, suppose that \(\delta=0.01\) (1% distortion for signals lasting \(\Delta t=100\ \mu\text{sec}\)) and that the amplifier contains five intermediate couplings (\(n=5\)). Then

\[ \tau=\frac{5\cdot 10^{-4}}{0.01}=0.05\ \text{sec}, \]

i.e., in the present case, for example, \(R=1\ \text{M}\Omega\) and \(C=0.005\ \mu\text{F}\) will be suitable. Calculation of the lower frequency corresponding to half power (the frequency at which the voltage gain decreases by 3 decibels, or by 30%) gives for it the value \(8.3\ \text{cps}\). If the amplifier must amplify signals having a rise time of about \(0.5\ \mu\text{sec}\) and a duration of about \(100\ \mu\text{sec}\), and the distortion due to the time constants of the amplifier must be...

is equal to only one percent, then the upper and lower frequencies corresponding to half power will be equal to \(8.3\,z_u\) and about \(3\cdot 10^6\,z_u\), respectively. We see that very stringent requirements are in fact imposed on the frequency characteristic of the amplifier.

Compensating \(RC\) circuits may be included in the amplifier in order to make the coefficient \(a_1\) in expressions (26) and (27) equal to zero. This is usually done in video amplifiers, although the method of analysis set forth above has hitherto not been applied in the literature for determining the optimal values of the component elements of compensating circuits[^3].

The traditional tendency to investigate the behavior of an amplifier by means of its frequency characteristic has the drawback that in this case one is interested in the amplification of established sinusoidal oscillations. In the case of amplifiers used for counting pulses arriving from an electrical detector, the partial characteristic in the low-frequency region must be chosen in such a way as to create short pulses from the small voltage impulses arising on the detector capacitance when each ionizing particle passes through the detector.

IV. RATIO OF SIGNAL TO INTERFERENCE IN AMPLIFIERS USED FOR COUNTING PULSES

1. Signal

In the following exposition it is assumed in most cases that the signal can be regarded as the sudden appearance of a certain charge \(Q_S\) on a capacitance \(C\), connected with the input grid of the amplifier. The capacitance \(C\) includes the capacitance of the electrical detector (counter, chamber, etc.), of the connecting leads, and of the input grid. It can be shown with a very good degree of approximation that any change in the effective input capacitance caused by feedback affects the signal and the interference in the same way. For this reason it is sufficient to confine ourselves to considering the case in which the magnitude \(C\) is not changed by the action of feedback. The charge \(Q_S\), obviously, causes a jump of voltage on the input grid. It is assumed that the rise time of the pulse is small in comparison with the rise time in the amplifier itself. If this condition is not fulfilled, then the signal-to-interference ratio will be smaller than that obtained from the calculation. The output signal of the amplifier (which includes both interference and the signal \(Q_S\)) is usually investigated in order to obtain information on the number of voltage pulses per second, often as a function of the magnitude of the pulse. Obviously, the amplifier must be constructed in such a way as to transform voltage jumps into pulses of the desired duration and, if possible, of the form most suitable for counting. The form of the pulse and its duration,

generally speaking, affect the magnitude of the signal-to-noise ratio. The most general method used for pulse shaping consists in including in the amplifier a single coupling circuit having a small time constant (see Sections V and XI).

2. Noise

The list of electrical noise in amplifiers given below is divided into two groups. The first group includes noise that can be eliminated by the proper arrangement and careful selection and mounting of the amplifier components. These are shocks, microphonic noise, random external noise, and noise caused by the presence of faulty elements in the circuit.

The second group of noise cannot be completely eliminated and determines the limiting value of the signal-to-noise ratio that can be achieved in a pulse amplifier intended for the detection of small charges. This includes noise produced by resistances, noise produced by grid current, noise produced by the shot effect, and noise produced by the flicker effect. The latter play a role only in the region of very low frequencies. Although many of the noises of the first group usually cause considerable trouble in practice, especially in those cases where the counting rate is low, we shall not consider them, but shall concentrate our attention on the noises of the second group. Also excluded from our consideration will be the noise produced by the flicker effect, which is significant only in amplifiers that must have appreciable gain at very low frequencies.

Since the signal of interest to us consists in the appearance of a charge on the capacitance \(C\), it is convenient to represent the action of all noise sources (both random and continuously acting) as the appearance on the capacitance \(C\) of some root-mean-square charge \(Q_N\), which would create at the amplifier input the same potential as the actual noise sources.

Our first task, therefore, is to transform the well-known expressions for the root-mean-square value of the potential into expressions for the root-mean-square charge produced by noise on the capacitance \(C\). In all cases (for voltage, current, or charge), a lowercase letter squared will hereafter be used to denote the mean square value per unit frequency interval (frequency being expressed in hertz).

A. Noise produced by resistances. The thermal motion of electrons in a resistance causes the appearance, at its open terminals, of a voltage (Fig. 14)

\[ v_R^2 = 4 k T R. \tag{29} \]

We are interested in the case when the resistance is shunted by the capacitance \(C\). To find the equivalent mean-square value of the charge that arises on the capacitance under the action of the noise voltage determined by equality (29), we carry out the two transformations indicated in Figs. 15 and 16.

Fig. 14.

The first transformation gives us:

\[ v_R^{\prime 2}=4kTR_{\mathrm{eff}}, \tag{29a} \]

where

\[ R_{\mathrm{eff}}=\frac{R}{1+\omega^2R^2C^2} \]

is the real part of the complex resistance

\[ Z=\frac{R}{1+i\omega RC}. \]

Then, with the aid of the second transformation, we find

\[ q_R^2=\frac{4kT}{\omega^2R}. \tag{30} \]

It should be emphasized that expression (30) is merely another form of expression (29), obtained by means of the usual analysis of the equivalent circuit. It is interesting to note that the capacitance \(C\) does not enter explicitly into expression (30).

B. Noise caused by grid currents. The principal noise caused by the grid current is due to statistical fluctuations in the number of ions arriving at the grid. The mean square of the noise current (per unit frequency interval) is given by the expression

\[ i_g^2=2eI_g, \tag{31} \]

where \(I_g\) is the sum of the absolute values of the positive and negative components of the grid current. If the resistance of the grid circuit is formed by \(R\) and \(C\) connected in parallel, then we can perform the transformations indicated in Fig. 17. In this case we obtain:

\[ v_g^2=2eI_gRR_{\mathrm{eff}}. \]

Fig. 15.

Fig. 16.

Fig. 17.

Using equation (29a), we obtain:

\[ v_g^2=\frac{2eI_gR}{4kT}\,v_R^{\prime 2}=\frac{R}{R_g}\,v_R^{\prime 2}, \]

where

\[ R_g=\frac{4kT}{2eI_g} \tag{32} \]

is called the equivalent noise resistance produced by the grid current. To obtain an expression for the equivalent charge, we must carry out the second transformation indicated in Fig. 16. As a result we obtain:

\[ q_g^2=\frac{4kT}{\omega^2R_g}. \tag{33} \]

Expression (33) is identical in form with expression (30), in which the physical resistance \(R\) has been replaced by the fictitious equivalent noise resistance \(R_g\). In calculations, both of these resistances may be regarded as connected in parallel.

B. Noise caused by the shot effect. Noise caused by the shot effect arises from statistical fluctuations in the value of the anode current of electron tubes, due to the fact that the electron charge has a finite value. The mean square of the anode current caused by an effect of this kind will be

\[ i_s^2=\eta\cdot 2eI_a; \tag{34} \]

\(\eta\) is a coefficient (\(\eta\leqslant 1\)) appearing because the effect under consideration is reduced under the influence of space charge. Equivalent circuits for a tube containing such a noise source are given in Fig. 18. It is evident that \(i_s/G_m\) represents the mean effective value of the equivalent noise voltage which, when applied to the grid, produces the same fluctuation of the anode current as does the shot effect.

Fig. 18.

It is convenient to introduce the equivalent noise resistance due to the shot effect

\[ R_s=\frac{1}{4kT}\,\frac{i_s^2}{G_m^2} =\frac{\eta\cdot 2eI_a}{4kTG_m^2}, \tag{35} \]

which, when connected in series with the grid, will produce in the anode circuit the same effect as the shot effect. The value of \(R_s\) varies from several hundred to several thousand ohms, depending on the type of tube.

If we now carry out the transformations indicated in Fig. 19, we find the equivalent charge of the noise caused by the shot effect. A simple analysis of the circuit gives:

\[ q_s^2=4kTR_sC^2. \tag{36} \]

It is obvious that, in order to reduce the noise under consideration, it is necessary that the value \(R_s C^2\) be as small as possible. This means that the quantity

\[ \frac{G_m}{C_{\text{input}}\sqrt{I_a}}, \]

which characterizes the tube, should be as large as possible. From this point of view it is desirable to use a 6AK5 as the input tube.

Fig. 19.

In deriving expression (36) we discarded the term

\[ \frac{4kT}{\omega^2R}\left(\frac{R_s}{R}\right), \]

which is in fact negligibly small in comparison with the quantity determined by equation (30)

\[ \left(\frac{R_s}{R}\simeq 10^{-7}\ \text{or less}\right). \]

3. Calculation of the signal-to-noise ratio

The total mean-square noise charge per unit frequency interval is equal to the sum of expressions (30), (35), and (36)

\[ q_N^2=4kT\left[\left(\frac{1}{R}+\frac{1}{R_s}\right)\frac{1}{\omega^2}+R_sC^2\right]. \tag{37} \]

The noise at the amplifier output depends on the shape of the amplifier amplitude characteristic and on the absolute value of the gain. Since the absolute value of the gain acts in the same way on the signal and on the noise, in calculating the noise it is sufficient to take into account the normalized amplitude characteristic \(|g(f)|\). We shall restrict our consideration to the case of an amplifier in which the amplification of low frequencies is determined by the time constant \(R_1C_1\), formed by the coupling element shown in Fig. 20, \(a\), while the amplification of high frequencies, in turn, is determined by the time constant \(R_2C_2\), formed by the resistance in the anode circuit, shown in Fig. 20, \(b\). It is assumed that all the other time constants affecting the trans—

Fig. 20.

transmission of low frequencies, is considerably greater than \(R_1 C_1\), and that all the remaining time constants affecting the transmission of high frequencies are considerably smaller than \(R_2 C_2\).

We have shown that for such an amplifier

\[ g(s)=\frac{s R_1 C_1}{1+R_1 C_1 s}\cdot \frac{1}{1+R_2 C_2 s}. \]

Putting \(s=i\omega\) \((\zeta=0)\) and introducing \(f=\dfrac{\omega}{2\pi}\), we find:

\[ |g(f)|^2=\frac{f^2}{f^2+f_1^2}\cdot \frac{f_2^2}{f^2+f_2^2}, \tag{38} \]

where

\[ f_1=\frac{1}{2\pi R_1 C_1}\quad \text{and}\quad f_2=\frac{1}{2\pi R_2 C_2} \]

are the upper and lower frequencies on the frequency characteristic of the amplifier corresponding to half the delivered power. The mean-square equivalent charge due to noise at the input, determining the noise in the signal at the amplifier output, can be written in the form:

\[ Q_N^2=\int_0^\infty q_N^2 |g(f)|^2\,df . \]

After substituting the values from expressions (37) and (38) and integrating, we obtain:

\[ Q_N^2=\frac{kT}{4\pi} \left\{ \left(\frac{1}{R}+\frac{1}{R_g}\right) \left(\frac{f_2}{f_1(f_1+f_2)}\right) +4\pi^2 R_s C^2 \left(\frac{f_2^2}{f_1+f_2}\right) \right\}. \tag{39} \]

We have already introduced the expression for the signal which appears at the amplifier output when a unit voltage impulse is applied to its input. (This signal is equivalent to the sudden appearance of charge on the input capacitance.) The pulse at the output is equal to

\[ V(t)=\frac{R_1 C_1}{R_1 C_1-R_2 C_2} \left[ \exp\left(-\frac{t}{R_1 C_1}\right) -\exp\left(\frac{t}{R_2 C_2}\right) \right], \tag{40} \]

and its amplitude is

\[ V(t)=\lambda^{\frac{1}{1-\lambda}}, \tag{41} \]

where

\[ \lambda=\frac{R_1 C_1}{R_2 C_2}=\frac{f_2}{f_1}. \]

The square of the ratio of signal to noise at the amplifier output is therefore equal to

\[ \left(\frac{V_S}{V_N}\right)^2 = \frac{ Q_S^2 \lambda^{\frac{2}{1-\lambda}} }{ \frac{kT}{2\pi} \left\{ \left(\frac{1}{R}+\frac{1}{R_g}\right)\frac{1}{f_1} +4\pi^2 R_s C^2 \lambda f_1 \right\} \frac{\lambda}{1+\lambda} }, \tag{42} \]

V. S. Elmor

where \(Q_S\) is the charge from the signal appearing at the input capacitance. The magnitude of the charge (signal) which, when applied to the input capacitance, gives a pulse equal in amplitude to the pulse produced by the mean probable charge from the disturbances, \((Q_N)_{\mathrm{av.\ prob.}}\), is obtained if we set

\[ \left(\frac{V_S}{V_N}\right)=0.6745. \]

Thus,

\[ (Q_N^2)_{\mathrm{av.\ prob.}} = (0.67)^2\frac{kT}{4\pi}\, \frac{ \left\{\left(\frac{1}{R}+\frac{1}{R_g}\right)\frac{1}{f_1} +4\pi^2R_sC^2\lambda f_1\right\}\frac{\lambda}{\lambda+1} }{ \lambda^{\frac{2}{1-\lambda}} }. \tag{43} \]

Let us now consider some consequences following from formulas (42) and (43). Of greatest interest are two cases:

1) The case in which the ratio \(\dfrac{V_S}{V_N}\) reaches the greatest possible value (without taking into account the pulse duration);

2) The case in which the ratio \(\dfrac{V_S}{V_N}\) reaches the greatest value for a specified pulse duration.

A third case, in which the ratio \(\dfrac{V_S}{V_N}\) reaches the greatest value for a specified rise time in the amplifier, is of only limited interest. In all cases we shall assume that the rise time of the signal is determined by the properties of the amplifier, and not by the charge-collection time in the detector.

Case 1. The signal-to-noise ratio given by equality (42) is a function of \(f_1\) and \(\lambda=\dfrac{f_2}{f_1}\).

By the usual methods we find that \(\dfrac{V_S}{V_N}\) has a maximum when

\[ f_1=f_2=\frac{1}{2\pi C\sqrt{R'_gR_s}}, \tag{44} \]

where

\[ \frac{1}{R'_g}=\frac{1}{R_g}+\frac{1}{R}. \]

Since \(\lambda=\dfrac{f_2}{f_1}=1\), the relative amplitude of the signal is equal to

\[ \frac{1}{\lambda^{\frac{1}{1-\lambda}}} = \frac{1}{e} = 0.3678. \]

The mean probable charge from the disturbances will therefore be equal (see equality (43)) to

\[ (Q_N)_{\mathrm{av.\ prob.}} = (0.67)\,e^2 kTC\sqrt{\frac{R_s}{R'_g}}. \tag{45} \]

Expressing this charge in terms of the number of electrons, we obtain: \((Q_N)_{\text{rms}} = 735 C^{1/2}\sqrt[4]{R_s/R_g}\) electrons, where \(C\) is expressed in micromicrofarads.

According to Terman (see\(^3\), p. 294), for a typical triode \(R_s = 3/G_m\) ohms, and for a pentode

\[ R_s=\frac{I_{\text{anode}}}{I_{\text{anode}}+I_{\text{scr}}} \left( \frac{2.5}{G_m}+\frac{20I_{\text{scr}}}{G_{m\text{scr}}} \right)\ \text{ohms}. \]

For 6AK5, with \(R_g\) equal to infinity (free grid),

\[ R_s=\frac{0.005}{0.005+0.002} \left( \frac{2.5}{0.004}+\frac{20\cdot0.002}{0.001} \right)=650\ \text{ohms} \]

and

\[ R'_g=10^9\ \text{ohms}. \]

Apparently, these values are correct only as to order of magnitude. If \(C=20\ \mu\mu\text{f}\), then \((Q_N)_{\text{rms}}=735\times5\times0.028\simeq100\) electrons, in agreement with calculations by other authors\(^5\). The frequencies corresponding to half the amplifier power are

\[ f_1=f_2=\frac{1}{2\pi\cdot20\cdot10^{-12}\sqrt{6.5\cdot10^{11}}}=10000\ \text{cps}. \]

The rise time in such an amplifier is approximately equal to \(\sqrt{2\pi R_2C_2}=2.5\cdot16=40\ \mu\text{sec}\), and \(R_1C_1=16\ \mu\text{sec}\) (decay time).

Two pulses will not overlap if they follow one another at intervals of about \(60\ \mu\text{sec}\). Consequently, if about 160 randomly distributed pulses are counted per second, only 1% of these pulses will appreciably overlap one another.

Case 2. It is obvious that in many practical cases it is desirable to obtain pulses whose duration is less than the duration of the pulses required to obtain the optimum signal-to-noise ratio. In order to consider this case, it is necessary to define the duration of a pulse. This can be done by a method analogous to that used to determine the rise time.

Fig. 21.

If \(g(s)\) is the normalized gain function of the amplifier, then the pulse

\[ V(t)=L^{-1}\left[\frac{1}{s}g(s)\right] \]

has the typical form shown in Fig. 21. It was found,

that it is convenient to define the pulse duration by the relation

\[ \tau^{2}= \frac{ 2\pi \displaystyle\int_{0}^{\infty} (t-t_{0})^{2} V(t)\,dt }{ \displaystyle\int_{0}^{\infty} V(t)\,dt }, \tag{46} \]

where

\[ t_{0}= \frac{ \displaystyle\int_{0}^{\infty} tV(t)\,dt }{ \displaystyle\int_{0}^{\infty} V(t)\,dt } \]

is the time coordinate of the “center of gravity” of the pulse.

From relation (46) it follows that the pulse duration is \(\sqrt{2\pi}=2.5\) times greater than the “standard deviation” of the pulse \(V(t)\). This definition has the advantage that it gives a ready method for estimating the pulse duration from the form of the system function \(g(s)\). We do not give the details of this calculation here, but it can be carried out easily if one applies the method used above for calculating the rise time.

For an amplifier of the “\(RC—RC\)” type we obtain:

\[ \tau = T_R(1+\lambda^{2})^{\frac{1}{2}}, \tag{47} \]

where

\[ T_R\left(= \sqrt{2\pi R_{2}C_{2}}\right) \]

is the rise time, and \(\lambda=\dfrac{R_{1}C_{1}}{R_{2}C_{2}}\).

We must now find the maximum of the signal-to-noise ratio [equality (42)] with simultaneous fulfillment of condition (47). Since in this case the frequencies are considerably higher than in case 1, the term containing the grid resistance in the denominator of (42) may be neglected. The maximum of the ratio \(\dfrac{V_S}{V_N}\) again occurs for \(\lambda=1\) \((R_{1}C_{1}=R_{2}C_{2})\). The curve of the dependence of the signal-to-noise ratio on \(\dfrac{R_{1}C_{1}}{T_R}=\dfrac{T_1}{T_R}=\dfrac{\lambda}{\sqrt{2\pi}}\) for this case is shown in Fig. 22 (curve I). This curve is useful for estimating the influence on the rise time of various ratios of the rise time to the time constant \(T_1\) (the pulse decay time) for a constant pulse duration. It is of interest to estimate the most probable

number of noise electrons (equivalent), which is a function of \(C_1\), \(R_s\), \(\tau\), and the parameter \(\dfrac{T_1}{T_R}=\dfrac{\lambda}{\sqrt{2\pi}}\). For the sake of simplification, all numerical coefficients in equation (43) have been combined into a single expression, which should be used together with the graph of the ratio \(\dfrac{V_s}{V_N}=\varphi\left(\dfrac{T_1}{T_R}\right)\) (Fig. 22). The final expression is

\[ (Q_N)_{\text{rms}} = 0.22 C \sqrt{\frac{R_s}{\tau}} \cdot \frac{1}{\varphi\left(\dfrac{T_1}{T_R}\right)} \text{ electrons} \tag{48} \]

(where \(C\) is expressed in \(\mu\mu\mathrm{F}\), \(R_s\)—in ohms, and \(\tau\)—in \(\mu\mathrm{sec}\)) and is in reasonable agreement with certain measurements of noise in an amplifier,

Fig. 22. Relative magnitude of the signal-to-noise ratio as a function of the ratio of the decay time to the rise time (at constant pulse duration).

which had a rise time of about \(0.15\,\mu\mathrm{sec}\) and a decay time of about \(5\,\mu\mathrm{sec}\). From the curve \(\varphi\left(\dfrac{T_1}{T_R}\right)\) it is seen that the optimum value of the ratio \(\dfrac{V_s}{V_N}\) is attained in the case when the rise time and the decay time are approximately equal.

It is practically incorrect to use an amplifier in which \(T_1\) is considerably larger than \(T_R\), although in practice precisely this is done. Curve 2 in Fig. 22 gives the value of the signal-to-noise ratio for the case when the signal has the form of a Gaussian error curve and its decay is formed by means of \(RC\). Fig. 22 also presents analogous curves for the case when the decay of the pulse is due to the action of a delay line. A still somewhat larger

the ratio \(\dfrac{V_S}{V_N}\) is obtained in the case when the signal fall is formed by a delay line, while at high frequencies the signal is represented by a Gaussian curve. These curves may be used together with equation (48). The use of a delay line for forming pulses will be discussed briefly in Section XI.

In those cases where it is necessary to preserve the rapid rise of the pulse (for example, in some coincidence circuits in time measurements), the signal-to-noise ratio proves to be greatest for \(T_1 \gg T_R\), if the amplitude of the pulse is still taken as the measure of the signal. This calculation is carried out analogously to the way it was done in the other cases. \(T_R\) is regarded as constant, and the maximum of the ratio \(\dfrac{V_S}{V_N}\) is sought as a function of \(T_1\) (or \(f_1\)). The values of the ratio \(\dfrac{V_S}{V_N}\) calculated in this way are substantially smaller than those which could be obtained for a pulse of the same duration but of another shape (for example, for a pulse not having a steep front). However, in most experiments on the study of coincidences the pulse amplitude is not of great importance, and therefore it is obvious that the results of case 2 are applicable to most coincidence-counting experiments.

V. SOME REMARKS ON THE DESIGN OF PULSE AMPLIFIERS\(^6\)

Figure 23 shows the block diagram of a typical apparatus often used in particle counting. We shall list some circumstances

Labels in Fig. 23: “Very short cable”; “Ionization chamber”; “Preamplifier 20–100×”; “Cable 6′ (low capacitance)”; “Main amplifier 10000×”; “Short cable (low capacitance)”; “Amplitude discriminator”; “Counting circuit”; “Register”; “One unit”; “Source.”

Fig. 23. Block diagram of an apparatus for particle counting.

which must be borne in mind in designing and constructing an amplifier intended for operation in such an apparatus.

1. The input pulse is usually negative.
2. The output pulse is always positive and is taken from a cathode follower.

1. The magnitude of the output pulse should usually lie within the range from 10 to 100 V.

2. The amplification should lie within the range from \(2 \cdot 10^5\) to \(10^6\) (in this case the noise is reduced to a few volts).

3. The amplifier must have coarse and fine gain control, permitting continuous (or with small intervals) variation of the gain by approximately a factor of 30.

4. It is necessary to specify in advance the degree of linearity that we wish to obtain in the circuit. For example, one may require that, at the greatest pulse amplitude, the deviation from linearity not exceed one percent.

5. The stability of the amplifier must be such that the change in gain caused by changes in the line voltage and by changes in other parameters be no greater than one percent.

6. If the amplifier is intended for coincidence measurements, it is necessary to know what resolving power we wish to attain. The magnitude of the resolving power will determine the rise time of the pulses.

7. It is necessary to take into account the duration of the pulses and the load on the amplifier, i.e., the number of pulses arriving at its input per second.

8. The location of the gain regulator must be chosen with such calculation as to avoid overloading the preceding stage.

9. The location of the pulse-shaping circuit must be chosen with such calculation as to avoid possible overloading of the preceding stage by the piling up of pulses and to reduce the level of noise arising in all preceding stages.

10. It is necessary to decide whether the pulses will be shaped by an \(RC\) circuit or by a delay line.

11. It is necessary to know what type of amplifier will be used: an ordinary amplifier or an amplifier with feedback.

In an ordinary amplifier it is easier to achieve a very short rise time; however, its linearity and gain stability are low. An amplifier with feedback possesses greater linearity and stability, but in an amplifier of this type it is difficult to obtain a rise time considerably less than \(0.1\) μsec (at a gain of \(3 \cdot 10^5\)). In the case when the rise time can be made small, it cannot be well stabilized.

1. The shielding and assembly of the amplifier must be such that external transient phenomena do not affect its operation.

2. For a signal in the form of a kick, the amplifier must give a monotonic “transient-response” curve.

The list given has been compiled somewhat arbitrarily and does not cover all the requirements imposed on a pulse amplifier. However, in designing it, these requirements are the principal ones and must be taken into account.

Before turning to a discussion of the principal types of circuits, it is useful briefly to consider the nature of the signal that is to be amplified. If \(n\) is the mean counting rate, then the probability that after a given pulse (at \(t=0\)) the second pulse will arrive in the time interval from \(t\) to \(t+dt\) is equal to

\[ p(t)\,dt = n e^{-nt}\,dt . \]

Let us note that \(p(t)\) has its greatest value for small \(t\). In fact, in this case \(p(t)\) is simply equal to \(n\,dt\). Thus, if \(n=10^3\) counts per second, then 1 percent of the counts will be separated by 10 microseconds or a smaller time interval.

Next, the voltage on the input grid consists of a random superposition of exponential signals of the form

\[ V(t)=\frac{Q_s}{C}\exp\left(-\frac{t}{RC}\right) =V_m\exp\left(\frac{-t}{RC}\right), \tag{49} \]

where \(R\) is the grid leakage resistance, \(C\) is the total effective input capacitance, and \(Q_s\) is the charge collected as a result of the gas ionization process in the ionization chamber (for each particle entering it). The mean voltage produced by the random superposition of voltage pulses of the form (49) is equal to

\[ \overline{V(t)}=n\int_0^\infty V(t)\,dt=nRCV_m, \tag{50} \]

where \(n\) is the mean rate of arrival of voltage pulses of the form (49). For the limiting case \(n=10\,000\) per second, \(R=10^8\ \Omega\), \(C=10^{-10}\ \mu\mu\mathrm{F}\), and \(V_m=10^{-2}\ \mathrm{V}\), we have
\[ V(t)=10^4\times 10^8\times 10^{-10}\times 10^{-2}=1.0\ \mathrm{V}. \]
This value of the voltage is sufficient to shift noticeably the operating point of the input grid in the case where it is directly connected to the electrode of the ionization chamber. Usually the indicated effect is negligibly small. The mean-square fluctuation of the voltage is found with the aid of the theorem\(^7\):

\[ \overline{\left(V(t)-\overline{V(t)}\right)^2} = n\int_0^\infty [V(t)]^2\,dt \]

\[ v_f^2=n\frac{RC}{2}V_m^2, \tag{51} \]

if \(V(t)\) is defined by equality (49).

For the case considered above,

\[ v_f^2=\frac{10^4\times 10^8\times 10^{-10}\times V_m^2}{2}, \]

i.e.

\[ v_f=7V_m. \tag{51a} \]

This voltage, created by the fluctuation, does not cause difficulties in the input network, but if the signal is amplified properly, then at some subsequent point of the amplifier an overload will occur (long before the arrival of individual pulses that might cause overload). The actual voltage peaks produced by fluctuations (often called “bunching”) will probably be 3 times larger than the root-mean-square value. Thus, it is advisable to assume that the voltage at overload is 20 times greater than the amplitude of the individual pulses.

The phenomenon just described affects the choice of location of the circuit that shapes the pulses and has a small time constant. Since the time constant is much smaller than the \(RC\) of the input circuit, the voltage produced by fluctuations at the output of the pulse-shaping circuit is practically zero. Usually the pulse-shaping circuit is placed between the preliminary amplifier (gain from 30 to 100) and the main amplifier (gain 10,000). It is desirable to place it as far from the amplifier input as is compatible with the fluctuation phenomena described above. This requirement is explained, of course, by the fact that this circuit acts as a filter for shocks, microphonic effects, and other low-frequency components of interference arising in the preliminary stages. Usually the most convenient and appropriate place in the amplifier for coarse gain adjustment (attenuator) is the part of the amplifier where the pulse-shaping circuit is located. In selecting the limits of gain variation, special attention should be paid to the problem of overload. (In doing so, it is necessary, of course, also to bear in mind voltage fluctuations.) The calculation of amplifiers without feedback requires no special explanation and will not be considered here. The chief difficulties arise in calculating stages having large gain. In this case great care is necessary in choosing operating points in order to avoid falling into the nonlinear part of the characteristic. Often, in order to ensure sufficient linearity, it proves necessary to use some feedback (for example, an unbypassed cathode resistance).

For a fast amplifier, critically compensated by inductance and containing about nine stages with two cathode followers, the following characteristics are typical. Gain—200,000. Reasonable output voltage with linear amplification (positive pulses)—40 V. Rise time—0.06 μsec. Stability: with a change of the voltage in the alternating-current line by 1%, the stability of the gain changes by 4% (meaning the change in stability caused by changes in the cathode temperature). Voltage source—250 V, 200 mA (stabilized).

Let us consider a feedback “loop” consisting of three tubes and used in pulse amplifiers. The basic

the circuit is shown in Fig. 24, where approximate values are indicated for some of the circuit parameters*). The magnitude of the voltage gain (with feedback) is usually limited to the range from 20 to 100.

Let us make a few remarks on the use of a feedback loop of this kind.

1. The gain is determined (to an accuracy of 5%) by the ratio $\dfrac{R_1+R_2}{R_2}$.

2. The small capacitance $C_1$ is used to set the level of feedback necessary to ensure a “good” (monotonic) “transient regime.”

3. The time constant $R_0C_0$ must differ substantially from the time constant $R_3C_3$ (if the latter is used).

Fig. 24. Typical circuit of a feedback loop. Voltage values are indicated approximately.

1. The resistance $R_2$ creates cathode bias for tube $T_1$, and also affects the magnitude of the gain.

2. The tubes usually operate in the regime recommended in radio-engineering handbooks (i.e. at the screen-grid and plate voltages, currents, etc., specified in the handbooks).

3. The stray capacitance of all leads through which the signal passes must be minimal.

4. In order for the gain to be stable, the resistances $R_1$ and $R_2$ must be wire-wound, but not substantially inductive.

5. The biases of the stages $L_2$ and $L_3$ must be asymmetric in the case where large signals of one sign pass through them.

*) In this figure, and in part of the following ones, an abbreviated system of notation for resistance and capacitance values is adopted: $K$ denotes the number of kilohms, $M$ megohms; the absence of a letter denotes the value of a resistance in ohms or a capacitance in microfarads.

1. The load in the stage \(L_3\) must be purely ohmic. A small capacitive load causes an undesirable phase shift in the feedback voltage, which leads to the establishment of a poor transient regime or even to the onset of oscillation. (Even the lead from an oscilloscope connected to the amplifier output affects the operation of the circuit!)

2. If the anode resistance \(L_3\) is made partly inductive (inductance compensation), then the properties of the circuit’s “transient regime” for fast and, especially, large signals are improved.

3. The sign of the signal can be reversed if a suitable resistance is placed in series with the anode of \(L_3\) and the output signal is taken from the anode. In this case the value of \(R_1\) will no longer be critical and may be reduced.

4. In constructing the amplifier, great attention must be paid to the placement of the grounding buses and to the wiring of the entire circuit, in order to avoid undesirable interactions.

5. If there is more than one feedback loop, then the additional loops must be decoupled by means of filters consisting of resistances and capacitances placed in their anode circuits (insufficient decoupling causes the appearance of “motor noises” at low frequency). The decoupled loops are, of course, designed for operation at a lower anode voltage.

VI. APPROXIMATE ANALYSIS OF TRANSIENT PHENOMENA IN A TWO-STAGE AMPLIFIER WITH FEEDBACK

Let us consider the transient regime of an idealized amplifier, shown in Fig. 25, and corresponding only approximately to the feedback loop just described.

The part of the input signal which is fed back to the amplifier input is equal to

\[ \beta(s)=\frac{R_2}{R_2+\dfrac{R_1}{1+\tau_1 s}} =\beta_0 \frac{1+\tau_1 s}{1+\tau_1\beta_0 s}, \tag{52} \]

where

\[ \beta_0=\frac{R_2}{R_1+R_2}\quad \text{and}\quad \tau_1=R_1C_1. \]

Fig. 25. Equivalent circuit of an amplifier with feedback.

The gain of an amplifier with feedback, according to the well-known theory, will be

\[ A(s)=\frac{g(s)}{1+\beta(s)\cdot g(s)}. \tag{53} \]

Let us consider the case of a two-stage amplifier characterized by a gain function (or system function) of the form

\[ g(s)=\frac{g_0}{(1+\tau s)^2}, \tag{54} \]

which corresponds to the gain of an equivalent high-frequency two-stage resistance amplifier, whose anode time constants \(RC=\tau\) are the same for both stages.

The gain \(g_0\) can be computed from the relation

\[ g_0=G_m^2 R^2, \tag{55} \]

where it is assumed that \(G_m\) is the same for each tube of the amplifier. (In a typical pulse amplifier \(g_0\simeq 10^4\) and \(\beta_0\simeq 10^{-2}\).) If \(\beta_0\) is small, then equation (52) can be simplified, since the term \(\beta_0\tau_1 s\) is small in comparison with unity and may be neglected. Then from expressions (52), (53), and (54) we find

\[ A(s)=\frac{A_0}{1+\left[\frac{2\tau+\beta_0 g_0\tau_1}{1+\beta_0 g_0}\right]s} +\frac{\tau^2 s^2}{1+\beta_0 g_0}, \tag{56} \]

where \(A=\dfrac{g_0}{1+\beta_0 g_0}\simeq \dfrac{1}{\beta_0}\) is the low-frequency gain in the presence of feedback.

For critical compensation the denominator of equation (56) must have two equal real roots:

\[ \left[\frac{2\tau+\beta_0 g_0\tau_1}{1+\beta_0 g_0}\right]^2 =\frac{4\tau^2}{1+\beta_0 g_0}. \tag{57} \]

It follows that \(\tau_1\) must have the following value:

\[ \tau_1=\frac{2\tau}{\beta_0 g_0} \left(\sqrt{1+\beta_0 g_0}-1\right) =\frac{2\tau}{\sqrt{\beta_0 g_0}} \left(1-\frac{1}{\sqrt{\beta_0 g_0}}+\cdots\right) \]

or

\[ \tau_1\simeq \frac{2\tau}{\sqrt{\beta_0 g_0}}. \tag{58} \]

Since in a typical amplifier \(\beta_0 g_0\simeq 100\), the time constant \(\tau_1\) should be approximately equal to \(\frac{1}{5}\tau\). If \(\tau=RC=10^2\times25\times10^{-12}=0.25\ \mu\mathrm{sec}\), then \(\tau_1\simeq 0.05\ \mu\mathrm{sec}\) and, consequently, \(C_1=5\ \mu\mu\mathrm{F}\), if \(R_1=10^4\). This calculation determines, with a fair degree of accuracy, the value of the compensating capacitance \(C_1\) that provides a monotonic transient response at the amplifier output, arising under the action of a unit voltage pulse at the amplifier input, shown in Fig. 24. (The value of \(C_1\) in the amplifier shown in Fig. 24 was found empirically.) The rise time in this amplifier, found by means of the method described in Section III, turns out to be equal to

\[ T_R=\frac{\sqrt{2\pi}\sqrt{2}\cdot\tau}{\sqrt{1+\beta_0 g_0}} \tag{59} \]

(in the case where \(\tau_1\) corresponds to critical compensation). For a ty-

of a typical amplifier possessing the constants given above; expression (59) gives for the rise time the value \(T_R = 0.087\ \mu\mathrm{sec}\), which is in reasonable agreement with the measured properties of the amplifier. It is evident that the value of the rise time is very sensitive to changes in \(g_0\). The quotient obtained by dividing the gain by the rise time is

\[ \frac{A_0}{T_R} = \frac{g_0}{1+\beta_0 g_0} \frac{\sqrt{1+\beta_0 g_0}}{\sqrt{2\pi}\sqrt{2\tau}} \simeq \sqrt{\frac{A_0}{2}} \left( \frac{G_m}{\sqrt{2\pi C}} \right). \tag{60} \]

For comparison let us compute the same quantity for a two-stage resistance amplifier having the same gain, but without feedback, i.e. in the present case the anode-load resistances are chosen so as to make \(G_m^2 R_0^2 = A_0\). Such an amplifier has the rise time \(T_R = \sqrt{2\pi}\sqrt{2}\,R_0 C\), and therefore

\[ \frac{A_0}{T_R} = \frac{G_m^2 R_0^2}{\sqrt{2}\sqrt{2\pi R_0 C}} = \sqrt{\frac{A_0}{2}} \left( \frac{G_m}{\sqrt{2\pi C}} \right). \tag{61} \]

This result is identical with expression (60). It is evident that the feedback considered:

1) does not increase the ratio of gain to rise time,
2) does not stabilize the rise time.

Feedback of this type improves linearity and stabilizes the amplification of signals whose duration is large in comparison with the rise time.

It is of interest to investigate the case when \(\tau_1\) is increased in order to increase the rise time. Let us therefore assume that

\[ \tau_1 \gg \frac{2\tau}{\sqrt{\beta_0 g_0}} \tag{62} \]

is an approximate value for critical compensation. Expression (56) then takes the form

\[ A(s)=A_0\frac{1}{1+\beta_0 A_0\tau_1 s}. \tag{63} \]

The normalized curve of the “transient regime” (arising under the action of a unit impulse) will have the form:

\[ V(t)=1-\exp\left(-\frac{t}{\beta_0 A_0\tau}\right) \simeq 1-\exp\left(-\frac{t}{\tau_1}\right), \tag{64} \]

and the rise time will be equal to

\[ T_R=\sqrt{2\pi}\,\beta_0 A_0\tau_1\simeq \sqrt{2\pi}\tau_1. \tag{65} \]

We see that it depends very little on the gain without feedback \(g_0=G_m^2R^2\).

The “transient regime” (64) coincides with the transient regime for a single-stage amplifier with resistive coupling. This analysis, carried out for a high-frequency equivalent amplifier, can also be applied to the consideration of a low-frequency equivalent amplifier. If such an amplifier has only one time constant \(T_2\) (in the grid—anode circuit), then it turns out that feedback increases it by \((1+\beta_0 g_0)\) times. If the amplifier has two time constants, then when a voltage step is applied to the amplifier input, slow damped oscillations arise at the amplifier output; these disappear if at least one of the time constants

\[ T_2 > 4\beta_0 g_0 T_1 \quad \text{(approximately).} \tag{66} \]

In practice it is not difficult to make one time constant approximately 50 times larger than the other. For example, the values \(T_1 = R_1 C_1 = 10^5 \times 10^{-9} = 10^{-4}\) sec., and \(T_2 = R_2 C_2 = 5 \cdot 10^5 \times 10^{-8} = 5 \cdot 10^{-3}\) sec. are satisfactory.

If \(\beta_0 g_0 \simeq 100\), then the effective time constant \(= 100 \times 10^{-4} = 10^{-2}\) sec., which corresponds to a lower frequency, corresponding to half power, of approximately 15 cps.

VII. DISCRIMINATORS AND SCALING CIRCUITS⁸

Pulses from a pulse amplifier may be counted (for subsequent analysis) by various methods. The most widespread method is the use for this purpose of a scaling circuit, followed by a mechanical counter. Other known methods include: a counting-rate meter and an oscilloscope with a camera, making it possible to photograph the pulse on a moving film.

In some special cases it is necessary to count pulses that stand in a definite time relation to other pulses, arriving either randomly or recurring with a definite frequency connected with the modulator of the primary source of nuclear particles (for example, a neutron spectrometer based on the use of the time of flight of neutrons).

1. Discriminators

In most counting devices the criterion by which pulses are selected is their amplitude: pulses with an amplitude smaller than some specified value are not registered.

A device that rejects or registers pulses depending on the magnitude of their amplitude may be called an amplitude discriminator, or simply a discriminator. Such a discriminator

Electronics in Nuclear Physics

is necessary in most counting circuits chiefly for two reasons:

1) to avoid counting pulses belonging to the background of the measurements and usually produced by sources of interference;
2) for selecting pulses according to the energy of the particles entering the electrical detector.

The discriminator circuit must have the following properties:

1) The circuit must separate pulses differing in amplitude by a small fraction of a volt, and be stable within approximately the same limits.

2) The circuit must operate from pulses of the shape and duration produced by pulse amplifiers. This means that, within certain limits, it must respond only to the amplitude, and not to the shape or duration of the pulses.

3) The circuit must be protected against overload, i.e., when large pulses, or pulses separated by small intervals of time, pass through the circuit, its biases must not change (of course, within certain limits).

4) The circuit must produce a standard trigger signal suitable for starting a counting circuit or some other device, for example a counting-rate meter. The shape and duration of such a trigger pulse must be matched to the resolving power of the counting circuit or other devices.

5) Adjustment of the bias voltages in the discriminator must be easily accomplished.

The simplest type of discriminator can be built by using a tube with a sharp current cutoff, for example a pentode, to which the appropriate bias is applied. A more complicated discriminator contains a diode (to which a bias is applied), used together with an amplifier and a trigger circuit. In designing discriminators that satisfy the requirements listed above, great care must be exercised; however, by using a biased diode, circuits that operate quite satisfactorily can be obtained. Apparently the most suitable discriminator for most practical cases is one in which a Schmitt trigger circuit\(^9\), or some modification of it, is used. A typical circuit of such a discriminator is shown in Fig. 26. The circuit shown is suitable for counters having a high resolving power. (Resolving time of the order of several microseconds.)

After certain changes, the circuit shown can be used for very fast counting (resolving time of the order of \(0.3\ \mu\mathrm{sec}\)). The circuit shown in Fig. 26 satisfies the basic requirements imposed on a discriminator. Its operation, characteristic of a Schmitt circuit, is based on the fact that the conducting tube may be either tube \(L_1\) or tube \(L_2\), depending on the value of the potential on the grid of \(L_1\). The resistance \(R\) in the anode circuit of \(L_1\)

determines the “hysteresis” of the Schmitt circuit. (By hysteresis here is meant the phenomenon that the grid potential \(L_1\) at which the circuit is actuated is greater for a rising pulse than for a falling pulse and than for the tail part of the pulse.) If the grid potential \(L_1\) is maintained at a value close to the triggering potential, then a hysteresis of the order of several volts is necessary in order to avoid the occurrence of oscillations (multivibration). Each pulse which raises the grid potential \(L_1\)

Fig. 26. Amplitude discriminator in which a Schmitt trigger circuit is used. In position 1 tube \(L_1\) is open; in position 2 it is cut off.

above the triggering value for \(0.1\ \mu\text{sec}\) or for a longer time produces at the anode a positive trigger pulse whose amplitude is sufficient to actuate the cell of a \(2:1\) counting circuit. (In this case, depending on how the counting circuit is started, an additional coupling stage may be required.) The amplitude of the input pulse may exceed the triggering potential by \(100\ \text{V}\) before grid current flows through \(L_1\), changing the constant bias voltage. In order that the discrimination be stable, the power supply must be well regulated. The heater voltage of both 6AC7 tubes need not be regulated, since variations in the difference of the (effective) grid–cathode potentials are minimized owing to the balanced character of the circuit. By taking precautions, it is possible to ensure that the discriminated voltage remains constant to within \(0.2\ \text{V}\) over a period of several days. However, when tubes are replaced it may change by a large amount. It is useful to note that the signal arising at the anode of \(L_2\) is suitable for intensifying the trace in an oscilloscope used for recording pulses by means of a moving-film camera. Intensification is often necessary in order to avoid undesirable exposure of the undeflected trace. It is usually necessary to include in the circuit a coupling stage with a cathode follower, in order to preserve the rapid rise of the pulse.

2. A Counting-Circuit Cell Performing 2:1 Counting

One of the counting circuits that has found wide application is the circuit performing 2:1 counting. The cell of a 2:1 counting circuit consists of two tubes connected so as to form a trigger circuit possessing two stable states (a “flip-flop”). The principal difficulty in using a flip-flop as a counting-circuit cell lies in the need to provide one-way coupling between adjacent flip-flops, and also between the discriminator and the first flip-flop of the counting circuit. In many older counting circuits, a special amplifying stage together with a circuit having a small time constant was used to accomplish this coupling. Such a device reduced undesirable interactions to a minimum and provided a pulse of suitable (but often critical) form for starting the flip-flop. For satisfactory operation even the best counting circuit of this type requires frequent adjustment. Figure 27 shows a cell of a counting circuit

Fig. 27. Counting cell 2:1.

2:1, whose operation at a moderately high counting rate proved quite satisfactory (resolving time 3–5 μsec). If the values shown in parentheses are used (including the 6SL7 tube), then the 2:1 counting element will consume less power and will have a resolving time of about 20 μsec. For counting 64:1, two elements with 6SN7 are usually used, followed by four elements with 6SL7. An unusual feature of the circuit presented is the use of a diode for one-way transmission

signal from one 2:1 scaling cell to another. Thanks to this, the output terminal of one element is connected directly to the input terminal of the next element, and so on, for all subsequent elements with 2:1 scaling. When the tube \(L_2\) of some preceding cascade begins to conduct current, a negative rectangular signal arrives at the anode of the nonconducting tube, which through the \(RC\) circuit is transmitted to the grid of the adjacent conducting tube. This process causes the flip-flop relay to pass into its other stable state.

Fig. 28. Cascade connecting a discriminator with a scaling circuit.

When the tube \(L_2\) of the preceding cascade becomes nonconducting, the resulting positive rectangular signal does not change the state of the trigger pair. Examination of the circuit shows that this method of triggering (or any other method serving the same purpose) will be successful only if the capacitances in the cross-coupling between grids and anodes are sufficiently large to overload the grids. In addition, the biases must be such as to ensure the presence of grid current in the conducting tube. In practice it has been found that scaling circuits of this type will always operate satisfactorily if the parameters of the cells differ from one another within 10%. In this case no special adjustment is necessary.

The discriminator shown in Fig. 26 may be connected to the 2:1 scaling circuit as indicated in Fig. 28. When such a coupling cascade is used, first, a negative trigger pulse is provided, and second, the trigger pulse is applied to the diode and to the right-hand cell of the scaling circuit at the proper value of the constant potential. In order to couple very narrow pulses with a 2:1 scaling circuit, it is advisable to replace the triode by a pentode possessing a sharp current cutoff. A typical circuit by means of which the output of a scaling circuit is connected to a register is shown in Fig. 29. The principal task in calculating a circuit of this kind is to provide a rectangular voltage pulse whose magnitude must be matched to the interval of time necessary for the operation of the register. The circuit must then recover faster than the register, i.e. the recovery time of the circuit must be less than the recovery time of the register. In this case the operating speed of the register is determined by its own properties. Study of the circuit shown in Fig. 29 indicates that usually

$\dfrac{1}{2}$ 6SL7 is in the conducting state, whereas the 6V6 is ordinarily cut off.

When a negative rectangular signal arrives from the last cell of the scaling circuit, $\dfrac{1}{2}$ 6SL7 is cut off and grid current passes through the 6V6. This continues for approximately $0.01$ microsec, i.e., until $\dfrac{1}{2}$ 6SL7 again becomes conducting and the 6V6 is again cut off. As $\dfrac{1}{2}$ 6SL7 there may be used the half of the 6SL7 tube that is in the stabilized anode-power supply. The time constants in the grid circuits of the circuit of Fig. 29 may be increased if the recorder used operates slowly. In other types of output circuits thyratrons are used (for example, the 2050 thyratron), or a cathode follower arranged so that it can produce a pulse with an amplitude of several hundred volts.

Fig. 29. Circuit for starting a mechanical counter.

VIII. DIFFERENTIAL AMPLITUDE DISCRIMINATORS

The distribution by amplitude of pulses from some electrical radiation detector can be measured by various methods.

1. If the intensity of the source is constant, then the integral distribution curve can be obtained with the aid of one discriminator and a scaling circuit.

2. If the intensity of the source varies, then to obtain the same curve it is necessary to use two discriminators. The bias of one discriminator is fixed, and its readings serve to normalize the readings of the other discriminator, whose bias is varied. In both cases, by calculation, the differential distribution curve can be obtained.

1. Since the differential distribution curve is usually of greater interest than the integral curve, it is desirable to obtain it directly. For this purpose a differential discriminator is used, which responds only to pulses whose amplitudes lie within specified limits. The third discriminator, whose bias is fixed, can serve as a source of normalized counts. Such a single-channel differential discriminator reduces the necessary number of measured counts, as well as the time required to obtain the differential distribution curve. It also ensures a more stable separation of the biases that determine the limits within which the pulses are counted.

2. A multichannel differential discriminator is an obvious improvement over a single-channel discriminator. With its aid one can substantially reduce the time required to obtain the differential distribution curve, since all the pulses arriving at it are sorted according to magnitude. The advantage of using this discriminator is not only the reduction in observation time, but also the fact that the data obtained with its aid are apparently more reliable, since the experimental conditions need remain constant only over a short interval of time.

The saving of time obtained when using a single-channel differential discriminator instead of a simple discriminator operating at two different biases may be estimated as follows:

Let \(N_1\) be the number of pulses with amplitude greater than \(A_1\), and \(N_2\) the number of pulses with amplitude greater than \(A_2\) \((A_2 > A_1)\), where in both cases the duration of the measurements is equal to \(T\). The root-mean-square error for \(N_1\) is equal to \(\sqrt{N_1}\), and the root-mean-square error for \(N_2\) is equal to \(\sqrt{N_2}\). The relative root-mean-square error for the difference of the counts \(N_1 - N_2\) is equal to

\[ \frac{\sqrt{N_1+N_2}}{N_1-N_2}. \]

In order to obtain such an accuracy of measurement, a time \(2T\) is required. If the measurements are made by means of a single-channel differential discriminator over a time \(t\), then \(\frac{t}{T}(N_1-N_2)\) counts are obtained with a relative root-mean-square error equal to

\[ \frac{1}{\sqrt{\frac{t}{T}(N_1-N_2)}}. \]

Equating the two root-mean-square errors, we find that

\[ \frac{t}{T}=\frac{N_1-N_2}{N_1+N_2}. \]

It should be remembered here that \(T\) is the time necessary for observing \(N_1\) (or \(N_2\)) counts. Consequently, the time necessary to ensure a certain specified statistical accuracy is reduced when using-

...of a differential discriminator by a factor of \(\dfrac{(N_1-N_2)}{2(N_1+N_2)}\). Differential discriminators usually consist of a series of simple amplitude discriminators whose biases are successively increased. Coincidence circuits controlling the counting of pulses are connected between adjacent discriminators. A count is obtained only in the coincidence channel connected between the discriminator that has operated and the one that has not operated. The chief difficulty in constructing a differential discriminator for fast pulses arises from the fact that successive discriminators are triggered at different instants of time, first during the rise of the pulse and then during its fall. Obviously, during the rise and the subsequent fall of the pulse, a certain number of coincidence circuits pass through the sensitive state. However, only that coincidence circuit which is located between the last discriminator to have operated and the next one not to have operated will send a pulse into the counting circuit. Difficulties of this kind have led to a discriminator design in which the discriminators that have operated remain temporarily in their second state, which provides the time necessary for the input pulse to fall to zero. The state of the coincidence circuits is controlled by a delayed pulse produced by the discriminator with the smallest bias. A second, still more prolonged pulse restores all the discriminators, and the circuit becomes ready to receive the next pulse.

Figure 30 presents a block diagram of a two-channel discriminator. Using this drawing, one can readily consider an arrangement with many channels.

Let us now consider the discriminator and coincidence circuit shown in Fig. 31, which are the principal parts of a well-functioning multichannel differential discriminator[^10]. Tubes \(L_1\), \(L_2\), and \(L_3\) are connected according to a circuit that is one variant of the Schmitt trigger circuit, with \(L_3\) acting in this circuit as a dc resistance and replacing the usual cathode resistance. The presence of \(L_3\) makes it possible to operate with pulses as large as \(150\ \mathrm{V}\), without grid current arising in tube \(L_1\) (grid current changes the bias value and therefore should be avoided). Tubes \(L_4\left(\dfrac{1}{2}\,6\mathrm{SN}7\right)\) and \(L_2\) are connected so that they form a univibrator. Thus, a positive pulse applied to the grid of \(L_1\) opens \(L_2\) and closes \(L_4\). A positive pulse appears on the grid of \(L_4\), which is connected with the anode of \(L_2\), and \(L_4\) opens. In this case the cathode potential of all three tubes remains sufficiently high that, when the value of the input pulse falls to zero, tubes \(L_1\) and \(L_2\) would be cut off.

Fig. 30. Block diagram of a multichannel discriminator (three channels shown).

Fig. 31. Simplified circuit of the discriminator and coincidence circuit. Tubes 1 and 6 are open; tubes 4, 5, 7, and 8 are blocked.

Then the positive restoring pulse applied to the grid \(L_5\) returns the discriminator to its initial state. However, even without this pulse the discriminator would automatically return to its initial state after several hundred microseconds. This last precaution ensures the operating state of the circuit when it is first switched on.

The signals arriving at the grids \(L_6\) and \(L_7\) of the coincidence circuit are taken from the anodes \(L_2\) and \(L_3\). Before triggering, the anode voltage of \(L_2\) is \(250\ \mathrm{V}\), and the anode voltage of \(L_3\) is \(300\ \mathrm{V}\). If both discriminators are triggered, these potentials interchange. It is obvious that only one coincidence circuit will receive two signals of different signs, which will cause it to operate. All the other coincidence circuits will have one of their two branches at a potential of \(+300\ \mathrm{V}\) and will not operate. Therefore, when the pulse to be recorded arrives at the grid of tube \(L_8\), the coincidence circuit that has operated will send into the corresponding scaling circuit a pulse that will cause it to operate. (The small negative pulses passing through the other coincidence circuits are too small to affect the corresponding scaling circuits.) The pulse from the coincidence circuit is generated against a background of constant voltage equal to \(+300\ \mathrm{V}\). This permits a direct connection of the diode to the first cell of the scaling circuit (see Fig. 27). The states in which the various tubes are found before a pulse enters the circuit input are indicated in the caption to Fig. 31 by the words “unlocked” and “locked.” In Fig. 32 a circuit is given for generating recorded and restoring pulses. A pulse (positive) is applied to tube \(L_2\) each time the discriminator with the smallest bias operates. After a certain selected time (for example, \(2.5\), \(5\), or \(10\ \mu\mathrm{s}\)), the blocking oscillator (tube \(L_7\)) generates the recorded pulse. The same pulse appears after \(0.5\ \mu\mathrm{s}\) at the cathode follower \(L_8\) and serves as the restoring pulse. The function of the various tubes may be defined as follows: \(L_1\) — diode restorer, serves to discharge the \(50\ \mu\mu\mathrm{F}\) capacitance into the differentiating circuit (for rapid restoration); \(L_2\) — cathode follower, applies the pulse to the grid of \(L_3\); \(L_3\) — coupling stage, locked in the normal state; this stage and the cathode follower prevent the powerful pulse from the blocking oscillator from penetrating back into the discriminator; \(L_4\) — blocking oscillator loaded by a delay line with lumped constants, which determines the delay of the recorded pulse; \(L_5\) — source of negative bias with low internal resistance for two blocking oscillators (the standard method for producing class-\(C\) bias); \(L_6\) — coupling stage, locked in the normal state; \(L_7\) — blocking oscillator generating the recorded pulse; this pulse passes through the “counting” key not shown in the diagram and switches on the differential

discriminator; \(L_8\) is the cathode-follower stage, whose task is to produce the restoring pulse.

The total number of counts is obtained by counting (with the aid of a counting circuit) the number of recorded pulses. The bias voltages applied to the various discriminators may be obtained by means of a voltage divider consisting of precision wire-wound resistors and connected between \(-150\) V and ground. It is convenient for the channel width of the discriminator to be set to certain definite values (for example, \(10\) V, \(5\) V, \(2\) V), and, in addition,

Fig. 32. Circuit generating pulses for a differential discriminator.

the entire device could be shifted up and down. It is also desirable to be able to make all the biases identical. If this can be done, then the zero adjustment for each discriminator can be carried out until all the discriminators are triggered by one and the same pulse. For calibration and checking the channel width it is necessary to use a pulse generator which generates a series of pulses uniformly distributed in time, whose envelope first increases, then decreases, etc., linearly in time. An equal number of counts obtained for each channel indicates that all channels have the same width. The pulses applied to the differential discriminator should, as far as possible, have an approximately flat maximum lasting not less than \(0.5\ \mu\text{sec}\), and should then fall rapidly. Such pulses are best obtained by means of a pulse-forming delay line and a highly stable feedback amplifier having a rise time of \(0.5\ \mu\text{sec}\) or greater.

IX. COUNTING-RATE METER

Counting-rate meters are often used in various instruments and, probably, can be used in laboratory practice to replace or supplement ordinary scaling circuits. To make sure that such an instrument is operating correctly, one can use an additional external circuit, which is controlled by a push-button switch and can feed to the meter pulses following one another at a known rate. In order to set this rate, one should use an alternating-current network giving a frequency of 50 cps. With the aid of such a device one can calibrate the meter. In order to understand the principle of operation of a counting-rate meter, let us turn to the circuit shown in Fig. 33.

Fig. 33. Schematic circuit of a counting-rate meter.

It is assumed that each pulse arriving from some discriminator actuates a circuit producing a standard rectangular pulse \(E\), the form of which is shown at the left in Fig. 33. Only the height \(E\) of the pulse must be constant; its duration is not so critical a quantity, since the capacitance \(C_1\) is fully charged by the end of the pulse. Suppose that the following conditions hold:

\[ \begin{aligned} \frac{1}{n} &\gg T > 5R_1C_1,\\ E &\gg E_0 > E_B,\\ C_2 &\gg C_1, \end{aligned} \tag{67} \]

where \(n\) is the counting rate. In this case each pulse transfers to the capacitance \(C_2\) a charge \(q = C_1E\), and therefore the mean potential \(E_0\), arising at the counting rate \(n\), is equal to:

\[ E_0 = nqR_2 = nC_1ER_2. \tag{68} \]

The scale of the meter can therefore be changed by switching in accurately measured resistances \(R_2\). A small negative bias \(E_B\) (at most several volts) is necessary in order to make the scale linear over the whole range of variation of the output voltage. If this bias is not applied, difficulties may arise connected with the contact potential difference in the diodes.

The voltage produced by each incoming pulse across capacitance \(C_2\) is equal to

\[ V(t)=\left(\frac{C_1}{C_2}\right)E\exp\left(-\frac{t}{R_2C_2}\right). \tag{69} \]

The mean square of the voltage fluctuations in the counting of random pulses can be calculated on the basis of Campbell’s theorem

\[ (\Delta E_0)^2=\bigl[\overline{V(t)}-E_0\bigr]^2 = n\int_{-\infty}^{+\infty} V^2(t)\,dt . \tag{70} \]

Integration gives:

\[ (\Delta E_0)^2=\frac{nC_1^2E^2R_2}{2C_2}. \tag{71} \]

Hence, for the relative probable error we obtain:

\[ \varepsilon=0.67\,\frac{\Delta E_0}{E_0} = \frac{0.67}{\sqrt{2nR_2C_2}} . \tag{72} \]

It follows from equation (68) that, for a given reading of the instrument, the quantity \(nR_2\) is fixed, and therefore a definite relative probable error (for deflection over the full scale) corresponds to a definite value of \(C_2\).

Fig. 34. Part of the circuit of a counting-rate meter.

Figure 34\(^{10}\) shows the circuit of that part of a typical counting-rate meter in which the pulse is generated. Each positive trigger pulse arriving from the discriminator causes the appearance at the anode of \(T_2\) of a negative rectangular pulse (of duration about \(15\ \mu\text{sec}\)). Since the time constant \(R_1C_1\) is approximately \(1.5\ \mu\text{sec}\), and the pulse amplitude at the anode is approximately \(100\ \text{V}\), conditions (67) are satisfied. The resistances \(R_2\) can be chosen in such a way that

cover, for example, the count-rate region from 2000 to 50,000 counts per minute. The capacitances \(C_2\) are chosen in such a way as to provide probable errors equal to 1, 2, 5, 10, and 20% of full scale. In other, simpler but somewhat less stable count-rate meters, a locked pentode is used as the source feeding the capacitance \(C_2\) (Fig. 33). Each input pulse produces a standard pulse, which is applied to the grid of the pentode. In this case the averaging circuit receives from the pentode a definite quantity of charge.

X. CIRCUITS FOR OBSERVING RANDOM PULSES¹¹

A cathode-ray oscilloscope, together with a triggered sweep and an amplifier containing a high-quality delay line, makes it possible to investigate any individual pulses arriving from the pulse amplifier, regardless of whether they are distributed in time randomly or regularly. A block diagram of a typical installation of this kind is shown in Fig. 35.

Fig. 35. Block diagram of an apparatus for observing the shape of random pulses on the screen of an oscilloscope tube.

The individual parts of this installation have the following purpose. The discriminator, which may be a simplified version of the discriminator shown in Fig. 26, serves to select for observation only those pulses whose amplitude is sufficiently large. In particular, it protects the triggered-sweep circuit from being triggered by pulses produced by interference. The triggered sweep, started by a pulse arriving from the discriminator, generates one pulse of linear sweep, the speed of which is known. At the same time a pulse controlling the illumination of the tube is generated, whose duration is equal to the duration of the linear sweep. Since, for good focusing, capacitive coupling between the deflecting plates of the oscilloscope tube and the circuit is usually necessary

sweep; the latter must contain diode restorers, owing to which each new sweep line starts from one and the same point. The sweep speeds usually employed are 0.3, 1, 3, 10, etc. μsec per inch. It should be noted that a device making it possible to vary the sweep speed smoothly offers no advantage unless a convenient method for calibrating the sweep speed is always at hand. Such a necessity causes an undesirable additional complication of the circuit, and therefore the use of smooth adjustment of the sweep speed cannot, as a rule, be recommended. The delay circuit, as its principal part, contains a high-quality delay line. (Delay lines will be considered in Section XI.) Most delay lines suitable for practical use have a small wave resistance (of the order of 1000 ohms and less). It is therefore advisable that the pulses passing along the delay line have an amplitude of only a few volts. Consequently, the delay-line circuit must contain devices by means of which the amplitude of the pulses arriving from the pulse amplifier could be reduced. These devices must not introduce distortions that may arise from the influence of the parasitic capacitance or inductance of the attenuator. Another solution of this problem is to connect the delay-line circuit to that point in the pulse amplifier at which the signal level does not exceed a few volts. These two connection methods are shown by dotted lines in Fig. 35.

The deflection coefficient of the cathode-ray tube 5CP1 (or of the newer tube 5CP1A) in an ordinary portable oscilloscope is approximately 45 V per inch. If, in order to facilitate observation of fast individual pulses, the total accelerating voltage is increased to 6 kV, then the deflection coefficient becomes approximately 110 V per inch. In any case, between the delay line and the vertical deflection plates a transition amplifier is necessary. For good focusing and for obtaining large deflections, this amplifier must have a push-pull output. In order to ensure good transmission of the fronts of fast pulses, the amplifier should be located immediately at the base of the cathode-ray tube. The cathode-ray tube 5CP1 is most suitable for visual observation of pulses whose rise time is 0.1 μsec or more. For photographing individual pulses, this tube should be replaced by a 5CP11. By applying 6 kV to this tube, it is possible to photograph a beam moving at a speed equal to 15 inches per microsecond. Another tube used in cases where it is important that the capacitance values be minimal (to obtain a shorter rise time at the amplifier output) is the 5JP11 tube. (In the 5JP11 tube the leads

deflecting plates are located on the neck of the tube.) The quality of focusing provided by this tube is, however, inferior to the focusing achieved in the 5CP11 tube. In those cases where the greatest light output is required, the 5RP11 tube may be used. By applying to this tube the maximum potential of 35 kV, one can photograph a beam moving at a speed equal to 200 inches per microsecond.

The problem of constructing an amplifier and sweep which would be sufficiently fast to provide such a beam speed is a very difficult one.

1. Linear sweep generators

One of the simplest methods of obtaining an approximately linearly increasing voltage for the sweep of an oscilloscope is to charge a capacitance with a constant current. The rate of rise of the voltage will then be determined by the relation \(dV/dt = i/C\). For example, if \(i = 5\) mA and \(C = 100\) micromicrofarads, then \(dV/dt = 50\) V per microsecond, which corresponds to approximately one inch per microsecond for an ordinary portable 5-inch oscilloscope. In order to attain a high degree of linearity, it is necessary to maintain as nearly constant as possible the current charging the capacitance as the voltage across the capacitance increases.

In practice the sweep voltage is usually the initial value of a voltage produced in a transient regime, which exponentially tends toward its maximum value. If this maximum is the positive voltage of the power source, for example \(+300\) V, then the signal must be amplified in order to obtain a sufficiently linear oscillation of the voltage. This method is usually used in thyratron sweep generators employed in modern commercial oscilloscopes. Another common method of obtaining a linear sweep is the use of a pentode as a source of constant current charging a capacitance. Although good linearity can be achieved with this method, the most satisfactorily operating sweep generators now use circuits with feedback. The circuit in Fig. 36 shows the basic sections of a feedback generator producing a sawtooth sweep. In the absence of a negative rectangular signal applied to the grid of \(Л_1\), current flows through diode \(Л_3\), resistance \(R\), and through pentode \(Л_1\). The anode voltage of \(Л_1\) at this moment is equal to 10–15 V. When pentode \(Л_1\) is cut off by a negative rectangular signal, current through resistance \(R\) enters capacitance \(C\) and thereby increases the grid potential of the cathode follower \(Л_2\). The cathode “follows” the

with the grid and is therefore a source of the output signal with a small internal resistance.

The output signal is fed back to the upper part of the resistor \(R\). Thus, the rise of the potential in the upper part of the resistor is almost equal to the rise of the potential on the grid (the diode serves to disconnect the upper end of the resistor from the battery). Consequently, the current flowing through the resistor \(R\) is maintained almost constant, as a result of which an almost linear sweep is obtained. The charge necessary to obtain the sweep comes, of course, from the capacitance \(C_1\). Its value is chosen so as to ensure the case of the greatest sweep duration. Simple analysis shows that the charging current is equal to:

Fig. 36. Sawtooth-signal generator (with feedback). The sweep speed is regulated by changing \(C\) and \(R\).

\[ I=I_0-\frac{V_C}{FR}, \tag{73} \]

where \(V_C\) is the voltage on the capacitance \(C\), and

\[ F=\frac{1}{1+\dfrac{C}{C_1}-A}, \tag{74} \]

where \(A\) is the gain of the cathode follower. In practice the values of \(F\) may lie between 10 and 25, depending on the chosen parameters of the circuit. It has been found that the voltage \(V_C\) is equal to

\[ V_C=\left(\frac{I_0}{C}\right)t \left[ 1-\frac{1}{2F}\left(\frac{t}{RC}\right)+\ldots \right]. \tag{75} \]

Analyzing this expression, one can estimate the deviation from linearity. The output signal is \(V_C\) times greater than the gain of the cathode follower (which, of course, is somewhat less than unity). Let us note that

the anode of the cathode follower is connected to \(+450\ \mathrm{V}\), which can be obtained from the output of the voltage stabilizer in the power-supply unit. This increases the permissible fluctuation of the sweep voltage and does not greatly impair the stability of the circuit. The sweep rates can be changed by switching in the corresponding values of \(C\) and \(R\) by means of a two-section rotary switch. The capacitance \(C_1\) must exceed the largest value of \(C\) used by at least a factor of 10 (in order to keep the ratio \(\dfrac{C}{C_1}\) in equation (74) sufficiently small). The recovery time of the circuit is determined primarily by the time required for the capacitance \(C_1\) to lose the charge acquired during the rise of the sweep voltage. The order of magnitude of this time can be estimated from the ratio \(\dfrac{I_0T}{0.006}\), where \(I_0\) is the current flowing through \(R\), \(T\) is the duration of the sweep, and \(0.006\) is the approximate current (in amperes) through \(R_b = 30\ \mathrm{k}\Omega\) during the recovery time of the circuit.

Fig. 37. Inverter stage for the sweep circuit.

If \(I_0 = 2\ \mathrm{mA}\) and \(T = 100\), then the recovery time is approximately \(30\ \mu\mathrm{s}\), and the filling factor (i.e., the part of the sweep time during which the voltage rises linearly), obviously, cannot be appreciably greater than 0.5; i.e., the circuit is most reasonably used in those cases where it can, under the experimental conditions, spend a considerable part of its sweep time waiting for the next trigger signal. In most practical cases of interest in nuclear physics, this circumstance is not a drawback of the circuit.

In the inverter stage shown in Fig. 37, voltage feedback is used in order to obtain good stability and linearity. This circuit makes it possible to invert the positive sawtooth sweep pulse. The negative sawtooth pulse obtained is applied to the second horizontal deflecting plate of the cathode-ray tube. The stage has voltage feedback effected by the direct connection of the anode with the grid through the parallel-connected \(R\) and \(C\). The circuit can invert both fast and slow sweep pulses equally well, provided only that the position of the small trimmer capacitor \(C\) is adjusted correctly. Since in such a circuit degeneration will inevitably take place (a gain of the order of 200 is fed back to a point of the circuit with a gain of the order of unity), the anode voltage

it is quite possible to take from the unstabilized side of the power supply. If then, by a corresponding change of \(R_g\), the voltage on the plate of the tube is chosen equal to \(+375\ \text{V}\), then at the output a negative signal exceeding \(200\ \text{V}\) is obtained. The grid current is then absent, and the linearity of the operation of such a circuit is excellent.

The most convenient type of circuit producing pulses for blanking is a circuit that is automatically restored as soon as the sweep reaches the specified length. Such a mode of operation of the circuit can be realized if one uses part

Fig. 38. Circuit of a generator producing the blanking pulse.

of the positive sawtooth pulse, taken from some point of the resistance \(R_k\) in Fig. 36, to return the flip-flop relay to the state in which it was before it was thrown over at the beginning of the sweep. A satisfactory circuit of this type is shown in Fig. 38. Tube \(L_2\) and part (cathode, grid, screen) of tube \(L_3\) form a flip-flop relay circuit having two stable states; in this circuit \(L_3\) is normally cut off, and a small current flows in its grid circuit. A fast positive trigger pulse applied to the grid of \(L_1\) (normally cut off) throws the flip-flop relay circuit into its second stable state. It is clear that, if the rise time of the trigger pulse is small, then the grid of \(L_3\) very quickly (approximately after \(0.1\ \mu\text{s}\)) becomes negatively charged. Thus, at the plate of \(L_3\) a fast positive rectangular blanking pulse is obtained, which then goes to the cathode follower \(L_5\).

The grid of \(L_3\) is directly connected to the grid of \(L_1\) in Fig. 36 (by means of a resistance and a capacitance, not shown in this diagram), and thus the sweep oscillation begins almost simultaneously with the blanking rectangular pulse. The anode of diode \(L_4\) ne-

directly connected with some point of the resistor \(R_k\), the voltage at which is substantially negative. As the sweep oscillation develops, the potential of this point increases, and at a certain value of the potential the diode begins to conduct. In this case the grid potential of \(L_3\) passes into that range of values at which \(L_3\) conducts, which returns the whole flip-flop circuit to its original state. At this moment both the sweep and the illuminating rectangular pulse end. The circuit that creates the illuminating pulse will be ready for the next operation as soon as the magnitude of the charges on the various capacitances of the circuit reaches its normal value. Other common variants of the circuit shown in Fig. 38 are based on different methods of terminating the sweep. For example, a tap from \(R_k\) may be connected to the grid of the pentode, which then plays the same role with respect to \(L_3\) as \(L_1\) does with respect to \(L_2\).

A more improved method is the use of a regenerative amplifier with a transformer from the blocking oscillator. The signal from the tap at the cathode resistor \(R_k\), applied through one winding of the transformer to the grid of the amplifier, opens the tube. A fast positive pulse, whose amplitude may be equal to 200 V or more, is applied from the third winding of the transformer through a cathode follower to the grid of \(L_3\) and thus quickly terminates the sweep. The second winding of the transformer is, of course, regeneratively connected with the anode circuit of the amplifier. Although the latter method is very effective for rapidly terminating the sweep, its realization requires a larger number of circuit elements than the method shown in Fig. 38. The diode shown in Fig. 38 may be the second half of the tube used in the feedback sweep circuit (Fig. 36), provided that the diode filament is at a potential of 300 V (for this purpose one may use, for example, the filament supply source of the second tubes in an electronic voltage stabilizer).

A complete sweep circuit, consisting of the various elements described above, should usually have the following adjustment possibilities:

1. A switch for selecting the sign of the input pulse.
2. A potentiometer for adjusting the bias on the discriminator.
3. A switch for the sweep speed.
4. Adjustment of the horizontal position of the beam.
5. Additional focusing control (see below).
6. Adjustment of the sweep length (optical).
7. Adjustment of the intensity of illumination (optical).

In modern oscillographs, along with adjustment relating to the triggered-sweep circuit, adjustment of the beam verticality, etc., is provided. It is often convenient to have, in addition to the triggered sweep, a generator of trigger pulses.

2. Bias circuit applied to the oscilloscope tube

It is necessary to make several comments concerning the construction of the bias circuit for the oscilloscope tube. Let us turn to the circuit in Fig. 39.

1) The voltages for centering the beam are obtained from a double potentiometer (one megohm each), connected to a \(+300\ \mathrm{V}\) power supply.

2) The potential on the second anode (additional focusing) can be adjusted in such a way that it is almost equal to the mean potential of the deflecting plates. A small change in the potential on the second anode creates a weak electrostatic lens, which can improve the focusing of the beam.

Fig. 39. Typical bias circuit for a 6SR11 tube operating at \(6\ \mathrm{kV}\).

3) Diode sweep restorers are shown in the sweep circuit. Note that the capacitances \(C_3\) are necessary for the operation of the diode restoration.

4) The time constant \(R_1C_1\) in the circuit of the illuminating device (grid) is matched to the time constant \(R_2C_2\) in the cathode circuit. The introduction of such decoupling circuits is very useful when there are voltage pulsations at the \(2\ \mathrm{kV}\) supply source, since these pulsations are distributed uniformly between the grid and the cathode if the grid is connected to a source with low internal resistance (for example, as is usually the case, to a cathode follower). If \(R_2\) is absent, then the voltage pulsations can cause modulation

intensity of the beam (with a frequency of 50 cycles). If the voltage source produces no pulsations, then the resistance \(R_2\) need not be included.

5) In some cases, in order to improve the focusing of the beam by means of a second weak electrostatic (cylindrical) lens (not shown in Fig. 39), it is highly desirable to provide the possibility of regulating the mean potential by means of one of the pairs of deflecting plates.

6) The time constants in the circuits of the deflecting plates must be made sufficiently large to ensure undistorted transmission of the longest pulses. (The considerations on which this requirement is based were examined by us in detail in connection with the question of transient phenomena in the amplifier.)

7) It is often desirable to have a diode rectifier in the grid circuit of the tube, so that no special adjustment of the intensity is required for sweeps with different filling factors.

The diode filament may be connected to the filament voltage source of the oscillograph tube (6.3 V).

3. Amplifiers for cathode oscillographs

For applying the deflecting voltage to the plates controlling the vertical displacement of the beam, it is desirable to use an amplifier with push-pull output. Such an amplifier provides not only better focusing, owing to the fact that the mean potential of the deflecting plates is kept constant, but also makes it possible to obtain large amplitudes of the amplified voltage while maintaining good linearity. The suitability of the tubes used at the amplifier output (understanding by the suitability of a tube the greatest attainable ratio of signal amplitude to rise time) is evidently determined by the value of the ratio \(\dfrac{I_a}{C_{\mathrm{out}}}\), since the maximum signal amplitude cannot be greater than \(R_{\mathrm{load}} I_a\), and the rise time cannot be less than the time determined by \(R_{\mathrm{load}} C_{\mathrm{out}}\). In practice the best tubes are powerful beam pentodes (or tetrodes) of the types 6AG7, 807, and 829-B. The transition from single-ended to push-pull operation requires the use of an inverting (in phase) stage. If feedback is used in the amplifier for stabilization, then the inverting stage shown in Fig. 40 is quite satisfactory. The output stages of modern oscillographs

Fig. 40. Phase-inverting amplifier.

DuMond’s are made according to similar circuits. Reversal of the pulse phase is produced by the presence of the cathode resistance \(R_k\), whose value must be taken large in comparison with \(\frac{1}{G_m}\), where \(G_m\) is the transconductance of the tube (for example, \(R_k \simeq \frac{10}{G_m}\)). Although the circuit of Fig. 40 is not stabilized by feedback, it can be used over a very wide frequency range if sufficient attention is paid to the choice of anode loads, possibly compensating them by inductances.

The gain provided by such a circuit is somewhat less than the gain that each tube of this circuit would give if \(R_k\) were bypassed. In those cases where greater stability and linearity of operation of the circuit are required, negative feedback may be used in the output stages. Figure 41 shows a type of circuit that has proved successful in such cases.

Fig. 41. Circuit of the output stages of a cathode-ray oscilloscope amplifier.

Tube \(L_1\) creates a symmetrical (push-pull) signal feeding two identical feedback loops, one of which is shown in the diagram. The gain in the feedback loop is somewhat less than \(\frac{R_F + R_k}{R_k}\). It should be noted that in this feedback circuit the blocking capacitor has been removed in order to avoid the appearance of a large parasitic capacitance with respect to ground. Removing this capacitance somewhat complicates the design of such an amplifier in the sense of selecting the dc components of current and voltage; however, usually this can still be done satisfactorily. Capacitances made in the form of trimmers are used when adjusting the amplifier in order to obtain good reproducibility of signals having the form of a pulse.

XI. PULSE FORMATION BY MEANS OF A DELAY LINE

We have already indicated that an amplifier used together with a radiation detector must convert voltage jumps into a series of separate pulses, which can be selected according to amplitude, frequency of occurrence, distribution in time, and other characteristics. The general method of pulse formation consists in reducing, in the amplifier, one of the time constants \(RC\) to a value at which a sudden voltage jump at the input is reproduced by the amplifier as a pulse of the required duration. However, such a pulse will not have an ideal shape because the tail part of the pulse has a long duration compared with the short duration of the sharpest part of the pulse. In some cases it becomes necessary to have pulses of approximately rectangular shape. This pulse shape is the best in all problems connected with counting pulses. The basic idea underlying the use of a delay line for pulse formation is as follows.

Let us imagine that at some point in an amplifier the pulse branches into two channels. One channel contains a delay line with a delay time \(T\) sec. and has gain \(A\). The other channel has no delay line and its gain is equal to \(-A\). Suppose that a unit voltage step is applied to the input of the amplifier and that the rise time in the amplifier is considerably less than the time \(T\). In this case the output signal obtained by adding the two signals arriving from both channels of the amplifier is a rectangular pulse of duration \(T\). In practice the same result can be achieved by simpler means, using a short-circuited delay line as a coupling element in the amplifier. To understand this method, let us assume that the delay line is an ideal line, i.e. that it introduces no losses and does not distort the shape of the signal. Suppose further that the line is connected to a circuit whose equivalent diagram is shown in Fig. 42, where \(V_i\) is the generator producing the voltage step \(V_i\), \(V_0\) is the output pulse, and \(Z_C\) is the characteristic impedance of the delay line. For a voltage step, at the initial instant of its rise the delay line represents a characteristic impedance (purely ohmic), and therefore \(V_0 = -\frac{1}{2} V_i\). The voltage wave, moving along the line, reaches its short-circuited end and is reflected back with a change of phase. After a time \(2T\) the reflected wave reaches

Fig. 42. Equivalent circuit for pulse formation by means of a delay line.

of the input end of the line, where it is completely absorbed by the series-connected resistance \(R\). Now the delay line will act as a short-circuited section of the circuit, and the output signal will become equal to zero. Thus, a voltage step is transformed into a rectangular pulse of duration \(2T\). This somewhat unusual way of considering the properties of a short-circuited line (the line is regarded as a resistance having at first one value and then another) can also be applied in the real case, when the line has a finite resistance \(R_0\) for direct current. It is evident that in this case, at the end of the time interval \(2T\), the magnitude of the output signal will fall not to 0, but to the value

\[ \left(\frac{R_0}{R_0 + Z_C}\right) V_i . \]

When using this method, the indicated deviation of the line from the properties of an ideal line must be compensated in some way. Two methods may be proposed for this:

1. The short-circuited delay line may be made an arm of a resistance bridge balanced for slowly varying signals.

2. The voltage step may be “differentiated” by means of an \(RC\) circuit. In this case the voltage drop will follow an exponential law, and with a suitable \(RC\) it is possible to make the decrease in the amplitude of the signal during the time \(2T\) equal to the decrease in amplitude occurring in the reflected wave. The subtraction of the pulses will take place as before, but the top of the formed rectangular pulse will have a slight negative slope. To compensate the direct-current resistance of the delay line, the second method is usually used. The finite rise time inherent in the delay line makes its contribution to the total rise time of the whole amplifying system.

The law according to which rise times are added was discussed in one of the preceding sections and requires no further explanation. For shaping pulses one may either use an ordinary high-quality line employed for signal transmission, or design a special line. In choosing a suitable line it is necessary to ensure as large a ratio as possible

\[ \frac{Z_C}{R_0}, \]

in order to reduce the amount of compensation required.

Fig. 43.

Let us consider in more detail the case of an ideal delay line. Such a consideration is of interest for clarifying the principles of operation of delay lines, but it is of little applicability to the practical design of high-quality delay lines, which is usually carried out

semirigorously. Consider the four-terminal network shown in Fig. 43, whose properties are such that the input and output voltages are related as follows:

\[ V_0(t)=V_i(t-\tau), \tag{76} \]

where it is assumed that \(V_i(t)=0\) for \(t<0\). Performing the Laplace transform on equation (76), we have:

\[ E_0(s)=\int_0^\infty e^{-st}V_i(t-\tau)\,dt =\int_0^\infty e^{-s(t+\lambda)}V_i(\lambda)\,d\lambda \]

or

\[ E_0(s)=e^{-s\tau}E_i(s). \tag{77} \]

It follows from this that an ideal delay line has as its image the function \(e^{-s\tau}\) (with the possible addition of a constant numerical factor allowing for attenuation of the signal). If it is assumed that a sinusoidal voltage is applied to the ideal four-terminal network under consideration (putting \(s=i\omega\)), then the image takes the form \(e^{-i\omega t}\), so that, if the input signal is \(e^{i\omega t}\), the output signal has the form \(e^{i\omega(t-\tau)}\). Therefore the phase shift introduced by the four-terminal network is a linear function of frequency, while the amplitude does not depend on frequency. We shall now show that a lossless transmission line also has as its image the function \(e^{-s\tau}\). Such a line is shown in Fig. 44, where \(L_1\) and \(C_1\) are the distributed inductance and capacitance per unit length of the line. The line connects the signal source to a load \(R\) equal to the wave resistance of the line \(\left(\dfrac{L_1}{C_1}\right)^{1/2}\). For an infinitely small section of the line \(dx\) we have the following equations:

Fig. 44. Diagram for analyzing a transmission line.

\[ \frac{\partial V}{\partial x}=-L_1\frac{\partial i}{\partial t},\qquad C_1V=-\int \frac{\partial i}{\partial x}\,dt, \tag{78} \]

from which one obtains the wave equation for the signal voltage

\[ \frac{\partial^{2}V}{\partial t^{2}}=\frac{1}{L_{1}C_{1}}\frac{\partial^{2}V}{\partial x^{2}}, \tag{79} \]

showing that the propagation velocity of the signal is equal to

\[ v=\frac{1}{\sqrt{L_{1}C_{1}}}, \]

i.e. that the delay introduced by a section of line of length \(1\) is equal to \(\sqrt{L_{1}C_{1}}\). Performing the Laplace transform on equation (79), we obtain:

\[ \frac{d^{3}E}{dx^{3}}-\frac{s^{3}}{v^{3}}\,E=0. \tag{80} \]

Equation (80) has the following general solution:

\[ E(x,s)=Ae^{\frac{x}{v}s}+Be^{-\frac{x}{v}s}, \tag{81} \]

where \(A\) and \(B\) are functions of \(s\) (but not of \(x\), of course). Subjecting the first of equations (78) to the Laplace transform, we find that the image of the current satisfying equation (81) has the form

\[ I(x,s)=-\sqrt{\frac{C_{1}}{L_{1}}}\left(Ae^{\frac{x}{v}s}-Be^{-\frac{x}{v}s}\right). \tag{82} \]

At the remote end of the line \(x=l\), and we have:

\[ \frac{E(l,s)}{I(l,s)}=R=\sqrt{\frac{L_{1}}{C_{1}}} = \frac{Ae^{\tau s}+Be^{-\tau s}} {-\sqrt{\frac{C_{1}}{L_{1}}}\left(Ae^{\tau s}-Be^{-\tau s}\right)}, \tag{83} \]

where \(\dfrac{l}{v}=\tau\) is the total delay.

Obviously, equation (83) requires that \(A=0\), i.e. if the line is loaded by its characteristic impedance \(\left(\dfrac{L_{1}}{C_{1}}\right)^{\frac12}\), then in this case the reflected wave \(A\exp\left(\dfrac{x}{v}s\right)\), which would travel back toward the source, is absent. Further, for \(x=0\) we obtain the image of the input voltage

\[ E(0,s)=B=E_i(s). \tag{84} \]

It is obvious that the image of the output signal will be

\[ E_0(s)=E(l,s)=Be^{-\tau s}=E_i(s)e^{-\tau s}. \tag{85} \]

This expression is identical with expression (77), i.e. an arbitrary input signal is related to the output signal by expression (76). The resis-

the impedance of a loaded line, measured at any point of it (between \(x=0\) and \(x=\ell\)), is equal to

\[ \frac{E(x,s)}{I(x,s)} = \frac{B e^{-\frac{x}{v}s}} {\sqrt{\frac{\overline{C}_1}{L_1}}\, B e^{-\frac{x}{v}s}} = \sqrt{\frac{L_1}{C_1}}. \]

The solution given shows that an ordinary coaxial cable can be successfully used as a delay line. For short delay times a coaxial cable is a sufficiently convenient form of delay line. However, for delays exceeding several tenths of a microsecond, a cable of excessively great length is required. The obvious method of creating a compact delay line consists in increasing the inductance and capacitance per unit length. This increase can be achieved by placing a certain number of strips of metal foil along the outer or inner surface of a narrow solenoid, as shown in Fig. 45. The distributed capacitance of these strips per unit length is equal to \(C_1\). Dividing the capacitive electrode into strips prevents the appearance of short-circuit currents induced by the solenoid. Another method of creating a delay line is the use of mutually unconnected inductances and capacitances, as shown in Fig. 46. In this case the delay produced by one section is equal to \(\sqrt{LC}\), and the characteristic impedance is equal to \(\sqrt{\frac{L}{C}}\). The line with lumped parameters obtained in this way distorts the shape of a fast pulse, superimposing damped oscillations on it. Therefore such a line is used mainly for delaying trigger pulses.

Fig. 45. Method of constructing a delay line.

Fig. 46. Delay line with lumped parameters.

For correcting the characteristic of a delay line, in some cases a carefully designed correcting circuit is used. But even in the presence of such correction, a signal in the form of a unit function cannot propagate along the delay line without the front of this signal gradually becoming smeared out. It can be shown that this fact is partly explained by the presence of inductive coupling between neighboring sections of the line (if the line is continuous). Therefore one should expect that the slope acquired by the front during the motion of the signal along the line is proportional

diameter of the solenoid. All real delay lines differ from an ideal delay line by the presence of a finite rise time for the front of a unit step. A high-quality line has, however, a monotonic “transient response.” The quality of a delay line is determined by the ratio of the delay time to the rise time. The usual values of this quantity lie between 10 and 50, i.e., one can obtain a delay line 2 μsec long with a rise time of less than 0.1 μsec.

It is desirable to feed the delay line from a voltage source having a small internal resistance (for example, a cathode follower may be used) and giving a signal not exceeding a few volts, since the characteristic impedance of the delay line is small. If difficulties arise, connected with reflection from the outer end of the line of the high-frequency components of a fast signal, then usually some improvement can be achieved by making the internal resistance of the source feeding the line equal to the characteristic impedance of the line. Such mutual matching of the input and output ends of the line requires parallel or series connection of additional resistances (provided that the signal source does not have the required characteristic impedance) and therefore causes a reduction in the signal amplitude.

The usual method of feeding a delay line consists in connecting the line as a load in the anode circuit of a pentode amplifier stage. In this case, for correct loading of the input end of the line, it is necessary to connect in parallel with the line a resistance equal to \(\left(\dfrac{L_1}{C_1}\right)^{\frac12}\), since the internal resistance of the tube is very large.

In this method the stabilization by feedback which occurs in a cathode follower is not used, and therefore it should not be applied in those cases where the output part of the amplifying system contains a feedback loop. Even in the case of a cathode follower, linearity and stability deteriorate if the cathode resistance is comparable with \(\dfrac{1}{G_m}\). Therefore it is sometimes desirable to feed the delay line from an amplifier at whose output there is a feedback loop similar to that indicated earlier.

Pulse shaping by means of a delay line is carried out with the greatest success if the duration of the pulses being shaped does not exceed approximately 10 μsec. Investigations of the signal-to-noise ratio, carried out by the author for pulse amplifiers, show that this ratio is greatest if the delay \(2T\) is approximately equal to the rise time of the amplifier. In this case the pulse has a shape approximately that of a Gaussian error curve, and not a rectangular one.

XII. PULSE GENERATION

Pulse generators are often used in nuclear experiments: 1) to produce periodic time signals, 2) to excite one or a series of processes that stand in a definite time relation to the exciting pulse (trigger-pulse generators), 3) to calibrate the gain of pulse amplifiers, 4) to study transient regimes of amplifiers, 5) to form separate elements of coincidence circuits.

We shall first consider several circuits used for pulse generation, and then describe a pulse generator intended for the investigation and calibration of pulse amplifiers.

1. Thyratron Circuit

Figure 47 shows a circuit (using a 2050 thyratron) intended for generating fast positive and negative pulses of very short duration and identical shape. The time constant \(RC\) determines the minimum time interval between two successive pulses; during a time \(3RC\) after the thyratron fires, capacitor \(C\) charges almost completely. The thyratron fires with some delay, which, however, is less than \(0.1\ \mu\text{sec}\), provided that the input trigger pulse on the thyratron grid is sufficiently large. The current through the thyratron can rise to its full value within \(0.01\ \mu\text{sec}\).

Fig. 47. Method of generating fast pulses by means of a thyratron.

Sometimes, to produce periodically recurring pulses, it is convenient to use a thyratron as a relaxation oscillator. For this purpose the circuit shown in Fig. 48 is used. The 884 thyratron (triode) does not have as steep a firing characteristic as the 2050 thyratron (tetrode), and therefore is a more stable relaxation oscillator. The shape of the output pulse is determined by the values of \(R_1\) and \(C_1\) and does not depend on \(C\). The value of \(C\), together with \(R\), at a given bias voltage, determines the pulse repetition frequency. (In both circuits of Figs. 47

Fig. 48. Thyratron circuit of a relaxation generator.

and 48 the anode resistance \(R\) must be chosen sufficiently large to ensure quenching of the thyratron.)

The shape of the output pulse produced by the thyratron circuit can be made almost rectangular. For this it is necessary to replace the capacitance whose discharge forms the pulse (\(C_1\) in Fig. 48) by an unloaded delay line. To avoid reflection from the input end of the delay line, the value of the resistance across which the pulse is formed (\(R_1\) in Fig. 48) must be made equal to the characteristic impedance of the line. This method is sometimes used for generating the rectangular pulses required in modulating magnetrons in radar. It is also sometimes used in laboratory pulse generators.

2. Blocking Oscillator

The blocking oscillator is a very convenient circuit for generating fast and narrow pulses, especially in the case of coincidence circuits with high resolving power. It can also be used in relaxation generators and in frequency-conversion circuits.

Fig. 49. Circuit of a blocking oscillator intended for triggering by an external trigger pulse.

A typical circuit operating from an external trigger pulse is shown in Fig. 49. Both tubes \(Л_1\) and \(Л_2\) are normally cut off. A fast positive trigger pulse applied to the input of the circuit (the grid of \(Л_1\)) produces a negative pulse in the anodes of tubes \(Л_1\) and \(Л_2\) and, by coupling through a special transformer, a positive pulse on the grid of \(Л_2\). Tube \(Л_2\) begins to conduct, and on its grid, owing to regeneration, a positive potential of the order of \(100\ \text{V}\) is temporarily established, which leads to an instantaneous increase of the anode current to several hundred milliamperes. The negative charge collected by the grid accumulates on capacitance \(C\) and prevents (blocks) repetition of the oscillation. The positive pulse may be taken from the resistance connected in the cathode, and the negative pulse from the anode resistance. By using a third winding of the transformer, a pulse of either sign can be obtained (accompanied by oscillations of smaller amplitude). Usually only one of these methods is used, depending on the requirements of the circuit with which the blocking oscillator is associated. The time constant \(RC\) determines the maximum repetition frequency of the pulses. There are several types of transformers commonly used in the blocking oscillator. Type 132AW may be used for generating pulses of approximately triangular form with a width of about \(0.3\ \mu\text{s}\) at the base, while type 145EW gives

pulses with a width of about 1 μsec. Other available transformers may be used to generate narrower (0.1 μsec) or broader pulses. In order for the oscillations to be sufficiently powerful, the capacitance \(C\) must usually be not less than 500 μμF. To obtain very narrow pulses one should take not less than 200 μμF. At a high counting rate, and in order to obtain narrower pulses, the 6SN7 should be replaced by tubes providing greater possibilities with respect to current and dissipated power (6AC7, 6AG7, 6V6, 6L6, etc., connected as triodes). The pulses can be made almost rectangular if 1) some resistance is connected in series with the grid, or 2) the capacitance \(C\) is replaced by an unloaded delay line. It is usually necessary to decouple the blocking oscillator from the source of plate voltage in order to avoid pickup on the supply line. The blocking oscillator can be put into such a mode of operation that free oscillations occur in it; for this it is sufficient to connect the resistance \(R\) to ground or, better, to \(+300\) V. The calculation of circuits containing blocking oscillators is usually carried out by a semi-empirical method, on the basis of accumulated experience. In order to study the behavior of a blocking oscillator, it is convenient to use an oscilloscope having a fast triggered sweep, which makes it possible to observe the shape of the pulses arising at various points of the circuit. The maximum frequency of oscillations that can be generated reliably by a blocking oscillator is of the order of \(10^6\) cycles/sec. However, the upper limit can be brought up to \(10^7\) cycles/sec. Blocking oscillators are very convenient sources of narrow standard pulses for coincidence circuits possessing high resolving power. In such applications they are triggered by pulses arriving from an amplitude discriminator.

3. Univibrator

Long pulses with a rise time of several microseconds are most conveniently obtained by means of a circuit called a univibrator. In Fig. 50 a circuit is given in which the univibrator is included as a component part. If a source of negative bias is absent, then the common cathode resistance may be used to create the bias in the univibrator. Tube \(L_1\), cut off in the normal state, serves to control the univibrator by means of a positive trigger pulse. Tube \(L_3\) is normally conducting; current flows in its grid circuit, and the plate potential of the tube is approximately \(+80\) V. Tube \(L_2\) is cut off, and the potential of its plate is 300 V. A positive trigger pulse applied to the grid of \(L_1\) lowers the plate potentials of \(L_1\) and \(L_2\). Since the grid of \(L_3\) is capacitively coupled to the plate of \(L_2\), its potential likewise decreases to the cutoff potential of the tube. The plate potential of \(L_3\) then rapidly ...

increases approximately to \(+300\) V. This change in potential is transmitted to the grid of tube \(L_2\) and causes the latter to open (current begins to flow in the circuit of the grid \(L_2\)). The new state is maintained until the capacitor \(C\) discharges through the resistance \(R\) to a potential sufficient to open tube \(L_3\) again. The circuit then returns to its initial state. The positive pulse (“pedestal”) may be taken from the anode of \(L_3\) (as shown in Fig. 50), and the negative pulse from the anode of \(L_2\). The duration of the “pedestal” is approximately equal to \(\dfrac{RC}{3}\). In practice the duration

Fig. 50. Univibrator with a decoupling tube and an output stage in the form of a cathode follower.

of the shortest pulses obtained in this circuit is \(3\)—\(5\ \mu\text{sec}\). A circuit of this type is often used as the basis for a trigger delay circuit; by differentiating a rectangular pulse, very narrow trigger pulses can be obtained. There are various methods suitable for controlling the magnitude of the delay, by means of which it is possible, for example, to make the delay depend linearly on the position of a potentiometer. It should be noted that by applying a very fast starting pulse to the grid of \(L_2\), one can obtain for the leading edge of the negative pulse at the anode of tube \(L_2\) a very short rise time (\(0.1\ \mu\text{sec}\)). This property of the circuit is often used in its various applications.

4. Generation of step pulses of prescribed amplitude

In the investigation and calibration of amplifiers used in nuclear physics, it is convenient to have a generator producing pulses whose shape is shown in Fig. 51. Let us note the following characteristic features of such pulses:

1) The rise time is several times shorter than the rise time in ordinary amplifiers.

2) The flat part of the pulse has a duration considerably greater than the small time constant in the amplifier used for shaping the pulse.

3) The pulse has a gradual fall. This fall is necessary in order to avoid the appearance at the amplifier output of a large pulse opposite in sign to the pulse arriving at the input. (Such a pulse arises owing to “differentiation” in the pulse-shaping circuits of the amplifier.)

Fig. 51. Pulses for calibrating a pulse amplifier.

If such a pulse were to arise, the amplifier would be overloaded, since the biases in the output stages are usually asymmetrical. The best method of producing step signals apparently consists in the sudden interruption of the current flowing through a resistance whose magnitude, together with the parasitic capacitance shunting it, provides a sufficiently short pulse rise time. If the voltage drop across the resistance (before the interruption of the current) is known exactly, then the amplitude of the generated step pulse, equal to this voltage drop, is also known. Fig. 52 shows a method of suddenly interrupting the current. If a pulse of the form indicated in Fig. 52 is applied to the grid, then a pulse of the form desirable for calibration arises at the anode. The screen grid of the pentode prevents distortions of the pulse shape that may arise because of the penetration of the fast pulse, by capacitive coupling, from the grid circuit of the tube into its anode circuit. If the same method is used to produce a negative pulse at the cathode, then the capacitive coupling between the grid and the cathode usually causes an undesirable distortion of the leading part of the negative output pulse. Therefore, if negative pulses are needed, it is better to generate positive pulses and then invert them with the aid of

Fig. 52. Generation of pulses for calibrating an amplifier.

of a wide-band amplifier having a negative gain equal to \(-1\) (stabilized by feedback). It is most convenient for the generated pulses to have an amplitude of from \(0.1\) to \(1.0\) V (the pulse amplitude is selected by changing \(R\) or by changing the current flowing through the pentode). Pulses of lower voltage are produced by means of a resistive attenuator, compensated by a capacitance which is best placed at the end of the cable connecting the pulse generator to the amplifier input. The frequency of the generated pulses should be chosen to be low (for example, 200 pulses per second) for the following reasons:

Fig. 53. Main part of the pulse generator.

1) To ensure the return of all potentials in the circuit to their normal state in the interval of time between two successive pulses.

2) To have a frequency convenient for the operation of an ordinary counting circuit.

3) To make it possible to measure the pulse amplitude (with a correction of only a few percent) from the value of the mean anode current in the pentode.

However, it is nevertheless desirable for the frequency of the generated pulses to be sufficiently high so that observations can be made at the amplifier output with a cathode-ray oscilloscope without resorting to a weakening of the illumination.

The generation of pulses applied to the grid of the pentode in Fig. 52 can be accomplished by means of the circuit shown in Fig. 53, which is part of a pulse generator developed by Sands\(^{10}\). The thyratron relaxation generator (tube \(L_1\)) generates about 200 pulses per second. Narrow positive trigger pulses pass through a delay line to the grid of tube \(L_2\), which in its normal state is cut off, since the potential of its cathode is higher than the grid potential by approximately 10 V. Each pulse that has passed through the delay line and arrives at the grid of this tube produces a rapid negative pulse at its anode. Through the capacitance \(C\)

this pulse is fed to diode \(L_3\), whose bias is chosen in such a way that the first part of the pulse (insufficiently fast) is not used. The voltage on the capacitance \(C\) thus has the form indicated in the drawing, with a recovery time \(RC = 100\ \mu\mathrm{sec}\). It is obvious that as soon as the pulse has been formed, the diode disconnects the capacitance \(C\) from the anode of tube \(L_2\). The positive pulse taken from the cathode of \(L_1\) through a cathode follower (not shown) serves as the trigger for starting the oscilloscope sweep. The base of the output pulse lies \(25\ \mathrm{V}\) above the potential \(-150\ \mathrm{V}\), owing to the fact that the cathode resistance \(R_k\) in Fig. 53 may be taken sufficiently large. A sufficiently large value of \(R_k\) ensures stabilization of the dc component of the current flowing through the tube serving as the pulse generator (owing to feedback through the cathode). The amplifier with gain \(-1\), used for inverting the output pulses, may be constructed according to the pattern of the three-stage feedback amplifier described in Section V. The negative pulses are then taken from the anode of the third tube of the amplifier.

XIII. STABILIZED POWER SUPPLIES

Power supplies for anode and grid circuits are stabilized in order to

1) make negligibly small the fluctuations of the dc voltage produced by fluctuations of the voltage in the ac line,

2) have voltage sources with low internal resistance (this is necessary for reducing mutual influences between different parts of the circuit connected to the source), and

3) reduce the ripple of the rectified voltage.

1. Sources with voltage regulators (“stabilovolts”)

The simplest form of a stabilized source, used when the current consumption is small, is a source stabilized by a glow-discharge tube, otherwise called a stabilovolt. Such tubes are, for example, the VR-75, VR-90, and VR-105 tubes. The equivalent circuit of such a source is shown in Fig. 54, where \(E\) is the equivalent voltage source (actually consisting of transformers, rectifiers, and filters), \(R_1\) is its equivalent internal resistance, and \(R_2\) is a series-connected resistance whose value is chosen so that the current through the stabilovolt has a suitable value. In order to clarify the factors affecting the operation of such a voltage source...

tion, we shall first introduce several quantities, and then obtain formulas for \(R\) and \(E\), into which the specified values of the variation of the load current and of the voltage in the a-c line enter. Let us denote:

\[ I_{\min} \text{ — the minimum expected load current,} \]

\[ I_{\max} \text{ — the maximum expected load current,} \]

\[ \bar I_L=\frac{1}{2}(I_{\min}+I_{\max}) \text{ — the average load current,} \]

\[ \Delta I_L=I_{\max}-I_{\min} \text{ — the change in load current,} \]

\[ V \text{ — the voltage drop across the stabilovolt.} \]

Since modern stabilovolts allow current variations from 5 to 40 ma, then with a load equal to \(I_L\), and for the mean value of the output voltage, the current through the stabilovolt should be made equal to \((5+40)/2=22.5\) ma. We specify some permissible change of voltage in the a-c line, equal, for example, to 10%. It is obvious that the current through the stabilovolt will vary from its lower limit, corresponding to the state when the voltage \(E\) is 10% less than the mean value and the load current has its maximum value, to the upper limit, at which \(E\) is 10% greater than the mean value and the load current is equal to \(I_{\min}\). From Ohm’s law it follows:

Fig. 54. Equivalent circuit of a power supply with a stabilovolt.

\[ \frac{E-V}{R}=0.0225+I_L, \tag{86} \]

where \(E\) is the voltage of the power source at the mean value of the voltage in the a-c line. Further, the change of current through the stabilovolt caused by the two causes mentioned will be

\[ \frac{\Delta E}{R}+\Delta I_L \leqslant 0.040-0.005=0.035, \tag{87} \]

where \(\Delta E=0.2E\) in the worst possible case.

Eliminating \(E\) from equations (86) and (87), we find that

\[ R>\frac{V}{0.1525-5\Delta I_L-I_L}. \tag{88} \]

As an example, let us suppose that the load current can vary from 5 to 25 ma. Then \(I_L=0.015\,a\), and \(\Delta I_L=0.020\,a\). If \(V=150\,v\), then equality (88) gives \(R>4.0\,\text{k}\Omega\), and from relation (86) we obtain \(E=300\,v\). Part of the resistance \(R\) is created by the internal resistance of the power source; the other part of this resistance must be created by inserting an additional series resistance.

resistance. If the value of \(R\), and consequently also \(E\), is less than the values calculated by us, then under unfavorable load conditions and changes of voltage in the AC mains the current through the voltage regulator will exceed the limits permissible for it.

2. Electronic voltage stabilizers

Voltage sources stabilized by one or by a number of series-connected voltage regulators do not meet their purpose in those cases where it is necessary to stabilize large currents, or where it is important to have a high degree of stabilization. In these cases electronic voltage stabilization is used, together with a voltage regulator or batteries as sources of fixed voltage, with which the stabilized voltage is compared (see below). It has been found that stabilized sources of the simple degenerated type (with negative feedback) are the most satisfactory, and therefore we shall confine ourselves to describing only such voltage sources. The basic circuit of a degenerated-type voltage source, containing a series-connected regulating electron tube, is shown in Fig. 55.

Fig. 55. Block diagram of a stabilized power supply with negative feedback.

Fig. 56.

The AC voltage \((115\ \mathrm{V} \pm 10\ \mathrm{V})\) is converted into DC voltage by means of the part of the circuit containing rectifiers, transformers, and filters. The simplest form of this part of the circuit is shown in Fig. 56. Good filtering is not necessary, since the voltage pulsations are eliminated by the stabilizing circuit. Part \(\beta\) of the output voltage is compared with a fixed voltage (a voltage regulator or battery), and the difference of these voltages is amplified by a differential amplifier, which is directly connected to the grid of the series-connected triode \(L_1\). If \(E_0\) tends to increase above its equilibrium value, then a negative signal arrives at the grid of \(L_1\), which

returns \(E_0\) to its equilibrium value. To estimate the action of such a source, it is convenient to introduce the following quantities:

the stabilization coefficient

\[ S=\left(\frac{E_0}{E_s}\right)\left(\frac{dE_s}{dE_0}\right), \tag{89} \]

the internal resistance

\[ R=-\frac{dE_0}{dI_0}=-\frac{V_0}{i_0}, \tag{90} \]

and the smoothing coefficient

\[ a=\frac{dE_0}{dE_1}=\frac{V_0}{V_1}. \tag{91} \]

Obviously, \(S\) is the coefficient by which the voltage fluctuations in the alternating-current network are reduced, \(R\) is the effective internal resistance of the source, and \(a\) is the coefficient indicating by how many times the stabilizing device reduces the ripple of the rectified voltage.

Fig. 57. Equivalent circuit of a stabilized power supply.

Let us now derive approximate formulas for the indicated quantities. The equivalent circuit of the power supply is given in Fig. 57. Here \(R_1\) is the internal resistance of the rectified-voltage source, measured at the output terminals of the filter, \(r_a\) and \(\mu\) are the internal resistance and amplification factor of tube \(L_1\), respectively, and \((1+\beta G)V_0\) is the magnitude of the signal applied between the grid and cathode of tube \(L_1\). The voltages \(V_1\) and \(V_0\) represent the changes in \(E_1\) and \(E_0\), respectively. By means of the elementary rules of circuit analysis we find

\[ S=\frac{E_0}{E_1}\,\mu\beta G, \tag{92} \]

\[ R=\frac{R_1+r_a}{\mu\beta G} \tag{93} \]

and

\[ a=\frac{1}{\mu\beta G}, \tag{94} \]

where it is assumed that \(\beta G \gg 1\). We shall apply these formulas for an approximate determination of the properties of two circuits of practical importance, which we shall now describe.

First consider the well-known stabilizing circuit containing a pentode differential amplifier (Fig. 58). It is necessary to note the following features of this circuit.

1) The plate resistance of the pentode \(R\) is connected to \(+450\ \mathrm{V}\) (not to \(+300\ \mathrm{V}\)), since tube \(L_2\) works well as an amplifier in the case where the grid potential of tube \(L_1\) is approximately equal to the potential of its cathode.

Fig. 58. Stabilized power supply with a pentode amplifier.

2) The voltage to the screen grid of tube \(L_2\) is supplied from a voltage divider fed by the current of \(L_3\).

3) The capacitance \(C_1\) (and the resistance \(R_1\)) allows the pulsations to pass by the feedback path to the grid of \(L_2\) without attenuation.

4) The current for \(L_2\) is taken from the stabilized part of the circuit.

If we take for the introduced quantities the following reasonable values:

\[ E_1 = 550\ \mathrm{V}, \qquad E_0 = 300\ \mathrm{V}, \qquad R_1 = 800, \qquad G = 260, \qquad \beta = \frac{1}{3} \]

(for direct voltages), \(\beta = 1\) (for pulsating voltages), \(\mu = 5\), \(r_a = 750\) (tube 6V6 connected as a triode), then

\[ S = \frac{300}{550}\times 5 \times \frac{1}{3}\times 250 = 230, \]

\[ R = \frac{750+800}{5\times \frac{1}{3}\times 250} = 3.7\ \Omega, \]

\[ \alpha = \frac{1}{5\times 1\times 250} = 8\cdot 10^{-4}\quad \text{(for pulsations).} \]

The properties of this circuit actually observed prove to be somewhat worse than those calculated by us. The reason for this is apparently that the voltage across the VR tube changes somewhat when

changes in the current passing through it (as a result of a change in the current flowing through the pentode).

In Fig. 59 a circuit is shown that has better properties. Tube \(L_4\) is a differential amplifier, very stable with respect to changes in the grid–cathode potential difference (because of the symmetry in voltages and currents and because of the degeneration due to the large cathode resistance, for signals applied to both grids). Such an amplifier produces in the circuit of its left anode a signal amplified approximately 30 times, as compared with the signal applied to the right grid. The left anode is connected to the grid of the next triode stage, having a gain of about 50, so that the total gain is somewhat greater than 1500, and is approximately 1700. On the basis of these data we find

Fig. 59. Stabilized power supply with a differential amplifier on a double triode.

\[ S = 1550, \]

\[ R = 0.55\ \text{ohm}, \]

\[ \alpha = 1.2 \cdot 10^{-4}\quad \text{(for ripple)}, \]

which is about a sevenfold improvement in comparison with the preceding simple circuit. For many applications this improvement does not compensate for the increased complexity of the circuit. For high frequencies or for fast signals the stabilizing circuit has poor characteristics, and therefore it is necessary to shunt the output of the stabilizer with a capacitance of not less than \(4\ \mu\text{F}\). Without such a capacitance the source shown in Fig. 59 usually oscillates. To increase the current drawn from such a source, one may use the parallel connection of any number of triodes. A reasonable precaution is to include

tion of a limiting resistance of the order of 100 ohms in series with each grid and with each screen. It should further be recommended that resistances of about 47 ohms be connected in series with each cathode, in order to ensure the proper division of current between the tubes connected in parallel. Usually a maximum current of 75 mA is drawn from one pentode. In those cases where this is insufficient, it is recommended to use the new tube 6AS7, specially intended for stabilizing circuits. The 6AS7 requires a larger change in grid voltage in order to cover a wide range of load variation. Thus, before replacing the 6V6 by a 6AS7 tube, the circuit shown in Fig. 59 should apparently be modified.

REFERENCES

1. M. F. Gardner, J. L. Barnes, “Transients in Linear Systems” (John Wiley and Sons, New York, 1942).
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4. E. B. Moullin, “Spontaneous Fluctuations of Voltage” (Oxford University Press, London, 1938).
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