CORRECTION OF THE SHAPE OF RECTANGULAR VOLTAGE PULSES TRANSFORMED BY PULSE TRANSFORMERS USING FREQUENCY-SELECTIVE CIRCUITS

N. I. VOROB’EV

Introduction

The correction method considered here makes it possible to improve the shape of pulses, and also to reduce the dimensions and weight of the pulse transformer (PT).

It is known [1–3] that, in transformed rectangular pulses, the leading edge is lengthened and droop appears in the flat top. The increase in the duration of the leading edge is due to the parasitic parameters of the winding, while the droop of the flat top is caused by the increase in the magnetizing current of the PT; moreover, the requirements of a minimal increase in the duration of the leading edge and of a reduced droop of the flat top are contradictory. For one and the same voltage and pulse duration, a PT designed for good reproduction of the flat top will considerably increase the leading edge, whereas a transformer designed for good reproduction of the leading edge will give a large droop of the flat top.

At the same time, a PT designed for good reproduction of the flat top will have larger dimensions and weight than a PT designed for greater distortion of the flat top and smaller distortion of the leading edge, since it must have a larger number of winding turns or a larger core cross section.

This correction method may be applied in all cases in which the equivalent circuit of the pulse generator can be represented as a source of constant emf with some internal resistance. An advantage of the proposed method is the possibility of more accurate correction of the droop of the pulse flat top than by other methods, since the distortion caused by the effect of eddy currents in the PT core can be corrected simultaneously. The method considered makes it possible to correct droop of the flat top of any magnitude. The technical implementation of the correction is simple: two resistors and two capacitors are required. If the distortion caused by the effect of eddy currents is relatively small, then one \(RC\) circuit may suffice.

Below, the theory is set forth and experimental data are presented. The case most important for practice is considered, when the load is matched with the generator.

Theory

In analyzing distortions of the flat portion of a rectangular pulse by a transformer, the equivalent circuit shown in Fig. 1 is used [1, 2].

In this circuit:

\(L_m\) is the inductance of the effective magnetization—

\[ L_m=\frac{0.4\pi w_1^2 S_c \mu_0 \cdot 10^{-8}}{e_c}\,[\mathrm{H}], \tag{1} \]

Fig. 1.

\(R_v\) and \(L_v\) are the resistance and inductance accounting for the effect of eddy currents,

\[ R_v=12\,\frac{\rho_c w_1^2 S_c}{e_c \delta_c^2}\,[\Omega], \tag{2} \]

\[ L_v \simeq 0.183 L_m . \tag{3} \]

As follows from the circuit, distortions of the flat portion are caused by the presence of branches containing the inductances \(L_m\) and \(L_v\). The currents \(i_m\) and \(i_v\) in these branches increase with time, and the operating point on the external characteristic of the generator \(u=f(i)\), which in real generators always has a falling character, shifts toward lower voltage values.

The task of correction is reduced to compensating the increasing components of the generator load current.

The current component \(i_v\) can be exactly compensated, as will be shown below, by connecting in parallel with the \(R_v\)—\(L_v\) branch a correcting branch \(R_{kv}\)—\(C_{kv}\) (Fig. 2). The current component \(i_m\) can also be compensated most effectively by an \(RC\)-type circuit (in Fig. 2, the branch \(R_{km}\)—\(C_{km}\)), but only over a limited time interval.

At \(t=0\), the conductance of the generator load is equal to

\[ \frac{1}{R_n}+\frac{1}{R_{km}}+\frac{1}{R_{kv}} . \tag{4} \]

When the load is matched to the generator, the internal conductance \(1/R_e\) of the generator is equal to the total conductance of the load,

\[ \frac{1}{R_{1e}}=\frac{1}{R_i}+\frac{1}{R_{kv}}+\frac{1}{R_{km}} . \tag{5} \]

\(R'_{n}\) — the load resistance referred to the primary winding,
\(R_{ig}\) — the internal resistance of the generator.

It is easy to see that, when correction is used, the internal resistance

Fig. 2.

of the generator is smaller, i.e., the generator must be more powerful.

Analysis of the transient process in the circuit (Fig. 2) is very cumbersome. It is more convenient to carry out the investigation in two stages. Let us determine the parameters of the circuit \(R_{kv} - C_{kv}\).

Correction of the distortion caused by the eddy-current effect

Let the parameters of the circuit \(R_{k.m} - C_{k.m}\) be chosen by us so that, during some time interval from \(t = 0\) to \(t = t_1\), the condition

\[ i_{k.m}(t) + i_m(t) = \mathrm{const}. \]

is satisfied.

Then the branches \(R_{k.m} - C_{k.m}\) and \(L_m\) may be replaced by the resistance \(R_{k.m}\). Combining this resistance with the resistance \(R'_n\), we obtain a simpler circuit (Fig. 3). It is obvious that the resistances \(R_v\) and \(R_{kv}\) must be equal, since in this case \(i_{kv}(0) = i_v(0)\).

Fig. 3.

Analysis of the transient process gives the following expression for the capacitance \(C_{kv}\):

\[ C_{kv} = \frac{L_v}{R_n^{\prime 2}} . \tag{6} \]

Formulas (1–3) make it possible to express \(C_{kB}\) through the parameters of the PT

\[ C_{kB}=16\,\frac{\delta_c^{4}\,\rho_c\,\mu_\Delta}{\rho_c^{2}\,\omega_1^{2}\,S_c}\,[n\phi], \tag{7} \]

where \(\delta_c\) is the thickness of the strip steel of the core (cm),
\(\mu_\Delta\) is the average value of the permeability over the partial cycle,
\(\rho_c\) is the resistivity of the steel (ohm\(\times\)cm).

Correction of the distortion caused by the magnetizing current

Now that it has been shown that \(i_B(t)+i_{kB}(t)=\mathrm{const}\), let us proceed to an analysis of the correction of the distortion of the flat portion by means of the circuit \(R_{kM}\)—\(C_{kM}\). For this purpose we shall use the equivalent circuit shown in Fig. 4. In the circuit equivalent to the branches \(R_B\)—\(L_B\) and \(R_{kB}\)—\(C_{kB}\), the resistance \(R_B\) is combined with the resistance \(R_n^{1}\) and denoted by \(R\), while the subscripts are omitted for the capacitance and inductance.

Fig. 4.

For obtaining a compact solution and constructing universal graphs, \(R_{1e}\) (see Fig. 2) is denoted by \(k_1R\), and \(R_{kM}\) by \(k_2R\), where \(k_1\) and \(k_2\) are arbitrary positive numbers. With the introduction of these designations, the generality of the problem is preserved.

When the load is matched to the generator (at \(t=0\)),

\[ k_1=\frac{k_2}{1+k_2}, \tag{8} \]

which makes it possible to eliminate the coefficient \(k_1\) from the solution, i.e., to reduce by one the number of solution parameters. We shall seek the solution in the form of a normalized generalized transient response

\[ h=\frac{u_n^{1}(t)}{u_n^{1}(0)}=f(bt), \tag{9} \]

where \(b=R/2L\). The graphs of function (9) will be universal and will make it possible, for a specified load resistance (\(R \simeq R_n^{1}\)) and permissible decrease of the flat portion, to determine the magnetizing inductance \(L_M\).

The transient response can be written in the following form:

\[ h(bt)=e^{-\frac{\alpha}{\delta}bt}\left[ \frac{2(1+k_2)-\alpha}{\sqrt{\alpha^2-\gamma^2}}\, \operatorname{sh}\frac{\sqrt{\alpha^2-\gamma^2}}{\delta}bt + \operatorname{ch}\frac{\sqrt{\alpha^2-\gamma^2}}{\delta}bt \right], \tag{10} \]

where

\[ \alpha = k_2^2 \tau b + k_2 + 0.5, \qquad \delta = 2k_2 \tau b (1+k_2), \]

\[ \gamma = 2k_2 \sqrt{\tau b(1+k_2)}, \qquad \tau = RC. \tag{11} \]

This expression has two parameters:

\[ k_2 \quad \text{and} \quad \tau b. \tag{12} \]

Figure 5 gives a family of generalized transient characteristics for several values of the parameter \(\tau b\) at \(k_2=1\). For comparison, the characteristic of the uncorrected system \((k_2=\infty)\) is given. It is evident from the graph that, for a specified \(k_2\), there exists an optimal value of the parameter \((\tau b)_{\text{opt}}\), for which \(h(bt)\) is constant over a larger region of \(bt\). When \(\tau b < (\tau b)_{\text{opt}}\), the voltage pulse will be overcorrected, and when \(\tau b > (\tau b)_{\text{opt}}\), it will be undercorrected. The case \(\tau b < (\tau b)_{\text{opt}}\) may be of interest when it is necessary to obtain a pulse with a rising flat portion, or when the pulse-transformer core operates with appreciable saturation. It is easy to see that an overcorrected pulse is obtained with a smaller capacitance.

Analysis has shown that the optimal transient characteristic for \(k_2 \ne 1\) is obtained under the condition

\[ \alpha - \gamma = 0, \tag{13} \]

whence

\[ \tau b_{1,2} = -\frac{1}{k_2^2} \left[ k_2 + 1.5 \pm \sqrt{2(1+k_2)} \right]. \tag{14} \]

Fig. 5.

Additional analysis shows that, of the two values of \(\tau b\), the optimal one will be

\[ (-\tau b)_{\text{opt}} = -\tau b_2 = \frac{1}{k_2^2} \left[ k_2 + 1.5 - \sqrt{2(1+k_2)} \right]. \tag{15} \]

For \(\tau b_1\), the coefficient preceding the hyperbolic sine becomes negative \((\alpha > 2(1+k_2))\), which gives

\[ h(bt) < 1. \tag{16} \]

The expression (10) corresponding to condition (13), after resolving the indeterminacy, takes the form

\[ h(bt) = e^{-\frac{\alpha}{\delta}bt} \left[ 1 + \frac{2(1+k_2)-\alpha}{\delta}\, bt \right]. \tag{17} \]

For \(k_1<1\) the optimal characteristics lie in the region of values of the parameter \(-b\) corresponding to the condition

\[ \alpha^2-\gamma^2<0, \tag{18} \]

i.e., to such values of \(-b\) for which an oscillatory process arises in the system. Expression (10), taking (18) into account, may be written in the form

\[ h(bt)= \sqrt{\frac{2(1+k_2)}{\gamma^2-\alpha^2}}\, e^{-\frac{\alpha}{\delta}bt} \sin\left[ \frac{\sqrt{\gamma^2-\alpha^2}}{\delta}bt +\operatorname{arctg} \left| \frac{\gamma^2-\alpha^2}{2(1+k_2)-\alpha} \right| \right]. \tag{19} \]

From the condition \(h(0)=1\), after a number of transformations we obtain

\[ (-b)_{\mathrm{opt}}=\frac{1}{2k_2^2}. \tag{20} \]

For the optimal values of \(-b\), formula (10) takes the very simple form

\[ h(bt)= \frac{2}{\sqrt{2(1-k_2)}}\, e^{-k_2bt}\, \sin\left[ k_2\sqrt{\frac{1-k_2}{1+k_2}}\,bt +\operatorname{arctg} \sqrt{\frac{1-k_2}{1+k_2}} \right]. \tag{21} \]

Fig. 6.

Fig. 7.

Figure 6 gives the optimal characteristics for several values of \(k_2\). The correcting properties of the circuit \(R_{k.m}C_{k.m}\) are illustrated by the curves shown in Fig. 7,

\[ \frac{(bt)_k}{(bt)_n}=f\left[\,h(bt)\,\right]. \]

The quantity \((bt)_k/(bt)_n\) shows how many times greater the magnetizing inductance of a PT without correction must be than with the use of correction, at unchanged load resistance, pulse duration, and specified reduction of the flat portion of the pulse, or how many times, all other conditions being equal, the duration can be increased.

increases if correction is applied. Correction is especially effective when it is necessary to obtain a pulse with a very small droop of the flat portion.

Energy consumption by the correcting circuits

The energy absorbed by the branch \(R_{kv} - C_{kv}\) is

\[ W_{kv} = \frac{U_1^2}{R_k} \int_0^{t_n} e^{-\frac{t}{R_{kv} C_{kv}}}\, dt = C_{kv} U_1^2 \left(1 - e^{-\frac{t_n}{R_{kv} C_{kv}}}\right). \tag{22} \]

The energy absorbed by the branch \(R_v - L_v\) is

\[ W_v = \frac{U_1^2}{R_v} \int_0^{t_n} \left(1 - e^{-\frac{R_v}{L_v}t}\right) dt = \frac{U_1^2}{R_v} \left[ t_n - \frac{L_v}{R_v} \left(1 - e^{-\frac{R_v}{L_v}t_n}\right) \right]. \tag{23} \]

Setting

\[ \tau_n = \frac{R_v}{L_v}\, t_n = 1.65\, \frac{t_n}{\Theta_v}, \tag{24} \]

where

\[ \Theta_v = \frac{\psi_c^2 S_c}{4\pi l_c}\,10^{-8} \ [\mathrm{sec}] \tag{25} \]

is the eddy-current time constant, taking (6) into account we obtain

Fig. 8.

\[ \frac{W_{kv}}{W_v} = \frac{1 - e^{-\tau_n}} {\tau_n\left(1 - e^{-\tau_n}\right)}. \tag{26} \]

The dependence \(W_{kv}/W_v = f(\tau_n)\) is given in Fig. 8. Taking into account that the core operating mode is usually chosen so that

\[ t_n > \Theta_v, \tag{27} \]

which corresponds to

\[ \tau_n > 1.65, \tag{28} \]

we obtain

\[ \frac{W_{k\mathrm{в}}}{W_n} < 0.5 . \tag{29} \]

The energy absorbed by the branches \(R_{\mathrm{в}} - L_{\mathrm{в}}\) and \(R_{k\mathrm{в}} - C_{k\mathrm{в}}\), taking (2) into account, is

\[ W_n = W_{\mathrm{в}} + W_{k\mathrm{в}} = \frac{U_1^2}{R_{\mathrm{в}}}\, t_n + 12\,\frac{\varepsilon^c U_1^2 \cdot t_n}{r_c W_1^2 s_c}\,\delta_c^2 . \tag{30} \]

It follows from formula (30) that \(W_n\) is proportional to the square of the thickness of the core steel \(\delta_c\). When changing to steel of thinner grade, \(W_n\) decreases rapidly; at the same time the energy absorbed by the correcting circuit decreases especially rapidly, since it is approximately proportional to the fourth power of \(\delta_c\), which follows directly from formulas (28) and (17).

The energy consumed by the branch \(R_{k.m} - C_{k.m}\), assuming \(R = R_n\) \((R_{\mathrm{в}} \gg R_n)\) and \(U_1 = \mathrm{const}\), is

\[ W_{k.m} = \frac{U_1^2}{k_2 R_n} \int_0^{bt} e^{-\frac{bt}{k_2(zb)_{\mathrm{opt}}}}\, d(bt) = \frac{U_1^2}{R_n} \left[ 1 - e^{-\frac{bt}{k_2(zb)_{\mathrm{opt}}}} \right]. \tag{31} \]

Taking into account that the energy consumed by the load is

\[ W_n = \frac{U_1^2}{R_n}\, t , \tag{32} \]

we obtain

\[ \frac{W_{k.m}}{W_n} = \frac{(zb)_{\mathrm{opt}}}{bt} \left[ 1 - e^{-\frac{bt}{k_2(zb)_{\mathrm{opt}}}} \right]. \tag{33} \]

Fig. 9.

In Fig. 8 the curves \(W_{k.m}/W_n = f(bt)\) are given for several values of the parameter \(k_2\). The curves are plotted in the range of values of \(bt\) corresponding to the condition \(h(bt) = 0.99\). As follows from the graph, \(W_{k.m}/W_n\) is almost

does not depend on \(bt\), but depends strongly on \(\kappa_2\). This dependence (Fig. 8) for maximum values of \(W_{k.m}/W_n\) (as \(bt \to 0\)) is expressed by the formula

\[ \frac{W_{k.m}}{W_n}=\frac{1}{\kappa_2}. \tag{34} \]

Although for \(\kappa_2<1\) the energy consumption by the circuit \(R_{k.m}—C_{k.m}\) is considerable and exceeds the energy consumed by the load, while for \(\kappa_2>1\) it is small, and it would seem expedient always to take \(\kappa_2>1\), it is more correct to judge the effectiveness, the “economy,” of the correction by the value of the ratio

\[ \eta=\frac{W_{k.m}}{W_n\cdot bt_{\max}}=\frac{1}{\kappa_2\cdot bt_{\max}}, \tag{35} \]

where \(bt_{\max}\) is the maximum value of \(bt\) at which \(h(bt)\) is not less than a certain value. We have taken \(h(bt)=0.995\).

In Fig. 9 a plot of the dependence \(\eta=f(\kappa_2)\) is given; from it it follows that the correction is most effective at \(\kappa_2=1\) and least effective at \(\kappa_2=2\). The complicated character of the curve \(\eta=f(\kappa_2)\), with a break at the point \(\kappa_2=1\), is explained by the fact that in the intervals \(0<\kappa_2<1\) (the region of an oscillatory process) and \(1<\kappa_2<\infty\) (the region of an aperiodic process) the value \(bt_{\max}\) is determined by different functions (17) and (21), and the curve \(bt_{\max}=f(\kappa_2)\) has a break at the point \(\kappa_2=1\) (Fig. 9).

It is necessary to bear in mind that our conclusions apply to the limiting capabilities of the correcting circuit at various \(\kappa_2\). If, however, the value of \(bt\) is such that correction is feasible for several values of \(\kappa_2\), it is expedient to choose the largest \(\kappa_2\).

The maximum relative voltage on the capacitor \(C_{k.m}\), for different values of the parameter \(\kappa_2\), can be determined from the formula

\[ \frac{U_{c.\max}}{U_1(0)}=1-e^{-\frac{bt_{\max}}{\kappa_2(z_h)_{\mathrm{opt}}}}. \tag{36} \]

In Fig. 9 a plot is given of

\[ \frac{U_{c.\max}}{U_1(0)}=f(\kappa_2), \]

from which it is evident that the highest voltage (at \(\kappa_2=1\)) does not exceed 18.7%, which is favorable from the standpoint of the technical realization of the correction.

The plots presented in Fig. 9 make it possible easily to calculate the correcting circuit \(R_{k.m}—C_{k.m}\) for a relative voltage decrease equal to 0.995. To illustrate the calculation method, an example is given.

Example. In designing a PT it was established that the required pulse front will be obtained if the magnetizing inductance does not exceed \(20\ \mathrm{mH}\). At the same time, a decrease in the flat part of the pulse is also inadmissible. The pulse duration is \(10\ \mu\mathrm{s}\); the reduced load resistance is \(200\ \Omega\).

1. We find the value of the product \(bt_n\):

\[ bt_n=\frac{R_n^{1}}{2L_m}\,t_n =\frac{2\cdot 10^2\cdot 10^{-5}}{2\cdot 2\cdot 10^{-2}} =0.05. \]

According to Fig. 6, for \(b/t_n = 0.05\), without correction,

\[ h(b/t) = 0.95. \]

1. From the graph in Fig. 9 we find the value of the correction parameter

\[ k_2 \leq 1.2. \]

1. Resistance of the correcting circuit

\[ R_{k.m}=k_2 R'_n = 1.2\cdot 200 = 240\ \Omega . \]

1. Internal resistance of the generator

\[ R_1 = 0.545\cdot R'_n = 0.545\cdot 200 = 109\ \Omega . \]

1. Relative energy consumption of the correcting circuit

\[ \frac{W_{k.m}}{W_n}=\gamma\cdot \frac{b}{t_n}=18.7\cdot 0.05=0.83. \]

1. Time constant of the correcting circuit

\[ \tau_{k.m}=\frac{k_2-b}{b}=\frac{k_2 b t_n}{b t_n} =\frac{0.5\cdot 10^{-5}}{0.05}=10^{-4}\ \text{sec}. \]

1. Capacitance of the capacitor of the correcting circuit

\[ C_{k.m}=\frac{\tau_{k.m}}{R_{k.m}} =\frac{10^{-4}}{2\cdot 10^2}=0.5\ \mu\text{F}. \]

1. Maximum relative voltage across the capacitor

\[ \frac{U_{c,\max}}{U_1}=8.5\%. \]

Experimental Results

An experimental verification of the proposed correction method was carried out. The experiments were conducted with two types of rectangular-pulse generators. In the first case, the pulse was applied to the input of the pulse transformer from a cathode follower that terminated a high-power rectangular-pulse generator. In the second case, the pulse was formed by a modulator with an artificial line. In both cases the pulse transformer had an active load. The pulse duration was \(5\ \mu\text{sec}\). Figure 10 shows oscillograms of the pulses at the load without correction (\(a\) and \(b\)) and with correction (\(c\) and \(d\)). The oscillograms (\(b\) and \(d\)) correspond to the experiment in which a modulator with an artificial line was used. The experiments showed complete agreement between theory and practice.

Fig. 10.

From the oscillograms shown in Fig. 10 it is also evident that the overshoot of opposite polarity, which occurs in all cases when there is a droop in the flat portion of the corrected pulse, is absent, which is very important in many pulse devices.

REFERENCES

1. Ya. S. Itskhokin. Pulse Transformers, Moscow, 1950.
2. V. F. Lukin. Transient Processes in Linear Elements of Radio-Engineering Devices, Oborongiz, 1950.
3. Parts and Elements of Radar Stations, Vol. II (translated from English), Sovetskoe Radio Publishing House, Moscow, 1952.