Phase Change Amplifiers with Electromagnetic Energy Storage Devices

L. M. Anan’ev, S. F. Vasilevskii

(Presented by the scientific seminar of the Department of Industrial Electronics)

When measuring small phase shifts, difficulties arise both in obtaining and in recording the averaged value of pulses of short duration; these difficulties increase when it is necessary to measure relatively small deviations of the phase from its mean value. One possible way to overcome these difficulties is to amplify the phase deviations. It is desirable to have a device in which phase changes at the input would produce phase changes at its output increased by a certain number of times, \(k_\phi\).

Preliminary investigations have shown that good results may be expected if, for amplifying phase changes, one uses the processes of charging (or discharging) a storage element under a stepwise change of the parameters of the charging circuit.

Fig. 1. Block diagram of a phase-change amplifier:
\(O\) — reference signal; \(I\) — interval signal; \(IP\) — power supply; \(KU\) — switching device; \(H\) — storage device; \(SrU\) — comparison device; \(SxV\) — recovery circuit.

With a change in the parameters of the charging circuit. The generalized block diagram of such a phase-change amplifier (Fig. 1) consists of an energy storage device \(H\), charged from the power supply \(IP\) through the switching device \(KU\), intended for a stepwise change of the para-

parameters of the charging circuit and is switched by the action upon it of the preliminarily transformed reference signal \(O\) and the signal under study (interval signal) \(I\). At the moment when the storage device \(H\) is charged to a preset level, a pulse appears at the output of the comparison device \(C_p U\). The comparison instant is determined by the parameters of the circuit for charging the storage device before and after switching by the interval signal, and by the instant at which the interval signal appears. When the switching instant changes, the instant at which the output pulse of \(C_p U\) appears also changes according to a definite law, and this change at the output is greater than the input change by a certain number of times. This can be shown as follows.

Let, from the instant \(t_n\) at which the reference signal is received, the energy storage device begin to charge at a certain rate determined by the parameters \(p_1, p_2, \ldots p_n\) of the charging circuit,

\[ \dot{u}_1(t)=\frac{du_1}{dt}=f_1(p_1,p_2\ldots p_i\ldots,p_n t). \tag{1} \]

At the instant \(t_0\) at which the interval signal \(I\) is received, some parameter \(p_i\) of the circuit changes its value abruptly by switching. From this instant \(t_0\), the rate of change of the voltage on the storage device will abruptly assume another value,

\[ \dot{u}_2(t)=\frac{du_2}{dt}=f_2(p_1,p_2,\ldots p'_i\ldots p_n t). \tag{2} \]

Hence the voltage on the storage device before, \(u_1(t)\), and after, \(u_2(t)\), the instant \(t_0\) at which the interval signal appears is

\[ u_1(t)=\int \dot{u}_1(t)\,dt+C_1, \tag{3} \]

\[ u_2(t)=\int \dot{u}_2(t)\,dt+C_2, \tag{4} \]

where \(C_1, C_2\) are constants of integration,

\[ C_1=u_{01}-\left[\int \dot{u}_1(t)\,dt\right]_{t=0}, \tag{5} \]

\[ C_2=u_1(t_0)-\left[\int \dot{u}_2(t)\,dt\right]_{t=t_0}, \tag{6} \]

and \(u_{01}\) is the initial voltage on the storage device.

\(u_1(t_0)\) is the voltage \(u_1\) on the storage device at the instant \(t_0\).

The comparison instant \(t_1\), when the energy storage device is charged to a certain value \(u_{\mathrm{comp}}\), which may be called the comparison level, and upon reaching which a signal appears at the output of the comparison device, is determined from (4) and (6):

\[ t_1=\left[\int \frac{1}{\dot{u}_2(t)}\,du_2\right]_{u_2=u_{\mathrm{comp}}}+C_4, \tag{7} \]

where

\[ C_4=t_0-\left[\int \frac{1}{\dot{u}_2(t)}\,du_2\right]_{u_2=u_1(t_0)}. \tag{8} \]

From expressions (7) and (8) it is evident that, when the instant \(t_0\) changes by some amount \(\Delta t_0\), the comparison instant \(t_1\) will receive an increment \(\Delta t_1\). Obviously, if \(\Delta t_1>\Delta t_0\), one may speak of amplification of the phase changes \(\Delta t_0\). This amplification may be estimated by the phase-change amplification coefficient \(k_f\), which is equal to the ratio of the increment \(\Delta t_1\) of the instant \(t_1\), when the voltage \(u_2(t)\) on the storage device reaches a certain level

comparison voltage \(u_{\mathrm{comp}}\), to an increment \(\Delta t_0\) of the instant \(t_0\) at which the interval signal appears (2, 3)

\[ k_\phi=\frac{\Delta t_1}{\Delta t_0} =1-\frac{d}{dt_0}\left\{\left[\int \frac{du_2}{\dot u_2}\right]_{u_2=u_1(t_0)}\right\}. \tag{9} \]

A sufficient condition for obtaining amplification of phase changes will be

\[ \left|\frac{d}{dt_0}\left\{\left[\int \frac{du_2}{\dot u_2}\right]_{u_2=u_1(t_0)}\right\}\right|>2. \tag{10} \]

For an exponentially varying voltage on the storage device we obtain

\[ k_\phi=1-\frac{\tau_2}{\tau_1} =1-\frac{\dot u_1(t_0)}{\dot u(t_0)}, \tag{11} \]

where \(\tau_1,\tau_2\) are the time constants of the storage-device charging circuit before and after the instant \(t_0\). Thus, the gain of phase changes is determined by the ratio of the charging rates \(\dot u_1(t_0)/\dot u_2(t_0)\) (or of the time constants of the charging circuit \(\tau_2/\tau_1\)) before and after the instant \(t_0\) at which the interval pulse appears, which determines the switching instant.

It is not difficult to see that the condition for amplification of phase changes is satisfied for different sequences of charge–discharge processes in the storage device before and after the switching instant \(t_0\). Four cases are possible: charge—charge, discharge—charge, charge—discharge, discharge—discharge, which are not equivalent with respect to the errors in amplification of phase changes.

It can be shown that the principal error is associated with instability of the comparison level and is equal to

\[ \delta k_\phi= \frac{\tau_1\cdot u_{\mathrm{comp}}\cdot \delta u_{\mathrm{comp}}} {\Delta t_0\cdot (E-u_{\mathrm{comp}})} . \tag{12} \]

From expression (12) it is seen that, in order to reduce the error due to instability of the comparison level, it is necessary to reduce the value of the comparison threshold \(u_{\mathrm{comp}}\). Therefore, to reduce \(\delta k_\phi\), it is advantageous to use UIF circuits in which the output voltage after the switching instant \(t_0\) decreases (for example, charge—discharge circuits).

Both an inductive and a capacitive storage device may be used as the energy storage device. A UIF with an inductive storage device is expedient to use for constant phase shifts \(t_0<10^{-5}\) sec. To obtain large gains of phase changes \(k_\phi\), it is expedient to use mixed-type UIFs, i.e., those in which storage devices \(L\) and \(C\) are combined.

As an example, Fig. 2 shows the circuit of the simplest \(RC\)-type UIF (with a capacitive storage device). In the initial state the tubes \(L_1\) and \(L_2\) are open; the resistance \(R_{a2}\) in the anode of tube \(L_2\) is adjusted so that in the initial state the voltage drops at the anodes of the tubes are equal and, consequently, the initial voltage on capacitor \(C\) is equal to 0. At the instant \(t_i\), a negative rectangular pulse of duration \(\tau_i \gg t_0\) from the pulse converter, triggered by the reference signal, is applied to input 1 m and cuts off tube \(L_1\). Owing to the voltage jump at the anode of \(L_1\), capacitor \(C\) begins to charge through

diode \(L_3\) and tube \(L_2\). Since the resistance \(R_0\) is much greater than the internal resistance \(R_{i2}\) of tube \(L_2\), its influence on the charging process may be neglected. At the instant \(t_0\) of commutation, the same negative rectangular pulse arrives at input 2, but now from the converter, triggered by an interval signal. Tube \(L_2\) is cut off, and capacitor \(C\) will continue to charge through the resistance \(R_0\). Since \(R_0 \gg R_{i2}+R_{i3}\), it follows that \(\tau_1 < \tau_2\), which is necessary for amplification of phase changes.

Fig. 2. Circuit of an \(RC\)-type PCA.

For this circuit,

\[ |k_{\phi}|=\left|1-\frac{R_0}{R_{i2}+R_{i3}}\right|>1. \]

The output voltage applied to the comparison device \(C_p U\) is proportional to the charging current, which makes it possible to use low values of the comparison threshold \(u_{\mathrm{comp}}\). The positive properties of the PCA circuit under consideration are self-restoration, grounded cathodes of both switching tubes, and the possibility of obtaining significant \(k_{\phi}\). The circuit of an \(RL\)-type PCA is shown in Fig. 3a.

Fig. 3. Circuit of an \(RL\)-type PCA (a) and its timing diagrams (b).

In the initial state, tubes \(L_2\) and \(L_1\) are closed; the current in the \(R_a - L - R_0\) circuit is maximal. At the instant \(t_n\) when the reference signal appears, the positive rectangular pulse applied to input 1 cuts off the tube...

…lamp \(L_1\), the potential of its anode is lowered, and the current through the inductance \(L\) begins to decrease with the time constant (1):

\[ \tau_1=\frac{L\left(R_{i1}+R_a\right)} {R_{i1}\cdot R_a+R_{i1}R_0+R_0R_a}. \]

At the instant \(t_0\) when the interval pulse arrives, the tube \(L_2\) is cut off. The current through the inductance continues to decrease, but now with a different time constant

\[ \tau_2=\frac{L\left(R_{i1}+R_a\right)} {R_{i1}R_a+R_{i1}\cdot R_{i2}+R_{i2}R_a}. \]

Therefore

\[ k_{\phi}=1-\frac{R_{i1}R_a+R_{i1}R_0+R_0R_a} {R_iR_a+R_{i1}R_{i2}+R_{i2}R_a}. \tag{14} \]

This UIF circuit has a low comparison level \(u_{\text{srav}}\), is self-restoring, gives considerable \(k_{\phi}\), and is simple in the implementation of switching. The UIF circuits considered are expedient for use in phase-measuring apparatus intended for measuring relatively small phase deviations from a certain constant value. A block diagram of such a phasemeter based on a UIF is shown in Fig. 4.

Fig. 4. Block diagram of a phasemeter based on an amplifier of phase changes:
\(OD\)—reference sensor, \(ID\)—measuring sensor, \(\Phi\)—pulse shaper, \(UIF\)—amplifier of phase changes, \(SrU\)—comparison device, \(Tr\)—trigger, \(IU\)—measuring device, \(DTs\)—differentiating circuit, \(FS\)—phase shifter.

Shifted in time in accordance with the phase shift, the reference and interval signals from the sensors are fed to the shaper \(\Phi\) of their respective channels. In investigations of pulse signals the shaper consists of an amplifier, a limiter, differentiating circuits, and a blocking multivibrator; for harmonic signals it consists of an amplifier, a limiter, and a Schmitt trigger.

The formed pulses with a phase shift are applied to the inputs of the \(UIF\). With the aid of the comparison device \(SrU\) and the trigger \(Tr\), the amplified phase changes are converted into voltage changes registered by the measuring device \(IU\). This is accomplished as follows. The leading pulse of the reference channel, after shaping and differentiation, switches the trigger \(Tr\) to state 1. At the moment when the voltage at the output of the \(UIF\) reaches the comparison level, the comparison device issues a signal that returns the trigger to its initial state. When the phase shift between

by the sensor signals by \(\Delta t_0\), the duration of the trigger output signal will change by \(k_\phi \cdot \Delta t_0\), and the changes in the mean value of the recorded voltage will be

\[ \Delta \bar{u}=\frac{u_m}{T}\,k_\phi\cdot \Delta t_0, \tag{15} \]

where \(u_m\) is the pulse amplitude at the trigger output,
\(T\) is the pulse repetition period.

The mean value of the voltage of the trigger output signal is recorded by means of a measuring device in which the measuring instrument is connected so as to compensate for the reading corresponding to the constant phase shift.

The phase-shifting \(FC\) device provided in the circuit is intended for “zero setting” at the beginning of the measurements.

The use of a \(UIF\) makes it possible to increase significantly the sensitivity and resolving power of phase-measuring equipment.

Notation

\(K_\phi\)—coefficient of amplification of phase changes,
\(\delta K_\phi\)—relative error of amplification of phase changes,
\(t\)—time,
\(t_n\)—instant of appearance of the reference signal,
\(t_0\)—instant of switching,
\(t_1\)—instant of comparison,
\(\Delta t_0\)—change in the switching instant,
\(\Delta t_1\)—change in the comparison instant,
\(T\)—pulse repetition period,
\(\tau_i\)—pulse duration,
\(\tau_1\)—constant of the charging circuit up to the instant \(t_0\),
\(\tau_2\)—constant of the charging circuit after the switching instant \(t_0\),
\(u_1,u_2\)—voltage before and after the switching instant \(t_0\),
\(\dot{u}_1,\dot{u}_2\)—rate of change of voltage before and after the switching instant \(t_0\),
\(u_{\mathrm{comp}}\)—comparison voltage,
\(\delta u_{\mathrm{comp}}\)—relative instability of the comparison level,
\(u_{01}\)—initial voltage on the storage device,
\(u_1(t_0)\)—voltage on the storage device at the instant \(t_0\),
\(u_m\)—pulse amplitude at the trigger output,
\(P\)—parameter of the storage-device charging circuit,
\(C_k\)—a certain integration constant,
\(R_a\)—anode resistance,
\(R_i\)—internal resistance of the tube,
\(R_0\)—resistance of the storage-device charging circuit after the instant \(t_0\),
\(C\)—capacitance,
\(L\)—inductance,
\(L\)—electron tube,

REFERENCES

1. L. R. Neiman, P. L. Kalantarov. Theoretical Foundations of Electrical Engineering. Gosenergoizdat, Moscow–Leningrad, part 2, 1959.

2. L. M. Anan’ev et al. Induction Electron Accelerator—Betatron. Gosatomizdat, 1961.

3. Registration certificate No. 51146, 1965.