On the Application of a Parametric Amplifier with Asynchronous Pumping in Ground-Fault Protection of Compensated Networks

R. A. Weinstein, L. I. Voronova

(Presented by the scientific seminar of the Department of Electrical Stations)

In [1] it was proposed to use a parametric amplifier as the responding element of ground-fault protection. The application of a PA in the synchronous mode made it possible to obtain responding elements having high sensitivity to current with a frequency of 25 Hz, with good detuning from industrial-frequency currents. However, as follows from the properties of a PA operating in the synchronous mode, in order to obtain maximum amplification it is necessary to apply to the pump circuit, in a definite manner, a phase-adjusted voltage.

If, at an electric station where voltage transformers are available, the need for phasing causes only additional difficulties in installation and operation of the protection, then at substations that do not have voltage transformers this drawback is insurmountable.

In the present work a method is proposed for eliminating the phase sensitivity of the PA by artificially taking the amplifier out of the synchronous mode. This can be achieved by changing the pump frequency corresponding to the synchronous mode, \(2\omega\), by a certain small amount \(\Delta\omega\) in comparison with \(2\omega\).

Fig. 1

In this case the phase of the signal relative to the periodically varying inductance will change continuously with frequency \(\Delta\omega\) and will assume all values, including those corresponding to maximum amplification.

The instantaneous value of the voltage on the PA circuit will be a beat voltage with an envelope of frequency \(\Delta\omega\).

Thus, the maximum value of the voltage on the PA circuit will not depend on the signal phase. In the present work an attempt is made to determine a rational value of the deviation of the pump frequency from the synchronous one and to investigate the change in the frequency-selective properties of the PA in the asynchronous mode.

As in [1], we analyze the operation of the PA using the equivalent circuit (Fig. 1) by the small-signal method, which assumes linearity of the system [2]. The law of variation of the inductance is

\[ l(t)=L_0(1+m\sin\omega_{\text{p}}t), \tag{1} \]

where \(L_0\) is the constant component of the inductance,
\(m_0 L\) is the amplitude of the first harmonic of the inductance, or the absolute modulation depth.

Let

\[ \omega_{\mathrm{н}}=2\omega+\Delta\omega . \]

The current of the amplified signal is

\[ i_0=I_\omega \sin(\omega t+\psi). \tag{2} \]

The voltage across the PA circuit, when the inductance varies with frequency \(2\omega+\Delta\omega\), for a signal with frequency \(\omega\), will contain, in addition to the fundamental frequency, components with combination frequencies:

\[ 2\omega+\Delta\omega-\omega=\omega+\Delta\omega, \]

\[ 2\omega+\Delta\omega+\omega=3\omega+\Delta\omega . \]

Assuming that the frequency deviation from synchronous is small, i.e. \(\Delta\omega \ll \omega\), we may assume that, with respect to the resonant frequency of the circuit, the frequencies \(\omega\) and \(\omega+\Delta\omega\) are located approximately in the same position, since the resonant frequency of the circuit is close to the signal frequency. The component with frequency \((3\omega+\Delta\omega)\), which differs greatly from the resonant frequency, is discarded.

On this basis, the law of variation of the voltage across the circuit can be written in the following form:

\[ U=U_{s_1}\sin\omega_1 t+U_{c_1}\cos\omega_1 t+U_{s_2}\sin\omega_2 t+U_{c_2}\cos\omega_2 t, \tag{3} \]

where

\[ \omega=\omega_1;\qquad (\omega+\Delta\omega)=\omega_2 . \]

Here and below, sinusoidal components are assigned the index “\(s\),” and cosine components the index “\(c\).”

The amplitudes and phases of the voltages with frequencies \(\omega\) and \((\omega+\Delta\omega)\) are to be determined next.

According to the equivalent circuit of the PA (Fig. 1),

\[ i=i_L+i_c+i_g, \tag{4} \]

\[ i_c=\omega C U_{1s}\cos\omega_1 t-\omega C U_{1c}\sin\omega_1 t+\omega C U_{s2}\cos\omega_2 t-\omega C U_{c2}\sin\omega_2 t, \tag{5} \]

\[ i_g=gU_{1s}\sin\omega_1 t+gU_{1c}\cos\omega_1 t+gU_{2s}\sin\omega_2 t+gU_{2c}\cos\omega_2 t. \]

The instantaneous values of the voltage across the circuit and of the current flowing through the inductance are related to each other by

\[ U=\frac{d\,[l(t)\cdot i_L]}{dt}. \tag{6} \]

The current \(i_L\) will contain components with frequencies

\[ \omega_1=\omega;\qquad \omega_2=\omega+\Delta\omega;\qquad \omega_3=3\omega+\Delta\omega;\qquad \omega_4=3\omega+2\Delta\omega, \]

that is,

\[ i_L=I_{s1}\sin\omega_1 t+I_{c1}\cos\omega_1 t+I_{s2}\sin\omega_2 t+I_{c2}\cos\omega_2 t+ \tag{7} \]

\[ +I_{s3}\sin\omega_3 t+I_{c3}\cos\omega_3 t+I_{s4}\sin\omega_4 t+I_{c4}\cos\omega_4 t. \]

Substituting (1) and (7) into (6), we obtain a system of equations relating the sine and cosine components of equal frequencies:

\[ \left\{ \begin{aligned} &- I_{c1}\omega_1 L_0-\frac{I_{s3}\omega_1 L_0 m}{2}-\frac{I_{s2}\omega_1 L_0 m}{2}=U_{s1},\\ &- I_{c2}\omega_2 L_0-\frac{I_{s1}\omega_2 L_0 m}{2}-\frac{I_{s4}\omega_2 L_0 m}{2}=U_{s2},\\ &- I_{c3}\omega_3 L_0+\frac{I_{s1}\omega_3 mL_0}{2}=0,\\ &I_{c4}\omega_4 L_0+\frac{I_{s2}\omega_4 mL_0}{2}=0,\\ &I_{s1}\omega_1 L_0+\frac{I_{c2}\omega_1 mL_0}{2}-\frac{I_{c3}\omega_1 mL_0}{2}=U_{c1},\\ &I_{s2}\omega_2 L_0-\frac{I_{c4}\omega_2 mL_0}{2}+\frac{I_{c1}\omega_2 mL_0}{2}=U_{c2},\\ &I_{s3}\omega_3 L_0+\frac{I_{c1}\omega_3 mL_0}{2}=0,\\ &I_{s4}\omega_4 L_0+\frac{I_{c2}\omega_4 mL_0}{2}=0. \end{aligned} \right. \tag{8} \]

The solution of the system of equations (8) gives:

\[ \left\{ \begin{aligned} I_{s1}&= \frac{ \frac{1}{2}m\omega_1 U_{s2} -\omega_2\left(\frac{1}{4}m^2-1\right)U_{c1} }{ \omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] },\\[6pt] I_{c1}&= \frac{ \omega_2\left(\frac{1}{4}m^2-1\right)U_{s1} -\frac{1}{2}\omega_1 mU_{c2} }{ \omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] }; \end{aligned} \right. \tag{9} \]

\[ \left\{ \begin{aligned} I_{s2}&= \frac{ \frac{1}{2}m\omega_2 \bar U_{s1} -\omega_1\left(\frac{1}{4}m^2-1\right)U_{c2} }{ \omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] },\\[6pt] I_{c2}&= \frac{ \omega_1\left(\frac{1}{4}m^2-1\right)U_{s2} -\frac{1}{2}m\omega_2 U_{c1} }{ \omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] }; \end{aligned} \right. \tag{10} \]

\[ \left\{ \begin{aligned} I_{s3}&= m\, \frac{ \frac{1}{2}\omega_1 mU_{c2} -\omega_2\left(\frac{1}{4}m^2-1\right)U_{s1} }{ 2\omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] },\\[6pt] I_{c3}&= m\, \frac{ \omega_1\left(\frac{1}{4}m^2-1\right)U_{s2} -\omega_2\left(\frac{1}{4}m^2-1\right)U_{c1} }{ 2\omega_1\omega_2 L_0 \left[ \left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2 \right] }; \end{aligned} \right. \tag{11} \]

\[ \left\{ \begin{aligned} I_{s4} &= m\, \frac{\frac{1}{2}m\omega_2 U_{c1} -\omega_1\left(\frac{1}{4}m^2-1\right)U_{s2}} {2\omega_1\omega_2 L_0 \left[\left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2\right]}, \\[6pt] I_{c4} &= m\, \frac{\frac{1}{2}m\omega_2 U_{s1} -\frac{1}{4}\omega_1\left(\frac{1}{4}m^2-1\right)U_{c2}} {2\omega_1\omega_2 L_0 \left[\left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2\right]} . \end{aligned} \right. \tag{12} \]

where the following notation has been introduced:

\[ \beta=\frac{1}{2}m;\qquad \gamma=\left(\frac{1}{4}m^2-1\right);\qquad \sigma=\gamma^2-\beta^2 . \]

Substituting (5), (9), and (10) into (2), we obtain:

\[ \left\{ \begin{aligned} \frac{\beta\omega_1 U_{s2}}{\omega_1\omega_2 L_0\sigma} -\frac{\omega_2\gamma U_{c1}}{\omega_1\omega_2 L_0\sigma} +gU_{s1}-\omega_1 C U_{c1} &= I_{\omega_1}\cos\psi, \\ \frac{\omega_2\gamma U_{s1}}{\omega_1\omega_2 L_0\sigma} -\frac{\omega_1\beta U_{c2}}{\omega_1\omega_2 L_0\sigma} +gU_{c1}+\omega_1 C U_{s1} &= I_{\omega_1}\sin\psi, \\ \frac{\beta\omega_2 U_{s1}}{\omega_1\omega_2 L_0\sigma} -\frac{\omega_1\gamma U_{c2}}{\omega_1\omega_2 L_0\sigma} +gU_{s2}-\omega_2 C U_{c2} &=0, \\ \frac{\omega_1\gamma U_{s2}}{\omega_1\omega_2 L_0\sigma} -\frac{\beta\omega_2 U_{c1}}{\omega_1\omega_2 L_0\sigma} +gU_{c2}+\omega_2 C U_{s2} &=0 . \end{aligned} \right. \tag{14} \]

Let us introduce the following notation:

\[ \frac{\omega_1}{\omega_2}=\Omega;\qquad \omega_1 L_0\cdot g=\delta_1;\qquad CL_0=\frac{1}{\omega_0^2};\qquad \omega_2 L_0 g=\delta_2; \]

\[ \left(\frac{\gamma}{\sigma}+\frac{\omega_1^2}{\omega_0^2}\right)=B_1;\qquad \left(\frac{\gamma}{\sigma}+\frac{\omega_2^2}{\omega_0^2}\right)=B_2 . \]

Then equations (14), after a number of transformations, take the form:

\[ \left\{ \begin{aligned} \left( \frac{\beta\Omega}{\sigma} -\frac{\delta_1\delta_2\sigma\Omega}{\beta} -\frac{B_1B_2\sigma\Omega}{\beta} \right)U_{s2} + \left( \frac{\delta_1\sigma\Omega B_2}{\beta} -\frac{B_1\sigma\Omega\delta_2}{\beta} \right)U_{c2} &= \Omega\omega_1 L_0 I_1\cos\psi, \\[4pt] \left( \frac{\delta_1\sigma\Omega B_2}{\beta} -\frac{B_1\delta_2\sigma\Omega}{\beta} \right)U_{s2} - \left( \frac{\beta\Omega}{\sigma}U_{c2} -\frac{\delta_1\delta_2\sigma\Omega}{\beta} -\frac{B_1B_2\sigma\Omega}{\beta} \right)U_{c1} &= \omega_1 L_0 I_1\sin\psi\,\Omega, \\[4pt] \left( \frac{\beta}{\sigma\Omega} -\frac{\delta_2\sigma\delta_1}{\beta\Omega} -\frac{B_1B_2\sigma}{\beta\Omega} \right)U_{s1} + \left( \frac{\delta_2 B_1\sigma}{\beta\Omega} -\frac{B_2\sigma\delta_1}{\beta\Omega} \right)U_{c1} &= -\left( \frac{\delta_2\sigma\omega_1 L_0 I\cos\psi +B_2\sigma\omega_1 L_0 I_1\sin\psi}{\beta\Omega} \right), \\[4pt] \left( \frac{\delta_2 B_1\sigma}{\beta\Omega} -\frac{B_2\sigma\delta_1}{\beta\Omega} \right)U_{s1} - \left( \frac{\beta}{\sigma\Omega} -\frac{\delta_1\delta_2\sigma}{\beta\Omega} -\frac{B_1B_2\sigma}{\beta\Omega} \right)U_{c1} &= -\frac{ B_2\sigma\omega_1 L_0 I_1\cos\psi +\delta_2\sigma\omega_1 L_0 I_1\sin\psi }{\beta\Omega}. \end{aligned} \right. \tag{15} \]

Solving system (15), we obtain expressions for the coefficients of series (3):

\[ \left\{ \begin{aligned} U_{s1} &= \frac{\omega_1 L_0 I_1 \big[(NP - TM)\cos\psi - (PM + NT)\sin\psi\big]} {M^2 + N^2},\\[4pt] U_{c1} &= \frac{\omega_1 L_0 I_1 \big[(MP + NT)\cos\psi - (MT - NP)\sin\psi\big]} {M^2 + N^2},\\[4pt] U_{s2} &= \frac{\omega_1 L_0 I_1 (M\cos\psi + N\sin\psi)} {\Omega (M^2 + N^2)},\\[4pt] U_{c2} &= \frac{\omega_1 L_0 I_1 (N\cos\psi - M\sin\psi)} {\Omega (M^2 + N^2)}; \end{aligned} \right. \tag{16} \]

\[ \operatorname{tg}\varphi_1 = \frac{(MP + NT)\cos\psi + (NP - MT)\sin\psi} {(NP + TM)\cos\psi - (PM + NT)\sin\psi}; \]

\[ \operatorname{tg}\varphi_2 = \frac{N\cos\psi - M\sin\psi} {M\cos\psi + N\sin\psi}, \]

where

\[ \left(\frac{\delta_1 \sigma B_2}{\beta} - \frac{\delta_2 \sigma B_1}{\beta}\right)=N; \]

\[ \left(\frac{\beta}{\sigma} - \frac{\delta_1\delta_2\sigma}{\beta} - \frac{B_1B_2\sigma}{\beta}\right)=M; \tag{17} \]

\[ \frac{B_2\sigma}{\beta}=P; \]

\[ \frac{\delta_2\sigma}{\beta}=T. \]

The amplitude values of the voltages with frequencies \(\omega_1\) and \(\omega_2\), respectively, are equal to:

\[ U_1=\sqrt{U_{s1}^{2}+U_{c1}^{2}} =\omega_1 L_0 I_1 \sqrt{\frac{P^2+T^2}{M^2+N^2}}, \tag{18} \]

\[ U_2=\sqrt{U_{s2}^{2}+U_{c2}^{2}} =\omega_2 L_0 I_1 \sqrt{\frac{1}{M^2+N^2}}. \]

As was to be expected, the voltage amplitudes do not depend on the phase of the signal. Only the phase of the envelope of the resultant voltage on the PA circuit depends on it.

Now let us investigate the expression for the amplitude of the beat voltage as a function of \(\Delta\omega\), setting \(\cos\Delta\omega t = 1\). The instantaneous value of the amplitude of the resultant voltage when \(\Delta\omega\) varies will be equal to

\[ U_{s\max} = -\frac{L_0 I T_1}{\sqrt{M^2+N^2}} \sqrt{(P^2+T^2)(\omega_1^2+2\omega_1\omega_2)+\omega_2^2}. \tag{19} \]

In constructing the dependence \(U_s=f(\Delta\omega)\), the following was adopted: modulation depth \(m=0.5\); the stability margin coefficient in the amplification mode

\[ \Delta_p = \frac{\delta_p-\frac{\beta}{\sigma}}{\delta_p} =0.1, \]

where \(\delta_p\) is the active conductance of the circuit referred to the inductive conductance, determined at the frequency \(\omega_p = 25\ \text{Hz}\), i.e., \(\delta_p = g\omega_p L_0\).

We also assume that the circuit is tuned to resonance at the frequency \(\omega_p\), i.e., the relative reactive conductance of the circuit at this frequency is

\[ B_p=\frac{\gamma}{\sigma}+\frac{\omega_p^2}{\omega_0^2}=0. \]

Under the adopted initial conditions, the relative active and reactive conductance of the circuit for any \(\Delta \omega\) will be determined by the relations:

\[ \delta_1=\delta_p\frac{\omega_1}{\omega_p}; \qquad \delta_2=\delta_p\frac{\omega_2}{\omega_p}; \]

\[ B_1=\frac{\gamma}{\sigma}\left(1-\frac{\omega_1^2}{\omega_p^2}\right); \]

\[ B_2=\frac{\gamma}{\sigma}\left(1-\frac{\omega_2^2}{\omega_p^2}\right). \]

Taking \(\Delta \omega\) successively equal to \(0.5\), \(1\), \(2\), and \(3\ \text{Hz}\), we calculate for each case the beat-voltage amplitude in relative units:

\[ U_{s*}=\frac{U_s}{U_{s\max}}. \]

The corresponding curve is given in Fig. 2.

Fig. 2

Fig. 3

Experimental data are also presented there. When the pump frequency deviates from the synchronous frequency, a decrease in gain is observed; consequently, the value of \(\Delta\omega\) is of fundamental importance, since at large deviations there is a loss of the necessary properties of the parametric amplifier.

Studies of the amplifier response in the region of the frequency \(50\ \text{Hz}\) show that the decrease in the frequency-detuning coefficient \((K_f)\) occurs not only because of a decrease in the voltage amplitude with fre-

with a frequency of 25 Hz, but also because of the increase in the voltage across the circuit at the signal frequency of 50 hertz.

The response of the amplifier to a current with a frequency of 50 Hz is illustrated in Fig. 2. Curve 2 was obtained from expression (19) by substituting, as the fundamental frequency, 50 Hz and \(\Delta\omega = 0.5;\ 1;\ 2;\ 3;\ 6\) hertz.

Comparing curves (1 and 2), one can construct the dependence

\[ K_f = f(\Delta\omega), \qquad \text{where} \qquad K_f = \frac{U_{S*25}}{U_{S*50}} \quad \text{(Fig. 3).} \]

This curve shows that the frequency deviation must be limited to such a value at which \(K_f\) remains sufficiently large. It is impractical to take a small frequency deviation \(\Delta\omega = 0\), since this will lead to a large delay in the operation of the protection when a fault occurs, when \(\cos \Delta\omega t = 0\). In addition, orientation toward the properties of the parametric amplifier at \(\Delta\omega = 0\) will require high frequency stability from the asynchronous-pumping generator. Apparently, acceptable values of \(\Delta\omega\) will be in the interval from 1 to 2 Hz. In this case \(K_f = 14 \div 11\).

Such a value of \(K_f\) is sufficient for the majority of practical cases.

Frequency oscillations of the asynchronous-pumping generator within the adopted range \(\Delta\omega\) are of no significance, which simplifies the problem of designing the pumping generator.

Conclusions

1. The operating properties of ground-fault protection at a frequency of 25 Hz can be improved by eliminating the phase sensitivity of the PA.

2. Phase sensitivity can be eliminated by using asynchronous pumping of the PA.

3. When the pumping frequency deviates from the synchronous one, some reduction in the frequency-selective properties of the PA occurs.

4. Suitable values of the deviation of the pumping frequency from the synchronous one may be \(\Delta\omega = 1\)—2 Hz.

Literature

1. R. A. Vainshtein, A. V. Shmoilov. Application of low-frequency parametric systems for ground-fault protection in compensated networks. Izvestiya vuzov. “Elektromekhanika”, No. 12, 1965.

2. L. Blekuel, K. Kotsebu. Parametric Amplifiers on Semiconductor Diodes. “Mir,” 1964.