TEMPERATURE STABILIZATION OF A PARAMETRIC AMPLIFIER

R. A. Vainshtein, L. I. Voronova, A. V. Shmoilov

(Presented by the scientific seminar of the Department of Electric Power Stations)

The study of the operation of a parametric amplifier (PA) under conditions of a strongly varying ambient temperature was undertaken because the responding elements of ground-fault protection using a PA had to be installed in an unheated room (the 6 kV switchgear of a power station).

It should be noted that the installation of responding elements of ground-fault protection for 6–10 kV consumer lines of power stations in the switchgear is typical. Therefore, the requirement for sufficiently high stability of characteristics when the temperature changes, imposed on the responding elements of ground-fault protection, may be considered general.

Under Siberian conditions, the maximum annual range of variation of the ambient-air temperature may be taken to be from \(-40^\circ\) to \(+40^\circ\).

Within this temperature range it is necessary to investigate and select measures ensuring stable operation of the responding element.

Causes of temperature instability of the PA

The voltage on the oscillatory circuit of the PA, according to [1], at the optimum signal phase, equal to \(90^\circ\), is determined by the expression

\[ U_{\omega}= \frac{\omega L_0 I_{\omega}\sqrt{B^2+\varepsilon_+^2}} {B^2+\delta_-\cdot\delta_+}, \tag{1} \]

where

\[ \omega \text{ — angular frequency of the signal,} \]

\[ L_0 \text{ — mean value of the inductance of the oscillatory circuit of the PA;} \]

\[ I_{\omega} \text{ — signal current;} \]

\[ B=\frac{\omega^2}{\omega_0^2}+\frac{\gamma}{\sigma} \text{ — relative reactive conductance of the circuit;} \]

\[ \frac{\gamma}{\sigma} = \frac{\frac{1}{4}m^2-1} {\left(\frac{1}{4}m^2-1\right)^2-\frac{1}{4}m^2} \text{ — relative inductive conductance, taking parametric action into account;} \]

\[ m \text{ — depth of inductance modulation;} \]

\[ \omega_0=\frac{1}{\sqrt{L_0C}} \]
is the resonant frequency of the circuit, determined from \(\alpha_0\),

\[ \delta_+=\delta+\frac{\beta}{s};\qquad \delta_-=\delta-\frac{\beta}{s}; \]

\[ \delta=g_{\mathrm{э}}\omega L_0 \]
is the relative equivalent active conductance of the circuit;

\[ \frac{\beta}{s}= \frac{\frac{1}{2}m}{(1/4\,m^2-1)^2-\frac{1}{4}m^2} \]
is the relative active conductance of the circuit introduced as a result of the parametric action.

Let us analyze the influence of temperature on the parameters entering formula (1).

The quantities \(\omega\) and \(I_a\) are, in principle, determined not by the parameters of the amplifier but by other elements; therefore, in what follows, when the temperature changes, we regard them as constant.

The mean value of the inductance \(L_0\) for an amplifier made according to the design described in [2] is determined by the expression

\[ L_0=\mu_{d\max}\frac{W_{\mathrm{k}}^2 q_1}{l_1} +\mu_0\frac{(W_{2\mathrm{k}})^2}{l_\delta}, \tag{2} \]

where

\[ \mu_{d\max} \]
is the maximum dynamic permeability of the ferromagnetic material of the PU cores;

\[ \mu_0 \]
is the magnetic permeability of vacuum;

\[ W_{\mathrm{k}};\ q_1;\ q_3;\ l_1;\ l_\delta \]
are design parameters of the amplifier, practically independent of temperature.

Fig. 1

The magnetic permeability of vacuum \(\mu_0\) likewise does not depend on temperature, while the maximum differential permeability of the silicon transformer steel that we use for manufacturing the PU, according to the data given in [3], changes only insignificantly in the temperature range \(-40\div +40^\circ\).

This is illustrated for steel E-42 in Fig. 1, a, borrowed from [3].

The relative active conductance introduced by the parametric action \(\beta/\sigma\) depends only on the modulation depth of the inductance \(m\), which, according to [2], is equal to

\[ m=\mu_{d\max}\frac{q_1 W_k^2}{l_1}\cdot \frac{1}{L_0}. \tag{3} \]

Comparing (2) and (3), we arrive at the conclusion that \(m\), and consequently also the introduced conductance \(\beta/\sigma\), practically do not depend on temperature. The relative active conductance of the circuit

\[ B=\frac{\omega^2}{L_0 C}+ \frac{\frac{1}{4}m^2-1} {\left(\frac{1}{4}m^2-1\right)-\frac{1}{4}m^2} \tag{4} \]

for \(m=\mathrm{const}\) and \(L_0=\mathrm{const}\) will remain constant under the condition that the capacitance \((C)\) of the capacitor of the oscillatory circuit is constant. The latter, for the metallized-paper capacitors used, changes over the adopted temperature range by no more than 3% (Fig. 1, б) [3].

Taking into account that the dependence of the gain coefficient of a parametric amplifier on tuning is flat in the region of its maximum, we may neglect changes in \(B\) under temperature fluctuations.

Fig. 2

The relative equivalent load of the amplifier \(\delta\) takes into account the losses in the external load of the oscillatory circuit of the PA and the losses in the copper of the circuit winding. Under the condition that \(\omega L_0\) is constant, \(\delta\) is proportional to the real equivalent active conductance of the circuit \(g_e\), which (Fig. 2) is determined by the expression

\[ g_e= \frac{ R_m+\dfrac{g_n}{(\omega C)^2+g_n^2} } { \left[ R_m+\dfrac{g_n}{(\omega C)^2+g_n^2} \right]^2 + \left[ \dfrac{\omega C}{(\omega C)^2+g_n^2} \right]^2 } \tag{5} \]

for \(R_d=g_d=0\), where
\(R_m\) is the active resistance of the circuit winding.

From (5) it is seen that \(g_e\) consists of two components, depending to one degree or another on temperature, since, even if \(g_n\) does not depend on temperature, \(R_m\) always changes very strongly under temperature fluctuations.

To facilitate the further analysis, let us simplify expression (5), taking into account the actual relations that occur in real PAs. Thus, for all constructed amplifiers \((\omega C)^2 \gg g_n^2\); therefore

\[ g_e \approx \frac{ R_m+g_n/(\omega C)^2 } { \left[R_m+g_n/(\omega C)^2\right]^2+\dfrac{1}{(\omega C)^2} }. \tag{6} \]

Further analysis of the ratios of the parameters entering into (6) makes it possible to conclude that its denominator changes very little under temperature fluctuations, if \(g_{\mathrm{n}}=\mathrm{const}\) is assumed.

On this basis, we shall represent \(g_{\mathrm{e}}\) in the form of a certain relative quantity consisting of two terms, the first of which depends only on \(R_{\mathrm{m}}\), and the second on \(g_{\mathrm{n}}\):

\[ g_{\mathrm{e}*}=R_{\mathrm{m}*}+\left[\frac{g_{\mathrm{n}}}{(\omega C)^2}\right]. \tag{7} \]

It is important to note that, under the assumptions adopted, the equivalent capacitive conductance \(b_{\mathrm{e}}\) in the circuit of Fig. 2 does not depend on temperature:

\[ b_{\mathrm{e}}= \frac{1/(\omega C)^2} {\left[R_{\mathrm{m}}+g_{\mathrm{n}}/(\omega C)^2\right]^2+1/(\omega C)^2}. \tag{8} \]

On the basis of the assumptions made, the relative change in the voltage across the PU circuit can be studied by putting \(B=0\) in (1) and \(\omega_0 L_0/\omega=1\), according to the expression

\[ \frac{U(t)}{U}=\frac{1}{\delta(t)-\beta/\sigma}. \]

The relative temperature changes \(\delta\) will be found from (7), taking \(R_{\mathrm{m}*}=1\) and several values of the ratio

\[ \frac{g_{\mathrm{n}}}{(\omega C)^2 R_{\mathrm{m}}}=P, \]

\[ g_{\mathrm{e}*}(t)=(1+\alpha \Delta t)+P_*. \]

Figure 3 presents curves of the dependence \(U(t)/U\) for \(m=0.6\) and stability margin coefficient in the amplifying mode at \(t=20^\circ\), \(\Delta=0.1\).

Fig. 3

From these curves it follows that, in the temperature range \(-40 \div +40^\circ\), completely inadmissible changes in the gain coefficient occur. Moreover, these changes decrease as \(P\) increases.

Large values of \(P\) occur in more powerful amplifiers. However, the comparatively small difference between the curves for different \(P\) makes it possible to conclude that the natural temperature stability, even with a considerable increase in amplifier power, will be low.

Method for increasing temperature stability

As follows from what has been said above, the temperature stability of a PA can be ensured if the constancy of \(g_e\) is ensured. Let us turn again to expression (6). The equivalent active conductance \(g_e\) will remain constant if the binomial \([R_m + g_n/(\omega C)^2]\), containing a parameter dependent on temperature, also remains constant. This can be achieved if, when the temperature changes, the increments of the quantities \(R_m\) and \(g_n/\omega^2 C^2\) are equal in magnitude and opposite in sign.

It is shown below that such a change in \(R_m\) and \(g_n/\omega^2 C^2\) is obtained with satisfactory accuracy when the external load resistance is made of metal wire. Denote the current value of the quantity \(R_m + g_n/\omega^2 C^2\) by \(A(t)\).

\[ A(t)=R_{m t}(1+\alpha_m \Delta t)+R_d+ \frac{g_n/(1+\alpha_n \Delta t)+g_d}{\omega^2 C^2}. \tag{11} \]

The resistance \(R_d\) and conductance \(g_d\), which do not depend on temperature, have been introduced into the circuit for generality. The expression \(g_n(1+\alpha_n \Delta)\) can be approximated, for a limited temperature range, by a straight line, so long as it may be considered that \((\alpha_n \Delta t)\ll 1\), since

\[ \frac{1}{1+\alpha_n \Delta t} = \frac{1-\alpha_n \Delta t}{1-(\alpha_n \Delta t)^2} \approx 1-\alpha_n \Delta t. \]

Taking (12) into account, expression (11) takes the form

\[ A(t)=R_m+R_d+g_n/\omega^2 C^2+g_d/\omega^2 C^2 +R_m\alpha_m \Delta t -\frac{g_n}{\omega^2 C^2}\alpha_n \Delta t. \tag{13} \]

Expression (13) will not depend on temperature if

\[ R_m\alpha_m \Delta t-\frac{g_n}{\omega^2 C^2}\alpha_n \Delta t=0. \tag{14} \]

The solution of equation (14) gives the required value of \(g_n\) at the given fixed temperature, ensuring constancy of the equivalent load of the amplifier:

\[ g_n=\frac{\alpha_m}{\alpha_n}\cdot R_m\omega^2 C^2. \tag{15} \]

From (15) it is evident that the material from which \(g_n\) is to be made must have a positive temperature coefficient of resistance, constant over the specified temperature range. In order that the power losses in the resistance \(g_n\), intended only for temperature stabilization, be small, \(\alpha_n\) should preferably be as large as possible. Unfortunately, the only available material from which the compensating conductance can be made is copper. Its drawback for this purpose is its low resistivity and, as a consequence, the comparatively high labor input required to manufacture large resistances.

The developed method of temperature stabilization has been applied in the frequency-selective reacting protection elements against ground faults at the Barnaul TPP-2. The active resistance of the windings of the oscillatory circuit of the parametric amplifiers is \(R_m=410\ \Omega\), and the capacitor capacitance is \(3\ \mu\mathrm{F}\).

The compensating resistance, under the condition that it is made of copper wire, according to (15), is equal to

\[ R_k=1/g_{\mathrm{n}}=\frac{1}{R_m\omega^2 C^2}=11.3\ \text{k}\Omega . \]

Figure 4 presents the experimental dependences of the voltage on the PA circuit on temperature, without stabilization and when a resistance \(R=11.3\ \text{k}\Omega\) is connected in parallel with the capacitance.

Fig. 4

As can be seen, this stabilization method is very effective, and condition (15) for determining the compensating resistance gives satisfactory accuracy.

The compensating resistance is wound bifilarly with copper wire of diameter \(0.05\ \text{mm}\) and has the following dimensions: diameter—\(15\ \text{mm}\), height—\(45\ \text{mm}\). The amplifier power in the presence of the compensating resistance is sufficient to actuate a polarized relay of type RP-5.

Conclusions

1. The dependence of the gain coefficient of a parametric amplifier on temperature is explained by the change in the resistance of the circuit winding.

2. High temperature stability of the gain coefficient can be achieved by connecting a metallic resistance in parallel with the oscillatory circuit, the value of which is chosen according to (15).

References

1. R. A. Vainshtein, A. V. Shmoilov. Application of a low-frequency parametric amplifier in protection against ground faults in compensated networks. Izvestiya VUZov, “Elektromekhanika,” No. 12, 1965.

2. R. A. Vainshtein. Parametric amplifier. Author’s certificate No. 178862 with priority of 15.12.1964.

3. N. M. Tishchenko. Stability of Magnetic Amplifiers. “Energiya,” 1964.