STUDY OF A LOW-FREQUENCY INDUCTIVE PARAMETRON WITHOUT DC BIAS

A. V. Shmoilov, A. Kh. Musin

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

At present, considerable attention is being devoted to the study of storage elements constructed on the parametric principle. The most widespread are high-frequency inductive Goto parametrons with output at the second subharmonic, used as binary elements of digital computers. The simplicity, high reliability, and specific properties of inductive parametrons make it possible to use them also in certain devices of automation, telemechanics, and relay protection.

The use here of high-frequency parametrons is inadvisable, since in a number of cases high speed is not essential; moreover, they require expensive power sources for supply. From the standpoint of maximum simplicity and low cost, attention is merited by an inductive parametron without DC bias with output at the fundamental frequency. When such a parametron is supplied by current of industrial frequency, its application is further simplified, since the signal for controlling the phase of the oscillations can be formed in any part of the installation, the individual elements of which are connected electrically or electromagnetically. It also appears possible to use such a parametron as an element of relay action in automation and telemechanics systems.

We have undertaken a theoretical and experimental investigation of a parametron without DC bias in order to clarify the physical nature of the phenomena and to establish quantitative relationships for the purpose of developing calculation methods.

The circuit used by us was investigated in works [1, 2] as an example of an essentially nonlinear system. However, the results obtained there are, to a large extent, qualitative in character.

Figure 1 shows the circuit of an inductive parametron without DC bias. In it the following are denoted: \(w_1\)—the number of turns of the excitation windings, \(w_2\)—the number of turns of the tank windings, \(i_1\)—the excitation current, \(u_{\mathrm{v}}\)—the excitation voltage, \(i_2\)—the current in the oscillatory circuit.

The field strengths in the cores of the chokes \(D_1\) and \(D_2\) are equal to

\[ h_1 = \frac{i_1 w_1 + i_2 w_2}{l}, \]

\[ h_2 = \frac{i_1 w_1 - i_2 w_2}{l}, \tag{1} \]

where \(l\) is the length of the mean power magnetic line.

The flux linkages of the excitation windings and of the tank windings, respectively, are:

\[ \Psi_{11}=\frac{w_1}{2}\left[2S\mu_1h_1+k_{01}(h_1+h_2)l\right], \]

\[ \Psi_{12}=\frac{w_1}{2}\left[2S\mu_2h_2+k_{01}(h_1+h_2)l\right], \tag{2} \]

\[ \Psi_{21}=\frac{w_2}{2}\left[2S\mu_1h_1+k_{02}(h_1-h_2)l\right], \]

\[ \Psi_{22}=\frac{w_2}{2}\left[2S\mu_2h_2+k_{02}(h_1-h_2)l\right], \tag{3} \]

where

\[ \begin{aligned} &\psi_{11}\ \text{and}\ \psi_{12} &&\text{are the flux linkages of the excitation windings of the first}\\ &&&\text{and second chokes, respectively;}\\ &\psi_{21}\ \text{and}\ \psi_{22} &&\text{are the same for the tank windings;}\\ &S &&\text{is the cross-sectional area of the active steel;}\\ &\mu_1\ \text{and}\ \mu_2 &&\text{are the instantaneous values of the magnetic permeabilities;}\\ &k_{01}\ \text{and}\ k_{02} &&\text{are the leakage permeances of the excitation winding and}\\ &&&\text{the tank winding of one choke.} \end{aligned} \]

The equations of electrical equilibrium are:

\[ i_1R_1+\frac{d\Psi_{11}}{dt}+\frac{d\Psi_{12}}{dt}=u_{\mathrm{B}}, \]

\[ i_2R_2+\frac{d\Psi_{21}}{dt}-\frac{d\Psi_{22}}{dt}+\frac{1}{C}\int i_2dt=0, \tag{4} \]

where

\[ \begin{aligned} &C &&\text{is the capacitance of the resonant-circuit capacitor,}\\ &R_1 &&\text{is the active resistance of the excitation circuit,}\\ &R_2 &&\text{is the active resistance of the resonant circuit.} \end{aligned} \]

Substituting into (4) the values of the quantities from expressions (1–3) and adopting the notation

\[ \frac{1}{2w_2}R_2=\rho;\qquad \frac{R_1w_2}{R_2w_1}=\lambda;\qquad \frac{1}{2w_2C}=\beta;\qquad 2w_1w_2k_{01}C=q; \]

\[ w_2S=\delta;\qquad \frac{w_1}{w_2}=k_T;\qquad \mu_1h_1=b_1;\qquad \mu_2h_2=b_2, \]

we obtain

\[ \left\{ \begin{aligned} &\left(\lambda\rho+q\beta\frac{d}{dt}\right) \left(\frac{b_1}{\mu_1}+\frac{b_2}{\mu_2}\right) +k_T\delta\frac{d}{dt}(b_1+b_2)=u_{\mathrm{B}},\\ &\left(\rho+\beta\int dt\right) \left(\frac{b_1}{\mu_1}-\frac{b_2}{\mu_2}\right) +\delta\frac{d}{dt}(b_1-b_2)=0. \end{aligned} \right. \tag{5} \]

The nonlinear equations (5) describe the entire diversity of oscillations in the system of Fig. 1.

Let us introduce into equations (5) a power function approximating the main magnetization curve \(b=\mu_k h^K\), where \(K\) is a fraction whose numerator and denominator are odd numbers, and \(\mu_k\) is a constant coefficient. The coefficients of the approximating function are determined from the main magnetization curve by the method of selected points.

Fig. 1

We neglect leakage and take the law of variation of the excitation voltage in the form

\[ u_{\mathrm{B}} = U_m \cos(\omega t + \psi), \]

where \(U_m\) and \(\psi\) are, respectively, the amplitude and phase of this voltage. Under the adopted conditions, equations (5) take the form:

\[ \frac{1}{\mu^{\frac{1}{K}}K}\lambda p \frac{d}{dt} \left(b_1^{\frac{1}{K}}+b_2^{\frac{1}{K}}\right) + k_T\delta \frac{d}{dt}(b_1+b_2) = U_m\cos(\omega t+\psi), \tag{6} \]

\[ \frac{1}{\mu^{\frac{1}{K}}K} \left(\rho+\beta\int dt\right) \left(b_1^{\frac{1}{K}}-b_2^{\frac{1}{K}}\right) + \delta \frac{d}{dt}(b_1-b_2) = 0. \tag{7} \]

We specify a particular solution of the system (6), (7) in the form

\[ b_1 = B_1\cos\omega t, \tag{8} \]

\[ b_2 = B_2\cos(\omega t+\theta), \tag{9} \]

where

\(B_1\) and \(B_2\) are the amplitudes of the inductions in the cores,

\(\theta\) is the angle between the amplitudes.

Differentiating (7) and then substituting the solutions (9) into (6) and (7), we obtain two identities. Assigning definite values to the arguments in the identities (for example, \(\omega t=0\) and \(\omega t=-\frac{\pi}{2}\)) and carrying out all the necessary transformations, we arrive at a system of three transcendental equations relating the coefficients of the solutions (9):

\[ B_2= \left[ \frac{ \rho \sin^{\frac{1}{K}-2}\theta \cos\theta }{ \mu^{\frac{1}{K}}K\delta\omega } + \frac{ \beta \sin^{\frac{1}{K}-1}\theta }{ \mu^{\frac{1}{K}}\delta\omega^2 } \right]^{\frac{K}{K-1}}, \tag{10} \]

\[ \frac{\beta}{\mu^{\frac{1}{K}}}B_1^{\frac{1}{K}} - \delta\omega^2 B_1 = \frac{\beta}{\mu^{\frac{1}{K}}}B_2^{\frac{1}{K}}\cos^{\frac{1}{K}}\theta - \delta\omega^2 B_2\cos\theta - \frac{\rho\omega}{\mu^{\frac{1}{K}}K} B_2^{\frac{1}{K}} \cos^{\frac{1}{K}-1}\theta\sin\theta, \tag{11} \]

\[ \left(\frac{\lambda\beta}{\mu_K^{1/K}}\right)^{2} \left[ B_1^{2/K}+2B_1^{1/K}B_2^{1/K}\cos^{1/K}\theta +B_2^{2/K}\left(\cos^{2/K}\theta+\sin^{2/K}\theta\right) \right]+ \]

\[ +\frac{2\lambda\rho k_T\delta\omega}{\mu_K^{1/K}} \left[ B_1B_2^{1/K}\sin^{1/K}\theta -B_1^{1/K}B_2\sin\theta +B_2^{1/K+1}\left(\sin^{1/K}\theta\cos\theta-\sin\theta\cos^{1/K}\theta\right) \right] +(k_T\delta\omega)^2(B_1^2+2B_1B_2\cos\theta+B_2^2)=U_m^2 \tag{12} \]

The system (10)–(12) is determined when the amplitude of the excitation voltage \(U_m\) is specified, at which oscillations arise in the oscillatory circuit (the excitation threshold). The excitation threshold is calculated on the basis of the assumption that the mean natural frequency of the nonlinear oscillatory circuit \((\Omega_{\mathrm{cp}})\) is equal to the frequency of the parametric oscillations \(\omega\). The calculation is carried out in the following sequence: 1) using the adopted approximation, we write an expression for the square of the natural frequency of the oscillatory circuit before oscillations arise \((b_1=b_2=b)\); 2) the mean natural frequency over a period is determined; 3) in the expression obtained, at \(\Omega_{\mathrm{cp}}=\omega\), the induction amplitude \(B_n\) corresponding to \(U_m\) is determined.

\[ B_n= \left[ \frac{ 2\Omega_{\mathrm{cp}}^{2}S w_2^{2}K\mu_K^{1/K}\sqrt{\pi}\,C\Gamma\left(\dfrac{1}{2K}+0.5\right) }{ l\Gamma\left(\dfrac{1}{2K}\right) } \right]^{\frac{K}{1-K}}, \tag{13} \]

where \(\Gamma\) is the gamma function.

Putting in (12) \(B_1=B_2=B_n\) and \(\Theta=0^\circ\) (parametric oscillations are absent), we determine \(U_m\)

\[ U_m=2 \left[ \left( \frac{\lambda\beta}{\mu_K^{1/K}}B_n^{1/K} \right)^2 +(k_T\delta\omega B_n)^2 \right]^{1/2}. \tag{14} \]

For a parametron with the following initial data \(l=0.182\ \text{m}\)

\[ w_2=1000\ \text{turns},\qquad S=0.176\cdot10^{-3}\ \text{m}^2,\qquad \omega=314\ \frac{1}{\text{s}}, \]

\[ \Omega_{\mathrm{cp}}=314\ \frac{1}{\text{s}},\qquad k_T=0.8,\qquad \delta=0.176\ \text{m}^2\ \text{turns}, \]

\[ R_1=39\ \Omega,\qquad R_2=31.4\ \Omega,\qquad \rho=2.86\cdot10^{-3}\ \frac{\text{m}\cdot\Omega}{\text{turns}}, \]

\[ \lambda=1.53,\qquad \mu_K=0.33,\qquad K=0.185,\qquad w_1=800\ \text{turns}. \]

the values of \(U_m\) were calculated by formulas (13), (14) in the range of capacitances \(C=(2\div40)10^{-6}\ \text{F}\). The results of the calculations are shown in Fig. 2 (solid

Fig. 2

curve). Experimental data are also given here (dashed curve). It is seen from Fig. 2 that the experimental values of \(U_{\mathrm{in}}\) exceed the calculated ones by no more than 8%. Increasing the resistance of the oscillatory circuit by a factor of 2 does not lead to any noticeable change in this difference.

Let us present the calculation of the amplitudes of the inductions \(B_1\), \(B_2\) and of the angle between them \(\Theta\) in the steady-state oscillation regime for a parametron whose data were given above, with capacitances of the oscillatory-circuit capacitor equal to 12, 16, and 20 \(\mu\mathrm{F}\). We shall carry out the calculations in the following order: 1) from expression (10) we calculate the values of \(B_2\) as a function of \(\Theta\); 2) substituting the known values of \(B_2\) and \(\Theta\) into (11), we calculate \(B_1\); 3) substituting \(B_1\), \(B_2\), \(\Theta\) into (12), we calculate the left-hand side (\(\mathrm{Л}\)) of equation (12). The results of calculating \(B_1\), \(B_2\), and \(\mathrm{Л}\) as functions of \(\Theta\) are shown in Fig. 3.

Fig. 3.

Plotting on the curve \(\mathrm{Л}\) the value \(U_m^2\), calculated from (14), we determine \(\Theta\), and from \(\Theta\) we find \(B_1\) and \(B_2\). The accuracy of the calculation depends on the accuracy with which \(U_m\) is determined. Since the calculated values of \(U_m\) differ from the experimental ones, the obtained values of the quantities must be refined. This refinement can be carried out with the aid of the equation of the energy balance of the oscillatory circuit over a period,

\[ W_{R2}+W_L=0, \tag{15} \]

where

\[ W_{R2}\text{ is the energy consumed in the active resistance of the oscillatory circuit during an oscillation period,} \]

\[ W_L\text{ is the energy introduced into the oscillatory circuit each period due to modulation of the inductance.} \]

With the adopted approximation of the main magnetization curve and the form of variation of the induction in the cores (9), equation (15) is expressed through the sought quantities as follows:

\[ \frac{1}{w_2 \mu K^{2/\kappa}\omega} \left[ N\left(B_1^{2/\kappa}+B_2^{2/\kappa}\right) - B_1^{1/\kappa}B_2^{1/\kappa}\varphi(\theta,K) \right] - \frac{18}{w_2 \mu K^{1/\kappa}} M\sin\theta \left(B_1^{1/\kappa}B_2-B_1B_2^{1/\kappa}\right) =0, \tag{16} \]

where

\[ N=\frac{\sqrt{\pi}\,\Gamma\left(1+\frac{2-K}{2K}\right)} {\Gamma\left(1.5+\frac{2-K}{2K}\right)}, \]

\[ M=\frac{\sqrt{\pi}\,\Gamma\left(1+\frac{1}{2K}\right)} {\Gamma\left(1.5+\frac{1}{2K}\right)}, \]

\[ \varphi(\theta,K)=\int_{0}^{2\pi}[\cos\omega t\cos(\omega t+\theta)]^{\frac{1}{K}}\,d\omega t. \]

The calculation of \(\varphi(\Theta,K)\) was carried out by means of a certain formal method. The results of the calculations are shown in Fig. 4.

[Figure: graph of \(\varphi(\theta,m)\) versus \(m\), with curves labeled \(\theta=0^\circ\), \(\theta=40^\circ\), \(\theta=50^\circ\), \(\theta=60^\circ\), \(\theta=70^\circ\), \(\theta=80^\circ\); marked values include \(K=5.4\) and \(0.155\).]

Fig. 4

Calculations according to (15) are presented in Fig. 3 (the curves intersecting the \(\Theta\) axis). For \(W_{R2}+W_L=0\), we obtain a refined value of the angle \(\Theta\), and from it, using the curves of Fig. 3, we determine refined values of \(B_1\), \(B_2\), and \(U_m\). Below, Table 1 gives the calculated results and experimental data for the parametron under consideration. As can be seen from the table, the calculated and experimental data agree quite closely. Consequently, the relations derived above can be used for calculating the parametron.

Table 1

REFERENCES

1. T. Hayashi. Forced Oscillations in Nonlinear Systems. IL, 1957.
2. L. A. Bessonov. Nonlinear Electric Circuits. “Vysshaya shkola,” 1964.
3. L. A. Bessonov. Theoretical Foundations of Electrical Engineering. “Vysshaya shkola,” 1964.