ANALYSIS OF THE EXCITATION CONDITIONS OF A PARAMETRIC GENERATOR POWERED BY A DC SOURCE THROUGH A CONTROLLED SWITCH

R. A. Vainshten, A. G. Chikunov

(Submitted by the Department of Electric Power Stations)

The circuit of the generator under study is shown in Fig. 1. As can be seen, this circuit of an electromagnetic parametric generator differs from the classical one described in [1] in that the excitation windings \((w_1)\) are powered from a DC source through a controlled switch, for example a transistor, and the excitation circuit is shunted by a circuit consisting of a diode and a resistor in order to limit the voltage on the switch at the moment the transistor is closed. The value of the resistor is selected from the condition

Fig. 1

\[ R \leq \frac{U_k - U}{I_m} \tag{1} \]

where

\(U\) — supply voltage;
\(U_k\) — permissible voltage on the switch;
\(I_m\) — maximum switched current.

In essence, the circuit is an autonomous inverter which, in comparison with known inverters, makes it possible, with a single power switch, to obtain from DC an AC voltage close in form to a sinusoid and stable in amplitude over a wide range of variation of the source voltage. Therefore it is of interest to investigate the conditions for excitation of oscillations in such an inverter circuit. The oscillations arise as a result of modulation of the nonlinear inductance of the ferromagnetic cores and are maintained autoparametrically at the expense of the energy of the DC source. The differential equations of the circuit are nonlinear and can be solved analytically in co-

… expressions only under certain approximations of the dependence relating the induction and field strength of ferromagnetic cores.

The condition for the excitation of oscillations may be taken to be fulfillment of the criterion of the square of the mean resonant frequency, proposed in [2]. According to this criterion, a circuit with a periodically varying nonlinear inductance is replaced by an equivalent oscillatory circuit with a time-constant inductance, in which the period of oscillations is equal to the period of free oscillations of the actual circuit. In view of the high quality factor of the oscillatory circuit of parametric generators, the natural frequency may be considered equal to the resonant frequency. To obtain an analytical expression for the linearized equivalent inductance, let us assume that the inductance varies according to some law \(l(t)\) and determines the period of free oscillations in the circuit. We divide the period into \(n\) parts, in each of which we take the inductance \(l_i\) and the resonant angular frequency \(\omega_i^2 = \dfrac{1}{l_i C}\) to be constant. During the time \(\dfrac{T}{n}\), the vector of the current or voltage of the free oscillations passes through the angle \(\Delta \varphi = \dfrac{T}{n}\omega_i\). Over the entire period,

\[ \sum_{n=1}^{n} \frac{T}{n}\omega_i = 2\pi . \tag{2} \]

The period of free oscillations in the equivalent circuit must also be equal to \(T\); therefore

\[ \sum_{n=1}^{n} \frac{T}{n}\omega_i = T\omega_{\mathrm{e}}, \tag{3} \]

where \(\omega_{\mathrm{e}}\) is the resonant frequency of the equivalent circuit. Letting \(n\) tend to infinity and passing to the integral, we obtain

\[ \omega_{\mathrm{e}} = \frac{1}{T}\int_{0}^{T} \omega_i\,dt . \tag{4} \]

Since the capacitance of the capacitor is constant, the expression for the equivalent inductance is

\[ l_{\mathrm{e}} = \frac{1}{ \left[ \frac{1}{T}\int_{0}^{T}\frac{1}{\sqrt{l(t)}}\,dt \right]^2 }. \tag{5} \]

Using formula (5) in most cases proves difficult because of the complexity of evaluating the integral in closed form. Numerous calculations with subsequent experimental verification, as well as comparison of calculated data for cases in which the integral in (5) can be evaluated exactly, have shown that the approximate expression can be used in practice:

\[ l_{\mathrm{e}} = \frac{1}{ \frac{1}{T}\int_{0}^{T}\frac{1}{l(t)}\,dt }. \tag{6} \]

Formulas (5) and (6) can strictly be applied only to synchronous oscillations, that is, to those for which the ratio of the periods of inductance variation and parametric oscillations is an integer, since in this case the period of oscillations is known to us in advance and, therefore, the limits of integration are known. The problem of studying the excitation conditions in

in the case of synchronous oscillations consists in finding the electrical quantities for which \(\omega_e\) is equal to the oscillation frequency of the synchronous mode.

In the circuit under study, oscillations whose frequency is half the switching frequency of the switch are of practical value, since in this case the above-mentioned properties of the circuit are manifested in the best way.

To apply formula (6), it is necessary to find an analytical expression for the integrand function \(\dfrac{1}{l(t)}\).

For the circuit under study in the quasi-stationary regime, when oscillations have not yet arisen, such an expression can be obtained if partial cycle 2 (Fig. 2) of magnetization of the cores is replaced by the single-valued curve 3 and approximated by a third-degree polynomial of the form

\[ h=\alpha b+\beta b^3 . \tag{7} \]

Partial cycle 2 of unipolar magnetization of the cores is assumed to lie inside symmetric cycle 1, and the coordinates \(b_0\) and \(h_0\) of its point \(m\) depend on the maximum value of the magnetic-field intensity \(h_m\), the value of resistor \(R\), and the duration of the closed state of the switch. The values \(b_0\) and \(h_0\) were measured experimentally by oscillographing the magnetization cycles. These measurements showed that, for constant \(R\) and variation of \(\Delta b\) (Fig. 2) within real limits, \(b_0\) changes by no more than 20%. This circumstance makes it possible, in the analysis, to regard \(b_0\) as a constant value equal to the average of the obtained measurements. Physically, \(b_0\) and \(h_0\) in the quasi-stationary regime are determined by the equality of the energies: that stored in the cores when the switch is open and that released in the resistor when the switch is closed. In addition, \(h_0 \ll h_m\) always, and one may take \(h_0=0\). Further, by the instantaneous value of the induction \(b\) we shall understand the induction increment \(\Delta b\), which is equivalent to shifting the origin of coordinates to point \(m'\). Owing to this shift of the origin, curve 3 is approximated quite accurately by expression (7), which makes it possible to solve the entire problem analytically.

Fig. 2

The function \(\dfrac{1}{l(t)}\) can be represented as:

\[ \frac{1}{l(t)} = \frac{1}{\mu_{\mathrm{d}}\dfrac{2w_2^2 q}{l_{\mathrm{cp}}}} = \frac{1}{\mu_{\mathrm{d}}K}, \tag{8} \]

where

\[ \mu_{\mathrm{d}} \text{ is the dynamic permeability of the core;} \]

\(w_2\) — number of turns of the tank winding;
\(q\) — cross-sectional area of one core;
\(l_{\mathrm{cp}}\) — length of the mean line of force of the core.

For the ratio \(\dfrac{1}{\mu_{\mathrm{д}}}\), from (7) one may write

\[ \frac{1}{\mu_{\mathrm{д}}}=\frac{dh}{db}=\alpha+3\beta b^2 . \tag{9} \]

The expression for \(b\) as a function of time can be found from the equations of electrical equilibrium for the supply-circuit loop in the quasistationary regime.

In setting up the differential equation for the supply-circuit loop, we neglect the active resistance of the excitation winding, take the switch to be ideal, and the power supply to have infinite power. We take the instant of switch closure as the origin of time \((t=0)\).

With the switch in the closed position, when \(0 \leq t \leq \dfrac{T_{\mathrm{k}}}{N}\),

\[ U=2qw_1\cdot\frac{db}{dt}, \tag{10} \]

where

\(T_{\mathrm{k}}\) — switching period of the switch;
\(N\) — duty factor of the control pulses;
\(w_1\) — number of turns of the excitation winding.

From (10) we find

\[ b=\frac{U\cdot t}{2qw_1}. \tag{11} \]

With the switch closed, when \(\dfrac{T_{\mathrm{k}}}{N}\leq t\leq T_{\mathrm{k}}\),

\[ -2qw_1\frac{db}{dt}=\frac{h l_{\mathrm{cp}}}{w_1}R, \tag{12} \]

where \(h\) is the magnetic-field strength in the core. Substituting (7) into (12) and separating the variables, we obtain

\[ t-\frac{T_{\mathrm{k}}}{N} = -\frac{2qw_1^2}{l_{\mathrm{cp}}R} \int \frac{db}{\alpha b+\beta b^3}. \tag{13} \]

After integration and transformations,

\[ b^2= \frac{\alpha}{ C_1 e^{\frac{t-T_{\mathrm{k}}/N}{\tau}}-\beta }, \tag{14} \]

where

\[ \tau=\frac{qw_1^2}{\alpha l_{\mathrm{cp}}R}. \]

We determine the integration constant from the initial conditions

\[ t=\frac{T_{\mathrm{k}}}{N},\qquad b_{\mathrm{н}}=\frac{U}{2qw_1}\cdot\frac{T_{\mathrm{k}}}{N}, \]

under which

\[ C_1=\frac{\alpha+\beta b_{\mathrm{н}}^2}{b_{\mathrm{н}}^2}. \tag{15} \]

Substitution of expressions (8), (9), and then (11) and (14), into (6) gives

\[ \frac{1}{l_{\mathrm e}}= \frac{1}{T_{\mathrm k}K} \left[ \int_{0}^{T_{\mathrm k}/N} \left( \alpha+\frac{3}{4}\beta\frac{U^{2}t^{2}}{q^{2}w_{1}^{2}} \right)dt+ \int_{T_{\mathrm k}/N}^{T_{\mathrm k}} \left( \alpha+3\beta \frac{\alpha}{C_{1}e^{-\frac{t-T_{\mathrm k}/N}{\tau}}-\beta} \right)dt \right]. \tag{16} \]

After integration,

\[ \frac{1}{l_{\mathrm e}}= \frac{1}{K} \left( \alpha+\frac{1}{4}\beta\frac{U^{2}T_{\mathrm k}^{2}}{q^{2}w_{1}^{2}N^{3}} + \frac{3\alpha\tau}{T_{\mathrm k}} \ln \frac{ C_{1}-\beta\cdot e^{-\frac{T_{\mathrm k}}{\tau}\left(1-\frac{1}{N}\right)} }{ C_{1}-\beta } \right). \tag{17} \]

The relation between the electrical quantities and the circuit parameters at the boundary of oscillation excitation is found from the equation

\[ \frac{1}{l_{\mathrm e}C}=\omega^{2}, \tag{18} \]

where

\[ \omega=\frac{\pi}{T_{\mathrm k}}. \]

Fig. 3

Figure 3 shows, as an example, the dependence of the supply-source voltage at the boundary of oscillation excitation on variation of the capacitance of the capacitor of the oscillatory circuit for an experimental sample of a parametric generator with the following design parameters: ferromagnetic material—ferrite NN-600, \(q=3.92\cdot10^{-4}\ \text{m}^{2}\), \(l_{\mathrm{av}}=0.18\ \text{m}\), \(w_{1}=80\), \(w_{2}=218\), \(T_{\mathrm k}=1.25\cdot10\ \text{s}\), \(R=100\ \Omega\).

According to the measurement results, \(b_{0}=0.09\ \text{T}\) was adopted. In this case, for the curve with a shifted origin of coordinates, \(\alpha=1000\), \(\beta=83000\).

Experimental data are also presented in the same figure. The discrepancy between the experimental and calculated data is explained by the fact that in the actual

... and the replacement of the cycle by a single-valued curve.

From the results of the calculation and the experiment it follows that the criterion of the square of the mean resonant frequency is applicable to the calculation of the minimum excitation voltage of a parametric generator also in the case when it is supplied from a direct-current source through a controlled switch.

REFERENCES

1. A. M. Bamdas, S. V. Shapiro, L. N. Davydova. Ferromagnetic frequency dividers. “Energiya,” 1967.

2. R. A. Weinstein, A. V. Shmoilov, A. Kh. Musin, A. P. Pushkov, A. G. Chikunov. Theoretical and experimental studies and development of methods for calculating nonlinear inductive parametric generators. Abstracts of papers of the Fourth All-Union Inter-University Conference on the Theory and Methods of Calculation of Nonlinear Electrical Circuits and Systems. Tashkent, 1971.