UDC 621.314:621.317

RESONANCE PHENOMENA IN ALTERNATING-CURRENT BRIDGES WITH TIGHT INDUCTIVE COUPLING

Yu. B. Volynskii, V. A. Raskolenko

(Presented by the scientific seminar of the Faculty of Physics and Engineering)

Let us consider an alternating-current bridge formed by two inductively coupled coils with numbers of turns \(W_1\) and \(W_2\) and two impedances \(\dot Z_3\) and \(\dot Z_4\) (Fig. 1). If losses in the inductance coils are neglected, then, for a coupling coefficient equal to unity, the following relations may be written for the voltages \(\dot U_3\) and \(\dot U_4\) [1]:

\[ \dot U_3 = U_c \frac{\dot A\cdot[1+\dot B(1+K)]} {\dot A+\dot B(K^2+\dot A)}, \tag{1} \]

\[ \dot U_4 = U_c\cdot \frac{\dot A+\dot B(1+K)K} {\dot A+\dot B(K^2+\dot A)}, \tag{2} \]

where

\[ \dot A=\frac{\dot Z_3}{\dot Z_4}, \qquad \dot B=\frac{j\omega L_2}{\dot Z_n}, \]

\(L_2\) is the inductance of the coil with number of turns \(W_2\),

\[ K=\frac{W_1}{W_2}. \]

Fig. 1. Circuit diagram of an alternating-current bridge with tight inductive coupling

Let us analyze relations (1, 2) for \(\dot A=\pm \alpha\); \(\dot A=\pm j\alpha\), \(\dot B=\pm \beta\), \(\dot B=j\beta\). Here \(\alpha=|\dot A|\), \(\beta=|\dot B|\). Special attention will be paid to those cases in which, when the coefficients \(\alpha\) and \(\beta\) are varied, either the moduli or the arguments of the voltages \(\dot U_3\) and \(\dot U_4\) assume maximum or minimum values. There can be five such cases.

1. \[ A=\alpha,\qquad \dot B=j\beta\;(Z_3=R_3,\; Z_4=R_4), \]

\[ U_3 = U_{mc} \sqrt{ \frac{\alpha^2+\beta^2\alpha^2(1+K)^2} {\alpha^2+\beta^2(K^2+\alpha)^2} } \times \]

\[ \times \sin\left[ \omega t+\operatorname{arctg} \frac{\beta K(\alpha-K)} {\alpha+\beta^2(1+K)(K^2+\alpha)} \right], \tag{3} \]

\[ U_4 = U_{mc} \sqrt{ \frac{\alpha^2+\beta^2K^2(1+K)^2} {\alpha^2+\beta^2(K^2+\alpha)^2} } \times \]

\[ \times \sin \left[\omega t+\operatorname{arctg}\frac{\alpha\beta(K-\alpha)}{\alpha^2+\beta^2K(1+K)(K^2+\alpha)}\right]. \tag{4} \]

A peculiarity of the first case is that, for \(\alpha=K\), the phase shift between the mains voltage and the voltages \(U_3\) and \(U_4\) becomes equal to zero. This peculiarity occurs both when \(R_3\) is varied and when \(R_4\) is varied.

2.
\[ A=-j\alpha,\quad B=j\beta \left(Z_3=\frac{1}{j\omega C_3},\quad Z_4=R_4\right), \]

\[ U_3=U_{mc}\sqrt{\frac{\alpha^2+\alpha^2\beta^2(1+K)^2}{(\alpha-\beta K^2)^2+\alpha^2\beta^2}}\times \]

\[ \times \sin \left[\omega t+\operatorname{arctg} \frac{\beta\alpha K-\beta^2K^2(1+K)} {\alpha(1+\beta^2K+\beta^2)-\beta K^2}\right], \tag{5} \]

\[ U_4=U_{mc}\sqrt{\frac{[\beta K(1+K)-\alpha]^2}{\alpha^2+(\beta K^2-\alpha)^2}} \sin \left[\omega t+\operatorname{arctg}\frac{\alpha\beta}{\beta K^2-\alpha}\right]. \tag{6} \]

In the second case, for \(\alpha=\beta K(1+K)\), the modulus of the voltage \(U_4\) and the argument of the voltage \(U_3\) become equal to zero. For \(\alpha=\beta K^2\), the voltage \(U_4\) leads the mains voltage in phase by \(\frac{\pi}{2}\), and the moduli of the voltages \(U_3\) and \(U_4\) may attain extreme values. For
\[ \alpha=\frac{\beta K^2}{1+\beta^2K+\beta^2}, \]
the voltage \(U_3\) lags in phase behind the voltage \(U_c\) by \(\frac{\pi}{2}\).

3.
\[ A=\alpha,\quad B=-\beta \left(Z_3=\frac{1}{j\omega C_3},\quad Z_4=\frac{1}{j\omega C_4}\right), \]

\[ U_3=U_{mc}\frac{\alpha[1-\beta(1+K)]}{\alpha(1-\beta)-\beta K^2}\sin\omega t, \tag{7} \]

\[ U_4=U_{mc}\frac{\alpha-\beta K(1+K)}{\alpha(1-\beta)-\beta K^2}\sin\omega t. \tag{8} \]

In the third case, the amplitudes of the voltages \(U_3\) and \(U_4\) assume extreme values for
\[ \alpha=\frac{\beta K^2}{1-\beta}. \]
For \(\beta=\frac{1}{1+K}\), \(U_3=0\), and \(U_4=0\) for
\[ \beta=\frac{\alpha}{K(1+K)}. \]

4.
\[ A=-\alpha,\quad B=-\beta \left(Z_3=j\omega L_3,\quad Z_4=\frac{1}{j\omega C_4}\right), \]

\[ U_3=U_{mc}\frac{\alpha[1-\beta(1+K)]}{\alpha(1-\beta)+\beta K^2}\cdot \sin\omega t, \tag{9} \]

\[ U_4=U_{mc}\frac{\alpha+\beta K(1+K)}{\alpha(1-\beta)+\beta K^2}\cdot \sin\omega t. \tag{10} \]

In the fourth case, resonance phenomena occur for
\[ \beta=\frac{\alpha}{\alpha-K^2}. \]
As in the preceding case, for \(\beta=\frac{1}{1+K}\), \(U_3=0\).

5.

\[ A=-\alpha,\quad B=\beta\left(Z_3=\frac{1}{j\omega C_3},\quad Z_4=j\omega L_4\right), \]

\[ U_3=U_{mc}\frac{x[1+\beta(1+K)]}{\alpha(1+\beta)-\beta K^2}\sin\omega t, \tag{11} \]

\[ U_4=U_{mc}\frac{\alpha-\beta K(1+K)}{\alpha(1+\beta)-\beta K^2}\sin\omega t. \tag{12} \]

For the fifth case, resonance phenomena arise when \(\alpha=\dfrac{\beta K^2}{1+\beta}\), and \(U_4=0\) when \(\alpha=\beta K(1+K)\).

A comparison of cases (1–5) shows that cases 3 and 5 are of greatest interest, since they make it possible to fix, with high accuracy, the parameters \(Z_3\) or \(Z_4\) corresponding to fulfillment of the condition

\[ \alpha=\frac{\beta K^2}{1+\beta}. \]

If the parameter \(\beta\) is a constant quantity (\(Z_4=\mathrm{const}\)), then, by measuring the voltage magnitudes \(U_3\) or \(U_4\), one can monitor the parameter \(Z_3\). If \(\beta\) is a variable quantity, then measurement of the parameter \(Z_4\) can be carried out by a compensation method, by varying the parameter \(Z_3\) and fixing its value at the instant resonance occurs in the system under consideration.

References

1. L. A. Bessonov, Theoretical Foundations of Electrical Engineering. Moscow, Vysshaya shkola, 1961.