Phase-Generator Circuit with Inductive-Capacitive Coupling Between Circuits

V. V. Volodarskii, V. K. Zhukov

(Presented by the scientific and technical seminar of the Department of Information-Measuring Technology)

The use of a phase-generator circuit (PGC) [1] for purposes of flaw detection provides high sensitivity, noise immunity, simplicity, and automatic implementation of the self-comparison principle. Moreover, the self-comparison principle is most effective when the half-windings of a differential eddy-current transducer are located close to one another, since in this case the influence of changes in the interfering parameters (for example, diameter, magnetic permeability) of the item being inspected is reduced. However, at small distances between the half-windings, the coupling of the generators becomes strong, and the sensitivity of the PGC to changes in inductance and active resistance drops sharply.

To increase sensitivity, a coupling capacitor tuned to resonance with the equivalent mutual inductance is connected between the PGC circuits, as a result of which the inductive coupling is compensated and the sensitivity increases.

In the present work, a PGC with inductive-capacitive coupling is analyzed, and analytical expressions are derived for determining the operating frequency, the capacitance of the coupling capacitor, and the sensitivity. The PGC (Fig. 1a) consists of two generators G1, G2 with circuits \(L1, C1, L2, C2\), operating in the frequency-locking mode. The generators are synchronized through the mutual inductance \(M\) of the half-windings. The generator circuits are connected by a coupling capacitor \(C_{\mathrm{c}}\).

To determine the resonant capacitance of the coupling capacitor, we shall find the operating frequency of the PGC that it had without it.

We represent the PGC in the form of an equivalent circuit (Fig. 1b) with the following designations: \(U_1, U_2\) are the output voltages of generators G1, G2; \(R_{i1}, R_{i2}\) are the internal resistances of the generators with respect to the circuits \(L1, C1\) and \(L2, C2\); \(R_1, R_2\) are the active resistances of the transducer half-windings.

We shall determine the operating frequency \(\omega\) of the PGC as the resonant frequency of the circuits, taking into account their coupling through mutual inductance. As the equivalent resistance of a generator circuit, we shall consider the resistance of the two-terminal network with respect to terminals 1–3 or 2–3. In doing so, we shall neglect the shunting effect of the internal resistances of the generators.

Taking the parameters of the circuits to be identical, \(L_1=L_2=L\), \(C_1=C_2=C\) and \(R_1=R_2=R\), let us write the expression for the complex admittance \(\dot Y\) of the equivalent circuit

\[ \dot Y=\frac{1}{\dot X}+ \frac{1}{ \dfrac{1}{\dot X_L(1-\kappa-jd)} + \dfrac{\dot X_L(1-\kappa-jd)+\dot X_C} {\dot X_L(1-\kappa-jd)+\dot X_C+\dot X_M} \cdot X_M }, \tag{1} \]

where \(\dot X=\dfrac{1}{j\omega C}\), \(\dot X_L=j\omega L\) are the reactances, respectively, of the capacitor and the half-windings; \(\kappa=\dfrac{M}{L}\) is the coupling coefficient of the half-windings; \(d=\dfrac{R}{\omega L}\) is the damping of the circuit; \(\dot X_M=j\kappa\omega L\) is the mutual-inductance reactance.

In the resonant mode the input resistance of the two-terminal network is purely active, i.e., the reactive admittance \(b\) is equal to zero. Performing elementary transformations of (1) and discarding infinitesimal terms of higher orders, with the above substitution taken into account, we obtain the equation

\[ \begin{aligned} b={}&(1+2d^2+d^4+2\kappa^2d^2-2\kappa^2)(\omega^2LC)^3\\ &-3(1+d^2+\kappa^2)(\omega^2LC)^2\\ &+(3+d^2-\kappa^2)\omega LC-1=0. \end{aligned} \tag{2} \]

Solving this equation and discarding roots that do not satisfy the physical meaning, we obtain the formula for determining the operating frequency of the PGC:

\[ \omega=\frac{1}{\sqrt{LC}}\left(1-\frac{1}{2}\kappa-\frac{1}{2}d^2\right). \tag{3} \]

To determine the resonant coupling capacitance, we transform the equivalent circuit (Fig. 1,b) into a Π-shaped one. Using the formulas for transforming from a star to a delta, we obtain the equivalent Π-shaped circuit shown in Fig. 1,c. At resonance, for the coupling circuit the condition [2] must be satisfied

\[ \frac{\omega^2 L_\mathrm{e}}{R_\mathrm{e}^2+\omega^2 L_\mathrm{e}^2} = \omega C_{\mathrm{coupl}}, \tag{4} \]

where \(L_\mathrm{e}=\dfrac{L}{\kappa}(1-d^2)\), \(R_\mathrm{e}=\dfrac{2R}{\kappa}\) are, respectively, the equivalent inductance and active resistance of the coupling circuit. Solving (4), taking into account (3), and discarding infinitesimal terms of higher orders, we obtain an expression for determining the resonant capacitance of the coupling capacitor.

Fig. 1. Phase-generator circuit with inductive-capacitive coupling between circuits:
a) block diagram of the PGC; b) equivalent circuit of the PGC; c) Π-shaped equivalent circuit; d) T-shaped equivalent circuit.

\[ C_{\mathrm{c}}=kC(1+k-2d^2). \tag{5} \]

Let us proceed to the derivation of the dependences of the phase and amplitude of the phase-generator circuit on the change in the inductances of the half-windings of the converter.

Let the inductance of the first circuit have decreased by the amount \(\frac{1}{2}\Delta L\) to the value \(L_1=L-\frac{1}{2}\Delta L\), and let the inductance of the second have increased by \(\frac{1}{2}\Delta L\) to \(L_2=L+\frac{1}{2}\Delta L\). In this case the natural frequencies of generators \(G1\) and \(G2\) will respectively increase and decrease, while the operating frequency of the phase-generator circuit will remain unchanged. A phase shift \(\varphi\) will arise between the output voltages of the generators, and the amplitudes of the circuit voltages will change.

We transform the circuit of Fig. 1,b into the equivalent T-shaped equivalent circuit (Fig. 1,c), where \(\dot U_{1\mathrm{e}}, \dot U_{2\mathrm{e}}\) are the equivalent output voltages of the generators; \(\dot Z_{i\mathrm{e}}\) are the equivalent internal impedances of the generators; \(\dot U_1, \dot U_2\) are the voltages across the circuits; \(\dot I_1, \dot I_2\) are the circuit currents; \(\dot Z_1, \dot Z_2, \dot Z_3\) are the impedances of the arms of the star.

The output voltages, internal impedances of the generators, and impedances of the star arms are determined, respectively, by the expressions

\[ \dot U_{1\mathrm{e}}=\frac{\dot X_C}{R_i+\dot X_C}\cdot \dot U'_1; \tag{6} \]

\[ \dot U_{2\mathrm{e}}=\frac{\dot X_C}{R_i+\dot X_C}\cdot \dot U'_2; \tag{7} \]

\[ \dot Z_{i\mathrm{e}}=\frac{\dot X_C R_i}{R_i+\dot X_C}; \tag{8} \]

\[ \dot Z_1= \frac{\dot X_{\mathrm{c}}\cdot \dot X_L\left(1-jd-k-\frac{1}{2}\delta_L\right)} {\dot X_{\mathrm{c}}+2\dot X_L(1-jd-k)}; \tag{9} \]

\[ \dot Z_2= \frac{\dot X_{\mathrm{c}}\cdot \dot X_L\left(1-jd-k+\frac{1}{2}\delta_L\right)} {\dot X_{\mathrm{c}}+2\dot X_L(1-jd-k)}; \tag{10} \]

\[ \dot Z_3= \frac{ \dot X_L^2\left[(1-jd-k)^2-\frac{1}{4}\delta_L^2\right] +\dot X_M\left[\dot X_{\mathrm{c}}+2\dot X_L(1-jd-k)\right] }{ \dot X_{\mathrm{c}}+2\dot X_L(1-jd-k) }, \tag{11} \]

where \(\dot X_{\mathrm{c}}=\frac{1}{j\omega C_{\mathrm{c}}}\) is the capacitive reactance of the coupling capacitor,

\[ \iota=\frac{\Delta L}{L} \]

is the relative magnitude of the inductive detuning.

Introducing the parameter \(\beta=\frac{X_L}{R_i}\ll 1\) and neglecting infinitesimals of higher orders, we represent expression (8) in the form

\[ \dot Z_{i\mathrm{e}}=\dot X_L\left[\beta(1+2k+2d^2)-j(1+k+d^2)\right], \tag{12} \]

By Kirchhoff’s second law, the equivalent output voltages of the generators are equal to

\[ \dot U_{1\mathrm{e}}=\dot F_1\dot I_1+\dot Z_3\dot I_2, \tag{13} \]

\[ \dot U_{2\mathrm{e}}=\dot F_2\dot I_2+\dot Z_3\dot I_1, \tag{14} \]

where

\[ \dot F_1=\dot Z_{i9}+\dot Z_1+\dot Z_3, \tag{15} \]

\[ \dot F_2=\dot Z_{i9}+\dot Z_2+\dot Z_3. \tag{16} \]

Then the loop currents are

\[ \dot I_1=\frac{\dot F_2 \dot U_{19}-\dot Z_3 \dot U_{29}} {\dot F_1 \dot F_2-\dot Z_3^{\,2}}, \tag{17} \]

\[ \dot I_2=\frac{\dot F_1 \dot U_{29}-\dot Z_3 \dot U_{19}} {\dot F_1 \dot F_2-\dot Z_3^{\,2}} . \tag{18} \]

Let us determine the voltages across the circuits and their ratio

\[ \dot U_1=\dot U_{19}-\dot I_1 \dot Z_{i9}, \qquad \dot U_2=\dot U_{29}-\dot I_2 \dot Z_{i9}, \tag{19} \]

\[ \frac{\dot U_1}{\dot U_2}= \frac{\dot U_{19}-\dot I_1 \dot Z_{i9}} {\dot U_{29}-\dot I_2 \dot Z_{i9}} . \tag{20} \]

We represent the output voltages and the voltages across the circuits in exponential form

\[ \dot U_1=U_{1m}e^{j\frac{\varphi}{2}}, \qquad \dot U_2=U_{2m}e^{-j\frac{\varphi}{2}}, \tag{21} \]

\[ \dot U_{19}=U_{19m}e^{j\frac{\varphi}{2}}, \qquad \dot U_{29}=U_{29m}e^{-j\frac{\varphi}{2}} . \tag{22} \]

Since the output voltages of the generators are stabilized, in the presence of detuning

\[ \frac{U_{19m}}{U_{29m}}=1, \]

whereas the ratio

\[ \frac{U_{1m}}{U_{2m}} \]

will differ from unity by the amount \(\delta_U\), i.e.

\[ \frac{U_{1m}}{U_{2m}}=1-\delta_U . \]

Taking the above into account and using (19), (20), we write (22) in the form

\[ (1-\delta_U)e^{j\varphi}= \frac{-[\dot F_1\dot F_2-\dot Z_3^{\,2}-\dot Z_{i9}\dot F_2]e^{j\varphi} +\dot Z_3\dot Z_{i9}} {[\dot F_1\dot F_2-\dot Z_3^{\,2}-\dot Z_{i9}\dot F_1] +\dot Z_3\dot Z_{i9}e^{j\varphi}}, \tag{23} \]

whence

\[ \dot Z_{i9}(\dot F_2-\dot F_1) +\dot Z_3\dot Z_{i9}(e^{j\varphi}-e^{-j\varphi}) -\delta_U(\dot F_1\dot F_2-\dot Z_3^{\,2} \]
\[ -\dot Z_{i9}\dot F_1+\dot Z_3\dot Z_{i9}e^{j\varphi})=0. \tag{24} \]

Solving equation (24) and equating the real and imaginary parts to zero, we obtain

\[ \beta\delta_L(1+2\kappa+2d^2)+4\kappa d(2d^2-\beta)\sin\varphi -\delta_U\left(\beta+2\kappa+\kappa\beta+5\beta d^2+d^3+ \frac{1}{2}\beta\delta_L+2\kappa d\cos\varphi\right)=0, \tag{25} \]

\[ \delta_L(1+\kappa+d^2)-4\kappa d\sin\varphi +\delta_U\left(2d^2-\beta d-\frac{1}{2}\delta_L-4\kappa d^2-2\kappa\beta d \right. \]
\[ \left. +2\kappa d\sin\varphi+2\kappa\beta d\cos\varphi-4\kappa d^3\cos\varphi \right)=0. \tag{26} \]

whence

\[ \varphi=\arcsin \frac{\beta(1+2d^2)+2\kappa-\beta^2 d-\frac{1}{2}\beta\delta_L}{4\kappa d(\beta+2\kappa)}\,\delta_L \approx \arcsin \frac{Q}{4\kappa}\,\delta_L, \tag{27} \]

\[ \delta_U=\frac{4\kappa d(2d^2+\kappa\beta+\beta d^2)}{\beta+2\kappa} \approx \frac{2d^2+\kappa\beta+\beta d^2}{\beta+2\kappa}\,\delta_L, \tag{28} \]

where \(Q\) is the Q factor of the circuit.

In Fig. 2, a, b, the dependences of the phase \(\varphi\) and of the relative amplitude \(\delta_u\) on the magnitude of the inductive detuning \(\delta_L\) are presented for various circuit parameters, plotted on the basis of equations (27) and (28) (solid line). The points denote experimentally obtained data; their discrepancy with the theoretical curves is explained by the fact that the amplitudes of the generators’ output voltages during the experiment were not strictly constant, since the coefficient of automatic amplitude stabilization was small (less than 15), and the sensitivity of the phase-generator circuit depends substantially on the degree of constancy of the output voltages.

Fig. 2. Graphs of the dependences of the phase shift (a) and of the relative change in amplitudes (b) on inductive detuning

Let us determine the sensitivities of the phase \(S_{\varphi}^{\mathrm{sv}}(L)\) and of the amplitude \(S_{U}^{\mathrm{sv}}(L)\) to a change in the inductance of the half-windings of the eddy-current transducer.

\[ S_{\varphi}^{\mathrm{sv}}(L)=\frac{d\varphi}{d\delta_L} =\frac{Q}{4\kappa\cos\varphi}, \tag{29} \]

\[ S_{U}^{\mathrm{sv}}(L)=\frac{d\delta_U}{d\delta_L} =\frac{2d^2+\kappa\beta+\beta d^2}{\beta+2\kappa}. \tag{30} \]

The graphs of the dependences \(S_{\varphi}^{\mathrm{sv}}(L)\) and \(S_{U}^{\mathrm{sv}}(L)\) are given in Fig. 3, a, b. Earlier we obtained expressions for determining the sensitivities \(S_{\varphi}(L)\) and \(S_u(L)\) of a phase-generator circuit without a coupling capacitor, which for the small-detuning mode \((\varphi\to 0)\) are written as follows:

\[ S_{\varphi}(L)=\frac{\beta+d}{2\kappa\left[(\beta+d)^2+\kappa\right]}, \tag{31} \]

\[ S_U(L)=\frac{1}{(\beta+d)^2+\kappa}. \tag{32} \]

On the basis of (29)—(32), for the small-detuning mode we find the sensitivity ratios

\[ \frac{S_{\varphi}^{\mathrm{sv}}(L)}{S_{\varphi}(L)} =\frac{(\beta+d)^2+\kappa}{2d(\beta+d)}, \tag{33} \]

\[ \frac{S_{U}^{\mathrm{sv}}(L)}{S_U(L)} =\frac{(2d^2+\kappa\beta+\beta d^2)\left[(\beta+d)^2+\kappa\right]}{\beta+2\kappa} \tag{34} \]

and plot the curves from them (Fig. 4, a, b).

Analysis of these curves shows that the presence of a coupling capacitance significantly increases the sensitivity of the phase to changes in the inductance of the converter half-windings, especially with strong coupling between the generators (Fig. 4a). However, at small values of the coupling coefficient \(k\) and low quality factors \(Q\) of the converter, the inclusion of the coupling capacitor

Fig. 3. Plots of the dependences of the phase sensitivities (a) and the relative amplitude (b) on the coupling coefficient

Fig. 4. Plots of the dependences of the ratio of phase sensitivities (a) and amplitudes (b) on the coupling coefficient

can lead to a decrease in the sensitivity of the phase-generator circuit. The amplitude sensitivity \(S_i^{\mathrm{sv}}(L)\) in the presence of the coupling capacitor decreases substantially (Fig. 4b).

Thus, the use of a phase-generator circuit with inductive-capacitive coupling is most effective for increasing phase sensitivity at large coupling coefficients of the generators.

REFERENCES

1. K. S. Poluyakh. On the theory of the phase-generator converter. Izmeritel’naya tekhnika, 1970, No. 1, 54.

2. G. I. Atabekov. Theoretical Foundations of Electrical Engineering, Part 1. Moscow—Leningrad, “Energiya,” p. 75, 1966.