THEORETICAL AND EXPERIMENTAL STUDIES OF A PHASE-GENERATING CIRCUIT

V. V. Volodarskii, V. K. Zhukov

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

At the present time the phase-generating circuit (PGC) is becoming increasingly widespread in monitoring and measuring technology. This is due to a number of its advantages: high sensitivity, noise immunity, simplicity, and—what is especially important for flaw detection—the automatic implementation of the self-comparison principle [1, 2, 3, 4].

In the present work, theoretical and experimental studies are given of a PGC with mutual inductive coupling of the generators.

The PGC is shown in Fig. 1a and consists of generators \(G_1\), \(G_2\) with circuits \(L_1, C_1\) and \(L_2, C_2\), coupled through the mutual inductance \(M\) of the sensors. The generators operate in the “locking” mode.

The PGC can be represented in the form of the equivalent T-shaped substitution circuit shown in Fig. 1b.

Fig. 1. Phase-generating circuit (a) and its equivalent substitution circuits (b, c).

The designations adopted in Fig. 1b are: \(\dot U'_1, \dot U'_2\) are the output voltages of generators \(G_1, G_2\); \(R_{11}, R_{12}\) are the internal resistances of the generators; \(r_1, r_2\) are the active resistances of the sensors.

We shall derive the analytical dependence of the phase shift between the output voltages of the generators for the case of symmetric inductive detuning in the circuits. In doing so, we shall assume that

the inductance of the first circuit is increased by an amount \(\Delta L\) (\(L_1=L+\Delta L\)), and the inductance of the second is decreased by \(\Delta L\) (\(L_2=L-\Delta L\)). In this case the operating frequency \(\omega\) of the PGC remains unchanged.

To simplify the analysis, we shall assume: \(R_{i1}=R_{i2}=R_i,\ r_1=r_2=r,\ C_1=C_2=C\).

Carrying out further transformations of the equivalent circuit of Fig. 1,b, we obtain the circuit shown in Fig. 1,c. Here \(\dot U_{1e}, \dot U_{2e}\) are the equivalent output voltages of the generators; \(\dot Z_{1e}\) are the equivalent output impedances of the generators; \(\dot X_1, \dot X_2\) are the reactances of the inductively coupled sensors; \(\dot X_M\) is the impedance of the mutual inductance; \(\dot I_1, \dot I_2\) are the loop currents; \(\dot U_1, \dot U_2\) are the voltages across the circuits.

The equivalent output voltages and impedances of the generators are expressed as follows:

\[ \dot U_{1e}=\dot U_1' \frac{\dot X_c}{R_i+\dot X_c}, \tag{1} \]

\[ \dot U_{2e}=\dot U_2' \frac{\dot X_c}{R_i+\dot X_c}, \tag{2} \]

\[ \dot Z_{1e}=\frac{\dot X_c R_i}{R_i+\dot X_c}, \tag{3} \]

where

\[ \dot X_c=\frac{1}{j\omega C}=-jX_c . \]

Let us introduce the notation \(\eta=\dfrac{X_c}{R_i}\); for the PGC \(\eta \ll 1\), since \(R_i \gg X_c\). Then (1)–(3) can be transformed:

\[ \dot U_{1e}=\dot U_1' \frac{-jX_c}{R_i-jX_c} =\dot U_1' \frac{-j\eta}{1-j\eta} \approx \dot U_1'\eta(\eta-j)\approx \eta e^{j\varphi_e}\dot U_1', \tag{4} \]

\[ \dot U_{2e}\approx \dot U_2'\eta(\eta-j)\approx \eta e^{j\varphi_e}\dot U_2', \tag{5} \]

\[ \dot Z_{1e}=\frac{-jX_cR_i}{R_i-jX_c} =\frac{-jX_c}{1-j\eta}\approx X(\eta-j), \tag{6} \]

where

\[ X=\frac{1}{\omega C}\approx \omega L,\quad \varphi_e=-\operatorname{arc\,tg}\frac{1}{\eta}. \]

Let us simplify the expressions for the reactances of the sensors and the mutual inductance.

The mutual inductance is \(M=k\sqrt{L_1L_2}\), where \(k\) is the coupling coefficient of the sensors. Introducing the notation \(\delta_L=\dfrac{\Delta L}{L}\) and neglecting an infinitesimal of second order, we obtain

\[ M=k\sqrt{(L+\Delta L)(L-\Delta L)} =kL\sqrt{1-\delta_L^2}\approx kL . \tag{7} \]

Using expression (7), we obtain

\[ \dot X_1=j\omega(L_1-M)=j\omega(L+\Delta L-kL)=jX(1+\delta_L-k), \tag{8} \]

\[ \dot X_2=jX(1-\delta_L-k). \tag{9} \]

Let us express the equivalent output voltages of the generators:

\[ \dot U_{1e}=[\dot Z_{1e}+r+jX_1+jX_M]\dot I_1+jX_M\dot I_2, \tag{10} \]

\[ \dot U_{2e}=[\dot Z_{1e}+r+jX_2+jX_M]\dot I_2+jX_M\dot I_1. \tag{11} \]

After simplification we have

\[ \dot U_{1\mathrm{e}}=\left[\dot Z_{1\mathrm{e}}+r+\mathrm{j}X(1+\delta_L)\right]\dot I_1+\mathrm{j}kX\dot I_2, \tag{12} \]

\[ \dot U_{2\mathrm{e}}=\mathrm{j}kX\dot I_1+\left[\dot Z_{1\mathrm{e}}+r+\mathrm{j}X(1-\delta_L)\right]\dot I_2. \tag{13} \]

The values of the currents \(\dot I_1,\dot I_2\) are determined by the expressions:

\[ \dot I_1= \frac{\dot U_{1\mathrm{e}}-\mathrm{j}kX\dot I_2} {\dot Z_{1\mathrm{e}}+r+\mathrm{j}X(1+\delta_L)} = \frac{\dot U_{1\mathrm{e}}-\mathrm{j}kX\dot I_2} {\dot Z+\mathrm{j}\Delta X}, \tag{14} \]

\[ \dot I_2= \frac{\dot U_{2\mathrm{e}}-\mathrm{j}kX\dot I_1} {\dot Z_{1\mathrm{e}}+r+\mathrm{j}X(1-\delta_L)} = \frac{\dot U_{2\mathrm{e}}-\mathrm{j}kX\dot I_1} {\dot Z-\mathrm{j}\Delta X}, \tag{15} \]

where \(\dot Z=\dot Z_{1\mathrm{e}}+r+\mathrm{j}X,\ \Delta X=X\delta_L\).

After elementary transformations we obtain the final expressions for the currents \(\dot I_1,\dot I_2\):

\[ \dot I_1= \frac{\dot U_{1\mathrm{e}}(\dot Z-\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{2\mathrm{e}}} {\dot Z^2+(\Delta X)^2+(kX)^2} = \frac{\dot U_{1\mathrm{e}}(\dot Z-\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{2\mathrm{e}}} {\dot F}, \tag{16} \]

\[ \dot I_2= \frac{\dot U_{2\mathrm{e}}(\dot Z+\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{1\mathrm{e}}} {\dot Z^2+(\Delta X)^2+(kX)^2} = \frac{\dot U_{2\mathrm{e}}(\dot Z+\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{1\mathrm{e}}} {\dot F}, \tag{17} \]

where \(\dot F=\dot Z^2+(\Delta X)^2+(kX)^2\).

The voltages on the generator circuits are related to the output equivalent voltages of the generators by the following relations:

\[ \dot U_1=\dot U_{1\mathrm{e}}-\dot I_1\dot Z_{1\mathrm{e}}, \tag{18} \]

\[ \dot U_2=\dot U_{2\mathrm{e}}-\dot I_2\dot Z_{1\mathrm{e}}. \tag{19} \]

Hence

\[ \frac{\dot U_1}{\dot U_2} = \frac{\dot U_{1\mathrm{e}}-\dot I_1\dot Z_{1\mathrm{e}}} {\dot U_{2\mathrm{e}}-\dot I_2\dot Z_{1\mathrm{e}}} = \frac{ \dot U_{1\mathrm{e}}\dot F- \left[\dot U_{1\mathrm{e}}(\dot Z-\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{2\mathrm{e}}\right]\dot Z_{1\mathrm{e}} }{ \dot U_{2\mathrm{e}}\dot F- \left[\dot U_{2\mathrm{e}}(\dot Z+\mathrm{j}\Delta X)-\mathrm{j}kX\dot U_{1\mathrm{e}}\right]\dot Z_{1\mathrm{e}} }. \tag{20} \]

Let us represent the generator voltages in complex form

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

\[ \dot U_{1\mathrm{e}}=U_{1\mathrm{e}m}e^{\mathrm{j}(\varphi+\varphi_\mathrm{e})},\qquad \dot U_{2\mathrm{e}}=U_{2\mathrm{e}m}e^{-\mathrm{j}(\varphi-\varphi_\mathrm{e})}. \tag{22} \]

In this case the ratio of the voltage amplitudes on the circuits will differ somewhat from unity by the amount \(2\delta_u\), i.e. \(\dfrac{U_{1m}}{U_{2m}}=1+2\delta_u\), while the amplitudes of the output voltages of the generators may, with a sufficient degree of accuracy, be regarded as equal. Then expression (20) is transformed to the form

\[ 1+2\delta_u= \frac{ e^{\mathrm{j}2\varphi}\dot F- \left[e^{\mathrm{j}2\varphi}(\dot Z-\mathrm{j}\Delta X)-\mathrm{j}kX\right]\dot Z_{1\mathrm{e}} }{ \dot F- \left[(\dot Z+\mathrm{j}\Delta X)-\mathrm{j}kXe^{\mathrm{j}2\varphi}\right]\dot Z_{1\mathrm{e}} }, \tag{23} \]

whence we have

\[ 2\dot p\delta_u-\mathrm{j}2\Delta X Z_{1\mathrm{e}} +\mathrm{j}kXZ_{1\mathrm{e}}\left(e^{\mathrm{j}2\varphi}-e^{-\mathrm{j}2\varphi}\right)=0, \tag{24} \]

where

\[ \dot p=\dot F-\left[(\dot Z+\mathrm{j}\Delta X)-\mathrm{j}kXe^{\mathrm{j}2\varphi}\right]\dot Z_{1\mathrm{e}}. \]

Neglecting infinitesimals of higher orders and introducing the damping coefficient \(d=\dfrac{r}{X}\), after elementary transformations we have

\[ \begin{aligned} &(d^2+d\eta-\delta_L+k\cos 2\varphi-k\eta\sin 2\varphi)\delta_u\\ &\quad+\mathrm{j}(d+\eta-\eta\delta_L+k\eta\cos 2\varphi+k\sin 2\varphi)\delta_u\\ &=\delta_L+k\eta\sin 2\varphi+\mathrm{j}(\eta\delta_L-k\sin 2\varphi). \end{aligned} \tag{25} \]

Equating the real and imaginary terms in the left- and right-hand sides,

Substituting expressions (25) and eliminating \(\delta_i\), we write the final expression for the inductive symmetric detuning of the circuits

\[ \delta_L=-k\sin 2\varphi \frac{(d+\eta)^2+k\cos 2\varphi}{\eta+d}. \tag{26} \]

Analysis of this expression shows that the sensitivity of the PGC

\[ S_{\varphi}(L)=\frac{d\varphi}{d\delta_L} \]

depends substantially on the coupling coefficient \(k\), the circuit damping \(d\), and the parameter \(\eta\).

In order to verify the theory of the PGC with the generators coupled through the mutual inductance of the sensors, an experimental study was carried out of the dependence of the sensitivity \(S_{\varphi}(L)\) under stabilization of the amplitudes of the generator output voltages, with \(d=0.01275\), \(\eta=0.0741\), \(k=0.001\), and \(k=0.01\). The discrepancy between the theoretical and experimental data for \(k=0.001\) does not exceed 20%, and for \(k=0.01\) it is 5%.

The studies showed the high sensitivity (1000–3000 deg/%) of the PGC with coupling through the mutual inductance of the sensors and the possibility of using it for metal detectors, detectors of metallic objects, and flaw detectors.

REFERENCES

1. B. Z. Mikhlin, High-Frequency Capacitive and Inductive Sensors. Moscow, Gosenergoizdat, 1960.

2. F. Klutke, “Archiv für technisches Messen,” Januar, 1961, No. 300.

3. K. S. Polulyakh. On the theory of a phase-generating transducer. Izmeritel’naya Tekhnika, 1970, No. 1, 54.

4. Nondestructive Testing. Handbook. Ed. R. McMaster. Translated from English. Moscow–Leningrad, “Energiya,” 1965.