MIXED CONNECTION OF AN EDDY-CURRENT SENSOR

V. K. ZHUKOV, D. V. MILYAEV

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

Depending on what is taken as the output quantity, a distinction is made between parametric and transformer connection of an eddy-current sensor [1]. In a parametric connection the output quantity is resistance; consequently, the measuring circuits that provide the parametric connection of the sensor must respond to some parameter of the output quantity: to the active and reactive components of the total resistance, to the quality factor, or to the modulus of the total resistance. Examples of such circuits are differential, resonant, bridge, and generator measuring circuits.

In the case of transformer connection, the output quantity of the sensor is the e.m.f. induced in the measuring winding by the magnetic flux of another winding, called the exciting winding. With such a connection of the sensor, the measuring circuit must respond to some parameter of the alternating voltage: amplitude, phase, and the real and imaginary components.

Besides the two known methods of connection named above, a third case is also possible—that of mixed connection. This case is characterized by the fact that the measuring circuit responds both to a change in the e.m.f. and to a change in the impedance of the measuring winding of the sensor. To illustrate this method of connection, let us consider the operation of the circuit shown in Fig. 1. This is a bridge circuit, in two arms of which the measuring windings of eddy-current sensors are connected. One sensor, with winding resistance \(Z_0\) and the e.m.f. \(\dot{E}_0\) induced in it, is the “reference” sensor, while the other, with e.m.f. \(\dot{E}=\dot{E}_0+\Delta \dot{E}\) and resistance \(Z=Z_0+\Delta Z\), is the measuring sensor. The increments of e.m.f. \(\Delta \dot{E}\) and resistance \(\Delta Z\) in the measuring sensor are caused by the object being inspected. If the bridge is balanced without the object, then

Fig. 1.

a change in the parameters of the measuring sensor by \(\Delta \dot E\) and \(\Delta Z\) will cause an unbalance of the bridge. The unbalance voltage applied to the indicator \(U\), for \(Z_1=Z_2=Z_0\), will be equal to:

\[ \dot U_{\text{н}}=\frac{2\Delta \dot E\cdot Z_0-\dot E\cdot \Delta Z}{4Z_0+\Delta Z}. \tag{1} \]

From the expression given we see that the bridge unbalance voltage is determined both by the impedance of the measuring winding and by the magnitude of the emf induced in it.

Dividing the numerator and denominator of expression (1) by \(|E_0|Z_0\), we transform it to the form:

\[ \frac{\dot U_{\text{н}}}{|\dot E_0|}= \frac{2\Delta \dot E/|E_0|-j\Delta Z/Z_0}{4+\Delta Z/Z_0}. \]

Since usually \(\Delta Z/Z_0\ll 1\) and \(Z_0\approx jX_0\), and, as shown in (2), \(\Delta \dot E/|E_0|=\Delta Z/Z_0=j\eta_{\text{к}}\), then

\[ U_{\text{н}}/|E_0|\approx \frac{1}{4}\Delta E/|E_0| \]

\[ U_{\text{н}}=\frac{1}{4}\Delta \dot E. \]

The last relations indicate the low sensitivity of the circuit considered (the sensitivity is 4 times lower than in the differential circuit); therefore its practical use is inexpedient. It can serve only as an example showing the possibility of a mixed connection of an eddy-current sensor.

As another example of mixed connection, a resonant measuring circuit may serve, in which one of the elements of the oscillatory circuit is the measuring winding of an eddy-current sensor (Fig. 2, a). This circuit includes: current generator 1; two-winding sensor \(D_{\text{и}}\); capacitor \(C\), used for tuning the secondary circuit of the sensor to resonance with the frequency of current 1; voltmeter \(V\). In the circuit under consideration, just as in the preceding one, the readings of the output instrument will depend both on the impedance of the measuring winding and on the emf induced in it by the exciting winding owing to the passage of current 1 through the latter. To establish this dependence, let us consider the equivalent circuit shown in Fig. 2, b. In it the sensor is represented by a voltage generator with an emf equal to that induced in the measuring—

Fig. 2.

measuring winding of the sensor an emf \(\dot E\) and an internal impedance \(Z = R + jX\). Here \(R\) and \(X\) denote, respectively, the active and reactive resistance of the measuring winding.

Let, in the absence of the article, the secondary circuit of the sensor be tuned to resonance; the resistance of the measuring winding is \(Z_0 = R_0 + jX_0\), and an emf equal to \(\dot E_0\) is induced in it. In this case the voltage across the capacitor, which is recorded by the voltmeter, will be

\[ \dot U_{c.0} = -j \dot E_0 \cdot Q_0, \tag{3} \]

where \(Q_0 = X_c/R_0 = X_0/R_0\) is the \(Q\)-factor of the secondary circuit of the sensor tuned to resonance.

If, when the article is introduced, the emf of the winding changes by \(\Delta \dot E\), and the impedance by \(\Delta Z\), then the voltage across the capacitor becomes equal to

\[ \dot U_{c.0}' = -j(\dot E_0 + \Delta \dot E) Q_0 \frac{R_0}{R_0 + \Delta Z}. \tag{4} \]

The change in voltage caused by introducing the article into the sensor is determined as the difference between \(\dot U_c'\) and \(\dot U_{c.0}\).

\[ \Delta \dot U_c = \dot U_c' - \dot U_{c.0} = \left( j\dot E_0 \cdot Q_0^2 \frac{\Delta Z}{Z_0} - \frac{\Delta E}{|\dot E_0|} \cdot \dot E_0 \cdot Q_0 \right) \frac{R_0}{R_0 + \Delta Z}. \]

On the basis of (2), we transform this expression to the form:

\[ \Delta \dot U_c = - U_{c.0}(Q_0 + j)\frac{R_0}{R_0 + \Delta Z}\frac{\Delta Z}{X_0}. \]

Hence we find the relative change in the voltage across the capacitor

\[ \frac{\dot U_c}{\dot U_{c.0}} = (Q_0 + j)\frac{R_0}{R_0 + \Delta Z}\frac{\Delta Z}{X_0}. \tag{5} \]

In the case of large \(Q\)-factors \((Q_0 \gg 1)\) and small introduced impedances \(|\Delta Z| \ll R_0\),

\[ \frac{\Delta \dot U_c}{\dot U_{c.0}} \approx Q_0 \frac{\Delta Z}{X_0} = Q_0 \frac{\Delta \dot E}{|\dot E_0|}; \tag{6} \]

i.e., the relative change in the voltage across the circuit capacitance is approximately \(Q_0\) times greater than the relative change in the emf of the measuring winding. The absolute change in the voltage across the capacitance, as follows from (3) and (6), is approximately \(Q_0^2\) times greater than the absolute change in the emf:

\[ \Delta \dot U_c \approx Q_0^2 \Delta \dot E. \]

Thus, the resonant circuit for the mixed connection of an eddy-current sensor provides high sensitivity to changes in its parameters and, consequently, in the parameters of the article. For a quantitative evaluation of the sensitivity of the circuit, one may use the quotient obtained by dividing the relative change in the voltage across the capacitance by the normalized change in the resistance of the measuring winding that caused it:

\[ A_1 = \frac{\Delta \dot U_c/\dot U_{c.0}}{\Delta Z/X_0} = (Q_0 + j)\frac{R_0}{R_0 + \Delta Z}. \tag{7} \]

For comparison, let us give the similarly defined sensitivities of the circuits of series and parallel resonance with a single-wind—

eddy-current sensor. The sensitivity of the circuit with voltage resonance (Fig. 3, a) is

\[ A_2 = Q_0 \frac{R_0}{R_0+\Delta Z}, \tag{8} \]

and with current resonance—

\[ A_3 = \frac{Q_0^2}{1+jQ_0}\cdot \frac{R_0}{R_0+\Delta Z}. \tag{9} \]

From the ratio of the sensitivities

\[ \left|\frac{A_1}{A_2}\right|=\sqrt{1+1/Q_0^2}\quad \left|\frac{A_1}{A_3}\right|=1+1/Q_0^2 \tag{10} \]

we see that the greatest sensitivity is provided by the mixed-connection circuit of the eddy-current sensor. The difference in sensitivities becomes especially noticeable at small \(Q_0\).

The conclusion obtained will be of practical significance for sensors with low quality factor, or when the secondary circuit of the sensor is tuned to resonance with an inserted article having a nominal value of the parameter being monitored. In general cases, the greatest sensitivity to the deviation of the monitored parameter from its nominal value will be provided by the mixed-connection circuit, and, consequently, it will be preferable for use.

Fig. 3.

REFERENCES

1. N. N. Shumilovskii, G. G. Yarmolchuk, V. P. Grabovetskii, M. A. Prusov. The Eddy-Current Method for Monitoring Production Parameters, Ilim Publishing House, Frunze, 1964.

2. V. K. Zhukov, V. E. Dreizin, I. G. Leshchenko. Induction-impedance sensor with an annular magnetic core, Defektoskopiya, No. 2, 1966.