Sensitivity Analysis of an Oscillatory Circuit with an Eddy-Current Sensor to Changes in the Electrical Conductivity of a Cylindrical Product
Abstract Generated abstract
This paper analyzes how connecting an eddy-current feed-through sensor to a parallel oscillatory circuit affects sensitivity to changes in the electrical conductivity of a cylindrical nonmagnetic workpiece, a case relevant to crack detection under weak skin-effect conditions. The authors derive expressions for the sensitivities of the equivalent active and reactive circuit resistances, impedance magnitude, and phase as functions of circuit detuning, damping, sensor quality factor, and sensor impedance sensitivity. Calculations and reported experimental comparisons indicate that circuit inclusion can substantially increase sensitivity, mainly through the loaded quality factor, and that optimal sensitivity depends strongly on detuning and the sensor operating point. The analysis suggests that monitoring the impedance modulus is generally preferable to monitoring phase, with negative detuning advantageous for certain parameter ranges.
Full Text
SENSITIVITY ANALYSIS OF AN OSCILLATORY CIRCUIT WITH AN EDDY-CURRENT SENSOR TO CHANGES IN THE ELECTRICAL CONDUCTIVITY OF A CYLINDRICAL WORKPIECE
V. K. Zhukov, A. V. Emelyanenko
(Presented by the joint scientific and technical seminar of the Department of Information and Measurement Engineering and the Electromagnetic Flaw-Detection Sector of the Research Institute of Electronic Introscopy)
Experimental investigations [1] have established that the effect of a longitudinal surface crack in a cylindrical workpiece on the output parameters of an eddy-current sensor under a weak skin effect is equivalent to the effect of changes in the electrical conductivity of the cylindrical workpiece placed in the sensor. This conclusion makes it possible, under certain assumptions, to reduce the analysis of the sensitivity of an eddy-current sensor to cracks to an analysis of its sensitivity to changes in the electrical conductivity of the rod being inspected.
It has also been observed experimentally that the sensitivity of an eddy-current feed-through sensor increases when it is included in an oscillatory circuit. In view of the need for a further increase in the sensitivity of electromagnetic flaw detectors, the possibility indicated above is of definite interest and requires more detailed investigation. To this end, we shall analytically determine the sensitivity of the resistance components of an oscillatory circuit to changes in the electrical conductivity of a cylinder placed in the sensor, which is an element of the oscillatory circuit.
The resistance of the oscillatory circuit is determined by the active
\[ R = X_0 \left( \frac{1}{Q_0} + R_e j\eta_k \right) \tag{1} \]
and reactive
\[ X = X_0 (1 + I_n j\eta_k) \tag{2} \]
components of the sensor impedance, and also by the impedance of the capacitor included in the oscillatory circuit,
\[ X_c = \frac{1}{j \omega C}. \tag{3} \]
The modulus and argument of the total impedance of the circuit can be expressed through its equivalent active
\[ R_k = \frac{R X_c^2}{R^2 + (X - X_c)^2} = R \frac{(1 - \delta)^2}{d^2 + \delta^2}, \tag{4} \]
and reactive
\[ X_k=\frac{X_c\left[(X_c-X)X-R^2\right]}{R^2+(X-X_c)^2} =X\frac{(\delta-1)(\delta+d^2)}{d^2+\delta^2} \tag{5} \]
resistances.
\[ Z_k=\sqrt{R_k^2+X_k^2}, \qquad \varphi_k=\operatorname{arc\,tg}\frac{X_k}{R_k}. \tag{6} \]
Let us determine the sensitivity of the components of the circuit resistance \(R_k\) and \(X_k\) to a change in electrical conductivity at different detunings of the circuit, defined by the relation:
\[ \delta=\frac{X-X_c}{X}=\frac{\Delta C}{C}, \tag{7} \]
in which \(X_c\) is the resistance of the capacitor at the operating frequency, and \(X\) is the reactive resistance of the sensor, equal to the resistance of the capacitor \(X_{\mathrm{res}}\) at the resonant frequency of the circuit. We assume that the detuning of the circuit is carried out by changing the capacitance of the capacitor.
We define the sensitivity of the equivalent active resistance of the circuit to a change in the electrical conductivity of the cylinder as the relative change in this resistance corresponding to a small relative change in electrical conductivity.
\[ S_\gamma(R_k)=\lim \frac{\Delta R_k}{R_k}\Big/ \frac{\Delta \gamma}{\gamma} =\frac{\partial R_k}{\partial \gamma}\cdot \frac{\gamma}{R_k}. \tag{8} \]
We determine the derivative using expression (4)
\[ \frac{\partial R_k}{\partial \gamma} = R'\frac{(\delta^2-d^2)(1-\delta)^2}{(d^2+\delta^2)^2} + \frac{2\delta d(1-\delta)^2}{(d^2+\delta^2)^2}X'. \tag{9} \]
The notation introduced here has a physical meaning: \(d\) is the damping of the circuit
\[ d=\frac{d_0+R_e j\gamma\tau_k}{1+I_m j\gamma\tau_k}. \tag{10} \]
Here \(d_0\) denotes the quantity reciprocal to the quality factor of the unloaded sensor.
The derivatives \(R'\) and \(X'\) can be expressed through the sensitivity of the active resistance of the sensor to a change in electrical conductivity \(S_\gamma(R)\) and the sensitivity of the reactive resistance \(S_\gamma(X)\)
\[ R'=\frac{\partial R}{\partial \gamma}=S_\gamma(R)\frac{R}{\gamma}, \tag{11} \]
\[ X'=\frac{\partial X}{\partial \gamma}=S_\gamma(X)\frac{X}{\gamma}. \tag{12} \]
Substituting (11) and (12) into (9), we obtain the final expression for the sensitivity of the equivalent active resistance of the circuit to changes in electrical conductivity.
\[ S_\gamma(R_k)=S_\gamma(R)\frac{\delta^2-d^2}{d^2+\delta^2} - S_\gamma(X_k)\frac{2\delta}{d^2+\delta^2}. \tag{13} \]
Reasoning in a similar way, we determine the sensitivity of the reactive resistance of the circuit to a change in electrical conductivity
\[ S_\gamma(X_k)= S_\gamma(X)\frac{(d^2-\delta^2)(1-\delta)}{(d^2+\delta^2)(\delta+d^2)} - S_\gamma(R)\frac{2\delta d^2(1-\delta)}{(d^2+\delta^2)(\delta+d^2)}. \tag{14} \]
For \(\delta = 0\)
\[ S_{\gamma}(X_k)=S_{\gamma}(X)Q^2. \tag{15} \]
The sensitivity of the magnitude and argument of the total resistance of the circuit is determined by the same method:
\[ S_{\gamma}(Z_k)=\frac{\partial Z_k}{\partial \gamma}\cdot\frac{\gamma}{Z_k} = \frac{R_k^2}{Z_k^2}\cdot S_{\gamma}(R_k) + \frac{X_k^2}{Z_k^2}\cdot S_{\gamma}(X_k), \tag{16} \]
\[ S_{\gamma}(\varphi_k)=\frac{\partial \varphi_k}{\partial \gamma}\cdot\gamma = \frac{R_k\cdot X_k}{Z_k^2}\left[S_{\gamma}(X_k)-S_{\gamma}(R_k)\right]. \tag{17} \]
Substituting in (16) and (17), in place of \(R_k\), \(X_k\), \(Z_k\), \(S_{\gamma}(X_k)\), \(S_{\gamma}(R_k)\), their values according to (4), (5), (6), (13), (14), and assuming that \(d^2 \ll 1\) and \(\delta^2 \ll 1\), for the required sensitivities we find
\[ S_{\gamma}(Z_k) = S_{\gamma}(X)\frac{(d^2-\delta)(1-\delta)}{d^2+\delta^2} - S_{\gamma}(R)\frac{d^2}{d^2+\delta^2}, \tag{18} \]
\[ S_{\gamma}(\varphi_k) = S_{\gamma}(R)\frac{d(\delta-d^2)(1-\delta)}{d^2+\delta^2} - S_{\gamma}(X)\frac{d}{d^2+\delta^2}. \tag{19} \]
From the expressions obtained it may be concluded that the sensitivity of an eddy-current sensor connected into a parallel oscillatory circuit is considerably higher than the sensitivity of the sensor itself and is determined mainly by the quality factor of the loaded sensor; i.e., in order to obtain high sensitivity it is necessary to increase the quality factor \(Q\) of the sensor with the article being tested.
Fig. 1. Dependence of the sensitivity of the magnitude of the total resistance of a resonant circuit with an eddy-current sensor to changes in electrical conductivity on the detuning value.
In addition, the sensitivity depends to a considerable extent on the value of the circuit detuning \(\delta\). Thus, the maximum sensitivity of the magnitude of the total resistance of the circuit \(S_{\gamma}(Z_k)\) corresponds to the detuning value
\[ \delta_{1,2} = -d^2 \left\{ \frac{S_{\gamma}(R)}{S_{\gamma}(X)} \pm \sqrt{ \left[ \frac{S_{\gamma}(R)}{S_{\gamma}(X)} \right]^2 + Q^2 } \right\}, \tag{20} \]
which depends on the position of the operating point on the complex plane of the total resistance of the sensor winding.
In order to clarify the nature of this dependence, calculations of the corresponding sensitivities were carried out from formulas (17) and (18) for various values of the generalized parameter \(m_k\) and detunings \(\delta\). The results of these computations are shown graphically in Figs. 1–2. For two values of the generalized parameter, 1.5 and 2, corresponding to two characteristic regions of the complex plane of the total impedance of the sensor, the experimental dependence \(S_\gamma(Z_k)\) was obtained, confirming the calculation results.
Analysis of the graphs makes it possible to conclude that it is more expedient to monitor the electrical conductivity of cylindrical nonmagnetic products by the change in the modulus of the total impedance of the circuit, rather than by the change in phase, since the phase increment \(\Delta \Phi_k = S_\gamma(\Phi_k)\dfrac{\Delta \gamma}{\gamma}\) amounts to fractions of a degree for a 1% change in electrical conductivity \(\dfrac{\Delta \gamma}{\gamma}\). The course of the curves also shows that the sensitivity maxima for various values of the generalized parameter \(m_k\) correspond to different detunings of the oscillatory circuit.
Fig. 2. Dependence of the sensitivity of the argument of the equivalent impedance of the circuit to changes in electrical conductivity on the detuning value
In particular, for \(m_k\) not exceeding 2.5, the sensitivity of the modulus of the equivalent impedance of the circuit is greatest at negative detunings. These conclusions should be borne in mind when designing flaw detectors and, where possible, the positive aspects obtained when the sensor is connected into an appropriately detuned oscillatory circuit should be used.
REFERENCES
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McMaster. Nondestructive Testing. Moscow, IL, 1965.
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V. G. Gerasimov. Analysis of the sensitivity and resolving power of an eddy-current sensor connected into an oscillatory circuit. — Proceedings of the Institute of Metal Physics, Academy of Sciences of the USSR, issue 26, 1967.