SENSITIVITY ANALYSIS OF AN OSCILLATORY CIRCUIT WITH AN EDDY-CURRENT TRANSDUCER TO THE PARAMETERS OF A CYLINDRICAL PRODUCT

A. V. EMELYANENKO, V. K. ZHUKOV, I. G. LESHCHENKO

(Submitted by the joint scientific-technical seminar of the Department of Information and Measurement Technology and the EMI Research Institute sector)

The increased requirements imposed on the sensitivity of instruments for electromagnetic quality control of materials and products make it necessary to seek new possibilities that provide an increase in the sensitivity both of the measuring circuit and of the primary-transducer circuit. From this point of view, one should consider the inclusion of a through-type eddy-current transducer in an oscillatory circuit; namely, one should analyze the sensitivity of the components of the impedance of a parallel oscillatory circuit to changes in the parameters of a cylinder (electrical conductivity, magnetic permeability, radius).

The impedance of the oscillatory circuit is determined by the impedance of the capacitor included in the oscillatory circuit, \(X_c\), and also by the active [2]

\[ R = X_0\left(\frac{1}{Q_0} + R_e j\eta_{\kappa}\right) \tag{1} \]

and reactive

\[ X = X_0(1 + I_m j\eta_{\kappa}) \tag{2} \]

components of the transducer impedance.

The equivalent active \(R_{\kappa}\) and reactive \(X_{\kappa}\) impedances of the circuit will be, respectively,

\[ R_{\kappa} = X_c \frac{2RX - RX_c}{R^2 + (X - X_c)^2}, \tag{3} \]

\[ X_{\kappa} = X_c \frac{XX_c - X^2 - R^2}{R^2 + (X - X_c)^2}. \tag{4} \]

Let us determine the sensitivity of the components of the circuit impedance \(R_{\kappa}\) and \(X_{\kappa}\) to a change in one of the product parameters for different detunings of the circuit, defined by the relation

\[ \delta = \frac{X - X_c}{X}, \tag{5} \]

in which \(X_c\) is the impedance of the capacitor at the operating frequency, and \(X\) is the reactive impedance of the sensor, equal to the impedance of the capacitor \(X_{c\,\mathrm{res}}\) at the resonant frequency of the circuit.

We define the sensitivity of the equivalent active resistance of the circuit to a change in some \(i\)-th parameter of the product as the relative change of this resistance per small relative change of this parameter,

\[ S_i(R_\kappa)=\lim_{\Delta i\to 0}\frac{\Delta R_\kappa}{R_\kappa}\bigg/\frac{\Delta i}{i} =\frac{\partial R_\kappa}{\partial i}\cdot\frac{i}{R_\kappa}. \tag{6} \]

We determine the derivative \(\dfrac{\partial R_\kappa}{\partial i}\) using expression (3)

\[ \frac{\partial R_\kappa}{\partial i} = R'\frac{(\delta^2-d^2)(1-\delta)^2}{(d^2+\delta^2)} + X'\frac{2d\delta(1-\delta)^2}{d^2+\delta^2}. \tag{7} \]

The notation \(d\) introduced here has a physical meaning—it is the damping of the circuit

\[ d=\frac{d_0+Re\,j\eta\tau_\kappa}{1+Im\,j\eta\tau_\kappa}. \tag{8} \]

Here \(\eta\) denotes the fill factor of the transducer, and \(d_0\) is the reciprocal of the quality factor of the unloaded transducer.

We express the derivatives \(R'\) and \(X'\) through the sensitivity of the active resistance of the eddy-current transducer to a change in the parameter being monitored:

\[ R'=\frac{\partial R}{\partial i}=S_i(R)\cdot\frac{R}{i}, \tag{9} \]

\[ X'=\frac{\partial X}{\partial i}=S_i(X)\cdot\frac{X}{i}. \tag{10} \]

Substituting (9) and (10) into (7) and (6), we obtain the final expression for the sensitivity of the equivalent active resistance of the circuit to a change in the \(j\)-th parameter of the product:

\[ S_i(R_\kappa)=S_i(R)\frac{\delta^2-d^2}{d^2+\delta^2} - S_i(X)\frac{2\delta}{d^2+\delta^2}. \tag{11} \]

Reasoning in a similar manner, we determine the sensitivity of the reactive resistance of the circuit:

\[ S_i(X_\kappa)=S_i(X)\frac{(d^2-\delta^2)(1-\delta)}{(d^2+\delta^2)(d^2+\delta)} - S_i(R)\frac{2d^2\delta(1-\delta)}{(d^2+\delta^2)(\delta+d^2)}. \tag{12} \]

If, as the output parameters of the transducer, one takes the modulus

\[ Z_\kappa=\sqrt{R_\kappa^2+X_\kappa^2} = \frac{X_c}{\sqrt{d^2+\delta^2}} \tag{13} \]

and argument

\[ \varphi_\kappa=\operatorname{arctg}\frac{X(X-X_c)+R^2}{RX_c} \tag{14} \]

of the total impedance of the circuit, then the expressions for the corresponding sensitivities, found by the same method, will have the form:

\[ S_i(Z_\kappa)=\frac{\partial Z_\kappa}{\partial i}\cdot\frac{i}{Z_\kappa} = \]

\[ = \frac{(1-\delta)(d^2-\delta)}{d^2+\delta^2}S_i(X) - S_i(R)\frac{d^2}{d^2+\delta^2}, \tag{15} \]

\[ S_i(\varphi_k)=\frac{\partial \varphi_k}{\partial i}\cdot i= \frac{d(1-\delta)(\delta-d^2)}{d^2+\delta^2}\cdot S_i(R) -\frac{d}{d^2+\delta^2}S_i(X). \tag{16} \]

To make the discussion concrete, we shall restrict ourselves to two monitored parameters of the product: the electrical conductivity \(\alpha\) and the radius \(r\), which is the case in the inspection of nonferromagnetic cylinders.

In accordance with this, we shall estimate the sensitivity of the components of the circuit impedance to changes in the electrical conductivity and radius of the product; for this purpose, in expressions (11), (12), (15), and (16), instead of \(S_i(R)\) and \(S_i(X)\) from [2], one must substitute, respectively, \(S_\gamma(R)\), \(S_r(R)\) and \(S_\gamma(X)\), \(S_r(X)\)—the sensitivities of the sensor impedance components to changes in the electrical conductivity and radius of the cylinder.

From the expressions obtained it is evident that the sensitivity of an eddy-current transducer connected into a parallel oscillatory circuit may be higher than the sensitivity of the transducer itself, and will be determined mainly by the quality factor of the sensor and by the detuning of the circuit. As an example, let us analyze the expression for the sensitivity of the modulus of the total impedance of the circuit to a change in electrical conductivity,

\[ S_\gamma(Z_k)= \frac{(d^2-\delta)(1-\delta)}{d^2+\delta^2}S_\gamma(X) -\frac{d^2}{d^2+\delta^2}S_\gamma(R) \tag{17} \]

for a maximum. The value of the optimum detuning in this case is determined from the expression

\[ \delta_{1,2}=-d^2 \left\{ \frac{S_\gamma(R)}{S_\gamma(X)} \pm \sqrt{ \left[ \frac{S_\gamma(R)}{S_\gamma(X)} \right]^2 +Q^2} \right\}. \tag{18} \]

It is also possible to show the possibility of improving the resolving power of a transducer connected into an oscillatory circuit when the parameters of the product are monitored separately; for this purpose, let us take the ratio of the sensitivities of the reactive impedance of the circuit to changes in the electrical conductivity and radius,

\[ \frac{S_\gamma(X_k)}{S_r(X_k)} = \frac{S_\gamma(X)}{S_r(X)}\cdot K, \tag{19} \]

where

\[ K= \frac{ 1-\dfrac{S_\gamma(R)}{S_\gamma(X)}\cdot \dfrac{2d^2\delta}{d^2-\delta^2} }{ 1-\dfrac{S_r(R)}{S_r(X)}\cdot \dfrac{2d^2\delta}{d^2-\delta^2} }. \]

The factor \(K\) characterizes the difference between the resolving power of an oscillatory circuit with a sensor and the resolving power of a single sensor.

Thus, expressions (18) and (19) show that the maximum sensitivity and resolving power of an eddy-current transducer correspond to certain detunings of the circuit, which, in turn, depend on the quality factor of the loaded sensor and on the position of the operating point on the complex plane of the total impedance of the sensor winding. In order to clarify the nature of this dependence, calculations of the corresponding sensitivities \(S_\gamma(Z_k)\), \(S_r(Z_k)\) were carried out using formula (15) for various values of the generalized parameter \(m_k\) and detunings \(\delta\), with a filling factor of 0.5. The results of these calculations are shown graphically in Fig. 1.

The curves obtained make it possible to choose correctly both the value of the optimal detuning and the generalized parameter that provide the best conditions for monitoring one or another parameter of the product. In particular, when monitoring the electrical conductivity of cylinders by the change in the modulus of the equivalent resistance of the circuit, it is advisable to choose the value of the generalized parameter no higher than 2.5. In this case the sensitivity is greatest for negative detunings.

Fig. 1. Dependence of the sensitivity of the modulus of the total resistance of the oscillatory circuit to changes in the radius \(r\) and electrical conductivity \(\gamma\) of a cylinder on the detuning \(\delta\) of the circuit at different values of the generalized parameter \(m_k\)

Analysis of the expressions and graphs obtained shows that, by including an eddy-current transducer in an ordinary parallel oscillatory circuit, it is impossible to obtain a significant increase in sensitivity because of the low quality factor of a real loaded eddy-current transducer. This circumstance considerably reduces the effectiveness of using an oscillatory circuit with an eddy-current transducer for electromagnetic inspection of the quality of products and materials. In this connection, the use for this purpose of a regenerative oscillatory circuit appears advisable, when a negative resistance is connected to its terminals, compensating the active losses of the transducer.

To estimate the sensitivity of such a circuit, we shall confine ourselves to considering the sensitivity of the modulus of the equivalent resistance of the regenerative oscillatory circuit to changes in electrical conductivity and radius; for this purpose we take the derivative of the expression

\[ Z_{k\,\mathrm{eq}}=\frac{XX_c\eta_p}{\sqrt{R^2+(X-X_c)^2\eta_p^2}}, \tag{20} \]

where

\[ \eta_p=\frac{1}{1-\dfrac{X^2}{RR_-}} \]

is the regeneration coefficient,

\[ R_- \text{ is the negative resistance.} \]

Since all the arguments set forth above for the ordinary circuit remain valid also for the regenerative circuit, the intermediate calculations may be omitted and the final expression for the sensitivity of the modulus of the equivalent resistance of the regenerative circuit to changes in the electrical conductivity and radius of the cylinder may be written as follows:

\[ S_{\gamma r}\left(Z_{k\,\mathrm{eq}}\right) = S_{\gamma r}(X)\, \frac{(2\eta_p-1)-Q^2\eta_p^2\delta(1+\delta)} {Q^2\eta_p^2\delta^2+1} - S_{\gamma r}(R)\, \frac{\eta_p}{1+Q^2\eta_p^2\delta^2}. \tag{21} \]

For \(\delta=0\)

\[ S_{\gamma}\left(Z_{k\,\mathrm{eq}}\right) = S_{\gamma r}\left(Z_{k\,\mathrm{eq}}\right) = S_{\gamma r}(X)\left(2\eta_p^{-1}\right) - S_{\gamma r}(R)\cdot \eta_p . \tag{22} \]

Figure 2 presents plots of the dependence of the sensitivity of the modulus of the equivalent resistance of the circuit to changes in electrical conductivity and radius for various \(\delta\), \(m_k\), and \(\eta_p\). Analysis of the plots

Fig. 2. Dependence of the sensitivity of the modulus of the equivalent resistance of the regenerated oscillatory circuit to changes in radius \(r\) and electrical conductivity \(\gamma\) on the magnitude of the circuit detuning, for different values of the generalized parameter \(m_k\) and the regeneration coefficient \(\eta_p\).

confirms the conclusions that, with an increase in the equivalent \(Q\)-factor, there is a significant increase in the sensitivity of the circuit. The value of the optimum detuning in this case approaches zero, which has its positive aspects from the standpoint of practical implementation of this transducer connection circuit. The use of the regeneration method to increase the sensitivity of an eddy-current transducer undoubtedly provides greater possibilities, especially for purposes of electromagnetic flaw detection.

REFERENCES

1. Mac-Master. Nondestructive Testing. IL, 1965.
2. V. K. Zhukov. Candidate’s dissertation. Tomsk Polytechnic Institute, 1964.
3. 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 Physics of Metals, Academy of Sciences of the USSR, issue 26, 1967.