STABILITY ANALYSIS OF A REGENERATIVE CIRCUIT FOR CONNECTING AN EDDY-CURRENT TRANSDUCER

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

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

It is shown in [1, 2] that compensation of the active losses of an eddy-current transducer connected in a parallel oscillatory circuit, by means of negative resistance, makes it possible to significantly increase the sensitivity and noise immunity of electromagnetic flaw detectors.

However, the use of the regeneration principle, along with increasing sensitivity to useful parameters, leads to a considerable increase in the influence of various destabilizing factors.

When a circuit element based on a transistor amplifier with positive feedback (PFB) is used as the negative resistance, it may be considered that the principal destabilizing effect will be exerted by the parameters of the active four-terminal network: the transmission coefficient \(K\), the input resistance \(R_{\mathrm{in}}\), and the output resistance \(R_i\).

To estimate the influence of these parameters on the equivalent parameters of the regenerated circuit, we shall analyze its stability.

Fig. 1. Circuit for connecting an eddy-current transducer into a regenerated oscillatory circuit

The influence of variations in the parameters of the circuit elements (Fig. 1) will be estimated by the relative instability of the circuit transmission coefficient \(\alpha\), which is written in the form

\[ \alpha=\frac{\rho |a|}{\sqrt{R_{\mathrm{eq}}^{2}+\left(X_{\mathrm{eq}}-X_c\right)^{2}}}=\frac{U}{e}, \tag{1} \]

where \(\rho\) is the characteristic resistance;

\[ R_{\mathrm{eq}}=r+R_{\mathrm{in}} \]

is the equivalent active resistance of the transducer winding;

\[ X_{\mathrm{eq}}=X+X_{\mathrm{in}} \]

is the equivalent reactance of the transducer winding.

The magnitude of the introduced active resistance \(R_{\mathrm{in}}\) is determined by the expression

\[ R_{\mathrm{in}}=\frac{X^{2}P(p-\kappa)}{R_{\mathrm{fb}}+R_i}+\frac{X^{2}}{R_{\mathrm{in}}}. \tag{2} \]

The first term characterizes the action of positive feedback; the second, the shunting action of the input resistance of the amplifier.

The introduced reactive resistance

\[ X_{\mathrm{in}}=-\frac{X^{2}p\kappa}{R_{\mathrm{fb}}+R_i}\cdot \tg \varphi_k \tag{3} \]

is determined by the magnitude of the phase shift between the output and input voltages of the four-terminal network.

In expressions (2, 3), \(p\) denotes the switching coefficient of the active element. \(R_{\mathrm{fb}}\) is the ballast resistance of the positive-feedback circuit.

The absolute instability \(\Delta \alpha\) will be found as the sum of the partial differentials with respect to \(R_{\mathrm{eqv}}\) and \(X_{\mathrm{eqv}}\):

\[ \Delta \alpha=\frac{\partial \alpha}{\partial R_{\mathrm{eqv}}}\,dR_{\mathrm{eqv}}+\frac{\partial \alpha}{\partial X_{\mathrm{eqv}}}\,dX_{\mathrm{eqv}}. \tag{4} \]

In this expression, the first term characterizes the stability of the height of the resonance curve (amplitude instability), and the second, the stability of the natural frequency of the oscillatory circuit, determined by the phase shift (phase instability).

The relative instability of the transmission coefficient after differentiating expression (4) is

\[ \frac{\Delta \alpha}{\alpha} = \frac{R_{\mathrm{eqv}}}{R_{\mathrm{eqv}}^{2}+\Delta X_{\mathrm{eqv}}^{2}}\,dR_{\mathrm{eqv}} + \frac{\Delta X_{\mathrm{eqv}}}{R_{\mathrm{eqv}}^{2}+\Delta X_{\mathrm{eqv}}^{2}}\,dX_{\mathrm{eqv}}, \tag{5} \]

where \(\Delta X_{\mathrm{eqv}}=X_{\mathrm{eqv}}-X_c\),

\[ X_c=\frac{1}{\omega C} \]

is the resistance of capacitor \(C\).

The instability of the resonant measuring circuit should be evaluated near its principal operating mode—near resonance tuning, i.e., when \(R_{\mathrm{eqv}}\gg \Delta X_{\mathrm{eqv}}\).

Then

\[ \frac{\Delta \alpha}{\alpha} = \frac{1}{R_{\mathrm{eqv}}}\,dR_{\mathrm{eqv}} + \frac{\Delta X_{\mathrm{eqv}}}{R_{\mathrm{eqv}}^{2}}\,dX_{\mathrm{eqv}}. \tag{6} \]

To determine the instability of the transmission coefficient, we find all partial derivatives depending on \(\kappa\), \(R_{\mathrm{in}}\), and \(R_i\):

\[ \begin{aligned} d_{(\kappa)}R_{\mathrm{eqv}} &= \frac{\partial R_{\mathrm{eqv}}}{\partial \kappa}\,d\kappa = R_{\mathrm{in}}\frac{\kappa}{p-\kappa}\frac{d\kappa}{\kappa}; \\[4pt] d_{(\kappa)}X_{\mathrm{eqv}} &= \frac{\partial X_{\mathrm{eqv}}}{\partial \kappa}\,d\kappa = X_{\mathrm{in}}\frac{d\kappa}{\kappa}; \\[4pt] d_{(R_i)}R_{\mathrm{eqv}} &= \frac{\partial R_{\mathrm{eqv}}}{\partial R_i}\,dR_i = -\frac{R_{\mathrm{in}}}{1+R_{\mathrm{fb}}/R_i}\frac{dR_i}{R_i}; \\[4pt] d_{(R_i)}X_{\mathrm{eqv}} &= \frac{\partial X_{\mathrm{eqv}}}{\partial R_i}\,dR_i = -\frac{X_{\mathrm{in}}}{1+R_{\mathrm{fb}}/R_i}\frac{dR_i}{R_i}; \\[4pt] d_{(R_{\mathrm{in}})}R_{\mathrm{eqv}} &= \frac{\partial R_{\mathrm{eqv}}}{\partial R_{\mathrm{in}}}\,dR_{\mathrm{in}} = \frac{X^{2}}{R_{\mathrm{in}}}. \end{aligned} \tag{7} \]

Substituting these derivatives into (6) and passing to finite increments, we obtain

\[ \frac{\Delta \alpha}{\alpha} = \left[ \frac{R_{\mathrm{in}}}{R_{\mathrm{eq}}}\, \frac{\kappa}{\rho-\kappa} + \frac{X_{\mathrm{in}}}{R_{\mathrm{eq}}}\, \frac{\Delta X_{\mathrm{eq}}}{R_{\mathrm{eq}}} \right]\frac{\Delta \kappa}{\kappa} + \left[ \frac{R_{\mathrm{in}}}{R_{\mathrm{eq}}} + \frac{X_{\mathrm{in}}}{R_{\mathrm{eq}}}\, \frac{\Delta X_{\mathrm{eq}}}{R_{\mathrm{eq}}} \right] \frac{1}{1+\dfrac{R_{\mathrm{fb}}}{R_i}}\, \frac{\Delta R_i}{R_i} + \frac{1}{1+rR_{\mathrm{in}}/X^2}\, \frac{\Delta R_{\mathrm{in}}}{R_{\mathrm{in}}}. \tag{8} \]

Let us introduce the notation: \(Q=\dfrac{X}{r}\), the quality factor of the transducer;

\[ \frac{Q_{\mathrm{eq}}}{Q}\simeq \frac{r}{R_{\mathrm{eq}}}=\xi_p, \]

the regeneration coefficient,

\[ \delta=\frac{\Delta X_{\mathrm{eq}}}{X}, \]

the initial detuning.

Taking these notations into account and carrying out some transformations, we obtain an expression for the total relative instability \(\dfrac{\Delta \alpha}{\alpha}\), caused by instability of the principal characteristics of the transistor amplifier \((K, R_i, R_{\mathrm{in}})\):

\[ \frac{\Delta \alpha}{\alpha} = \frac{\xi_p-1}{\rho-\kappa}\cdot \kappa\,\frac{\Delta \kappa}{\kappa} + \frac{\xi_p-1}{1+\gamma_1}\,\frac{\Delta R_i}{R_i} + \frac{\xi_p Q}{\xi_p Q+\gamma_2}\,\frac{\Delta R_{\mathrm{in}}}{R_{\mathrm{in}}} + \]

\[ + Q^2\xi_p^2\delta\,(m-1)\,\frac{\Delta \kappa}{\kappa} + Q^2\xi_p^2\delta\, \frac{m-1}{1+\gamma_1}\, \frac{\Delta R_i}{R_i}; \tag{9} \]

where \(\gamma_1=\dfrac{R_{\mathrm{fb}}}{R_i}\); \(\gamma_2=\dfrac{R_{\mathrm{in}}}{X}\) are coefficients characterizing, respectively, the matching at the amplifier output with the measuring circuit; the coefficient \(m\) characterizes the ratio \(\dfrac{X_{\mathrm{eq}}}{X}\).

The first three terms of the expression obtained characterize the amplitude instability of the circuit, with the first term making the largest contribution to the change in the transfer coefficient, since the parameter \(K\) is the least stable. The only condition for ensuring smallness of this component at a specified regeneration coefficient \(\xi_p\) is the smallness of \(\Delta \kappa/\kappa\), which can be achieved by rigid stabilization of the gain coefficient.

To reduce the second and third terms it is necessary to provide the best matching conditions, when \(\gamma_1 \gg 1\) and \(\gamma_2 \gg Q_{\mathrm{eq}}\). This requires the use of an amplifier with a large input resistance \(R_{\mathrm{in}}\) and a small output resistance \(R_i\). When, however, the amplifier has a sufficient gain margin, the value of \(\gamma_1\) can be increased considerably by including a stable high-resistance ballast resistor in the positive-feedback circuit, and \(\gamma_2\) by choosing a small value of \(\rho\).

The last two terms of expression (9) represent the phase instability of the circuit. These components will be the smaller, the smaller the reactive resistance introduced into the transducer winding. The condition of maximum stability for these components coincides with the condition \(m=1\). In order to reduce, to the maximum extent, the influence of these components on the total instability of the transfer coefficient, it is necessary, first, to ensure a minimum of phase shifts,

caused both by the amplifying element itself and by the coupling circuits, and, secondly, (when the measurement conditions permit this) to operate at detunings of the circuit, \(\delta\), close to zero.

References

1. A. V. Emel’yanenko, V. K. Zhukov. Increasing the sensitivity of circuits in electromagnetic quality control of materials and products. Metrologiya, No. 5, 1972.

2. A. V. Emel’yanenko, V. K. Zhukov, I. G. Leshchenko. Application of a vortex-current transducer with compensation of its active losses for defectoscopy. In: Proceedings of the First Interuniversity All-Union Conference on Electromagnetic Methods of Quality Control of Materials and Products, Part II, MEI, 1972.