A Bridge for Monitoring the Modulus and Q Factor of Complex Impedances

M. S. Roitman

(Presented at the scientific and technical conference of the Radio Engineering Faculty)

Under conditions of mass inspection, what is usually of interest is not the absolute value of the quantity being measured, but its percentage deviation from a certain nominal value, or merely the establishment of a difference from a specified limit.

In this case, from the standpoint of minimum expenditure of time, the use of direct-reading instruments is the most expedient.

Below a relatively simple AC bridge is described, which makes it possible to monitor the magnitudes of the deviations of the modulus and argument of the tested complex impedance \(Z_x\) from the nominal value \(Z\).

Fig. 1.

Fig. 2.

Figure 1 shows the circuit of a four-arm bridge with the tested complex impedance \(Z_x\) and the standard impedance \(Z\) connected in one branch. The vector diagram of the bridge is given in Fig. 2.

The quantity \(Z_x\) can be determined by comparing the voltage moduli \(U_{ac}\) and \(U_{cb}\):

\[ \frac{Z_x-Z}{Z}=\frac{\Delta Z}{Z}=\frac{U_{ac}-U_{cb}}{U_{cb}} . \tag{1} \]

If a phase-sensitive indicator is connected at the bridge output, and the vector of its control voltage coincides in direction with the vector of the voltage \( \dot U_{ab}\) feeding the bridge, then we can compare the projections of \( \dot U_{ac}\) and \( \dot U_{cb}\) on \( \dot U_{ab}\).

The vector of the current through the branch \(abc\) is shifted with respect to the bridge supply voltage \(U_{ab}\) by an angle determined by the impedance of the branch

\[ \dot Z_{abc}=Z_x e^{j\alpha_x}-Z e^{j\beta}. \]

If \(\alpha_x=\beta\), then the point \(c\) will lie on the straight line \(ab\); but if \(\alpha_x\ne\beta\), then the point \(c\) will be above or below this straight line, depending on the sign of the deviation \(\alpha_x\) from \(\beta\).

Consequently, when \(\alpha_x\ne\beta\), monitoring of \(Z_x\) is carried out with a certain error \(\delta_M\). The magnitude \(\delta_M\) is equal to

\[ \delta_M= \frac{(Z+\Delta Z)\cos\psi-Z\cos\gamma}{Z\cos\gamma} - \frac{(Z+\Delta Z)-Z}{Z}. \]

Omitting the intermediate calculations, because of their relative cumbersome nature, we obtain

\[ \delta_M \simeq \frac{1}{4}\cdot\frac{\Delta Z}{Z}\,\Theta^2 \simeq \frac{(Q_x-Q)^2}{4Q^4}\cdot\frac{\Delta Z}{Z}, \tag{2} \]

where

\[ \Theta=\psi-\gamma=\alpha-\beta \simeq \frac{Q_x-Q}{Q}. \]

For example, for the data

\[ \frac{\Delta Z}{Z}=0.2;\qquad \frac{Q_x-Q}{Q}=0.4 \quad\text{and}\quad Q=4 \]

\[ \delta_M \le 0.05\%. \]

As we see, the error of the method lies within permissible limits.

The total error of comparing \(Z_x\) with \(Z\) is also determined by the error in measuring, with the indicator, the voltage \((U_{ac}\cos\psi-U_{cb}\cos\gamma)\) and by the instability of \(U_{ab}\).

Consequently, monitoring \(Z_x\) by direct measurement of the difference voltage leads to the need to stabilize the bridge supply voltage and to use a phase-sensitive indicator with an admissible error.

Meeting the indicated requirements does not present any special difficulties, although it is associated with a certain complication of the instrument circuit, which in itself is already undesirable.

However, what is decisive is that, when monitoring \(Z_x\) by the above method, it is very difficult to monitor the quality factor \(Q_x\) (for \(Q>5\)) or small values of the argument.

As already indicated, the position of the point \(c\) will be above or below the straight line \(ab\), depending on the sign of the deviation \(\alpha\) from \(\beta\). By rotating the vector of the indicator control voltage by \(90^\circ\) with respect to \(\dot U_{ab}\),

one can perform the sorting of complex impedances by argument or by \(Q\) factor.

The relation of the magnitude of the projection of the bridge unbalance voltage to \(Z_x - Z\) is determined by the dependence

\[ -\frac{\operatorname{Re} j U'_{cd}}{U_{ab}} = \frac{1}{2}\tan \frac{\alpha-\beta}{2} = \frac{1}{2}\tan \frac{\theta}{2}. \]

A change in the modulus \(Z_x\) by \(\Delta Z\) leads to the appearance of an error in determining the quantity \(\theta\)

\[ \delta_\theta \simeq \frac{1}{4}\left(\frac{\Delta Z}{Z}\right)^2 . \]

For the data \(\theta = 6^\circ\), \(\dfrac{\Delta Z}{Z}=0.2\), the error \(\delta_\theta\) does not exceed \(1\%\).

However, the phase error of the indicator \(\varepsilon\) leads to an error in checking the argument \(\alpha\)

\[ \Delta\theta > \frac{\Delta Z}{Z}\,\varepsilon . \]

The minimum difference of the arguments \(\alpha\) and \(\beta\) must be greater than \(\Delta\theta\). Hence, in a rough approximation, we obtain

\[ \alpha-\beta = \frac{Q_x-Q}{Q^2} > \frac{\Delta Z}{Z}\cdot \varepsilon . \]

For example, at \(Q=10\), \(\dfrac{\Delta Z}{Z}=0.3\), and \(\varepsilon=3^\circ\), the minimally distinguishable relative change in \(Q\) is greater than \(15\%\).

However, if the component of the unbalance voltage \(\operatorname{Re} U'_{cd}\) is always reduced to a small value, then the accuracy of checking complex impedances by \(Q\) factor can be sharply increased. From consideration of the vector diagram of the bridge (Fig. 2) it follows clearly that \(\operatorname{Re} U'_{cd}\) can be reduced and even brought to zero by shifting point \(d\).

For the purpose of shifting point \(d\), a motor may be used. But such a solution, though used exceptionally often, has a number of substantial drawbacks, the chief of which are: low speed of operation; limited service life and low reliability of operation of the bridge owing to the presence of moving elements; relatively great complexity and cost.

It is considerably better to use, for this purpose, an electrically controlled resistance. As the latter, it is most expedient to use a photoresistor in combination with an incandescent lamp.

In this case the elementary circuit of the sorting device will have the form shown in Fig. 3.

The voltage from the output of the bridge measuring circuit is fed to electronic amplifier 1. After amplification it is fed to phase-sensitive detectors 2 and 5. The control voltage of detector 2 is shifted with respect to the vector of the bridge supply voltage \(U_{ab}\) by \(90^\circ\) by phase-shifting circuit 3.

The control voltage of detector 5 coincides in phase with \(U_{ab}\).

Thus, the voltage at the output of phase-sensitive detector 2 is determined by the projection \(\dot U_{cd}\) on \(\dot U_{k}\), i.e., by the \(Q\) factor \(Q_x\), while at the output of detector 5 it is determined by the projection \(\dot U_{cd}\) on \(U_{ab}\), i.e., by the modulus \(Z_x\).

The reduction of the voltage \(Re\dot U_{cd}\) is achieved by feeding the voltage \(\kappa Re\dot U_{cd}\) from the output of detector 5 into the feedback circuit 7, which regulates the position of point \(d\).

Fig. 3.

In essence, we have a static automatic-control system for the position of point \(d\). Let, for example, \(Z_x < Z\). At the output of detector 5 there will appear a voltage \(\kappa Re\dot U_{cd}\), causing a current \(I_y\) in circuit 7. In this case, through lamp \(L_1\) there will flow a current \(I_0 + \frac{1}{2} I_y\), and through \(L_2\),

\[ I_0 - \frac{1}{2} I_y . \]

The resistance of FS-1 will decrease, while that of FS-2 will increase. Point \(d\) will move to point \(f\) (Fig. 2). As a result, the projection of the unbalance-voltage vector on the vector \(\dot U_{ab}\) will become equal to

\[ \frac{Re\dot U_{cd}}{k_{\text{syst.}}}, \]

where \(k_{\text{syst.}}\) is the overall gain of the automatic-control system.

Since the quantity

\[ \frac{Re\dot U_{cd}}{k_{\text{syst.}}} \]

can be made very small, the accuracy of monitoring \(Q_x\) is then practically limited only by the relative sensitivity \(S'\) of the indicator.

The magnitude of the current \(I_y\) is a function of \(\frac{\Delta Z}{Z}\), and the direction of the current \(I_y\) makes it possible to judge the sign of the deviation of the modulus of the complex impedance being checked from its nominal value. Consequently, by measuring with electrical measuring instrument 6 the current in feedback circuit 7, we can judge the magnitude \(Z_x\).

The error in measuring \(Z_x\) is equal to

\[ \delta_z = \sqrt{ \left( \frac{Re\dot U_{cd}}{U_{ab} k_{\text{syst.}}} \cdot \frac{\Delta Z}{Z} \right)^2 + \left( \varphi \frac{Re\,j\dot U_{cd}}{U_{ab}} \right)^2 + \left( \delta_{\theta}\frac{\Delta Z}{Z} \right)^2 + \delta_M^2 + \delta_{\text{fs}}^2 } \simeq \]

\[ \approx \sqrt{\left(\xi \frac{Q_x-Q}{4Q^2}\right)^2+\left(\delta_{\mathrm{M}}\frac{\Delta Z}{Z}\right)^2+\delta_{\mathrm{b}}^{\,2}+\delta_{\mathrm{fs}}^{\,2}}, \tag{4} \]

where \(\xi\) is the error of the vector-meter angle indicator

\[ (\xi \approx 2 \div 3); \]

\(\delta_{\mathrm{b}}\) is the error of electrical measuring instrument 6

\[ (\delta_{\mathrm{b}} \approx 1.5\%); \]

\(\delta_{\mathrm{M}}\) is the error of the method for measuring \(Z_x\), (2);

\(\delta_{\mathrm{fs}}\) is the error due to instability of the luminous flux of the incandescent lamps and of the resistance of the photoresistors.

The incandescent lamps and photoresistors (after preliminary aging for 200–300 hours) have high parameter stability. According to preliminary data, changes in the parameters of the SN-3 incandescent lamps and the FS-K2N photoresistors (with shunting resistors \(R_1\) and \(R_3\) absent) over 50 hours of continuous operation, with an instability of \(U_0\) equal to 0.1%, led to an unbalance voltage \(\operatorname{Re}\dot U_{\mathrm{cd}}\) corresponding to

\[ \frac{\Delta Z}{Z}=0.1\%. \; {}^{1)} \]

For \(Q \ge 4\),

\[ \frac{Q_x-Q}{Q}=|0.4|;\qquad \frac{\Delta Z}{Z}=|0.2|,\qquad \delta_{\Sigma}<0.5\%. \]

Thus, the relatively simple bridge described above makes it possible to monitor the \(Q\) factor and the modulus of a complex impedance with high accuracy.

It should be noted that, when it is necessary to record deviations of the modulus \(Z_x\) from \(Z\), a self-recording milliammeter may be used as meter 6.

Owing to the advantages indicated above, and to the absence of moving elements in the bridge, it may find wide application in monitoring-and-measuring technology and automation.

\(^{1)}\) If it is necessary to monitor small relative changes of \(Z_x\) from \(Z\), the photoresistors may be shunted by small resistances \(R_1\) and \(R_3\), and the sensitivity of the circuit will correspondingly decrease.