A SIMPLE CIRCUIT FOR INDICATING A 90° PHASE SHIFT

(Presented by the scientific seminar of the AVTF)

A. I. Kramnyuk, V. N. Studenikin

The indicator described makes it possible, over a wide frequency range, to register a phase shift \(\varphi = 90^\circ\). The block diagram is shown in Fig. 1.

The voltages under test, \(\dot U_1\), \(\dot U_2\), pass through matching identical amplifiers (in the simplest case, cathode followers) and enter the measuring circuit.

Fig. 1. Block diagram of the indicator

The existing phase shift between the voltages \(\dot U_1\) and \(\dot U_2\) is supplemented by an additional angle \(\gamma = 90^\circ\). For this purpose the voltage \(\dot U_1\) is applied to the circuit \(R_1C_1\), and the voltage \(\dot U_2\), respectively, to the circuit \(R_2C_2\).

The voltage vector \(\dot U_{23}\) across the capacitor \(C_1\) lags behind the vector \(\dot U_1\) by the angle \(\gamma_1\) (Fig. 2), while the voltage vector \(\dot U_{46}\) leads by the angle \(\gamma_2\). If the conditions \(C_1 = C_2\) and \(R_1 = R_2\) are satisfied, then \(\gamma = \gamma_1 + \gamma_2 = 90^\circ = \mathrm{const}\), and this condition is valid for any frequency. If the angle \(\varphi = 90^\circ\), then the phase shift between the vectors \(\dot U_{23}\), \(\dot U_{46}\) is equal to \(180^\circ\), although the amplitudes of these voltages are in general not equal to one another. On the potentiometer \(R_{\mathrm{p}}\), connected between these voltages, it is always possible to find a point at which the potential relative to ground is equal to 0, i.e., \(U_{R_5} = 0\), which will indicate the presence of quadrature of the voltages \(\dot U_1\) and \(\dot U_2\). If, however, it is impossible to find such a position at which \(U_{\mathrm{out}} = U_{R_5} = 0\), then the angle differs from \(90^\circ\). This dev-

...the device can serve only as an indicator, but it cannot provide a quantitative estimate of the deviation from \(90^\circ\), since the value of the voltage \(U_{R_5}\) is a complicated function of frequency, of the amplitude ratio, and of the magnitude of the phase shift.

Fig. 2.

To increase the accuracy of indication, the voltage \(U_{R_5}\) is amplified and applied to a null indicator.

The main factors determining the indication error are:

1. The spread of the parameters of the networks \(R_{1,2}C_{1,2}\) and the shunting action of \(R_{\mathrm{p}}\) — \(\Delta\varphi_\varphi\).
2. The resolving power of the null indicator — \(\Delta\varphi_0\).
3. The nonidentity of the phase characteristics of the cathode followers — \(\Delta\varphi_{\mathrm{k}}\).

Since the values \(\Delta\varphi_\varphi\), \(\Delta\varphi_0\), \(\Delta\varphi_{\mathrm{k}}\) are small, their mutual influence may be neglected, and the resulting error is equal to the sum:

\[ \Sigma\Delta\varphi=\Delta\varphi_\varphi+\Delta\varphi_0+\Delta\varphi_{\mathrm{k}}. \]

Under the condition of quadrature of the input voltages \((U_{R_5}=0)\), for analyzing the error of the indicator one may (using the equivalent circuits in Figs. 3 and 4) short-circuit the resistance \(R_5\). The adopted notation is:

\[ R_{\mathrm{eqv}_1}=\frac{(R_{\mathrm{out}_1}+R_1)R_3}{R_{\mathrm{out}_1}+R_1+R_3}; \qquad R_{\mathrm{eqv}_2}=\frac{R_2\cdot R_4}{R_2+R_4}+R_{\mathrm{out}_2}, \]

where

\[ R_3=R_{\mathrm{p}}\frac{1}{\omega RC}\cdot \frac{1}{1+\dfrac{|\dot U_1|}{|\dot U_2|}\cdot\dfrac{1}{\omega RC}}, \]

\[ R_4=R_{\mathrm{p}}\frac{|\dot U_1|}{|\dot U_2|}\cdot \frac{1}{1+\dfrac{|\dot U_1|}{|\dot U_2|}\cdot\dfrac{1}{\omega RC}}. \]

The additional angle

\[ \gamma=|\gamma_1|+|\gamma_2|=\operatorname{arctg}\omega_1 R_{\mathrm{eqv}}C_1+\operatorname{arctg}\frac{1}{\omega R_{2\mathrm{eq}}C_2}. \]

To reduce the influence of the output resistance of the matching amplifiers and the shunting action of \(R_{\mathrm{p}}\), the inequalities must be satisfied:

\[ R_{\mathrm{out}_{1,2}}\ll R_{1,2}\ll R_{\mathrm{p}}. \]

In the simplest case, a cathode follower is used as the amplifier; its \(R_{\mathrm{out}}=100\ \Omega\) at a transconductance \(S=10\ \mathrm{mA/V}\).

If the circuit is symmetrical, then a sufficient condition for satisfying the above inequality is \(R_1=R_2=3\ \mathrm{k}\Omega\) and \(R_{\mathrm{p}}=1\ \mathrm{M}\Omega\). The resistance \(R_{\mathrm{p}}\) should not be chosen large, in order to avoid the influence of distributed capacitances in the upper-frequency region. When choosing the values of \(C_1\) and \(C_2\), the inequality must be satisfied—

with \(C_{1,2} \ll C_{\text{par}}\), so that the value of the latter has less effect on the accuracy of the measuring circuit. Thus, for \(C_1 = C_2 = 1000\ \text{pF}\) and a range-coverage coefficient \(Q = 3\), the upper cutoff frequency is determined as follows:

\[ f_{\mathrm{v}}=\sqrt{Q}\,\frac{1}{2\pi RC} \]

for \(Q = 3,\ R_1 = R_2 = 3\ \text{k}\Omega,\ C = 1000\ \text{pF},\ f_{\mathrm{v}} = 100\ \text{kHz}.\)

If it is assumed that the amplitudes of the input voltages differ by no more than a factor of two, the spread of \(R_{\text{out}}\) is \(\pm 10\%\), and the tolerances on the resistances and capacitances of the measuring circuit do not exceed \(\pm 0.3\%\),

Fig. 3. Equivalent circuit of the indicator

Fig. 4. Reduced equivalent circuit

then the maximum indication error due to the listed causes will be \(\Delta \varphi_{\varphi} = 1^\circ 10'\). If the amplitudes of the input voltages are equal, \(\Delta \varphi_{\varphi} = 50'\).

When calculating the error, the nonidentity of the phase characteristics of the cathode followers may be neglected.

The error introduced by the null indicator can be estimated from the expression:

\[ \Delta \varphi_0 = \operatorname{arctg}\frac{2\Delta U_0}{K U_{\min}}, \]

where \(\Delta U_0\) is the resolving power of the null indicator;
\(K\) is the gain of the amplifier;
\(U_{\min}\) is the minimum value of the input voltage.

For \(\Delta U_0 = 0.05\ \text{V},\ K = 200,\ U_{\min} = 1\ \text{V}\), we have \(\Delta \varphi_0 = 20'\).

Thus the maximum total error will be

\[ \Sigma \Delta \varphi = 1^\circ 30'. \]

Conclusions

1. The indicator circuit presented makes it possible to assess, with sufficiently high accuracy, the quadrature of two voltages.

2. To extend the frequency range and improve the accuracy of indication, it is desirable to use elements in the measuring circuit with a tolerance of no more than 0.1%.

3. The amplifiers at the input of the device must have identical phase characteristics and low output resistance.

4. To improve the accuracy of indication, equality of the input amplitudes is desirable.

5. A drawback of this type of indicator is the need to switch the capacitances \(C_1\) and \(C_2\), owing to the small range-overlap factor. Increasing it, however, leads to an increase in the error at the extreme portions of the range.

References

1. B. P. Aseev. Phase Relationships in Radio Engineering. Svyazizdat, 1951.