On the Adjustment of Signal Amplitude in Phase Measurements

K. M. Shulzhenko

(Presented by the seminar of the Radio Engineering Faculty)

In phase measurements one often has to deal with the need to adjust the amplitude of signals, the phase shift between which is to be measured. The principal requirement imposed on amplitude regulators in this case consists in maintaining the constancy of the phase shift between the voltages at the input and output of the regulators, as well as the equality of the phase shifts produced by them in both channels.

Stepwise Adjustment of Signal Amplitude

Input attenuators widely used in measuring instruments [1, 2, 3] make it possible to carry out stepwise adjustment of the signal amplitude over a wide frequency range without introducing frequency-phase distortions, if the internal resistance of the signal source is \(R_i = 0\). Finite values of the internal resistance lead to the appearance of a phase shift of the output voltage \(U_2\) relative to the input voltage \(U_1\). This phase shift will depend on frequency. For the equivalent circuit of the attenuator (Fig. 1) we have:

\[ \overline{U}_2 = \overline{U}_1 \frac{1}{ 1+\dfrac{R_i}{R_2} + j\omega C_2 R_i + \dfrac{R_1(1+j\omega C_2R_2)}{R_2(1+j\omega C_1R_1)} }. \tag{1} \]

For \(R_1 C_1 = R_2 C_2\), formula (1) takes the form:

\[ \overline{U}_2 = \overline{U}_1 \frac{1}{ 1+\dfrac{R_i}{R_2} + \dfrac{R_1}{R_2} + j\omega C_2 R_i }. \tag{2} \]

The phase shift between the voltages \(U_2\) and \(U_1\) is

\[ \varphi = -\operatorname{arctg} \frac{\omega C_2 R_2}{ 1+\dfrac{R_1+R_2}{R_i} }. \tag{3} \]

From (3) we see that, for \(R_i = 0\), the phase shift between \(U_2\) and \(U_1\) will always be equal to zero.

In real circuits \(R_i \ne 0\); consequently, there will always be some value of the phase shift between \(U_2\) and \(U_1\). The maximum phase shift will occur at the upper frequency of the specified range.

If the input attenuators of both channels introduce constant, i.e., independent of the division ratios, and equal phase shifts, then the error in measuring the phase shift between two harmonic signals will be eliminated. At a given frequency, constancy of the phase shift produced by the input attenuator will be ensured if, when adjusting the amplitude, the constancy of the sum of the resistances \(R_1 + R_2 = R_{2n-1} + R_{2n}\) and of the products \(R_1 C_1 = R_2 C_2 = R_n C_n\) is maintained. Consequently, in calculating the attenuator it is necessary to proceed from the condition of equality not only of the time constants of both arms of the divider at a given switch position, but also of the time constants of all arms of the attenuator. For example, for the attenuator shown in Fig. 2, the initial conditions for the calculation will be:

Fig. 1.          Fig. 2.

a) the condition of equality of the time constants of all arms of the attenuator

\[ R_1 C_1 = R_2 C_2 = R_3 C_3 = R_4 C_4 = \ldots = R_n C_n = \ldots RC, \tag{4} \]

where \(C_2 = C'_2 + C\) and \(C_4 = C'_4 + C\);

b) the condition of constancy of the input active resistance

\[ R_1 + R_2 = R_3 + R_4 = \ldots = R_{2n-1} + R_{2n} = R, \tag{5} \]

where

\[ R_2 = \frac{R'_2 \cdot R}{R'_2 + R} \quad \text{and} \quad R_4 = \frac{R'_4 \cdot R}{R'_4 + R}. \]

Starting from the specified load resistance and capacitance of the attenuator, \(R\) and \(C\), and the division ratios

\[ \kappa_1 = \frac{R_2}{R}, \quad \kappa_2 = \frac{R_4}{R}, \ldots \quad \kappa_n = \frac{R_{2n}}{R}, \]

we determine all elements of the attenuator satisfying conditions (4) and (5).

For the arms of the divider in the first position of the switch we shall have:

\[ R_1=(1-\kappa_1)R,\qquad C_1=\frac{C}{1-\kappa_1}, \]

\[ R'_2=\frac{\kappa_1 R}{1-\kappa_1}\quad \text{and}\quad C'_2=C\frac{1-\kappa_1}{\kappa_1}. \]

For the divider in the second position of the switch:

\[ R_3=(1-\kappa_2)R,\qquad C_3=\frac{C}{1-\kappa_2}, \]

\[ R'_4=\frac{\kappa_2 R}{1-\kappa_2}\quad \text{and}\quad C'_4=C\frac{1-\kappa_2}{\kappa_2}. \]

For the divider in the \(n\)-th position of the switch:

\[ R_{2n-1}=(1-\kappa_n)R,\qquad C_{2n-1}=\frac{C}{1-\kappa_n}, \]

\[ R'_{2n}=\frac{\kappa_n R}{1-\kappa_n}\quad \text{and}\quad C'_{2n}=C\frac{1-\kappa_n}{\kappa_n}. \]

Fulfillment of conditions (4) and (5) preserves the constancy of the input capacitance and the input active resistance of the attenuator and ensures independence of the phase shift from the division coefficient of the attenuator, i.e., from the switch position. Depending on the frequency, the phase shift will change, increasing as the signal frequency increases.

Equality of the phase shifts of the input attenuators of both channels of the phase-measuring device when measuring the phase shift between signals of the same frequency can, in practice, be achieved when the internal resistances of the signal sources are equal; the phase shift between them is to be measured. The latter can be ensured by using transformation of the internal resistance of the signal sources by means of cathode followers at the input of both channels of the phase-measuring device. The attenuators in this case are connected at the output of the input cathode followers (Fig. 3). The use of cathode followers at the channel input makes it possible to obtain a sufficiently large input resistance and small input capacitance, which are necessary for the input circuits of any measuring device, especially a phase-measuring one.

Fig. 3. Block diagram of the input circuits of a phase-measuring device. 1—input cathode follower, 2—attenuator, 3—second cathode follower, 4—smooth amplitude-adjustment circuit.

To reduce the error in measuring the phase shift due to possible nonidentity of the attenuators, it is desirable to have \(R_i \ll R_1 + R_2\), which can be achieved by using tubes with high transconductance in the input cathode followers.

Smooth Adjustment of Signal Amplitude

The performance of smooth adjustment of signal voltage over a wide frequency range without introducing a phase shift is a very difficult problem. A low-resistance potentiometer at the output of a cathode follower (Fig. 3) permits adjustment of the signal amplitude over a considerable frequency range if the load capacitance \(C_2\) (Fig. 4), connected to the smooth-adjustment potentiometer, is negligibly small. To reduce this capacitance, another cathode follower or an amplifier stage with a series negative-feedback circuit may be connected at the output of the smooth adjustment.

To reduce the frequency-phase distortions of the smooth gain adjustment (Fig. 4), a correcting capacitance \(C_1 = C_2\) is sometimes connected. The output voltage in this circuit, using the notation adopted in Fig. 4, is determined by formula (1), which after transformation may be written in the form

\[ U_2 = U_1 \frac{1}{ 1 + \dfrac{R_i}{R_2} + \dfrac{R_1\left(1+\omega^2 C_1 C_2 R_1 R_2\right)}{R_2\left(1+\omega^2 C_1^2 R_1^2\right)} + j\omega\left[ C_2 R_i + \dfrac{R_1}{R_2}\left(C_2 R_2 - C_1 R_1\right) \right] }. \tag{6} \]

The phase shift between \(\overline{U}_2\) and \(\overline{U}_1\), determined from (6), will be:

\[ \varphi = -\operatorname{arctg} \frac{ \omega\left[ C_2 R_i + \dfrac{R_1}{R_2}\left(C_2 R_2 - C_1 R_1\right) \right] }{ 1 + \dfrac{R_i}{R_2} + \dfrac{R_1\left(1+\omega^2 C_1 C_2 R_1 R_2\right)} {R_2\left(1+\omega^2 C_1^2 R_1^2\right)} }. \tag{7} \]

Fig. 4.

In the middle position of the potentiometer slider, when \(R_1 = R_2 = \dfrac{R_n}{2}\), with \(C_1 = C_2\), the phase shift produced by the smooth-adjustment circuit is

\[ \varphi_{\mathrm{cp}} = -\operatorname{arctg} \frac{ \dfrac{1}{2}\omega C_2 R_n }{ 1 + \dfrac{R_n}{R_i} }. \tag{8} \]

For the extreme upper position of the potentiometer slider, when \(R_1=0\) and \(R_2=R_n\), we obtain the phase shift

\[ \varphi_{v}=-\operatorname{arctg}\frac{\omega C_2 R_n}{1+R_n/R_i}. \tag{9} \]

For small values of the phase shift produced by the smooth-control circuit, it may be assumed that \(\varphi_v \approx 2\varphi_{\mathrm{av}}\).

Moving the potentiometer slider to the extreme lower position, when \(R_2 \to 0\) and \(R_1 \to R_n\), will cause the phase shift

\[ \varphi_{n}\to \operatorname{arctg}\frac{\omega C_1 R_n}{1+\dfrac{R_i}{R_n}}. \tag{10} \]

Here it is assumed that \(\omega^2 R_n^2 C_1^2 \ll 1\), i.e., it is supposed that the smooth control under consideration is used for such a frequency range in which the adopted inequality holds.

If \(R_i=R_n\), then \(\varphi_n \approx -2\varphi_{\mathrm{av}}\), while for \(R_i<R_n\) the phase shift \(|\varphi_n|>2|\varphi_{\mathrm{av}}|\).

It is not expedient to use \(R_i>R_n\), since in this case \(\varphi_v\) and \(\varphi_{\mathrm{av}}\) increase.

From the foregoing we see that moving the slider over the upper half of the potentiometer gives a three times smaller change in the phase shift \((\Delta\varphi_v=\varphi_v-\varphi_{\mathrm{av}}\approx\varphi_{\mathrm{av}})\) than the change in phase shift when moving the slider over the lower half of the potentiometer \((\Delta\varphi_n=\varphi_n-\varphi_{\mathrm{av}}\approx-3\varphi_{\mathrm{av}})\).

Consequently, it is expedient to limit the range of smooth adjustment of the signal amplitude, carrying out a reduction of the amplitude to 0.5 of the maximum value. The change in phase shift in this case will occur within the limits \(\Delta\varphi=\Delta\varphi_v\approx\varphi_{\mathrm{av}}\). Thus, the potentiometric smooth adjustment of the amplitude of signals in phase-measuring devices considered here is applicable in such a frequency range that, at the maximum signal frequency, the change in phase shift

\[ \Delta\varphi \approx \varphi_{\mathrm{av}} = -\operatorname{arctg} \frac{\dfrac{1}{2}\omega C_2 R_n}{1+\dfrac{R_n}{R_i}} \]

will not exceed the permissible value.

In some cases of phase measurements it is necessary to adjust the amplitude of signals of a certain fixed frequency. For this purpose, the smooth-control circuit shown in Fig. 5 is proposed. The operating principle of the circuit is based on the property of a series oscillatory circuit to change the voltage transfer coefficient when its quality factor is changed.

The voltage at the output of the circuit (Fig. 5)

\[ \overline{U}_2=\overline{U}_1 \frac{1}{1-\omega^2LC+j\omega C(R_i+R_0+R_n)}. \tag{11} \]

The phase shift between \(U_2\) and \(U_1\) will be

\[ \varphi=-\operatorname{arctg}\frac{\omega C(R_i+R_0+R_n)}{1-\omega^2LC}. \tag{12} \]

If \(\omega=\dfrac{1}{\sqrt{LC}}\), then \(\varphi=-\dfrac{\pi}{2}\), and the modulus of the transfer coefficient of the circuit is

\[ K=\frac{U_2}{U_1}= \frac{1}{\omega C(R_i+R_0)\left(1+\dfrac{R_n}{R_i+R_0}\right)}. \tag{13} \]

Fig. 5.

Adopting the notation:

\[ R'_0=R_i+R_0 \]

—the total loss resistance of the circuit at \(R_n=0\), and

\[ K_0=\frac{1}{\omega C R'_0} \]

—the maximum transfer coefficient of the circuit, from (13) we obtain:

\[ \frac{K}{K_0}=\frac{1}{1+\dfrac{R_n}{R'_0}}. \tag{14} \]

Fig. 6.

The dependence of the relative change in the modulus of the transfer coefficient on the resistance ratio \(\dfrac{R_n}{R'_0}\), plotted according to (14), is shown in Fig. 6. A smoother change of the coefficient

transmission over the range \(\dfrac{R_n}{R'_0}=0\div 1\) can be obtained if type B resistors are used as the variable resistance \(R_n\).

Tuning the circuit to resonance can be achieved by adjusting the capacitance \(C\), which includes the load capacitance and the capacitance of the trimming capacitor.

Obviously, it is desirable to obtain \(K_0=1\), which can be achieved by an appropriate choice of the circuit parameters: \(L\), \(C\), \(R_i\), and \(R_0\). The resistance \(R_0\) consists of the loss resistance of the coil \(R_1\) and an additional resistance \(R_{\text{add}}\) (i.e., \(R_0=R_1+R_{\text{add}}\)), if it is necessary to include it in the circuit in order to obtain the required \(K_0\) or to reduce the shunting effect of the amplitude-control circuit on the signal source to which it is connected.

Fig. 7.

For a large internal resistance of the signal sources, it is proposed to use the smooth amplitude-control circuit shown in Fig. 7. At the frequency of parallel resonance of the circuit \(\omega \simeq \dfrac{1}{\sqrt{LC}}\), the transmission coefficient of the circuit

\[ K=\frac{\overline{U}_2}{\overline{U}_1} = \frac{R_{\mathrm{eq}}} {(R_i+R_{\mathrm{eq}})\left(1+\dfrac{R_n}{R_i+R_{\mathrm{eq}}}\right)} \tag{15} \]

will be real. Here \(R_{\mathrm{eq}}=\dfrac{\rho^2}{r}\), and \(\rho=\sqrt{\dfrac{L}{C}}\), \(r=r_L+r_C\).

Consequently, the phase shift between \(U_2\) and \(U_1\) will be equal to zero.

Let us denote by \(K_0=\dfrac{R_{\mathrm{eq}}}{R_i+R_{\mathrm{eq}}}\) the transmission coefficient of the circuit at \(R_n=0\). If \(R_i\ll R_{\mathrm{eq}}\), then \(K_0\simeq 1\). The relative change in the transmission coefficient

\[ \frac{K}{K_0} = \frac{1} {1+\dfrac{R_n}{R_i+R_{\mathrm{eq}}}} \tag{16} \]

for this circuit has the same character as for the circuit shown in Fig. 5.

The advantage of the circuit of Fig. 7 in comparison with the circuit of Fig. 5 is the zero phase shift between \(U_2\) and \(U_1\) and the larger value of the input resistance, which in many cases is of substantial importance.

In a certain frequency range, when the capacitance of the parallel circuit \(C\) is sufficiently large and the effect of the parasitic capacitance connected to the circuit is neglected, it is possible to obtain a circuit with a non-

by a different resonance, choosing \(r_L = r_C = \sqrt{\dfrac{L}{C}}\). In this case the resistance of the parallel circuit

\[ Z_e = \rho = \sqrt{\dfrac{L}{C}} \]

will be purely active and independent of frequency. Consequently, within a certain frequency range it is possible to achieve smooth adjustment of the signal amplitude. The phase shift introduced by the circuit of Fig. 7 can then be reduced to an allowable value.

Experimental verification of the adjustment circuits was carried out at frequencies up to \(2\) MHz. The experimental data confirm the correctness of the conclusions of the analysis presented. The maximum phase error arising when adjusting the amplitude of signals at frequencies not exceeding \(2\) MHz was no more than \(0.05^\circ\). By more careful tuning of the amplitude-adjustment circuits considered, the phase error can be reduced.

References

1. O. B. Lur’e, Video-Frequency Amplifiers. Sov. Radio Publishing House, 1955.
2. V. L. Kreitser, Video Amplifiers. Sov. Radio Publishing House, 1952.
3. B. A. Varshaver, Design of Pulse Amplifiers, MEI, 1953.
4. S. N. Krize, Amplifying Devices, Svyazizdat, 1958.