ON THE POSSIBILITY OF USING PHASE RELATIONSHIPS TO ESTIMATE THE FREQUENCY ERROR OF SYMMETRIC ATTENUATION BOXES

M. S. Roitman, V. N. Studenikin

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

A symmetric attenuation box is a ladder-type four-terminal network made with active resistances. If its transmission coefficient at direct current depends on the spread of the parameters of the circuit elements, then at higher frequencies an additional frequency error begins to appear because of the presence of reactive conductances. In this case the equivalent circuit has a complicated form, and the frequency properties of the four-terminal network will be determined by the ratio of the reactive quantities in the elements of which the sections are made.

The frequency error of attenuation boxes amounts to \(1 \div 3\%\), and its measurement presents certain difficulties. However, it is known that in minimum-phase circuits there is a one-to-one relationship between the phase and frequency characteristics. The frequency error for such four-terminal networks is much easier to obtain by using phase relationships, since the angle of slope of the phase characteristic is greater than the angle of slope of the frequency characteristic.

In order to use phase relationships, it is necessary to determine the influence of parasitic conductances at different attenuation values, i.e., what relationship is observed in this case between the real and imaginary components of the transmission coefficient of the four-terminal network.

Known methods of analysis [1] establish the relationship between the phase and frequency characteristics of a circuit by drawing up equivalent circuits and studying the constant coefficients in the equations of multiterminal networks of general form. In doing so, complicated determinants are obtained and cumbersome calculations are performed on them. Obviously, such an approach to the analysis of four-terminal networks may lead to incorrect conclusions, since the structure of the circuit will differ from the real one.

On the other hand, in order to determine the character of the circuit, it is sufficient, by experiment, to obtain the phase and frequency characteristics and, on the basis of the experimental data, comparing the law of their variation, to draw the corresponding conclusion.

Before studying a symmetric attenuation box, let us first consider an ideal minimum-phase circuit and, for it, find an expression for the relationship between the frequency error and the phase characteristic.

The transmission coefficient of the circuit in Fig. 1 is equal to:

\[ K_{\omega} = \frac{r_2}{(r_1 + r_2) + i \omega \tau}; \]

its argument:

\[ \varphi_\omega=\operatorname{arctg}\frac{r_1}{r_1+r_2}\,\omega\tau; \]

where \(\tau=Cr_2\);

the modulus of the transfer coefficient has the form:

\[ |K_\omega|=\frac{K_0}{\sqrt{1+\frac{r_1^2}{(r_1+r_2)^2}(\omega\tau)^2}}. \]

Fig. 1.

If we introduce the quantity \(m=\dfrac{r_1}{r_1+r_2}\) as a scale factor, then we obtain an expression for the frequency distortions:

\[ \frac{\kappa_0}{\kappa_\omega}=\sqrt{1+(m\omega\tau)^2}. \]

Expanding the right-hand side of the resulting expression in a series and, neglecting higher-order terms, limiting ourselves to only the first two terms, we obtain

\[ \frac{\kappa_0}{\kappa_\omega}\simeq 1+\frac{(m\omega\tau)^2}{2}. \tag{1} \]

The transfer coefficient of the four-terminal network, taking the frequency error into account, is equal to

\[ \kappa_\omega=\kappa_0\pm\Delta\kappa_\omega \]

or

\[ \frac{\kappa_0}{\kappa_\omega}=1\pm\frac{\Delta\kappa_\omega}{\kappa_\omega}. \tag{2} \]

Let us introduce the notation of relative error in (1) and (2) and find the value of the frequency error and phase for a single ideal four-terminal network:

\[ \delta_\omega=\frac{(m\omega\tau)^2}{2}; \]

\[ \varphi_\omega=\operatorname{arctg}\sqrt{2\delta_\omega}. \tag{3} \]

For \(n\) identical ideal circuits of minimum phase shift, decoupled by amplifiers with a transfer coefficient equal to unity, we shall have

\[ \delta_{\omega n}=\frac{n}{2}(m\omega\tau)^2; \]

\[ \varphi_{\omega n}=n\operatorname{arctg}\sqrt{2\delta_\omega}; \tag{4} \]

From expressions (3) and (4) follows the well-known conclusion that, for minimum-phase circuits, the phase characteristic is considerably steeper than the characteristic of the frequency error.

Let us denote by \(\delta_0\) the error at the frequency \(\omega_0\) and, using relations (3), find the ratio of the normalized frequencies, i.e.,

\[ \frac{\omega}{\omega_0}=\sqrt{\frac{\delta_\omega}{\delta_0}}, \]

then

\[ \begin{aligned} \delta_\omega &= \left(\frac{\omega}{\omega_0}\right)^2 \sqrt{\delta_0};\\ \varphi_\omega &= \operatorname{arctg}\frac{\omega}{\omega_0}\sqrt[4]{4\delta_0}. \end{aligned} \tag{5} \]

It follows from this that, for a given error, the phase characteristic varies according to a linear law, whereas the amplitude-frequency characteristic has a quadratic character.

Let us proceed to consideration of the circuit diagram in Fig. 2, according to which the phase and frequency characteristics of symmetric attenuation boxes were measured. As can be seen from the diagram presented, the voltage from the output of generator \(G\) is applied to the balancing transformer \(\mathrm{Tr}_1\) and to the uncalibrated phase shifter \(NF\), which serves for the initial alignment of the beam on the screen of the electronic balance indicator \(ELUR\).

Fig. 2.

The resistance \(R\), connected into the secondary winding of the transformer, must be equal to the characteristic resistances of the calibrated section \(N_0\) and the attenuation box under test \(N_x\). The comparison of the values of the transmission coefficient of the four-terminal networks is carried out by means of the double switch \(K_1\).

The attenuated voltage at the output of the four-terminal network is fed, through the second balancing transformer \(\mathrm{Tr}_2\), to the input of the amplifier and to the calibrated phase shifter \(GF\) connected in parallel with it. The voltage at the amplifier output is applied to a bridge balancing circuit, in one of whose diagonals a sensitive instrument is connected.

In contrast to existing circuits [3], here a substitution method was used with the same load resistances and a constant insertion attenuation for both paths being checked.

The magnitude of the measured attenuation is compared with the readings of calibrated sections whose frequency error has a known distribution law.

Calibrated sections for fixed attenuation were used as reference dividers. To reduce electromagnetic coupling in them, electrostatic shielding is provided. The frequency error and phase shifts of the sections in the operating frequency range are so small that they can be neglected without noticeable error. Thus, for example, for a section with attenuation of THREE nepers, the frequency error at \(f_b = 600\) kHz is \(0.05\%\), and the corresponding phase shift is no more than \(\varphi \leq 1^\circ 30'\). Consequently, the calibrated sections are purely active dividers and serve as standard elements in the verification circuit.

The substitution method is simple in measurement, gives high reading accuracy, and makes it possible not to take into account the errors introduced by transformers, amplifiers, and other auxiliary elements.

Balancing of the verification circuit was carried out by means of balancing transformers TP1 and TP2 with multifilar winding of the windings [4]. The roll-off of the frequency characteristic of the transformers at the limiting operating frequencies is no more than \((2 \div 3)\) dB, and the total phase shift introduced by them does not exceed on the order of \(15^\circ\).

Recording of the frequency and phase characteristics was carried out simultaneously with the aid of amplitude and phase indicators.

Fig. 3.

As the first indicator, a bridge balancing circuit with a photoresistor of type SF-2 and an electric measuring instrument at the output was used. The sensitivity of the indicator is not less than \(0.3\%\). The role of the amplifier load is performed by an incandescent lamp illuminating the resistor. The amplifier device provides the required signal level at the load for different attenuation values of the attenuator under test and of the reference attenuator.

The phase characteristics were recorded by the method of interference figures with the aid of two phase shifters and a cathode-ray indicator with a sensitivity of not less than \(0.5 \dfrac{\text{mm}}{\text{V}}\)—at maximum gain of the oscilloscope sweep channels along the \(x\) and \(y\) axes. The magnitude of the phase angle was calculated by the following formula:

\[ \varphi_\omega = \operatorname{arctg} \frac{\omega c_2(\tau_2-\tau_1)} {\omega^2 \tau_2(\tau_1 c_2+\tau_2 c_1)+(c_1+c_2)'}, \tag{6} \]

where \(\tau_1, \tau_2\) are the time constants of the transient circuits of the calibrated phase shifter.

For the attenuation box, the phase shifts will be expressed as the difference of readings obtained on the calibrated phase shifter, i.e.,

\[ \varphi = \alpha_1-\alpha_2, \]

Fig. 4

Visible plot labels: \(f\) (kHz); curves \(1, 2, 3\) and \(1', 2', 3'\).

Fig. 5

Visible plot labels: \(\varphi^\circ\), \(\delta_a(\%)\); \(K\) (nepers); \(f = 500\) kHz; \(f = 400\) kHz; \(f = 300\) kHz.

where \(\alpha_1\) is the value of the resistance \(R\) for zero phase shifts of the auxiliary elements in switch position P1;

\(\alpha_2\) is the value of the resistance \(R\) that also corresponds to a zero phase shift in position P2.

From the vector diagram presented in Fig. 3 it is evident that the phase shifter makes it possible to compensate phase angles for different division ratios.

The results of measurements for different attenuations in the operating frequency range are shown in Fig. 4. Curves 1, 2, and 3 represent the amplitude–frequency characteristics, while curves \(1'\), \(2'\), and \(3'\) represent, respectively, the phase–frequency characteristics. Fig. 5 gives plots of the variation of the phase angle and of the frequency error as functions of the division ratio at constant frequency values. As follows from the plots (Fig. 4), the frequency and phase characteristics measured in the frequency range from 50 to 900 kHz have different forms of increase.

Thus, for frequencies not exceeding 150 kHz, the phase characteristics are steeper than the characteristics of the frequency error and obey the expressions obtained for the transfer coefficient and its argument as applied to minimum-phase circuits. In this frequency region the reactive conductances have only a weak influence on the constant transmission of the four-terminal network, and phase relationships can be used to check the frequency error of symmetrical attenuators.

In the frequency range above 150 kHz, the frequency characteristics are located at a larger angle of inclination than the phase characteristics; that is, the attenuator, in its properties, belongs to non-minimum-phase circuits.

The frequency error of the four-terminal network in this frequency range is considerably easier to estimate from the frequency characteristic.

References

1. G. V. Voishvillo. Low-Frequency Amplifiers Using Electron Tubes. Svyazizdat, 1959.

2. N. N. Solov’ev. Fundamentals of Measurement Engineering in Wire Communication. Gosenergoizdat, 1963.

3. G. Bode. Network Theory and Feedback Amplifier Design. IL Publishing House, 1948.

4. A. L. Grokhol’skii. Method of manufacturing transformers with distributed parameters, Author’s Certificate No. 164892, 1962.