ON THE APPLICABILITY OF INTERPOLATION FOR ESTIMATING FREQUENCY DISTORTIONS OF PASSIVE THREE-TERMINAL NETWORKS

M. S. ROITMAN

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

At the present time, in connection with the improvement of measuring equipment and the extension of its frequency range into the region of ever higher frequencies, the reliable determination of very small frequency distortions (FD) of multi-terminal networks is urgently necessary. Especially high requirements are imposed on the frequency properties of voltage dividers of standard calibration devices, where the permissible distortions are estimated in thousandths of a percent. Direct determination of such quantities for multi-terminal networks with a transmission coefficient different from 1 does not appear possible. This has led to the fact that, for estimating small FD, the interpolation method is most often used. Its essence is that the FD are assumed proportional to the square of the frequency and the distortions are determined at higher frequencies, after which the error at the given frequency is found by recalculation.

For aperiodic minimum-phase circuits this approach is entirely justified and very effective. However, before using it, in order to avoid errors one should make sure, and make sure experimentally, that the given circuit is aperiodic and minimum-phase.

The necessity of experimentally determining the character of the circuit is dictated by the fact that the structure of even a comparatively simple circuit becomes complex at high frequencies. Any three-terminal network (practically all unbalanced resistance dividers are three-terminal networks) with a complex internal structure can be represented in the form of series-connected sections. The total phase shift of the circuit is

\[ \varphi_\Sigma=\sum_{n=1}^{m}\varphi_n, \]

and the frequency distortions

\[ \alpha_\Sigma=M_\Sigma-1=\prod_{n=1}^{m} M_n-1=\prod_{n=1}^{m}(1+\alpha_n)-1, \tag{1} \]

where \(\varphi_n\) is the phase shift of the \(n\)-th section,
\(M_n\) is the coefficient of frequency distortions of the \(n\)-th section,
\(m\) is the number of sections.

Since we are considering circuits with small frequency and phase distortions, in (1) the product may be replaced by a sum

\[ \alpha_{\Sigma}=\prod_{n=1}^{m}(1+\alpha_n)-1 \simeq \left(\sum_{n=1}^{m}\alpha_n+1\right)-1=\sum_{n=1}^{m}\alpha_n. \]

But for any section (aperiodic or resonant),

\[ |\alpha_n|\simeq \frac{1}{2}\varphi_n^2. \]

In the case where the circuits are aperiodic and minimum-phase, then

\[ \alpha_{\Sigma}=\sum_{n=1}^{m}\frac{1}{2}\varphi_n^2 =\sum_{n=1}^{m}\frac{1}{2}(\omega\tau_n)^2 =\frac{1}{2}\omega^2\sum_{n=1}^{m}\tau_n^2, \tag{2} \]

\[ \varphi_{\Sigma}=\sum_{n=1}^{m}\varphi_n =\omega\sum_{n=1}^{m}\tau_n, \tag{3} \]

where \(\tau_n\) is the time constant of the \(n\)-th section.

It follows directly from (2) that the generally known conclusion that the frequency-response distortions of a minimum-phase aperiodic circuit (MPA) have a quadratic character.

The phase shift varies according to a linear law and, consequently, for small frequency-response distortions is expressed much more distinctly. And although for minimum-phase circuits there is a one-to-one relationship between the phase-frequency and amplitude-frequency characteristics, unfortunately from the phase shift \(\varphi_{\Sigma}\) (3) one cannot determine \(\alpha_{\Sigma}\) (2), since the order of the system \(m\) is unknown. There is no contradiction here.

The amplitude-frequency characteristic can be found with acceptable accuracy only when the phase-frequency characteristic is known over a substantial interval, which is equivalent to knowing the order of the system. Let us note once again that (2) is valid only for small distortions.

Thus, in order to make justified use of the interpolation method for finding frequency-response distortions, one should first make sure that the system is MPA. For this purpose it is necessary, at several frequency points (at least two), to determine experimentally \(\varphi_{\Sigma}\) and \(\alpha_{\Sigma}\), and, after comparing the obtained results, to judge the legitimacy of interpolation.