I want to find the transfer function with a certain number of poles of an electronic amplifier circuit based on the same number of phase measurements.
For example, I have 3 phase measurements:
\begin{align}
\phi_{1}&=-45° @ f_{1}&=&&6.3\text{ kHz}\\
\phi_{2}&=-135° @ f_{2}&=&&58.8\text{ kHz}\\
\phi_{3}&=-225° @ f_{3}&=&&211\phantom{.8}\text{ kHz}
\end{align}
And I'd like to find the transfer function in the following form:
$$
H(\omega) = \frac{1}{(1+j\omega\tau_{1})(1+j\omega\tau_{2})(1+j\omega\tau_{3})}
$$
If the poles of the transfer function are far enough apart I could simply approximate the poles as follows:
$$
\tau_{1} = \frac{1}{2\pi f_{1}}
$$
or in general:
$$
\tau_{n} = \frac{1}{2\pi f_{n}}
$$
But if the poles are close together the measured frequency at -45° won't match with the first pole for example.
Since I can get the angle of a transfer function by:
$$
\angle H(\omega) = \tan^{-1}\left(\frac{\Im\left\{H(\omega)\right\}}{\Re\left\{H(\omega)\right\}}\right)
$$
There should be a way to match the curve with my measured points. However, this would involve solving a system of equations containing arctan functions. So what would be the best way to get the poles?
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1\$\begingroup\$ Why make life so difficult; just plot a few more phase angles and add to the list of values the amplitude responses at the extra frequencies too. \$\endgroup\$– Andy akaCommented Jan 20, 2022 at 13:37
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\$\begingroup\$ You have to keep in mind that phase measurement relate to magnitude only if you deal with a minimum-phase system. Should you have RHP zeroes or RHP poles or a pure delay, phase measurements no longer correlate with magnitude (see Kramer-Kronig theory). \$\endgroup\$– Verbal KintCommented Jan 20, 2022 at 21:22
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1 Answer
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On a log-log scale graph paper, the phase of each pole changes +/-45 degrees over a span of +/-1 decade. You can draw this linear asymptotic slope that flattens at the end of each +/- 1 decade.
Using superposition show the overlap within 1 decade of each pole, you can estimate the center of each range which is the -3dB breakpoint by predicting the influence of shifting the phase measurement at f3.
This is because f2 to f3 are less than a decade of f apart.
