# How to find the complete transfer function based on phase measurements

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?

• 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. Commented Jan 20, 2022 at 13:37
• 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). Commented Jan 20, 2022 at 21:22