Determining transfer function from Bode plot

A Bode plot can be read "backwards" to find the transfer function of a minimum-phase system. The magnitude asymptotes give the gain, type and corner frequencies; the phase plot confirms them.

Procedure

1. Draw straight-line asymptotes on the measured magnitude curve, using slopes that are multiples of ±20 dB/decade.
2. Initial (low-frequency) slope gives the type:
   • 0 dB/dec: type 0 (no pole at the origin);
   • −20 dB/dec: one pole at the origin;
   • −40 dB/dec: two poles at the origin.
3. Find the gain K:
   • Type 0: the low-frequency level is $20\log K$.
   • Type 1: the −20 dB/dec line (extended if needed) meets 0 dB at ω = K. Equivalently, $K = \omega \cdot 10^{M/20}$ at any point ω on it, where M is the magnitude in dB.
   • Type 2: the −40 dB/dec line meets 0 dB at $\omega = \sqrt K$.
4. Corner frequencies: at every break in slope:
   • −20 dB/dec change: simple pole $1/(1 + s/\omega_c)$;
   • +20 dB/dec change: simple zero $(1 + s/\omega_c)$;
   • −40 dB/dec change: double pole or quadratic $1/(1 + 2\zeta s/\omega_n + s^2/\omega_n^2)$, with $\omega_n$ = corner.
5. Damping of a quadratic factor: measure the resonant peak above the asymptote at $\omega_n$. Then $M_r = 1/(2\zeta\sqrt{1 - \zeta^2})$, or approximately $20\log\frac{1}{2\zeta}$ dB at $\omega_n$.
6. Check with the phase plot: for a minimum-phase system the phase must agree with the factors found. For example, the phase approaches $-90^\circ \times (\text{number of poles} - \text{zeros})$ at high frequency. Extra phase lag with no extra magnitude change indicates a transport delay $e^{-sT}$ (the slope of phase vs ω gives T) or an RHP zero.
7. Write the result in time-constant form, then convert to pole-zero form if needed.

Short example

The initial slope is −20 dB/dec, and its extension crosses 0 dB at ω = 10; the slope breaks to −40 dB/dec at ω = 5:

$$G(s) = \frac{10}{s(1 + s/5)} = \frac{50}{s(s+5)}$$

Limitations

The method assumes minimum-phase behaviour, and asymptotes give only approximate corner frequencies. Accurate values come from fitting the exact curve (e.g. −3 dB at a simple-pole corner).