Relationship between bode plot and pole zero diagram
What is the relationship between the pole zero diagram and bode plot?
Case example: a 2nd order RLC low pass filter: -
$$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{\frac{1}{j\omega C}}{R+j\omega L +\frac{1}{j\omega C}} = \dfrac{1}{j\omega RC +j^2\omega^2 LC +1} = \dfrac{\frac{1}{LC}}{\frac{1}{LC}-\omega^2 +j\omega\frac{RC}{LC}} $$
Or, in "s" terms: -
$$\dfrac{V_{OUT}}{V_{IN}} =\dfrac{\frac{1}{LC}}{s^2 + s\frac{RC}{LC} + \frac{1}{LC}} $$
Therefore we know the poles are defined when the denominator equals zero and: -
$$s = \dfrac{-R}{2L} ± j\sqrt{\dfrac{1}{LC} - \dfrac{R^2}{4L^2}}$$
But how does that affect the bode plot?
1 answer
For any 2nd order low pass filter, the components (R, L and C for electrical and k, M and c for mechanical) can be reduced into two more meaningful quantities: -
- $\zeta$ (the damping ratio)
- $\omega_n$ (the natural resonant frequency)
Concentrating on the RLC circuit in the question, these relationships hold: -
- $\zeta = \dfrac{R}{2}\sqrt{\dfrac{C}{L}}$ and,
- $\omega_n = \dfrac{1}{\sqrt{LC}}$
Bode plot
The amplitude section of the bode plot will be similar to this: -
3-D picture introducing pole-zero diagram
The bode plot and pole zero diagram are combined. Only the positive pole is shown for clarity: -
Traditional pole zero diagram
Looking down onto the above 3-D picture shows the traditional pole zero diagram (both poles shown): -
Pole zero geometry and |H(s)|
If you know the pole positions (or the zero positions) you can predict the bode plot magnitude by calculating the distance from each pole (or zero) to any particular point on the bode plot. This reveals the magnitude along the jω axis. Note that the red dot below is a variable point on the jω axis that is required to be calculated. The geometry of the pole zero diagram is examined: -
The reciprocal of d1⋅d2 gives you the magnitude of the bode plot at any point on the jω axis. If there was a zero involved (at a distance n1 to the particular point on the jω axis), the magnitude would be the reciprocal of: -
$$\dfrac{d_1⋅d_2}{n_1}$$
Some pictures from here.
Pole zero diagram and phase response
The phase response is derived from the pole zero diagram. Here it is using the conjugate pole example from above: -
But, of course, you could just take the transfer function and mathematically derive phase and magnitude response without reference to the geometry of the pole zero diagram.
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