Bode Plot - Calculating ωgc and ωpc analytically

[phiby]

I am learning to draw Bode Plots. I am able to figure out ω_gc, ω_pc, Phase Margin & Gain Margin graphically from the Bode Plot. But I was wondering if there is a way to calculate ω_gc and ω_pc mathematically with some formulas - how do I do this?

 

[dimensionless]

There is the Laplace transform that converts a time domain equation to an "s" domain equation, where

$$s = i\omega$$
or
$$s = \sigma + i\omega$$

In this case, the time domain equation might be some differential equation that defines the behavior of a filter.

 

[phiby]

I calculated ω_gc analytically & compared it to the one I got from a Bode plot.

When Gain is rather low (say 2), then the calculated ω_gc varies a lot from the one obtained from an asymptotic Bode Plot.

For eg.

let's take
G(s)H(s) = 80/(s)(s+2)(s+20)

= 2/(s)(1 + 0.5s)(1 + 0.05s).

In this case the ω_gc is very close to where the approximation error happens for the first cornering frequency (2 rad/s - corresponding to (1 + 0.5s)).

My calculated ω_gc = 1.57 rad/s.
The one on the graph (where the Magnitude plot intersects 0) is around 2 rad/s.

On a semilog paper the horizontal distance in the 2 lines (ω = 1.57 & ω = 2) is rather big.

Is this a known issue?

 

[yungman]

I use Bode plot for years to design all sort of closed loop control systems and I consider myself pretty good at taming them. I never get into the s-plane stuff. Bode plot and the ωgc is almost two different thing, you draw your Bode plot from knowing the pole and zero frequency of the system, not the other way around. When you use Bode Plot, you try to avoid all the calculation and use graph to design the system. If you want to do calculation, don't use Bode Plot.

When you design a closed loop system, you measure the system poles and zeros and design the amplifier circuit with poles and zeros to get single pole cross over with enough phase margin. Identifying the system poles and zeros are the most difficult part.

When you identify all the poles and zeros of the system, your job is over 90% done, the rest is just defining which part belong to the forward gain and which part be the reverse feedback and draw the Bode plot and add your own gain, poles and zeros to get the single pole cross over with phase margin!

 

[reddvoid]

i remember calictlating phase cross over frequency by equating argument of open loop transfer function to -180 degrees and gain cross over frequency by equating magnitude to 1. . .