# Inverted Pole : Pole-Zero Diagram

When analyzing a Bode plot or its transfer function the technique of "inverted poles" is sometimes used. I first became aware of this at lecture decades ago by Dr. R. David Middlebrook. While the basic method of using inverted poles in the mathematics is somewhat straightforward I don't recall ever seeing how you would itemize that on a pole-zero diagram. Anyone seen this?

## Answers and Replies

I have the Middlebrook "Technical Therapy for Structured Analytical Design" DVD and the GFT DVD at home - in Florida. However I am presently at camp in SW NY. But as I was reviewing material on Bode Plots, especially using online information from Professor Erickson at Colorado it occurred to me that while I see how the equations and bode plots work with this method I never saw a Pole-Zero plot when using the inverted pole/inverted zero method.

Erickson gives an example of this technique: http://ecee.colorado.edu/~ecen2270/materials/Bodenotes.pdf

Basically having a transfer function such as Erickson uses for example: A = A0 (1 + s/w1)/(1 + s/w2) ; A0 being the DC gain we can re-write with a little algebra
A = Ahf (1 + w1/s)/(1 + w2/s); here Ahf is A high frequency or A infinity. The numerator is an inverted zero and the denominator is a inverted pole, inversion being w1/s instead of the non-inverted s/w1. Well, all this makes good sense and bode plots and math are relatively easy to follow. You are referencing gain to the high frequency gain instead of the low frequency gain. Sometimes you will have mixed poles and inverted poles and mixed zeros and inverted zeros.

So I muse - how do you put both poles and inverted poles on the same pole-zero diagram?

I have the Middlebrook "Technical Therapy for Structured Analytical Design" DVD and the GFT DVD at home - in Florida. However I am presently at camp in SW NY. But as I was reviewing material on Bode Plots, especially using online information from Professor Erickson at Colorado it occurred to me that while I see how the equations and bode plots work with this method I never saw a Pole-Zero plot when using the inverted pole/inverted zero method.

Erickson gives an example of this technique: http://ecee.colorado.edu/~ecen2270/materials/Bodenotes.pdf

Basically having a transfer function such as Erickson uses for example: A = A0 (1 + s/w1)/(1 + s/w2) ; A0 being the DC gain we can re-write with a little algebra
A = Ahf (1 + w1/s)/(1 + w2/s); here Ahf is A high frequency or A infinity. The numerator is an inverted zero and the denominator is a inverted pole, inversion being w1/s instead of the non-inverted s/w1. Well, all this makes good sense and bode plots and math are relatively easy to follow. You are referencing gain to the high frequency gain instead of the low frequency gain. Sometimes you will have mixed poles and inverted poles and mixed zeros and inverted zeros.

So I muse - how do you put both poles and inverted poles on the same pole-zero diagram?
Well, I guess I will answer my own question. There is no special symbol that I know off for an inverted pole on a pole-zero diagram. However, an inverted pole is equivalent to a real pole and a real zero at the origin. Apparently no one goes to that detail but instead just use the method with transfer functions and bode plots to simplify the analysis. So I think this thread can be closed.