Poles & Zeroes: I Understand, Mostly

When I was in college, I was taught about AC circuits, Bode plots, Laplace transforms, and pole-zero plots on the s-plane — all to help me analyze those AC circuits.

Sadly, only the Bode plots analysis really stuck. Every now and then, I take out one of my college texts and read up on the things I should know but don't quite. Mostly, this means I try to understand pole-zero plots.

My brother, who is a mechanical engineer and therefore not as well developed as I am from an engineering perspective, sent me a document that was published by MIT. It explains the pole-zero plots and ties the plots into real-world systems. It also shows system responses to step-function stimuli. With this, I started to really understand why some systems were not stable. And I developed a real appreciation of that joke about poles on the right side of the plane.

Unfortunately, I don't use this knowledge very often, so that understanding that I had seems to seep away. Much like a foreign language that you learned back in high school, if you don't use it, you lose it.

The MIT document to which I referred can be found here. One figure that helped a lot is shown below:

Pole and zero locations and the corresponding transient response

Pole and zero locations and the corresponding transient response

Just this one figure helped me a ton. I intuitively understood the transient waveforms shown and the sort of circuitry that will produce those waveforms. Now I understood the associated s-plane graphs, too.

I am not doing as much circuit analysis now as I was 20 years ago, so again, this knowledge is slipping away. I'll need to reread the document cited above about once every six months to stay sharp.

Let me know in your comments what you wish you could understand better. And let me know what resources you've found that help explain things or help you stay sharp.

9 comments on “Poles & Zeroes: I Understand, Mostly

  1. shovelit
    January 28, 2013

    True, I haven't used the concept of poles and zeroes since I no longer do analog filter design. Basically, it was a great tool for visualizing the stability of a circuit.Adding a capacitor would move the poles into a stable region. The only thing that really stuck in my mind is don't go outside the unit circle.

  2. Clyde
    January 29, 2013

    Digital filtering requires the use of poles and zeros in the Z-domain. Or, am I incorrect?

  3. shovelit
    January 29, 2013

    Yes. The Z transform is done to obtain the co-effecients to be used in the digital implementation of the filter. It would be complete if we design an anolog hi -pass filter. Let's say 60 Hz. Then bring the design over into the digital realm with a sampling rate of 40khz. This meets the nyguist criteria for the audio range, but we might have to use a decimated digital filter due to the lo range. OK, you start! Lets make it a 2 pole filter with 12 db roll off.

  4. PierreBTOL
    January 30, 2013

    Thank you Brad for this article and valuable MIT link !

  5. Man21
    January 31, 2013

    Dear Brad,

    Many years ago I realised the need to understand circuit analysis better and found the best way to learn, understand and remember the techniques was to write a book. This really does illuminate ones lack of understanding. When you have to write a coherent account the gaps become apparent. The present version is found under ISBN 978-0-521-69780-5 and you will find a slightly expanded diagram, similar to your quote from the MIT note, on p67. There are examples of how use of this representation of circuit properties helps in understanding of circuit function and design.

    Scott Hamilton.

  6. davebirdieee
    January 31, 2013

    The MIT link is a really good link, especially in understanding sinudoidal oscillators. But, many oscillators depend on the natural limitations of the non-linear circuit elements to limit the amplitude. That's fine when it happens, but sometimes one would like to design in some form of guaranteed starting, and amplitude regulation. One concept is to regulate the position of the real part of the system response back and forth across the imaginary axis to make the response maintain itself exactly on the axis. Question is, this is inherently non-linear. At least one can't usefully use the existing system respsonse analytically to do it. At least I don't know how to do that. Maybe others do.

    For those who design analog/linear sinusoidal oscillators, anyone who could point out some useful links as to how to handle the amplitude regulation would be highly appreciated.



    BTW, we need some more tags to get linear oscillators. This seems to be a forgotten topic.

  7. Brad Albing
    January 31, 2013

    Dave – interesting idea. Don't know if it can be done beyond tweaking the gain of the amplifier (that is part of the oscillator) with an active element (think JFET or incandescent lightbulb). We'll see if anyone else has a suggestion.

    Meanwhile, I do plan on tweaking my list of key words to make it more comprehensive and suitably specific.

  8. davebirdieee
    February 1, 2013

    Yes, I've done bulb, and JFET to stabilize a Wien Bridge. The quadrature oscillator is easy to regulate with a hard limit and the design equations are so easy that you know exactly what you are doing. Discrete transistor type phase shift oscillators and RC oscillators can be a bear though. Those are low Q, which has something to do with it…


  9. Brad Albing
    March 28, 2013

    That's good advice. There are easy to use sim-programs that let you tinker w/ component values and watch what happens to the pole and zero locations and to the freq and phase response. Sweet.

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