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Given a system with poles at 0.01 Hz, 1 Hz, and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz, the approximate phase of the system response at 20 Hz is

Question

Given a system with poles at 0.01 Hz, 1 Hz, and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz, the approximate phase of the system response at 20 Hz is

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Solution

To calculate the phase response of a system at a specific frequency, we need to consider the phase contributions from each pole and zero at that frequency. The phase contribution from a pole or zero at a frequency f is given by the phase angle, which is the arctangent of the ratio of the frequency to the pole or zero frequency.

Here are the steps to calculate the phase response at 20 Hz:

  1. Calculate the phase contributions from the poles:

    • Pole at 0.01 Hz: arctan(20/0.01) = 89.43 degrees
    • Pole at 1 Hz: arctan(20/1) = 86.41 degrees
    • Pole at 80 Hz: arctan(20/80) = 14.04 degrees

  2. Calculate the phase contributions from the zeros:

    • Zero at 5 Hz: arctan(20/5) = 75.96 degrees
    • Zero at 100 Hz: arctan(20/100) = 11.31 degrees
    • Zero at 200 Hz: arctan(20/200) = 5.71 degrees

  3. Add up the phase contributions from the poles and subtract the phase contributions from the zeros:

    • Total phase contribution from poles = 89.43 + 86.41 + 14.04 = 189.88 degrees
    • Total phase contribution from zeros = 75.96 + 11.31 + 5.71 = 92.98 degrees
    • Phase response at 20 Hz = 189.88 - 92.98 = 96.9 degrees

So, the approximate phase of the system response at 20 Hz is 96.9 degrees.

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