Zeros and poles and phase shift

Effects on the deflection points in the gain plot to the phase plot

• Poles and Zeros: In the transfer function of a linear system, poles are values of \( s \) (in the Laplace domain) that make the transfer function go to infinity, while zeros are values of \( s \) that make the transfer function zero.

• Effect of Poles on Phase: Each pole in the transfer function typically contributes a -90-degree phase shift as frequency increases. This phase shift is gradual and starts before the pole's frequency, reaching -90 degrees some distance after the pole's frequency. If there are multiple poles, their phase contributions stack. So, two poles can contribute up to -180 degrees, three poles up to -270 degrees, and so on.

• Deflection Points in the Gain Plot: A deflection point on the gain plot typically corresponds to the frequency of a pole or zero. At a pole, the gain plot starts to slope downwards more steeply, showing an increase in attenuation with frequency.

• Phase Plot Reaching -180 Degrees: When the phase plot indicates -180 degrees at certain frequencies, it means that at these frequencies, the output signal of the system is inverted relative to the input signal. If the system's gain is still high (above 0 dB) at these frequencies, it can result in positive feedback if used in a feedback loop, leading to potential instability.

• Lack of Compensating Zeros: If the deflection points in the gain plot (indicating poles) correspond to where the phase plot reaches -180 degrees, and there are no zeros to counteract this phase shift, it suggests that the system's output will be significantly out of phase with the input at these frequencies. Zeros can contribute positive phase shifts, partially negating the phase lag introduced by poles. Without zeros to compensate, the system is at risk of instability because the total phase lag at these frequencies can lead to a situation where positive feedback occurs, meeting the criteria for instability (gain > 0 dB when the phase shift is -180 degrees).

Phase shift of -180 degree cause the instability

1. Feedback Loop: In a typical feedback control system, a portion of the output is fed back and subtracted from the input. This negative feedback is intended to stabilize the system, improve accuracy, and reduce sensitivity to parameter variations.

2. Phase Shift and Signal Inversion: A phase shift of -180 degrees means that the output signal is inverted relative to the input signal. If this inverted signal is fed back and subtracted from the input, it effectively becomes positive feedback because the inverted signal (180 degrees out of phase) and the negative sign from the feedback loop combine to add the output back to the input.

3. Barkhausen Stability Criterion: For a system to oscillate, it needs to satisfy two conditions: the loop gain must be equal to 1 (or 0 dB in logarithmic scale), and the net phase shift around the loop must be an integer multiple of 360 degrees (or 0 degrees modulo 360). This implies that a signal can circulate around the feedback loop and come out with the same amplitude and phase as it started with, thus sustaining the oscillation.

4. Instability at -180 Degrees: When the phase shift reaches -180 degrees, and the system also has a gain greater than or equal to 1, any small disturbance or noise can be amplified and inverted in phase each time it circulates through the loop. This creates a condition where the signal can build up indefinitely, leading to sustained oscillations or even to a runaway condition where the output grows without bounds, signifying instability.

5. Practical Implications: In practice, to ensure stability, designers aim for a phase margin that is greater than 0 degrees at the frequency where the gain crosses 1 (or 0 dB). A phase margin of 45 degrees is often considered a minimum for stability, with a larger phase margin providing a more robust system.

Why does the phase shift of -180 degrees cause the invert of the output signal relative to the input signal

1. Phase Shift Interpretation:
• Phase shift is a measure of the relative timing between two periodic signals.
• A phase shift of 0 degrees means the two signals are in sync; their peaks and troughs align perfectly.
• A phase shift of 180 degrees means the two signals are in complete opposition; when one signal is at a peak, the other is at a trough.

2. Waveform Behavior:
• Consider a sinusoidal waveform, which is a common representation of AC signals.
• If the waveform is shifted by 180 degrees, every point on the wave is effectively flipped across the time axis. Where there was a peak, there is now a trough, and where there was a zero crossing from positive to negative, there is now a zero crossing from negative to positive.

3. Signal Inversion:
• In the time domain, this phase shift translates to an inversion of the signal. For example, a sine wave described by \( A\sin(\omega t) \) when shifted by 180 degrees becomes \( A\sin(\omega t + \pi) \), which is equivalent to \( -A\sin(\omega t) \) because \( \sin(\theta + \pi) = -\sin(\theta) \).
• This inversion is particularly meaningful in electronic circuits where the phase relationship between voltage and current determines power flow direction.

4. Implications in Feedback Loops:
• In a feedback loop, the expected outcome of negative feedback is that any deviation in the output from the desired condition is fed back into the system in such a way that it corrects the error.
• If the feedback signal is inverted (180-degree phase shift), then instead of correcting the error, it reinforces it, leading to a potential increase in the error.

5. Stability and Oscillation:
• If this inverted signal is not properly accounted for, it can lead to oscillations because the system tries to correct in the wrong direction, thereby magnifying the deviation with each loop cycle.
• If the system also has a gain of 1 or more at the frequency where the phase shift is -180 degrees, the conditions for oscillation (as per the Barkhausen criterion) are met, and the system can become unstable.