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Basic Electronics » Frequency Response and stability, how do you determine if your system is stable

April 01, 2012
by Akeshish
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Hello everyone,

From my understanding you system will not be stable(start to oscillate) if you have a 180 degree phase shift when your gain is 0 (or 1db).

My question is why does this happen (physically with-in the circuit) i understand the math behind it. Im just trying to visually understand whats happening within the circuit to cause this effect.

Im guessing it has to do with the capacitance and inductance!

April 01, 2012
by hevans
(NerdKits Staff)

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Hi Akeshish,

This is a very deep question. If you understand why the math tells you that that a system is unstable with the above criteria then you are doing really well.

If you want a physical idea of why this happens here is a slightly hand wavy explanation. Assume you have a circuit with negative feedback where the feedback path has 180 degrees of phase at unity gain at frequency 100hz. Now put a 100hz sine wave into it. If you take a sine wave, shift it by 180 degrees and subtract it from the original (because its negative feedback) you end up adding the same sine wave to itself. Since you are in a feedback loop your sine wave now twice as large goes around the feedback path again, gets shifted and added to itself again.

From a circuit perspective it might help to explore how capacitors and inductors introduce a phase shift to a voltage (or current signal). This simple circuitlab example https://www.circuitlab.com/circuit/64ayuv/rc-time-domain-phase-shift-analysis/ can help you explore that. Run the time domain simulation, and notice how there is a 90 degree phase shift from the input to the output. If you sit and play with the equation of a capacitor (I=C*dv/dt) you can see the same phase shift come out of the math. So yes, capacitors and inductors do in fact introduce a phase shift, but they are certainly not the only elements that can change the phase of your signal.

April 01, 2012
by Akeshish
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thank you very much

April 01, 2012
by Akeshish
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ohh, by the way from your example wouldnt having a phase shift great than 180 to 359 degrees at unity gain also cause your system to be unstable? This is where im getting mixed up.

April 02, 2012
by hevans
(NerdKits Staff)

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Hi Akeshish,

That's correct, having a phase of more than 180 degrees at crossover will result in an unstable system. I encourage you to not get too caught up thinking of the feedback loop in the time domain (my example was only mean to give you an idea of why the feedback loop blows up) and instead think of these systems in the frequency domain, and understand the math!

Humberto

April 02, 2012
by Akeshish
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thanks for the response, i think i get it. Lets say i have a circuit with three capacitors. I know that R = 1 / jwC or R= 1/SC. I can find the magnitude and phase of each capacitor based on frequency of the system.

The one thing im having trouble understand is that if the capacitors are in parallel does the phase shift add together? If the capacitors are in series what happens to the phase shift (add, subtract, are in parallel)?

April 02, 2012
by Akeshish
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Never mind i think i got it. you can take the magnitude and phase of your transfer function in order to find what im looking for.

Thanks again

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