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I am trying to understand how loop gain of system determines the stability of a complete system.
Loop gain is given by 1+GH where G is forward transfer function (TF) and H is feedback factor.
If we make loop gain stable then how it is possible that my transfer function is also stable? because the poles of H becomes zeros of transfer function. So the complete systerm response changes?
How I can determine the transfer function shape from loop gain/Characteristic equation?
Consider a G of 1,000,000,000 at 500 degrees, at 100Hz. (actually the frequency does not matter).
Implement an H of 10 (or 100, or 1,000)
This will be stable, despite the huge phase shift. Why?
Running the math on this, we find
The Numerator, of course, is 1,000,000,000 at 500 degrees phase shift
The Denominator, of course (1 + G * H) = (1 + 1,000,000,000/_ 500 * 0.1) which becomes 100,000,001 /_ 500
and the combined transfer function is stable, because the phaseshifts cancel.
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Notice the mag/phase is nowhere near the Nyquist point.
Quote: If we make loop gain stable then how it is possible that my transfer function is also stable? because the poles of H becomes zeros of transfer function. So the complete systerm response changes? How I can determine the transfer function shape from loop gain/Characteristic equation?
In most cases, the loop gain (gain around the complete loop when it is open) is stable. But this does not mean that the closed-loop will also be stable. Stability properties (stability margin) must be checked using the stability criterion which can be applied in different ways: Bode diagram or Nyquist diagram or Nichols diagram).
It is to be noted that there are some rare cases where the lopp gain is NOT stable. In this case, the closed-loop can be made stable.
Concerning your last question:
Yes, the system response changes when the loop is closed. In general, it is not possible to guess how the "shape" of the closed-loop transfer function (magnitude) will look like.
However, in many cases this will be possible (approximately) when the gain of the forward function is pretty large (example: Operational amplifier) because in this case we have [G/(1+GH)]=1/[(1/G)+H] which equals 1/H for G>>1. Hence, the closed-loop function is just the inverse of the feedback function H.
Take, for example, a non-invering amplifier with a beta equal to 0.1 (R1 = 1k, R2 = 9k) operating at a frequency where its open loop gain is equal to 10. Its noise gain = 1/beta = 10 and it is operating at its closed loop -3dB frequency because the open loop gain is equal to the noise gain. Therefore the closed loop gain is equal to 10 * 0.707 = 7.07. If Vin is 1V pk to pk the output voltage will be 7.07V pk to pk. The voltage across the lower arm resistor is equal to 7.07*0.1 = 0.707V pk to pk. The voltage between the amplifier's inputs, Vdiff will be (output voltage)/(open loop gain) = 7.07/10 = 0.707V pk to pk. The loop gain, with out the inversion which occurs in the open loop loop gain measurement, is equal to the voltage across the lower arm resistor divided by Vdiff = 0.707/0.707 = 1.
The feedback fraction is 0.1 and the overall negative feedback factor (1 + beta*Aol) = Vin/Vdiff = 1.414 (Assuming open loop phase lag is -90 degrees).
Also of interest may be the closed loop phase lag which is -45 degrees.
The loop gain is equal to 1 (unity) and because the feedback network is purely resistive it means that the loop phase is equal to the open loop phase which is approx -90 degrees. The phase margin is 180-90 = 90 degrees and therefore the amp is very stable.
