how to determine oscilation frequency?

[liusupeng]

for a multiple pole system, it will oscilate if its phase is more negative than -180 degree. if this happens, it will oscilate at the frequency where the phase is exactly -180 degree?

 

[LvW]

for a multiple pole system, it will oscilate if its phase is more negative than -180 degree. if this happens, it will oscilate at the frequency where the phase is exactly -180 degree?

No, that's not the correct description.

An active circuit of at least second order with feedback will (probably) oscillate if and only if there is one single frequency for which the loop gain (that is the gain around the open loop - not the closed loop gain) has a phase of exactly 360 deg and a magnitude of (slightly) above 0 dB.
However, this is only a necessary condition (Barkhausen criterion), but not a sufficient one. This fact often is not mentioned correctly in textbooks.

 

[liusupeng]

for a multiple pole system, it will oscilate if its phase is more negative than -180 degree. if this happens, it will oscilate at the frequency where the phase is exactly -180 degree?

No, that's not the correct description.

An active circuit of at least second order with feedback will (probably) oscillate if and only if there is one single frequency for which the loop gain (that is the gain around the open loop - not the closed loop gain) has a phase of exactly 360 deg and a magnitude of (slightly) above 0 dB.
However, this is only a necessary condition (Barkhausen criterion), but not a sufficient one. This fact often is not mentioned correctly in textbooks.

Hi LvW,
Thanks a lot. so it oscillates at the frequency where the phase is exactly 360 degree?

 

[flatulent]

This is a simple, useful way of looking at it. More mathematically rigorous, do the root locus plot by varying the gain of the amplifier element from zero to full value. If a pair of poles crosses over into the right half plane before the gain gets to the operating value of the amplifier element, there will be oscillations. The frequency will be the point where the locus crosses the imaginary axis and the amplifier or some other part of the circuit will limit to make the loop gain be the value at the crossing point.

 

[LvW]

The above description from Flatulent is 100% correct.
However, the original question was how to "determine the oscillation frequency".
And in this case it is appropriate and wise to use the loop gain vs. frequency as a criterion. Thus, you can see if and at which frequency self-sustained oscillations could occur.

Added after 11 minutes:

Hi LvW,
Thanks a lot. so it oscillates at the frequency where the phase is exactly 360 degree?

Yes, the phase of the loop gain (which at the desired frequency also must have a magnitude of slightly above unity).

 

[liusupeng]

Hi LvW,
suppose in the plot, at a particular frequency f1, the phase is -360 degree and its magnitude is 3 which is above unity. At another frequency f2 which is slightly above f1, the phase is -365 and its magnitude is 2 which is also above unity. In this case, is it going to oscillate at f1 or f2?

---------- Post added at 09:18 AM ---------- Previous post was at 09:17 AM ----------

The above description from Flatulent is 100% correct.
However, the original question was how to "determine the oscillation frequency".
And in this case it is appropriate and wise to use the loop gain vs. frequency as a criterion. Thus, you can see if and at which frequency self-sustained oscillations could occur.

Added after 11 minutes:



Yes, the phase of the loop gain (which at the desired frequency also must have a magnitude of slightly above unity).

Hi LvW,
suppose in the plot, at a particular frequency f1, the phase is -360 degree and its magnitude is 3 which is above unity. At another frequency f2 which is slightly above f1, the phase is -365 and its magnitude is 2 which is also above unity. In this case, is it going to oscillate at f1 or f2?

 

[FvM]

I see at least two points:

There can't be a stationary oscillation under the said conditions. After an initial stimulation (e.g. switch-on transient, arbitrary non-zero initial condition, noise) the amplitude of the transient solution rises until it reaches a stable condition, where a more general non-linear oscillation condition can be met.

Due to the non-linear system properties, the final solution (t-> infinity) isn't necessary stationary, also the oscillation frequency must not be determined unequivocally. For a particular system with multiple oscillation frequency candidates, there's most likely a preferred frequency, but you'll have to examine the loop's complete phase and magnitude frequency characteristic to find it.

My preferred method to analyse the behaviour is assuming an initial condition and watching the differential equation solution in time domain.

 

[liusupeng]

I see at least two points:

There can't be a stationary oscillation under the said conditions. After an initial stimulation (e.g. switch-on transient, arbitrary non-zero initial condition, noise) the amplitude of the transient solution rises until it reaches a stable condition, where a more general non-linear oscillation condition can be met.

Due to the non-linear system properties, the final solution (t-> infinity) isn't necessary stationary, also the oscillation frequency must not be determined unequivocally. For a particular system with multiple oscillation frequency candidates, there's most likely a preferred frequency, but you'll have to examine the loop's complete phase and magnitude frequency characteristic to find it.

My preferred method to analyse the behaviour is assuming an initial condition and watching the differential equation solution in time domain.

Hi FVM,
thanks for your reply! what i mean is that f1 and f2 are very very close. and f1 and f2 all satisfy phase and gain requirement for oscillation. We can say that the oscillation frequency is around f1 or f2 because they are very close. but the question is how to determine the exact oscillation frequency.

 

[FvM]

the question is how to determine the exact oscillation frequency

As said it can't be determined without knowing the phase and magnitude characteristic and the non-linear properties of the system.

If both frequencies are very close, your phase and magnitude specification sounds very unlikely. If it's a real system though, you should give a circuit or an equivalent circuit of it.

is it going to oscillate at f1 or f2

Neither of it, most likely.