Hi,

I have a transfer function in the s-domain that looks like

\(TF = \frac{e^{\frac{-s}{25000}(s+2)}}{(s+9)}\)

I'm having trouble determining what is the meaning of the term \(e^{\frac{-s}{25000}\). Is this a zero?

Furthermore, I can't quite get a grasp on how I would plot this in MATLAB using the transfer function call.

Any and all suggestions are appreciated.

tia

 

It's a transport delay term - or pure time delay in time domain. One approach in analysis is to approximate the pure delay term with a function such as the Pade approximation.

 

Check the attachment ...

 

The effective dead time in your TF is quite small - the order of 40uS. I doubt this is going to have any significance given your pole & zero are at quite low frequencies. I'd need to check with some actual plots however before saying that the dead time could be ignored in an analysis when this system was used in a closed loop feedback control case.

An additional problem when using the Pade approximation is deciding whether the resulting closed loop behavior truly reflects the case when pure dead time is the reality.

 

I had a go at using a 4th order Pade approximation type R(n-1),Rn(s) with T=40usec as the pure delay approximation in your TF.

The open loop gain margin for the TF is -0.68dB and the phase margin -149.6°. This suggests an unstable system would result with a case of simple closed loop unity feedback. This was confirmed by plotting the closed loop response.

Just goes to show one can't make any generalizations without undertaking a careful analysis.

 

I had a go at using a 4th order Pade approximation type R(n-1),Rn(s) with T=40usec as the pure delay approximation in your TF.

The open loop gain margin for the TF is -0.68dB and the phase margin -149.6°. This suggests an unstable system would result with a case of simple closed loop unity feedback. This was confirmed by plotting the closed loop response.

Just goes to show one can't make any generalizations without undertaking a careful analysis.

Can a system be unstable with loop gain<1?

 

Yes - I know conventions can often be a distraction. In the software I use, a negative gain or phase margin in the open loop transfer function bode plot indicates the likelihood of an instability in direct [-ve] feedback closed loop operation. A positive value in both cases [gain & phase margin] would normally indicate a conditionally stable case in the aforementioned closed loop operation.

So a gain margin of -0.68dB [as I quoted] means the gain is 0.68dB above 0dB at the open loop transfer function -180° phase transition point.

Hope that makes sense.