# Linearisation of electrical model in simulink using simulink control design toolbox

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Pamela on 26 Nov 2022
Answered: Joel Van Sickel on 2 Dec 2022
I am trying to linearize a simple system containing a synchronous machine vs an infinite bus (see figure below) using the model linearizer from simulink control design toolbox. I am using the continuous PowerGui solver with variable step.
When I linearise my model by taking the mechanical power as an input and the rotor speed as an output, I obtain the results below, we can see that the linearized model is more damped than the nonlinear simulation.
Whereas if I use the phasor mode, the linearized model is similar to the Nonlinear simulation.
Anyone have an idea? Why I have different linearization results with the continuous mode? I want to absolutely use the continuous solver for my study.
I tried to use different solvers in the model settings (fixed step, variable step, …) but I always obtain the same result. And I checked if my model contains blocks that are linearized to 0, but there is not.
Is there a specific solver for the model lineariser?

Joel Van Sickel on 2 Dec 2022
Hello Pamela,
I think a classical linearization of the PMSM in the time domain around those variables is simply too non-linear of a system for a basic linearizaiton analysis to be useful. The results you are getting confirms this. The issue is that this model is being analytically linearized based on the physics equations in a specific moment of time. This is not a good thing in AC systems, particularly with a motor attatched. The rotor angle, electrical phase, and every other variable are playing into this calculation, so the linearization while technically accurate, is not useful over an extended time frame because as soon as the electrical phase, or rotor angle change, your dynamic equations have changed. You'd have to linearize around all of the different operating points using this approach and basically just end up with the original model you started with. To abstract out much of the system behavior without using phasor techniques, you could try using a frequency response estimation and then fitting a transfer function to that system using system identification. This would give you a behavioral representation of the variables that you are interested in while abstracting out the AC nature of the electrical network. It would have limited accuracy aorund the operating point you linearize at as it true for all linearizations.
Phasor works because the underlying mathematical model is highly linear when converted to Phasor, so linearizing it doesn't change the behavior much. In general, if you are using classical linearizaiton techniques on an AC circuit, you would want to work with phasors.
Regards,
Joel

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