Transfer Function Conversion

Question

I am working on a system which comprises an inverter with output LC filter feeding an induction motor

Its trivial to calculate the resonant frequency of an LC filter, when the filter is loaded with an induction motor the frequency response is altered (no surprise here!)

Looking at this paper

https://pdfs.semanticscholar.org/40b4/76cc10d10cad506870fad6c4baad82a87bd2.pdf

They treat the motor as purely reactive and calculate the Thevenin equivalent which works quite well

I am trying to calculate the effects more accurately, so I use the equivalent circuit for the motor to derive a transfer function which is then put in parallel with the filter components with the clearf goal of reducing it all to one R one L and one C which then allows me to calculate things like the -3db point

I have written a Matlab script which generates a transfer function Z3 but no matter what I try to do I can not reduce the transfer function to R + Xj

In fact I am confused by the fact that if I substitute s = jw I end up with the equivalent resistance depending upon the supply frequency

clc
clear all
close all

syms    C L R Lm Le1 Le2 Rs Rr 
s=tf('s');

V = 72;
f = 1200;

 Rs = 1.4;              % Stator resistance (ohms)
 Rr = 0.918;            % Rotor resistance (ohms)
 Rc = 23.87;            % Core loss resistance (ohms)
Le1 = 2.32e-3;          % Stator leakage reactance (H)
Le2 = 2.32e-3;          % Rotor leakage reactance (H)
 Lm = 25.25e-3;         % Magnetising reactance (H)
 Lr = Lm + Le2;
 Ls = Lm + Le1;
 P = 4;                  %Poles
Rn = 100000;


% Filter
R  = 0.05;               %Parasitic resistance
C  = 10e-6;              %Set C 
L = 2.94e-3;             %Set L

Z1 = Rs + s*Le1 + s*Lm*(s*Le2 + Rr)/(s*Lm + s*Le2 + Rr);

Z2 = R + s*L;

Z3 = Z1*Z2/(Z1 + Z2);

Zc = 1/(s*C);

Hs = Zc/(Z3 +Zc);

Vos = minreal(Hs)



W=logspace(0,7,40000);
h=bodeplot(Vos,W); grid;
p = getoptions(h);           %Create a plot options handle p.
p.FreqUnits = 'Hz';          %Modify frequency units.
setoptions(h,p);             %Apply plot options to the Bode plot and 
                             %render.  

And if we type into Matlab

minreal(Z3)

  0.00177 s^3 + 0.9529 s^2 + 34.25 s + 315.6
  ------------------------------------------
             s^2 + 333.9 s + 6538

It gives us the transfer function for everything except the capacitor and my question is how do I convert this transfer function into its equvalent R and L. I tried replacing s with jw but I got no where

Thanks in advance

Answer 1

I am trying to calculate the effects more accurately, so I use the equivalent circuit for the motor to derive a transfer function which is then put in parallel with the filter components with the clearf goal of reducing it all to one R one L and one C which then allows me to calculate things like the -3db point

Why don't you just draw the LC and equivalent load of the induction motor as a circuit for a simulator then ask the sim to do an AC analysis.

The formula you derive will be complex and you should forget about your: -

goal of reducing it all to one R one L and one C

Because it won't happen. Use more appropriate tools.

On another point you equate the induction motor TF to be: -

Z1 = Rs + sLe1 + sLm*(sLe2 + Rr)/(sLm + s*Le2 + Rr)

But I see nowhere in this equation the resistance that represents the mechanical output of motor and slip-factor.