Electronic – Clarke transform of pwm signals

controlinduction motorpower electronics

I have been developing a simulation of field oriented control of three phase induction motor. During this process I have encountered strange behavior of the Clarke transform. In case I have three phase voltage source consisting of three smooth sinewaves with 120° phase shifts between each other

the Clarke transform gives two sinewaves with 90° phase shift

That's what I have expected.

In case I have three phase voltage source consisting of three pulse width modulated signals
(fpwm = 1 kHz) produced by three phase voltage source inverter

the Clarke transform applied to those signals gives following outcomes

Those outcomes are strange for me because I have expected again two signals with the same shape
and with 90° phase shift as was the case for the smooth sinewaves. In both cases I have been using below given matrix multiplication for Clarke transformation

$$
\begin{bmatrix}
x_{\alpha} \\
x_{\beta}
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 \\
0 & \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}}
\end{bmatrix}
\cdot
\begin{bmatrix}
x_{a} \\
x_{b} \\
x_{c}
\end{bmatrix}
$$

Can anybody tell me whether the outcomes of the Clarke transformation in the second case are correct or whether are caused by some mistake in my simulation? In case they are correct why are not in accordance with my expectation i.e. two signals with same shape and with 90 ° phase shift? Thanks in advance for explanation.

EDIT:

In case I use simple moving average filter with window width equal N samples before applying Clarke transform to the pwm signals I get

N = 8
N = 16
N = 32
N = 64

Best Answer

The outcomes of the Clarke transform are correct. The alpha component corresponds directly to the \$u_a\$ voltage and the beta component corresponds to the line-to-line voltage \$u_{bc}\$ scaled by the factor \$\frac{1}{\sqrt{3}}\$. This is also answer to the second part of the question because the line-to-line voltage of the voltage source inverter has different shape than the phase voltage.

Related Solutions

Electronic – How to use Luenberger observer for induction motor rotor flux estimation

I agree that using the \$\omega_m\$ to produce a time-varying observer is a valid and intuitive approach. The "theoretical correctness" of it will mostly depend on two aspects, and both will require some reading and research on your part.

(1) the criterias you want to guarantee for the closed-loop system. This mostly concerns stability guarantees and perhaps some pole-placement objectives.

(2) the design method you will be using (i.e. how the gain coefficients of closed loop are obtained).

My suggestion is that you start by modeling \$\omega_m\$ as an uncertain parameter within \$\omega_{min}\$ and \$\omega_{max}\$, so now your system is "bounded by a polytope". This will produce LMI (linear matrix inequality) constraints that can be used to guarantee quadratic stability of the system. Such constraints can also easily be coupled with \$H_2\$ or \$H_\infty\$ design objectives to produce a convex optimization problem. There are plenty of papers available on the topic if you search using these keywords. Good luck!

Additional question in comments: eigenvalues of the above mentioned system matrix in symbolic form?

You can always use a symbolic calculator to figure this out. Just did it using WolframAplha:

Just replace \$a=-\alpha\$, \$b=\beta\$, \$c=\gamma\omega_m\$, \$d=R_r\frac{L_h}{L_r}\$, \$e=-\frac{R_r}{L_r}\$ and \$f=-\omega_m\$

Electronic – Unexpected behavior of the Luenberger observer for three phase induction motor

Example observer simulation of your motor system + observer, with Scilab XCos:

// Numeric values of model
Rs=1; Rr=1; Ll=1; Lm=1; Wm=1;

// Matrices for states space model
a11 = -(Rs+Rr)/Ll; A11 = [a11 0; 0 a11];
a121 = Rr/(Lm*Ll); a122 = Wm/Ll; A12 = [a121 a122; -a122 a121];
a21 = Rr; A21 = [a21 0; 0 a21];
a221 = -Rr/Lm; a222 = -Wm; A22 = [a221 a222; -a222 a221];
b1 = 1/Ll; B1 = [b1 0; 0 b1]; B2 = [0 0; 0 0];
c1 = 1; C1 = [c1 0; 0 c1]; C2 = [0 0; 0 0];

A=[A11 A12; A21 A22]; B=[B1; B2]; C=[C1 C2];

// Plot transfer poles of the system
poles = spec(A);
plzr(motor_sys)
// 4 poles plotted, therefore motor_sys is fully observable.
// All poles have negative real parts, therefore motor_sys is stable.

// Observer gains using pole placement at 10 times the sys poles real parts.
obs_pp = 10*real(poles);
L = ppol(A', C', obs_pp)';

Append observer gains to the observer system as so:

You will see that, with a null initial state and no disturbances, estimation error will always be zero:

Changing initial state, you should see an initial estimation error, which should quickly decay. Adding random disturbances to the motor system, or intentionally adding modeling errors/non-linearities, you will notice the observer start presenting some steady-state estimation errors, which can be reduced by increasing observer gain, with transient errors (peaking) as a trade-off.

I don't know what went wrong in your simulation, hope this example serves as good starting point.