# Clarke transform of pwm signals

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
• Apply a suitable LPF before the Clarke transforms.
– user16324
Commented Sep 30, 2020 at 13:23
• @BrianDrummond thank you for your reaction. I have done an experiment with moving average filter (please see the edit). Neither in this case the outcomes are as I expected. Can you tell me what is the idea behind your suggestion? Commented Sep 30, 2020 at 13:31
• @Steve As mentioned above, you need better filtering, or try an FFT -- that will surely spell out the components (a simple test with LTspice). Normally this is not done for PWM waveforms. If you meant sampled waveforms, while PWM is a form of sampling, that's not what it refers to. Commented Sep 30, 2020 at 13:54
• That's not much of a filter. Nevertheless you can see the waveform start to emerge and the LF content is substantially what you expect. The HF is irrelevant.
– user16324
Commented Sep 30, 2020 at 13:55
• @BrianDrummond the reason why I have decided to use the moving average filter is its linear phase characteristics. I will do some experiments with wider window. As far as I understand correctly your main idea the outcomes of the Clarke transform for the unfiltered pwm signals are correct. The deformation of the beta component is caused by the harmonics? Commented Sep 30, 2020 at 14:06

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.