Estimation of an orthogonal component for a single-phase PLL

Question

I'm trying to implement a single-phase PLL, for which operation I need an orthogonal component of the grid voltage.

Let the grid voltage be:

$$v_g(t)=A\sin(\omega t) + \tilde{v}(t),$$

where the latter component is a noise.

I need an estimator that would produce:

$$v_\alpha (t) = A \sin (\omega t), \quad v_\beta (t) = A \cos (\omega t)$$

Now, I've read that it is really challenging to estimate the orthogonal component. Can you tell me why a 1st order filter is not a good idea:

$$G_\alpha(s) = \frac{1}{1+sT}, \quad G_\beta (s) = \frac{1}{\omega}\frac{s}{1+sT},$$

where the filter time constant should be (at least):

$$\frac{1}{T} \geq 10\omega.$$

Note that I have to implement these filters on a microcontroller.

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I've simulated the system in PLECS, and everything seems fine.

Here is a response of the estimation error:

Answer 1

Can you tell me why a 1st order filter is not a good idea

Quite simply, a 1st order filter will only produce a phase shift of 90 degrees at either infinite or zero frequency.

An integrator will produce 90 degrees at all frequencies but amplitude falls ten times for each rising decade in frequency. Thus it won't achieve what you want.

Answer 2

Run the sin into a LPF, then into the CA3011 limiter; that, with 60dB gain, will give a squarewave output. Can you use squarewave?

Or use a state-variable oscillator, where Sin and Cos are inherent to the oscillator.