Approach to an unknown orthogonal Beta Signal with a known alfa in single phase dq transformation

Question

I am trying to implement a PLL controller to the MCU for tracking single phase line voltage . I get samples via an opamp circuit with a DC offset and the samples' raw values vary between |-244 , +244| when removed the DC offset . The key problem I'm trying to overcome is that the DQ transformation needed in the design requires a made up second orthogonal signal (Beta) other than the alpha . As far as I read the papers written for single phase PLL implementation , we need a 90 degrees time delay of the original signal , so it is our alpha .. Let's say an instant adc value is 128 and that is the alpha , so how to create an imaginary orthogonal Beta signal using that ADC value ? What I need for a park transformation is the alpha and beta , I got the alpha but what is the beta ?

D = Alpha*cosTETA  + Beta*sinTETA
Q = Beta*cosTETA - Alpha*sinTETA

Alpha= Instant ADC value

Beta = ?

Answer 1

Sounds like you seek to phase_track a signal, for which you have some inphase and some quadrature information.

And you need to preserve the polarity of the information, so you can have a Negative Feedbac control system instead of a Positive Feedback system.

Show all your information_extarction maths, and the sensor input to the system.

Then think about how to preserve the needed polarity of the control output.

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If you want to phase_track a sinusoid, then yes you can generate a quadrature signal using a differentiator. This will be noisy.

If you want to phase_track a non_sinusoid, you need to decide which edge to align with, and use a phase_frequency digital_phase_frequency circuit; this has 2 FlipFlips, a ORing gate, and (usually) a tiny delay needed to ensure healthy RESET pulse behavior.

Answer 2

Generating a quadrature signal from a direct signal can be accomplished with a SOGI (Second Order Generalized Integrator). The conceptual schematic is this:

V1 generates the input and the two integrators enclosed in a feedback loop (an electronic ouroboros) generate the quadrature signal. V2, through the two multipliers, provides the frequency. Here, I chose to use f, frequency, as opposed to ω, pulsation, in which case the two time constants for the integrators would have been 1. The gain sets the damping, 1.41 (√2) providing a good compromise between the bandwidth, flatness, and response time. The latter is what matters more in the case of PLLs. This is usually achieved in software, and multiple SOGIs can be used, if needed.

The transfer functions for each output are:

$$\begin{align} D(s)&=\frac{ks}{s^2+ks+\omega^2} \\ Q(s)&=\frac{k\omega^2}{s^2+ks+\omega^2} \end{align}$$

so you can see that there is a lowpass and a bandpass, whose magnitudes coincide at ω and their phases are 90^o appart (k is the gain block):