Transfer function of phase change controlled with capacitance

Question

We have an RLC series circuit with sinusoidal voltage source of frequency \$\omega\$. The normal transfer function of current to voltage is: $$ G(s) = \frac{I(s)}{V(s)} = \frac{1}{R+s L+ \frac{1}{s C}}$$

We would like to find out the transfer function for phase shift of current to the capacitance. We can control the capacitance and therefore also control the phase shift.

$$ G(s) = \frac{\phi(s)}{C(s)} = \,? $$

The nonlinear function of phase shift from vector diagram is $$ \phi(C) = \arctan\frac{\omega L- \frac{1}{\omega C}}{R}$$

We could linearize this function with Taylor series, but this equation doesn’t have any dynamics.

Is there a way to get the linearized transfer function (how small step change of the capacitance changes the phase shift of the current) so we can do the stability analysis for a control loop?

We have a black box with a value of capacitance as input and phase shift of the current as the output with RLC circuit inside. Lets now try to do identification of this system. I can set a value (operating point) of the phase shift with the value capacitance. Now I will do small sinusoidal perturbation around the operating point of C at some frequency range, and measure the amplitude and phase shift of the output phi. I know it's confusing phase shift if phi is the phase shift of the phase shift. What I get is the Bode plot and from this, I can approximate the transfer function. This is the transfer function that I want to drive mathematically.

This is a classical PLL

The VCO is controlled with voltage \$ V_{cont}\$, the output frequency of VCO is $$\omega_{out} = K_O V_{cont}, $$ where \$ K_O \$ is the gain coefficient [rad/s/V] and the output phase is $$\phi_{out} = \int \omega_{out} dt = \int K_O V_{cont}. $$ The transfer function VCO is then $$\frac{\phi_{out}(s)}{V_{cont}(s)} = \frac{K_O}{s} $$ Instead of the VCO is the RLC circuit. I also can’t control the frequency like in the VCO, only the phase shift. This is the block diagram of the control system

This is the working principle scheme

Here \$ C(s) \$ is the compensator (regulator, loop filter)

The PLLs are used for measurement of the currents in the main and auxiliary phase, they also have AGC (automatic gain control) to ensure the same amplitudes of the current signals, because I only want to compare the phase difference. What I want to control is the phase shift of the current in the auxiliary phase to the current in the main phase, let's say to get 90 degrees phase shift. I have tested this in a simulation and it works.

Answer 1

The phase change due to capacitance is not set in stone for all applied frequencies. For instance at resonance (current is maximum) the phase rapidly changes 180 degrees over a very small band of frequencies.

At higher or lower frequencies (nowhere near resonance) the phase change hardly moves at all unless the quality factor (Q) of the circuit is very small. Altering C at these extremes has very little effect on phase change.

Answer 2

If I understand correctly, you want to be able to precisely control the phase by controlling the capacitor (if I am wrong, please disregard my answer). If so, you can take the next step:

$$ \phi=\arctan\frac{\omega L-\frac{1}{\omega C}}{R} $$

solving for C results in (corrected after edit):

$$C=\frac{1}{\omega^2 L-\omega R\tan\phi}$$

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I'll try going on, maybe it leads somewhere. \$\phi\$ will result in a different formula if the complex Laplace transfer function is taken into account:

$$ \phi=\arctan\frac{\frac{1}{s C}-s L}{R} $$ $$C=\frac{1}{s^2 L+s R\tan\phi}$$ $$ G(s)=\frac{s}{2 s^2 L+sR+R\tan\phi} $$

This could be solved for both \$s\$ and \$\phi\$ if the angle is calculated step by step.

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Hardly helpful, not a "control freak", sorry I can't be of more assistance. Still, there's a problem I see: C's value can be calculated from L, R and \$\tan \phi\$, but there is a discontinuity for negative \$\phi\$ where \$\omega^2 L=\omega R\tan\phi\$, and around the point the curve is quite sharp. Here's an example with L=0.1 and R=10, for varying \$\phi\$:

L would have to be very much larger than R. Although, you're probably trying to compensate the negative \$\phi\$ from the other branch, so it's unlikely you'll need that.

I tried making it in LTspice, but, no matter what dead-time I chose, the control voltage always blows up (hopefully it's a readable mess):

In your schematic you chose cos*sin, I made it cos*sin-sin*cos, for a smoother waveform due to quadrature. It doesn't work. I gave up the PWM part, using only LTspice's behavioural capacitor, using the control voltage directly for its Q=x*v(ctl), still, it doesn't work. For now, I admit defeat, but this shows a very nice way of creating a 3-phase from a 1-phase. And, with a SOGI, I suspect this might turn into a nice quadrature source. Or, without the aux LR branch (just the C), it could make a beautiful compensator for RL loads.