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Steady State of periodically switched circuit

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CataM

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Hello everybody,

In DC-DC converters (SMPS) for example, in order to focus on the steady state, we assume the inductor for example has an initial current through it if the converter is working in CCM. In order to find equations and that initial value, we then say that as it is in steady state, current(t=0) = current (t=Period) i.e. Δcurrent in a period = 0 .

Another example is a 1st order low pass filter.
Let us consider a simple 1st order RC low pass filter with the input a square wave and the output across the capacitor with NO load.
Focusing on the steady state, we would say Vcapacitor(0)=Vinitial and solve its differential equations for each period of time.

When Vin is positive:
-its differential equation
-Vcapacitor(0)=Vinitial

When Vin is negative:
-its differential equation
-Vcapacitor(t=T)=Vinitial

By making Vcapacitor(t=T)=Vinitial=Vcapacitor(0) we are able to find Vinitial which is the voltage in the steady state.

The steady state is the following one:


My question is: For more complex circuits which make the differential equations more difficult to solve, what would be the approach ?
Example is this circuit: The Switches changes position as the voltage of the capacitor "C" crosses 0.


Any hint or comment is welcomed !
 

I spent a lot of time trying to derive solutions to circuits almost exactly like the one you posted. For that circuit it's not very difficult, so long as there's no overlap in the switches. However if the switching causes the voltage/current to be discontinuous anywhere, then things become far more messy and it's very difficult to get exact closed form solutions.
 

, so long as there's no overlap in the switches. However if the switching causes the voltage/current to be discontinuous anywhere, then things become far more messy and it's very difficult to get exact closed form solutions.
There are no 2 switches closed at the same time. Switches changes position when the voltage across the capacitor (VBA) crosses 0, one switch closes while the other one opens.

The circuit is easy to solve if there are no initial conditions i.e. exactly at startup. However, I am trying to find its steady state which makes me assume there is already some current through inductors (no voltage across the capacitor because the switch has just changed and the switches changes when the voltage crosses 0).

The circuit is called Royer Oscillator and a similar version of the circuit I am trying to find its equations uses only 1 RF choke and is described in this IEEE paper: Rigorous modeling of mid-range wireless power transfer systems based on royer oscillators

What I want to find is: What is the peak voltage across the capacitor and how do I know on what terms does it depend ?

I have solved the differential equations for when the switch 1 = closed and switch 2 = open and with the voltage and current polarities as follows in this picture:


I have arrived to this solutions:


Making the voltage across the capacitor = 0 in order to find the time when the transition takes place gives me 2 solutions ... I do no think an engineer would design such circuit with my approach... but how would he do it ?
 

Okay so it's a self oscillating circuit. But for the purpose of analyzing it, I would look at it as if the switches are driven in a manner determined a priori. Then after finding the results you can check that the switching happens at the same time the switch voltage reaches zero.

If you assume that two choke inductors are much larger than the resonant tank inductor, and that the circuit is driven at the tank's resonant frequency, you can assume that the tank sees a sinusoidal voltage with amplitude pi*Vdc, and this is also the peak voltage seen by each switch. But once you introduce resistance into the tank circuit or switches (which is necessary to make it meaningful) then this becomes a less valid approximation.
 
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    CataM

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If you assume that two choke inductors are much larger than the resonant tank inductor, and that the circuit is driven at the tank's resonant frequency, you can assume that the tank sees a sinusoidal voltage with amplitude pi*Vdc, and this is also the peak voltage seen by each switch. But once you introduce resistance into the tank circuit or switches (which is necessary to make it meaningful) then this becomes a less valid approximation.
You are absolutely right. How did you know that it is pi*Vdc ? What equivalent circuit should I draw in order to get to that ?
If choke coils are >>> than tank coil, what assumption can you make ?

Peak voltage = 36,615 = pi*12 (approximately)
See schematics below. (frequency = 20 kHz)

Schematic


Voltage across capacitor and gate of mosfet M1


Current through choke coil
 

You are absolutely right. How did you know that it is pi*Vdc ? What equivalent circuit should I draw in order to get to that ?
If the drain chokes have no resistance, than at steady state the average voltage across them must be zero. So the drain voltage waveform must have an average of Vdc. For the half sinusoid waveform expected on the drains, its peak-to-average ratio is pi.
If choke coils are >>> than tank coil, what assumption can you make ?
That the resonant frequency of the circuit is the same as the tank's resonant frequency. Otherwise the chokes pull the resonant frequency up a little bit.
 
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