Control to output transfer function

Question

I am trying to derive control to output transfer function of phase shift full bridge converter.

Applying small signal analysis, I reach to the point/schematic as shown in the picture below:

Document calculated the control to output transfer fucntion as shown in the picture below. When I equate vin = 0 V, both sides of the transformer becomes short, so it turns out to point as shown in the picture below. But here there are 2 dependent and 2 independent source. I dont know how to deal with 4 source. Generally if I had 2 source, I was making superposition technique. How should I proceed with 4 source?

enter image description here
Source: https://www.researchgate.net/publication/3279135_Small-signal_analysis_of_the_phase-shifted_PWM_converter

enter image description here
Source: https://www.researchgate.net/publication/3279135_Small-signal_analysis_of_the_phase-shifted_PWM_converter

Considering my above question, control to output transfer function given in the picture below. I think they both connected to each other. I think first I need to understand my above question, than I can derive the transfer function as stated in the picture below. But here I couldn't understand the Zf/ (Zf+Rd). How should I come to that point?

enter image description here
Source: https://www.researchgate.net/publication/3279135_Small-signal_analysis_of_the_phase-shifted_PWM_converter

By the way, dv and di are given as below. These are just equations derived for the small signal model. Ho is also given below I have no question about dv, di or Ho. I just put them to show what are they.

enter image description here
Source: https://www.researchgate.net/publication/3279135_Small-signal_analysis_of_the_phase-shifted_PWM_converter

enter image description here
Source: https://www.researchgate.net/publication/3279135_Small-signal_analysis_of_the_phase-shifted_PWM_converter

enter image description here Source: https://www.youtube.com/watch?v=45zMW6TYpnA

enter image description here Source: My own calculation

enter image description here Source: my simulation

• Verbal Kint: What you zero for your analysis, is \$v_{in}\$ which is an ac component. \$V_{in}\$ in the other sources is the dc value of the input source acting as coefficient. You keep it of course ^_^ If you like deriving transfer functions, you may want to look at my book from 2021 dedicated to the topic.
• Mhan: Sir, what do you suggest me at this stage shortly, I will be very happy if you can give some advice at this stage @VerbalKint
• Mhan: Yes sir, vin will be zero as you said. but ı dont now the rest, how should I apply superimposition then ? @VerbalKint

Answer 1

When I derive a transfer function, I usually qualify the model I want to use. In your case, you should build this equivalent ac model in LTspice, for instance, and check that you obtain the curves given in the paper. This is key to understand the architecture. It will also let you understand how the equations are built.

In 2009, I published the PWM switch model version for the phase-shifted full-bridge converter. The paper is here. I used this model to test the equivalent ac model we need for the transfer function:

enter image description here

In this example, I ac-modulate the \$d\$ input which is the ac part of the duty ratio and see how it propagates to generate a response at both outputs. As confirmed by the plots, all responses in magnitude and phase are the same so the model is ok.

Now, this model is complicated to solve because of the various controlled sources and the generator. I prefer merging all these sources, simplify the expression and obtain the simplest possible model. This is what I've done below:

enter image description here

You can see now how the model looks simpler to deal with. Now, you can either apply the Fast Analytical Circuits Techniques or FACTs or apply brute-force KVL and KCL to that circuit. Then rearrange the final expression in a low-entropy form, meaning you can see a resonant frequency and a quality factor as they've done in the paper.

If I apply the real brute-force - what I never do : ), I end up with the same expression they found, neglecting the capacitor ESR:

enter image description here

• Mhan: Sir, your comment is very valuable to me. As far as I know, you put VIL in order to represent the vs reduction due to the small amount of filter inductor current. If so how should I define VIL and I(VIL) in Ltspice in terms of equation @VerbalKint
• Verbal Kint: Hello, no, the VIL source is a dummy 0-V voltage source. It is only there to measure the inductor current \$i_L\$ with I(VIL) and use it in the equations. Just curious, why are looking into this converter small-signal response? By the way, if my answer meets your expectations, I would appreciate if you could kindly acknowledge it, thank you!
• Mhan: Sir, firstly I am a very big fan of you and your studies :) I am trying for this converter to develop feedback compansation sir. I am a self paced learner, I have no one to ask these kind of questions. @VerbalKint
• Mhan: I have a couple of question. You mensioned 'merging all these sources'. I simply add the voltage sources together, let the vin= 0 and I have same result as yours. For the current sources, again simply add together, let vin= 0, this time I think you ignored the R because it is much smaller than Rd, and you reverse the current source direction by reversing the equation (instead of N*V(d)*Vin/Rl – I(VIL)*Rd) I think ? Am I true sir ? And we only let vin= 0, because il of di is in the control loop isn’t it sir ? @VerbalKint
• Verbal Kint: Yes, I did reverse the source because of the neg. sign and two sources in series are merged into one. \$v_{in}\$ is zero because when you ac-modulate the duty ratio \$d\,ドル the input voltage source is constant to a dc value \$V_{in}\$ and is thus 0 V in ac.