# Determining $Z_N$ and $Z_D$ for a flyback converter

According to page 448 of "Fundamentals of Power Electronics" (pdf) the small signal model of a flyback converter is

In my case $$\R_\text{on} = 0 \: \Omega\$$ so it should be ignored. Now, to find $$\Z_N \$$ you supposedly apply a test current $$\i_\text{test} \$$, nullify the output and measure the voltage across the test source $$\v_\text{test} \$$, $$\Z_N = \frac{v_\text{test}}{i_\text{test}}\bigg|_{V_o=0} \$$:

simulate this circuit – Schematic created using CircuitLab

To find $$\Z_D \$$ one has to nullify the generator and then find the equivalent impedance looking into the generator terminals.

simulate this circuit

I don't know how to analyse a circuit that looks like this. I have been searching through google to find guidance but without luck. So now I'm asking here, what is $$\Z_N \$$ and $$\Z_D \$$ for an ideal flyback converter? I need those two parameters in order to design an input filter.

• Maybe you should explain what Zn and Zd are on the circuit? Commented Nov 1, 2022 at 15:22

This is a complex matter and not often documented. First off, you can have a look at my APEC 2017 seminar dedicated to the subject. You can also check the summary answer I gave on SE here. The whole thing is to minimize the impact of the EMI filter on the converter's characteristics once it is installed.

It is possible to show that the control-to-output transfer function (TF) of a converter featuring a front-end EMI filter obeys the following expression:

$$\H(s)=\frac{V_{out}}{D(s)}\frac{1+\frac{Z_{th}(s)}{Z_N(s)}}{1+\frac{Z_{th}(s)}{Z_D(s)}}\$$

in which the first term represents the TF of the converter without filter, affected by a coefficient implying three terms: the EMI filter output impedance ($$\Z_{th}\$$), the converter open-loop input impedance ($$\Z_D\$$) and the input impedance when the output is nulled ($$\Z_N\$$). The coefficient is derived using the extra-element theorem (EET) forged by Dr. Middlebrook.

The terms $$\Z_D\$$ and $$\Z_N\$$ are determined as below:

For $$\Z_D\$$, the converter under study - your flyback converter - runs open-loop and you determine the input impedance in this mode. For $$\Z_N\$$, you have to apply a null double injection or NDI where the stimulus is back but it does not produce a response, this is a nulled output response: you determine the input impedance in this condition. Needless to say this is quite complicated and rather abrupt if you are not familiar with the fast analytical circuit techniques or FACTs as described in my book on the subject for instance.

A few remarks there:

1. I don't use the small-signal model proposed by the CoPEC folks but, rather, I use the PWM switch model derived by Vatché Vorpérian in 1986. It is a more compact and simpler approach in my opinion even if, in voltage-mode control, you'll obtain similar results. You can find the comprehensive small-signal version of an isolated buck-boost converter operated in voltage- or current-mode control in my last book on small-signal modeling.

2. You have two approaches to design an EMI filter. The first one is described by the book you referred to: design the filter together with the converter so that you minimize, from the beginning, the impact on the control-to-output transfer function and the small-signal output impedance. This is the best way to truly make the EMI filter addition transparent for the final closed-loop characteristics. In my experience, nobody does that because it is overly complicated and difficult to manage. What people do is the following: once the converter has been stabilized, you extract the closed-loop input impedance and plot it in magnitude and phase. Then, you superimpose on the same graph, the output impedance of the EMI filter you have designed to pass the standard you target. If no overlap occurs with some margin, then you don't have the conditions for oscillations and can safely turn the converter on. If you don't have margin or if overlap occurs, then you need to damp the filter so that sufficient margin exists. For the closed-loop input impedance, most designers I know simply plot a flat line equal to $$\20\cdot log_{10}(\frac{V_{in}^2}{P_{out}})\$$ and check for overlap with the filter output impedance. If sufficient margin exists, then this is a safe design.

Let's assume you have a working flyback converter that is properly compensated. You can find one from one of 80+ ready-made simulation templates I built to work with the SIMPLIS free demo version. The below figure shows how to measure the input impedance in the worst-case (low line):

Then, sweep the output impedance of the filter you have designed like in the below circuit as a possible example:

When you have $$\|Z_{out}|\$$, superimpose it on the closed-loop input impedance magnitude graph of the converter and check if there is overlap:

As you can see on this quick example, the EMI filter does peak quite a bit and may interfere with the converter: damping is necessary to avoid operating issues. Damping means dissipating power to lower the quality factor $$\Q\$$ and a tradeoff is often necessary to limit the filter interaction and overall efficiency. The control-to-output transfer function of the converter when the filter is installed does show signs of magnitude and phase distortion but the filter seems damped enough to limit the effect.

Interactions between a converter and the front-end filter is a vast territory and I could only scratch the surface here. It is however an important subject and often overlooked because people rarely realize that their design works because the filter is naturally damped with parasitics associated with the selected passive elements, PCB traces etc. Sometimes, the margin is too thin and oscillations occur in the field. So better look closely to this topic and understand the mechanisms at work during the design phase, not once the product is released.

• What a great answer ! Do you think that measuring the closed loop input impedance is the same as calculating ZN and ZD in terms of rigouroouness ? As ZD and ZN are not the closed loop input impedance ?
– Jess
Commented Apr 7 at 17:24
• Hello Jess, see slide 25 of my APEC 2017 seminar where I explain the differences. You can with $Z_N$ and $Z_D$ on hand, determine $Z_{in,CL$ but it is simpler and faster to measure the latter and, if necessary, damp the filter. Commented Apr 7 at 20:58

I'm working on a homework problem that is almost identical to the circuit you show with the exception of the loss resistance, Ron, and the Vg*dhat source.

To find Zd, you set all dhat sources to zero - this greatly simplifies the circuit.

I've derived Zd for an ideal flyback. The question asks for the value in front of the standard form function, (RDp^2/D^2n^2) which is correct. I don't believe I've made any errors in the algebra, so the entire function should be correct and in standard pole/zero form. I'm still working on Zn.

Dp = 1-D, n = transformer turns ratio

I solved it, consider that the small signal aproximation model of the flyback looks exactly the same as the buck boost, so the expressions of ZD and ZN are the same but adding a n^2 under