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I'm trying to understand the NEET from the Fast analytical techniques for electrical and electronic circuits book by Vatché Vorpérian. The book defines the following terms for the development of NEET: reference diagrams

Reference network: The network in Fig. 4.3 is called the reference network, which is obtained from the original network in Fig. 4.2 by opening ports (1) and (2). Four reference networks can be obtained from the original network in Fig. 4.2. Reference transfer function: The transfer function of the reference network in Fig. 4.3 is called the reference transfer function and is written as:

reference transfer function

defining function final function

in which summation over repeated indices of e_i e_j · · · e_m e_ij...m is assumed.

My problem: I try solving the transfer function for a damped LC filter of a converter using the above method, but the dc value, H_o, of Z(s) is zero, so that the whole transfer function will be equal to zero regardless of the numerator or denominator. Here is the circuit I'm referring to

damped LC filter

Clearly, if I use the other method where the transfer function is given as the ratio of capacitor impedance to its addition with parallel connection of inductor L_2 to the resistor connected in series with the other inductor L_1, the result won't be zero.

What is the solution to this scenario?

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1 Answer 1

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The difficulty with NEET approach, is that you have to choose a so-called reference state which combines all energy-storing elements set in a particular state: dc (caps are open and inductors shorted) or high-frequency (caps are shorted, inductors open). For example, if you consider a 2nd-order circuit, you have two energy-storing elements and four possible states: all elements are in their dc state, all in high-frequency state, 1st in dc - second in HF and 1st in HF - second in dc. You will choose the reference state so that it brings a meaningful configuration for you, easy to analyze. You realize how many combinations you then have when facing higher-order circuits. And the complexity is that you need to remember the state of the component in its reference state when you select the opposite.

I have adopted a slightly different approach in my first book on FACTs. The reference state is always dc (\$s=0\$) and I justify this choice considering my experience with SPICE which always computes a bias point with caps open and shorted inductors. It is easy to retain and the opposite state is always high-frequency. I recognize that, in some situations, it is not the most elegant approach compared to what Vatché describes, but I recommend beginners to first go that way and acquire the skill. Once they understand the process, they can mix various reference states if they want, but it is perilous to do it in the beginning.

Also, beside the null double injection (NDI) process, I often resort to the generalized form, in which the zeroes are obtained by determining high-frequency gains. This is the method developed by Pr. Ali Ajimiri and this paper is a good read. It truly helps in complicated circuits and very often, the high-frequency gains are zero which tremendously simplifies the analysis. It is good to know and master both approaches, NDI and generalized form.

One trick you could also apply in your circuit if you want to consider the resistance at dc, is to not consider zero but add a small resistance in series with \$L_2\$ (which exists anyway) so that you can have a non-zero leading term at dc. Later on, when all the TF is determined, you can have this small resistance approach zero and simplify the whole expression. In the same spirit, add a high-value resistance in parallel with a port returning infinity and have that resistance go infinite after factorization.

In your case, below is the determination of the time constants:

enter image description here

Then the zeroes appear after determining high-frequency gains:

enter image description here

and, finally, you test the expression versus the brute-force version to check for any deviation. Then, and that is usually the difficulty, you have to rewrite the expression to unveil the poles and zeroes in a low-entropy way. It is important as you will use this factored expression to design your EMI filter. Below is what I obtained:

enter image description here

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