IV curve arithmetic - reference request

When playing around with electrical concepts in my head, I realized that the IV characteristic of any combination of elements, in series or in parallel, can be computed by simply adding the IV curves for individual elements -- in the V direction for elements in series, and in the I direction for elements in parallel. This works for linear as well as non-linear elements, and any combination of both.

So far I have been able to use this method for combinations of one-port elements, and independent sources. I was trying to generalize this method to general n-port networks, possibly containing dependent sources, but I can't wrap my head around how one would generalize this method. It would be nice if I could see transistors analyzed "intuitively" this way. To some extent it's possible in the large-signal sense, with VTC curves instead of IV curves, when analyzing only the output of the transistor at one of its ports, by considering everything to the left a one-port active element, and everything to the right an arbitrary load, whose VTCs together sum to the DC supply or whatnot. I feel like this idea could be made more sophisticated, and I'm sure I'm not the only one to explore it.

Is anyone aware of such a method? Is there a book or a paper that talks about this exact thing?

• There is a technique that focuses on using I-V curves (black-box) for components used in circuit analysis. I skimmed over such a PDF about a year and a half ago. It was a research paper and it was clear that this was an active area of research and that this paper was showing a generalized approach and illustrating it with specific application examples. It was long and detailed and was completely novel to my experience. I skimmed it for a bit and then decided that it was "technically interesting" but only something to put to the back of my mind for some future date. Can't recall the field name. Commented Dec 4, 2023 at 22:55
• @periblepsis Do you recall what might have been the vague ballpark title or source of this paper? Or at least a few keywords that might make me more likely to find it myself? I mostly don't even know how to type it into a search engine, because I don't know what terminology to use Commented Dec 4, 2023 at 23:06
• If it comes back to me I'll say so. Its substance would have been published at some time in "IEEE Transactions on Circuits and Systems". That would be the "go to" place to look. But it may have been at a university page that I found it. It's really frustrating me. Because the paper read to me very very much like your question does, as I interpreted both. So I would love to match you up with it. But I'm going to have to allow my brain to recall some details so I can search. I wasn't immersed in it, just skimming, so words aren't coming to me now. But the networks are very clear to mind. Commented Dec 4, 2023 at 23:18

I was trying to generalize this method to general n-port networks

Let's focus the attention on a BJT that can be modeled as 2-port network.

Port 1: Base and Emitter Port 2: Collector and Emitter

The main math problem you have is that IV curve at Port 2 is a non linear function of the IV curve at Port 1.

That leads to a non-linear algebraic system of equations.

You can solve it graphically in a 3D or 4D space.

• Sure, but I was looking for a work/treatment on this. I know that it would roughly be a multidimentional manifold, just don't know how to work through the details. Commented Dec 8, 2023 at 15:35

This generalization is the idea of “n-ports”, examples are two port networks or three port networks. The relation between the voltage across each port and the current into each port, instead of a single equation, becomes two or three equations, each involving the two or three port variables.

By using matrix notation, the math for linear devices looks the same as the regular circuit elements you have mentioned. For example, v=RI is the voltage-current relation of a resistor, and V=ZI could be the voltage-current relation of a three port device, where V is a vector [v1,v2,v3], ( the port voltages), I is a vector of the port currents; [i1, i2, i3], and Z is a three by three matrix of numbers ( for linear devices). For nonlinear devices, matrix notation is not as convenient.

• You just talked about the idea of n-ports, which I'm familiar with already Commented Dec 8, 2023 at 15:36
• Then you know that ports in series and parallel add vi curves as you mentioned for one port devices. To analyze complicated circuits, though, you have to use kirchoff’s laws, of course, because usually things are not always in series or parallel. I guess I don’t understand your question: is the point that you want to analyze circuits graphically instead of with equations? Commented Dec 9, 2023 at 19:34