# Looking for a general, rigorous definition of “impedance”

Forget capacitors. Forget inductors. Forget resistors. Assume a generous background in elementary complex analysis and algebra. I'm trying to arrive at an elegant, minimalist, mathematically simplified, linear circuit theory. So let's assume all we know is that the voltage or current at any point on a geometrically lineal conductor (i.e., exclude current junctions) in the circuit is a mathematically linear function of the voltage or current at any (other or same) point on a geometrically lineal conductor in the circuit. (Formally, the linear function would be called a "linear operator"; and mathematically a "domain" and a "range" would need to be defined for the operator; the domain would consist of real-valued functions of time; same for the range; but we can skip further detail on this for now.) Assume an arbitrary reference point in the circuit for voltage ground.

There are two results that I suppose are true about this circuit. I would like to find a simple, mathematically elegant, rigorous proof of these results. Hopefully the proofs will be intuitively meaningful -- but this is not strictly necessary. (The intuitive aspects can be a subsequent development if necessary.)

First, a proof is needed that the voltage or current at any point in the circuit has the form Acos(wt+theta) if the voltage or current at any single (same or other) point has the form Bcos(wt+phi), for a fixed, given angular frequency w. (Some simplifying assumptions might be needed to, say, preclude additional inputs, say with different frequencies. But hopefully this can be done in a simple, elegant manner without the proof degenerating, for example, into many sub-cases.)

Second, a proof is needed that the ratio of any voltage in its complex form [Aexp(jwt+theta)] to any current also in its complex form [Bexp(jwt+phi)] in the circuit is a constant complex number (which we can call "impedance") independent of the amplitudes A, B and phases theta, phi. It seems such a result needs to be proved before we can talk meaningfully about impedance. (And by the way, does this accurately capture the usual meaning of "impedance" in electrical engineering?)

The second result may be a very simple consequence of the first. But also of interest here is how can we modify the assumptions to make them as simple and minimal as possible and yet still have a "powerful" set of theorems for the development of a mathematically simple, elegant (and useful) linear circuit theory.

To illustrate the relevance of this question and where it could lead, I'm thinking that a fairly simple, linear circuit theory should be possible for a very complicated, arbitrary interconnection of coaxial cable segments in a complicated topology involving arbitrary interconnections between shields and center conductors. And hopefully a useful linear circuit theory for such a circuit would not have to degenerate first to a discussion of capacitance, inductance and resistance, although a theory of capacitance, inductance and resistance (say based on empirical electrical sampling of the circuit) might be a nice offshoot derived from the main theory.

A subsequent development might be to ask similar questions about an electrically linear 3D continuum, say like cookie dough, with a few wires stuck into it. But we don't need to go there yet.

I have not been able to find anything that comes close to such a development. So I'm asking this question because I don't want to reinvent the wheel, so to speak. Does anyone know of any theoretical development of this nature? (And while I'm at it, do (rigorous) mathematicians and electrical engineers talk to each other?)

I'm a Amateur Extra ham radio operator, and commentary about "impedance" exudes all over the place in ham radio; I also have a strong background in the rigors of pure mathematics. So, in particular, I'd feel a whole lot more comfortable if I could find a rigorous, mathematically elegant (i.e., mathematically "simple") definition of impedance. Circuit theory textbooks, physics textbooks, and E&M textbooks I've looked at and comments I've found on the Internet just don't cut it for me. Invariably, any definition of impedance I've found first degenerates to a discussion of discrete capacitors and inductors and then impedance is defined only for very specific (and typically very simple) circuits involving resistors, capacitors, and inductors. That seems to create a huge limitation in the formal idea of impedance. But it seems to me that impedance is probably a much more general concept, especially, for example, when people start talking about impedance at various points on say a dipole antenna. In that case there are typically no discrete capacitors and inductors in the antenna itself aside from potential such elements of sorts in tuners, RF chokes, capacitance hats, and loading coils.

• I could not read all that mumbo jumbo. Impedance is a concept that applies to a device with two nodes. The impedance is the ratio of voltage to current at those two nodes. It is that simple. Have a nice day. – mkeith Mar 5 '16 at 6:59
• Start with Maxwell's equations and derive everything from there. – John D Mar 5 '16 at 6:59
• I read a bit of it. When studying circuits and systems in electrical engineering, we usually start by saying that the system is linear and time invariant (and usually causal too). Linear implies that impedance is independent of voltage and current (fixed). Diodes and transistors are not linear devices, but we find ways to use them where they behave linearly because otherwise it is too hard to do analysis (have to do simulation instead). Transmissions lines can be simulated using lumped elements (look it up). – mkeith Mar 5 '16 at 7:15

What you say is trivially correct, correct by a tautology, because impedance is defined as the ratio of a complex voltage to a complex current.

The reason nobody talks about it in the terms you do, is that the very general case has no real application to circuit design or analysis.

What is interesting for design engineers is the impedance of a component, idealised so that both its terminals are at the same point, the impedance of a transmission line and the way it transforms an impedance at its terminals depending on its length. That's the zero dimension and one dimensional case.

I am sure that the mathematical treatment of a 3D arbitrary cookie dough would have great relevance in impedance tomography, when an engineering community find a useful way to use such a thing. For instance, diffusion tensor imaging uses tensor mathematics to interpret the results of an MRI scan sequence as the rate of fluid movement between parts of the brain. So when a need arises, the rigourous mathematics appears. Until then, it's just interesting pure maths.

When people do electromagnetic field solving in 3D for antennae and the like, you could interpret the results as impedance ratios from point to point if you liked, but you would gain no greater insight into the problem at hand than from the fields, currents etc of the standard output.

• what particularly intrigued me about impedance was comments I read that impedance at the ends of a dipole antenna is very high because there is no place there for the current to go and so the current I would be very small, making Z = V/I very large provided V does not also become very small. The writer claimed that the voltage would therefore be very large at the ends of a dipole, but his reasoning was unclear. Of course rigorous proofs of the ideas would be needed in order to form a reliable method of analysis for extensive application to a variety of antenna systems. – Richard Haney Mar 6 '16 at 3:07
• While impedance can be defined as Z = V/I across two points in even a nonlinear circuit, apparently linearity is necessary and perhaps even sufficient in order to prove that the impedance is an intrinsic property of the circuit independent of voltage and current levels and phases. It would be so much nicer to be able to do that by, if possible, using only linearity without reverting to discrete components such as capacitors, etc, and therefore do that without invariably devolving seemingly into an inelegant (i.e., "messy" and perhaps confusing) discussion of multiple cases. – Richard Haney Mar 6 '16 at 3:15

As fare as I understand your explanation and your first question is one of the impedance definition Z=U/I, where U is the complex voltage on a measurement point and I is the complex current at the same point. If you increase the frequency to measure voltage and current will be more and more difficult and in the end impossible. To overcome this problem you measure the power flow in forward and backward direction and you calculate what you want from them.

Coming to the second point

To illustrate the relevance of this question and where it could lead, I'm thinking that a fairly simple, linear circuit theory should be possible for a very complicated, arbitrary interconnection of coaxial cable segments in a complicated topology involving arbitrary interconnections between shields and center conductors.

With the transmission line theory you calculate the impedance of an open, shorted or loaded with an impedance line of a specific length. If you connect different transmission transmission lines that it is similar to connect the calculated impedances. It isn't exact because the connection has an influence of the fields and if you want to know it you need most likely a field simulation software.