Why does electrical impedance have as many parameters as it has?

Question

The impedance of a circuit is written:

$$ Z(R,L,C,f) = R + j ( 2 \pi f L - \frac{1}{2 \pi f C} ). $$

What is the line of thinking that frequency, resistance, capacitance, and inductance are the only parameters? From a physical perspective, why is the definition of impedance parameterized exclusively by these four variables? This model can be verified with experiments, but what goes to show that I couldn't make another circuit element that affects the inductance without changing any of those four properties?

Answer 1

Good questioning. I don't know for sure what eventually lead us to adopt the paradigm of RLC circuits, but do know that the physics of electrical circuits deals with the way energy flows. In a circuit energy can be stored in the form of magnetic or electrical fields, and so the inductor and capacitor manage that capability. Energy can also be lost, and thus the resistor provides that means of modeling.

In a generalized way, the capacitor and inductor have a conceptual symmetry about how energy is stored. Why not the resistor regarding loss? Actually there is a conceptual element that fits the bill: the memristor. Although memristors have been realized, I don't believe they have been the subject of practical electrical circuits.

If we could conceive of other ways in which energy can flow in an electrical circuit, perhaps we could invent new components to consider.

I should also mention that the modeling in terms of RLC is idealized in another manner - as 'lumped' parameters. We deal with R, L, and C in our circuit models as pure elements of one or another. In real components and circuits things are not so pure. Capacitors have a little inductance & vice versa. Resistors have capacitance at high frequencies. Depending on what we seek to predict with our models determines whether we can get by with the lumped parameter models or if we need to model more precisely.

Answer 2

Only some circuits can have such a representation. Such circuits can be represented as a resistor in series with an inductor in series with a capacitor. This is hardly the only circuit possible. For instance, an electrolytic capacitor is sometimes (depending on the use being considered) represented as a cascade of resistors and capacitors, with each capacitor being in parallel with a resistor in series with another composite capacitor. This is a useful model for dealing with "capacitance creep" caused by slow relaxation of molecules in the capacitor dielectric.