Meaning of complex impedance

Complex voltage and current have clear physical meaning: they represent amplitude and phase shift of a sine wave.

I was wondering if complex impedance has any physical meaning as well. Can I plot it as a sine wave,or is it just a number that we are supposed to treat as a mathematical abstraction without any relation to the real world?

• Complex impedance also implies frequency sensitive. Dec 20 '20 at 20:35
• Are you looking to understand why complex numbers ever crossed anyone's mind as a way to re-state the differential equations describing inductors and capacitors?
– jonk
Dec 20 '20 at 20:56
• @jonk Haven't thought of it that way, but it would be interesting to hear your opinion on this. Dec 20 '20 at 21:32
• @RodionDegtyar I'm not thinking about re-writing text, equations, and diagrams today. It's not hard to grasp, if I supply the right diagrams that make it easy to see. But it really does need to be supplemented with the rigor of showing the equations and theory, as well. And that's more than I want to write, today. I studied Heaviside's two papers (1893) on the topic. But also Steinmetz published a paper on phasors that same year. Also worth reading. Between that, and knowing how Euler's helix "tracks" the orthogonal plots of voltage and current, you have what you need. Complex analysis helps.
– jonk
Dec 20 '20 at 22:11
• @RodionDegtyar To get a jump on it, there is a nice segue paper called "Steinmetz and the Concept of Phasor: A Forgotten Story", 2013. It doesn't provide the details you need to see. But its references section at the bottom does refer to the specific papers you should find and read: those from Heaviside and from Steinmetz, primarily. You may someday later want to add the Dirac $\delta$-function and Green Functions to your reading, though.
– jonk
Dec 20 '20 at 22:16

The question can look obscure, but it isn't. Complex impedance is a way to present how certain measurable quantities depend of each other in a certain system in certain conditions. It's a physical quantity as well as complex phasors of current and voltage are physical quantities. These all are defined in circuit theory.

Voltage and current (sinusoidal or other) present things which are considered to be a part of lower level physical theory (=electrodynamics) than the circuit theory. Circuit theory is an abstraction which hides such things as what current and voltage are. In circuit theory we formulate concepts "sinusoidal voltages and currents" and present them with complex phasors. In circuit theory we assume that resistors, capacitors and inductors obey certain equations that bind voltage and current to depend on each other. Those equations we have got from electrodynamics. We can prove a circuit theory fact: Complex impedance presents how sinusoidal voltage and current depend on each other.

Difference: Complex phasors of voltages and currents are built in circuit theory on voltage and current which are defined in electrodynamics. Complex impedance is built on concepts which are defined purely in circuit theory, but it's still a physical quantity and can be measured.

A complex impedance is an impedance that will have a phase shift between the voltage across it and the current through it. It's that simple.

A completely real impedance (in other words, a resistance) has no phase shift between the current through it and the voltage across it.

Complex voltage and current have clear physical meaning: they represent amplitude and phase shift of a sine wave.

Impedance is the ratio of voltage to current, so a complex impedance means that voltage will be shifted relative to current.

• So the imaginary part of the impedance specifies the phase shift of voltage relative to current? This makes sense. Dec 20 '20 at 21:33

The impedance itself is a complex number. The real part of it is resistance, the imaginary part is the reactance due to inductance or capacitance. If yo do understand the meaning of complex representation of current and voltage, then you would agree that a simple formula,

$$Z=\dfrac{V}{I}$$

can't be nothing else than complex number.

Can I plot it as a sine wave?

No. A complex number is a vector. Where have you seen complex number drawn as a sine wave?

• What I meant is that complex voltage and current have a sine wave associated with them. Is there a sine wave associated with complex impedance as well? Dec 20 '20 at 21:31