Imaginary Capacitance and Imaginary Inductance interpretation and relations with circuit theory

Today I came across with Complex Capacitances and Inductances for the first time: $$L(j\omega)=L_{real}+jL_{imaginary}$$ $$C(j\omega)=C_{real}+jC_{imaginary}$$

So I started looking at their meaning and how they relate to the basic circuit theory that I've already learned. Let's start with the inductors.

Inductors:

I found this article, that states (pages 7 and 8):

After that, I found that this "Power Loss" is associated with "Core Losses", like hysteresis losses, Eddy/Foucault current losses. So, my first question is:

• If $$\L_{imaginary}<0\$$, we have $$\Power\space Loss >0\$$, so we have a dissipation of energy by the magnetic core of the inductor due to (according to what the article says) eddy currents;
• If $$\L_{imaginary}=0\$$, we have $$\Power\space Loss =0\$$, so no eddy current and no energy dissipation;
• But, what if $$\L_{imaginary}>0\$$ ? Then we will have $$\Power\space Loss <0\$$, i. e., the magnetic core will supply power to the circuit! How is this possible? Or is the value of $$\L_{imaginary}\$$ somehow restricted by some equation, so that it cannot assume positive values?

(Another way to state this is considering the inductor impedance:$$Z_L=j\omega L=j\omega\cdot(L_r+jL_i\ )=-\omega L_i+j\omega L_r\ \$$ The imaginary part of $$\L\$$ contributes to the real part of inductor impedance. This contribution is given in the form of a frequency dependent resistance $$\-\omega L_i\$$. So, if we have $$\L_i>0\$$ then $$\-\omega L_i<0\$$, a negative resistance!)

Capacitors:

I found this article. It states that (page 2) $$\C_{imaginary}\$$ is associated with dielectric loss. Another source states that this dielectric loss can be by Joule's heating effect, hysteresis losses and dielectric absorption. I guess we can model this dielectric loss in a analogous way to the losses in the inductor core: $$Power\space Loss=-\frac\omega2 C_i v^2$$

Thus, a similar question arises:

• If $$\C_{imaginary}<0\$$, we have $$\Power\space Loss >0\$$, so we have a dissipation of energy by the dielectric;
• If $$\C_{imaginary}=0\$$, we have $$\Power\space Loss =0\$$, so no energy dissipation;
• But, what if $$\C_{imaginary}>0\$$ ? Then we will have $$\Power\space Loss <0\$$, i. e., the dielectric of the capacitor will supply power to the circuit! Again, how is this possible? Or is the value of $$\C_{imaginary}\$$ somehow restricted by some equation, so that it cannot assume positive values?

Relations with basic circuit theory:

I've already learned that real circuit elements can be modeled using parasitic elements. For example, a real capacitor can be modeled like this: Image source: Wikipedia

My final questions are:

• The imaginary component of capacitance and inductance were used to "account losses" that occur in real components: hysteresis losses, Eddy/Foucault current losses, dielectric absorption ... Are all those losses counted using complex values also counted when we model a real component with parasitic elements?
• Which of the two models - using complex L and C or using parasitic elements - is the most accurate if we want a representation as close as possible to reality? Or are they equivalent?
• It bothers me that you never use the word "impedance", and that you seem to only consider parasitic effects that are frequency dependent. For example, the parasitic resistors in your capacitor model will obviously cause power loss that is not proportional to $\omega$. – Elliot Alderson Apr 1 at 0:08
• @ViniciusACP Have you identified any papers that discuss the basis for using a complex valued inductance and how that value is worked out? (I have. I'm just wondering if you have.) By the way, it is always negative. (And it is never the same, twice, so to speak.) There's a motivation for this approach, but if I had to guess (and I do) I'd say you haven't understood where and why it arrives in the paper you cite. The topic is too complex to present here, by the way. (No pun intended.) – jonk Apr 1 at 2:31
• @ViniciusACP If I were well-versed on the subject, I'd of course provide the seminal papers for you. I've found some that tell me enough about why and it has to do with modeling the Eddy Currents in a convenient mathematical way -- it's not physics, it's modeling with a parameter that cannot be directly coupled to a physical reality. It's just convenience. And it is different at different speeds for the same motor/whatever. So there's a lot of "fitting" going on, as well. Look up Kapjin Lee and Kyihwan Park and see about papers using FEM formulations, as well. – jonk Apr 1 at 18:29
• @ViniciusACP Euler's and complex notation can be quite useful for modeling anything that rotates or spirals or depends upon such behaviors in a plane. Just as quaternions may be useful for very compact modeling of continual rotation in three dimensions. In pure math, they are interesting of their own right. As they apply in reality, you have to keep some things in mind. (For example, devices are real-valued and yet electronics engineers use complex-valued voltages, currents, and impedances all the time.) $e^{s=\sigma+j \omega}$ is quite compact and easy to work with using Euler's. – jonk Apr 1 at 18:39
• @ViniciusACP Rather than papers, I did find a web site that may help a little. It includes a table of real and imaginary inductance values and it deals with a similar subject (I think.) It's here. – jonk Apr 1 at 21:01