Recently I have been trying to understand complex magnetic impedance and when I try to formulate the equations for it based off the similar equations for electrical circuits I keep coming up with some weird answers that don't make sense. I realize that two of the equations regarding electrical fields I found on Wikipedia seem logically inconsistent and this may be the root of the problem as I depend on them in my formulations.
On this page on Wikipedia two related equations are defined relative to the tangent loss angle:
https://en.wikipedia.org/wiki/Dielectric_loss#Electromagnetic_field_perspective
The two equations are:
$$\tan(\delta) = \frac{\omega\epsilon'' + \sigma}{\omega\epsilon'}$$
$$tan(\delta) = \frac{\epsilon''}{\epsilon'}$$
However I notice if I set these equations equal to each other and simplify I get the following.
$$\frac{\omega\epsilon'' + \sigma}{\omega\epsilon'} = \frac{\epsilon''}{\epsilon'}$$
$$(\omega\epsilon'' + \sigma) \epsilon' = (\omega\epsilon')\epsilon''$$
$$\omega\epsilon'\epsilon'' + \sigma\epsilon' = \omega\epsilon'\epsilon''$$
$$\sigma\epsilon' = \omega\epsilon'\epsilon'' - \omega\epsilon'\epsilon''$$
$$\sigma\epsilon' = 0$$
But this can't possibly be correct can it? This would imply that either $\epsilon'$ or $\sigma$ must always be 0. However if $\epsilon'$ is ever 0 then both equations becomes undefined. Also if $\sigma$ is always 0 then the material would have to be a perfect resistor for the equations to make sense and also why even including conductivity as a variable at all in that case?
I am convinced I must be doing something wrong but I have no idea what that could be.