Impedance can be complex!

Put your complex numbers and calculus to the test with this impedance calculation.
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Problem



 


An AC generator of rms voltage E is connected in series with two reactive impedances,

$Z_1$ and $Z_2$, as shown below:

$\theta$ is the argument of $Z_1$ and $\phi$ is the argument of $Z_2$.

Image
Impedance can be complex!


Find the power dissipated in $Z_2$

If the phase of $Z_2$ ($\phi$) is fixed whilst allowing the magnitude to vary, find the relationship between $Z_1$ and $Z_2$ when we achieve maximum power transfer to $Z_2$.