# Solving for the impedance and phase shift in a series-parallel AC RCL Circuit with Complex Numbers

Having a lot of trouble working through an AC circuit, asking for some help.

The circuit is relatively simple - I have a capacitor of capacitance $C$ and resistor of resistance $R$ hooked in parallel, which is hooked in series with a coil loop of impedance $L$ to an alternating voltage source of $e_0\sin(\omega t)$.

I'll work through the steps to show you where I'm having my problems.

So, first, solving the parallel combination. $$\frac{1}{Z_t} = \frac{1}{Z_c} + \frac{1}{Z_r}$$

and we know that $Z_c = \frac{-i}{\omega C}$ and $Z_r = R$

So $$\frac{1}{Z_t} = \frac{1}{R} - \frac{\omega C}{i} = \frac{1}{R} + \omega Ci$$

and thus $$Z_t = \frac{R}{1+i \omega CR}$$

Now, to find the total impedance, $Z = Z_t + Z_l$

$$Z = \frac{R}{1+i \omega cR} + i \omega L = \frac{R(1- \omega^2 LCR) + i \omega L}{1+i \omega CR}$$

Is that right?

Additionally, the next problem I then have is the phase shift. By definition, the phase shift of a complex circuit is $$\phi = \arctan\left(\frac{\text{Imaginary}}{\text{Real}}\right)$$

How the heck am I supposed to get an imaginary and a real out of that equation? What parts are imaginary and what are real?

Assuming this equation is correct: $$Z = \frac{R}{1+i \omega cR} + i \omega L$$

We multiply the first term's nominator and the denominator by the complex conjugate of the denominator: $$.. = \frac{R(1-i\omega c R)}{(1+i \omega cR)(1-i\omega c R)} + i \omega L$$ and thus getting rid of imaginary stuff in the denominator: $$.. = \frac{R(1-i\omega c R)}{(1+ \omega^2 c^2R^2)} + i \omega L$$ And this thing is easily separated to real and imaginary parts: $$.. = \frac{R}{1+ \omega^2 c^2R^2} + i \left(\omega L - \frac{\omega c R^2}{1+ \omega^2 c^2R^2} \right)$$

Easiest method to get the overall phase angle is:

Phi = (phase angle of numerator) - (phase angle of denominator)

Phi = arctan[wL/R(1-w^2LCR)] - arctan(wCR)

This saves a lot of algebra.

Generally, if you have a complex impedance, Z, in the form: Z = (A+jB)/(C+jD), the best way to get modulus (or 'magnitude') and phase angle is:

Magnitude = SQRT(A^2 + B^2)/SQRT(C^2 + D^2)

Phase angle = arctan(B/A) - arctan(D/C)

(Note, in engineering we use 'j' as the imaginary operator, since 'i' is reserved for current)