# Schematics: Effect factor and impedance

impedanceschematics

I have a questions regarding a simple schematic.
I would like to calculate the total angle (φ) and power factor (cos φ) of this circuit and the total impedance (Z).

Not using the complex-method (jω)

Values are as follows:

\$R_{L} = 120 Ω \$

\$L = 800 mH => X_{L} = 251 Ω \$

\$R_{C} = 40 Ω \$

\$C = 16 µF => X_{C} = 199 Ω \$

\$f = 50 Hz \$

My initial thoughts were to do a normal calculation for parallel connections, where I would only take the length of the vectors.

\$Z = {\frac{Z_{1}*Z_{2}}{Z_{1}+Z_{2}}} \$

Where \$Z_{1} = \sqrt{R_{L}² + (X_{L})²} \$ ;
\$Z_{2} = \sqrt{R_{C}² + (X_{C})²} \$

Which gives me a total of \$Z = 117 Ω \$

And then calculating the angle as \$arcsin(\frac{X_{L}-X_{C}}{Z}) => φ = 26.4° => cosφ = 0.89\$

Which is WRONG. The correct effect factor should be \$cosφ = 0.85\$.

Could you please explain what parts I've misunderstood and give me any ideas to solve it. Im able to solve the circuit using the jω-method but by using that I feel like I'm missing fundamental parts that I would be better off learning and applying with the regular vector calculations.

simulate this circuit – Schematic created using CircuitLab

EDIT:
I think I could do something like
\$Z = {\frac{\sqrt{X_{L}² + R_{L}²}*\sqrt{X_{C}² + R_{C}²}}{\sqrt{(X_{L}+X_{C})²+(R_{L}+R_{C})²}}} \$

In that way, I'm splitting the real and imaginary parts and taking them seperatly. And then for the angle, I'm thinking something along the lines of \$arg(Z_{1}) – arg(Z_{2}) + arg(\frac{\sqrt{(X_{L} – X_{C})²}}{\sqrt{(R_{L} + R_{C})²}}) \$ but unfortunately with that I get a degree of φ = 56.25° => cosφ = 0.55.
How should I do to calculate the correct effect factor, angle and impedance? What am I missing?

\$\dfrac{Z_1 Z_2}{Z_1+Z_2}\$
Imagine what sort of rubbish answer you'd get if you did this for an inductor in parallel with a capacitor at resonance. The right answer is infinite impedance and how you'd get that without respecting \$j\omega\$ (or s) is impossible.