Parallel RLC Admittance

Series circuits add impedances; parallel circuits add admittances. Admittance $Y = 1/Z$ is measured in siemens and splits into conductance G (the resistive part, $1/R$) and susceptance B (the reactive part), with capacitive and inductive susceptance in opposition. So they subtract first and then join G by Pythagoras. With G = 0.03 S, B_C = 0.08 S and B_L = 0.04 S, the net susceptance is 0.04 S and $Y = \sqrt{0.03^2 + 0.04^2} = 0.05$ S — the 3-4-5 triangle again, and an impedance of 20 Ω.

Working in admittance is not an affectation; it is what stops parallel problems from becoming reciprocal soup. The alternative is $1/Z = 1/R + 1/jX_L + 1/jX_C$ with complex reciprocals throughout, which is exactly the calculation people get wrong. Note that $B_C = 2\pi f C$ rises with frequency while $B_L = 1/2\pi f L$ falls, so the two cross at resonance and cancel — leaving $Y = G$, a minimum admittance and therefore a maximum impedance. That is the opposite of the series case, and it is why a parallel tank is used to reject a frequency rather than pass it.

Because only the magnitude $|B_C - B_L|$ matters, solving backwards for a susceptance has two answers; this solver returns the capacitive branch, so subtract the root instead if you know the circuit is inductive. The practical warning mirrors the series one: never add susceptances to conductance arithmetically. And watch the ideal-component assumption — a real parallel tank's Q is limited by the coil's own series resistance, which appears as a small extra conductance and stops the impedance at resonance from going anywhere near infinite.