Impedance and Admittance Formulas for RLC Combinations

Here is an extensive table of impedance, admittance, magnitude, and phase angle equations (formulas) for fundamental series and parallel combinations of resistors, inductors, and capacitors. All schematics and equations assume ideal components, where resistors exhibit only resistance, capacitors exhibit only capacitance, and inductors exhibit only inductance.

For those unfamiliar with complex numbers, the "±j" operator signifies a phase of ±90°. Voltage across a capacitor lags the current through it by 90°, so -j is used along with its capacitive reactance (-j/ωC). Voltage across an inductor leads the current through it by 90°, so +j is used along with inductive reactance (jωL).

"M" is the mutual inductance between inductors.

"ω" is frequency in radians/second, and is equal to 2π times frequency in cycles/second.

This is probably one of the most comprehensive collections you will find on the Internet.

Z = R + jX           |Z| = (R² + X²)^½           ϕ = tan^-1(X/R)           Y = 1/Z

Circuit Configuration	Impedance Z = R + jX	Magnitude |Z| = (R2 + X2)½	Phase Angle ϕ = tan-1(X/R)	Admittance Y = 1/Z
	R	R	0	1/R
	jωL	ωL	+π/2	-j/ωL
	-j/ωC	1/ωC	-π/2	jωC
	jω(L1+L2±2M)	ω(L1+L2±2M)	+π/2	-j/[ω(L1+L2±2M)]
	-(j/ω)(1/C1+1/C2)	(1/ω)(1/C1+1/C2)	-π/2	jωC1C2/(C1+C2)
	R+jωL	(R2+ω2L2)½	tan-1(ωL/R)	(R-jωL)/(R2+ω2L2)
	R-j/ωC	(1/ωC)(1+ω2C2R2)½	-tan-1(1/ωCR)	(R+j/ωC)/(R2+1/ω2C2)
	j(ωL-1/ωC)	(ωL-1/ωC)	±π/2	jωC/(1-ω2LC)
	R+j(ωL-1/ωC)	[R2+(ωL-1/ωC)2]½	tan-1[(ωL-1/ωC)/R]
	R1R2/(R1+R2)	R1R2/(R1+R2)	0	1/R1+1/R2
			+π/2
	-j/ω(C1+C2)	1/ω(C1+C2)	-π/2	jω(C1+C2)
		ωLR/(R2+ω2L2)½	tan-1(R/ωL)	1/R-j/ωL
	R(1-jωCR)/(1+ω2C2R2)	R/(1+ω2C2R2)½	-tan-1(ωCR)	1/R+jωC
	jωL/(1-ω2LC)	ωL/(1-ω2LC)	±π/2	j(ωC-1/ωL)
		[(1/R)2+(ωC-1/ωL)2]-½	tan-1[R(1/ωL-ωC)]	1/R+j(ωC-1/ωL)
	Impedance Z
	Magnitude |Z|
	Phase Angle ϕ
	Admittance
	Impedance Z
	Magnitude |Z|
	Phase Angle ϕ
	Admittance
	Impedance Z
	Magnitude |Z|
	Phase Angle ϕ
	Admittance
	Impedance Z
	Magnitude |Z|
	Phase Angle ϕ
	Admittance
	Impedance Z
	Magnitude |Z|
	Phase Angle ϕ	tan-1(X1/R1)+tan-1(X2/R2)-tan-1[(X1+X2)/(R1+R2)]
	Admittance	1/(R1+jX1)+1/(R2+jX2)

 

Note: Corrections made to RLC Magnitude and Admittance formulas, and to RL||R Admittance formula on 7/3/2014. Thanks to Bob N. for catching the errors.

 

(source: Reference Data for Engineers, 1993)