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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
RF Cafe: Schematic Symbol - ResistorRR01/R
RF Cafe: Schematic Symbol - jωLωL+π/2-j/ωL
RF Cafe: Schematic Symbol - Capacitor-j/ωC1/ωC-π/2jωC
RF Cafe: Schematic Symbol - Series Inductorsjω(L1+L2±2M)ω(L1+L2±2M)+π/2-j/[ω(L1+L2±2M)]
RF Cafe: Schematic Symbol - Series Capacitors-(j/ω)(1/C1+1/C2)(1/ω)(1/C1+1/C2)-π/2jωC1C2/(C1+C2)
RF Cafe: Schematic Symbol - Series Resistor / InductorR+jωL(R22L2)½tan-1(ωL/R)(R-jωL)/(R22L2)
RF Cafe: Schematic Symbol - Series Resistor / CapacitorR-j/ωC(1/ωC)(1+ω2C2R2)½-tan-1(1/ωCR)(R+j/ωC)/(R2+1/ω2C2)
RF Cafe: Schematic Symbol - Series Inductor / Capacitorj(ωL-1/ωC)(ωL-1/ωC)±π/2jωC/(1-ω2LC)
RF Cafe: Schematic Symbol - Resistor / Inductor / CapacitorR+j(ωL-1/ωC)[R2+(ωL-1/ωC)2]½tan-1[(ωL-1/ωC)/R]RLC equations 1 - RF Cafe
RF Cafe: Schematic Symbol - Parallel ResistorsR1R2/(R1+R2)R1R2/(R1+R2)01/R1+1/R2
RF Cafe: Schematic Symbol - Parallel InductorsRLC equations 1 - RF CafeRLC equations 2 - RF Cafe+π/2RLC equations 3 - RF Cafe
RF Cafe: Schematic Symbol - Parallel Capacitors-j/ω(C1+C2)1/ω(C1+C2)-π/2jω(C1+C2)
RF Cafe: Schematic Symbol - Parallel Resistor / InductorRLC equations 4 - RF CafeωLR/(R22L2)½tan-1(R/ωL)1/R-j/ωL
RF Cafe: Schematic Symbol - Parallel Resistor / CapacitorR(1-jωCR)/(1+ω2C2R2)R/(1+ω2C2R2)½-tan-1(ωCR)1/R+jωC
RF Cafe: Schematic Symbol - Parallel Inductor / CapacitorjωL/(1-ω2LC)ωL/(1-ω2LC)±π/2j(ωC-1/ωL)
RF Cafe: Schematic Symbol - Parallel Resistor / Inductor / Capacitorparallel rlc[(1/R)2+(ωC-1/ωL)2]tan-1[R(1/ωL-ωC)]1/R+j(ωC-1/ωL)
parallel rl / rImpedance ZRLC equations 5 - RF Cafe
Magnitude |Z|RLC equations 6 - RF Cafe
Phase Angle ϕRLC equations 7 - RF Cafe
AdmittanceRLC equations 8 - RF Cafe
parallel rl / cImpedance ZRLC equations 9 - RF Cafe
Magnitude |Z|RLC equations 10 - RF Cafe
Phase Angle ϕRLC equations 11 - RF Cafe
AdmittanceRLC equations 12 - RF Cafe
parallel lc / rlcImpedance ZRLC equations 13 - RF Cafe
Magnitude |Z|RLC equations 14 - RF Cafe
Phase Angle ϕRLC equations 15 - RF Cafe
AdmittanceRLC equations 16 - RF Cafe
parallel series rl / rcImpedance ZRLC equations 17 - RF Cafe
Magnitude |Z|RLC equations 18 - RF Cafe
Phase Angle ϕPhase angle 19 - RF Cafe
AdmittanceRLC equations 20 - RF Cafe
paralel, series rlcImpedance ZRLC equations 21 - RF Cafe
Magnitude |Z|RLC equations 22 - RF Cafe
Phase Angle ϕtan-1(X1/R1)+tan-1(X2/R2)-tan-1[(X1+X2)/(R1+R2)]
Admittance1/(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)

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