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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 - Resistor R R 0 1/R
RF Cafe: Schematic Symbol - jωL ωL +π/2 -j/ωL
RF Cafe: Schematic Symbol - Capacitor -j/ωC 1/ωC -π/2 jωC
RF Cafe: Schematic Symbol - Series Inductors jω(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/2 jωC1C2/(C1+C2)
RF Cafe: Schematic Symbol - Series Resistor / Inductor R+jωL (R22L2)½ tan-1(ωL/R) (R-jωL)/(R22L2)
RF Cafe: Schematic Symbol - Series Resistor / Capacitor R-j/ωC (1/ωC)(1+ω2C2R2)½ -tan-1(1/ωCR) (R+j/ωC)/(R2+1/ω2C2)
RF Cafe: Schematic Symbol - Series Inductor / Capacitor j(ωL-1/ωC) (ωL-1/ωC) ±π/2 jωC/(1-ω2LC)
RF Cafe: Schematic Symbol - Resistor / Inductor / Capacitor R+j(ωL-1/ωC) [R2+(ωL-1/ωC)2]½ tan-1[(ωL-1/ωC)/R] RLC equations 1 - RF Cafe
RF Cafe: Schematic Symbol - Parallel Resistors R1R2/(R1+R2) R1R2/(R1+R2) 0 1/R1+1/R2
RF Cafe: Schematic Symbol - Parallel Inductors RLC equations 1 - RF Cafe RLC equations 2 - RF Cafe +π/2 RLC equations 3 - RF Cafe
RF Cafe: Schematic Symbol - Parallel Capacitors -j/ω(C1+C2) 1/ω(C1+C2) -π/2 jω(C1+C2)
RF Cafe: Schematic Symbol - Parallel Resistor / Inductor RLC equations 4 - RF Cafe ωLR/(R22L2)½ tan-1(R/ωL) 1/R-j/ωL
RF Cafe: Schematic Symbol - Parallel Resistor / Capacitor R(1-jωCR)/(1+ω2C2R2) R/(1+ω2C2R2)½ -tan-1(ωCR) 1/R+jωC
RF Cafe: Schematic Symbol - Parallel Inductor / Capacitor jωL/(1-ω2LC) ωL/(1-ω2LC) ±π/2 j(ωC-1/ωL)
RF Cafe: Schematic Symbol - Parallel Resistor / Inductor / Capacitor parallel rlc [(1/R)2+(ωC-1/ωL)2] tan-1[R(1/ωL-ωC)] 1/R+j(ωC-1/ωL)
parallel rl / r Impedance Z RLC equations 5 - RF Cafe
Magnitude |Z| RLC equations 6 - RF Cafe
Phase Angle ϕ RLC equations 7 - RF Cafe
Admittance RLC equations 8 - RF Cafe
parallel rl / c Impedance Z RLC equations 9 - RF Cafe
Magnitude |Z| RLC equations 10 - RF Cafe
Phase Angle ϕ RLC equations 11 - RF Cafe
Admittance RLC equations 12 - RF Cafe
parallel lc / rlc Impedance Z RLC equations 13 - RF Cafe
Magnitude |Z| RLC equations 14 - RF Cafe
Phase Angle ϕ RLC equations 15 - RF Cafe
Admittance RLC equations 16 - RF Cafe
parallel series rl / rc Impedance Z RLC equations 17 - RF Cafe
Magnitude |Z| RLC equations 18 - RF Cafe
Phase Angle ϕ Phase angle 19 - RF Cafe
Admittance RLC equations 20 - RF Cafe
paralel, series rlc Impedance Z RLC equations 21 - RF Cafe
Magnitude |Z| RLC equations 22 - RF Cafe
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)

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