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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 - jωLωLπ/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(R2+ω2L2)½tan-1(ωL/R)(R-jωL)/(R2+ω2L2)
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)[R+(ωL-1/ωC)]½tan-1[(ωL-1/ωC)/R]
RF Cafe: Schematic Symbol - Parallel ResistorsR1R2/(R1+R2)R1R2/(R1+R2)01/R1+1/R2
RF Cafe: Schematic Symbol - Parallel Inductorsπ/2
RF Cafe: Schematic Symbol - Parallel Capacitors-j/ω(C1+C2)1/ω(C1+C2)-π/2jω(C1+C2)
RF Cafe: Schematic Symbol - Parallel Resistor / InductorωLR/(R2+ω2L2)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 / Capacitor[(1/R)2+(ωC-1/ωL)2]-½tan-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 ϕtan-1(X1/R1)+tan-1(X2/R2)-tan-1[(X1+X2)/(R1+R2)]
Admittance1/(R1+jX1)+1/(R2+jX2)


(source Reference Data for Engineers, 1993)

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 RF Cascade Workbook is a very extensive system cascaded component Excel workbook that includes the standard Gain, NF, IP2, IP3, Psat calculations, input & output VSWR, noise BW, min/max tolerance, DC power cauculations, graphing of all RF parameters, and has a graphical block diagram tool. An extensive User's Guide is also included. - Only $35.
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