Single Phase AC Circuits

(1) Reactance:


Inductive Reactance ($X_L$) = ωL = 2πfL

Capacitive Reactance ($X_C$) = 1/{ωC}

(2) Impedance (Z):
Z = R + jX
Z = |Z|∠ϕ

Magnitude, |Z| = $√{R^2 + X^2}$
Phase angle, ϕ= $tan^{-1}[X/R]$

(3) Purely resistive AC Circuit:

$ v=V_m sinωt $

$ V_m = peak voltage $

$ i = I_m sinωt $
$ I_m = peak current $

$ i = I_m ∠0° $

Avg. Power, $P_{av} = V_{rms} I_{rms} $ watt

Z = R i.e. Z = R∠0°

(4) Purely inductive AC Circuit:
$ v=V_m sinωt $

$ i = I_m sin(ωt – π/2) $
Avg. Power, $P_{av} = 0 $

Z = j$X_L$ i.e. Z = $X_L ∠90°$

(5) Purely capacitive AC Circuit:
$ v=V_m sinωt $

$ i = I_m sin(ωt + π/2) $
Avg. Power, $P_{av} = 0 $

Z = -j$X_C$ i.e. Z = $X_L ∠-90°$

(6) Series RL Circuit:

Z = R + j$X_L$
Z = |Z|∠ϕ

Magnitude, |Z| = $√{R^2 + X_L_^2}$
Phase angle, ϕ= $tan^{-1}[X_L/R]$


$ i = I_m sin(ωt – ϕ) $
Avg. Power, $P_{av} = V_{rms} I_{rms} cosϕ $ watt

Apparant Power (S) = V x I Volt-ampere

Real/True/Active Power (P) = $P_{av} = V_{rms} I_{rms} cosϕ $ watt

Reactive/Imaginary Power (Q) = VI sinϕ kVAR

Power factor = cosϕ = $R/{|Z|}$

(7) Series RC Circuit:
Z = R – j$X_C$
Z = |Z|∠ϕ

Magnitude, |Z| = $√{R^2 + X_C_^2}$
Phase angle, ϕ= $tan^{-1}[-X_C/R]$


Power formulae are same as that of series RL circuit

(8) Series RLC Circuit
Z = R + j$(X_L – X_C)$
Z = |Z|∠ϕ

Magnitude, |Z| = $√{R^2 + {(X_L – X_C)}^2}$
Phase angle, ϕ= $tan^{-1}[{X_L – X_C}/R]$


R = |Z| cosϕ
X = $(X_L – X_C)$ = |Z| sinϕ



Power formulae are same

(9) Resonance in Series RLC Circuit
$X_L = X_C$

$ f_r = {1}/{2π√{LC}} $ Hz