PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Alternating Current Analysis 233

Observe that the phase angle θ between the current iR(t) and the voltage vR(t) in a

resistive circuit is zero, a condition that is referred to as in phase.

R.3.26 Let the current through an inductor L be i(t) = Im cos(ωt), then its voltage is given by

vt L

di t

dt

L

d

dt

Lm() It

()

[cos()]

vtLm()
LIsint

vtLm()
LIcos t



2









then

vtLm() 
Vcos t



2









Clearly, if vL(t) is a sinusoidal wave, then iL(t) is also sinusoidal with the same

frequency, but with a phase shift of π/2 rad.

Observe that the inductor voltage vL(t) leads its current iL(t), by an angle of

π/2 rad.

R.3.27 From R.3.26, the following relations can be observed: Vm = ωLIm, then by Ohm’s law,

ωL is the inductive reactance or the impedance of the inductor L in ohms, expressed

as XL(ω) = jωL, where j indicates a phase angle of π/2 rad. The inductive reactance

opposes the fl ow of current, which results in the interchange of energy between the

source and the magnetic fi eld of the inductor.

R.3.28 In R.3.26, a current through the inductor was assumed and the voltage was then

evaluated across the inductor L. The same result can be obtained by assuming a

voltage across L, and solving for the current through iL(t) as illustrated as follows:

Let

vL(t) = Vm sin(t)

then

it

L

vtdt

L

()

Lm() Vtdtsin( )

11

∫∫



it

V

L

()
m( cos( ))tK



 ( where is the initial current)K

it

V

L

()
msin tK








2









(as uming 0 w hsitout any loss of geenerality)

Again, the reader can appreciate that by letting XL(ω) = jωL, then

I

V

L X

L

L

()

()

()