0195136047.pdf

108 TIME-DEPENDENT CIRCUIT ANALYSIS

−

+

v(t) = Vm cos ωt i(t) C

R L

(a)

−

+

−

+

I 1 = I 1 ejθ^1

R

+ I 2 = I 2 ejθ^2

R −jωL

(c)

−

+

ejωt

i(t)

C

R L

(b)

Vm

2

−

+

V 2 me−jωt

Vm

2

Vm

jωC 2

1

−jωC

1

jωL

Figure E3.1.3RLCseries circuit with sinusoidal excitation.(a)Time-domain circuit with a sinusoidal

excitation.(b)Time-domain circuit with exponential excitations.(c)Transformed networks (one withs=jω

and the other withs=−jω).

In view of the redundancy that is found in the information contained inI ̄ 1 andI ̄ 2 as seen

from Example 3.1.3, only one component needs to be considered for the purpose of finding

the sinusoidal steady-state response. Notice that an exponential excitation of the formv(t)=

(Vmejφ)ejωt=V ̄mejωtproduces an exponential responsei(t)=

(

Imejθ

)

ejωt=I ̄mejωt, whereas

a sinusoidal excitation of the formv(t)=Vmcos (ωt+φ) produces a sinusoidal response

i(t)=Imcos (ωt+θ). The complex termsV ̄m=VmejφandI ̄m=Imejθare generally known as

phasors, with the additional understanding that a function such asv(t)ori(t) can be interpreted

graphically in terms of a rotating phasor in the counterclockwise direction (considered positive

for positiveωand positivet). When the frequency of rotation becomes a constant equal toω

rad/s, the projection of a rotating phasor on the real (horizontal) axis varies as cosωt, whereas its

projection on the imaginary (vertical) axis varies as sinωt.

The use of a single exponential function withs=jωto imply sinusoidal excitation (and

response) leads to the following volt–ampere relations.

ForR:

V ̄R=RI ̄R (3.1.21)

I ̄R=GV ̄R (3.1.22)

ForL:

V ̄L=jωLI ̄L=jXLI ̄L (3.1.23)

I ̄L=( 1 /j ω L)V ̄L=jBLV ̄L (3.1.24)

ForC:

V ̄C=( 1 /j ω C)I ̄C=jXCI ̄C (3.1.25)