0195136047.pdf

3.1 SINUSOIDAL STEADY-STATE PHASOR ANALYSIS 123

(a)

v(t)

Vm

t

T

Period T ==

4

T

2

3T

4

2 π

ω

1

f

−

+

−

+

V VC

20 Ω

=

(c)

−

• 
  

−

+

v(t)

i(t)

C = 0.1 F vC(t)

R = 20 Ω

(b)

I

1

jωC Ω

10

jω

Figure E3.1.10(a)Square wave.(b)Time-domain circuit.(c)Frequency-domain circuit.

a 0 =

1

T

∫T

0

f(t)dt=

1

T

(∫T/ 4

0

Vmdt+

∫T

3 T/ 4

Vmdt

)

=

Vm

2

From Equation (3.1.47),

an=

2

T

(∫T/ 4

0

Vmcosnωt dt+

∫T

π/ 4

Vmcosnωt dt

)

and

an=







0 , for evenn

±

2 Vm

nπ

, for oddn, where the algebraic sign is+forn= 1

and changes alternately for each successive term.

From Equation (3.1.48),

bn=

2

T

(∫T/ 4

0

Vmsinnωt dt+

∫T

3 T/ 4

Vmsinnωt dt

)

= 0

which can also be seen fromsymmetryof the square wave with respect to the chosen

origin.

The Fourier series is then

v(t)=

Vm

2

+

2 Vm

π

cosωt−

2 Vm

3 π

cos 3ωt+

2 Vm

5 π

cos 5ωt−

2 Vm

7 π

cos 7ωt+···

(b) The time-domain and frequency-domain circuits are shown in Figures E3.1.10(b) and

(c). Note that the capacitive impedance is expressed in terms ofω, since the frequency

of each term of the Fourier series is different. The general phasor expressions forI ̄and

V ̄Care given by