0195136047.pdf

1.2 LUMPED-CIRCUIT ELEMENTS 25

w(t)=

∫t

−∞

p(τ) dτ=C

∫t

−∞

v(τ)

dv(τ )

dτ

=

1

2

Cv^2 (t)−

1

2

Cv^2 (−∞) (1.2.21)

Assuming the capacitor voltage to be zero att=−∞, the stored energy in the capacitor at some
timetis given by

w(t)=

1

2

Cv^2 (t) (1.2.22)

which depends only on the voltage of the capacitor at that time, and represents the stored energy
in the electric field between the plates due to the separation of charges.
If the voltage across the capacitor does not change with time, no current flows, as seen from
Equation (1.2.17). Thus the capacitor acts like an open circuit, and the following relations hold:

C=

Q

V

; I= 0 ,W=

1

2

CV^2 (1.2.23)

An ideal capacitor, once charged and disconnected, the current being zero, will retain a potential
difference for an indefinite length of time. Also, the voltage across a capacitor cannot change
value instantaneously, while an instantaneous change in the capacitor current is quite possible.
The student is encouraged to reason through and justify the statement made here by recalling
Equation (1.2.17).
Series and parallel combinations of capacitors are often encountered. Figure 1.2.6 illustrates
these.
It follows from Figure 1.2.6(a),
v=vAC=vAB+vBC
dv
dt

=

dvAB

dt

+

dvBC

dt

=

i

C 1

+

i

C 2

=i

(

C 1 +C 2

C 1 C 2

)

=

i

Ceq

or, whenC 1 andC 2 are in series,

Ceq=

C 1 C 2

C 1 +C 2

or

1

Ceq

=

1

C 1

+

1

C 2

(1.2.24)

Referring to Figure 1.2.6(b), one gets

i=i 1 +i 2 =C 1

dv

dt

+C 2

dv

dt

=(C 1 +C 2 )

dv

dt

=Ceq

dv
dt
or, whenC 1 andC 2 are in parallel,

Ceq=C 1 +C 2 (1.2.25)

Note that capacitors in parallel combine as resistors in series, and capacitors in series combine as
resistors in parallel.

B

C 1

i 1 i 2

C 2

D C

A

B

C 1

C 2

A

C

vv

i i

+

−

+

−

(a) (b)

Figure 1.2.6Capacitors in series and in parallel.(a)C 1

andC 2 in series.(b)C 1 andC 2 in parallel.