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Contents
• Capacitors
• Series and Parallel Capacitors
• Inductors
• Series and Parallel Inductors
Capacitors
• The capacitor consists of two parallel plates
separated by the insulator or known as dielectric.
The symbol of the basic capacitor is shown in
Figure 5.1

Figure 5.1

• The capacitance is defined as the value of capacitor when the

potential difference across the capacitor is one volt and is charged
by one coulomb of electricity.

Q
C=
V
3
 Capacitors
• A capacitor consists of two conducting plates
separated by an insulator (or dielectric).
• The energy is stored in its electric field.

q = Cv
A
C=
d
Unit = farad (F)
I-V Characteristic of Capacitor

 q = Cv
dq dv
i= =C
dt dt
1 t
v =  idt
C −
 Power and Energy in Capacitor
The instantaneous power delivered to the capacitor is
 dv  dv
p = vi = v C  = Cv
 dt  dt
The energy stored in the capacitor is
t t dv t
w =  pdt = C  v dt = C  vdv
− − dt −

1 2t
t = −
1
(
= Cv | = Cv t − Cv − )2 1
( )2

2 2 2
where v(−  ) = 0 because the capacitor was uncharged at t = -.
1 2 q2
 w = Cv =
2 2C
 Cont’d
+ i + i
IS v IS v
_ _
v Energy absorbed
Charging Discharging
during charging mode
= Energy supplied
t during discharging mode
i
IS
t
−I S
 Properties of Capacitor
• A capacitor is an open circuit to dc.
dv(t )
i =C = 0 (no dc current)
dt

Capacitors
replaced by
open circuits
Parallel Capacitors

Applying KCL gives

i = i1 + i2 + i3 +    + iN
dv dv dv dv
i = C1 + C2 + C3 +    + CN
dt dt dt dt
N  dv dv
=   Ck  = Ceq
 k =1  dt dt
where Ceq = C1 + C2 + C3 +    + C N
Series Capacitors
Applying KVL gives
v = v1 + v2 + v3 +    + v N

v =  i (t )dt + v1 (t0 ) +  i(t )dt + v (t )

1 t 1 t
2 0
C1 t0 C2 t0

 i(t )dt + v (t )
1 t
+  + N 0
CN t0

N 1  t
 i (t )dt +  vk (t0 )
N

k =1 C   t 0
=  
k =1
 k 

i (t )dt + v(t0 )
1 t
=
Ceq 0  t

1 1 1 1 1
where = + + +  +
Ceq C1 C2 C3 CN

v(t0 ) =  vk (t0 )
N

k =1
 Example 1
Find Ceq.
 Example 2
Find v1, v2, v3. 1
=
1
+
1
+
1
Ceq 20m 30m 60m
 Ceq = 10mF
q = Ceq v = 10  10 −3  30 = 0.3 C
Because q = C1v1 = C2 v2 = C3v3 ,
q q
 v1 = = 15 V, v2 = = 10 V
C1 C2
q
v3 = =5V
C3
 Example

Find the equivalent capacitance seen at the

terminal of the circuit.

Ans: 40 µF
 Inductors
• An inductor consists of a coil of conducting wire.
• The energy is stored in its magnetic field.

di
v=L
dt
Unit = henry (H)

N 2 A
L=
l
I-V Characteristic of Inductor

di 1
v = L , di = vdt
dt L
1 t 1 t
 i =  v ( t ) dt =  v ( t ) dt + i ( t0 )
L − L t0
Power and Energy in Inductor
The power delivered to the inductor is
 di  di
p = vi =  L i = Li
 dt  dt
The energy stored in the inductor is
t t di t
w =  pdt =  Li dt = L  idi
− − dt −

= Li(t ) − Li(−  )
1 2t 1 1
= Li |
2 2

2 t = − 2 2
1 2 0
w = Li
2
Series Inductors

Applying KVL gives

v = v1 + v2 + v3 +    + v N
di di di di
v = L1 + L2 + L3 +    + LN
dt dt dt dt
 N  di di
=   Lk  = Leq
 k =1  dt dt
where Leq = L1 + L2 + L3 +    + LN
Parallel Inductors
Applying KCL gives
i = i1 + i2 + i3 +    + iN

i =  v(t )dt + i1 (t0 ) +  v(t )dt + i (t )

1 t 1 t
2 0
L1 t0 L2 t0

 v(t )dt + i (t )
1 t
+  + N 0
LN t0

N 1  t
 v(t )dt +  ik (t0 )
N

k =1 L   t 0
=  
k =1
 k 

v(t )dt + i (t0 )

1 t
=
Leq 0  t

1 1 1 1 1
where = + + +  +
Leq L1 L2 L3 LN

i (t0 ) =  ik (t0 )
N

k =1
 Example 1

Find Leq.
 Example
• A capacitor is open circuit to dc.
• An Inductor is short circuit to dc.

Find i, iL, vC.

v 12
i = iL = = =2A
R 1+ 5
vC = 5i = 10 V
 Summary
Series connections Parallel connections

1 1 1 1 1
Req = R1 + R2 + R3 +    + RN = + + +  +
Req R1 R2 R3 RN

1 1 1 1 1
= + + +  + Ceq = C1 + C2 + C3 +    + C N
Ceq C1 C2 C3 CN

1 1 1 1 1
Leq = L1 + L2 + L3 +    + LN = + + +  +
Leq L1 L2 L3 LN