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ECA1212
Introduction to Electrical &
Electronics Engineering
Chapter 3: Capacitors and Inductors
by Muhazam Mustapha, October 2011
Learning Outcome
By the end of this chapter students are
expected to:
• Understand the formula involving
capacitors and inductors and their duality
• Be able to conceptually draw the I-V
characteristics for capacitors and inductors
Chapter Content
• Units and Measures
• Combination Formula
• I-V Characteristics
Units and Measures
CO2
Capacitors
• Capacitors are electric devices that store static
electric charge on two conducting plates when
voltage is applied between them.
• Energy is stored as static electric field between
the plates.
Electrostatic
Field
CO2
+++++++++++++++++++++++++++
−−−−−−−−−−−−−−−−−−−−−−−−−−−
Capacitance, Charge & Voltage
• Capacitance: The value of a capacitor that
maintains 1 Coulomb charge when applied a
potential difference of 1 Volt across its
terminals.
Q = CV
Q = charge, C = capacitance (farad),
V = voltage
CO2
Inductors
• Inductors are electric devices that hold
magnetic field within their coils when current is
flowing through them.
• Energy is stored as the magnetic flux around
the coils.
Magnetic
Field
CO2
Inductance, Magnetic Flux &
Current
• Inductance: The value of an inductor that
maintains 1 Weber of magnetic flux when
applied a current of 1 Ampere through its
terminals.
Φ = LI
Φ = magnetic flux, L = inductance (henry),
I = current
CO2
Combination Formula
– Duality Approach
CO2
Inductors Combination
• Inductors behave (or look) more like resistors.
• Hence, circuit combination involving inductors
follow those of resistors.
• Series combination:
LEQ = L1 + L2 + L3
L1
L2
L3
• Parallel combination:
L1
L2
L3
CO2
1
1
1
1
 

L EQ L1 L 2 L3
Inductors Combination
• Series:
– Current is the same for all
inductors
– Equivalent flux is simple
summation
I1
I2
I3
Φ1
Φ2
Φ3
IEQ = I1 = I2 = I3
ΦEQ = Φ1 + Φ2 + Φ3
• Parallel:
– Equivalent current is simple
summation
– flux is the same for all inductors
Φ1
I1
Φ2
I2
Φ3
I3
IEQ = I1 + I2 + I3
CO2
ΦEQ = Φ1 = Φ2 = Φ3
Capacitors Combination
• The inverse of resistors are conductors; and the
dual of inductors are capacitors.
• If inductors behave like resistors, then
capacitors might behave like conductors – in
fact they are.
• Series combination:
C1 C2 C3
• Parallel combination:
1
1
1
1
CEQ

C1
C1
C2
C3
CO2
CEQ = C1 + C2 + C3

C2

C3
Capacitors Combination
• Series:
– Charge is the same for all
capacitors
– Equivalent voltage is
simple summation
• Parallel:
– Equivalent charge is simple
summation
– Voltage is the same for all
capacitors
Q1
Q2
Q3
V1
V2
V3
QEQ = Q1 = Q2 = Q3
VEQ = V1 + V2 + V3
Q1
V1
Q2
V2
Q3
V3
QEQ = Q1 + Q2 + Q3
VEQ = V1 = V2 = V3
CO2
I-V Characteristics
CO2
Capacitors
• At the instant of switching on, capacitors
behave like a short circuit.
• Then charging (or discharging) process starts
and stops after the maximum charging
(discharging) is achieved.
• When maximum charging (or discharging) is
achieved, i.e. steady state, capacitors behave
like an open circuit.
• Voltage CANNOT change instantaneously, but
current CAN.
CO2
Capacitors
• I-V relationship and power formula of a
capacitor
dq
dv
i
C
dt
dt
2
q
1Q
1
1
2
W   Vdq  
dq 
 CV  VQ
q 0
q 0 C
2 C 2
2
Q
CO2
Q
Capacitors
• Charging Current:
i
V
R
V t / RC
i e
R
i
C
V
τ
CO2
R
time constant, τ = RC
2τ
3τ
4τ
Charging period finishes after 5τ
5τ
t
Capacitors
• Charging Voltage:
v
R
C
V
v
V
v  V (1  et / RC )
τ
CO2
time constant, τ = RC
2τ
3τ
4τ
Charging period finishes after 5τ
5τ
t
Capacitors
• Discharging Current:
R
i
C
V
i
Discharging period finishes after 5τ
τ
2τ
V
i   e t / RC
R
V

R
CO2
3τ
4τ
5τ
time constant, τ = RC
t
Capacitors
• Discharging Voltage:
v
R
C
V
v
V
v  Ve  t / RC
τ
CO2
time constant, τ = RC
2τ
3τ
4τ
Discharging period finishes after 5τ
5τ
t
Inductors
• At the instant of switching on, inductors behave
like an open circuit.
• Then storage (or decaying) process starts and
stops after the maximum (minimum) flux is
achieved.
• When maximum (or minimum) flux is achieved,
inductors behave like a short circuit.
• Current CANNOT change instantaneously, but
voltage CAN.
CO2
Inductors
• I-V relationship and power formula of a inductor
d
di
v
L
dt
dt

1
1 2 1
W   Id  
d 
 LI  I
 0
 0 L
2 L 2
2

CO2

2
Inductors
• Storing Current:
i
V
R
i
V
(1  e t /( L / R ) )
R
i
L
V
τ
CO2
R
time constant, τ = L/R
2τ
3τ
4τ
Storing period finishes after 5τ
5τ
t
Inductors
• Storing Voltage:
v
R
L
V
v
V
v  Vet /( L / R )
τ
CO2
time constant, τ = L/R
2τ
3τ
4τ
Storing period finishes after 5τ
5τ
t
Inductors
• Decaying Current:
i
i
τ
V t /( L / R )
e
R
i
L
V
V
R
CO2
R
time constant, τ = L/R
2τ
3τ
4τ
Decaying period finishes after 5τ
5τ
t
Inductors
• Decaying Voltage:
R
L
V
v
i
Discharging period finishes after 5τ
τ
2τ
v  Ve
−V
CO2
 t /( L / R )
3τ
4τ
5τ
time constant, τ = L/R
t