ECA1212Introduction to Electrical &

Electronics EngineeringChapter 3: Capacitors and Inductors

by Muhazam Mustapha, October 2011

Learning Outcome

• Understand the formula involving capacitors and inductors and their duality

• Be able to conceptually draw the I-V characteristics for capacitors and inductors

By the end of this chapter students are expected to:

Chapter Content

• Units and Measures

• Combination Formula

• I-V Characteristics

Units and Measures

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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

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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

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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

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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

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Combination Formula– Duality Approach

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Inductors Combination

• Inductors behave (or look) more like resistors.• Hence, circuit combination involving inductors

follow those of resistors.• Series combination:

L1 L2 L3

LEQ = L1 + L2 + L3

• Parallel combination:L1

L2

L3321EQ L

1

L

1

L

1

L

1

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Inductors Combination

I1 I2 I3

IEQ = I1 = I2 = I3

Φ1

Φ2

Φ3

• Series:– Current is the same for all

inductors– Equivalent flux is simple

summation

• Parallel:– Equivalent current is simple

summation– flux is the same for all inductors

Φ1 Φ2 Φ3

ΦEQ = Φ1 + Φ2 + Φ3

I1

I2

I3

ΦEQ = Φ1 = Φ2 = Φ3

IEQ = I1 + I2 + I3

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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:• Parallel combination:

C1 C2 C3

CEQ = C1 + C2 + C3

C1

C2

C3

321EQ C

1

C

1

C

1

C

1

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Capacitors Combination• Series:

– Charge is the same for all capacitors

– Equivalent voltage is simple summation

Q1 Q2 Q3

QEQ = Q1 + Q2 + Q3

Q1

Q2

Q3

QEQ = Q1 = Q2 = Q3

V1 V2 V3

• Parallel:– Equivalent charge is simple

summation– Voltage is the same for all

capacitors

V1

V2

V3

VEQ = V1 = V2 = V3

VEQ = V1 + V2 + V3

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I-V Characteristics

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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.

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Capacitors

• I-V relationship and power formula of a capacitor

dt

dvC

dt

dqi

VQCVC

Qdq

C

qVdqW

Q

q

Q

q 2

1

2

1

2

1 22

00

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Capacitors• Charging Current:

i

t

RCteR

Vi /

V

R

C

i

time constant, τ = RC

τ 2τ 4τ3τ 5τ

R

V

Charging period finishes after 5τ

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Capacitors• Charging Voltage:

v

t

)1( / RCteVv

V

R

C v

time constant, τ = RC

τ 2τ 4τ3τ 5τ

Charging period finishes after 5τ

V

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Capacitors• Discharging Current:

i

t

RCteR

Vi /

V

R

C

i

time constant, τ = RC

τ 2τ 4τ3τ 5τ

R

V

Discharging period finishes after 5τ

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Capacitors• Discharging Voltage:

v

t

RCtVev /

V

R

C

time constant, τ = RC

τ 2τ 4τ3τ 5τ

Discharging period finishes after 5τ

V

v

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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.

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Inductors

• I-V relationship and power formula of a inductor

dt

diL

dt

dv

ILIL

dL

IdW

2

1

2

1

2

1 22

00

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Inductors• Storing Current:

i

t

)1( )//( RLteR

Vi

V

R

L

time constant, τ = L/R

τ 2τ 4τ3τ 5τ

Storing period finishes after 5τ

i

R

V

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Inductors• Storing Voltage:

v

t

)//( RLtVev time constant, τ = L/R

τ 2τ 4τ3τ 5τ

Storing period finishes after 5τ

V

V

R

L v

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Inductors• Decaying Current:

i

t

)//( RLteR

Vi time constant, τ = L/R

τ 2τ 4τ3τ 5τ

Decaying period finishes after 5τ

V

R

L

i

R

V

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Inductors• Decaying Voltage:

i

t

)//( RLtVev time constant, τ = L/R

τ 2τ 4τ3τ 5τ

Discharging period finishes after 5τ

−V

V

R

L v

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