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ECE2262 Electric Circuits

Chapter 6: Capacitance and Inductance

Capacitors

Inductors

Capacitor and Inductor Combinations

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CAPACITANCE AND INDUCTANCE

Introduces two passive, energy storing devices: Capacitors and Inductors

LEARNING GOALS

CAPACITORS Store energy in their electric field (electrostatic energy) Model as circuit element

INDUCTORS Store energy in their magnetic field Model as circuit element

CAPACITOR AND INDUCTOR COMBINATIONS Series/parallel combinations of elements

RC OP-AMP CIRCUITS Integration and differentiation circuits

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6.1 Capacitors

Electronic Symbol:

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• A capacitor consists of two conductors separated by a non-conductive region. The non-conductive region (orange) is called the dielectric Charge separation in a parallel-plate capacitor causes an internal electric field. • A dielectric reduces the electric field and increases the capacitance. Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than if the conductors were separated, giving the capacitor a large capacitance.

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• In the hydraulic analogy, a capacitor is analogous to a rubber membrane sealed inside a pipe. This animation illustrates a membrane being repeatedly stretched and un-stretched by the flow of water, which is analogous to a capacitor being repeatedly charged and discharged by the flow of charge.

https://en.wikipedia.org/wiki/File:CapacitorHydraulicAnalogyAnimation.gif

• In the hydraulic analogy, charge carriers flowing through a wire are analogous to water flowing through a pipe. A capacitor is like a rubber membrane sealed inside a pipe. Water molecules cannot pass through the membrane, but some water can move by stretching the membrane. The analogy clarifies a few aspects of capacitors. •The more a capacitor is charged, the larger its voltage drop; i.e., the more it "pushes back" against the charging current. This is analogous to the fact that the more a membrane is stretched, the more it pushes back on the water.

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• The capacitance describes how much charge can be stored on one plate of a capacitor for a given "push" (voltage drop). A very stretchy, flexible membrane corresponds to a higher capacitance than a stiff membrane.

• A charged-up capacitor is storing potential energy, analogously to a stretched membrane. C : Capacitance of 1 Farad = 1 Coulomb of charge on each conductor causes a voltage of 1 Volt across the device.

1C = 1F !1V

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Current!Voltage Relations

C !Ad

The charge on the capacitor is proportional to the voltage across

q = C v

C : Capacitance in Farads (F) - common values µF , pF pico = 10!12( )

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Example

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Voltage !Current Relation

Since q = C v and i = dqdt

then we have i =d C v( )dt

= C d vdt

.

i t( ) = C dv t( )

dt

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i t( ) = C dv t( )dt

In circuits with with time-independent voltage ( DC circuits) the capacitor behaves as open circuit (capacitor blocks dc).

In analyzing a circuit containing dc voltage sources and capacitors, we can replace the capacitors with an open circuit and calculate voltages and currents in the circuit using our many analysis tools.

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Example

C = 5µF

i = Cdvdt

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Current ! Voltage Relation

i t( ) v t( )

Since i t( ) = C d v t( )dt

then

i !( )d!t0

t

" = C d v !( )d!

d!t0

t

" = C dv !( )t0

t

" = = C v t( )! v t0( ){ }

v t( ) = v t0( )+ 1C

i !( )d!t0

t

" t ! t0

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v t( ) = v t0( )+ 1C

i !( )d!t0

t

" for any time t ! t0

v t0( ) - initial value (initial state) representing the voltage due to the charge that accumulates on the capacitor from the past to t = t0 .

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Example Suppose that C = 1F and v 0( ) = 1V . The current source i t( ) = e! t /2 , t > 0 . Determine v t( ) .

i t( ) v t( )

v t( ) = v 0( ) + 1C

i !( )d!0

t

" = 1+ e!" /2 d"0

t

# = 1! 2 e!" /2{ }0

t

= 1+ 2 1! e! t /2{ } for t ! 0 ! v t( ) = 1+ 2 1! e! t /2{ } t > 0

1 t " 0

#$%

&%

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v t( ) = 1+ 2 1! e! t /2{ } t > 0

1 t " 0

#$%

&%

Note: i t( ) has a jump, but v t( ) is continuous !

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Power Delivered to Capacitor

Since i t( ) = C d v t( )dt

or v t( ) = v t0( ) + 1C

i !( )d!t0

t

" then we have

p t( ) = v t( )i t( )

= C v t( ) d v t( )dt

= i t( ) v t0( ) + 1C

i !( )d!t0

t

"#$%

&'(

Note: If v t( ) would have a jump then d v t( )dt

= ! and p t( ) = ! but this is

impossible. Hence, the voltage response of a capacitor must be continuous !

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If we encounter circuits containing switches then the idea of “continuity of voltage” for a capacitor tells us that the voltage across the capacitor just after a switch moves is the same as the voltage across the capacitor just before that switch moves.

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Energy Stored in the Capacitor Electric Field

p t( ) = C v t( ) d v t( )dt

= C 12d v2 t( )dt

! ddt

v2 t( ){ } = 2v t( ) ddt

v t( ){ }

= ddt

12C v2 t( )!

"#

$%&

w t( ) = 1

2C v2 t( )

(J)

Since q t( ) = Cv t( ) then also w t( ) = 12q2 t( )C

.

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Example An uncharged capacitor 2 F is driven by a rectangular current pulse.

1

1i t( )t

(a)The capacitor voltage

v t( ) = v 0( ) + 12

i !( )d!0

t

" = 121 0 ! " !1( )d"0

t

# = t / 2 0 ! t !11/ 2 t > 1

!"#

.

(b) The capacitor power: p t( ) = i t( )v t( ) = t / 2 0 ! t !10 else

!"#

.

(c) The capacitor energy: w t( ) = 12C v2 t( ) = v2 t( ) = t 2 / 4 0 ! t !1

1/ 4 t > 1!"#

$#.

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Note that power is always positive for the duration of the current pulse, which means that energy is continuously being stored in the capacitor. When the current returns to zero, the stored energy is trapped because the ideal capacitor offers no means for dissipating energy. Thus, a voltage remains on the capacitor after i t( ) returns to zero.

If at some later time an energy-absorbing device (e.g., a flash bulb) is connectedacross the capacitor, a discharge current will flow from the capacitor and, therefore, the capacitor will supply its stored energy to the device.

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Example

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• Energy is being stored in the capacitor whenever the power is positive

• Energy is being delivered by the capacitor whenever the power is negative

• p t( )dt = 4µJ0

1

! - energy stored

p t( )dt = !4µJ1

"

# - energy delivered

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6.2 Inductors

An inductor, also called a coil, choke or reactor, is a passive two terminalelectrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in a magnetic field in the coil.

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When the current flowing through an inductor changes, the time-varying magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic induction.An inductor is characterized by its inductance, the ratio of the voltage to the rate of change of current, which has units of henries (H). Inductors have values that typically range from 1 !H (10!6H) to 1 H.

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A capacitor is an integrator of its input current: v t( ) = v t0( ) + 1C

i !( )d!t0

t

"

L Inductance in henrys( )

An inductor produces a current response that is related to the integral of the voltage applied to its terminals.

i t( ) = i t0( ) + 1L

v !( )d!t0

t

"

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Current ! Voltage Relation

L v t( ) = L di t( )dt

Voltage ! Current Relation

L i t( ) = i t0( )+ 1L

v !( )d!t0

t

"

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Power Delivered to an Inductor p t( ) = v t( )i t( )

= Li t( )di t( )dt

= v t( ) i t0( ) + 1L

v !( )d!t0

t

"#$%

&'(

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Energy Stored in the Inductor Magnetic Field

Since p t( ) = i t( )L di t( )dt

! p t( ) = 12L di

2 t( )dt

! p t( ) = ddt

12Li2 t( )!

"#

$%&

w t( ) = 1

2Li2 t( )

(J)

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DC Circuits

Consider the case of a dc current flowing through an inductor. Since

v t( ) = L di t( )dt

we see that the voltage across the inductor is directly proportional to the time rate of change of the current flowing through the inductor. A dc current does not vary with time, so the voltage across the inductor is zero. We can say that an inductor is “a short circuit to dc.”

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In analyzing a circuit containing dc sources and inductors, we can replace any inductors with short circuits and calculate voltages and currents in the circuit using our many analysis tools.

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Continuity of i t( ) ! p t( ) = Li t( ) di t( )dt

Due to p t( ) = Li t( ) di t( )dt

we note that an instantaneous change in inductor

current would require infinite power. Since we don’t have any infinite power sources, the current flowing through an inductor cannot change instantaneously.

This will be a particularly helpful idea when we encounter circuits containing switches. This idea of “continuity of current” for an inductor tells us that the current flowing through an inductor just after a switch moves is the same as the current flowing through an inductor just before that switch moves.

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Example The independent current source generates zero current for t < 0 and a pulse i t( ) = 10te!5t A for t > 0 in the following circuit

(a) i t( )

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(b) v t( ) = L di t( )dt

= 0.1!10e"5t 1" 5t( ) = e!5t 1! 5t( ) , t > 0

• At t = 0.2s the voltage changes polarity • At t = 0 the voltage has a jump, i.e., the voltage can change instantaneously across the terminals (but not current !)

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(c) p t( ) = i t( )v t( ) = 10te!5t{ }" e!5t 1! 5t( ){ } = 10te!10t ! 50t 2e!10t W

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(d) w t( ) = 12Li2 t( ) = 1

2! 0.1! 10te"5t( )2 = 5t 2e!10t

• An increasing energy curve indicates that energy is being stored. Thus, energy is being stored in the time interval 0 to 0.2 sec. This corresponds to the interval when p t( ) > 0 .

• A decreasing energy curve indicates that energy is being extracted. This takes place in the time interval 0.2 to ! . This corresponds to the interval when p t( ) < 0

• Energy is max when i t( ) is max: wmax = 27.07mJ at t = 0.2 sec.

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(e) Integrals p t( )dt0

0.2

! , p t( )dt0.2

!

"

•! p t( )dt = 27.07mJ0

0.2

!

• p t( )dt0.2

!

" = #27.07mJ

The area p t( )dt0

0.2

! represents the energy stored in the inductor during 0,0.2[ ]

The area p t( )dt0.2

!

" represents the energy extracted

Energy Stored + Energy Extracted = 0

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Example Find the total energy stored in the circuit (DC circuit)

wC = 12Cv2 , wL =

12Li2

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DC circuit ! capacitors ! open circuit, inductors ! short circuit

We need Vc1 , Vc2 and IL1 , IL2

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• KCL at A: IL1 + 3! IL2 = 0 ! IL2 = IL1 + 3

• KVL (big loop): !9 + 6IL1 + 3IL2 + 6IL2 = 0 ! 6IL1 + 9IL2 = 9

! IL1 = !1.2A , IL2 = 1.8A

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• KVL (first loop): !9 + 6IL1 +VC1 = 0 ! VC1 = 9 ! 6IL1 = 16.2 V

• VC2 = 6IL2 = 10.8 V

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IL1 = !1.2A , IL2 = 1.8A , VC1 = 16.2A , VC2 = 10.8A

• wC1 =12C1VC1

2 = 2.62 mJ, wC2 =12C2VC2

2 = 2.92 mJ

• wL1 =12L1IL1

2 = 1.44 mJ, wL2 =12L2IL2

2 = 6.48 mJ

The total stored energy = 13.46 mJ

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6.3 Capacitor and Inductor Combinations

Series Capacitors

v = v1 + ....+ vN The charge gained by a plate of any capacitor must come from a plate of an adjacent capacitor, i.e.,

QCS

= QC1

+ ....+ QCN

1CS

=1C1

+ ....+1CN

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Example Find C2

C1 = 30µF v1 = 8V

C2v =12V

The charge on both capacitors must be the same

! Q = v1C1 = 8V( )! 30µF( ) = 240 µC

! Q = v2C2 ! C2 =Qv2

= Qv ! v1

= 240µC4V

= 60µF

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Example

• the capacitors have been charged before they were connected Q1 !Q2 !Q3

• Equivalent capacitance: 1Ceq

= 1C1

+ 1C2

+ 1C3

! Ceq = 1µF

• Total energy stored: w == 12C1VC1

2 +C2VC22 +C3VC 3

2{ } = 31µJ

• !v + 2 ! 4 !1= 0 ! v = !3V ! wCeq== 12Ceqv

2 = 4.5µJ

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Parallel Capacitors

The total stored charge: QT =Q1 + ....+QN ! vCT = vC1 + ....+ vCN

Cp = C1 + ....+CN

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Example Find each capacitor voltage

+

!+

++

!!

!400V

V1

V2 V3 V4

20µF

40µF

30µF

70µF9µF

• 20 || 40 ! C1 = 20 + 40 = 60µF

• C2 = 30! 70( ) || 9 = 30 ! 7030 + 70

+ 9 = 21 + 9 = 30µF

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+

!+

++

!!

!400V

V1

V2V3 V4

20µF

40µF

30µF

70µF9µF

C2 = 30µFC1 = 60µF

• Ceq = C1!C2 = 60 ! 3060 + 30

= 20µF !• Qeq = Ceq !V = 20 !10"6( ) 400( )=8mC

• V1 =8mC60µF

= 133V • V2 =8mC30µF

= 267V

•charge on 30µF & 70µF = Qeq ! charge on 9µF = 8 ! 267V " 9µF= 5.6mC

• V3 =5.6mC30µF

= 187 V, V4 =5.6mC70µF

= 80 V

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Series Inductors

v t( ) = v1 t( ) + ...+ vN t( ) = L1di t( )dt

+ ...+ LNdi t( )dt

= L1 + ....+ LN( ) di t( )dt

LS = L1 + ....+ LN

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Parallel Inductors

i t( ) = i1 t( ) + ...+ iN t( ) = i1 0( ) + 1L1

v !( )d!0

t

"#$%

&'(+ ...+ iN 0( ) + 1

LNv !( )d!

0

t

"#$%

&'(

i t( ) = i1 0( ) + ...+ iN 0( ){ }+ 1L1

v !( )d!0

t

" + ...+ 1LN

v !( )d!0

t

"#$%

&'(

i t( ) = i 0( ) + 1L1

+ ...+ 1LN

!"#

$%&

v '( )d'0

t

(

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1Lp

=1L1

+ ...+1LN

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Example • Initial values: i1 0( ) = 3A , i2 0( ) = !5A • The voltage at the terminal v t( ) = !30e!5t mV for t ! 0

(a) Equivalent inductance: Leq =L1 ! L2L1 + L2

= 60 ! 24060 + 240

= 48mH

(b) Initial current: i 0( ) = i1 0( ) + i2 0( ) = 3+ !5( ) = !2A (current goes up)

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(c) Find i t( ): i t( ) = i 0( ) + 1Leq

v !( )0

t

" d!

= !2 + 148 "10!3 !30e!5# "10!3{ }

0

t

$ d# = !2 + 18e!5t !1( )

= !2.125 + 0.125e!5t A, t ! 0

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(d) Find i1 t( ) , i2 t( )

i1 t( ) = i1 0( ) + 1L1

v !( )0

t

" d! = 3+ 160 !10"3 "30e"5# !10"3{ }

0

t

$ d#

= 2.9 + 0.1e!5t A, t ! 0

i2 t( ) = i2 0( ) + 1L2

v !( )0

t

" d! = !5 + 1240 "10!3 !30e!5# "10!3{ }

0

t

$ d#

= !5.025 + 0.025e!5" A, t ! 0

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SUMMARY

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■ In dc steady state, a capacitor looks like an open circuit and an inductor looks like a short circuit.

■ • The voltage across a capacitor cannot change instantaneously

• The current flowing through an inductor cannot change instantaneously.