Chapter 31B - Transient Currents and Inductance

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Chapter 31B

-

_-

Transient

_Transient

Currents and Inductance

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

©

2007

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Objectives:

After completing this

module, you should be able to:

•

• Define and calculate

Define and calculate

inductance

in

terms of a changing current.

•

• Discuss and solve problems involving

Discuss and solve problems involving

the

rise

and

decay

of current in

capacitors

and

inductors

.

•

• Calculate the

Calculate the

energy

stored in an

inductor

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Self

-

_-

Inductance

_Inductance

R

Increasing

I

Consider a coil connected to resistance

R

and voltage

V

. When switch is closed, the rising current

I

increases flux, producing an internal back emf in the coil. Open switch reverses emf.

Consider a coil connected to resistance

R

_R

and and voltage

voltage

V

. When switch is closed, the rising . When switch is closed, the rising current

current

I

_I

increases flux, producing an internal increases flux, producing an internal back emf in the coil. Open switch reverses emf.

back emf in the coil. Open switch reverses emf.

R

Decreasing

I

Lenz

Lenz’’s Law:s Law: The

The back emf back emf (red arrow) (red arrow)  must oppose must oppose change in flux: change in flux:

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Inductance

The back emf

The back emf E_E induced in a coil is proportional induced in a coil is proportional to the rate of change of the current

to the rate of change of the current



_

I/



t

.

;

inductance

i

L

t



 





E

An

An inductance inductance of one of one henry henry (H)

(H) means that current means that current changing at the rate of

changing at the rate of one one ampere per second

ampere per second will induce will induce a back emf of

a back emf of one voltone volt..

R Increasing i/ t 1 V 1 H 1 A/s 

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Example 1:

Example 1: A coil having A coil having 20 turns 20 turns has an has an induced emf of

induced emf of 4 mV 4 mV when the current is when the current is changing at the rate of

changing at the rate of 2 A/s2 A/s. What is the . What is the inductance? inductance?

;

/

i

L

t

i

t











 

E

( 0.004 V)

2 A/s

L



 

_L

_{= 2.00 mH}_{= 2.00 mH}

Note: We are following the practice of using lower case

i

for transient or changing current and upper case I for steady current.

Note:

Note: We are following the practice of using We are following the practice of using lower case

lower case

i

_i

for transient or for transient or changing current changing current and upper case

and upper case I I for for steady currentsteady current..

R

i/ t = 2 A/s 4 mV

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Calculating the Inductance

Recall two ways of finding

Recall two ways of finding E:_E:

i

L

t



 



E

N

t



 



E

Setting these terms equal gives:

i

N

L

t









Thus, the inductance

L

can be found from:

Thus, the inductance

L

can be found from:

N

L

I





Increasing i/ t R Inductance

L

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Inductance of a Solenoid

The

B

_B

--field created by a field created by a current

current

I

for length for length l _lis:is: 0 NI B 



 and



= BA

0

NIA

N

L

I





 





Combining the last

two equations gives:

2 0

N A

L







R Inductance

L

l

B

Solenoid

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Example 2:

Example 2: A solenoid of area A solenoid of area 0.002 m0.002 m2 2 _and_and

length

length 30 cm30 cm, has , has 100 turns100 turns. If the current . If the current increases from

increases from 0 0 to to 2 A 2 A in in 0.1 s0.1 s, what is the , what is the inductance of the solenoid?

inductance of the solenoid?

First we find the inductance of the solenoid:

-7 2 2 2 _{T m} 0

(4 x 10

A

)(100) (0.002 m )

0.300 m

N A

L











R

l

A

L

= 8.38 x 10= 8.38 x 10-5 -5 _HH

Note:

L

does NOT depend on current, but on physical parameters of the coil.

Note:

L

_L

does does NOT NOT depend depend on

on currentcurrent, but on , but on physical physical parameters

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Example 2 (Cont.):

Example 2 (Cont.): If the current in the If the current in the 83.883.8- -

H H solenoid increased from solenoid increased from 0 0 to to 2 A 2 A in in 0.1 s0.1 s, , what is the induced emf?

what is the induced emf?

R

l

A

L

= 8.38 x 10= 8.38 x 10-5 -5 _HH

i

L

t



 



E

-5 (8.38 x 10 H)(2 A - 0) 0.100 s   E

E

 

_{1.68 mV}

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Energy Stored in an Inductor

At an instant when the current

is changing at



i/



t

, we have:, we have:

;

i

L

P

i

Li

t







E

Since the power

P

= Work/t

, ,

Work = P



t

. Also . Also the average value of

the average value of

Li

is is

Li/2

during rise to the during rise to the final current

final current

I

.

Thus, the total energy stored is:Thus, the total energy stored is: Potential energy stored in inductor: 2 1 2

U



Li

R

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Example 3:

Example 3: What is the potential energy What is the potential energy stored in a

stored in a 0.3 H 0.3 H inductor if the current rises inductor if the current rises from 0 to a final value of

from 0 to a final value of 2 A2 A??

2 1 2

U



Li

2 1 2

(0.3 H)(2 A)

0.600 J

U



U

= 0.600 J This

This energy energy is equal to the is equal to the work work done in done in reaching the

reaching the final current final current

I

; it is returned ; it is returned when the current decreases to zero.

when the current decreases to zero.

L = 0.3 H

I

= 2 A

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Energy Density (Optional)

R

l

A

The energy density

u

_u

is the is the energy

energy

U

per unit volume per unit volume

V

2 2 0 1 2 ; ; N A L 



U  LI V  A  Substitution gives

Substitution gives

u = U/V :

_{u = U/V :}

2 2 0 2 2 0 1 2 2 ; N AI N A U U I u V A         _ _  _ _        2 2 0 2

2 N I

u







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Energy Density (Continued)

R

l

A

2 2 0 2

2 N I

u







Energy Energy density: density:

Recall formula for B

Recall formula for B--field:field: 0 0

NI

B











2 ₂ 0 0 2 0

2

2 NI

B

u













_

_



_

_







_

_

2 0

2 B

u





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Example 4:

Example 4: The final steady current in a The final steady current in a

solenoid of 40 turns and length 20 cm is 5 A.

What is the energy density?

R

l

A

-7 0 (4 x 10 )(40)(5 A) 0.200 m NI B 







 B = 1.26 B = 1.26 mTmT 2 -3 2 -7 T m 0 A

(1.26 x 10 T)

2 2(4

x 10

)

B

u









u

= 0.268 J/m3

u

= 0.268 J/m3 Energy density is Energy density is

important for the

study of electro

study of electro- -magnetic waves.

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The R

-

_-

L Circuit

_{L Circuit}

R

L

S

₂

S

₁

V

E An inductor

An inductor

L

_L

and resistor and resistor

R

are connected in series are connected in series and

and switch 1 switch 1 is closed:is closed:

i

V

–

E

_E

=

iR

_L i t    E i V L iR t    

Initially,



i/



t

is large, making the back emf large and the current

i

small. The current rises to its

maximum value

I

when rate of change is zero.

Initially,



i/



t

is large, making the is large, making the back emf back emf large large and the current

and the current

i

small. The current rises to its small. The current rises to its maximum value

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The Rise of Current in L

( / )

(1

R L t

)

V

i

e

R







At t = 0, I = 0 At t = 0, I = 0 At t = At t = , I = V/R, I = V/R The time constant

The time constant



L

R

 

In an inductor, the current will rise to 63% of its maximum value in one time constant



= L/R.

In an inductor, the current will rise to

In an inductor, the current will rise to 63% 63% of its of its maximum value in one time constant

maximum value in one time constant



= L/R= L/R.. Time,

t

I

i

Current Current Rise Rise



0.63 I

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The R

-

_-

L Decay

_{L Decay}

R

L

S

₂

S

₁

V

Now suppose we close

S

_S

₂₂

after energy is in inductor:

E

=

iR

L i t    E i L iR t  _ 

Initially,



i/



t

is large and the emf E driving the current is at its maximum value

I

. The current decays to zero when the emf plays out.

Initially,



i/



t

is large and the is large and the emf emf E_E driving the driving the current is at its maximum value

current is at its maximum value

I

. The current . The current decays

decays to zero when the emf plays out.to zero when the emf plays out. For current For current decay in L: decay in L: _E

i

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The Decay of Current in L

( / )R L t

V

i

e

R





At t = 0, At t = 0,

i

= V/R= V/R At t = At t = , ,

i

= 0= 0 The time constant

The time constant



L

R

 

In an inductor, the current will decay to 37% of its maximum value in one time constant



In an inductor, the current will decay to 37%

of its maximum value in one time constant



Time,

t

I

i

Current Current Decay Decay



0.37 I

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Example 5:

Example 5: The circuit below has a The circuit below has a 4040--mH mH inductor connected to a

inductor connected to a 55-- resistor and a resistor and a 16

16--V V battery. What is the time constant and battery. What is the time constant and what is the current after one time constant?

what is the current after one time constant?

5



L

= 0.04 H 16 V

R

0.040 H

5 L

R







Time constant:



= 8 ms Time constant:



= 8 ms ( / )

(1

R L t

)

V

i

e

R







After time After time



i = 0.63(V/R)

16V

0.63

5 i





_



_







i

= 2.02 A

(20)

The R

-

_-

C Circuit

_{C Circuit}

R

C

S

₂

S

₁

V

E Close

Close

S

_S

₁₁. Then as charge . Then as charge Q

Q builds on capacitor builds on capacitor CC, a , a back emf

back emf E_E results:results:

i

V

–

E

_E

=

iR

Q C  E Q V iR C  

Initially,

Q/C

is small, making the back emf small and the current

i

is a maximum

I.

As the charge Q builds, the current decays to zero when E_b=

V.

Initially,

Q/C

_Q/C

is small, making the is small, making the back emf back emf small small and the current

and the current

i

is a maximum is a maximum

I

.

As the charge As the charge Q

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Rise of Charge

t = 0, Q = 0, t = 0, Q = 0, I = V/R I = V/R t = t =  , i = , i = , , QQ_m_m = C V= C V

The time constant



R C

 

In a capacitor, the charge Q will rise to 63% of its maximum value in one time constant



In a capacitor, the charge

Q

Q will rise to will rise to 63% 63% of its of its maximum value in one

maximum value in one

time constant time constant





Q V iR C   /

(1

t RC

)

Q



CV



e



Of course, as charge rises, the current

i

will will decaydecay.. Time,

t

Q

_max

q

Increase in Increase in Charge Charge Capacitor



0.63 I

(22)

The Decay of Current in C

/ t RC

V

i

e

R





At t = 0, At t = 0,

i

= V/R= V/R At t = At t = , ,

i

= 0= 0 The time constant

The time constant



R C

 

The current will decay to 37% of its maximum value in one time constant



the charge rises.

The current will decay to

The current will decay to 37% 37% of its maximum of its maximum value in one time constant

value in one time constant



the charge rises.the charge rises. Time,

t

I

i

Current Current Decay Decay Capacitor



0.37 I As charge Q increases

(23)

The R

-

_-

C Discharge

_{C Discharge}

Now suppose we close

S

_S

₂₂

and allow

C

to discharge:to discharge:

E

=

iR

Q C  E Q iR C 

Initially,

Q

is large and the emf E driving the current is at its maximum value

I

. The current decays to zero when the emf plays out.

Initially,

Q

is large and the is large and the emf emf E_E driving the driving the current is at its maximum value

current is at its maximum value

I

. The current . The current decays

decays to zero when the emf plays out.to zero when the emf plays out. For current For current decay in L: decay in L:

R

S

₂

S

₁

V

i

C

E

(24)

Current Decay

At t = 0, I = V/R At t = 0, I = V/R At t = At t = , I = 0, I = 0

In a discharging capacitor, both current and

charge decay to 37% of their maximum values in one time constant



= RC.

In a discharging capacitor, both current and

charge decay to 37% of their maximum values

in one time constant



= RC.= RC.

/ t RC

V

i

e

R





Time,

t

I

i



0.37 I R C  

As the current decays,

the charge also decays:

(25)

Example 6:

Example 6: The circuit below has a The circuit below has a 44--F F capacitor connected to a

capacitor connected to a 33-- resistor and a resistor and a 12

12--V V battery. The switch is opened. What is battery. The switch is opened. What is the current after one time constant

the current after one time constant



??

Time constant:



= 12 s Time constant:



= 12 s /

(1

t RC

)

V

i

e

R







After time After time



i = 0.63(V/R)

12V

0.63

3 i





_



_







i

= 2.52 A

3



C = 4

F 12 V

R



= RC = (3 = RC = (3 )(4 )(4 F)F)

(26)

Summary

;

inductance

i

L

t



 





E

R

l

A

N

L

I





2 0

N A

L







Potential Energy Energy Density: 2 1 2

U



Li

2 0

2 B

u





(27)

Summary

( / )

(1

R L t

)

V

i

e

R







L R  

In an inductor, the current will rise to 63% of its maximum value in one time constant



= L/R.

In an inductor, the current will rise to

In an inductor, the current will rise to 63% 63% of its of its maximum value in one time constant

maximum value in one time constant



= L/R= L/R.. Time,

t

I

i

Current Current Rise Rise  0.63I Inductor

The initial current is zero due to fast-changing current in coil. Eventually, induced emf becomes zero, resulting in the maximum current V/R.

(28)

Summary (Cont.)

( / )R L t

V

i

e

R





The current will decay to 37% of its maximum value in one time constant



= L/R.

The current will decay to

The current will decay to 37% 37% of its maximum of its maximum value in one time constant

value in one time constant



= L/R.= L/R.

Time,

t

I

i

Current Current Decay Decay  0.37I Inductor

The initial current, I = V/R, decays to zero as emf in coil dissipates.

The initial current,

I = V/R

I = V/R, decays to , decays to zero as emf in coil

zero as emf in coil

dissipates.

(29)

Summary (Cont.)

When charging a capacitor the charge rises to 63% of its maximum while the current decreases to 37% of its maximum value.