Chapter 31B Chapter 31B -- Transient Transient Currents and InductanceCurrents and Inductance

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 2007

Objectives: Objectives: After completing this After completing this module, you should be able to:module, you should be able to:

•• Define and calculate Define and calculate inductanceinductance in in terms of a changing current.terms of a changing current.

•• Discuss and solve problems involving Discuss and solve problems involving thethe riserise and and decaydecay of current in of current in capacitors capacitors and and inductorsinductors..

•• Calculate the Calculate the energyenergy stored in an stored in an inductor inductor and find the and find the energy densityenergy density..

SelfSelf--InductanceInductance

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 Consider a coil connected to resistance RR and and voltage voltage VV. 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 ILenzLenz’’s Law:s Law:

The The back emfback emf (red arrow)(red arrow)

must oppose must oppose change in flux:change in flux:

InductanceInductanceThe back emfThe back emf EE

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/I/tt..

; inductanceiL Lt

E

An An inductanceinductance of one of one henryhenry (H)(H) means that current means that current changing at the rate of changing at the rate of one one ampere per secondampere per second will induce will induce a back emf of a back emf of one voltone volt..

R

Increasing i/ t

1 V1 H1 A/s

Example 1:Example 1: A coil having A coil having 20 turns20 turns has an has an induced emf of induced emf of 4 mV4 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?

; /

iL Lt i t

EE

( 0.004 V)2 A/s

L L = 2.00 mHL = 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 currentchanging current and upper case and upper case II for for steady currentsteady current..

R

i/ t = 2 A/s4 mV4 mV

Calculating the InductanceCalculating the Inductance

Recall two ways of finding Recall two ways of finding E:E:

iLt

EN

t

E

Setting these terms equal gives:Setting these terms equal gives:

iN Lt t

Thus, the inductance L can be found from: Thus, the inductance L can be found from:

NLI

Increasing i/ t

R

Inductance L

Inductance of a SolenoidInductance of a Solenoid

The The BB--field created by a field created by a current current II for length for length l l is:is:

0 NIB

and = BA

0 NIA NLI

Combining the last Combining the last two equations gives:two equations gives:

20N AL

R

Inductance L

lB

Solenoid

Example 2:Example 2: A solenoid of area A solenoid of area 0.002 m0.002 m22 and and length length 30 cm30 cm, has , has 100 turns100 turns. If the current . If the current increases from increases from 00 to to 2 A2 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:First we find the inductance of the solenoid:-7 2 22 T m

0 A(4 x 10 )(100) (0.002 m )0.300 m

N AL

R

l

AL = 8.38 x 10-5 HL = 8.38 x 10-5 H

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

Note: Note: L L does does NOTNOT depend depend on on currentcurrent, but on , but on physical physical parametersparameters of the coil.of the coil.

Example 2 (Cont.):Example 2 (Cont.): If the current in the If the current in the 83.883.8-- HH solenoid increased from solenoid increased from 00 to to 2 A2 A in in 0.1 s0.1 s, , what is the induced emf?what is the induced emf?

R

l

AL = 8.38 x 10-5 HL = 8.38 x 10-5 H

iLt

E

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

E 1.68 mV E

Energy Stored in an InductorEnergy Stored in an InductorAt an instant when the current At an instant when the current is changing at is changing at i/i/tt, we have:, we have:

; i iL P i Lit t

E E

Since the power Since the power PP = Work/t= Work/t, , Work = P Work = P tt. Also . Also the average value of the average value of LiLi is is Li/2Li/2 during rise to the during rise to the final currentfinal current II. . Thus, the total energy stored is:Thus, the total energy stored is:

Potential energy stored in inductor:

212U Li

R

Example 3:Example 3: What is the potential energy What is the potential energy stored in a stored in a 0.3 H0.3 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??

212U Li

212 (0.3 H)(2 A) 0.600 JU

U = 0.600 J

This This energyenergy is equal to the is equal to the workwork done in done in reaching the reaching the final current final current II; 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

R

Energy Density (Optional)Energy Density (Optional)

R

l

A

The energy density The energy density uu is the is the energy energy U U per unit volume per unit volume VV

220 1

2; ; N AL U LI V A

Substitution gives Substitution gives u = U/V :u = U/V :

2 20

2201

2

2;

N AIN A UU I u

V A

2 20

22N Iu

Energy Density (Continued)Energy Density (Continued)

R

l

A

2 20

22N Iu

Energy Energy density:density:

Recall formula for BRecall formula for B--field:field:

0

0

NI NI BB

2 20 0

202 2

NI Bu

2

02Bu

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. solenoid of 40 turns and length 20 cm is 5 A. What is the energy density?What is the energy density?

R

l

A

-70 (4 x 10 )(40)(5 A)

0.200 mNIB

B = 1.26 B = 1.26 mTmT2 -3 2

-7 T m0 A

(1.26 x 10 T)2 2(4 x 10 )Bu

u = 0.268 J/m3u = 0.268 J/m3

Energy density is Energy density is important for the important for the study of electrostudy of electro-- magnetic waves.magnetic waves.

The RThe R--L CircuitL Circuit

RL

S2

S1

V

E

An inductor An inductor LL and resistor and resistor RR are connected in series are connected in series and and switch 1switch 1 is closed:is closed:

iiV V –– E E = = iRiR iLt

E

iV L iRt

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, Initially, i/i/tt is large, making the is large, making the back emfback emf large large and the current and the current ii small. The current rises to its small. The current rises to its maximum value maximum value II when rate of change is zero.when rate of change is zero.

The Rise of Current in LThe Rise of Current in L( / )(1 )R L tVi e

R

At t = 0, I = 0At t = 0, I = 0

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

The time constant The time constant LR

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

Ii

Current Current RiseRise

0.63 I

The RThe R--L DecayL Decay

RL

S2

S1

V

Now suppose we close Now suppose we close SS22 after energy is in inductor:after energy is in inductor:

E E = = iRiRiLt

E

iL iRt

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, Initially, i/i/tt is large and the is large and the emf emf EE

driving the driving the current is at its maximum value current is at its maximum value II. The current . The current decaysdecays 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

ii

The Decay of Current in LThe Decay of Current in L

( / )R L tVi eR

At t = 0, At t = 0, ii = V/R= V/R

At t = At t = , , ii = 0= 0

The time constant The time constant LR

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

Time, t

Ii

Current Current DecayDecay

0.37 I

Example 5:Example 5: The circuit below has a The circuit below has a 4040--mHmH inductor connected to a inductor connected to a 55--

resistor and a resistor and a

1616--VV 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 H5

LR

Time constant:

= 8 msTime constant:

= 8 ms

( / )(1 )R L tVi eR

After time After time

i = 0.63(V/R)i = 0.63(V/R)

16V0.635

i i = 2.02 A

The RThe R--C CircuitC Circuit

RC

S2

S1

V

E

Close Close SS11 . Then as charge . Then as charge QQ builds on capacitor builds on capacitor CC, a , a back emf back emf EE

results:results:

iiV V –– E E = = iRiR QC

E

QV iRC

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 Eb = V.

Initially, Initially, Q/CQ/C is small, making the is small, making the back emfback emf small small and the current and the current ii is a maximum is a maximum II.. As the charge As the charge QQ builds, the current decays to zero when builds, the current decays to zero when EEbb = = V.V.

Rise of ChargeRise of Charget = 0, Q = 0, t = 0, Q = 0,

I = V/RI = V/R

t = t =

, i = , i = , , QQmm = C V= C V

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

QV iRC

/(1 )t RCQ CV e

Of course, as charge rises, the current Of course, as charge rises, the current ii will will decaydecay..

Time, t

Qmaxq

Increase in Increase in ChargeCharge

Capacitor

0.63 I

The Decay of Current in CThe Decay of Current in C

/t RCVi eR

At t = 0, At t = 0, ii = V/R= V/R

At t = At t = , , ii = 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

Ii

Current Current DecayDecay

Capacitor

0.37 I

As charge Q increases

The RThe R--C DischargeC DischargeNow suppose we close Now suppose we close SS22 and allow and allow CC to discharge:to discharge:

E E = = iRiRQC

E

Q iRC

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, Initially, QQ is large and the is large and the emf emf EE

driving the driving the current is at its maximum value current is at its maximum value II. The current . The current decaysdecays to zero when the emf plays out.to zero when the emf plays out.

For current For current decay in L:decay in L:

RS2

S1

V

iiC

E

Current DecayCurrent Decay

At t = 0, I = V/RAt 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 In a discharging capacitor, both current and charge decay to 37% of their maximum values charge decay to 37% of their maximum values in one time constant in one time constant = RC.= RC.

/t RCVi eR

Time, t

I i

Current Current DecayDecay

Capacitor

0.37 IR C

As the current decays, As the current decays, the charge also decays:the charge also decays:

/t RCQ CVe

Example 6:Example 6: The circuit below has a The circuit below has a 44--FF capacitor connected to a capacitor connected to a 33--

resistor and a resistor and a

1212--VV 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 sTime constant:

= 12 s

/(1 )t RCVi eR

After time After time

i = 0.63(V/R)i = 0.63(V/R)

12V0.633

i i = 2.52 A

3

C = 4 F

12 V

R

= RC = (3 = RC = (3 )(4 )(4 F)F)

SummarySummary

; inductanceiL Lt

E

R

l

ANLI

2

0N AL

Potential Energy Energy Density:

212U Li

2

02Bu

SummarySummary( / )(1 )R L tVi e

R

LR

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 RiseRise

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.

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

Summary (Cont.)Summary (Cont.)

( / )R L tVi eR

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 DecayDecay

0.37I

Inductor

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

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

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

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

Time, t

Qmaxq

Increase in Increase in ChargeCharge

Capacitor

0.63 I

Time, t

I i

Current Current DecayDecay

Capacitor

0.37 I

/(1 )t RCQ CV e RC /t RCVi e

R

CONCLUSION: Chapter 31BCONCLUSION: Chapter 31B Transient Current Transient Current -- InductanceInductance