Inductance Chapter 30• mutual inductance

• self-inductance

• magnetic-field energy

• R-L circuits

• L-C circuits

• L-R-C circuits

1

http://physci.kennesaw.edu/javamirror/CCP/21-5/CircuitiE.html

A straight wire has little inductance. Coil the wire an inductance increases.

A change in current in a coil induces an emf in an adjacent coil

Only time varying currents can induce and emf

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indcur.html

Mutual InductancePotential induced in Coil-2

Flux in Coil-2 is proportional to the current in Coil-1, M12 is the mutual inductance

Assumed that magnetic material has a constant Km, so flux is directly proportional to current and M21 only depends on geometry

Even when the coils are not symmetric:

M=Henry= 1Wb/A

2

22 2

BdN

dt

2 2 12 1BN M i

2 12 21

Bd diN M

dt dt

12 21

diM

dt

12 21M M M

2 2 1 1

1 2

B BN NM

i i

Derivative with time

Mutual inductance—examples

3

M = mutual inductance is proportional to the of the turns N1N2

Self-inductance

4

5

Self-Inductance and Inductors

il

NKi

l

NB m 0

il

NAKBA mB 0 l

ANKL m

2

0

where L is the inductance in Henries

=

Liil

ANKN mB

2

0

Self-induced emf opposes changes in current• Self-inductance, L, depends upon on

size, shape, and turns, N• For N turns close together, L is

proportional to N2

• L depends upon magnetic material• If the core is not air, • For soft iron Km =5000 producing an L

5000 times greater than an air-core coil• Ferromagnetic material produce s L that

are not totally linear with current

6

0mK

7

Real Inductor is a combination of L and R

Vab

+

–

Ideal Inductor

+–Vab

Real Inductor

c

bcabac VVV

Ridt

diLVac

8

i

t

Inductor

Current and Voltage in an Inductive Circuit

Current can not change instantaneouslyvab

t

Inductor

i

t

vab

t

t0

t0

Inductor

Inductor

impossible

Theoretically the voltage can change instantaneously

9

Inductance (Chapter 30)Example 30-3 and 30-4 Calculating Self-Inductance and Induced EMF

Figure 30-8 Toroidal Solenoid

ir

NKi

r

NB m

22 0

ir

NAKBA mB

20

dt

diL

dt

di

r

ANK

dt

dNV m

BL

2

2

0

i

N

r

ANKL Bm

2

2

0

N = 200 turnsA = 5.0 cm2 = 5.0 x 10-4 m2

r = 0.1 m

i increases uniformly from 0 to 6.0 Ain 3.0 sec.

Determine L

Determine the magnitude anddirection of the induced emf ()

Km = 1

Inductance Application

– A charged coil can create a field that will induce a current in a neighboring coil.

– Inductance can allow a sensor to trigger the traffic light to change when the car arrives at an intersection. Circuit counts how many cars pass over the coil (how many changes in inductance).

– Bikes may not trigger circuit

– Drive back and forth in crease the car count?

10

11

Inductance (Chapter 30)The RL Circuit (30.4)

Figure 30-11

Figure 30-12 Increasing Current(S1 closed and S2 open)

Figure 30-13 Decreasing Current(S1 open and S2 closed)

0.37I0

0.63I

dt

diLVbc iRVab

dt

diLiRVV bcab

iL

R

LL

iR

dt

di

RLdt

di

t

0R

L

12

Details of current growth in an R-L circuit

0di

iR Ldt

iL

R

LL

iR

dt

di

Rdi i dt

L L

didt

Ri

L L

di Rdt

LiR

0

0 0

''

'

ln

ln ' | ' |

ln ln

ln

i t

o

i t

RtL

di Rdt

LiR

duu

uR

i tR L

Ri tR R L

i RR tL

R

iR e

R

Take exponent of both sides

13

1

RtL

RtL

RtL

RtL

iR e

R

i eR R

i eR R

i eR

L

R

Details of current growth in an R-L circuit -continued

0

RtLi e

R

IR

Decay current

R-L circuit current decay

Current thru L reaches I0 and S1 opens and S2 closes

14

0

RtLi e

R

IR

L-C oscillating circuit

• Consider Figure 30.14.

15

L-C Circuit

16

Mechanical Analog

17

Mechanical Analog L m mass1/C k spring constant

18

Inductance (Chapter 30)Energy Stored in an Inductor (30.3)

Figure 30-9

idt

diLiVp ab

pdtdw

i

o

t t

idiLdtdt

diLipdtw

0 0

dt

dwp

0

22

2

0

2 iL

iLw

i

2

2

1Liw joules

where w = energy stored in inductor i = current in inductor in amperes

(30.9)

L-R-C Circuit

19

20

21

22