Prof. D. Wilton ECE Dept. Notes 27 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston.

Prof. D. WiltonECE Dept.

Notes 27

ECE 2317 ECE 2317 Applied Electricity and MagnetismApplied Electricity and Magnetism

Notes prepared by the EM group,

University of Houston.

Mutual InductanceMutual Inductance

1n

I1

2n

I2

Two coils are in proximity of each other.

Current reference directions and unit normals are defined on both coils

(The unit normals are determined from the current reference directions, by the right-hand rule.)

Mutual Inductance (cont.)Mutual Inductance (cont.)

2

1 221

S

B n dS

2n

1n

21I1

In general, if coil 2 has multiple turns

2121

1

MI

21 2 2121

1 1

NM

I I

Coil 1 is energized. Coil 2 is left open-circuited

Define mutual inductance:

1

2 112

S

B n dS

2n

1n

12 I2

In general, if coil 1 has multiple turns,

1212

2

MI

12 1 1212

2 2

NM

I I

Define mutual inductance:

Coil 2 is energized. Coil 1 is left open-circuited

Mutual Inductance (cont.)Mutual Inductance (cont.)

Property: 12 21M M M

Mutual Inductance (cont.)Mutual Inductance (cont.)

Circuit Law for Coupled CoilsCircuit Law for Coupled Coils

1 21 1 12

2 12 2 21

di div L M

dt dtdi di

v L Mdt dt

+- v1

i1

+- v2

i2

M

L1L2

1 11 12

1 1 12 2L i M i

2 22 21

2 2 21 1L i M i

total flux through coil 1:

total flux through coil 2:

ExampleExample

1 1

12 1 1212

2 2

2 112 2

22 1

2 220 2 2 1 0 2 1

z

S S

z

s

NM

I I

B n dS B dS

B R

Nn I R I R

L

2212 1 0 1 H

s

NM N R

L

Bz2

R1

Find M12 , M21

Ls = length

R2

z

R1

I2

I1

Example (cont.)Example (cont.)

2 2

21 2 2121

1 1

21 221 1 1 1

2 2 11 0 1 1 1 0 1

z z

S S

s

NM

I I

B n dS B dS R B

NR n I R I

L

Bz1

R1

2 121 2 0 1

s

NM N R

L

R2

z

R1

I2

I1

Example (cont.)Example (cont.)

2212 1 0 1 H

s

NM N R

L

2 121 2 0 1

s

NM N R

L

2

112 21 0 1 2

s

RM M N N

L

R2

z

R1

I2

I1

Force on WireForce on Wire

charge:

wire:

F q v B

C

F I d B

qB

v

F

B

Idl

FThe contour C is in the direction of the reference direction for the current.

d dqdq v dq d Id

dt dt

ExampleExample

I2

I1

1

2

z

h

z = 0 z = L

x

2

2

2 12

12 0 2

C

C

F I z dz B

II z dz y

h

Note: there is no self-force on wire 2 due to the magnetic field produced by the current on wire 2.

Example (cont.)Example (cont.)

F

F

1 22 0

02

LI IF x dz

h

2

12 2 0 2C

IF I z dz y

h

1 22 0 N

2

I IF x L

h