Prof. David R. JacksonECE Dept.

Fall 2014

Notes 31

ECE 2317 Applied Electricity and Magnetism

1

Inductance

WbˆS

B n dS n̂ unit normal to loop

The unit normal is chosen from a “right hand rule for the inductor”:

HLI

n̂

I

S

The fingers are in the direction of the current reference direction, and the thumb is in the direction

of the unit normal.

Definition of inductance:

Single turn coil

2

Note: Please see the Appendix for a summary of right-hand rules.

The current I produces a flux though the loop.

Magnetic flux through loop:

Note: L is always positive

N-Turn Solenoid

N

total flux

NL

I I

flux through one turn

The definition of inductance is now

3

NI

We assume here that the same flux cuts though each of the N turns.

Example

2

2

20

20

ˆ

z

r z

r

a B n

a B

a H

a nI

Find LL N

I

20 /rL N a nI I

N turns

z r

I

a

Ls

so ˆ ,

0,

H z nI a

a

4

Note: We neglect “fringing” here and assume the

same magnetic field as if the solenoid were

infinite.

:

ˆ ˆn z

RHrule for inductor flux

/ sn N L

Example

2

20 Hr

s

NL a

L

N turns

z r

I

a

Ls

Final result:

5

Note: L is increased by using a high-permeability core!

Example

Find H inside toroid

Find L

6

Note: This is a practical structure, in which we

do not have to neglect fringing and assume that the length is infinite.

Toroidal Inductor

r

IN turns

R

Example (cont.)

7

Assume

ˆH HA is the cross-sectional area:

aA

r

8

2A a

The radius R is the average radius (measured to the center of the toroid).r

IN turns

R

Example (cont.)

Example (cont.)

ˆencl

C

H d I

A/mˆ [ ]2

NIH

encl

C

H dr I

2enclI

H

Hence2

0

enclH d I

so

9

We then have

enclI NIRH rule in Ampere's law :

N turns

r

I

C r

Example (cont.)

0

0

ˆ

2

R

r R

r

NL B n A

IN

B AIN

H AIN NI

AI R

2

0 [H]2r

NL A

R

NL

I

Hence

A/mˆ [ ]2

NIH

10

ˆn̂

RH rule for inductor flux :

S

a

2A a

r

I N turns

R

Example

Coaxial cableIgnore flux inside wire and shield

Find Ll

(inductance per unit length)

A short is added at the end to form a closed loop.

S

h

+-

I

I

z

Center conductor

Note: S is the surface shaded in green.11

I

I

h

a

bz

Example (cont.)

LI

ˆ

S S S

B n dS B dS H dS

n̂

12

S

h

+-

I

I z Center conductor

ˆn̂

RH rule for inductor flux :

Example (cont.)

S

H dS

ln2 2

b bb

aa a

I Ih H d h d h

n̂

a

b

13

S

h

+-

I

I z Center conductor

Example (cont.)

ln2

ln2

b

a

Ih

I bh

a

1ln

2

bL h

I a

0 H/mln [ ]2

rl

bL

a

Hence

Per-unit-length:

14

I

I

h

a

bz

Example (cont.)

0 H/mln [ ]2

rl

bL

a

Hence:

15

Recall:

0l

l

LZ

C

0 /m2

[F ]ln

rlC

ba

From previous notes:

00 ln [ ]

2r

r

bZ

a

00

0

376.7603 [ ]

(intrinsic impedance of free space)

Note: r =1 for most practical coaxial cables.

Voltage Law for Inductor

ˆC S

B dE dr n dS

t dt

C

+ -

v (t)

B

n̂

16

ˆS

B n dS

Apply Faraday’s law:

C

v t E dr Note: There is no electric field inside the wire.

PEC wire:

dv t

dt

Hence

Note: The unit normal (and hence the direction of the flux) is chosen from the right-hand rule

according to the direction of the path C.

Note: We are using the “active” sign convention.

Li d Li

vdt

Voltage Law for Inductor (cont.)

dv

dt

div L

dt

i

C

+ -v

B

n̂so

or

17

Note: The direction of the

current is chosen to be consistent with the

direction of the defined flux, according to the RH

rule for the inductor.

Note: This assumes a passive sign convention.di

v Ldt

To agree with the usual circuit law (passive sign convention), we change the

reference direction of the voltage drop.

(This introduces a minus sign.)

i

+-v

B

Voltage Law for Inductor (cont.)

Passive sign convention (for loop)

18

Energy Stored in Inductor

0 0

2 2

00

1 1

2 2

t t

i t i t

ii

div t L

dtdi

P t vi L idt

diW t P t dt L i dt

dt

L i di L i Li

21J

2HU Li

i t = 0

L

+ v(t) -

+ -

Hence we have

19

+ v -

Energy Formula for Inductor

2

2 HUL

I

2

1

V

L B H dVI

1

2H

V

U B H dV

We can write the inductance in terms of stored energy as:

Next, we use

We then have

20

This gives us an alternative way to calculate inductance.

Example

22 2

0

1

2H rs

NU a I

L

Find L

2

20 Hr

s

NL a

L

Solenoid

From a previous example, we have

2

2 HUL

IEnergy formula:

Hence, we have

21

N

Ls

za

Example

Coaxial cable

2

02

202

22

020 0

2

0

1

1

1

2

1 12

2

r

V

r

V

h b

r

a

b

r

a

L H dVI

H dVI

Id d dz

I

h d

Find Ll 2

1

V

L B H dVI

22

We ignore magnetic stored energy from fields inside the conductors.

We would have these at DC, but not at high frequency due to the skin effect.

I

I

hbz

a

Example (cont.)

2

0

0

1 12

2

ln2

b

r

a

r

L h d

h b

a

0 H/mln2

rl

bL

a

Per-unit-length:

23

Note: We could include the inner wire region if we had a solid wire at DC. We could even include the energy stored inside the shield at DC.

These contributions would be called the “internal inductance” of the coax.

I

I

hbz

a

Appendix

24

In this appendix we summarize the various right-hand rules that we have seen so far in electromagnetics.

Summary of Right-Hand Rules

Faraday’s law:C

dE dr

dt

Stokes’ theorem: ˆC S

V dr V n dS Fingers are in the direction of the path C, the thumb gives the direction of the unit normal.

Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the flux).ˆ

S

B n dS (fixed path)

Ampere’s law: encl

C

H dr I Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the current enclosed).ˆencl

S

I J n dS 25

The following two RH rules follow from Stokes’s theorem.

Summary of Right-Hand Rules (cont.)

Inductor definition:

LI

Fingers are in the direction of the current I and the thumb gives the direction of the unit normal (the reference direction for the flux).

ˆS

B n dS

Magnetic field law:

For a wire or a current sheet or a solenoid, the thumb is in the direction of the current and the fingers give the direction of the magnetic field. (For a current sheet or solenoid the fingers are simply giving the overall sense of the direction.)

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