Download Notes 31 3318 Inductance

ECE 3318
Applied Electricity and Magnetism
Spring 2017
Prof. David R. Jackson
ECE Dept.
Notes 31
1
Inductance
Magnetic flux through loop:
   B  nˆ dS
n̂
S
 Wb
S
n̂  unit normal to loop
I
The unit normal is chosen from a
“right hand rule for the inductor flux”:
The fingers are in the direction of the current
reference direction, and the thumb is in the direction
of the unit normal.
Single turn coil
The current I produces a
flux though the loop.
Definition of inductance:
L

I
H
Note:
Please see the Appendix for a
summary of right-hand rules.
Note: L is always positive
2
N-Turn Solenoid
I
N
  total flux
 N
We assume here that the
same flux cuts though
each of the N turns.
  flux through one turn
The definition of inductance is now
 N
L 
I
I
3
Example
Find L
LN
Ls
a
r
z
I
I
Note:
We neglect “fringing”
here and assume the
same magnetic field as
if the solenoid were
infinite.
N turns
   a 2  B  nˆ
n  N / Ls
  a 2  Bz
  a 2   0  r H z
RH rule for inductor flux:
  a 2  0 r  nI 
nˆ  zˆ
From before:
H  zˆ  nI  ,   a
 0,   a

so
L  N  a 2  0 r  nI  / I
4
Example (cont.)
Ls
a
r
z
I
N turns
Final result:
N2
2
L

a
0  r


Ls
 H
Note: L is increased by using a high-permeability core!
5
Example
Toroidal Inductor
N turns
I
r
 Find H inside toroid
 Find L
R
Note:
This is a practical structure, in which we
do not have to neglect fringing and
assume that the length is infinite.
6
Example (cont.)
7
Example (cont.)
N turns
I
r
r
R
The radius R is the average radius
(measured to the center of the toroid).
Assume
A
a
H  ˆ H
A is the cross-sectional area: A   a 2
8
Example (cont.)
 H  dr  I
encl
C
I

r
 H  ˆ  d   I
encl
C
C
r
N turns
so
2
 H   d  I
Hence
encl
0
H 
I encl
2
We then have
RH rule in Ampere's law :
I encl  NI
 NI 
ˆ
H 
 [A/m]
 2 
9
Example (cont.)
N
L
I
RH rule for the inductor flux :
n̂  ˆ
N turns
I
R
A   a2
N
L   B  nˆ  A
I
N
B   R A
I
N

0  r H    R A
I
N
 NI 
 0  r 
A
I
 2 R 


r

 NI 
ˆ
H 
 [A/m]
 2 
a

Hence
 N2 
L  0 r A 
 [H]
 2 R 
10
Example
Coaxial cable
Ignore flux inside wire and shield
(PEC, f > 0)
I
a
Find Ll
(inductance per unit length)
I
h
z
b
A short is added at the end
to form a closed loop.
I
+
z
I
Center conductor
S
h
Note: We calculate the flux through the surface S shaded in green.
11
Example (cont.)
RH rule for the inductor flux :
n̂  ˆ
I
+
I
z
Center conductor
S
n̂
h
L

I
   B  nˆ dS   B dS    H dS
S
S
S
12
Example (cont.)
I
+
I
z
Center conductor
n̂
S
h
 a
 b
    H dS
S
b
b
I
I
b
   h  H d    h 
d   h
ln  a
2
2
a
a
13
Example (cont.)
I
a
z
I
h
b
Hence
I
b
   h  ln  a
2
I
b
 h
ln  
2  a 

1 b
L   h
ln  
I
2  a 
Per-unit-length:
0  r  b 
Ll 
ln   [H/m]
2
a
14
Example (cont.)
Recall:
Ll 
Ll
Z0 
Cl
0  r  b 
ln   [H/m]
2
a
Note: r =1 for most practical coaxial cables.
From previous notes:
Hence:
2 0 r
Cl 
[F/m]
b
ln  
a
0
Z0 
2
r  b 
ln   []
r  a 
0 
0
 376.7603 []
0
(intrinsic impedance of free space)
15
Voltage Law for Inductor
Apply Faraday’s law:
C
B
B
d
C E  d r  S  t  nˆ dS   dt
n̂
A
+
B
S
-
v t 
   B  nˆ dS
Note:
The unit normal is chosen from the righthand rule in Faraday’s law, according to
the direction of the path C.
PEC wire:
B
v t    E  d r 
A
Hence
 E dr
C
d
v t   
dt
Note: There is no electric field inside the wire.
16
Voltage Law for Inductor (cont.)
B
d
v
dt
C
so
d  Li 
v
dt
or
di
v  L
dt
i
+
v t 
-
n̂
From the inductor
definition we have:
  Li
Note:
The direction of the current is
chosen to be consistent with the
direction of the flux, according to
the RH rule for the inductor flux.
Note: We are using the “active” sign convention here.
17
Voltage Law for Inductor (cont.)
B
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
+
Passive sign convention (for loop)
di
vL
dt
Note: This assumes a passive sign convention.
18
Energy Stored in Inductor
di
v t   L
dt
L
 di 
P  t   vi   L  i
 dt 
+
i
i  0  0
v
-
t
t
0
0
W  t    P  t  dt  L  i
t 0
it 
di
dt
dt
it 
1
1
 L  i di  L i 2
 Li 2
2 i 0 2
i 0
+-
v t 
Hence we have
For DC we have
1 2
U H  t   Li  t   J 
2
1 2
U H  LI  J 
2
19
Energy Formula for Inductor
We can write the inductance in terms of stored energy as:
2U H
L 2
I
Next, we use
We then have
1
U H   B  H dV
2
V
1
L  2  B  H dV
I V
This gives us an alternative way to calculate inductance.
20
Example
Find L using the energy formula
z
N
a
Solenoid
Ls
From a previous example, we have
Energy formula:
Hence, we have
L
2
1
2 N  2
U H  0  r  a 
I
2
 Ls 
2U H
I2
2


N
2
L  0 r a 

L
 s 
H
21
Example
Find Ll using the energy formula
I
a
z
1
B  H dV
2 
I V
1
2
L  2 0 r  H dV
I
V
I
b
L
h
1
 2 0 r  H2 dV
I
V
h 2 b
Coaxial cable
Note:
We ignore the magnetic stored energy
from fields inside the conductors.
We would have this at DC, but not at
high frequency due to the skin effect.

1
0  r 
2
I
0

0
2
 I 
a  2   d  d dz
2
1
 1 
 0  r 
  h  2   d 

 2 
a
b
22
Example (cont.)
2
1
 1 
L  0  r 
  h  2   d 

 2 
a
I
a
z
I
b
h
Per-unit-length:
b
0  r h  b 

ln  
2
a
0 r  b 
Ll 
ln  
2
a
 H/m
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.
23
Appendix
In this appendix we summarize the various right-hand
rules that we have seen so far in electromagnetics.
24
Summary of Right-Hand Rules
Stokes’s theorem:
 V  d r   V   nˆ dS
C
Faraday’s law:
(stationary path)
S
d
C E  d r   dt
   B  nˆ 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).
S
Ampere’s law:
 H  dr  I
I   J  nˆ dS
encl
C
encl
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).
S
25
Summary of Right-Hand Rules (cont.)
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.)
Inductor flux rule:
L

I
   B  nˆ dS
S
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).
26