Transcript Unit 5: Day 8 – Mutual & Self Inductance

Unit 5: Day 8 – Mutual & Self Inductance
• Mutual Inductance
• Mutual Inductance of Coaxial
Solenoids
• Self Inductance
• Inductive Reactance & Impedance
• Inductance / Unit Length of a Coaxial Cable
Mutual Inductance
• If 2 coils are placed near
each other (in the same plane),
a changing current in one coil will induce an EMF in the
other coil
• From Faraday’s law, the EMF E2 induced in coil 2, is
proportional to the rate of change of the magnetic flux
passing through it
Mutual Inductance
• Let  B - be the magnetic flux in each loop of coil 2,
2 ,1
created by the current in coil 1. If coil 2 contains N2 turns,
the N 2 B is the total flux in coil 2
2 ,1
Define M 2,1 
N 2  B2 ,1
But  2   N 2
 B2 ,1  M 2,1
I1
d B2 ,1
dt
the Mutual Inductance
from Faraday’s Law
d B2 ,1 M 2,1 dI1
I1
and

N2
dt
N 2 dt
M 2.1 dI1
dI
 2  N2
  M 2,1 1
N 2 dt
dt
• This relates the change in current in coil 1 to the EMF
induced in coil 2
Mutual Inductance
• The mutual inductance of coil 2 with respect to coil 1 is a
constant and depends upon the geometry of the two
coils
• The reverse situation can also take place when a change
in current in coil 2 induces an EMF into coil 1
1   M 1, 2
dI 2
dt
• The mutual inductances M1,2 = M2,1 so that
dI 2
dt
dI
 2  M 1
dt
1   M
the SI Units for mutual inductance is
the Henry (H)
1H = 1V·s/A = 1Ω·s
Mutual Inductance of a Coaxial Solenoid
• A long thin solenoid of length l, radius r1, contains N1
turns
• Coil 2 is wrapped around coil 1, with a radius r2, contains
N2 turns
• The mutual inductance of the coils is
M   0 n1n2lr
2
1
N
where n 
l
Self Inductance
• The concept of inductance applies also to a single
isolated coil of N turns
• When a changing current passes through a coil
(solenoid), a changing magnetic flux is produced in the
coil. This in turn, induces an EMF in the same coil.
• This EMF opposes the change in flux (Lenz’s Law)
Self Inductance
• The magnetic flux ΦB passing through N-turns of the coil
is proportional to the current I inn the coil. This
proportionality is now called self inductance
B  L  I
B
LN
B
dI

L
I
dt
dt
• The induced EMF E, due to the self-inductance of the
coil is (from Faraday’s Law):
d B
dI
  N
 L
dt
dt
• Self Inductance is also measured in Henrys
Since   B  A   0 NI
A
then L   0 N
l
2
A
l
Inductance
• The inductance, L, is dependent on the geometry and
the presence of a core made out of ferro-magnetic
material
• The symbol for inductance is:
• Every electronic component, such as a resistor, or a
wire, has some amount of inductance called “parasitic
inductance”, which is usually unwanted.
Inductive Reactance
• Large inductance tends to oppose alternating current.
The greater the inductance, the less AC current that can
pass (ie: it impedes the flow of AC current similar to how
resistance impedes the flow of DC current)
• The opposition to AC current is called inductive
reactance (XL)
• The term impedance is used to represent the vector
magnitude of resistance and inductive reactance
Z  R 2  X L2
Inductance /Unit Length of a Coaxial Cable
• Conductors in a coaxial cable are thin hollow tubes.
There is no magnetic field within the inner conductor.
The magnetic field within the thin conductor tubes can be
ignored. The conductors carry equal currents in opposite
directions
L  0  r1 
• The inductance per unit length is:

ln  
l
2
 r2 