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 - be the magnetic flux in each loop of coil 2, created by the current in coil 1. If coil 2 contains N2 turns, the is the total flux in coil 2

the Mutual Inductance

from Faraday’s Law

• This relates the change in current in coil 1 to the EMF induced in coil 2

1,2B

1,22 BN

1

2

1,21,2

I

NMDefine

B

dt

dNBut

B 1,2

22

dt

dIM

dt

dI

N

MN

dt

dI

N

M

dt

dand

N

IM

B

B

11,2

1

2

1.222

1

2

1,2

2

11,2

1,2

1,2

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

• The mutual inductances M1,2 = M2,1 so that

the SI Units for mutual inductance is

the Henry (H)

1H = 1V·s/A = 1Ω·s

dt

dIM 2

2,11

dt

dIM

dt

dIM

12

21

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

l

NnwhererlnnM 2

1210

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

• The induced EMF E, due to the self-inductance of the coil is (from Faraday’s Law):

• Self Inductance is also measured in Henrys

INL B

dt

dIL

dt

IL

B

B

dt

dIL

dt

dN B

l

ANLthen

l

ANIABSince

20

0

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

22LXRZ

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

• The inductance per unit length is:

2

10 ln2 r

r

l

L