ee132 mutual inductance

EE 132Electric Circuit Theory II

Magnetically-coupled circuits

Background So far, we have considered conductively coupled circuits,

i.e. one loop affects the neighboring loop through current conduction.

Definition. When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, they are said to be magnetically coupled.

Background Study of magnetically-coupled circuits - electrical circuits

linked not by hard electrical connections but by the flux lines of a magnetic field.

Transformers Transformers to step-up/step-down voltages Transformers to electrically isolate parts of a circuit Transformers for impedance matching Inadvertent coupling

Review of self-inductance, mutual inductance

Mutual Inductance The ability of one inductor to induce a voltage across a

neighboring inductor, measured in Henrys.

Consider a single conductor with, a coil with N turns.

When current i flows through the coil, a magnetic flux is produced around it.

Mutual Inductance Faraday’s Law. The voltage v induced in the coil is

proportional to the number of turns N and the time rate of change of the magnetic flux ; i.e.

Self inductance, L. The inductance that relates the voltage induced in a coil by a time-varying current in the same coil.

dN

dt

d di

Ndi dt

diL

dt For the inductor,

Thus, dL N

di

Mutual Inductance Consider two coils with self inductances L1 and L2 in close

proximity with each other with

an open-circuited secondary

For i2 = 0

where

Note: Although the two coils are physically separated, they are said to be magnetically coupled.

1 11 12

11

12

leakage flux

mutual flux

Mutual Inductance Since the entire flux links only coil 1, the voltage induced

in coil 1 is

Only flux links coil 2, thus

1

11 1

dN

dt

12

122 2

dN

dt

Mutual Inductance Since the entire flux links only coil 1, the voltage induced

in coil 1 is

Only flux links coil 2, thus

Since the fluxes are cause by the current i1 flowing in coil 1,

1

11 1

dN

dt

12

122 2

dN

dt

Mutual Inductance Since the entire flux links only coil 1, the voltage induced

in coil 1 is

Only flux links coil 2, thus

Since the fluxes are cause by the current i1 flowing in coil 1,

1

11 1

dN

dt

12

122 2

dN

dt

1 1 11 1 1

1

d di diN L

di dt dt

L1 = self inductance of coil 1

Mutual Inductance Since the entire flux links only coil 1, the voltage induced

in coil 1 is

Only flux links coil 2, thus

Since the fluxes are cause by the current i1 flowing in coil 1,

1

11 1

dN

dt

12

122 2

dN

dt

12 1 12 2 21

1

d di diN M

di dt dt

M21 = mutual inductance of coil 2 with respect to coil 1

Open-circuit mutual voltage across coil 2 (by current in

coil 1)

Mutual Inductance Suppose we now let current i2 flow in coil 2, while coil 1 is

open-circuited.

The magnetic flux emanating from coil 2 is

where

2 21 22

22

21

leakage flux

mutual flux

Mutual Inductance The entire flux links coil 2, so the voltage induced in coil

2 is

2

2 2 2 22 2 2 2

2

d d di diN N L

dt di dt dt

L2 = self inductance of coil 2

The entire flux links coil 2, so the voltage induced in coil 2 is

Since only flux links coil 1, the voltage induced in coil 1 is

Mutual Inductance

2

2 2 2 22 2 2 2

2

d d di diN N L

dt di dt dt

L2 = self inductance of coil 2

21

21 21 2 21 1 1 12

2

d d di diN N M

dt di dt dt

M12 = mutual inductance of coil 1 with respect to coil 2

Open-circuit mutual voltage across coil 1

Mutual Inductance Note:

Detailed sketch of the windings determine the algebraic sign of M.

In practice, the DOT CONVENTION designates the polarity of the mutual voltage.

.2112 MMM

Dot Convention Dot convention. Dots are placed beside each coil

(inductor) so that if the currents are entering (or leaving) both dotted terminals, then the fluxes add.

Right hand rule says that curling the fingers (of the right hand) around the coil in the direction of the current gives the direction of the magnetic flux based on the direction of the thumb.

Dot ConventionIf current enters the dotted terminal of one coil, the

reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coi.

If current leaves the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is negative at the dotted terminal of the second coil.

Dot Convention

Dot Convention

Dot Convention

Dot Convention

Dot Convention

i1(t)

+

–

v1(t) L1

i2(t)

+

–

v2(t)L2

M

dt

diL

dt

diMv

dt

diM

dt

diLv

22

12

2111

Dot Convention

dt

diL

dt

diMv

dt

diM

dt

diLv

22

12

2111

i1(t)

+

–

v1(t) L1

i2(t)

+

–

v2(t)L2

M

Dot Conventioni1(t)

+

–

v1(t) L1

i2(t)

+

–

v2(t)L2

M

dt

diL

dt

diMv

dt

diM

dt

diLv

22

12

2111

Dot Conventioni1(t)

+

–

v1(t) L1

i2(t)

+

–

v2(t)L2

M

dt

diL

dt

diMv

dt

diM

dt

diLv

22

12

2111

Mutually-coupled AC coils The frequency-domain model of the coupled circuit

is essentially identical to that of the time domain.

I1

+

–

V1 sL1

I2

+

–

V2sL2

sM

ssMIsIsLsV

ssMIsIsLsV

1222

2111

At complex frequency,I1(s)

+

–

V1(s) sL1

I2(s)

+

–

V2(s)sL2

sM

sIsLssMIsV

ssMIsIsLsV

2212

2111

Example 1 Calculate the phasor currents I1 and I2.

Solution

Example 2 Determine V0 in the circuit below.

Example Find the phasor voltage V2 in the network below

due to the complex forcing function V1. Ω

V12 Ω

3 Ω2ss

•

•

I1 I2

+

V2

_

+

_

s

Energy Storage The energy stored in an inductor at time t is

The stored energy is the sum of the energies supplied to the primary & secondary terminals.

tLitw 2

2

1

Mi1(t)

+

–

v1(t) L1

i2(t)

+

–

v2(t)L2

Energy Storage Instantaneous powers are

Suppose that at t0, i1(t0)= 0 & i2(t0) = 0,

then w(t0) = 0.

22

21

21222

12

121

111

idt

diL

dt

diMivP

idt

diM

dt

diLivP i

Energy Storage

Assume that beginning at t0, keep i2 = 0 and increase i1 until at time t1,

From

.0, 12111 tiIti

2110 111

111211

2210

1

1

0

1

0

2

1

.00,

ILdiiL

dt

diiLdtppw

dt

ditittt

I

t

t

t

t

Energy Storage

Then we maintain i1=I1 and increase i2, until at time t2,

Since i1 is held constant at I1,

.222 Iti

2222112

0 222112

222

21122

2

1

2

2

1

ILIIM

diiLIM

dtdt

diiL

dt

diIMw

I

t

t

.0 102 ttt

dt

di for

Energy Storage

At time t = t2,

If we reverse the order in w/c we increase i1 and i2, i.e. from t0 < t < t1, increase i2, so that i2 (t1) = I2, i1 = 0;

from t1 < t < t2, keep i2 = I2, while increasing i1 so that at t2, i1(t2) = I1.

2222112

211

2102

2

1

2

1ILIIMIL

wwtwtw

2222121

2112 2

1

2

1ILIIMILtw

Energy StorageFor both cases,

Then w(t2) should be the same.

Equal only if

.222121 ItiIti and

2222121

2112 2

1

2

1ILIIMILtw

2222112

2112 2

1

2

1ILIIMILtw

.2112 MMM

Energy Storage In general,

where (+) if both currents enter the dotted (undotted) terminal, (-) if otherwise.

The coefficient of coupling, k, between the inductors is given by

22221

211 2

1

2

1ILIMIILw

21LL

Mk

Coefficient of Coupling If k = 0, no coupling exists, M = 0.

We can write

since then

122

21

11

12

21

2112

21

LL

MM

LL

Mk

.1,122

21

11

12

.10 k

Coefficient of Coupling If k = 1, all of the flux links all of the turns of both

windings. Thus we have a Unity-Coupled Transformer.

If k < 0.5, loosely coupled. e.g air-core transformers

If k > 0.5, tightly coupled. e.g. iron-core transformers

Note: Value of k and M depends on the physical dimensions, no. of turns of each coil, their relative positions to one another, and the magnetic properties of the core.