Magnetically coupled circuits
Examples
+
The
rest
of
the
circuit
i(t)
L
R
v(t)
–
Electrically
ect ca y coupled
coup ed
Magnetically coupled
Inductance
 Inductance occurs when current flows through a (real)
conductor.
conductor
 The current flowing through the conductor sets up a
magnetic field that is proportional to the current.
 The voltage difference across the conductor is
proportional
p
opo t o a to the
t e rate
ate of
o change
c a ge of
o the
t e magnetic
ag et c field.
ed
 The proportionality constant is called the inductance,
denoted L. Units of Henrys (H) - V
V·s/A.
s/A.
5
Self-inductance
Self
inductance in AC
 An inductor is a two-terminal device that consists of a
coiled conducting wire around a core.
 A current flowing through the device produces a
magnetic flux Φ forms closed loops threading its coils.
 Total flux linked by N turns of coils,
coils flux linkage
λ=NΦ
 For a linear inductor, λ=Li (L is the inductance).
Self-inductance
Self
inductance and induced voltage
Faraday’s law of electromagnetic induction:
Th EMF (Electromotive
The
(El t
ti Force)
F
) induced
i d d in
i a magnetic
ti circuit
i it is
i
equal to the rate of change of flux linked with the circuit.
d
d ( N )
d
e

 N
dt
dt
dt
 Li  N
e  
dLi
di
 L
dt
dt
λ is total flux linkage and L the self inductance.
Mutual inductance in AC
 Two coil arrangement:


When the flux produced by coil 1 links with coil 2, a voltage
is generated across the terminals of that coil.
The arrangement shown below an AC voltage applied to the
terminals of coil 1 that produced an AC current which
produces an AC flux.
flux This flux links the second coil through
the centre from bottom to top. This gives rise to V2(t).
1
i2
i1

N1 turns

N2 turns
v2(t)
v1(t)
Coil 1
Coil 1
Coil 2
Mutual inductance in AC
 Consider the coupled coils: coil1 and coil 2







The total flux Φ1 threading coil 1 is the sum of components due to
i1 and i2: Φ1 = Φ11 + Φ12
The net flux linkage for coil 1 is λ1= N1Φ1 = N1Φ11 + N1Φ12
The first term is due to coil’s own current
L1 is
i called
ll d the
th self
lf inductance
i d t
The second term is due to current in the other coil N1Φ12 =M12i2
M12 is called the mutual inductance
The sign is determined by the direction of the flux
 By Faraday’s law:
d 1
di1
di2
v1 
 L1
 M 12
dt
dt
dt
Energy analysis
 Similar analysis, we have
d 1
di
di
d 2
di2
di1
 L1 1  M 12 2



v
L
M
2
2
21
dt
dt
dt
dt
dt
dt
 The instantaneous power delivered to the coils:
di
di
di
di
p1  v1i1  L1i1 1  M 12i1 2
p2  v2i2  L2i2 2  M 21i2 1
dt
dt
dt
dt
v1 
 The energy stored in the coils:
1 2
1 2
L
i
t

L2i2 (t )  M 12i1 (t )i2 (t )
(
)
2 11
2
w(t )  
1
1
 L2i22 (t )  L1i12 (t )  M 21i1 (t )i2 (t )
2
2
apply i1 first, then i2
apply i2 first, then i1
 Initial and final conditions are the same. The mutual
inductances are identical:
M 12  M 21  M
Formulas for coupled coils
 i-v laws for coupled coils:
di1
di
M 2
dt
dt
di2
di1
v2  L2
M
dt
dt
v1  L1
 The energy stored w in a pair of coupled coils:
1 2 1 2
w  L1i1  L2i2  Mi1i2
2
2
Various flux definitions
1
2
i1
12

N1 turns
ell1
v1(t)
21
Coil 1


i2

N2 turns
v2(t)
ell2
C il 2
Coil 2
Note that ell1 and ell2 "ell" means l as little but will only
confuse if it is in the diagram as l.
These flux terms represent flux that is produced in coils 1 (ell1)
and 2 (ell2) which do not link coils 2 and 1 respectively. They
can be seen as leakage terms. But they are assumed
negligible for the purposes of the analysis.
analysis
Various flux definitions

Φ11 is the flux produced in coil 1 by the current flowing in coil 1
Φ11 = Φell1 + Φ21

Φ22 is the flux produced in coil 2 by the current flowing in coil 2
Φ22 = Φell2 + Φ12

Φ21 is the flux linking coil 2 produced by coil 1

Φ12 is the flux linking coil 1 produced by coil 2

Φ1 - The total flux threading coil 1 is the sum of components due to i1
and i2:
Φ1 = Φ11 + Φ12
d 1
d1
d11 d12
di1
di2
v1 
 N1
 N1 (

)  L1
 M 12
dt
dt
dt
dt
dt
dt
Phasor format
 From the time-domain equations:
di1
di2
M
v1  L1
dt
dt
di2
di1
v2  L2
M
dt
dt
 In phasor analysis (used for AC steady-state
steady state
response to sinusoidal excitations)
V1  j L1 I1  j MI 2
V2  j L2 I 2  j MI1
Coefficient of coupling
 The total flux Φ11 resulting from i1 through N1 turns
consists of leakage flux Φell1 and coupling flux Φ21.
 The coupling coefficient, k, is defined as the ratio of
l k
linking
fl
flux to totall fl
flux. Also,
l
the
h coupling
l
is b
bilateral:
l
l
 21
12
k 

11
 22
Coefficient of coupling
 Coefficient of coupling k


A measure off the
th degree
d
off coupling
li
Defined by
k 
M
L1 L 2
 If there is no coupling between the coils, M=k=0

There is equivalent to two simple, uncoupled coils
 Since M21<L1 and M12<L2, we have
 M2  L1 L2
0≤ K ≤ 1
 If k is close to 1, the coils are said to be tightly coupled
Self evaluation
 When one coil of a magnetically coupled pair
has a current of 5.0A, the resulting fluxes ell1
and 21 are 0.2mWb and 0.4mWb, respectively.
If the turns are N1 = 500 and N2 = 1500,, find
L1, L2, M and the coefficient of coupling k.
[60mH, 540mH, 120mH, 0.667]
Dot convention
 The dot convention is used to determine whether the
fluxes add and therefore the mutual inductance adds.
adds
 The rules are:




Assume positive M
Place a dot where i1 enters coil 1
Determine the direction of the flux produced in coil 1 due to i1
Consider coil 2 and the direction of the current i2. If i2 provides a
flux in the same direction as the flux due to coil 1 and i1, a dot is
placed on coil 2 where i2 enters. If the flux in coil 2 opposes the
flux due to coil 1 and i1, a dot is placed on coil 2 where i2 leaves.
Examples
• a
i1
–
1
• a
b
•b
+
v21
2
–
Positive M
•
i1
1
+
v21
–
Negative M
2
Self evaluation
The physical construction of two mutually coupled coils. From a
consideration of the direction of magnetic flux produced by each
coil, it is shown that dots may be placed either on the upper
terminal of each coil or on the lower terminal of each coil.
•
•
Example 1
 Two circuits magnetically coupled with positive M
i1
M
+ •
v1 L1
_
R1
v1  I1R1  L1

i2
+
•
L2 v2
_
R2
di1
di
M 2
dt
dt
di1
di2
v2  I 2 R2  M
 L2
dt
dt
+

V1
I1
j M
•
j  L1

I2
+
•
j L2 V
2
_
_
R1
R2
V1  I1R1  jLI
1 1  jMI2
V2  I2R2  jMI1  jL2I2
Example 2
 Two circuits magnetically coupled with negative M
i1
+ •
u1 L1
_
R1
i2
M
L2
•
+
u2
_
R2
V1  I1R1  jLI
1 1  jMI2
V2  I2R2  jMI1  jL2I2
Example 3
i
 Series adding
R
+
v
–
i
vR v
1
+
•
v
L1
–
+ •
v2 L2
R
M
–
L
.
V  VR +V1  V2
  (j L I  j MI)  ( j L I  j MI)
=IR
1
2
 RR
L  L1  L2  2 M
Example 4
 Series opposing
i
+ +
R1
v1
v
– –
.
v2
R2
i
–
+
•
L1
M
L2
•
+
R
v
–
L
V  2 j MI  j ( L1  L2 ) I  I( R1  R2 )
 R  R1  R2
L  L1  L2  2 M
Mutual inductance measurement
 If you do practical tests on the coils for series adding and
opposing you can calculate the mutual inductance if the
self-inductances are known.
LA  L1  L2  2 M
Series adding
LB  L1  L2  2 M
S i opposing
Series
i
LA  LB
M
4
Mutual inductance
k 
LA  LB
M

L1 L 2
4 *
L1L 2
Coupling
p g coefficient
Example 5
 Mutual inductance in parallel and dotted terminals at the
same side
M
i
+
u
–
i1 *
L1
*
i2
L2
d i1
u  L1
M
dt
di
u  L2 2  M
dt
d i2
dt
d i1
dt
i = i1 +i2
Solve the relation of u, i：
( L1L2  M 2 ) di
u
L1  L2  2 M dt
( L1L2  M 2 )
Leq 
L1  L2  2 M
0
Example 5 cont
cont’d.
d.
M
i
º+
u
_
º
i1
L1
* *
di1
di 2
u  L1
M
dt
dt
i2
di1
 ( L1  M )
 M di
dt
dt
L2
Draw equivalent
q
circuit:
i
º
+
u
_
º
M
i1
L1-M
i2 = i - i1
i2
L2-M
di 2
di1
u  L2
M
d
dt
dt
d
i1 = i - i2
di 2
 ( L2  M )
 M di
dt
dt
T circuit
i1
12

N1 turns
ell1
v1(t)
N2 turns
v2(t)
ell2

i2
21
Coil 2
Coil 1
di
di
V1  L1 1  M 2
dt
dt
i1(t)
v1(t)
V 2  L2
d i2
di
M 1
dt
dt
L2 ‐ M
L1 ‐ M
M
i1(t) + i2(t)
i2(t)
v2(t)
T circuit
V1  ( L1  M )
d i1
d( i1  i2 )
di
di
M
 L1 1  M 2
dt
dt
dt
dt
V 2  ( L2  M )
i1(t)
v1(t)
d i2
d( i1  i2 )
di
di
M
 L2 2  M 1
dt
dt
dt
dt
L2 ‐ M
L1 ‐ M
M
i1(t) + i2(t)
i2(t)
v2(t)
Mutual inductance and transformers
 If we have two coils wound on the same magnetic (iron)
core, a changing current in one coil would cause a
voltage in both coils.
 The effect of changing current in one coil inducing a
voltage in another is mutual inductance and is the basis
of transformer.
 Transformers are one of the most common devices to
convert from one voltage to another with high (>99%)
efficiency.
 Since the current must continuously change for these
devices to operate, they are generally only used with AC
supplies.
 This ease of voltage conversion is one of the main
reasons why the majority of power is distributed in AC
rather than DC.
Review of example 2: Two circuits magnetically
coupled with negative M, ignore leakage fluxes
 Two coils with opposing fluxes (Φ21- Φ12) and N1≠N2.
i1
12

N1 turns
N2 turns
ell1
v1(t)
()
v2(t)
ell2

i2
21
Coil 2
Coil 1
v1 (t )  N1
d
[21  12 ]
dt
Dividing the two equations:
v1 (t ) N1

v2 ( t ) N 2
d
v2 (t )  N 2 [21  12 ]
d
dt
q
defines the ideal transformer.
This equation