This document discusses magnetically coupled circuits and inductors. It begins by defining key concepts like mutual inductance, self-inductance, and the ideal transformer. It then provides examples of analyzing circuits with two mutually coupled inductors by writing loop equations in terms of the inductors' voltages and currents. Specific cases are examined where the inductors are connected in different configurations. Phasor analysis is also introduced for modeling coupled inductors in the frequency domain.

1. Magnetically Coupled Circuit
Circuit Analysis II (EE-201)

2. MAGNETICALLY COUPLED NETWORKS
LEARNING GOALS
Mutual Inductance
Behavior of inductors sharing a common magnetic field
Energy Analysis
Used to establish relationship between mutual reluctance and
self-inductance
The ideal transformer
Device modeling components used to change voltage and/or
current levels
Safety Considerations
Important issues for the safe operation of circuits with transformers

3. BASIC CONCEPTS – A REVIEW
Magnetic field
Total magnetic flux
linked by N-turn coil
Ampere’s Law
(linear model)
Faraday’s
Induction Law
Ideal Inductor
Assumes constant L
and linear models!

4. MUTUAL INDUCTANCE
Overview of Induction Laws
Magnetic
flux
webers)
linkagefluxTotal
( N
Li
If linkage is created by a current flowing
through the coils…
(Ampere’s Law)
The voltage created at the terminals of
the components is
dt
di
Lv  (Faraday’s Induction Law)
Induced links
on second
coil 2
( )
What happens if the flux created by the
current links to another coil?
One has the effect of mutual inductance

5. TWO-COIL SYSTEM (both currents contribute to flux)
Self-induced Mutual-induced
Linear model simplifying
notation

7. THE ‘DOT’ CONVENTION
COUPLED COILS WITH DIFFERENT WINDING CONFIGURATION
Dots mark reference polarity
for voltages induced by
each flux

9. j
i
i
ij
i
n
j
iji
circuitin
currentabycausedcircuitlinkingFlux
circuitlinkingfluxTotal
1


 



Assume n circuits interacting
jijij iL
modelsinductorlinearFor
ji
LL
iL
jiij
ii
and
circuitsbetweeninductanceMutual
circuitof"inductanceself"


Special case n=2
2221212
2121111
iLiL
iLiL




2112 LL 
:ModelLinear
A GENERALIZATION

10. THE DOT CONVENTION REVIEW
Currents and voltages follow
passive sign convention
Flux 2 induced
voltage has + at dot
)()()(
)()()(
2
2
1
2
21
11
t
dt
di
Lt
dt
di
Mtv
t
dt
di
Mt
dt
di
Ltv


LEARNING EXAMPLE
)(1 ti )(2 ti



))(( 2 tv
For other cases change polarities or
current directions to convert to this
basic case













dt
di
M
dt
di
Ltv 21
11 )(













dt
di
L
dt
di
Mtv 2
2
1
2 )(
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11



11. LEARNING EXAMPLE
Mesh 1 Voltage terms

13. Mesh 2
LEARNING EXAMPLE - CONTINUED
Voltage Terms

14. 1i



1v
dt
di
L
dt
di
Mv
dt
di
M
dt
di
Lv
2
2
1
2
21
11


)()()(
)()()(
2
2
1
2
21
11
t
dt
di
Lt
dt
di
Mtv
t
dt
di
Mt
dt
di
Ltv


Equivalent to a
negative mutual
inductance
2i



2v dt
di
M
dt
di
Lv 21
11 
dt
di
L
dt
di
Mv 2
2
1
2 
More on the dot convention

15. LEARNING EXTENSION


)()()( 2
11 t
dt
di
Mt
dt
di
Ltv 
)()()( 2
2
1
2 t
dt
di
Lt
dt
di
Mtv 
)()()( 2
11 t
dt
di
Mt
dt
di
Ltv 
)()()( 2
2
1
2 t
dt
di
Lt
dt
di
Mtv 
Convert to
basic case
)(),( 21 tvtvforequationstheWrite
PHASORS AND MUTUAL INDUCTANCE
)()()(
)()()(
2
2
1
2
21
11
t
dt
di
Lt
dt
di
Mtv
t
dt
di
Mt
dt
di
Ltv


2212
2111
ILjMIjV
MIjILjV



 Phasor model for mutually
coupled linear inductors
Assuming complex
exponential sources

16. LEARNING EXAMPLE The coupled inductors can be connected in four different ways.
Find the model for each case
CASE I
 1V  2V ILjMIjV
MIjILjV
22
11




Currents into dots
CASE 2
 1V  2V
Currents into dots
I I
I I 21 VVV 
21 VVV 
ILjMIjV
MIjLjV
22
11




ILjMLLjV eq  )2( 21
ILMLjV )2( 21  
eqL
M
Leq
ofvaluetheon
constraintphysicalaimposes0

17. CASE 3
Currents into dots
1I 2I
 V  V
21 III 
221
211
ILjMIjV
MIjILjV




12 III 
I
MLL
MLL
jV
221
2
21


 
CASE 4
Currents into dots
1I 2I
 V  )( V
1I 2I 21 III 
221
211
ILjMIjV
MIjILjV




I
MLL
MLL
V
221
2
21




18. LEARNING EXAMPLE
0VVOLTAGETHEFIND


1V


2V
2I
1123024 VI :KVL
022 222  IIjV-:KVL
)(62
)(24
212
211
IjIjV
IjIjV


CIRCUITINDUCTANCEMUTUAL
20 2IV 
SV
21
21
)622(20
2)42(
IjjIj
IjIjVS


1I
42/
2/
j
j


  2
2
)42(42 IjVj S 
168
2
2
j
Vj
I S

 



j
j
j
VS
816
2

j
V
IV S
24
2 20





57.2647.4
3024
 42.337.5
1. Coupled inductors. Define their
voltages and currents
2. Write loop equations
in terms of coupled
inductor voltages
3. Write equations for
coupled inductors
4. Replace into loop equations
and do the algebra