K. A. Saaifan, Jacobs University, Bremen
13. Magnetically Coupled Circuits
The change in the current flowing through an inductor induces (creates) a
voltage in the conductor itself (self-inductance) and in any nearby conductors
(mutual inductance)
The mutual inductance forms the basis for an extremely important device
called a transformer
The transformer consists of two coils of wire separated by a small distance,
and is used to convert ac voltages to higher or lower values depending on the
application
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13.1 Mutual Inductance
The inductor's voltage and current relationship is
v t=L
dit
dt
where L is the self-inductance of the inductor
The current flowing through the inductor creates magnetic flux
The time rate of change the current induces a voltage in the inductor
Coefficient of Mutual Inductance
A current flowing in one coil establishes a magnetic flux about that coil and also
about a second coil nearby
The time-varying flux surrounding the second coil produces a voltage across the
terminals of the second coil
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v2 t=M 21
di1 t
dt
v 1 t=M 12
di 2 t
dt
M21 and M12 are the coefficients of mutual inductance
The subscripts on M21 indicates that the voltage response at L2 is produced by
a current source at L1
The subscripts on M12 indicates that the voltage response at L1 is produced by
a current source at L2
The double headed arrow indicates that these inductors are coupled
K. A. Saaifan, Jacobs University, Bremen
The Dot Convention
M21=M21=M and is always a positive quantity
The induced voltage M di/dt may be positive or negative
The dot convention determines the sign of the mutual voltage as follows:
If the current enters the dotted terminal of one coil, the voltage will be positive
at the dot on the second coil
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A current entering the undotted terminal of one coil provides a voltage that is
positively sensed at the undotted terminal of the second coil
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Example: (a) determine v1 if i2 = 5 sin 45t A and i1 = 0;
(b) determine v2 if i1=−8e−t A and i2 = 0
The current i2 (entering undotted terminal) results in a positive reference for the
voltage induced across the left coil is the undotted terminal
v1 t=−M
di2 t
=− 450 cos 45 t V
dt
The current i1 (entering dotted terminal) results in a positive reference for the
voltage induced across the right coil is the dotted terminald
di1 t
v2 t=−M
=− 16 e−t V
dt
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Combined Mutual and Self-Induction Voltage
This mutual voltage is present independently of and in addition to any voltage of
self-induction
Since the pairs v1, i1 and v2, i2 each satisfy the
passive sign convention, the voltages of selfinduction are both positive
Since i1 and i2 each enter dotted terminals,
and since v1 and v2 are both positively sensed
at the dotted terminals, the voltages of
mutual induction are also both positive
di1 t
di2 t
v1 t=L1
M
dt
dt
v2 t=L2
di2 t
di t
M 1
dt
dt
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K. A. Saaifan, Jacobs University, Bremen
Since the pairs v1, i1 and v2, i2 are not sensed
according to the passive sign convention, the
voltages of self-induction are both negative
Since i1 enters the dotted terminal and v2 is
positively sensed at the dotted terminal
v2 t=−L 2
di2 t
di t
M 1
dt
dt
Since i2 enters the undotted terminal and v1 is
positively sensed at the undotted terminal
di1 t
di2 t
v1 t=−L1
M
dt
dt
The same considerations lead to identical
choices of signs for excitation by a sinusoidal
source operating at frequency ω
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Physical Basis of the Dot Convention
The dot convention is made by examining the way in which both coils are
physically wound and applying Lenz’s law in conjunction with the right-hand-rule
Using the right hand rule, the flux direction at the left-hand side
is upward and the right-hand side is downward
Since the flux at both coils has the same direction, the upper
terminals of coils will be of the same dot sign
Dots may be placed either on the upper terminal of each coil or
on the lower terminal of each coil
Using the right hand rule, the flux direction at the left-hand side
is upward and the right-hand side is upward
Since the flux at both coils has the apposite direction, the upper
terminals of coils will be of the opposite dot notation
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Example: find the ratio of the output voltage across the 400 Ohm resistor to the
source voltage, expressed using phasor notation
Transform the circuit into phasor
ZL=j  L
ZM =j M
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Mesh 1
Since I2 enters the undotted terminal of L2, the mutual voltage across L1 must
have the positive reference at the undotted terminal. Thus,
1 I1 j10 I1 −j 90 I 2 =10 0
o
Mesh 2
Since I1 enters the dot-marked terminal of L1, the mutual voltage across L2 must
have the positive reference at the dotted terminal. Thus,
400 I 2j 1000 I 2 −j 90 I 1=0
By solving the equations, we find
Thus,
I2 =0.172 −16.70
o
A
V2 4000.172 −16.70o 
=
V1
10 0o
K. A. Saaifan, Jacobs University, Bremen
P1: Calculate the mesh currents in the shown circuits
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K. A. Saaifan, Jacobs University, Bremen
P2: Calculate the mesh currents in the shown circuits
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K. A. Saaifan, Jacobs University, Bremen
13.2 Energy in a Coupled Circuit
The energy stored in an inductor is given by
1
wL= L i2L
2
We now consider the energy stored in magnetically coupled coils, which leads to
1
1
W = L1 I 12 L 2 I 22M I 1 I 2
2
2
If one current enters a dot-marked terminal while the other leaves a dotmarked
terminal, the sign of the mutual energy term is reversed
1
1
W = L1 I 12 L 2 I 22−M I 1 I 2
2
2
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K. A. Saaifan, Jacobs University, Bremen
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Since I1 and I2 are arbitrary values, they may be replaced by i1(t) andi2(t), which
gives the instantaneous energy stored in the circuit
1
1
wt= L1 i21 t L2 i22 t±M i12 ti22 t
2
2
Since w(t) represents the energy stored within a passive network, it cannot be
negative for any values of i1, i2, L1, L2, or M
this implies that
M   L1 L 2
The ratio M /  L1 L 2 represents the coupling coefficient, which is a measure of
the magnetic coupling between two coils
k=
or
M
 L1 L 2
M =k  L1 L 2
where 0k1 or equivalently 0M   L1 L2
For k < 0.5, coils are said to be loosely coupled; and for k > 0.5, they are said to be
tightly coupled
K. A. Saaifan, Jacobs University, Bremen
Example: Determine the coupling coefficient. Calculate the stored energy in the
coupled inductors at t=1 s if v=60 cos(4t+30) V
The coupling coefficient is
Thus, the inductors are tightly coupled
To find the energy stored, we need to obtain the frequency-domain equivalent
of the circuit.
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We now apply mesh analysis. For mesh 1,
For mesh 2,
or
Solving the two equations
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In the time-domain,
At time t = 1 s, 4t = 4 rad = 229.2◦, and
The total energy stored in the coupled inductors is,
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P3: Determine the coupling coefficient. Calculate the stored energy in the coupled
inductors at t=1.5 s
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13.3 The Linear Transformer
A transformer is generally a four-terminal device comprising two (or more)
magnetically coupled coils
The coil that is directly connected to the voltage source is called the primary
winding
The coil connected to the load is called the secondary winding
The resistances R1 and R2 are included to account for the losses (power
dissipation) in the coils
The transformer is said to be linear if the coils are wound on a magnetically linear
material
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Transformer: Reflected Impedance
We shall compute the input impedance Zin as seen from the source
Applying KVL to the two meshes
We get the input impedance as
The input impedance comprises two terms:
The first term, Z11=R1 + jωL1, is the primary impedance
The second term is due to the coupling between the primary and secondary
windings
The second term is known as the reflected impedance ZR
Z22=R2 + jωL2+ZL
K. A. Saaifan, Jacobs University, Bremen
Transformer: The T Equivalent Network
The voltage-current relationships for the primary and secondary coils give the
matrix equation
For the T (or Y) network, mesh analysis provides the terminal equations as
Then,
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Transformer: The Delta Equivalent Network
The current-voltage relationships for the primary and secondary coils give the
matrix equation
For the Delta network, nodal analysis provides the terminal equations as
Then,
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P4: Find the input impedance and the current from the voltage source
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K. A. Saaifan, Jacobs University, Bremen
13.4 Ideal Transformers
An ideal transformer is one with perfect coupling (k = 1)
It consists of two (or more) coils with a large number of turns wrapped on a
common core of high permeability
The ideal transformer is the limiting case of two coupled inductors where the
inductances approach infinity and the coupling is perfect
In the frequency domain,
thus,
Since M =  L1 L 2
where n =√L2/L1 and is called the turns ratio
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K. A. Saaifan, Jacobs University, Bremen
1. Coils have very large reactances (L1, L2, M → ∞)
2. Coupling coefficient is equal to unity (k = 1)
3. Primary and secondary coils are lossless (R1 = 0 = R2)
The vertical lines between the coils indicate an iron core as distinct from the air
core used in linear transformer
According to Faraday’s law, the voltage across the primary
winding and the secondary winding are
Then, we have
where n is, again, the turns ratio or transformation ratio
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K. A. Saaifan, Jacobs University, Bremen
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Using phasor voltages V1 and V2, we may write
The energy supplied to the primary equals to the energy absorbed by the secondary
Thus, we have
The primary and secondary currents are related to the turns ratio as
n = 1, we have an isolation transformer
n > 1, we have a step-up transformer
n < 1, we have a step-down transformer
K. A. Saaifan, Jacobs University, Bremen
Voltage Polarities and Current Directions
1. If V1 and V2 are both positive or both negative at the dotted terminals, use +n.
Otherwise, use −n
2. If I1 and I2 both enter into or both leave the dotted terminals, use −n.
Otherwise, use +n
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Ideal Transformer: The Complex Power
Express V1 in terms of V2 and I1 in terms of I2
The complex power in the primary winding is
Ideal Transformer: Reflected Impedance
The input impedance as seen by the source
Since ZL=V2/I2, we have
(Impedance Matching)
K. A. Saaifan, Jacobs University, Bremen
Ideal Transformer: Thevenin Equivalent
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K. A. Saaifan, Jacobs University, Bremen
P5: Find the Thevenin equivalent of the circuit to the left of the terminals c-d
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