Magnetically coupled circuits
Magnetically coupled electric circuits are central to the
operation of transformers and electric machines. In the
case of transformers, stationary circuits are magnetically
coupled for the purpose of changing the voltage and
current levels.
Transformer
Transformer

http://en.wikipedia.org/
Basic_principlse
Transformer
In general, the flux produced by each coil can be
separated into two components:


a leakage component denoted with ILand
a magnetizing component Im
Each of these components is depicted by a single
Streamline with the positive direction determined
by applying the right-hand rule to the direction of
current flow in the coil. Often, in transformer
analysis, i2 is selected positive out of the top of
coil 2, and a dot is placed at that terminal.
Flux in Transformer
The leakage flux l1 is produced by current flowing in coil
1, and it links only the turns of coil 1. Likewise, the
leakage flux  l2 is produced by current flowing in coil 2,
and it links only the turns of coil 2. The magnetizing flux
m1 is produced by current flowing in coil 1, and it links all
turns of coils 1 and 2. Similarly, the magnetizing flux  m2 is
produced by current flowing in coil 2, and it also links all
turns of coils 1 and 2.
Basic Principles
The transformer is a static device working on the
principle of Faraday’s law of induction. Faraday’s
law states that a voltage appears across the
terminals of an electric coil when the flux linkages
associated with the same changes. This emf is
proportional to the rate of change of flux linkages.
Putting mathematically:
Where, e is the induced emf in volt
and  is the flux linkages in Weber
turn.
dj
e=
dt
Transformer Model
Equivalent Circuit
Mechanical Analogy
Transformer in action
Transformer Core Design
Transformer Model
Voltage Equation of a transformer in matrix form is:
where r = diag [r1 r2], a diagonal matrix, and
The resistances r1 and r2 and the flux linkages l1 and l2
are related to coils 1 and 2, respectively. Because it is
assumed that 1 links the equivalent turns of coil 1 and 2
links the equivalent turns of coil 2, the flux linkages may
be written as
Where
Linear Magnetic System
Reluctance is impossible to measure
accurately, could be determined using:
l
Â=
mA
N1i1 N1i1 N2i 2
f1=
+
+
Âl 1
Âm
Âm
f
2
N2 i 2
N2 i 2
N1i1
=
+
+
Âl 2
Âm
Âm
2
1
2
1
N
N
N1N2
l1=
i1 +
i1 +
i2
Âl1
Âm
Âm
2
2
2
2
N
N
N1N2
l2=
i2 +
i2 +
i1
Âl 2
Âm
Âm
Flux Linkage of a Coil
Fig. 1 shows a coil of N turns. All these N turns link flux
lines of Weber resulting in the N flux linkages.
In such a case:
y = Nf
df
e= N
dt
Where
N is number of turns in a coil;
e is emf induced, and
 is flux linking to each coil
Change in Flux
The change in the flux linkage can be
brought about in a variety of ways:
1. coil may be static and unmoving but the flux
linking the same may change with time
2. flux lines may be constant and not changing in
time but the coil may move in space linking
different value of flux with time.
3. both 1 and 2 above may take place. The flux
lines may change in time with coil moving in
space.
Magnetically coupled M/C
In the case of electric machines, circuits in relative motion
are magnetically coupled for the purpose of transferring
energy between mechanical and electrical systems.
Because magnetically coupled circuits play such an
important role in power transmission and conversion, it is
important to establish the equations that describe their
behavior and to express these equations in a form
convenient for analysis.
Experiment LHR
RHR & LHR
Generating
Fig. 2 shows a region of length L m, of uniform flux density
B Tesla, the flux lines being normal to the plane of the
paper. A loop of one turn links part of this flux. The flux
linked by the turn is L B X Weber. Here X is the length of
overlap in meters as shown in the figure.
If now B does not change with time
and the loop is unmoving then
no emf is induced in the coil as the
flux linkages do not change. Such a
condition does not yield any useful
machine. On the other hand if the
value of B varies with time a voltage
is induced in the coil linking the same
coil even if the coil does not move.
Change in Flux Linkage
The magnitude of B is assumed to be varying
sinusoidal, and can be expressed as:
B = Bm sin wt
Where
Bm is the peak amplitude of the flux density.  is the
angular rate of change with time. Then, the
instantaneous value of the flux linkage is given by:
= N = NLXBm sin t
Which of electrical machine that is applicable?
Rate of change of Flux Linkage
 Instantaneous
flux:
j = Nf = NLXBm sin wt
Moving Coil

Instantaneous emf :
dj
p
e=
= Nf m w cos wt = Nf m w sin(wt + )
dt
2
(MatLab) Flux and Emf
EMF induced

The Peak emf induced:
em = Nf m w

rms value of induced emf is:
E=
Nf m w
2
volts