Post on 24-Dec-2015
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
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:d
edt
j=Where, e is the induced emf in volt
and is the flux linkages in Weber turn.
Transformer Model
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
Voltage Equation of a transformer in matrix form is:
Where
Linear Magnetic System
Reluctance is impossible to measure
accurately, could be determined using:
1 1 1 1 2 21
1
2 2 2 2 1 12
2
l m m
l m m
l
A
N i N i N i
N i N i N i
m
f
f
 =
= + +Â Â Â
= + +Â Â Â
2 21 1 1 2
1 1 1 21
2 22 2 1 2
2 2 2 12
l m m
l m m
N N N Ni i i
N N N Ni i i
l
l
= + +Â Â Â
= + +Â Â Â
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:
Where
N is number of turns in a coil;
e is emf induced, and
is flux linking to each coil
N
de N
dt
y f
f
=
=
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.
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 thenno 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:
sinmB B tw=
Which of electrical machine that is applicable?
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
Rate of change of Flux Linkage
Instantaneous flux:
sinmN NLXB tj f w= =
Instantaneous emf :
cos sin( )2m m
de N t N t
dt
j pf w w f w w= = = +
Moving Coil