1. 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.

2. Transformer

3. Transformer

4. 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.

5. 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.

6. 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.

7. Transformer Model

8. Equivalent Circuit

9. Mechanical Analogy

10. Transformer in action

11. Transformer Core Design

12. 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

13. Linear Magnetic System
Reluctance is impossible to measure
accurately, could be determined using:

15. 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

16. Change in Flux The change in the flux linkage can be
brought about in a variety of ways:
coil may be static and unmoving but the flux linking the same may change with time
flux lines may be constant and not changing in time but the coil may move in space linking different value of flux with time.
both 1 and 2 above may take place. The flux lines may change in time with coil moving in space.

17. 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.

18. Experiment LHR

19. RHR & LHR

20. 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.

21. Change in Flux Linkage The magnitude of B is assumed to be varying
sinusoidal, and can be expressed as:
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?

22. Rate of change of Flux Linkage
Instantaneous flux:
Moving Coil 
Instantaneous emf :

23. (MatLab) Flux and Emf

24. EMF induced
The Peak emf induced:
rms value of induced emf is: