1. Summary of the last lecture

2. Magnetic circuits
Compare this formula with Ohm’s law in electric circuits:

3. Another useful formula in a series situation

4. Problem 1-8 Assume 4% increase in effective Cross-sectional area
for fringing effect
Calculate the flux density in each of the legs

8. Magnetic Behaviour of Ferromagnetic Materials
We talked about Ferromagnetic materials, such as iron, steel, cobalt, nickel and some alloys. They have a high relative permeability ( ).
In magnetic circuit theory we assumed:
where is a constant.

9. By this assumption we have assumed a linear relation, i.e.:
This means in a coil if I increase the current to twice as much, the flux will be twice as much as well. This is what we call linear relation.

10. But the reality is different. These are typical curves
But the reality is different. These are typical curves.We have saturation.

11. Typical B-H curve of steel

12. Typical ur-H curve of steel

13. Now, can we solve the problems knowing these curves ?
In some problems, instead of giving the relative permeability, they produce the B-H curve.
In these cases, the operating condition (Bo-Ho), should be known somehow. If it is known, there would be no problem.

14. Remember the example:
Given : i=1 A, N=100, lc=40 cm, A= 100 cm2 r =5000
Calculate : F, H, B, , and

15. Now consider this example:
Given : i=1 A, N=100, lc=40 cm, A= 100 cm2 given the above B-H curve
Calculate : F, H, B, , and
From the magnetizing curve:

16. Hysteresis When we increased the current we observed, saturation.
What would happen if I decrease the current after saturation?
The flux for a given H is higher when decreasing

17. Can we explain the hysteresis phenomena?
All materials consist of small magnetic domains.
When they are in a magnetic field the domains are intended to be in line with the field.

18. The domains
after applying
magnetic field
The domains
before applying
magnetic field
When the magnetic field is removed, not all domains
are randomized again

20. Hysteresis loss
Hysteresis is not a serious problem when we have DC excitation (the examples considered so far). It causes some loss when we have AC excitation, called hysteresis loss.
If we have AC excitation, e.g. the current i is sinusoid, the hysteresis happens at each cycle. The hysteresis loss is proportional to the frequency and also depends on the area of the hysteresis loop.

21. Other losses
- Copper loss:
- Eddy Current loss:
- Core losses:

22. Eddy Current Eddy current:
As we saw, a flux induces a voltage on a coil.
Q: Why not inducing a voltage on the core itself?
A: It actually does.
The result is eddy current. That is why the transformers core are laminated.