Download 74. Leakage field of the transformer

Historical burdens on physics
74 Leakage field of the transformer
Subject:
“…, the magnetic flux Φ should be completely confined to the interior of the
iron core, i.e. run through both windings with the same intensity (no leakage
flux).” [1]
“When measuring the secondary voltage more precisely it turns out to be
smaller, than what would be expected from the calculus: This fact is due on
the one hand to Joule losses,…. The second cause is that, due to leakage,
only a part of the induction flux in the primary winding crosses the secondary winding.” [2]
Deficiencies:
Students learn that leakage fields or stray fields are something that should
be avoided. In principle, they are not necessary, and no fundamental physical principle is violated if we imagine a physical world without them. It is
similar to mechanical friction. Also friction often appears only as a nuisance,
which one tries to prevent. A somewhat rougher analogy is a leak in a garden hose. Admit the hose has some perforations or the fittings are not watertight. These leaks can, in principle, be completely tamped. And indeed,
this can also be said for certain “stray fields”. A metallic shielding prevents
the leakage of an electric field, a Mu-metal shielding encloses a magnetic
field or holds it off from some other device.
Now, the same name “leakage field” or “stray field” is also used in cases
where where the working principle of an apparatus depends on this field.
Among several examples there is the transformer, which we will discuss in
the following.
Fig. 1. The magnetic field strength H is different from zero only on that part of the integration
path which runs outside of the iron core.
We consider a transformer in its most simple form: an iron core that forms a
closed rectangle, with the two windings sitting on the opposite shorter
sides, Fig. 1. We make the conventional assumptions:
–! the ohmic resistance of the coils is small compared with the respective
inductive resistances;
–! the load resistance is small compared with the inductive resistance of the
secondary coil;
–! the load resistance is great compared with the ohmic resistances of either
coil;
–! the permeability μ of the core material is much greater than one.
We now apply Ampere’s law, and first integrate along path A:
 
∫ H dr = n1I1
A
The value of the integral is equal to the total current n1 I1, that is looped by
the integration path. (n is the number of windings, the indices refer to the
primary and secondary windings, respectively.) Now, the magnetic field
strength H inside the iron core is smaller than outside by the factor μ. Since
typical values of μ are greater than 1000, the contribution of the path inside
the iron to the integral is negligible. Thus, only the “leakage field” contributes to the value of the integral.
We now consider integration path B. It loops through both of the coils.
Since it passes on its entire length within the iron core the integral is equal
to zero:
 
∫ H dr = n1I1 – n2 I 2 = 0
B
We thus get the well-known relation:
n1 I1 = n2 I2 .
We see that this relation could not hold if there were no “leakage field”.
Fig. 2. The fields between the legs of a transformer. The energy flow is from left to right.
The importance of the denigrated field can be seen in yet another way.
Fig. 2 shows schematically the H field lines as well as the electric field lines.
The change of the magnetic flux within the iron core is the cause of an electric eddy field, whose field lines loop around the legs of the iron core. In addition, the figure shows the Poynting vector:
  
S = E × H ,!
(1)
i.e. the energy flow density within the field. It is seen that the energy gets
from the primary to the secondary circuit through the field.
The situation is analogous to that of the energy transport by means of an
electric cable. The only difference is that the electric and the magnetic fields
are interchanged, Fig. 3. Since the conductors have different electric potentials, the electric field lines run from one conductor to the other, and since
an electric current is flowing in the conductors, they are surrounded by a
magnetic eddy field. The energy flow distribution is the same as in the
transformer.
Fig. 3. The fields between the two conductors of an electric cable. The energy flow is from
left to right.
The stray field of the transformer is not more responsible for the losses of
the transformer than the electric field between the conductors of a cable are
responsible for the losses of the cable. In both cases the efficiency is essentially limited by the dissipation within the “conductors”. In the transformer we have energy dissipation in the coils and in the iron core due to
the steady change of the magnetization. A measure of this dissipation is the
magnetic field strength within the iron core. Ideally it should be zero, just as
the electric field strength within the electric conductors of the cable should
be zero. Since the dissipation in an iron core is rather high, in technical
transformers the distance between the primary and the secondary coil is
made as short as possible.
Origin:
In the common discussion of the working principle of the transformer one
does not argue with the magnetic field strength H, but only with the flux
density B. Since B is much greater inside the iron core than outside the impression results that the field outside does not play an essential role. This is
only one of many examples which show that the exclusive use of B for the
description of magnetic phenomena causes misconceptions. Another cause
may be that one avoids the discussion of the local energy balance.
Disposal:
1. Do not throw all the “stray fields” into the same pot. Since the terms
“stray field” and “leakage field” have a negative connotation, it would be
better not to employ these words to the field between the legs of a transformer.
2. Do not limit the discussion of the magnetic field of the transformer to B.
Discuss also H.
3. Ask as often as possible the questions: Where is the energy? Which way
does the energy go?
[1] Gerthsen: Physik, 21. Auflage, Springer-Verlag Berlin 2002, S. 414.
[2] Handbuch der experimentellen Schulphysik, Elektrizitätslehre III, Aulis
Verlag Deubner & Co KG Köln 1965, S. 70.
Friedrich Herrmann, Karlsruhe Institute of Technology