EE201
Transformers
Oct. 9, 2007
Denard Lynch
Page 1 of 7
The inductance we have studied previously is correctly know as the
self inductance
of a
coil, commonly called just
inductance
.
Mutual inductance
is the inductive effect (the reaction to a changing flux predicted by
Faraday’s Law) in
one
coil due to the flux created in
another
.
(Refer to notes on mutual
inductance in previous notes on “Inductance”.)
This mutual inductance is used to
advantage in a device called a transformer, which is used extensively in the electrical and
electronics industries.
A transformer is a device in which two coils are deliberately arranged so that the flux
from one coil is maximally coupled to a second coil, usually on the same core.
Assume for the moment that there is a flux established in one of two closely coupled
coils,
Φ
p
, then the voltage induced in this coil, e
p
, is given by:
e
p
=
N
p
d
Φ
p
dt
=
L
p
di
p
dt
Similarly, the voltage induced in the second coil would be:
e
s
=
N
s
d
Φ
s
dt
If we define a coefficient of coupling,
k
, as
k
=
Φ
s
Φ
p
, where
Φ
p
is the originally induced
flux and
Φ
s
is the portion of the flux which couples the second coil.
Clearly
Φ
s
can never
be greater than
Φ
p
, so
k
max
≤
1.
Modern transformers are “tightly coupled”, with
k
≈
1.
Most of the useful formulas describing transformer behaviour are developed here
assuming that
k
= 1, but they could easily be modified to reflect the performance of a
“loosely coupled” coil (
k
< 1).
The first useful operational formula:
if k = 1,
Φ
p
=
Φ
s
=
Φ
e
p
=
N
s
d
Φ
dt
, and
e
s
=
N
s
d
Φ
dt
and
e
p
e
s
=
N
p
N
s
, where
N
s
N
p
is called the turns ratio,
n
therefore,
e
s
=
ne
p
(1)
Note that a turns ratio, n > 1 is called a step-up transformer, a turns ratio < 1 is called a
step-down transformer, and an n = 1 is called an isolation transformer.
While the voltage depends on
changing
flux, the flux depends only on the
current
(Ampere’s Law for magnetic circuits).