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Custom Transformer Math Proof/JACKSON ENGINEERING
A step-up transformer has its secondary voltage 
greater than its
primary voltage 
,
and a step-down transformer has its secondary voltage 
less than its primary
voltage 
.
This article
demonstrates (using Algebra) that given a fixed primary voltage 
, the secondary (step-up or step-down) voltage 
produced by a
transformer is mathematically determined by the number of its primary and
secondary windings 
and 
respectively.
Let's first review some fundamental transformer concepts. Transformers
transfer electrical power (energy) from one electrical circuit to another,
through its coils. Each primary and secondary coil has a set number of
windings (wound about two opposite sides of a magnetic core) 
and 
respectively.
Through the study of Physics - Electricity
& Magnetism, given a fixed primary voltage 
, a changing current passing through a magnetic cores
primary coil winding creates a magnetic field through the cores secondary coil
winding. The magnetic field "induces” an electro-magnetic force
(emf) or voltage in the secondary winding. This new voltage is called the
"induced voltage", and is used to produce the electrical power
(current) used in our everyday life. See (fig 1).
When a load is connected to the “secondary”, electric current flows through the
secondary winding. Therefore electrical energy is transferred
(transformed using a transformer) from the primary circuit through the
transformer to the load.
In an optimal transformer, the induced secondary voltage 
is proportional
to the fixed primary voltage
.
Mathematically this means 
(where k is the
constant of proportionality).
(Equation 1)
It is also true that k equals the ratio
of the number of winding turns in the
secondary coil 
, to the number of winding turns in the primary coil 
.
Mathematically this means
.
(Equation 2)
Next substituting 
for k in (Equation 1)
makes 
= [
][
].
= [
] [
] (Equation
3)
Next notice that using (Equation 3), since the primary voltage 
is a fixed
voltage, the secondary voltage 
depends on the
number of windings 
and 
of the
transformer.
CASE 1)
Using (Equation 1) and making 
< 
, 
< 1. Therefore the secondary voltage 
is a proper
fraction of 
. This means 
is stepped-down
from 
. Therefore the secondary step-down voltage 
produced by the
transformer is mathematically determined by the number of its primary and
secondary windings 
and 
respectively.
CASE 2)
Again using (Equation 1) and this time by making 
> 
, 
> 1.
Therefore the secondary voltage 
is the product of
and a number
greater than one. This means 
is stepped-up
from 
.
Therefore the secondary step-up voltage 
produced by the
transformer is again mathematically determined by the number of its primary and
secondary windings
and 
respectively.
In conclusion, by changing 
and 
with a fixed
primary voltage 
, the secondary
voltage 
can be changed to
a greater or lesser amount than the primary voltage 
,
hence the terms step-up or step-down
transformers.