1 Introduction to Transformers
Transformer design and test are sometimes viewed as an
art rather than a science.
Transformers are imperfect devices, and there will be
differences between a transformer's design values, its test measurements,
and its performance in circuit.
By going back to basics, this tech note will help design
and test engineers understand how a transformer's electrical
characteristics are the result of physical properties of the core and
windings.
A partner tech note, "Transformer Testing Basics",
describes how these characteristics are measured in order to confirm that
a transformer has been correctly manufactured and how to avoid some of the
common pitfalls in making those measurements.
2 Basic Transformer Theory
The above figure represents the essential elements for a
transformer: a magnetic core with a primary and secondary coil wound on
the limbs of the magnetic core.
An alternating voltage (V_{p}) applied to the
PRIMARY creates an alternating current (I_{p}) through the
primary.
This current produces an alternating magnetic flux in the
magnetic core. This alternating magnetic flux induces a voltage in each
turn of the primary and in each turn of the secondary.
As the flux is a constant, i.e. the same in both primary
and secondary:
This equation shows that a transformer can be used to
step up or step down an ac voltage by controlling the ratio of primary to
secondary turns. (Voltage transformer action).
It can also be shown that:
Primary Volt Amperes = Secondary Volt Amperes
This equation shows that a transformer can be used to
step up or step down an ac current by controlling the ratio of primary to
secondary turns. (Current transformer action)
It will be noted that there is no electrical connection
between the primary and secondary windings. A transformer, therefore,
provides a means of isolating one electrical circuit from another.
These features  voltage/current transformation and
isolation  cannot be obtained efficiently by any other means, with the
result that transformers are used in almost every piece of electrical and
electronic equipment in the world.
3 BH curves
When the primary of a transformer is energized with the
secondary unloaded, a small current flows in the primary. This current
creates a 'magnetizing force' that produces the magnetic flux in the
transformer core.
The magnetizing force (H) is equal to the product of
magnetizing current and the number of turns, and is expressed as Ampere 
Turns.
For any given magnetic material, the relationship between
magnetizing force and the magnetic flux produced can be plotted. This is
known as the BH curve of the material.
From the BH curve it can be seen that, as the
magnetizing force is increased from zero, the flux increases up to a
certain maximum value of flux.
Above this level, further increases in magnetizing force
result in no significant increase in flux. The magnetic material is said
to be 'saturated'.
A transformer is normally designed to ensure that the
magnetic flux density is below the level that would cause saturation. The
flux density can be determined using the following equation:
Where:
E represents the rms value of the applied voltage.
N represents the number of turns of the winding.
B represents the maximum value of the magnetic flux density in the core
(Tesla).
A represents the crosssectional area of the magnetic material in the core
(sq. meters).
f represents the frequency of the applied volts.
Note
1 Tesla = 1 Weber/meter²
1 Weber/m² = 10,000 Gauss
1 Ampereturn per meter = 4p x 103 Oersteds
In practice, all magnetic materials, once magnetized,
retain some of their magnetization even when the magnetizing force is
reduced to zero.
This effect is known as 'Remanence' and results in the
BH curve for the material exhibiting a response to a decreasing
magnetizing force that is different to the response to an increasing
magnetizing force.
In practice, then, real magnetic materials have a BH
curve as follows:
The curve shown above is termed the 'hysteresis' loop of
the material, and it represents the true BH response of the material.
(The BH curve shown in figure 2 represented the average or mean of the
true BH loop response).
The slope of the BH curve, the saturation level, and the
size of the hysteresis loop are dependent on the type of material used,
and on other factors. This is illustrated using the following examples:

Lowgrade iron core
Highsaturation flux density
Large loop = large hysteresis loss
Suitable for 50/60Hz 

Highgrade iron core
Highsaturation flux density
Medium loop = medium hysteresis loss
Suitable for 400Hz transformers 

Ferrite core  no air gap
Mediumsaturation flux density
Small loop = small hysteresis loss
Suitable forhigh frequency transformers 

Ferrite core  large air gap
Small loop = small hysteresis loss
Suitable for highfrequency Inductors with large DC current 
4 Hysteresis Loss
Hysteresis loss is the result of cycling the magnetic
material along its BH curve.
It represents the energy taken as the applied voltage,
aligns magnetic dipoles first in one direction, and then in the other.
The loss increases with the area of the BH curve
enclosed. As the material is driven closer to saturation, both the area
within the curve, and the corresponding energy loss each cycle, increase
substantially.
5 Eddy Current Loss
Eddy current loss is caused by small currents circulating
within the core material, stimulated by the alternating flux in the core.
The I^{2}R power loss associated with these
currents produces heating of the core known as eddy current loss.
In ironcored transformers, insulated iron sheets known
as laminations are used to minimize this effect by restricting the path
for circulating currents.
Ferrite cores restrict these paths even further.
6 Transformer Equivalent Circuit
An ideal transformer with one primary winding and two
secondary windings can be represented as shown below
Such a transformer has the following characteristics:
• No losses
• Perfect coupling between all windings
• Infinite open circuit impedance (i.e., no input current when secondaries
are opencircuited).
• Infinite insulation between windings
In reality, practical transformers show characteristics
that differ from those of an ideal transformer. Many of these
characteristics can be represented by a transformer equivalent circuit:
Where:
• R1, R2, R3 represent the resistance of the winding wire.
• C1, C2, C3 represent the capacitance between the
windings.
• Rp represents the losses which are due to the eddy
current and hysteresis losses. These are the real power losses, sometimes
called the core loss, that may be measured by performing an opencircuit
power measurement. Because there is no load current, there is very little
I^{2}R copper loss in the energized winding, and the watts
measured at no load are nearly all due to the core.
• Lp represents the impedance due to the magnetizing
current. This is the current that generates the magnetizing force, H, used
in the BH loop diagrams. Note that this current may not be a simple sine
wave, but can have a distorted, peaked shape, if the transformer is
operated in the nonlinear region of the BH curve. This is usually the
case for linefrequency, laminate type transformers.
• L1 L2 L3 represents the leakage inductance of all the
windings. (This is discussed in detail in Voltech Note 104105, "Leakage
Inductance".)
7 Conclusions
The equivalent circuit of a transformer reflects the real
properties of the magnetic circuit comprising the core and windings.
The equivalent circuit can therefore be used with
confidence to understand and predict the transformer's electrical
performance in a variety of situations.
8 Further Reading
The equivalent circuit can also be used to help
understand and optimize the tests and test conditions that can be used to
check that a transformer has been constructed correctly.
Further technical notes in this series discuss how the
equivalent circuit parameters are used to derive practical tests for
transformers to guarantee their quality in a manufacturing environment.
See also:
Leakage Inductance Tech Note (VPN 104105)
Turns Ratio Tech Note (VPN 104113)
Ferrite Transformer Testing Tech Note (VPN 104128)
Laminate Transformer Testing Tech Note (VPN 104127)