Ideal
Transformer
A transformer
is a device in which two circuits are coupled by a magnetic
field that is linked to both. There is no conductive connection
between the circuits, which may be at arbitrary constant potentials.
Only changes in one circuit affect the other. The circuits
often carry at least approximately sinusoidal currents, and
the effect of the transformer is to change the voltages, while
transferring power with little loss. Sinusoidal excitation
is not necessary, and transformers may handle arbitrary signals,
in which the action can be considered as a transformation
of impedances. The magnetic field coupling the circuits can
be in air, but is usually in a ferromagnetic material, the
core, in which the field can be thousands of times greater
than it would be in air, making the transformer efficient
and small. The transformer is an honorary electrical "machine"
in which the flux changes occur by variation in currents with
time, instead of by motion.
Most transformers
with iron cores can be considered as ideal when you use them.
An ideal transformer has no losses, an aim that is closely
attained in practice, so the energy transfer from the primary
circuit to the secondary circuit is perfect. The diagram represents
such a transformer, showing the core with magnetic flux ?,
the primary winding of N1 turns, and the secondary winding
of N2 turns. The reference directions for the voltages and
currents at the terminals are shown. All of these quantities
are to be considered as phasor amplitudes, varying sinusoidally
with time. Note the dots at one or the other of the terminals
of each winding. Currents entering the dotted terminals produce
flux in the same direction, the direction shown. The current
and voltage ratios are equal to the turns ratio. This means
that the power factor (cosine of the phase angle), and the
power, are the same at input (primary) and output (secondary).
These things you probably already know, and we will not explore
their consequences further.
The diagram
shows the usual schematic way to represent a transformer.
In an actual transformer, the windings are wound on top of
each other, not on separate legs, to reduce leakage flux.
In the usual shelltype transformer, both primary and secondary
are on one leg, and are surrounded by the core. A coretype
transformer has windings covering the core legs.
In order
to design a transformer, or to examine in more detail how
it departs from ideality, it is necessary to understand how
a transformer works, not just how to express its terminal
relations in an approximate way. It is also important to know
how the properties of the iron core affect the performance
of the transformer. A real transformer becomes hot because
of losses, and the ouput voltage may vary with load even when
the primary voltage is held constant.
The
mutual flux ? is the means of transfer of energy from primary
to secondary, and links both windings. In an ideal transformer,
this flux requires negligibly small ampereturns to produce
it, so the net ampereturns, primary plus secondary, is about
zero. When a current is drawn from the secondary in the positive
direction, ampereturns decrease substantially. This must
be matched by an equal increase in primary ampereturns, which
is caused by an increase in the current entering the primary
in the positive direction. In this way, the backemf of the
primary (the voltage induced in it by the flux ?) equals the
voltage applied to the primary, as it must. This fundamental
explanation of the operation of a transformer must be clearly
understood before proceeding further.
Magnetic
Fields
Since
magnetic fields play an essential role in transformers, we
must review them first. They exert forces on magnetic poles,
but here the important thing is that they induce electric
fields when they move. A magnetic field is produced surrounding
any flow of electric charge, or current. The relation between
the magnetic field and the electric current can be most compactly
expressed by Ampère's Law, which states that the line
integral of the component of the magnetic field along a closed
curve is proportional to the current linked by the closed
curve. The value of this line integral is called the magnetomotive
force, or mmf, around the curve. There are equivalent formulas
that give the magnetic field at a point directly in terms
of the currents. These also require integration, and are much
more difficult to use in the simple cases we require.
The question
of units always comes up, and the question is not a pretty
one. We'll treat the matter in some detail here, because it
is a frequent source of confusion and anguish. If we wish
simple formulas, without a lot of constants, we have to massage
the units so that it all comes out straight. If we make the
choice shown in the box, and measure current in units called
absolute amperes, or abamperes, then the magnetic field H
comes out in units called oersteds if distance is in centimetres.
The value of the mmf can be called so many gilberts if you
like, and then oersteds are gilberts per centimeter. The basic
concept is current linked per unit length of path, however
expressed. We presume that there is some way to define the
absolute ampere independently of Ampère's Law.
The unit
for current you know is probably ordinary amperes. Then, if
we use an even simpler expression for the mmf, omitting the
4?, and use metres for distance, we get a magnetic field in
amperes per metre. Here, amperes means the total current.
If we have N turns of wire carrying I amperes, we have NI
amperes total linking the magnetic field. Hence, people usually
speak of ampereturns in this case. It is the omission of
the 4? that makes things confusing, though you can easily
work out the relation between ampereturns/metre and oersted
by calculating the field for the same current in the two cases,
and comparing the results.
Less weird,
perhaps, was the practice in American electrical engineering
of using the inch as the length unit. Now we have gilberts
per inch for the magnetic field, which is not an oersted,
but 1/2.54 of an oersted. One can also speak of ampereturns
per inch in the same way.
Now there
is the matter of magnetic flux. Flux is another manifestation
of the magnetic field, defined by its ability to induce an
electric field when it changes. The basic relation is Faraday's
Law. Note the analogy with Ampère's Law. The negative
sign (another fertile source of confusion) is there so energy
will be conserved, and depends on the positive directions
shown in the figure. This requires that if ? is increasing,
any current that flows due to the induced voltage must act
to decrease ? (Lenz's Law). Check that this is so in the diagram.
The integral
of the electric field around a closed curve is called the
electromotive force, emf, and is probably a more familiar
quantity than the mmf. As in the case of Ampère's Law,
we presume that there is some independent way to define an
emf, and the unit here is the abvolt. This unit is, in fact,
defined by Coulomb's Law, at least in principle. Now, we choose
the simplest relation for Faraday's Law, emf = d?/dt. The
unit of flux is such that a change of one unit per second
induces an emf of one abvolt. The unit is called a line or
a maxwell. Let's use line here, and remember that a magnetic
field is not made of physical lines of anything. "Line"
is just a conventional name, recalling the representation
of a field by lines of force. The flux density B is in lines
per square centimetre, units called gauss with G as the symbol.
We now
have two quantities, the abampere of current or mmf, and the
abvolt of potential or emf, defined in terms of the forces
between currents, and between charges, respectively. We also
have two definitions of magnetic field, one in terms of mmf
per unit length, and the other in terms of emf times seconds.
There is really only one magnetic field with two manifestations
or properties, though early mechanical theories of electromagnetism
considered that one caused the other, like stress caused strain.
This interpretation has been fossilized in the Giorgi system
of units that is now standard in engineering, and which has
caused a great deal of confusion and unnecessary anxiety,
especially in magnetism. Things are actually simpler than
they have been made out to be.
The big
advantage of the definitions of the magnetic field in terms
of the abampere and abvolt is that the magnetic field H in
oersted comes out to be exactly the magnetic flux density
B in gauss. Naturally: they are the same thing, after all.
In all the other systems we have skirted around, this is not
true. You already know that B = ?0H in Giorgi units. The ?0
is simply a number, not a physical quantity, that adjusts
for arbitrary choices of units. It has no physical interpretation
whatsoever.
We are
rightly very attached to our volt and ampere, the practical
units arbitrarly defined by the potential of a Daniell cell
and the resistance of a column of mercury early in the history
of electromagnetism. Giorgi found that if we took 10 ampere
= 1 abampere, and 1 volt = 108 abvolt, approximately, that
everything came out wonderfully consistently, so long as the
metre was the length unit, not the centimetre. There were
some small adjustments in the previous practical units, but
this was a small price to pay for having them join an absolute,
very scientific, system of units. The opportunity was also
taken to remove the factors of 4? from some equations, and
add them to other equations, which has complicated things.
For any overall view, it is best to consider Maxwell's Equations
and the force equations, but what we have done here will do
us.
The older
way was to use oersted and gauss, with volts and amperes,
and inches as the length unit, at least in Englishspeaking
countries. The gauss and oersted have not disappeared, even
with Giorgi units everywhere else. In fact, 104 G = 1 Weber,
and 1 Tesla = 108 lines. Obviously, 1 T/s = 1 V. If it weren't
for the damned 4?, all the new units would differ only by
powers of 10 from the old! The Giorgi system is just the old
absolute electromagnetic system, tarted up, really. All the
fuss over units is actually a blessing; when you understand
the units, you will understand electromagnetism.
We
all are comfortable with an electric circuit. The current
stays in the wires, and the applied emf is equal to the sum
of the voltage drops, and the sum of the currents entering
a junction is zero. A resistive element has a voltage drop
that is proportional to the current. This analogy to friction
is actually rather complicated, but at least it looks simple,
and all we need is the resistance, R = l / ? A, where ? is
the conductivity of the material of uniform cross section
A and length l. What happens is that the electric field resulting
from the emf causes free electrons to move along the wire.
There is an electric field outside the wire as well, but it
finds no movable electrons.
An mmf
also causes a magnetic field to occur, and it usually just
spreads out in space. There are no little free magnetic poles
for it to act on, but in iron there are electrons that can
be called to attention, so that they all point in roughly
the same direction, and hold themselves there by atomic forces.
The electrons are spinning charges, and the net effect is
that of a lot of small current loops causing a very strong
magnetic field. That is, the H field associated with the mmf
calls up a total magnetic field in the same direction that
we must call B, because it will certainly induce an electric
field according to its magnitude. H is only a partial field,
produced by whatever makes the mmf, while B is the total field.
This is the usual interpretation of H and B, which is quite
useful. H and B are not really different kinds of field, except
in the imagination. In the Giorgi system, they are expressed
in different units, however, which has no physical significance.
So, an
mmf is produced by ampereturns, and in a uniform iron core
creates a uniform H field, which snaps the electrons into
order making a flux density B ? times larger, and a total
flux ? = BA. Just as in the electrical case, there is a magnetic
field outside the iron core, but no electrons are there to
join the army, so not much flux results. The small flux that
does go this way is called leakage flux. Leakage flux is important
in transformers. The whole thing is remarkably like a circuit,
with mmf in place of emf, and flux in place of current. The
analog of resistance is called reluctance, given by the same
formula as resistance, but with permeability replacing conductance.
The ratio of the permeability in the iron to that in air is
about 1000. For electric currents, the ratio of conductivities
is a million times greater.
The ideal
is to have all the magnetic flux produced by one winding link
the other. This is not attained in practical transformers.
Flux that does not link both windings is called leakage flux
and has the effect of adding an inductance that produces a
voltage drop when current is present. Leakage is affected
by the shape of the core and by the arrangement of the windings.
To reduce leakage, the core must be compact and the windings
close together.
Iron
The conductivity
of a conductor is practically constant, independent of the
current. The permeability of iron is by no means constant.
The flux density B of a sample is a nonlinear function of
H, given by the magnetization curve. Worse than this, the
flux density depends on what has happened in the past, as
well as to the current value of H. If H is varied sinusoidally,
B follows an Sshaped curve called the hysteresis loop, which
is different for every different maximum value of H. B lags
behind H in its variation, and energy is lost in a kind of
internal friction. The energy lost in each cycle is proportional
to the area inside the hysteresis loop. The slope of the approximately
linear portions of this loop is the value to be used in estimating
the reluctance, which we will discuss later. It is not, incidentally,
the slope of the magnetization curve. Refer to books on magnetism
for a full discussion of magnetization and hysteresis. There
is a lot of lore here.
An ideal
core material would have a hysteresis loop with no area inside
it. In this case, B would be a singlevalued function of H,
as shown in the diagram on the left. At high values of H,
when all the electrons were aligned, the curve would bend
over and saturate. Although the iron saturates, the flux continues
to increase as in air, so the B curve is never horizontal.
It is most economical to operate magnetic materials at as
high a flux density as possible, usually near the knee of
the curve. Note that the curve in the figure is a hysteresis
curve (alternating flux), not a magnetization curve.
In actual
magnetic materials, the flux does not drop to zero when the
mmf returns to zero, but there is a remanent flux. To reduce
the flux to zero, an mmf in the reverse direction called the
coercive force must be applied. Now the B vs H curves are
moved left and right, without changing their shapes much.
The horizontal distance between the curves is twice the coercive
force. The ends of the loop depend on the maximum flux density,
so there are different hysteresis curves for each maximum
flux density. Since the flux lags the current, there is now
component of the voltage in phase with the applied current,
which means an energy loss, the hysteresis loss. The amount
of loss, in ergs per cycle per cubic centimeter, is the area
within the hysteresis loop divided by 4?. (Axes are gauss
and Oersted.)
If a sinusoidal
alternating voltage is applied to the winding creating the
emf, the flux is sinusoidal, but the current is not. The current
must vary in just such a way as to produce the sinusoidal
flux, which is 90° out of phase with the applied voltage.
The figure shows a sketch of the current waveform that results.
This current, called the magnetization current, will be almost
90° out of phase with the applied voltage, as shown, with
a small inphase component representing hysteresis loss. The
magnetizing current shows a strong third harmonic component.
The prominent peaking at maximum B is the result of the bending
over of the magnetization curve. Note that the flux is a maximum
when the voltage is zero, and that the coercive force can
be observed in the current waveform, showing that there is
a component of the current in phase with the voltage, representing
an energy loss. A transformer is designed so that the magnetizing
current is a small fraction of the normal load current, usually
only a few per cent. Since the flux is accurately sinusoidal,
the secondary voltage (and current) will also be sinusoidal.
The peculiarities of the magnetizing current will not be reflected
in the transformer output, and will not cause distortion.
The magnetizing
current can be observed with an oscilloscope using the circuit
at the right. The voltage across the 10? currentsensing resistor
will be negligible. Channel 1 will be set to 50 V/division,
Channel 2 to perhaps 0.2 V/division. If you feed the circuit
with a Variac, make QUITE sure that the connection shown grounded
is indeed grounded, so it will not conflict with the oscilloscope
ground. To be safe, use an isolation transformer. If you know
the constructional details of the transformer tested (see
the paragraph below on dissecting a transformer), you can
find B from the voltage, number of turns and core area, while
H can be found from the current, number of turns, and magnetic
circuit length. This makes it possible to sketch a hysteresis
curve for the transformer core.
As
the flux ? varies with time, there will certainly be an induced
electric field in the core, and if it is conducting, currents
will flow. These are called eddy currents, and are precisely
analogous to a shortcircuited secondary winding. They will
cause heating, and will represent an energy loss. They are
minimized by decreasing the conductivity of the iron (silicon
iron has a smaller conductivity), and by making the core from
a pile of thin laminations insulated from each other. They
cannot be reduced to zero, but can be made quite small. Like
hysteresis, eddy currents will produce a current in phase
with the applied voltage, that is included in the magnetization
current. Iron losses, as they are called, are reflected in
the inphase component of the magnetization current.
Ferromagnetic
materials saturate at a certain flux, as has been mentioned,
and are usually used at fluxes not far below saturation. If
the primary of a transformer carries direct current, which
is not resisted by inductance but only by the resistance of
the winding, which is always low, it is easy to drive the
core into saturation for at least part of the cycle. The result
is excessive current, first appearing as current spikes, that
causes the transformer to heat, and, of course, the transformer
action to fail. Transformers must not be subjected to direct
currents. This includes secondary windings as well as primary
windings, of course.
Some ironcore
inductors are designed to carry direct currents. These are
chokes that block alternating currents. To prevent saturation,
the core is provided with an air gap. Even a small air gap
dominates the reluctance and prevents saturation at reasonable
currents (the flux is proportional to the ampereturns). The
flux density is kept lower than the saturation value for the
iron, of course, but now the inductance is independent of
the direct current. In audio transformers for pushpull amplifiers,
direct currents flow in the divided primary in opposite directions,
so that their mmf's cancel, while the alternating audio signal
flows in the whole primary.
Losses
and Iron
Early
transformers used pure iron for their cores. Pure iron has
a hysteresis loss of about 600 ergs per cycle per cubic centimeter,
a relatively small value, and a coercive force of 0.2 Oe from
10 kG, so it is magnetically quite "soft." Commercial
sheet iron has a larger, but still small, hysteresis loss.
However, it is a fairly good electrical conductor, with a
resistivity of 7.64 ??cm. Therefore, it acts like a shortcircuited
turn, and currents are induced that circle round the magnetic
field as it changes. These eddy currents dissipate large amounts
of energy, making a solid core very inefficient. This can
be overcome by laminating the core, and interrupting the possible
current paths. Bundles of wire were used as cores for this
purpose, but thin sheet laminations are now the universal
practice. The eddycurrent loss is proportional to the square
of the thickness of the laminations. It's proportional to
the squares of the frequency and the maximum flux density
as well.
Around
1903, Sir Robert Hadfield discovered that adding silicon to
steel greatly increased its resistivity, while affecting the
magnetic properties very little. A silicon steel with 4.25%
Si has a resistivity of 60 ??cm. The hysteresis loss is about
1800 ergs per cycle per cubic centimeter, and the coercive
force is about 0.4 Oe. The only drawback of silicon is that
it makes the steel brittle in such concentrations, making
punching and other mechanical operations more difficult. The
laminations must be insulated from each other, of course.
It seems that originally rust was the insulation, but since
rust contributes greatly to hysteresis loss, laminations are
now pickled and insulated by a compound. The density of silicon
steel is 7.5 g/cc for Si < 2%, and 7.7 g/cc for Si >
2%.
Serendipitously,
it was found that steel containing more than 1% Si was not
subject to the phenomenon of "ageing" in which the
coercive force (and so the hysteresis loss) increased as much
as 100% with time, when the iron operated at temperatures
above 150 °F (as is often the case) or was subject to
mechanical actions. All transformer cores for power frequencies,
and parts of rotating machinery subject to alternating fields,
are now made of silicon steel, for this and the other reasons
mentioned.
With
the eddycurrent loss problem solved by silicon steel and
laminations, the larger part of core loss is hysteresis loss,
which is proportional to the frequency (since a fixed amount
of energy is lost in each hysteresis cycle). Steinmetz found
that the hysteresis loss was W = A B1.6, where B is the maximum
flux density. The constant A is about 0.001 for silicon steel,
0.013 for cast iron, and 0.004 for sheet iron. The large value
for cast iron should be noted. Magnetic cores for early rotating
machines were typically cast iron, which has a relatively
high electrical resistance (reducing eddycurrent losses)
but large hysteresis loss. The eddycurrent and hysteresis
losses can be separated experimentally by tests at different
frequencies, because of their distinct frequency dependences.
Core losses
generally determine the power ratings of large transformers,
which often must be artificially cooled by forced ventilation
or circulating fluids. It is more efficient to design transformers
for high maximum flux densities, but this also increases the
core losses.
Copper
losses are the I2R losses in the windings, and are relatively
more important in small transformers than in large. It was
the practice to design windings at 700 c.m./A (c.m. = circular
mils, the square of the diameter in thousandths of an inch),
or alternatively at 500 c.m./A. Many transformers are now
designed with as little copper as 340 c.m./A.
Transformer
Operation
In a real
transformer, we must include the magnetizing current, as well
as the effects of leakage flux. Leakage flux links only one
winding, and is completely independent of the other. Its effect
is to act like an inductance in series with a winding. In
addition, there is the IR drop in the copper of a winding
when it carries current. These are the things that make a
real transformer behave differently from an ideal transformer.
If we just talk about these things, we soon are mired in a
swamp of phase angles and relative directions that make understanding
very difficult. The best way to understand things is to draw
a phasor diagram of the currents, voltages and fluxes. In
doing this, we draw the diagram in stages, showing how to
proceed step by step so that it would be possible to make
accurate numerical estimates of each quantity. The magnetizing
current is not wellrepresented by a phasor, since it is nonsinusoidal,
but this causes no serious inaccuracy, since most of the quantities
in a transformer under load are sinusoidal.
Before
a phasor diagram can be drawn, a circuit diagram showing the
connections and the positive directions of each circuit quantity
is necessary. Such a diagram is shown at the right. It is
not really complete, since it does not show the connection
through the flux ? that reflects equal induced voltages and
currents from secondary to primary. This is hinted at by the
generators marked E. When the phasor diagram is drawn, we
will reverse the positive directions in the primary to separate
the two parts of the diagram so that they will not fall on
top of each other. The diagram shows the winding resistances
R, and the reactances due to the leakage fluxes X, as well
as a current generator for the magnetizing current I0. We
will assume that the turns ratio is unity for convenience,
so the vectors turn out to be about the same lengths on the
same scale. If the turns ratio were not unity, the voltages
and currents would be reflected into the primary multiplied
by the proper ratios.
The phasor
diagram for the transformer at full load is shown at the left.
This complicated diagram can be understood by following through
how it was constructed. We start by assuming that the secondary
is supplying a current I2 at a terminal voltage V2 with phase
angle ?. These are the first two phasors drawn. Now, to V2
we add I2R2 in phase with I2, and I2X2 in quadrature, to find
the induced voltage E in the secondary. The voltage E is
induced in the primary. The flux ? is at right angles to E,
as shown. The current I0 is necessary to create the flux,
and is drawn with its proper relation to ?. The current I2
is reflected to the primary as I2, and added to I0 to find
the total primary current I1. Now that we know I1, we can
add I1R1 and I1X1 to E to find the primary terminal voltage,
V1. In this diagram, the magnetizing current, and the voltages
due to leakage flux and winding resistance, are greatly magnified
so their effects can easily be seen. As an exercise, draw
a phasor diagram for an open secondary, when I2 is zero, and
find V1 for this case. Also, draw diagrams for various phase
angles ?.
Try to
visualize what happens when the secondary current decreases.
It may help to draw additional phasor diagrams. The difference
between the terminal voltage V2 and the induced voltage E
becomes less, and they approach one another. The magnetizing
current becomes a larger part of the smaller primary current,
which changes in phase accordingly. The flux, of course, remains
the same. The input voltage decreases, and becomes equal to
V2. One of the important characteristics of a transformer
is its regulation, the change in output voltage between no
load and full load, divided by the fullload output voltage
(or some such definition), expressed as a percentage. This
is practically the same as the difference between the fullload
and no load input voltages divided by the noload input voltage,
holding the output voltage constant, and this latter quantity
can be read off the fullload vector diagram. The noload
input voltage is V2, and the fullload input voltage is V1.
Typical power transformer regulation is around 1% or even
less at unity power factor, falling off to a few percent at
0.8 pf. Such transformers are called constantpotential transformers.
Regulation is improved by decreasing leakage flux and winding
resistance. If poor regulation is desirable, it can be obtained
by increasing the leakage flux without added resistive heating.
The efficiency
of a transformer is quite high, usually over 98% at full load.
It increases at partial load, sometimes to over 99%. A real
transformer is not very far from an ideal transformer, in
fact. These figures for efficiency and regulation apply to
large power and distribution transformers, say above about
5 kVA capacity (product of rated voltage and current) with
silicon steel cores. Smaller transformers used in electronics
and consumer products are slightly less ideal, largely because
of economy in use of copper and the proportionately greater
leakage flux. It is difficult to measure the efficiency of
a transformer directly (by comparing output with input), so
the copper and core losses are usually measured or estimated
instead. Core losses are usually only about 1/3 of the total.
Transformer
Design
Transformer
design is a welldeveloped subject, and the reader is referred
to the many textbooks that treat it. It is important not simply
to design a transformer that will do, but one that is economical,
efficient and makes the best use of available materials. However,
it is not difficult to design a serviceable transformer, and
by understanding how this is done we will know better why
transformers are as they are. As in all design processes,
there are numerous tradeoffs between competing requirements,
and some sort of optimum is sought. On the other hand, the
properties of available materials rigidly limit the possibilities.
The size,
shape and material of the core must be chosen, and the number
of turns of the primary and secondary windings. The size of
the wire and the insulation determines if the windings will
fit in the space available. The windings must be arranged
for minimum leakage flux. The insulation determines the permissible
temperature rise, and means of cooling must balance the loss
of energy under load. We will treat only the core and the
number of turns, the fundamental parameters, in our analysis.
A complete design must, of course, include all these features.
Two basic
equations are used in transformer design. The first is essentially
Faraday's law, E = d?/dt x 108 V, where ? is maxwells (gauss
times square centimeters). For transformers, this is written
Bmax = ?2 E 108 / 2?fNAK, where E = rms voltage in a winding,
N = number of turns, f = frequency (Hz), A = area of core
(cm2), K = stacking factor (the proportion of A occupied by
iron). A sinusoidal variation in flux is assumed, which is
a reasonable assumption when the core does not saturate, but
by no means exact. Bmax is generally assumed as a design parameter,
as well as a value for E/N in volts per turn, and the necessary
A results. A typical value for a small transformer is E/N
= 0.1 V/turn.
Two basic
equations are used in transformer design. The first is essentially
Faraday's law, E = dΦ/dt x
10^{8} V, where Φ is maxwells
(gauss times square centimeters).
For transformers, this is written B_{max} = √2 E 10^{8} / 2πfNAK,
where E = rms voltage in a winding,
N = number of turns, f = frequency (Hz), A = area of core
(cm^{2}), K = stacking factor (the proportion of A
occupied by iron). A sinusoidal variation in flux is assumed,
which is a reasonable assumption when the core does not saturate,
but by no means exact. B_{max}
is generally assumed as a design parameter, as well as a value
for E/N in volts per turn, and the necessary A
results. A typical value for a small transformer is E/N =
0.1 V/turn.
The second
is Ampere's law, H = 0.4πNI/l, where l is the length
of the magnetic circuit. From the value of B_{max},
the value of H_{max} can be found from the magnetization
curve of the core material, and this equation used to determine
N, when a reasonable value of I is assumed (say, 5% of the
fullload current).
Let's
try to proportion a transformer for 120 V, 60 Hz supply, with
a fullload current of 10 A. All the AC values we use will
be effective values. The core material is to be siliconsteel
laminations with a maximum operating flux density B_{max}
= 12,000 gauss. This is comfortably less than the saturation
flux density, B_{sat}. The first requirement is to
ensure that we have sufficient ampereturns to magnetize the
core to this level with a permissible magnetizing current
I_{0}. Let's choose the magnetizing current to be
1% of the fullload current, or 0.1
A. The exact value is not sacred; this might be thought of
as an upper limit. From past experience, we should have some
idea of the size of core that will be required. Here, we will
assume a simple, uniform magnetic circuit for simplicity.
In an actual case, a more complicated magnetic circuit would
have to be considered. If l is the length of the magnetic
circuit, H is 0.4πN(√2I_{0})/l,
and the magnetization curve for the core iron gives the H
required for the chosen B_{max}. From this, we can
find the number of turns, N, required for the primary.
We could
also estimate the ampereturns required by using an assumed
permeability μ. Experience will furnish a satisfactory
value. It is not taken from the magnetization curve, but from
the hysteresis loop. Let's take μ = 1000. Then, N = B_{max}l
/ 0.4π√2 μI_{0}. If we estimate l
= 20 cm, the number of primary turns required is N = 1350.
The rms voltage induced per turn is determined from Faraday's
Law: √2 e = (2πf)B_{max}A
x 10^{8}. Now, e must be 120 / 1350 = 0.126 V/turn,
f is 60, and B_{max} = 12,000 gauss. We know everything
but A, the crosssectional area of the core. We find A = 2.8
cm^{2}.
This may,
or may not, be an acceptable result. If not, we simply change
our assumptions and try again. Design, after all, is an iterative
process. By considering the above calculations, we can appreciate
the changes that a different frequency, voltage and current
rating, permeability or maximum flux density would produce.
The number of secondary turns is now easily found from the
volts/turn and the desired ratio. A 24 V secondary would have
24 / 0.126 = 190 turns.
The number
of primary turns is determined so that the magnetizing current
is limited to an acceptable value, and depends on the length
of the core. The area of the core is determined by the required
volts per turn, now that the total number of turns is known.
These are the things that determine the size and weight of
a transformer.
Toroidal
ferrite or powdered iron cores are now easily available. Ferrite
is a highpermeability, highresistance material that has
acceptable losses at high frequency. Powdered iron has granules
insulated from each other for high resistivity (low eddycurrent
losses) and is good at moderate frequencies. Such cores can
be used to experiment with transformer design, using a signal
generator as a power source. The length, area and permeability
of the core is now known at the beginning, making the above calculations
somewhat easier. An oscilloscope can be used for measurements.
Powdered
iron and ferrite cores have low B_{sat} and permeability
values. A type 43 ferrite has B_{sat} = 2750 gauss,
but a maximum permeability of 3000, and is recommended for
frequencies from 10 kHz to 1 MHz. Silicon iron is much better
magnetically, but cannot be used at these frequencies. The
approximate dimensions of an FT114 ferrite core (of any desired
material) are OD 28 mm, ID 19 mm, thickness
7.5 mm. The magnetic dimensions are l = 74.17 mm, A = 37.49
mm^{2}, and volume 2778 mm^{3}. Similar information
is available for a wide range of cores. There are tables showing
how much wire can be wound on them, and even the inductance
as a function of the number of turns. Consult the ARRL Radio
Amateur's Handbook, or specifications from Amidon Associates.
Transformers
with air cores are used in radio work. They have no core losses,
and are not limited in frequency (which is why they are used).
A little calculation will show how hopeless an aircore transformer
is at power frequencies. Air core transformers have large
leakage fluxes, which cannot be avoided, and therefore poor
regulation. Windings must be carefully designed to give the
largest mutual flux possible.
Dissecting
a Transformer
There
is a silver lining if you happen to burn out a small transformeryou
can take the transformer apart to see what is inside, and
analyze its design. I disassembled a transformer with a 24
VCT, 1 A secondary and 117 V primary. The primary had a resistance
of 48.8Ω, the secondary 2.7Ω. The first step is
to remove the mounting U bracket by bending back the tabs
holding it in place. This leaves you with the core and windings,
with the core tightly held by the plastic bobbin holding the
windings. The windings on this transformer are wound one over
the other, the secondary on top of the primary. Some transformers
now use separate bobbins for primary and secondary, which
are stacked on the core. This arrangement offers advantages
in manufacture, but the leakage flux is probably greater than
with windings one over the other.
The
next step is to remove the core, which is done lamination
by lamination. The core consists of E and I pieces, alternating
so that successive I pieces do not fall one over the other.
A suitable tool for the disassembly, which is not easy, is
a small, sharp cold chisel driven by a light hammer. The first
E lamination can be driven out by using the chisel at the
edge of the bobbin. The laminations are glued together by
the insulating substance, and have to be pried apart before
they can be removed. After the first E lamination on each
side is removed, things become a little easier.
The
idea is to chisel off the I piece, then pry the E piece away
so that it comes out. I discovered 37 laminations in the core
(37 E, 35 I). The area of the magnetic circuit was A = 2.88
cm2, and its length l = 9.6 cm, found by measuring the laminations.
The stacking factor K could not be determined accurately,
so we shall assume that it is 1.00 without much error. The
laminations were 0.50 mm thick, close to USS gauge #26. Use
the DMM to verify that the surfaces of the laminations are
not conducting. The total weight of the core was 224 g, about
half a pound.
The two general configurations of iron and copper in a transformer
are shown at the right. Most small power transformers are
shelltype, in which the iron surrounds the copper. The ideal
would be for the iron to completely surround the windings,
but this is impractical. The compromise is to divide the magnetic
circuit into two return paths on opposite sides of the core,
as is done with the E and I laminations, as shown. Another
configuration, the core type, in which the copper surrounds
the iron, is common with larger highvoltage transformers,
as well as with small ferritecore toroidal transformers.
Note that both primary and secondary windings are placed on
each leg. The laminations are Lshaped to make a core of uniform
crosssection with a central rectangular window. Core transformers
are easier to insulate and to cool. If the primary winding
were placed on one leg and the secondary on the other, as
in the diagrams of transformers in texts, leakage flux would
be excessive, and the transformer would have poor regulation
(variation of voltage with load). There are other special
types of construction, but all can be classified as shell
or core.
Note how connections are made, and the provision for insulation.
When the outside insulating material has been removed, the
secondary winding can be unwound. Note the number of turns,
which came out at 240 for my transformer, so that there are
10 turns per volt rms. It is now possible to estimate Bmax
from Faraday's law, with the result 13 000 G, a quite reasonable
result. The primary should have 1170 turns (I did not unwind
the primary to find outthis would result in far too much
fine wire). The magnetization curve shows that about 9.5 Oe
is required to produce this flux. From Ampere's law, the primary
magnetizing current should be 44 mA (rms), a reasonable figure.
The primary inductance is about 7 H, from this result. The
actual transformer had a measured magnetizing current of 63
mA rms. This is a relatively large value, but of little consequence
in a transformer of this size.
The primary winding consisted of 0.50 mm enameled wire, about
AWG #25, while the primary was 0.21 mm, about AWG #31. From
the current rating of the transformer, these sizes correspond
to 340 c.m./A, a somewhat adventurous figure. Note that the
size of the wire has nothing to do with the magnetic design
of the transformer, only with the copper loss and heating,
and determines what will fit in the space provided by the
core. The matter of insulation between the windings is of
importance. In this small transformer, the critical point
appeared to be where the primary wires were led out past the
secondary winding. An extra piece of insulating cloth was
used to protect the leads.
Using
Transformers; Phasing
Three
rules should always be observed in using transformers: first,
a voltage greatly in excess of the rating should not be applied
to any winding; second, a significant direct current should
not be allowed to flow through any winding not designed for
it; third, the frequency should not be significantly less
than the design frequency. If you make the primary of a small
120 V transformer carry, say, 50 mA direct current, and then
connect it to the 120 V line, you will have smoke and combustion!
Applying 120 V to a 12 V secondary in hopes of getting 1200
V at the primary will have a similar result, accompanied by
insulation failure which will add further fireworks. Merely
overloading a transformer is not so seriousit will simply
get hot and eventually burn up, or, more likely, a wire will
fuse. There is normally little chance of running across a
lower power frequency, such as 25 Hz. A 60 Hz transformer
will draw too much magnetizing current on 25 Hz, and will
run quite hot.
However,
you can always apply less than the rated voltage to any winding,
and use any winding as primary or secondary, so long as the
current rating is not exceeded. A few milliamperes of direct
current is no problem, either. Any frequency higher than the
rated frequency is also safe.
Many
transformers are made with duplicate windings, to permit flexibility
in operation. Two 110 V primaries can be connected in parallel
for 110 V, or in series for 220 V. Even more common are dual
secondaries, which can be connected in series or parallel.
In parallel, there will be a center tap as well. The windings
cannot arbitrarily be connected to each other, because of
the problem of phasing. The voltages across the windings will
either be in phase, which is the desired state, or in antiphase.
If connected in antiphase, the result is a short circuit.
Two windings on a core are shown at the right. The black dot
at one terminal or the other of each winding is called a phase
or polarity mark. Currents entering the marked terminals cause
magnetic flux in the same direction in the core. An increasing
current entering the marked terminal as shown will cause a
positive voltage at the marked terminal of the other winding.
If the unmarked terminal of the winding on the left is connected
to the marked terminal of the winding on the right, the two
windings will be in phase and their ampereturns will add.
If they are connected in the opposite sense, their ampereturns
will cancel, and there will be no inductance. Unfortunately,
transformers are not provided with phase marks, and you have
to figure them out for yourself. This process is called phasing.
Of course, most transformers come with numbered terminals
and wiring diagrams so that phasing is not required. Still,
it is a good idea to check the phasing in any case, and to
know how to do it in case you have a strange transformer without
instructions. An obvious method is to use an oscilloscope
and a function generator, which is easy and completely safe.
However,
it is possible to phase a transformer with a voltmeter alone.
I use a Variac to apply a low voltage, but a series resistance
can be used to limit the current in case of an unexpected
short. Apply a voltage to one of the windings, here winding
12. We can arbitrarily assign a phase mark to one end of
this winding, say terminal 1. Connect the other end of this
winding to one end of the other winding. There are only two
possibilities, that we have connected to the marked or the
unmarked terminal, as shown. Now measure the voltage between
terminal 1 and the free terminal of the other winding (this
cannot result in a short circuit because of the high resistance
of the voltmeter). If you find twice the applied voltage,
then conditions are as at the left. If you find zero volts,
then conditions are as at the right. Now the second winding
can be marked, and the proper connections determined. To phase
the secondaries, measurements can be made in exactly the same
way. One winding is marked arbitrarily, the other connected
to its other end, and the voltage measured to the open terminal.
The secondaries can be phased relative to the primary just
as if they were more primary windings, provided low voltages
are used and the turns ratio taken into consideration. In
all these
measurements,
we make use of the fact that the phase relations can be only
in phase or in antiphase, which give distinctly different
resultant voltages.
Autotransformers
In the
diagram at the right, a stepdown transformer is shown. The
secondary voltage is half the primary, so the currents are
doubled, in this example. The ampereturns of the secondary,
carrying 20 A, are opposed by the same number of ampereturns
of the whole primary. In the middle diagram, the secondary
load is shown attached to the primary winding at its midpoint.
Now, the ampereturns below the connection are opposed by
the ampereturns above the connection, each excited by 10
A, but in opposite directions. The currents add at the node
to give 20 A in the secondary load, as required. Conditions
are just as in the case of separate windings, but without
the secondary winding carrying 20 A. This is called an autotransformer,
and the savings involved in its use are obvious. The disadvantage,
of course, is that the primary and secondary are no longer
isolated and share a metallic connection. In many applications,
this is no matter, and an autotransformer can be used.
At the right is shown a "Variac" (a trade name belonging
to General Radio Corporation, but now used like Kleenex and
Vaseline) or variable transformer, a useful device that provides
an adjustable AC voltage. For safety, the common connection
must be connected to the grounded or white wire of the AC
lines. If, somehow, the connection is reversed, then everything
connected to the Variac becomes "hot" and hazardous.
The common 120/240V converters are autotransformers, and present
the same hazard.
Some early AC/DC radios (the name for those without power
transformers) had only one wire in the line cord, the hot
wire. The other connection was to be made to a handy pipe,
which was the service ground. If the plug were inserted incorrectly,
no power would be supplied, and everything would be safe.
With today's polarized plugs, this is not necessaryunless
an amateur electrician wired the receptacle improperly.
