Mutual inductance and basic operation
Suppose we were to wrap a coil of
insulated wire around a loop of ferromagnetic material and energize this
coil with an AC voltage source:
As an inductor, we would expect this
iron-core coil to oppose the applied voltage with its inductive
reactance, limiting current through the coil as predicted by the
equations XL = 2πfL and I=E/X (or I=E/Z). For the purposes of
this example, though, we need to take a more detailed look at the
interactions of voltage, current, and magnetic flux in the device.
Kirchhoff's voltage law describes how the
algebraic sum of all voltages in a loop must equal zero. In this
example, we could apply this fundamental law of electricity to describe
the respective voltages of the source and of the inductor coil. Here, as
in any one-source, one-load circuit, the voltage dropped across the load
must equal the voltage supplied by the source, assuming zero voltage
dropped along the resistance of any connecting wires. In other words,
the load (inductor coil) must produce an opposing voltage equal in
magnitude to the source, in order that it may balance against the source
voltage and produce an algebraic loop voltage sum of zero. From where
does this opposing voltage arise? If the load were a resistor, the
opposing voltage would originate from the "friction" of electrons
flowing through the resistance of the resistor. With a perfect inductor
(no resistance in the coil wire), the opposing voltage comes from
another mechanism: the reaction to a changing magnetic flux in
the iron core.
Michael Faraday discovered the
mathematical relationship between magnetic flux (Φ) and induced voltage
with this equation:
The instantaneous voltage (voltage
dropped at any instant in time) across a wire coil is equal to the
number of turns of that coil around the core (N) multiplied by the
instantaneous rate-of-change in magnetic flux (dΦ/dt) linking with the
coil. Graphed, this shows itself as a set of sine waves (assuming a
sinusoidal voltage source), the flux wave 90o lagging behind
the voltage wave:
Magnetic flux through a ferromagnetic
material is analogous to current through a conductor: it must be
motivated by some force in order to occur. In electric circuits, this
motivating force is voltage (a.k.a. electromotive force, or EMF). In
magnetic "circuits," this motivating force is magnetomotive force,
or mmf. Magnetomotive force (mmf) and magnetic flux (Φ) are
related to each other by a property of magnetic materials known as
reluctance (the latter quantity symbolized by a strange-looking
letter "R"):
In our example, the mmf required to
produce this changing magnetic flux (Φ) must be supplied by a changing
current through the coil. Magnetomotive force generated by an
electromagnet coil is equal to the amount of current through that coil
(in amps) multiplied by the number of turns of that coil around the core
(the SI unit for mmf is the amp-turn). Because the mathematical
relationship between magnetic flux and mmf is directly proportional, and
because the mathematical relationship between mmf and current is also
directly proportional (no rates-of-change present in either equation),
the current through the coil will be in-phase with the flux wave:
This is why alternating current through
an inductor lags the applied voltage waveform by 90o: because
that is what is required to produce a changing magnetic flux whose
rate-of-change produces an opposing voltage in-phase with the applied
voltage. Due to its function in providing magnetizing force (mmf) for
the core, this current is sometimes referred to as the magnetizing
current.
It should be mentioned that the current
through an iron-core inductor is not perfectly sinusoidal (sine-wave
shaped), due to the nonlinear B/H magnetization curve of iron. In fact,
if the inductor is cheaply built, using as little iron as possible, the
magnetic flux density might reach high levels (approaching saturation),
resulting in a magnetizing current waveform that looks something like
this:
When a ferromagnetic material approaches
magnetic flux saturation, disproportionately greater levels of magnetic
field force (mmf) are required to deliver equal increases in magnetic
field flux (Φ). Because mmf is proportional to current through the
magnetizing coil (mmf = NI, where "N" is the number of turns of wire in
the coil and "I" is the current through it), the large increases of mmf
required to supply the needed increases in flux results in large
increases in coil current. Thus, coil current increases dramatically at
the peaks in order to maintain a flux waveform that isn't distorted,
accounting for the bell-shaped half-cycles of the current waveform in
the above plot.
The situation is further complicated by
energy losses within the iron core. The effects of hysteresis and eddy
currents conspire to further distort and complicate the current
waveform, making it even less sinusoidal and altering its phase to be
lagging slightly less than 90o behind the applied voltage
waveform. This coil current resulting from the sum total of all magnetic
effects in the core (dΦ/dt magnetization plus hysteresis losses, eddy
current losses, etc.) is called the exciting current. The
distortion of an iron-core inductor's exciting current may be minimized
if it is designed for and operated at very low flux densities. Generally
speaking, this requires a core with large cross-sectional area, which
tends to make the inductor bulky and expensive. For the sake of
simplicity, though, we'll assume that our example core is far from
saturation and free from all losses, resulting in a perfectly sinusoidal
exciting current.
As we've seen already in the inductors
chapter, having a current waveform 90o out of phase with the
voltage waveform creates a condition where power is alternately absorbed
and returned to the circuit by the inductor. If the inductor is perfect
(no wire resistance, no magnetic core losses, etc.), it will dissipate
zero power.
Let us now consider the same inductor
device, except this time with a second coil wrapped around the same iron
core. The first coil will be labeled the primary coil, while the
second will be labeled the secondary:
If this secondary coil experiences the
same magnetic flux change as the primary (which it should, assuming
perfect containment of the magnetic flux through the common core), and
has the same number of turns around the core, a voltage of equal
magnitude and phase to the applied voltage will be induced along its
length. In the following graph, the induced voltage waveform is drawn
slightly smaller than the source voltage waveform simply to distinguish
one from the other:
This effect is called mutual
inductance: the induction of a voltage in one coil in response to a
change in current in the other coil. Like normal (self-) inductance, it
is measured in the unit of Henrys, but unlike normal inductance it is
symbolized by the capital letter "M" rather than the letter "L":
No current will exist in the secondary
coil, since it is open-circuited. However, if we connect a load resistor
to it, an alternating current will go through the coil, in phase with
the induced voltage (because the voltage across a resistor and the
current through it are always in phase with each other).
At first, one might expect this secondary
coil current to cause additional magnetic flux in the core. In fact, it
does not. If more flux were induced in the core, it would cause more
voltage to be induced voltage in the primary coil (remember that e = dΦ/dt).
This cannot happen, because the primary coil's induced voltage must
remain at the same magnitude and phase in order to balance with the
applied voltage, in accordance with Kirchhoff's voltage law.
Consequently, the magnetic flux in the core cannot be affected by
secondary coil current. However, what does change is the amount
of mmf in the magnetic circuit.
Magnetomotive force is produced any time
electrons move through a wire. Usually, this mmf is accompanied by
magnetic flux, in accordance with the mmf=ΦR "magnetic Ohm's Law"
equation. In this case, though, additional flux is not permitted, so the
only way the secondary coil's mmf may exist is if a counteracting mmf is
generated by the primary coil, of equal magnitude and opposite phase.
Indeed, this is what happens, an alternating current forming in the
primary coil -- 180o out of phase with the secondary coil's
current -- to generate this counteracting mmf and prevent additional
core flux. Polarity marks and current direction arrows have been added
to the illustration to clarify phase relations:
If you find this process a bit confusing,
do not worry. Transformer dynamics is a complex subject. What is
important to understand is this: when an AC voltage is applied to the
primary coil, it creates a magnetic flux in the core, which induces AC
voltage in the secondary coil in-phase with the source voltage. Any
current drawn through the secondary coil to power a load induces a
corresponding current in the primary coil, drawing current from the
source.
Notice how the primary coil is behaving
as a load with respect to the AC voltage source, and how the secondary
coil is behaving as a source with respect to the resistor. Rather than
energy merely being alternately absorbed and returned the primary coil
circuit, energy is now being coupled to the secondary coil where
it is delivered to a dissipative (energy-consuming) load. As far as the
source "knows," it's directly powering the resistor. Of course, there is
also an additional primary coil current lagging the applied voltage by
90o, just enough to magnetize the core to create the
necessary voltage for balancing against the source (the exciting
current).
We call this type of device a
transformer, because it transforms electrical energy into magnetic
energy, then back into electrical energy again. Because its operation
depends on electromagnetic induction between two stationary coils and a
magnetic flux of changing magnitude and "polarity," transformers are
necessarily AC devices. Its schematic symbol looks like two inductors
(coils) sharing the same magnetic core:
The two inductor coils are easily
distinguished in the above symbol. The pair of vertical lines represent
an iron core common to both inductors. While many transformers have
ferromagnetic core materials, there are some that do not, their
constituent inductors being magnetically linked together through the
air.
The following photograph shows a power
transformer of the type used in gas-discharge lighting. Here, the two
inductor coils can be clearly seen, wound around an iron core. While
most transformer designs enclose the coils and core in a metal frame for
protection, this particular transformer is open for viewing and so
serves its illustrative purpose well:
Both coils of wire can be seen here with
copper-colored varnish insulation. The top coil is larger than the
bottom coil, having a greater number of "turns" around the core. In
transformers, the inductor coils are often referred to as windings,
in reference to the manufacturing process where wire is wound
around the core material. As modeled in our initial example, the powered
inductor of a transformer is called the primary winding, while
the unpowered coil is called the secondary winding.
In the next photograph, a transformer is
shown cut in half, exposing the cross-section of the iron core as well
as both windings. Like the transformer shown previously, this unit also
utilizes primary and secondary windings of differing turn counts. The
wire gauge can also be seen to differ between primary and secondary
windings. The reason for this disparity in wire gauge will be made clear
in the next section of this chapter. Additionally, the iron core can be
seen in this photograph to be made of many thin sheets (laminations)
rather than a solid piece. The reason for this will also be explained in
a later section of this chapter.
It is easy to demonstrate simple
transformer action using SPICE, setting up the primary and secondary
windings of the simulated transformer as a pair of "mutual" inductors.
The coefficient of magnetic field coupling is given at the end of the "k"
line in the SPICE circuit description, this example being set very
nearly at perfection (1.000). This coefficient describes how closely
"linked" the two inductors are, magnetically. The better these two
inductors are magnetically coupled, the more efficient the energy
transfer between them should be.
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 100
l2 3 5 100
** This line tells SPICE that the two inductors
** l1 and l2 are magnetically "linked" together
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
Note: the Rbogus resistors are
required to satisfy certain quirks of SPICE. The first breaks the
otherwise continuous loop between the voltage source and L1
which would not be permitted by SPICE. The second provides a path to
ground (node 0) from the secondary circuit, necessary because SPICE
cannot function with any ungrounded circuits.
freq v(2) i(v1)
6.000E+01 1.000E+01 9.975E-03 Primary winding
freq v(3,5) i(vi1)
6.000E+01 9.962E+00 9.962E-03 Secondary winding
Note that with equal inductances for both
windings (100 Henrys each), the AC voltages and currents are nearly
equal for the two. The difference between primary and secondary currents
is the magnetizing current spoken of earlier: the 90o lagging
current necessary to magnetize the core. As is seen here, it is usually
very small compared to primary current induced by the load, and so the
primary and secondary currents are almost equal. What you are seeing
here is quite typical of transformer efficiency. Anything less than 95%
efficiency is considered poor for modern power transformer designs, and
this transfer of power occurs with no moving parts or other components
subject to wear.
If we decrease the load resistance so as
to draw more current with the same amount of voltage, we see that the
current through the primary winding increases in response. Even though
the AC power source is not directly connected to the load resistance
(rather, it is electromagnetically "coupled"), the amount of current
drawn from the source will be almost the same as the amount of current
that would be drawn if the load were directly connected to the source.
Take a close look at the next two SPICE simulations, showing what
happens with different values of load resistors:
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 100
l2 3 5 100
k l1 l2 0.999
vi1 3 4 ac 0
** Note load resistance value of 200 ohms
rload 4 5 200
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 4.679E-02
freq v(3,5) i(vi1)
6.000E+01 9.348E+00 4.674E-02
Notice how the primary current closely
follows the secondary current. In our first simulation, both currents
were approximately 10 mA, but now they are both around 47 mA. In this
second simulation, the two currents are closer to equality, because the
magnetizing current remains the same as before while the load current
has increased. Note also how the secondary voltage has decreased some
with the heavier (greater current) load. Let's try another simulation
with an even lower value of load resistance (15 Ω):
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 100
l2 3 5 100
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 15
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 1.301E-01
freq v(3,5) i(vi1)
6.000E+01 1.950E+00 1.300E-01
Our load current is now 0.13 amps,
or 130 mA, which is substantially higher than the last time. The primary
current is very close to being the same, but notice how the secondary
voltage has fallen well below the primary voltage (1.95 volts versus 10
volts at the primary). The reason for this is an imperfection in our
transformer design: because the primary and secondary inductances aren't
perfectly linked (a k
factor of 0.999 instead of 1.000) there is "stray" or "leakage"
inductance. In other words, some of the magnetic field isn't linking
with the secondary coil, and thus cannot couple energy to it:
Consequently, this "leakage" flux merely
stores and returns energy to the source circuit via self-inductance,
effectively acting as a series impedance in both primary and secondary
circuits. Voltage gets dropped across this series impedance, resulting
in a reduced load voltage: voltage across the load "sags" as load
current increases.
If we change the transformer design to
have better magnetic coupling between the primary and secondary coils,
the figures for voltage between primary and secondary windings will be
much closer to equality again:
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 100
l2 3 5 100
** Coupling factor = 0.99999 instead of 0.999
k l1 l2 0.99999
vi1 3 4 ac 0
rload 4 5 15
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 6.658E-01
freq v(3,5) i(vi1)
6.000E+01 9.987E+00 6.658E-01
Here we see that our secondary voltage is
back to being equal with the primary, and the secondary current is equal
to the primary current as well. Unfortunately, building a real
transformer with coupling this complete is very difficult. A compromise
solution is to design both primary and secondary coils with less
inductance, the strategy being that less inductance overall leads to
less "leakage" inductance to cause trouble, for any given degree of
magnetic coupling inefficiency. This results in a load voltage that is
closer to ideal with the same (heavy) load and the same coupling factor:
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
** inductance = 1 henry instead of 100 henrys
l1 2 0 1
l2 3 5 1
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 15
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 6.664E-01
freq v(3,5) i(vi1)
6.000E+01 9.977E+00 6.652E-01
Simply by using primary and secondary
coils of less inductance, the load voltage for this heavy load has been
brought back up to nearly ideal levels (9.977 volts). At this point, one
might ask, "If less inductance is all that's needed to achieve
near-ideal performance under heavy load, then why worry about coupling
efficiency at all? If it's impossible to build a transformer with
perfect coupling, but easy to design coils with low inductance, then why
not just build all transformers with low-inductance coils and have
excellent efficiency even with poor magnetic coupling?"
The answer to this question is found in
another simulation: the same low-inductance transformer, but this time
with a lighter load (1 kΩ instead of 15 Ω):
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 1
l2 3 5 1
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 2.835E-02
freq v(3,5) i(vi1)
6.000E+01 9.990E+00 9.990E-03
With lower winding inductances, the
primary and secondary voltages are closer to being equal, but the
primary and secondary currents are not. In this particular case, the
primary current is 28.35 mA while the secondary current is only 9.990 mA:
almost three times as much current in the primary as the secondary. Why
is this? With less inductance in the primary winding, there is less
inductive reactance, and consequently a much larger magnetizing current.
A substantial amount of the current through the primary winding merely
works to magnetize the core rather than transfer useful energy to
the secondary winding and load.
An ideal transformer with identical
primary and secondary windings would manifest equal voltage and current
in both sets of windings for any load condition. In a perfect world,
transformers would transfer electrical power from primary to secondary
as smoothly as though the load were directly connected to the primary
power source, with no transformer there at all. However, you can see
this ideal goal can only be met if there is perfect coupling of
magnetic flux between primary and secondary windings. Being that this is
impossible to achieve, transformers must be designed to operate within
certain expected ranges of voltages and loads in order to perform as
close to ideal as possible. For now, the most important thing to keep in
mind is a transformer's basic operating principle: the transfer of power
from the primary to the secondary circuit via electromagnetic coupling.
- REVIEW:
- Mutual inductance
is where the magnetic flux of two or more inductors are "linked" so
that voltage is induced in one coil proportional to the rate-of-change
of current in another.
- A transformer is a device made
of two or more inductors, one of which is powered by AC, inducing an
AC voltage across the second inductor. If the second inductor is
connected to a load, power will be electromagnetically coupled from
the first inductor's power source to that load.
- The powered inductor in a transformer
is called the primary winding. The unpowered inductor in a
transformer is called the secondary winding.
- Magnetic flux in the core (Φ) lags 90o
behind the source voltage waveform. The current drawn by the primary
coil from the source to produce this flux is called the magnetizing
current, and it also lags the supply voltage by 90o.
- Total primary current in an unloaded
transformer is called the exciting current, and is comprised of
magnetizing current plus any additional current necessary to overcome
core losses. It is never perfectly sinusoidal in a real transformer,
but may be made more so if the transformer is designed and operated so
that magnetic flux density is kept to a minimum.
- Core flux induces a voltage in any
coil wrapped around the core. The induces voltage(s) are ideally in
phase with the primary winding source voltage and share the same
waveshape.
- Any current drawn through the
secondary winding by a load will be "reflected" to the primary winding
and drawn from the voltage source, as if the source were directly
powering a similar load.
Step-up and step-down transformers
So far, we've observed simulations of
transformers where the primary and secondary windings were of identical
inductance, giving approximately equal voltage and current levels in
both circuits. Equality of voltage and current between the primary and
secondary sides of a transformer, however, is not the norm for all
transformers. If the inductances of the two windings are not equal,
something interesting happens:
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 10000
l2 3 5 100
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vi1)
6.000E+01 9.962E-01 9.962E-04 Secondary winding
Notice how the secondary voltage is
approximately ten times less than the primary voltage (0.9962 volts
compared to 10 volts), while the secondary current is approximately ten
times greater (0.9962 mA compared to 0.09975 mA). What we have here is a
device that steps voltage down by a factor of ten and current
up by a factor of ten:
This is a very useful device, indeed.
With it, we can easily multiply or divide voltage and current in AC
circuits. Indeed, the transformer has made long-distance transmission of
electric power a practical reality, as AC voltage can be "stepped up"
and current "stepped down" for reduced wire resistance power losses
along power lines connecting generating stations with loads. At either
end (both the generator and at the loads), voltage levels are reduced by
transformers for safer operation and less expensive equipment. A
transformer that increases voltage from primary to secondary (more
secondary winding turns than primary winding turns) is called a
step-up transformer. Conversely, a transformer designed to do just
the opposite is called a step-down transformer.
Let's re-examine a photograph shown in
the previous section:
This is a step-down transformer, as
evidenced by the high turn count of the primary winding and the low turn
count of the secondary. As a step-down unit, this transformer converts
high-voltage, low-current power into low-voltage, high-current power.
The larger-gauge wire used in the secondary winding is necessary due to
the increase in current. The primary winding, which doesn't have to
conduct as much current, may be made of smaller-gauge wire.
In case you were wondering, it is
possible to operate either of these transformer types backwards
(powering the secondary winding with an AC source and letting the
primary winding power a load) to perform the opposite function: a
step-up can function as a step-down and visa-versa. However, as we saw
in the first section of this chapter, efficient operation of a
transformer requires that the individual winding inductances be
engineered for specific operating ranges of voltage and current, so if a
transformer is to be used "backwards" like this it must be employed
within the original design parameters of voltage and current for each
winding, lest it prove to be inefficient (or lest it be damaged
by excessive voltage or current!).
Transformers are often constructed in
such a way that it is not obvious which wires lead to the primary
winding and which lead to the secondary. One convention used in the
electric power industry to help alleviate confusion is the use of "H"
designations for the higher-voltage winding (the primary winding in a
step-down unit; the secondary winding in a step-up) and "X" designations
for the lower-voltage winding. Therefore, a simple power transformer
will have wires labeled "H1", "H2", "X1",
and "X2". There is usually significance to the numbering of
the wires (H1 versus H2, etc.), which we'll
explore a little later in this chapter.
The fact that voltage and current get
"stepped" in opposite directions (one up, the other down) makes perfect
sense when you recall that power is equal to voltage times current, and
realize that transformers cannot produce power, only convert it.
Any device that could output more power than it took in would violate
the Law of Energy Conservation in physics, namely that energy
cannot be created or destroyed, only converted. As with the first
transformer example we looked at, power transfer efficiency is very good
from the primary to the secondary sides of the device.
The practical significance of this is
made more apparent when an alternative is considered: before the advent
of efficient transformers, voltage/current level conversion could only
be achieved through the use of motor/generator sets. A drawing of a
motor/generator set reveals the basic principle involved:
In such a machine, a motor is
mechanically coupled to a generator, the generator designed to produce
the desired levels of voltage and current at the rotating speed of the
motor. While both motors and generators are fairly efficient devices,
the use of both in this fashion compounds their inefficiencies so that
the overall efficiency is in the range of 90% or less. Furthermore,
because motor/generator sets obviously require moving parts, mechanical
wear and balance are factors influencing both service life and
performance. Transformers, on the other hand, are able to convert levels
of AC voltage and current at very high efficiencies with no moving
parts, making possible the widespread distribution and use of electric
power we take for granted.
In all fairness it should be noted that
motor/generator sets have not necessarily been obsoleted by transformers
for all applications. While transformers are clearly superior
over motor/generator sets for AC voltage and current level conversion,
they cannot convert one frequency of AC power to another, or (by
themselves) convert DC to AC or visa-versa. Motor/generator sets can do
all these things with relative simplicity, albeit with the limitations
of efficiency and mechanical factors already described. Motor/generator
sets also have the unique property of kinetic energy storage: that is,
if the motor's power supply is momentarily interrupted for any reason,
its angular momentum (the inertia of that rotating mass) will maintain
rotation of the generator for a short duration, thus isolating any loads
powered by the generator from "glitches" in the main power system.
Looking closely at the numbers in the
SPICE analysis, we should see a correspondence between the transformer's
ratio and the two inductances. Notice how the primary inductor (l1) has
100 times more inductance than the secondary inductor (10000 H versus
100 H), and that the measured voltage step-down ratio was 10 to 1. The
winding with more inductance will have higher voltage and less current
than the other. Since the two inductors are wound around the same core
material in the transformer (for the most efficient magnetic coupling
between the two), the parameters affecting inductance for the two coils
are equal except for the number of turns in each coil. If we take
another look at our inductance formula, we see that inductance is
proportional to the square of the number of coil turns:
So, it should be apparent that our two
inductors in the last SPICE transformer example circuit -- with
inductance ratios of 100:1 -- should have coil turn ratios of 10:1,
because 10 squared equals 100. This works out to be the same ratio we
found between primary and secondary voltages and currents (10:1), so we
can say as a rule that the voltage and current transformation ratio is
equal to the ratio of winding turns between primary and secondary.
The step-up/step-down effect of coil turn
ratios in a transformer is analogous to gear tooth ratios in mechanical
gear systems, transforming values of speed and torque in much the same
way:
Step-up and step-down transformers for
power distribution purposes can be gigantic in proportion to the power
transformers previously shown, some units standing as tall as a home.
The following photograph shows a substation transformer standing about
twelve feet tall:
- REVIEW:
- Transformers "step up" or "step down"
voltage according to the ratios of primary to secondary wire turns.
-
- A transformer designed to increase
voltage from primary to secondary is called a step-up
transformer. A transformer designed to reduce voltage from primary to
secondary is called a step-down transformer.
- The transformation ratio of a
transformer will be equal to the square root of its primary to
secondary inductance (L) ratio.
-
Electrical isolation
Aside from the ability to easily convert
between different levels of voltage and current in AC and DC circuits,
transformers also provide an extremely useful feature called
isolation, which is the ability to couple one circuit to another
without the use of direct wire connections. We can demonstrate an
application of this effect with another SPICE simulation: this time
showing "ground" connections for the two circuits, imposing a high DC
voltage between one circuit and ground through the use of an additional
voltage source:
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
v2 5 0 dc 250
l1 2 0 10000
l2 3 5 100
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
DC voltages referenced to ground (node 0):
(1) 0.0000 (2) 0.0000 (3) 250.0000
(4) 250.0000 (5) 250.0000
AC voltages:
freq v(2) i(v1)
6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vi1)
6.000E+01 9.962E-01 9.962E-04 Secondary winding
SPICE shows the 250 volts DC being
impressed upon the secondary circuit elements with respect to ground,
but as you can see there is no effect on the primary circuit (zero DC
voltage) at nodes 1 and 2, and the transformation of AC power from
primary to secondary circuits remains the same as before. The impressed
voltage in this example is often called a common-mode voltage
because it is seen at more than one point in the circuit with reference
to the common point of ground. The transformer isolates the common-mode
voltage so that it is not impressed upon the primary circuit at all, but
rather isolated to the secondary side. For the record, it does not
matter that the common-mode voltage is DC, either. It could be AC, even
at a different frequency, and the transformer would isolate it from the
primary circuit all the same.
There are applications where electrical
isolation is needed between two AC circuit without any transformation of
voltage or current levels. In these instances, transformers called
isolation transformers having 1:1 transformation ratios are used. A
benchtop isolation transformer is shown in the following photograph:
- REVIEW:
- By being able to transfer power from
one circuit to another without the use of interconnecting conductors
between the two circuits, transformers provide the useful feature of
electrical isolation.
- Transformers designed to provide
electrical isolation without stepping voltage and current either up or
down are called isolation transformers.
Phasing
Since transformers are essentially AC
devices, we need to be aware of the phase relationships between the
primary and secondary circuits. Using our SPICE example from before, we
can plot the waveshapes for the primary and secondary circuits and see
the phase relations for ourselves:
legend:
*: v(2) Primary voltage
+: v(3,5) Secondary voltage
time v(2)
(*)----------- -10 -5 0 5 10
(+)----------- -10 -5 0 5 10
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . x . .
1.000E-03 3.675E+00 . . . + * . .
2.000E-03 6.803E+00 . . . . + * .
3.000E-03 9.008E+00 . . . . +* .
4.000E-03 9.955E+00 . . . . x
5.000E-03 9.450E+00 . . . . *+.
6.000E-03 7.672E+00 . . . . * + .
7.000E-03 4.804E+00 . . . *.+ .
8.000E-03 1.245E+00 . . . * + . .
9.000E-03 -2.474E+00 . . * + . . .
1.000E-02 -5.864E+00 . *+ . . .
1.100E-02 -8.390E+00 . *+ . . . .
1.200E-02 -9.779E+00 .x . . . .
1.300E-02 -9.798E+00 +* . . . .
1.400E-02 -8.390E+00 . +* . . . .
1.500E-02 -5.854E+00 . + *. . . .
1.600E-02 -2.479E+00 . . + * . . .
1.700E-02 1.246E+00 . . .+ * . .
1.800E-02 4.795E+00 . . . + *. .
1.900E-02 7.686E+00 . . . . + * .
2.000E-02 9.451E+00 . . . . x .
2.100E-02 9.937E+00 . . . . x
2.200E-02 9.025E+00 . . . . *+ .
2.300E-02 6.802E+00 . . . . *+ .
2.400E-02 3.667E+00 . . . * + . .
2.500E-02 -1.487E-03 . . * + . .
2.600E-02 -3.658E+00 . . * + . . .
2.700E-02 -6.814E+00 . * + . . . .
2.800E-02 -9.026E+00 . *+ . . . .
2.900E-02 -9.917E+00 *+ . . . .
3.000E-02 -9.511E+00 .x . . . .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
legend:
*: i(v1) Primary current
+: i(vi1) Secondary current
time i(v1)
(*)---------- -2.000E-04 -1.000E-04 0 1.000E-04 2.000E-04
(+)---------- -1.000E-03 -5.000E-04 0 5.000E-04 1.000E-03
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
0.000E+00 0.000E+00 . . x . .
1.000E-03 -2.973E-05 . . + *. . .
2.000E-03 -6.279E-05 . + . * . . .
3.000E-03 -8.772E-05 . + . * . . .
4.000E-03 -1.008E-04 + * . . .
5.000E-03 -9.954E-05 .+ * . . .
6.000E-03 -8.522E-05 . + . * . . .
7.000E-03 -5.919E-05 . +. * . . .
8.000E-03 -2.500E-05 . . + *. . .
9.000E-03 1.212E-05 . . . * + . .
1.000E-02 4.736E-05 . . . * .+ .
1.100E-02 7.521E-05 . . . * . + .
1.200E-02 9.250E-05 . . . *. +.
1.300E-02 9.648E-05 . . . *. +
1.400E-02 8.602E-05 . . . * . + .
1.500E-02 6.362E-05 . . . * . + .
1.600E-02 3.177E-05 . . . * + . .
1.700E-02 -4.998E-06 . . x . .
1.800E-02 -4.136E-05 . . + * . . .
1.900E-02 -7.246E-05 . + . * . . .
2.000E-02 -9.331E-05 . + .* . . .
2.100E-02 -1.019E-04 + * . . .
2.200E-02 -9.651E-05 . + * . . .
2.300E-02 -7.749E-05 . + . * . . .
2.400E-02 -4.842E-05 . . + * . . .
2.500E-02 -1.275E-05 . . x. . .
2.600E-02 2.428E-05 . . . * + . .
2.700E-02 5.761E-05 . . . * .+ .
2.800E-02 8.261E-05 . . . * . + .
2.900E-02 9.514E-05 . . . *. +.
3.000E-02 9.487E-05 . . . *. +.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
It would appear that both voltage and
current for the two transformer windings are in phase with each other,
at least for our resistive load. This is simple enough, but it would be
nice to know which way we should connect a transformer in order
to ensure the proper phase relationships be kept. After all, a
transformer is nothing more than a set of magnetically-linked inductors,
and inductors don't usually come with polarity markings of any kind. If
we were to look at an unmarked transformer, we would have no way of
knowing which way to hook it up to a circuit to get in-phase (or 180o
out-of-phase) voltage and current:
Since this is a practical concern,
transformer manufacturers have come up with a sort of polarity marking
standard to denote phase relationships. It is called the dot
convention, and is nothing more than a dot placed next to each
corresponding leg of a transformer winding:
Typically, the transformer will come with
some kind of schematic diagram labeling the wire leads for primary and
secondary windings. On the diagram will be a pair of dots similar to
what is seen above. Sometimes dots will be omitted, but when "H" and "X"
labels are used to label transformer winding wires, the subscript
numbers are supposed to represent winding polarity. The "1" wires (H1
and X1) represent where the polarity-marking dots would
normally be placed.
The similar placement of these dots next
to the top ends of the primary and secondary windings tells us that
whatever instantaneous voltage polarity seen across the primary winding
will be the same as that across the secondary winding. In other words,
the phase shift from primary to secondary will be zero degrees.
On the other hand, if the dots on each
winding of the transformer do not match up, the phase shift will
be 180o between primary and secondary, like this:
Of course, the dot convention only tells
you which end of each winding is which, relative to the other winding(s).
If you want to reverse the phase relationship yourself, all you have to
do is swap the winding connections like this:
- REVIEW:
- The phase relationships for voltage
and current between primary and secondary circuits of a transformer
are direct: ideally, zero phase shift.
- The dot convention is a type of
polarity marking for transformer windings showing which end of the
winding is which, relative to the other windings.
Winding configurations
Transformers are very versatile devices.
The basic concept of energy transfer between mutual inductors is useful
enough between a single primary and single secondary coil, but
transformers don't have to be made with just two sets of windings.
Consider this transformer circuit:
Here, three inductor coils share a common
magnetic core, magnetically "coupling" or "linking" them together. The
relationship of winding turn ratios and voltage ratios seen with a
single pair of mutual inductors still holds true here for multiple pairs
of coils. It is entirely possible to assemble a transformer such as the
one above (one primary winding, two secondary windings) in which one
secondary winding is a step-down and the other is a step-up. In fact,
this design of transformer was quite common in vacuum tube power supply
circuits, which were required to supply low voltage for the tubes'
filaments (typically 6 or 12 volts) and high voltage for the tubes'
plates (several hundred volts) from a nominal primary voltage of 110
volts AC. Not only are voltages and currents of completely different
magnitudes possible with such a transformer, but all circuits are
electrically isolated from one another.
A photograph of a multiple-winding
transformer is shown here:
This particular transformer is intended
to provide both high and low voltages necessary in an electronic system
using vacuum tubes. Low voltage is required to power the filaments of
vacuum tubes, while high voltage is required to create the potential
difference between the plate and cathode elements of each tube. One
transformer with multiple windings suffices elegantly to provide all the
necessary voltage levels from a single 115 V source. The wires for this
transformer (15 of them!) are not shown in the photograph, being hidden
from view.
If electrical isolation between secondary
circuits is not of great importance, a similar effect can be obtained by
"tapping" a single secondary winding at multiple points along its
length, like this:
A tap is nothing more than a wire
connection made at some point on a winding between the very ends. Not
surprisingly, the winding turn/voltage magnitude relationship of a
normal transformer holds true for all tapped segments of windings. This
fact can be exploited to produce a transformer capable of multiple
ratios:
Carrying the concept of winding taps
further, we end up with a "variable transformer," where a sliding
contact is moved along the length of an exposed secondary winding, able
to connect with it at any point along its length. The effect is
equivalent to having a winding tap at every turn of the winding, and a
switch with poles at every tap position:
One consumer application of the variable
transformer is in speed controls for model train sets, especially the
train sets of the 1950's and 1960's. These transformers were essentially
step-down units, the highest voltage obtainable from the secondary
winding being substantially less than the primary voltage of 110 to 120
volts AC. The variable-sweep contact provided a simple means of voltage
control with little wasted power, much more efficient than control using
a variable resistor!
Moving-slide contacts are too impractical
to be used in large industrial power transformer designs, but multi-pole
switches and winding taps are common for voltage adjustment. Adjustments
need to be made periodically in power systems to accommodate changes in
loads over months or years in time, and these switching circuits provide
a convenient means. Typically, such "tap switches" are not engineered to
handle full-load current, but must be actuated only when the transformer
has been de-energized (no power).
Seeing as how we can tap any transformer
winding to obtain the equivalent of several windings (albeit with loss
of electrical isolation between them), it makes sense that it should be
possible to forego electrical isolation altogether and build a
transformer from a single winding. Indeed this is possible, and the
resulting device is called an autotransformer:
The autotransformer depicted above
performs a voltage step-up function. A step-down autotransformer would
look something like this:
Autotransformers find popular use in
applications requiring a slight boost or reduction in voltage to a load.
The alternative with a normal (isolated) transformer would be to either
have just the right primary/secondary winding ratio made for the job or
use a step-down configuration with the secondary winding connected in
series-aiding ("boosting") or series-opposing ("bucking") fashion.
Primary, secondary, and load voltages are given to illustrate how this
would work.
First, the "boosting" configuration.
Here, the secondary coil's polarity is oriented so that its voltage
directly adds to the primary voltage:
Next, the "bucking" configuration. Here,
the secondary coil's polarity is oriented so that its voltage directly
subtracts from the primary voltage:
The prime advantage of an autotransformer
is that the same boosting or bucking function is obtained with only a
single winding, making it cheaper and lighter to manufacture than a
regular (isolating) transformer having both primary and secondary
windings.
Like regular transformers,
autotransformer windings can be tapped to provide variations in ratio.
Additionally, they can be made continuously variable with a sliding
contact to tap the winding at any point along its length. The latter
configuration is popular enough to have earned itself its own name: the
Variac.
Small variacs for benchtop use are
popular pieces of equipment for the electronics experimenter, being able
to step household AC voltage down (or sometimes up as well) with a wide,
fine range of control by a simple twist of a knob.
- REVIEW:
- Transformers can be equipped with more
than just a single primary and single secondary winding pair. This
allows for multiple step-up and/or step-down ratios in the same
device.
- Transformer windings can also be
"tapped:" that is, intersected at many points to segment a single
winding into sections.
- Variable transformers can be made by
providing a movable arm that sweeps across the length of a winding,
making contact with the winding at any point along its length. The
winding, of course, has to be bare (no insulation) in the area where
the arm sweeps.
- An autotransformer is a single, tapped
inductor coil used to step up or step down voltage like a transformer,
except without providing electrical isolation.
- A Variac is a variable
autotransformer.
Voltage regulation
As we saw in a few SPICE analyses earlier
in this chapter, the output voltage of a transformer varies some with
varying load resistances, even with a constant voltage input. The degree
of variance is affected by the primary and secondary winding
inductances, among other factors, not the least of which includes
winding resistance and the degree of mutual inductance (magnetic
coupling) between the primary and secondary windings. For power
transformer applications, where the transformer is seen by the load
(ideally) as a constant source of voltage, it is good to have the
secondary voltage vary as little as possible for wide variances in load
current.
The measure of how well a power
transformer maintains constant secondary voltage over a range of load
currents is called the transformer's voltage regulation. It can
be calculated from the following formula:
"Full-load" means the point at which the
transformer is operating at maximum permissible secondary current. This
operating point will be determined primarily by the winding wire size (ampacity)
and the method of transformer cooling. Taking our first SPICE
transformer simulation as an example, let's compare the output voltage
with a 1 kΩ load versus a 200 Ω load (assuming that the 200 Ω load will
be our "full load" condition). Recall if you will that our constant
primary voltage was 10.00 volts AC:
freq v(3,5) i(vi1)
6.000E+01 9.962E+00 9.962E-03 Output with 1k ohm load
freq v(3,5) i(vi1)
6.000E+01 9.348E+00 4.674E-02 Output with 200 ohm load
Notice how the output voltage decreases
as the load gets heavier (more current). Now let's take that same
transformer circuit and place a load resistance of extremely high
magnitude across the secondary winding to simulate a "no-load"
condition:
transformer
v1 1 0 ac 10 sin
rbogus1 1 2 1e-12
rbogus2 5 0 9e12
l1 2 0 100
l2 3 5 100
k l1 l2 0.999
vi1 3 4 ac 0
rload 4 5 9e12
.ac lin 1 60 60
.print ac v(2,0) i(v1)
.print ac v(3,5) i(vi1)
.end
freq v(2) i(v1)
6.000E+01 1.000E+01 2.653E-04
freq v(3,5) i(vi1)
6.000E+01 9.990E+00 1.110E-12 Output with (almost) no load
So, we see that our output (secondary)
voltage spans a range of 9.990 volts at (virtually) no load and 9.348
volts at the point we decided to call "full load." Calculating voltage
regulation with these figures, we get:
Incidentally, this would be considered
rather poor (or "loose") regulation for a power transformer. Powering a
simple resistive load like this, a good power transformer should exhibit
a regulation percentage of less than 3%. Inductive loads tend to create
a condition of worse voltage regulation, so this analysis with purely
resistive loads was a "best-case" condition.
There are some applications, however,
where poor regulation is actually desired. One such case is in discharge
lighting, where a step-up transformer is required to initially generate
a high voltage (necessary to "ignite" the lamps), then the voltage is
expected to drop off once the lamp begins to draw current. This is
because discharge lamps' voltage requirements tend to be much lower
after a current has been established through the arc path. In this case,
a step-up transformer with poor voltage regulation suffices nicely for
the task of conditioning power to the lamp.
Another application is in current control
for AC arc welders, which are nothing more than step-down transformers
supplying low-voltage, high-current power for the welding process. A
high voltage is desired to assist in "striking" the arc (getting it
started), but like the discharge lamp, an arc doesn't require as much
voltage to sustain itself once the air has been heated to the point of
ionization. Thus, a decrease of secondary voltage under high load
current would be a good thing. Some arc welder designs provide arc
current adjustment by means of a movable iron core in the transformer,
cranked in or out of the winding assembly by the operator. Moving the
iron slug away from the windings reduces the strength of magnetic
coupling between the windings, which diminishes no-load secondary
voltage and makes for poorer voltage regulation.
No exposition on transformer regulation
could be called complete without mention of an unusual device called a
ferroresonant transformer. "Ferroresonance" is a phenomenon
associated with the behavior of iron cores while operating near a point
of magnetic saturation (where the core is so strongly magnetized that
further increases in winding current results in little or no increase in
magnetic flux).
While being somewhat difficult to
describe without going deep into electromagnetic theory, the
ferroresonant transformer is a power transformer engineered to operate
in a condition of persistent core saturation. That is, its iron core is
"stuffed full" of magnetic lines of flux for a large portion of the AC
cycle so that variations in supply voltage (primary winding current)
have little effect on the core's magnetic flux density, which means the
secondary winding outputs a nearly constant voltage despite significant
variations in supply (primary winding) voltage. Normally, core
saturation in a transformer results in distortion of the sinewave shape,
and the ferroresonant transformer is no exception. To combat this side
effect, ferroresonant transformers have an auxiliary secondary winding
paralleled with one or more capacitors, forming a resonant circuit tuned
to the power supply frequency. This "tank circuit" serves as a filter to
reject harmonics created by the core saturation, and provides the added
benefit of storing energy in the form of AC oscillations, which is
available for sustaining output winding voltage for brief periods of
input voltage loss (milliseconds' worth of time, but certainly better
than nothing).
In addition to blocking harmonics created
by the saturated core, this resonant circuit also "filters out" harmonic
frequencies generated by nonlinear (switching) loads in the secondary
winding circuit and any harmonics present in the source voltage,
providing "clean" power to the load.
Ferroresonant transformers offer several
features useful in AC power conditioning: constant output voltage given
substantial variations in input voltage, harmonic filtering between the
power source and the load, and the ability to "ride through" brief
losses in power by keeping a reserve of energy in its resonant tank
circuit. These transformers are also highly tolerant of excessive
loading and transient (momentary) voltage surges. They are so tolerant,
in fact, that some may be briefly paralleled with unsynchronized AC
power sources, allowing a load to be switched from one source of power
to another in a "make-before-break" fashion with no interruption of
power on the secondary side!
Unfortunately, these devices have equally
noteworthy disadvantages: they waste a lot of energy (due to hysteresis
losses in the saturated core), generating significant heat in the
process, and are intolerant of frequency variations, which means they
don't work very well when powered by small engine-driven generators
having poor speed regulation. Voltages produced in the resonant
winding/capacitor circuit tend to be very high, necessitating expensive
capacitors and presenting the service technician with very dangerous
working voltages. Some applications, though, may prioritize the
ferroresonant transformer's advantages over its disadvantages.
Semiconductor circuits exist to "condition" AC power as an alternative
to ferroresonant devices, but none can compete with this transformer in
terms of sheer simplicity.
- REVIEW:
- Voltage regulation
is the measure of how well a power transformer can maintain constant
secondary voltage given a constant primary voltage and wide variance
in load current. The lower the percentage (closer to zero), the more
stable the secondary voltage and the better the regulation it will
provide.
- A ferroresonant transformer is
a special transformer designed to regulate voltage at a stable level
despite wide variation in input voltage.
Special transformers and applications
Because transformers can step voltage and
current to different levels, and because power is transferred
equivalently between primary and secondary windings, they can be used to
"convert" the impedance of a load to a different level. That last phrase
deserves some explanation, so let's investigate what it means.
The purpose of a load (usually) is to do
something productive with the power it dissipates. In the case of a
resistive heating element, the practical purpose for the power
dissipated is to heat something up. Loads are engineered to safely
dissipate a certain maximum amount of power, but two loads of equal
power rating are not necessarily identical. Consider these two 1000 watt
resistive heating elements:
Both heaters dissipate exactly 1000 watts
of power, but they do so at different voltage and current levels (either
250 volts and 4 amps, or 125 volts and 8 amps). Using Ohm's Law to
determine the necessary resistance of these heating elements (R=E/I), we
arrive at figures of 62.5 Ω and 15.625 Ω, respectively. If these are AC
loads, we might refer to their opposition to current in terms of
impedance rather than plain resistance, although in this case that's all
they're composed of (no reactance). The 250 volt heater would be said to
be a higher impedance load than the 125 volt heater.
If we desired to operate the 250 volt
heater element directly on a 125 volt power system, we would end up
being disappointed. With 62.5 Ω of impedance (resistance), the current
would only be 2 amps (I=E/R; 125/62.5), and the power dissipation would
only be 250 watts (P=IE; 125 x 2), or one-quarter of its rated power.
The impedance of the heater and the voltage of our source would be
mismatched, and we couldn't obtain the full rated power dissipation from
the heater.
All hope is not lost, though. With a
step-up transformer, we could operate the 250 volt heater element on the
125 volt power system like this:
The ratio of the transformer's windings
provides the voltage step-up and current step-down we need for
the otherwise mismatched load to operate properly on this system. Take a
close look at the primary circuit figures: 125 volts at 8 amps. As far
as the power supply "knows," it's powering a 15.625 Ω (R=E/I) load at
125 volts, not a 62.5 Ω load! The voltage and current figures for the
primary winding are indicative of 15.625 Ω load impedance, not the
actual 62.5 Ω of the load itself. In other words, not only has our
step-up transformer transformed voltage and current, but it has
transformed impedance as well.
The transformation ratio of impedance is
the square of the voltage/current transformation ratio, the same as the
winding inductance ratio:
This concurs with our example of the 2:1
step-up transformer and the impedance ratio of 62.5 Ω to 15.625 Ω (a 4:1
ratio, which is 2:1 squared). Impedance transformation is a highly
useful ability of transformers, for it allows a load to dissipate its
full rated power even if the power system is not at the proper voltage
to directly do so.
Recall from our study of network analysis
the Maximum Power Transfer Theorem, which states that the maximum
amount of power will be dissipated by a load resistance when that load
resistance is equal to the Thevenin/Norton resistance of the network
supplying the power. Substitute the word "impedance" for "resistance" in
that definition and you have the AC version of that Theorem. If we're
trying to obtain theoretical maximum power dissipation from a load, we
must be able to properly match the load impedance and source (Thevenin/Norton)
impedance together. This is generally more of a concern in specialized
electric circuits such as radio transmitter/antenna and audio
amplifier/speaker systems. Let's take an audio amplifier system and see
how it works:
With an internal impedance of 500 Ω, the
amplifier can only deliver full power to a load (speaker) also having
500 Ω of impedance. Such a load would drop higher voltage and draw less
current than an 8 Ω speaker dissipating the same amount of power. If an
8 Ω speaker were connected directly to the 500 Ω amplifier as shown, the
impedance mismatch would result in very poor (low peak power)
performance. Additionally, the amplifier would tend to dissipate more
than its fair share of power in the form of heat trying to drive the low
impedance speaker.
To make this system work better, we can
use a transformer to match these mismatched impedances. Since we're
going from a high impedance (high voltage, low current) supply to a low
impedance (low voltage, high current) load, we'll need to use a
step-down transformer:
To obtain an impedance transformation
ratio of 500:8, we would need a winding ratio equal to the square root
of 500:8 (the square root of 62.5:1, or 7.906:1). With such a
transformer in place, the speaker will load the amplifier to just the
right degree, drawing power at the correct voltage and current levels to
satisfy the Maximum Power Transfer Theorem and make for the most
efficient power delivery to the load. The use of a transformer in this
capacity is called impedance matching.
Anyone who has ridden a multi-speed
bicycle can intuitively understand the principle of impedance matching.
A human's legs will produce maximum power when spinning the bicycle
crank at a particular speed (about 60 to 90 revolution per minute).
Above or below that rotational speed, human leg muscles are less
efficient at generating power. The purpose of the bicycle's "gears" is
to impedance-match the rider's legs to the riding conditions so that
they always spin the crank at the optimum speed.
If the rider attempts to start moving
while the bicycle is shifted into its "top" gear, he or she will find it
very difficult to get moving. Is it because the rider is weak? No, it's
because the high step-up ratio of the bicycle's chain and sprockets in
that top gear presents a mismatch between the conditions (lots of
inertia to overcome) and their legs (needing to spin at 60-90 RPM for
maximum power output). On the other hand, selecting a gear that is too
low will enable the rider to get moving immediately, but limit the top
speed they will be able to attain. Again, is the lack of speed an
indication of weakness in the bicyclist's legs? No, it's because the
lower speed ratio of the selected gear creates another type of mismatch
between the conditions (low load) and the rider's legs (losing power if
spinning faster than 90 RPM). It is much the same with electric power
sources and loads: there must be an impedance match for maximum system
efficiency. In AC circuits, transformers perform the same matching
function as the sprockets and chain ("gears") on a bicycle to match
otherwise mismatched sources and loads.
Impedance matching transformers are not
fundamentally different from any other type of transformer in
construction or appearance. A small impedance-matching transformer
(about two centimeters in width) for audio-frequency applications is
shown in the following photograph:
Another impedance-matching transformer
can be seen on this printed circuit board, in the upper right corner, to
the immediate left of resistors R2 and R1. It is
labeled "T1":
Transformers can also be used in
electrical instrumentation systems. Due to transformers' ability to step
up or step down voltage and current, and the electrical isolation they
provide, they can serve as a way of connecting electrical
instrumentation to high-voltage, high current power systems. Suppose we
wanted to accurately measure the voltage of a 13.8 kV power system (a
very common power distribution voltage in American industry):
Designing, installing, and maintaining a
voltmeter capable of directly measuring 13,800 volts AC would be no easy
task. The safety hazard alone of bringing 13.8 kV conductors into an
instrument panel would be severe, not to mention the design of the
voltmeter itself. However, by using a precision step-down transformer,
we can reduce the 13.8 kV down to a safe level of voltage at a constant
ratio, and isolate it from the instrument connections, adding an
additional level of safety to the metering system:
Now the voltmeter reads a precise
fraction, or ratio, of the actual system voltage, its scale set to read
as though it were measuring the voltage directly. The transformer keeps
the instrument voltage at a safe level and electrically isolates it from
the power system, so there is no direct connection between the power
lines and the instrument or instrument wiring. When used in this
capacity, the transformer is called a Potential Transformer, or
simply PT.
Potential transformers are designed to
provide as accurate a voltage step-down ratio as possible. To aid in
precise voltage regulation, loading is kept to a minimum: the voltmeter
is made to have high input impedance so as to draw as little current
from the PT as possible. As you can see, a fuse has been connected in
series with the PTs primary winding, for safety and ease of
disconnecting the PT from the circuit.
A standard secondary voltage for a PT is
120 volts AC, for full-rated power line voltage. The standard voltmeter
range to accompany a PT is 150 volts, full-scale. PTs with custom
winding ratios can be manufactured to suit any application. This lends
itself well to industry standardization of the actual voltmeter
instruments themselves, since the PT will be sized to step the system
voltage down to this standard instrument level.
Following the same line of thinking, we
can use a transformer to step down current through a power line so that
we are able to safely and easily measure high system currents with
inexpensive ammeters. Of course, such a transformer would be connected
in series with the power line, like this:
Note that while the PT is a step-down
device, the Current Transformer (or CT) is a step-up
device (with respect to voltage), which is what is needed to step
down the power line current. Quite often, CTs are built as
donut-shaped devices through which the power line conductor is run, the
power line itself acting as a single-turn primary winding:
Some CTs are made to hinge open, allowing
insertion around a power conductor without disturbing the conductor at
all. The industry standard secondary current for a CT is a range of 0 to
5 amps AC. Like PTs, CTs can be made with custom winding ratios to fit
almost any application. Because their "full load" secondary current is 5
amps, CT ratios are usually described in terms of full-load primary amps
to 5 amps, like this:
The "donut" CT shown in the photograph
has a ratio of 50:5. That is, when the conductor through the center of
the torus is carrying 50 amps of current (AC), there will be 5 amps of
current induced in the CT's winding.
Because CTs are designed to be powering
ammeters, which are low-impedance loads, and they are wound as voltage
step-up transformers, they should never, ever be operated with an
open-circuited secondary winding. Failure to heed this warning will
result in the CT producing extremely high secondary voltages, dangerous
to equipment and personnel alike. To facilitate maintenance of ammeter
instrumentation, short-circuiting switches are often installed in
parallel with the CT's secondary winding, to be closed whenever the
ammeter is removed for service:
Though it may seem strange to
intentionally short-circuit a power system component, it is
perfectly proper and quite necessary when working with current
transformers.
Another kind of special transformer, seen
often in radio-frequency circuits, is the air core transformer.
True to its name, an air core transformer has its windings wrapped
around a nonmagnetic form, usually a hollow tube of some material. The
degree of coupling (mutual inductance) between windings in such a
transformer is many times less than that of an equivalent iron-core
transformer, but the undesirable characteristics of a ferromagnetic core
(eddy current losses, hysteresis, saturation, etc.) are completely
eliminated. It is in high-frequency applications that these effects of
iron cores are most problematic.
One notable example of air-core
transformer is the Tesla Coil, named after the Serbian electrical
genius Nikola Tesla, who was also the inventor of the rotating magnetic
field AC motor, polyphase AC power systems, and many elements of radio
technology. The Tesla Coil is a resonant, high-frequency step-up
transformer used to produce high voltages that are relatively harmless
to human beings (the "skin effect" of high-frequency alternating current
precluding electric shock, although capable of producing skin burns).
One of Tesla's dreams was to employ his coil technology to distribute
electric power without the need for wires, simply broadcasting it in the
form of radio waves which could be received and conducted to loads by
means of antennas. The basic schematic for a Tesla Coil looks like this:
The capacitor in parallel with the
transformer's primary winding forms the tank circuit needed for
resonance. The secondary winding is wound in close proximity to the
primary, usually around the same nonmagnetic form. Several options exist
for "exciting" the primary circuit, the simplest being a high-voltage,
low-frequency AC source and spark gap:
With each cycle peak of the high-voltage
AC source, the current will jump across the spark gap, briefly
energizing the tank circuit. The tank circuit, tuned for a resonant
frequency far in excess of the AC source, will oscillate for many cycles
before the next cycle peak of the source, when it will receive another
"kick" to keep the oscillations going. The secondary of the Tesla Coil
will output a fairly constant high voltage at very high frequencies,
usually producing a spark discharge into the surrounding air at the
discharge terminal.
Tesla Coils find application primarily as
novelty devices, showing up in high school science fairs, basement
workshops, and the occasional low budget science-fiction movie.
So far, we've explored the transformer as
a device for converting different levels of voltage, current, and even
impedance from one circuit to another. Now we'll take a look at it as a
completely different kind of device: one that allows a small electrical
signal to exert control over a much larger quantity of electrical
power. In this mode, a transformer acts as an amplifier.
The device I'm referring to is called a
saturable-core reactor, or simply saturable reactor.
Actually, it is not really a transformer at all, but rather a special
kind of inductor whose inductance can be varied by the application of a
DC current through a second winding wound around the same iron core.
Like the ferroresonant transformer, the saturable reactor relies on the
principle of magnetic saturation. When a material such as iron is
completely saturated (that is, all its magnetic domains are lined up
with the applied magnetizing force), additional increases in current
through the magnetizing winding will not result in further increases of
magnetic flux.
Now, inductance is the measure of how
well an inductor opposes changes in current by developing a voltage in
an opposing direction. The ability of an inductor to generate this
opposing voltage is directly connected with the change in magnetic flux
inside the inductor resulting from the change in current, and the number
of winding turns in the inductor. If an inductor has a saturated core,
no further magnetic flux will result from further increases in current,
and so there will be no voltage induced in opposition to the change in
current. In other words, an inductor loses its inductance (ability to
oppose changes in current) when its core becomes magnetically saturated.
If an inductor's inductance changes, then
its reactance (and impedance) to AC current changes as well. In a
circuit with a constant voltage source, this will result in a change in
current:
A saturable reactor capitalizes on this
effect by forcing the core into a state of saturation with a strong
magnetic field generated by current through another winding. The
reactor's "power" winding is the one carrying the AC load current, and
the "control" winding is one carrying a DC current strong enough to
drive the core into saturation:
The strange-looking transformer symbol
shown in the above schematic represents a saturable-core reactor, the
upper winding being the DC control winding and the lower being the
"power" winding through which the controlled AC current goes. Increased
DC control current produces more magnetic flux in the reactor core,
driving it closer to a condition of saturation, thus decreasing the
power winding's inductance, decreasing its impedance, and increasing
current to the load. Thus, the DC control current is able to exert
control over the AC current delivered to the load.
The circuit shown would work, but it
would not work very well. The first problem is the natural transformer
action of the saturable reactor: AC current through the power winding
will induce a voltage in the control winding, which may cause trouble
for the DC power source. Also, saturable reactors tend to regulate AC
power only in one direction: in one half of the AC cycle, the mmf's from
both windings add; in the other half, they subtract. Thus, the core will
have more flux in it during one half of the AC cycle than the other, and
will saturate first in that cycle half, passing load current more easily
in one direction than the other. Fortunately, both problems can be
overcome with a little ingenuity:
Notice the placement of the phasing dots
on the two reactors: the power windings are "in phase" while the control
windings are "out of phase." If both reactors are identical, any voltage
induced in the control windings by load current through the power
windings will cancel out to zero at the battery terminals, thus
eliminating the first problem mentioned. Furthermore, since the DC
control current through both reactors produces magnetic fluxes in
different directions through the reactor cores, one reactor will
saturate more in one cycle of the AC power while the other reactor will
saturate more in the other, thus equalizing the control action through
each half-cycle so that the AC power is "throttled" symmetrically. This
phasing of control windings can be accomplished with two separate
reactors as shown, or in a single reactor design with intelligent layout
of the windings and core.
Saturable reactor technology has even
been miniaturized to the circuit-board level in compact packages more
generally known as magnetic amplifiers. I personally find this to
be fascinating: the effect of amplification (one electrical signal
controlling another), normally requiring the use of physically fragile
vacuum tubes or electrically "fragile" semiconductor devices, can be
realized in a device both physically and electrically rugged. Magnetic
amplifiers do have disadvantages over their more fragile counterparts,
namely size, weight, nonlinearity, and bandwidth (frequency response),
but their utter simplicity still commands a certain degree of
appreciation, if not practical application.
Saturable-core reactors are less commonly
known as "saturable-core inductors" or transductors.
- REVIEW:
- Transformers can be used to transform
impedance as well as voltage and current. When this is done to improve
power transfer to a load, it is called impedance matching.
- A Potential Transformer (PT) is
a special instrument transformer designed to provide a precise voltage
step-down ratio for voltmeters measuring high power system voltages.
- A Current Transformer (CT) is
another special instrument transformer designed to step down the
current through a power line to a safe level for an ammeter to
measure.
- An air-core transformer is one
lacking a ferromagnetic core.
- A Tesla Coil is a resonant,
air-core, step-up transformer designed to produce very high AC
voltages at high frequency.
- A saturable reactor is a
special type of inductor, the inductance of which can be controlled by
the DC current through a second winding around the same core. With
enough DC current, the magnetic core can be saturated, decreasing the
inductance of the power winding in a controlled fashion.
Practical considerations
Power capacity
As has already been observed,
transformers must be well designed in order to achieve acceptable power
coupling, tight voltage regulation, and low exciting current distortion.
Also, transformers must be designed to carry the expected values of
primary and secondary winding current without any trouble. This means
the winding conductors must be made of the proper gauge wire to avoid
any heating problems. An ideal transformer would have perfect coupling
(no leakage inductance), perfect voltage regulation, perfectly
sinusoidal exciting current, no hysteresis or eddy current losses, and
wire thick enough to handle any amount of current. Unfortunately, the
ideal transformer would have to be infinitely large and heavy to meet
these design goals. Thus, in the business of practical
transformer design, compromises must be made.
Additionally, winding conductor
insulation is a concern where high voltages are encountered, as they
often are in step-up and step-down power distribution transformers. Not
only do the windings have to be well insulated from the iron core, but
each winding has to be sufficiently insulated from the other in order to
maintain electrical isolation between windings.
Respecting these limitations,
transformers are rated for certain levels of primary and secondary
winding voltage and current, though the current rating is usually
derived from a volt-amp (VA) rating assigned to the transformer. For
example, take a step-down transformer with a primary voltage rating of
120 volts, a secondary voltage rating of 48 volts, and a VA rating of 1
kVA (1000 VA). The maximum winding currents can be determined as such:
Sometimes windings will bear current
ratings in amps, but this is typically seen on small transformers. Large
transformers are almost always rated in terms of winding voltage and VA
or kVA.
Energy losses
When transformers transfer power, they do
so with a minimum of loss. As it was stated earlier, modern power
transformer designs typically exceed 95% efficiency. It is good to know
where some of this lost power goes, however, and what causes it to be
lost.
There is, of course, power lost due to
resistance of the wire windings. Unless superconducting wires are used,
there will always be power dissipated in the form of heat through the
resistance of current-carrying conductors. Because transformers require
such long lengths of wire, this loss can be a significant factor.
Increasing the gauge of the winding wire is one way to minimize this
loss, but only with substantial increases in cost, size, and weight.
Resistive losses aside, the bulk of
transformer power loss is due to magnetic effects in the core. Perhaps
the most significant of these "core losses" is eddy-current loss,
which is resistive power dissipation due to the passage of induced
currents through the iron of the core. Because iron is a conductor of
electricity as well as being an excellent "conductor" of magnetic flux,
there will be currents induced in the iron just as there are currents
induced in the secondary windings from the alternating magnetic field.
These induced currents -- as described by the perpendicularity clause of
Faraday's Law -- tend to circulate through the cross-section of the core
perpendicularly to the primary winding turns. Their circular motion
gives them their unusual name: like eddies in a stream of water that
circulate rather than move in straight lines.
Iron is a fair conductor of electricity,
but not as good as the copper or aluminum from which wire windings are
typically made. Consequently, these "eddy currents" must overcome
significant electrical resistance as they circulate through the core. In
overcoming the resistance offered by the iron, they dissipate power in
the form of heat. Hence, we have a source of inefficiency in the
transformer that is difficult to eliminate.
This phenomenon is so pronounced that it
is often exploited as a means of heating ferrous (iron-containing)
materials. The following photograph shows an "induction heating" unit
raising the temperature of a large pipe section. Loops of wire covered
by high-temperature insulation encircle the pipe's circumference,
inducing eddy currents within the pipe wall by electromagnetic
induction. In order to maximize the eddy current effect, high-frequency
alternating current is used rather than power line frequency (60 Hz).
The box units at the right of the picture produce the high-frequency AC
and control the amount of current in the wires to stabilize the pipe
temperature at a pre-determined "set-point."
The main strategy in mitigating these
wasteful eddy currents in transformer cores is to form the iron core in
sheets, each sheet covered with an insulating varnish so that the core
is divided up into thin slices. The result is very little width in the
core for eddy currents to circulate in:
Laminated
cores like the one shown here are standard in almost all low-frequency
transformers. Recall from the photograph of the transformer cut in half
that the iron core was composed of many thin sheets rather than one
solid piece. Eddy current losses increase with frequency, so
transformers designed to run on higher-frequency power (such as 400 Hz,
used in many military and aircraft applications) must use thinner
laminations to keep the losses down to a respectable minimum. This has
the undesirable effect of increasing the manufacturing cost of the
transformer.
Another, similar technique for minimizing
eddy current losses which works better for high-frequency applications
is to make the core out of iron powder instead of thin iron sheets. Like
the lamination sheets, these granules of iron are individually coated in
an electrically insulating material, which makes the core nonconductive
except for within the width of each granule. Powdered iron cores are
often found in transformers handling radio-frequency currents.
Another "core loss" is that of magnetic
hysteresis. All ferromagnetic materials tend to retain some
degree of magnetization after exposure to an external magnetic field.
This tendency to stay magnetized is called "hysteresis," and it takes a
certain investment in energy to overcome this opposition to change every
time the magnetic field produced by the primary winding changes polarity
(twice per AC cycle). This type of loss can be mitigated through good
core material selection (choosing a core alloy with low hysteresis, as
evidenced by a "thin" B/H hysteresis curve), and designing the core for
minimum flux density (large cross-sectional area).
Transformer energy losses tend to worsen
with increasing frequency. The skin effect within winding conductors
reduces the available cross-sectional area for electron flow, thereby
increasing effective resistance as the frequency goes up and creating
more power lost through resistive dissipation. Magnetic core losses are
also exaggerated with higher frequencies, eddy currents and hysteresis
effects becoming more severe. For this reason, transformers of
significant size are designed to operate efficiently in a limited range
of frequencies. In most power distribution systems where the line
frequency is very stable, one would think excessive frequency would
never pose a problem. Unfortunately it does, in the form of harmonics
created by nonlinear loads.
As we've seen in earlier chapters,
nonsinusoidal waveforms are equivalent to additive series of multiple
sinusoidal waveforms at different amplitudes and frequencies. In power
systems, these other frequencies are whole-number multiples of the
fundamental (line) frequency, meaning that they will always be higher,
not lower, than the design frequency of the transformer. In significant
measure, they can cause severe transformer overheating. Power
transformers can be engineered to handle certain levels of power system
harmonics, and this capability is sometimes denoted with a "K factor"
rating.
Stray capacitance and inductance
Aside from power ratings and power
losses, transformers often harbor other undesirable limitations which
circuit designers must be made aware of. Like their simpler counterparts
-- inductors -- transformers exhibit capacitance due to the insulation
dielectric between conductors: from winding to winding, turn to turn (in
a single winding), and winding to core. Usually this capacitance is of
no concern in a power application, but small signal applications
(especially those of high frequency) may not tolerate this quirk well.
Also, the effect of having capacitance along with the windings' designed
inductance gives transformers the ability to resonate at a
particular frequency, definitely a design concern in signal applications
where the applied frequency may reach this point (usually the resonant
frequency of a power transformer is well beyond the frequency of the AC
power it was designed to operate on).
Flux containment (making sure a
transformer's magnetic flux doesn't escape so as to interfere with
another device, and making sure other devices' magnetic flux is shielded
from the transformer core) is another concern shared both by inductors
and transformers.
Closely related to the issue of flux
containment is leakage inductance. We've already seen the detrimental
effects of leakage inductance on voltage regulation with SPICE
simulations early in this chapter. Because leakage inductance is
equivalent to an inductance connected in series with the transformer's
winding, it manifests itself as a series impedance with the load. Thus,
the more current drawn by the load, the less voltage available at the
secondary winding terminals. Usually, good voltage regulation is desired
in transformer design, but there are exceptional applications. As was
stated before, discharge lighting circuits require a step-up transformer
with "loose" (poor) voltage regulation to ensure reduced voltage after
the establishment of an arc through the lamp. One way to meet this
design criterion is to engineer the transformer with flux leakage paths
for magnetic flux to bypass the secondary winding(s). The resulting
leakage flux will produce leakage inductance, which will in turn produce
the poor regulation needed for discharge lighting.
Core saturation
Transformers are also constrained in
their performance by the magnetic flux limitations of the core. For
ferromagnetic core transformers, we must be mindful of the saturation
limits of the core. Remember that ferromagnetic materials cannot support
infinite magnetic flux densities: they tend to "saturate" at a certain
level (dictated by the material and core dimensions), meaning that
further increases in magnetic field force (mmf) do not result in
proportional increases in magnetic field flux (Φ).
When a transformer's primary winding is
overloaded from excessive applied voltage, the core flux may reach
saturation levels during peak moments of the AC sinewave cycle. If this
happens, the voltage induced in the secondary winding will no longer
match the wave-shape as the voltage powering the primary coil. In other
words, the overloaded transformer will distort the waveshape from
primary to secondary windings, creating harmonics in the secondary
winding's output. As we discussed before, harmonic content in AC power
systems typically causes problems.
Special transformers known as peaking
transformers exploit this principle to produce brief voltage pulses
near the peaks of the source voltage waveform. The core is designed to
saturate quickly and sharply, at voltage levels well below peak. This
results in a severely cropped sine-wave flux waveform, and secondary
voltage pulses only when the flux is changing (below saturation levels):
Another cause of abnormal transformer
core saturation is operation at frequencies lower than normal. For
example, if a power transformer designed to operate at 60 Hz is forced
to operate at 50 Hz instead, the flux must reach greater peak levels
than before in order to produce the same opposing voltage needed to
balance against the source voltage. This is true even if the source
voltage is the same as before.
Since instantaneous winding voltage is
proportional to the instantaneous magnetic flux's rate of change
in a transformer, a voltage waveform reaching the same peak value, but
taking a longer amount of time to complete each half-cycle, demands that
the flux maintain the same rate of change as before, but for longer
periods of time. Thus, if the flux has to climb at the same rate as
before, but for longer periods of time, it will climb to a greater peak
value.
Mathematically, this is another example
of calculus in action. Because the voltage is proportional to the flux's
rate-of-change, we say that the voltage waveform is the derivative
of the flux waveform, "derivative" being that calculus operation
defining one mathematical function (waveform) in terms of the
rate-of-change of another. If we take the opposite perspective, though,
and relate the original waveform to its derivative, we may call the
original waveform the integral of the derivative waveform. In
this case, the voltage waveform is the derivative of the flux waveform,
and the flux waveform is the integral of the voltage waveform.
The integral of any mathematical function
is proportional to the area accumulated underneath the curve of that
function. Since each half-cycle of the 50 Hz waveform accumulates more
area between it and the zero line of the graph than the 60 Hz waveform
will -- and we know that the magnetic flux is the integral of the
voltage -- the flux will attain higher values:
Yet another cause of transformer
saturation is the presence of DC current in the primary winding. Any
amount of DC voltage dropped across the primary winding of a transformer
will cause additional magnetic flux in the core. This additional flux
"bias" or "offset" will push the alternating flux waveform closer to
saturation in one half-cycle than the other:
For most transformers, core saturation is
a very undesirable effect, and it is avoided through good design:
engineering the windings and core so that magnetic flux densities remain
well below the saturation levels. This ensures that the relationship
between mmf and Φ is more linear throughout the flux cycle, which is
good because it makes for less distortion in the magnetization current
waveform. Also, engineering the core for low flux densities provides a
safe margin between the normal flux peaks and the core saturation limits
to accommodate occasional, abnormal conditions such as frequency
variation and DC offset.
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