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.
|
