Magnetics for Morons
From henry–(at)–acconnect.com Tue Dec 1 13:31:08 CST 1998
From: henry–(at)–acconnect.com (Henry Pasternack)
Newsgroups: rec.audio.tubes
Subject: Magnetics for morons.
Date: Fri, 27 Nov 1998 17:07:24 -0500
Xref: geraldo.cc.utexas.edu rec.audio.tubes:97662
Electromagnetics for Morons.
Copyright (c) 1998, by Henry A. Pasternack
This article is a review of the basic principles of electromagnetics.
It covers some of the basic terminology encountered in discussion of
transformer theory. I’ve tried to be intuitive and avoid the use of
any mathematics. Still, it would be good to read this article along
with a standard electromagnetics text or especially the RDH. Some
of the ideas are challenging and are better understood with the aid
of graphs and math.
I may find the inspiration to write subsequent articles discussing
important concepts missing here, namely electromagetic induction and
the operation of chokes and transformers.
Introduction.
Electromagnetics is the branch of Physics having to do with the
properties and relationships of electric currents and magnetic fields.
A magnetic field is a “phenomenon” that arises whenever electricity
is on the move. Of course, moving electricity implies the flow of
electrons in a wire, but it need not be so. For instance a permanent
magnet gets its field from tiny atomic current loops formed by electrons
whizzing about in their orbits.
A magnetic field is hard to describe because it cannot normally
be seen, felt, or sensed directly in any way. The field is only
tangible through the force it exerts on real-world objects. It can
be mapped by probing the field with a tiny magnet. The field will
exert a twisting force on the magnet, the strength and orientation
of which vary with position in the field.
Field lines.
Probing with a test magnet lets us draw a three-dimensional plot
of the field. This consists of a set of smoothly-flowing field lines.
The field lines always form continuous loops. Where the field is
intense the lines are bunched tightly together and where the field
is weak they are more widely separated. The magnetic field is
directional. The direction of the force exerted on the test probe
depends on the orientation of the field lines at the point of interest.
Field lines cannot be broken. If a field line were to break, the
ends would have to terminate on positive and negative particles called
magnetic monopoles. Physicists have searched for years for proof
that magnetic monopoles exist. As far as I know, they haven’t found
any, or if they have, magnetic monopoles are so hard to produce
and so short-lived as to be of no practical consequence.
Magnetic flux.
A magnetic field line is called a line of flux. The symbol for
flux is the Greek letter “phi” (a circle with a slash through it),
and the unit of flux is the Maxwell. One Maxwell corresponds to
one line of flux. The flux line is an abstraction; the field is
continuous and doesn’t actually form into spaghetti-like strands.
The flux line is useful because it quantifies “units” of magnetism.
Drawing a loop in the vicinity of a magnetic field encloses
lines of flux. A flux line is enclosed when it enters one face
of the loop and exits the other. For a loop of given area the
total enclosed flux depends on the shape of the loop and its
orientation relative to the field. The most flux is enclosed
when the plane of the loop is perpendicular to the orientation
of the field.
Magnetic flux density.
Dividing the number of flux lines enclosed by a loop by the
loop area gives the flux density, that is, the flux per unit area.
The symbol for flux density is “B” and the unit of flux density
is the Gauss. One Gauss is equal to one Maxwell (one line of
flux) per square centimeter.
Flux density is a point quanity which means the the loop used
to compute it is supposed to be vanishingly small. This means
we technically need 3D vector calculus to solve the problem, but
we can often get away with algebra in simple cases.
Whereas flux is a measure of a quantity of magnetism, flux
density relates to the strength of the magnetic field at a
particular point. Both concepts are important and it’s helpful
(and necessary) to learn and understand each of them.
Steady electromagnetic fields.
The steady flow of current creates a fixed magnetic field in the
space surrounding a wire. If the wire is straight, the field lines
form concentric rings about wire. Near the wire the field lines are
more tightly bundled and the flux density is higher. The field
reaches its maximum density at the surface of the wire and diminishes
to zero at its center.
Forming the wire into a loop consolidates magnetic field lines
and increases the density of the flux through the center of the
loop. Winding many loops to form a coil increases the flux density
further still. The more compact the coil, the denser the field
for a given current flow.
Magnetic circuits.
The path of a flux line, which necessarily forms an unbroken
loop, describes a magnetic circuit that is analgous to an electric
circuit. The magnetic analogy to Ohm’s law relates magnetomotive
force to flux and reluctance. Here are the correspondences:
Electric Circuit Magnetic Circuit
—————- —————-
Voltage (EMF) Magnetomotive Force
Current Flux
Resistance Reluctance
The electric circuit model says that the current that flows in a
circuit depends on the applied voltage and the total resistance
in the loop. Similarly, the magnetic circuit model says that the
flux developed in a magnetic circuit depends on the applied
magnetomotive force and the reluctance of the loop. The greater
the magnetomotive force or the lower the reluctance, the greater
the flux developed.
Magnetomotive force and reluctance.
The symbol for magnetomotive force is “F” and it is measured in
Gilberts. “F” developed by a coil is proportional to the number of
turns of the coil and the magnitude of the current flowing through
it. One Ampere flowing through one turn is equivalent to 1.257
Gilberts. Sometimes the term Ampere-turns is used. Ampere-turns
and Gilberts are equivalent except for the scale factor. Magneto-
motive force is a measure of the “drive” in a magnetic circuit.
Reluctance, “R”, is analous to electrical resistance. It indicates
the opposition of the core to deve. The higher the reluctance, the
lower the flux developed for a given number of Ampere-turns. Reluctance
depends on the total contribution of all the materials in the magnetic
circuit. Volume-for-volume, air has more much more reluctance than
transformer steel. Putting a steel core through a coil results in
a much stronger magnetic field for a given current flow. Similarly,
cutting an air gap in a core increases the reluctance of the magnetic
circuit and reduces the generated flux.
To figure out the reluctance of a core, you must know the geometry
of the magnetic path and the magnetic properties of all the substances
(i.e., steel and air) in the path. That means specifically knowing
the value of a quantity called permeability which I will discuss in
just a few moments.
Magnetic field intensity (magnetizing force).
Given a coil producing a constant magnetomotive force (i.e., for
a fixed number of Ampere-turns), the shorter the length of the
magnetic circuit, the more intense will be the magnetic field.
Magnetic field intensity is defined as magnetomotive force divided
by path length. It is also known as magnetizing force. The symbol
for magnetizing force is “H” and its unit is the Oersted. One
Oersted is equivalent to one Gilbert per centimeter.
It makes sense that if you have a coil with some number of turns,
spreading the coil out (making it longer) will decrease the intensity
of the field produced, while squeezing the turns more closely will
increase the intensity. This assumes the coil is in air so that
the field lines are free to follow the shortest possible loops.
If the coil is wound on a core if fixed size, the magnetizing force
doesn’t change much with coil geometry because the length of the
magnetic circuit doesn’t change.
Magnetic field intensity is closely related to flux density, as
we will see in the next section.
Permeability.
Permeability is analgous to electrical conductivity. It is a
property of a material and does not depend on sample shape or size
like reluctance. The symbol for permeability is the Greek letter
“mu” (or “u” for the font-impaired). Technically the units of
permeability are Gauss per Oersted (flux density per magnetizing
force), but “u” is usually given as a pure number, the ratio of the
permeability of a material to that of air. The permeability of air
is defined as unity, so whichever interpretation you choose, it
doesn’t change the numbers when doing math.
Permeability isn’t a constant, but varies with flux density. At
low flux densities, the permeability is relatively low, but it
increases to some maximum value as the flux density increases, and
then drops again. For silicon steel, used in making transformers,
the initial permeability is about 450 and the maximum permeability
is about 8000.
Since permeability relates flux density to magnetizing force,
if we know how much magnetizing force a coil produces, and we know
the permeability of the core, we can compute the resulting core
flux density. Plotting flux density versus magnetizing force gives
us the all-important B-H curve which is in a sense the “transfer
function” of a transformer. It also paves the way for hours of
silly speculation and arguments among devotees of single-ended
and push-pull amplifiers.
Air gaps and distortion.
Air is a neutral magnetic material. The permeability of air
is low and it doesn’t change with applied magnetic fields. Putting
some air in a transformer core increases the reluctance and lowers
the flux density generated for a given magnetizing force. But it
also makes the effective permeability of the core much less
sesitive to changes in field strength, and it reduces the tendency
of the core to saturate (exceed the point where increases in
magnetizing force cause the permeability to the core to decline).
This helps linearize the core, reducing distortion.
Putting it all together.
All this terminology is getting really confusing, so let’s go back
and run through it one more time. If you have a magnetic core made of
some material, you can calculate the reluctance, “R”, if you know the
permeability, the core cross-section, and the length of the magnetic
path. Let’s assume for now the core has a simple shape, like a solid
steel donut, with no airgap.
Now, wind some turns on the core and apply a current. The number
of turns times the current gives us the magnetomotive force, “F”, give
or take a constant multiplier. Divide the magnetomotive force by the
average circumference of the core and we get the magnetizing force,
“H” (that’s Oersteds per centimeter). Multiply “H” by “u” (the permea-
bility) and we get the flux density, “B” (Gauss). Multiply “B” by the
core cross-sectional area and you get flux, “phi” (Maxwells).
Alternatively, divide “F” by “R” and you’re back to “phi” again.
The magnetic circuit concept is a short-cut that saves a lot of
monkeying around with “B” and “H” and “u”.
Now, put an air gap in the core. Air has much lower permeability
than steel, and much higher reluctance per unit volume. A tiny bit
of air suffices to raise the total reluctance of the core. This
reduces “H” for a given coil current, and therefore “B” in the steel
part of the core. This helps keep the core froms saturating when we
run DC in the coil.
Our coil produces some amount of flux in the core. Since flux
lines form continuous loops, the number of lines in the steel part
of the core must be the same as the number crossing the airgap.
Since the permeability of air is lower than that of steel, the
flux density in the airgap must be lower than it is in the rest
of the core. This means the lines of flux must spread out or
“fringe” when they enter the gap. Why? Because lower flux density
(flux per unit area) means the flux lines aren’t so close together.
When the flux lines cross back into the steel part of the core,
they squeeze back down to their original density. High-permeability
materials act like flux vacuum cleaners, “sucking up” and channeling
lines of magnetic flux. This is the principle behind mu-metal, a
high-permeability material used for magnetic shielding.
From henry–(at)–artier.raleigh.ibm.com Tue Dec 1 13:32:51 CST 1998
From: henry–(at)–artier.raleigh.ibm.com (Henry A. Pasternack)
Newsgroups: rec.audio.tubes
Subject: Magnetics for Morons, Part 2.
Date: 1 Dec 1998 19:21:37 GMT
Xref: geraldo.cc.utexas.edu rec.audio.tubes:98158
Electromagnetics for Morons, Part 2.
Copyright (c) 1998, by Henry A. Pasternack
In the first part of this article, I introduced the fundamental
terminology of electromagnetics — flux, flux density, magnetomotive
force, magnetizing force, permeability, and reluctance. I explained
in a nutshell how these quantities relate to one another. And I
mentioned but did not elaborate on the concept of transformer core
linearity and the B-H curve.
These concepts are enough to explain how the flow of electric
current in the primary windings establishes a magnetic field in
the core of a transformer. Now I’d like to talk about how we get
electricity back out of a magnetic field. This will eventually
lead to the subject of electromagnetic induction. I will then be
in a position to introduce and explain inductance and inductive
reactance, and the operation of chokes and transformers.
Moving electrons in magnetic fields.
An electron placed in an electric field experiences a force in
the positive direction of the field. An electron placed in a
magnetic field, on the other hand, experiences no force as long
as it remains at rest. Only if the electron is set in motion will
the field exert a force on it. The magnitude of the force depends
on the speed of the electron, the flux density, and the angle of
motion relative to the orientation of the field.
Imagine two vectors (arrows of specified length) with their
tails anchored to the moving electron. One vector points in the
direction of the field and the other along the line of motion.
The two vectors define a plane. That is to say, there is exactly
one flat plane in space in which both of these vactors lie flat.
The direction of the force on the electron is always perpendicular
to this plane. The magnitude of the force is greatest when the
vectors are at ninety degrees to one another and diminishes as
the angle between the two vectors closes. If the directions of
the field and path of motion coincide, the plane is undefined and
the electromagnetic force is zero. An electron moving directly
along a flux line is not influenced by the field.
A familiar phenomenon that also obeys this rule is the force
exerted by a spinning gyroscope. Undisturbed, the gyroscope sits
perfectly still. Try to rotate it and the gyroscope twists at
ninety degrees both to the spin axis and the axis of the applied
torque. The magnitude of the reaction depends on the angle between
the two axes. It’s zero if you rotate the gyro along its spin axis.
Mathematically, we say the reaction is the cross-product of two
vectors. The term “cross-product” comes from vector algebra. By
definition the cross-product is three-dimensional. This means we
have to think in three-dimensions if we want to study electro-
magnetism.
Permnent magnets and compass needles.
Now we can understand why a small permanent magnet twists to
align itself with an external magnetic field. A good example of
this is a magnetic compass needle. The needle gets its magnetism
>from electrons orbiting inside the material it’s made from. The
earth’s magnetic field exerts a force on these moving electrons.
The force is transferred to the body of the needle causing it to
rotate on its pivot. When the needle is aligned with the field
the net torque is zero and the rotation stops.
Electron circling in a steady magnetic field.
Imagine we have a uniform, constant magnetic field coming up
vertically out of a horizontal work surface. Suppose we roll
the electron through the field like a marble on a desktop. The
electron will experience a steady force in the horizontal plane
perpendicular to the direction of motion. The force will deflect
the electron into a circular path. The radius of the circle will
depend on the charge of the electron and its mass (both physical
constants), the strength of the field, and the speed of motion.
The deflecting force is always perpendicular to the velocity
vector. This means that the field does no work on the electron.
Provided there is no friction, no energy is lost or gained and
the electron circles indefinitely, neither gaining nor losing
speed.
Electron in a moving wire.
Now imagine a wire lying flat on the desktop. Our electron
is trapped in the wire, like a marble in a straw. Draw the wire
broadside from back to front through the field. Once again the
electron experiences a force perpendicular to the direction of
motion. It would like to go into a circular orbit as before
but it hasn’t enough energy to escape the surface of the wire.
Instead, as it is swept through the field the electron is pushed
lengthwise along the wire by the electromagnetic force.
The electron doesn’t accelerate indefinitely. Two things tend
to slow it down. First, if the wire has some resistance the force
of friction opposes the force due to the magnetic field. Second,
the moving electron creates its own magnetic field that cancels
the external field. The net result is that the electron rapidly
reaches a constant linear velocity in the wire.
Viewed from above the electron follows a diagonal path relative
to the desktop. The back-to-front component of velocity equals
the speed of the wire, and the left-to-right component equals the
speed of drift along the length of the wire.
The force exerted on the electron by the field is always at
right angles to the motion of the electron. Since the electron
moves both forward and to the side, it experiences a force to the
side and to the back. The sideways force is balanced by friction
within the wire. The backwards force is transfered to the wire
as the electron bangs into and presses against the wire’s inner
surface. This force is felt as mechanical drag on the wire as
it is drawn through the field.
Induced voltage.
There are actually many electrons in the wire. All of them
begin to drift sideways as the wire cuts through the magnetic field
lines. There is no external circuit connected to the wire. The
electrons, unable to jump into free space, pile up when they arrive
at the free end. The separation of charge sets up an electric
potential gradient across the wire’s length.
A voltmeter connected to the ends of the wire as it moves will
register a DC voltage. The strength of the voltage will vary with
the speed of motion and the polarity will depend on the direction
of motion. We say that when the moving wire cuts the magnetic field
lines a voltage is induced across the wire. This is the principle
of operation of an electric generator.
The potential gradient in the wire creates an electric field that
exerts a force on the electrons and opposes their lengthwise motion.
Shortly after the wire begins to move all electron drift ceases. The
sideways component of electron velocity disappears and so does the
mechanical drag on the wire. Some of the physical work done on the
wire when it is just starting to move ends up as heat dissipated in
the wire’s electrical resistance. The rest ends up in the electric
field. Essentially we are charging a small capacitor here. The
capacitance is very, very tiny, and so is the momentary drag force
on the wire.
Electric generator.
If we connect an external circuit to the free ends of the wire,
electrons will flow out of the wire into the load. An equal number
of electrons will flow back into the wire at the other end. The
number of electrons moving out of the wire is small compared to
the total population. For every electron that finds its way out
of the wire, another one is ready to pop instantly into its place.
For this reason the overall equilibrium is essentially undisturbed.
The foregoing assumes the load resistance is high compared to the
resistance of the wire. If a heavy load is applied, the electron
population at the end of the wire will be depleted. This will reduce
the electric field in the wire and encourage more electrons to drift
under influence of the electromagnetic force. A new equilibrium will
be established with a lower induced voltage but a higher current flow.
The drop in terminal voltage is directly due to the internal resistance
of the generator. Some energy is lost to this resistance and shows
up as heating of the wire.
Current flowing in the load consumes energy that is supplied by
the mechanical force dragging the wire. This energy is converted
to electric current and is dissipated as power in the generator wire
and the load.
Square wire loop.
Let’s take the wire and form it into a square loop lying flat on
the desk. Position the loop so its edges are aligned with the cardinal
(left-right, front-back) axes of the desktop. In the nearest edge of
of the loop, cut a small gap and insert a tiny meter so we can read the
voltage induced in the wire as it moves. The uniform magnetic field
still emerges vertically out of the work surface.
Slide the loop left and right. Electrons in the front and back
edges don’t drift because the force shoves them sideways against the
inner surface of the wire and they have nowhere to go. Electrons in
the left and right edges try to circulate around the loop. But while
electrons on the left circulate clockwise, electrons on the right
circulate counter-clockwise, and vice-versa. The net circulation is
zero and no voltage reading appears on the meter.
In fact, regardless of how the loop moves in the field the net
voltage is zero. This is true if it slides side-to-side, up and down,
diagonally, or in a circle. In every case the motion of electrons in
one part of the loop is canceled out by motion in another part of the
loop. The only constraint is that the loop must remain wholly in the
field, and must remain flat in the horizontal plane.
Changing flux in a loop.
Consider what happens if we narrow the cross-section of the field.
Instead of covering the whole surface, assume it is restricted to a
small patch six inches square in the center of the desktop. Let’s
make the loop larger — a foot on each side — and position it so it
is centered about the field with a three inch margin all around.
Repeat the experiment of sliding the loop, but make the movements
small enough that the wire never touches the field. As we expect, no
voltage registers on the meter. No lines of flux are cut, no electron
drift occurs, and no induced voltage appears.
The meter will deflect if we allow the loop to cut into the field.
This is equivalent to the original experiment with the straight wire.
Because the loop is not fully immersed in the field there is a net
imbalance in electron circulation and a net voltage induced around
the loop. The exception is the case where you put one edge of the
loop in the field and slide the loop along the length of this edge.
Since no flux lines are cut, no voltage appears.
You can also generate a voltage by picking up the loop and moving
it about in three dimensions. How much voltages you produce depends
on the relative orientations of the loop, the field, and the motion.
If you attach the loop to a stick like a lolly-pop or a road sign and
spin it about this axis, you will generate a nice sinusoidal AC
waveform.
Of course, if you slide the loop completely out of the field,
you’ll be back to the condition of zero induced voltage.
Rule of electromagnetic induction.
We can think of this situation in terms of enclosed flux. When
the loop is horizontal and completely immersed in the field the net
enclosed flux is constant regardless of its position. For every
flux line that leaves the loop, another one slips in on the opposite
side. A voltage is induced only when the loop moves in such a way
that the net enclosed flux varies. This is the case when one side
of the loop is leaving the region of the field. It’s also true if
the loop remains in the field but rotates so that the number of flux
lines passing through it changes.
This leads to the all-important rule of electromagnetic induction.
The rule states that the voltage induced in a loop is proportional to
the rate of change of flux in the loop. There is a minus sign in
there as well; a positive change in flux creates a negative voltage,
and vice-versa. (This minus sign is very important, by the way, as
we will see later on.)
The rule of induction applies when the loop is cutting lines of
flux in a constant field. It also applies if the physical extent
of the field is constant but its strength is changing. This has a
profound implication. It means you don’t need physical motion (as
between a loop and a field) to generate electricity. All you need
to do is modulate the strength of a field enclosed by a loop to
induce a voltage in that loop.
