[The following text essay was yanked from the newsgroup rec.audio.tubes. My thanks go to Henry for yet another killer explanation of esoteric, ephemeral concepts. It appears in its entirety, with minor HTML formatting done by me.]
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.
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.
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.
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.
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 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.
[Here begins the second installment of the article.]
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.
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.
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.
[That's all he wrote.]