Theory & Practice of Electromagnetic Design of DC Motors & Actuators
George P. Gogue & Joseph J. Stupak, Jr.
4.1 Force and emf generation:
If electric current flows in a long straight wire, an odd magnetic field is set up in a circle around the wire (no poles), as shown in Figure 4.1a. The strength of the field is:
i = current in the wire, amperes
H = coercive force at a point (amps/meter)
r = radius from the wire to the point, meters
If the wire is wrapped into a long cylinder, the field inside the coil (for an infinitely long coil) is
The magnetomotive force produced over a length of this coil is:
where n = total number of turns in the coil
In convenient engineering terms,
1 Oersted-inch = 2.0213 amp-turns (4.4)
If an electron is injected into a magnetic field, with its path at right angles to the field, a force is exerted on the electron, at right angles to its direction of travel and to the field, proportional to the strength of the field, the charge on the electron and its velocity. If instead of a single electron, a steady current is passed through a wire, which is in a magnetic field, a force is created on the wire:
In US engineering units, for a wire perpendicular to the magnetic field, with the resultant force perpendicular to both the field and the wire (i.e., sideways on the wire),
It seems at first as if there were two different magnetic forces at work, one in line with the magnetic flux and proportional to the square of the field, and the other at right angles to the field and proportional to it. The two are, in fact, a single force, in spite of their apparent differences.
A steady magnetic field does not affect the voltage or current in a wire at rest, in or near it. If the magnetic field changes with time however, a circular electric potential (EMF, or voltage) is caused around the path of the flux. If a wire surrounds the path, voltage will be caused in it, and if the wire forms a closed loop, electric currents will be induced in it. This induced voltage and current (called eddy currents, in some circumstances) can be very useful or very troublesome, depending on circumstances. The current is in a direction so as to cause a magnetic field, which opposes the change, in the original field which caused it. If it is able to flow, the result is to slow down or reduce the rate of current change (either increase or decrease).
The voltage caused by changing flux linking a coil of number of turns n is:
As before, a wire moving at right angles to a magnetic field shows the same effect in a somewhat disguised form (Figure 4.2). The wire "cuts" flux lines at a rate related to its speed, length and field strength, and a back EMF is caused in it. For a wire perpendicular to the surrounding magnetic field, moving at right angles to it and to its length (i.e., sideways), the induced voltage is:
If a wire is free to move and is initially at rest in a magnetic field, and a fixed voltage is imposed across it, the wire will initially accelerate at a rate determined by its mass, its resistance and the force from Equation (4.6). As its speed increases however, a back EMF will be induced in it. The back EMF opposes the applied voltage and with less net voltage across the resistance, less current flows. The rate of acceleration decreases. Eventually if the magnet is long enough, the wire will approach the speed at which the back EMF caused by its motion just equals the applied voltage. At this speed, called the terminal speed, no further acceleration is possible, as shown in Figure 4.3. If the speed is disturbed by outside forces, causing either an increase or decrease, the electromagnetic system will act to oppose the change. Referring to Figure 4.4, if electric current is caused in a wire which links a magnetic circuit, the magnetic field will change with the mmf (amp-turns) caused by the current. The magnetic field contains stored energy, which had to come from the electric circuit. The energy is removed by slowing down the current due to back EMF.
The constant L is called the inductance. Inductance is a magnetic effect, related to the permeance (or 1/reluctance) by:
The permeance of the gap (which is where almost all of the energy is stored, if the circuit is not saturated) is from Equations (3.8) and (3.11):
For this reason, it is often convenient to use units of which are consistent with electrical units. In this form:
As an example, the inductance of an electromagnetic circuit with 28 turns, having a gap 0.25" wide X 4" X 4" on a side, ignoring leakage, is:
The calculation is only approximate because we have ignored leakage flux. If the turns are wound close to the gap, the approximation will be fairly good, since the gap width is relatively small compared to its cross-sectional dimensions.
The relationship holds regardless of the nature of the magnetic circuit but, of course, P is dependent on the permeability of the pole material. In the example, it was assumed that the permeability of the steel in the surrounding structure was high enough that it could be neglected (which is equivalent to infinite permeability). If the magnetic structure is driven into saturation however, (either entirely, or only in certain locations of dense flux), the permeability and thus the inductance will change, and no longer will be well represented by a single constant.
4.2 Transformer operation:
The emphasis of these notes is primarily on motors and actuators but it nonetheless seems appropriate to add a brief description of electrical inductors and transformers. Those components are used to affect electric circuits, rather than to produce useful force and motion but the principles involved are often encountered in actuator and motor design. In the past, the analysis of some types of motor was done by regarding them as rotating transformers.
In an electric circuit, such as that shown in Figure 5.1, capacitors act something like the electrical analog of mechanical springs. They store up electric charge and release it later. An inductor, on the other hand, acts like the electric equivalent of a mass. It resists the change of current through it, independent of the amount of current, as a mass resists acceleration from rest, or deceleration once it is moving. The voltage across an inductor of fixed inductance L is:
This energy equation assumes, however, that L (and, therefore, P or R) is a constant. If a core material is used, the material will saturate above some coil current value, bringing about a dramatic reduction in P and L. At much lower current, core materials begin to become non-linear, and L is no longer a constant. Since inductance often must operate at relatively high frequencies, eddy current behavior becomes very important. Coil materials should therefore, have high electrical resistance as well as low hysteresis. To meet these requirements, high saturation density often has to be sacrificed.
An inductor limits current flow for an ac signal by storing up energy during part of the cycle and retrieving it later in the cycle. The voltage E and current I are out of phase by some angle. When the phase angle equals 90°, power flows into the inductor during one part of the cycle and an equal amount flows back to the source during the next part, resulting in (almost) no dissipated power. This is how a transformer with an open secondary avoids waste power even though it has low internal resistance and is connected across the power line.
A simple transformer is shown in Figure 4.4b. The shape shown is easy to understand but is not a likely one for actual use, because it would have high flux-leakage. An alternating electrical current flows in the primary circuit of turns . An alternating magnetomotive force mmf = is then set up and flux flows through the magnetic loop in response. Since is varying sinusoidally, mmf is varying in the same way and therefore, is an approximately sinusoidal function of time. Since links each turn on the primary as well as the secondary winding, a back emf or voltage is caused, which is the same for each turn in both coils. In order for the system to be at equilibrium:
i.e., the output voltage is proportional to the input voltage and to the turns ratio.
If we ignore the small losses in the transformer core and windings, the power into the primary circuit must be equal to the power out of the secondary circuit:
It can be seen therefore, that the output current is proportional to the input current times the reciprocal of the turns ratio:
In practice, the actual turns ratio is increased somewhat from the desired ratio of voltage out/voltage in. This is done to account for the voltage "droop" caused by the resistance drop iR across the secondary winding at rated current.
The first step in the design of a transformer is the design of a magnetic circuit which is capable of blocking the primary voltage with a current flow small enough that the power lost is acceptable. An appropriate number of turns and wire size is chosen for the primary coil. The number of turns of the secondary is then determined to provide the required output voltage. of course, the "window" in the magnetic circuit must be large enough to permit both windings.
The waveform of voltage input to the primary winding is usually sinusoidal. The time intergral of the voltage, in volt-seconds, reaches a maximum as the voltage decreases to zero.
i.e., the flux which must pass through the magnetic circuit is equal to this time integral:
1 weber = 1 volt-sec = 10,000 Gauss-meter = 15.5 M Gauss-in (approximately)
In order to avoid flux-leakage in a single-phase power transformer, the primary and secondary coils are wound on top of each other. The design of 3-phase transformers and the various methods of connecting them is beyond the scope of these notes (see Reference 14).
Pulse transformers are electromagnetic sensors which give an output voltage proportional to electric current passing through them. Since they rely on d/dt for this voltage, they cannot operate with dc current, and the fidelity of the output voltage depends on the rate of change or frequency of the linking current.
The primary is just a single turn and the current to be sensed passes straight through the magnetic circuit only once. An integral of mmf around this current-carrying wire is the same for any complete path. This mmf causes a voltage in each turn of the toroidal winding about the core. Therefore, the current in the secondary is less than that in the primary by this ratio. For example, if the current to be sensed is 100 A and there are 200 turns on the toroid, the current in the sensor winding will be 500 mA.
A resistor, usually of small value, is connected across the two sensor-coil leads. The voltage measured across this resistor is the output. It is proportional to the current in the main circuit but is isolated electrically from it and scaled as required by the number of turns and the sensor resistor value.
4.3 Instruments for magnetics:
If an electron moves in a magnetic field, a force is exerted on it at right angles to the direction of motion. In 1879, Dr. Edwin Hall discovered that if an electric current were passed through a flat strip of conductive material (he used gold) in a magnetic field, the electrons would be forced to one side, creating a voltage difference between the two sides (Figure 4.5).
For the same current, electrons must travel faster in a poor conductor than in a good one and the force (and thus the voltage) is proportional to speed. Modern Hall sensors use silicon, not for its switching and control capabilities but just as a high-resistance material. A typical Hall sensor of the linear variety may have a resistance of 1000 ohms or so across the reference current leads and perhaps a little less across the sense leads. If 10 V is connected across the reference leads, causing a current of 10 mA, the sense leads may indicate 50 mV when the sensor is placed in a 100 G field. With Hall sensors which have no internal processing, the direction of reference current may be reversed, reversing the polarity of the output. The active site is very small, perhaps 0.040 inch on a side, so the field is measured nearly at a point. Only the component of B which is normal to the surface is measured, so care must be taken to align the sensor properly to the field. Proper alignment causes the maximum reading at that point. The values given here are only to convey an approximate idea of what may be expected; actual components vary widely from these numbers. Another type of Hall sensor has electronic processing built in and might contain, for example, a constant-current source, a switch with hysteresis and an output driver with relatively high current capability (Reference 15). The voltage cannot be reversed across this type of device, of course.
Hall type Gauss meters are relatively easy to use but the making of accurate measurements is complicated by several factors. The earth's magnetic field varies in strength and direction from place to place; it averages about 0.4 Gauss. In a room containing electric lights and other electrically-powered equipment and walls carrying electric wires, the field may be higher. The presence of tools, desk parts, lamps and filing cabinets of steel may warp and concentrate the stray field enough to cause reading errors of tens of Gauss.
Hall sensors have a slight offset, indicating a reading with no magnetic field present, and the offset may vary with time and temperature. The offset is caused by slight inaccuracies in manufacturing. Gauss meters are usually provided with a shielding tube and an offset adjustment to allow for zero correction. They may also have a means to adjust the reference current (the current through the Hall sensor which is deflected by the field to cause the output), if it should drift.
Another problem is the scale adjustment. It is a common experience among workers in magnetics to place a Hall probe at a marked location, with one side of the probe flat against the surface. The flux is known to be normal to the surface. A reading is made. As a check, the probe is turned over, in the same location, and the second reading is found to be somewhat different. The operator is then unsure as to which to use, or whether to take an average. A means is needed to check the output at a known high reading, near that which the operator is trying to make. Some magnet companies sell standard magnets for this purpose. However, the operator finds that he gets different readings at different places in the supposedly uniform reference gap. Some of these devices are variable with temperature. If one compares several of the standard magnets from different vendors, they may not agree.
It is possible to generate a magnetic field, the strength of which is known from basic, traceable standards up to perhaps 100 Gauss. One way to do this is with a Helmholtz coil, which is really two coils with axes aligned, separated by the distance of one coil radius. For higher fields however, the coils overheat and the attainable field is too low to give a good check on measurements made at thousands of Gauss.
While a Hall-effect Gauss meter reads field strength nearly at a point, a flux meter indicates total flux over an area. A search coil is used, with one or more turns. The coil may be handmade on the spot to conform to the magnet or region being measured. A flux meter reads the time integral of voltage induced across the coil when it is introduced into a field, or withdrawn from it. If the coil is turned over in place, the reading is doubled. Some flux meters are difficult to read because they drift with time and are sensitive to stray fields in the area. It is possible to make an impromptu flux meter by using a coil and a storage oscilloscope and integrating the area under the curve (voltage vs. time) by hand.
There is a form of plastic sheet on the market which is very useful for checking the uniformity of magnetization and pole shape. Microscopic flakes of nickel, which is highly permeable, are encapsulated in oil inside plastic spheres too tiny to be visible. They are then bonded to a dark-colored sheet and given an overcoat layer of a transparent colored plastic. When exposed to a magnetic field with flux normal to the sheet (going out or in), the flakes stand on end, revealing the dark color below. When the flux density is low, or when the flux is parallel to the paper surface, the flakes lie flat, reflecting the color of the sheet above them. When this sheet is placed on a magnet which is charged, poles and transitions are clearly visible through the sheet (although the sheet is nearly opaque to light). The material does not indicate strength of field, but rather direction, and it is useful to keep in mind that the light-colored areas do not necessarily represent low field intensity but could be horizontal fields instead.