What do you need to know about motors to design motor-drive electronics? This article presents basic motor theory from a circuit-design viewpoint.

The motor theory presented here is based on vectors, not phasors (as found in most electric-machine textbooks), for one major category of motors, permanent-magnet synchronous (PMS) motors. "Servo motors," "brushless dc," and most step motors are of this kind. Induction or "ac" motors are somewhat more complicated and, along with variable-reluctance and dc or "brush" motors, produce torque differently.

Let's start from common childhood experience, that of pulling a magnet on a table with another magnet in our hand. The attractive force between the magnets lets us pull the free magnet along a straight path. Our hand supplies the energy that moves the free magnet on the table, and it is coupled via the interaction of the magnetic fields of the two magnets.

This is a two-pole, linear motor, though not an electric-powered one. Control consists in keeping the spacing between magnets about right. If they come too close, the free magnet will stick to the hand magnet – a minimum-energy "equilibrium" state that produces no force. And if the magnets are too widely separated, the field interaction is too weak to produce significant force. Control consists in maintaining the right distance. The two poles of the hand-powered motor are the two ends of the free magnet.

An obvious improvement over the hand-powered motor is to use electric power instead. The hand magnet is made into an electromagnet by winding turns of wire on an iron core, which is an excellent conductor of magnetic flux. The core is shaped like brake-shoes (called "teeth") to distribute the flux around the rotor, with a small air gap between stator teeth and rotor.

Instead of linear motion, the free magnet will be attached to a free-turning shaft, called a rotor. Instead of a single magnet, we will use two opposite-polarity magnets shaped as half-cylinders mounted on the shaft, as shown below. The polarity of the "N" magnet is positive in that its flux points in an outward direction from its outer surface. This is a two-pole (two rotor magnets) electric machine.

Electromagnets are also placed on the opposing sides of the rotor, as shown, for symmetry. They are connected so that their fields aid – that is, their field vectors point in the same direction. The electromagnets are stationary (do not move) and are called the stator; the iron structure on which they are wound is the armature. When the magnets are rotationally positioned as shown above, the magnetic field vectors for the stator winding and rotor magnets can be drawn, as shown below. The stator-field vector orientation assumes current out of the top end of the right-side winding conductors (using the right-hand rule).

The relative orientation of the stator and rotor magnetic fields is critical for torque production. If the fields are aligned, no torque is produced – like motor # 1 magnets stuck together. When the angle from rotor to stator field vector is 90 degrees (and not 45° as shown), the maximum torque is produced. A motor with such fields is "field oriented." The goal of motor phase control is to keep the stator field vector 90° ahead of the rotor vector in the direction of motion.

Our simple motor # 2 is too limited to achieve such vector control because the stator design allows only one-dimensional control of the stator field vector. It is stuck horizontally, at zero degrees (when positive) or 180 degrees (when negative). Another pair of windings is needed to produce a vertical field.

Motor # 3, shown below, is a simple, workable PMS machine because it can generate a stator field in the two dimensions needed for rotation in a plane (that of the rotor). The two additional (vertical) windings are connected as a pair, to aid each other in producing flux in the vertical direction. By controlling the currents in the two winding-pairs, or phase-windings, a stator field vector of controlled magnitude and orientation can be produced. As these currents are varied in time, the stator field vector can be rotated. By driving the phase-windings with sinusoidal waveforms 90 degrees out of phase, a vector of constant magnitude is rotated in the rotor plane. If the x-axis windings are driven with a cosine-wave and the y-axis windings are driven with a sine-wave, a stator field vector will rotate at the frequency of the sinusoids. If this drive can be phased so that the resultant stator vector stays at right-angles ahead of the rotor vector, maximum torque will be produced for the given current amplitude of the windings. Of course, we need to know the rotor angle to do this. That is a major aspect in the design of the motor controller.

What control do we have of motor # 3? We can control the stator magnetic-field vector – a vector with two components, magnitude and phase angle. We already covered phase control as maintaining field orientation. The angle between rotor and stator field vectors is the torque angle, d . It affects torque according to:

where is the maximum torque that can be produced with the given current amplitude. At d  = 0° , the fields are in equilibrium and no torque is produced. This rotor position aligns the rotor field vector – the direct, or d axis – with the stator field vector. For field orientation, the rotor field vector must be –90° from (90° behind) the stator field vector, which is aligned with the quadrature or q axis. Our goal is to keep the rotor vector aligned with the d axis by keeping the stator q axis 90° ahead of it. Motors operated with d  ¹  90° (not field-oriented) not only produce less than the maximum torque, but can exhibit undesirable side-effects due to torque generation by stator current in the d axis.

The other component of control is magnitude control. Phase control maintains field orientation while the amount of resulting motor torque is controlled by the amplitude of current applied to the windings. The basic torque equation is:

where i_qs is the q-axis stator current and L is a motor parameter representing the average total flux linkage of the motor magnetic circuit (in the d axis of the rotor). It is popularly referred to as the "torque constant," K_T, and is related to motor geometry and winding turns. (See derivation of L in boxed section, "Derivation of Motor Flux-Linkage Constant".)

The previous motor was a two-phase machine because it had two phase-windings, driven 90° out of phase. This is the minimalist motor in that at least two phase-windings are required for 2-D control. For reasons of circuit minimization and control simplification, most PMS motors have three phase-windings, spaced so that their field vectors are 120° apart, as shown in the vector diagram below. By driving the three phase-windings with equal-amplitude, 3-phase (120° -electrical spaced) currents, a constant-magnitude stator-field vector rotates at the frequency of the current.

The phase-winding field vectors A, B, and C vary in magnitude with time, and their vector sum is the resultant field vector. From a circuit standpoint, these vectors correspond to the stator currents (and their orientations). In time, the sequence ABC occurs in the positive (CCW) rotational direction. By reversing the sequence as ACB, the field vector rotates in the opposite direction, as will the rotor.

A four-pole motor has four rotor magnets in the order: NSNS. Each NS pair in the sequence around the rotor is a full magnetic cycle, where N is the positive half-cycle. Therefore, with four poles, one mechanical revolution of the rotor corresponds to two magnetic cycles. This has the effect of making the electrical frequency twice the mechanical frequency. For p poles generally,

w _el = (p/2)× w _me

For each complete electrical cycle, a four-pole rotor has moved a half revolution.

Multiple poles distribute torque production spatially around the rotor. Step motors have many poles – typically 100 – so that for each electrical cycle, the rotor moves 360° me/(100/2) = 7.2° me. When stepped a quarter-cycle per step ("half-stepping"), the mechanical resolution is 1.8° . With many poles, torque production is also high.

The most important electrical design parameter of a PMS motor besides its power rating is its flux-linkage constant, L . This constant is the scaling factor that relates torque to (q-axis) current and motor induced voltage to speed:

A general expression for L can be derived from basic magnetics theory and motor geometry, as follows. Torque is defined as a vector,

or, for F at right angles to the armature radius, r, then the torque magnitude is T = r× F. Second, force and current are related by the basic equation:

where in this case, i× l and B, the magnetic field density, are at right angles because the loops of the stator windings produce a field that goes through the loops at right angles to the current path. Consequently, the force-current relation simplifies to

Substituting from the torque equation,

T = r× (i× l× B)

Stator windings have N turns, which effectively multiplies the terminal current, i_s, by N:

With N current loops, a field N times that of one current loop is generated. Substituting into the above torque equation,

T = (N× r× l× B)× i_s  = l _m× i_s

With p poles distributed around the stator, and the two conductors per turn of a winding producing torque, the final expression relating torque to stator terminal current is:

T = (p× N× 2× r× l× B)× i_s = L × i_s

The motor flux-linkage constant has been derived, where l is the armature conductor length (approximately the inside length dimension of the motor), B is the average magnetic-field density (or induction), and 2× r is the rotor diameter, including the air gaps – that is, the armature inside diameter.

In Part 2, a motor electrical model will be developed, which can be used in electronic design. Also, motor control circuitry will be discussed.