CONSTANT FLUX CONTROL
Motor Induced Voltage
In Part 1, we found that motor torque is proportional to motor current, related by L . Besides torque production, the other major behavior of a motor is the generation of a winding voltage due to rotor motion. If the shaft of motor # 4 is spun, the magnets move across the windings, and by Lenz's law, induce a voltage into them:

where l is the flux linkage of the rotor magnets mutually coupling the stator windings. The induced voltage is also called the speed voltage. (Several misleading and/or obsolete expressions also continue to float around the industry, such as "back emf" or "motor emf.") The change of l can be due to a change in winding current (stator field vector) or due to the rotating rotor flux change (rotor field vector) with rotor position, q .
A stator winding of N turns will have a flux linkage of

where f is the (per-turn) winding magnetic circuit flux. l is a terminal-referred flux in that it corresponds to the winding terminal current, is. Put simply,
f = B× A
where B is the magnetic field density (in V× s/m2) and A is the cross-sectional area of the winding flux path – the area of the winding loops. For magnets with flux density of B, an N to S flux reversal of 2× B occurs during magnet movement through A, or
D f = (2× B)× A
Combining the above equations,

The rate of change of area depends on the rotational speed, fme, of the rotor and the rotor air-gap radius, r. Then the speed, u, of the magnets past the windings is:
u = (2× p × fme)× r = w me× r
where wme is the radian speed (in s–1). Rotor movement is at right angles to the winding conductors of length, l, and

With p windings – one per pole – in series, the induced voltage is found by substituting dA/dt, which results in an expression for vw in terms of motor parameters:
vw = (p× N× 2× r× l× B)× wme = L × wme
Interestingly enough, L is also the proportionality factor that relates motor torque to stator current. (See derivation in Part 1.) Note that N is the number of turns per pole and that N× p is the total turns of a phase-winding.
Motor Model
An electric circuit model for a field-oriented PMS motor is shown below.

The model has electrical and mechanical sides, linked by the two basic relations - the torque and induced-voltage equations – through the common motor parameter, L . If phase control can maintain field orientation, then this model applies – the same model as for dc brush motors.
The mechanical side of the model uses electrical circuit elements, though the quantities are actually mechanical. Because linear electrical and mechanical systems are described by the same equations, the corresponding electrical and mechanical quantities become analogs of each other. Two analogs are possible: the first by relating torque to voltage, and the second, torque to current. Speed and voltage are "across" quantities because they are measured with respect to some other reference point. Torque (or force) and current are "through" quantities. To relate the same kind of quantities to each other, the torque-current analogy is used here. Analogous quantities are given in the following table.
electrical mechanical
current, i force, F
torque, T
voltage, v speed, u, 
flux linkage,  distance, r, 
capacitance, C mass, M
inertia, J
inductance, L compliance, K, K
conductance, G damping, D, D
Analogous laws are, for example, Kirchhoff's current law (the sum of the currents at a node is zero), and D'Lambert's principle (the sum of forces or torques acting on masses in a system is zero).
Power can be transferred in either direction in the model. Mechanical-to-electrical power flow is the generator mode of operation, and electrical-to-mechanical flow is the motor mode. For field orientation, l r lags the stator current by 90° . The induced voltage is dl r/dt, which leads l r by +90° , in phase with the stator current. Current and voltage in phase produces real (not reactive) power and is the power of electrical-mechanical conversion. In other words, a circuits view of field orientation is that the rotor induced voltage is in phase with the stator drive current.
For motoring, terminal current flows into the induced voltage source of the model, opposing its voltage and delivering negative power (power sink) on the electrical side. On the mechanical side, the torque (current) source develops positive speed (voltage) across the mechanical impedance of motor mechanics and external mechanical load.
Torque-Speed Characteristics
Electronic active devices, such as transistors, can be characterized by parametric plots on a curve tracer. Motors can be similarly characterized by their torque-speed plots for quasistatic operation (L = 0 and no mechanical reactances). What is needed is an equation relating torque and speed variables, with source voltage amplitude, Vs, as a parameter.
The electrical side of the motor model can be analyzed as follows. A terminal-current expression is written from Kirchhoff's current law:

Substituting into the torque-current equation,

A family of torque-speed curves is plotted below.

The second term of the torque-speed equation is the stall torque,

At stall, or zero speed, the induced voltage is zero and the terminal current is simply Vs/R. Usually, this current is excessive and current-limiting occurs in the motor drive. On the speed axis, the maximum speed for a given terminal voltage is the no-load speed,

At maximum speed,  0, the induced voltage equals the supply voltage and no current flows through R. No torque can be produced at this speed.
The torque-speed equation is linear, as is the T- plot, with a slope of . (The square-root of is the motor constant.) The slope is an important motor parameter. A load requirement of high torque at low speed implies a large negative slope, and high-speed, low-torque applications require a shallow slope. Because slope is expressed in parameters related to motor geometry and winding resistance, geometric and electrical requirements of the motor are constrained by the required motor performance, as expressed by the torque-speed relation.
Motor design applies the relationships between motor geometry and magnetic and electrical quantities to determine optimal magnetic-path design, electrical winding "window" area, magnet B-field strength, number of poles, whether the rotor magnets rotate inside the stator (as previously shown) or outside, and how to trade off number of turns and wire diameter to match the intended Vs. It is useful for the motor-drive designer to be aware of motor design issues so that both motor and drive designer (and mechanical/thermal designer too) have an overall understanding of the motion system, so that each can optimize a part of it with system-level optimization in mind.
Induced Voltage Waveforms from Motor Geometry
The induced voltage waveform is the derivative of the flux linkage of the rotor magnets as they sweep past a stator winding. The waveshape can be determined by inspection of the rotor magnet rotation as N-S edges sweep over the area of a stator winding. As an edge moves across the face of a winding loop (or stator tooth), the flux linkage within the winding-loop area changes at a constant rate (for constant flux N and S magnet sections and speed), and the induced voltage is constant. For a full-pitch motor design, the magnets span a rotational distance equal to that of the stator winding area. As the leading magnet edge leaves the loop area, a new edge enters the beginning of the loop. If the winding area is now covered by an N-pole magnet, the flux has reached its positive (N) maximum. The new edge is followed by an S pole, causing flux to decrease linearly with rotation angle. The induced voltage changes polarity, as the derivative of the flux, to a constant negative value. A full-pitch motor consequently produces a square-wave induced-voltage waveform in a winding, as shown below.

Rotor Position Sensing
To maintain field orientation, the position of the rotor field vector must be known so that the stator vector can be oriented 90 el ahead of it. One common method of position sensing is to place Hall-effect devices (HEDs) on a circuit-board mounted to the end of the motor, so that the end of the rotor magnets can switch the HEDs. Three HEDs are required for  30 el phase resolution. This is sufficient for many applications, for this amount of phase error will cause (1cos(30)) or about  13% torque ripple. Sufficient load inertia (rotational mass) will low-pass filter this vibration to an acceptable level. Sometimes other kinds of sensors than HEDs, such as optical-path sensors are used to sense position.

Rotor position can be sensed with 3 digital position sensors. When positioned for 120 el spacings, the following waveforms will be generated by a rotating rotor.
A more elegant approach to rotor position sensing is to use the built-in electrical-position variable, the induced voltage. A winding that is undriven during part of its cycle can be sensed for induced voltage. The advantage of sensors is that they indicate phase even when the rotor is stopped, thereby eliminating the need for a start-up scheme.
Winding Configurations
For three-phase motors, the three phase windings can be connected in two ways, wye, Y (or T) and delta,  (or  ). They are dual configurations. A Y configuration is shown below, without winding inductances. Drive voltage, v, is applied across terminals Y and Z. The induced voltage at terminal Y with respect to Z is Y  Z, or YZ. The X terminal is open. For field-oriented phase control, the phase of v must be that of the induced-voltage vector YZ.

For symmetrical phase-windings, a Y circuit with phase-winding impedance, Z, is equivalent to a delta circuit with phase impedance of 3 Z and a 30 el phase shift. When the rotor is spun, the induced voltage in the phase-windings produces three sinusoids spaced at 120 el, represented as vectors below.

The X, Y, and Z vectors rotate together with the rotor. If one of the phase-windings has inverted polarity, the vectors will be apart by 60 el instead of 120 el.
CONCLUSION
Drive and induced-voltage sensing circuits will be described, and a winding-sensed motor-drive design will be developed for a Y-configured, 3-phase PMS motor using 6-step phase control.