Synchronous and DC motors

The induction motor covers most of industry. Two other families still matter: the synchronous motor, for exact speed and high efficiency, and the DC motor, which you will meet in older plants and which explains how every modern drive controls torque.

The synchronous motor

The rotor carries its own magnetic field, from a DC-fed winding or from permanent magnets. It locks onto the rotating stator field and turns at exactly the synchronous speed, $n_s = 120 f / p$ , whatever the load, as long as the load stays below a limit.

Load makes the rotor fall behind the stator field by the torque angle δ. With terminal voltage $V_t$ , excitation EMF $E_f$ and synchronous reactance $X_s$ :

T = \frac{3 V_t E_f}{\omega_s X_s}\sin\delta

Torque is largest at δ = 90°: this is the pull-out torque. Beyond it, more angle gives less torque, the rotor falls further behind, slips a pole and loses synchronism. The motor must then be tripped.

Try it: synchronous motor under load

Load torque100 N·m

Field excitation E_f100 %

Locked to the field, lagging by δ = 29.7°.

dashed: stator field · bar: rotor (dot = north pole)

Motor
Load

Pull-out torque: 202.1 N·m
Torque angle δ: 29.7°
Margin to pull-out: 51 %
Speed: 1,500 rpm

FoundationStart here if this is new to you

Picture a dog on a lead following its owner around a circular track. With no pull, the dog walks right beside the owner. Give the dog a heavy cart and it drops back, the lead stretches at an angle, and the lead pulls harder. Too heavy a cart and the lead cannot pull hard enough any more: the dog falls behind a whole lap. That lap is "slipping a pole".

Predict first

With the Lab I motor at a 100 N·m load, you raise the field excitation from 100 % to 120 %. What happens?

The speed rises by 20 %Pull-out torque rises by 20 % and δ gets smallerNothing: excitation only changes power factor

Excitation and power factor. An under-excited synchronous motor draws lagging reactive power, like an induction motor. Over-excited, it supplies reactive power to the grid, so a large synchronous motor can correct the power factor of a whole plant. Plotting the stator current against excitation gives the classic "V-curves".

Starting. At standstill the stator field sweeps past the rotor too fast for it to follow, so the average torque is zero. Line-fed synchronous motors carry a cage (damper) winding and start as induction motors, then pull into step when the field is switched on. On a VFD the frequency simply ramps up from zero.

Today the most common synchronous machine is the permanent-magnet synchronous motor on a VFD: no rotor current, so no rotor copper loss, which is how it reaches the highest efficiency classes.

DC motors

The field (from a winding or magnets) makes a flux Φ; the armature current $I_a$ in that flux makes torque; the rotating armature makes a back-EMF:

T = k\Phi\,I_a, \qquad E = k\Phi\,\omega, \qquad V = E + I_a R_a \;\Rightarrow\; \omega = \frac{V - I_a R_a}{k\Phi}

How the field is connected decides the character of the motor:

Shunt (or separately excited): the field is fed at constant voltage, so Φ is constant. Speed falls only by the $I_a R_a$ drop: nearly constant speed. Legacy use: precision machine tools (lathes, milling machines).
Series: the field carries the armature current, so $\Phi \propto I_a$ and $T \propto I_a^2$ . Huge starting torque, but as the load falls the flux collapses and speed climbs without limit. Legacy use: traction (trains, trams) and cranes.
Compound (cumulative): both windings, with fluxes that add. Strong starting torque, speed regulation close to a shunt motor, and no runaway at no load. Legacy use: stamping presses, shears, large reciprocating pumps.

Try it: shunt, series and compound DC motors

Load torque29 N·m (102 %)

Shunt
Series
Compound

All three run at a safe speed.

Shunt: 1,498 rpm · 20.3 A
Series: 1,460 rpm · 20.2 A
Compound: 1,466 rpm · 20.3 A

The example motor is the Lab I DC motor: 240 V, $R_a = 0.8$ Ω (from the booklet). The booklet leaves the field constants to you; this site assumes a rated point of 20 A at 1500 rpm, a 0.2 Ω series field and a 70/30 compound. At rated torque (28.5 N·m) all three run near 1500 rpm. Press "Remove the load": the shunt motor speeds up by about 7 %, the compound motor to about 2200 rpm, and the series motor races towards 6000 rpm.

Speed control. Below base speed, vary the armature voltage at full flux (constant torque). Above base speed, weaken the field: speed rises, available torque falls as $1/\omega$ (constant power). A modern VFD with field-oriented control does exactly this to an induction motor, which is why the DC motor is still the model every drive engineer thinks with.

Servo, stepper and crane-duty motors

Three more names you will meet. They are defined by how the motor is built and used, not by a new principle.

Servomotor. Not a separate kind of motor but a closed loop: a low-inertia motor (permanent-magnet synchronous, brushless DC or induction), a high-resolution encoder or resolver, and a servo drive that cancels the position error at every instant. Very precise and very fast: CNC axes, robots, packaging machines.
Stepper motor. Moves in fixed steps, usually open loop, as the stator phases are energised one after another. With $N_p$ rotor poles and $N_{ph}$ stator phases the step angle is $\beta = 360° / (N_p N_{ph})$ : 50 rotor poles and 4 phases give 1.8°, 200 steps a turn. Cheap and accurate without a sensor, but its torque falls off with speed: 3D printers, medical equipment, indexing tables.
Crane-duty motor. A wound-rotor or special cage motor built for lifting: starting torque of 200–300 % of rated, an intermittent rating (S3 to S5), an oversized shaft, a totally enclosed fan-cooled frame and an integrated fail-safe brake (Chapter 4).

	Servo system	Stepper motor	Standard cage motor
Control loop	closed	open	open (unless on a VFD with encoder)
Positioning	very accurate	accurate, errors do not accumulate	not for positioning
Torque	high over a wide speed range	high at low speed, falls off	peaks near synchronous speed
Typical use	robots, CNC	3D printers, indexing	pumps, fans, conveyors

Which one, when

	Induction (cage)	Synchronous	DC
Speed	slightly below $n_s$ , falls with load	exactly $n_s$	set by armature voltage
Maintenance	very low	low (brushless) or slip rings	brushes and commutator
Efficiency	good (IE3, IE4)	best (PM: IE4, IE5)	moderate
Typical use	pumps, fans, conveyors, hoists: the default	large compressors, PF correction, PM servo and traction drives	legacy: machine tools (shunt), traction and cranes (series), presses and shears (compound); PM and brushless DC servos today

ExplorerGo deeper: derivations and open questions

Power angle and stiffness. Around an operating angle $\delta_0$ , a small change gives $\Delta T = T_\text{max}\cos\delta_0\,\Delta\delta$ . Since δ is an electrical angle, $\delta = (p/2)\,\theta_\text{mech}$ for a $p$ -pole machine, and the rotor with its inertia $J$ behaves like a spring-mass system oscillating about $\delta_0$ at $\omega_n = \sqrt{(p/2)\,T_\text{max}\cos\delta_0 / J}$ . Why does the stiffness vanish at 90°, and what does the damper winding do?

Series motor on AC. Since $T \propto I_a^2$ , the torque keeps its sign when the current reverses. That is the "universal motor" of drills and vacuum cleaners. What limits its use at 50 Hz?