Fractional Kilowatt Motors

Double field revolving theory (DFRT)

A single-phase winding carrying alternating current produces a pulsating (alternating) flux along a fixed axis: $\phi = \phi_m\cos\omega t$. According to the double field revolving theory, this pulsating flux is equal to two rotating fluxes, each of constant magnitude $\phi_m/2$, rotating at synchronous speed $N_s = \frac{120f}{P}$ in opposite directions.

$$\phi_m\cos\omega t = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t}$$

• $\phi_f$: forward-rotating component (in the direction of the rotor).
• $\phi_b$: backward-rotating component (opposite).

 phi_f = phi_m/2 -->  (anticlockwise)
          \   /
   phi_b   \ /  resultant = phi_m cos(wt)
 = phi_m/2  X   along fixed axis (pulsating)
 (clockwise)

At every instant the vertical components cancel and the horizontal components add up, giving the pulsating flux $\phi_m\cos\omega t$.

Rotor behaviour

Each field induces rotor currents and produces its own torque. For rotor speed $N_r$ in the forward direction:

• Slip with respect to the forward field: $s_f = s = \frac{N_s - N_r}{N_s}$
• Slip with respect to the backward field: $s_b = 2 - s$

The net torque is $T = T_f - T_b$.

• At standstill ($s = 1$): $s_f = s_b = 1$, so $T_f = T_b$ and net torque is zero. The motor is not self-starting.
• If the rotor is started in either direction, the forward and backward torques differ. The rotor-resistance and reactance seen by the backward field cause its torque to fall. So $T_f > T_b$ in the direction of rotation and the motor keeps accelerating to near $N_s$.
• At $N_s$ ($s=0$): $T_f = 0$, but $T_b$ is not zero (slip 2), so the net torque is slightly negative; the motor runs at a speed a little below $N_s$.

  T
  |   Tf              net T = Tf - Tb
  |   .--.            _.-'-._
  |  /    \          /       \
  +-/------\--------/---------\---> N
  -Ns  Tb   0      (start)     Ns

The net torque curve passes through zero at standstill and is positive at positive speeds (and the reverse for negative speeds).

Universal motor

A universal motor is a series-wound motor (commutator type) designed to run on both ac and dc supplies, giving similar performance on both. It is a small motor, up to about 750 W, and has high speed (3000 to 20000 rpm).

Construction

• Stator: laminated (to reduce eddy currents on ac) salient-pole field with the field winding connected in series with the armature.
• Armature: laminated slotted rotor with a winding connected to a commutator, and carbon brushes.
• Large motors may have a compensating winding to reduce armature reaction and commutation problems on ac.

 Supply ---[Field]---[Armature]--- Supply
              (series connection)

Operation

1. On dc: the field flux $\phi \propto I$, and the torque $T \propto \phi I_a \propto I^2$, as in a dc series motor.
2. On ac: the current and flux reverse at the same time, so the product $\phi I_a$ remains unidirectional. The torque $\propto I^2$ is always in the same direction but pulsates at twice the supply frequency, so the motor rotates continuously.
3. Speed varies strongly with load: high speed at light load (can reach dangerous speed at no load) and low speed at heavy load. Speed can be controlled by changing the voltage, using a regulator or a tapped field.
4. On ac the input is less efficient: the field reactance drops the voltage, the power factor is lower, and more losses occur than on dc. At the same voltage the ac speed is lower than the dc speed.

Applications

Because of high speed, high starting torque and compact size: vacuum cleaners, mixer-grinders, hair dryers, portable drills, sewing machines, food mixers, electric shavers, blenders, and power tools.

A stepper motor is a brushless motor that converts electrical pulses into discrete angular steps of shaft rotation. Each pulse moves the shaft through a fixed step angle, so position can be controlled without feedback (open loop).

Types and construction

1. Variable reluctance (VR) stepper: the stator has several salient poles with windings (phases), and the rotor is a soft-iron toothed cylinder with no winding or magnet.
2. Permanent magnet (PM) stepper: the rotor is a permanent magnet.
3. Hybrid stepper: combines both; the rotor has toothed cups with a permanent magnet between them (step angles 0.9° to 3.6°).

Operating principle (variable reluctance type)

Take a 3-phase VR motor with 6 stator poles (phases A, B, C on pole pairs) and a 4-tooth rotor.

        A
     C     B'
   B   [ R ]   C'      stator poles with phases
     A'    ...         rotor: 4 teeth

1. When phase A is energised, a magnetic flux is set up and the rotor turns so that its nearest pair of teeth aligns with the A poles (minimum reluctance position).
2. Phase A is switched off and phase B is energised: the nearest rotor teeth now align with the B poles, so the rotor steps through the step angle.
3. Then C is energised, and the sequence A-B-C repeats for continued rotation. Reversing the sequence reverses the direction.

Step angle:

$$\beta = \frac{N_s - N_r}{N_s N_r}\times360^\circ \quad\text{or}\quad \beta = \frac{360^\circ}{m N_r}$$

For $N_s = 6$, $N_r = 4$, $m=3$: $\beta = \frac{360}{3\times4} = 30^\circ$.

Speed (rpm) $= \frac{\beta\times f_{pulse}\times 60}{360}$, so the speed depends on the pulse rate.

Features and uses

Accurate positioning, no feedback, good holding torque at standstill. Used in printers, robotics, CNC machines, disk drives, and camera positioning.

Double field revolving theory

A single-phase winding carrying alternating current produces a pulsating (alternating) flux along a fixed axis: $\phi = \phi_m\cos\omega t$. According to the double field revolving theory, this pulsating flux is equal to two rotating fluxes, each of constant magnitude $\phi_m/2$, rotating at synchronous speed $N_s = \frac{120f}{P}$ in opposite directions.

$$\phi_m\cos\omega t = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t}$$

• $\phi_f$: forward-rotating component (in the direction of the rotor).
• $\phi_b$: backward-rotating component (opposite).

 phi_f = phi_m/2 -->  (anticlockwise)
          \   /
   phi_b   \ /  resultant = phi_m cos(wt)
 = phi_m/2  X   along fixed axis (pulsating)
 (clockwise)

At every instant the vertical components cancel and the horizontal components add up, giving the pulsating flux $\phi_m\cos\omega t$.

Rotor behaviour

Each field induces rotor currents and produces its own torque. For rotor speed $N_r$ in the forward direction:

• Slip with respect to the forward field: $s_f = s = \frac{N_s - N_r}{N_s}$
• Slip with respect to the backward field: $s_b = 2 - s$

The net torque is $T = T_f - T_b$.

• At standstill ($s = 1$): $s_f = s_b = 1$, so $T_f = T_b$ and net torque is zero. The motor is not self-starting.
• If the rotor is started in either direction, the forward and backward torques differ. The rotor-resistance and reactance seen by the backward field cause its torque to fall. So $T_f > T_b$ in the direction of rotation and the motor keeps accelerating to near $N_s$.
• At $N_s$ ($s=0$): $T_f = 0$, but $T_b$ is not zero (slip 2), so the net torque is slightly negative; the motor runs at a speed a little below $N_s$.

  T
  |   Tf              net T = Tf - Tb
  |   .--.            _.-'-._
  |  /    \          /       \
  +-/------\--------/---------\---> N
  -Ns  Tb   0      (start)     Ns

The net torque curve passes through zero at standstill and is positive at positive speeds (and the reverse for negative speeds).

Making a single-phase induction motor self-starting

Since a single-phase winding gives no starting torque, a rotating field must be created at starting by adding an auxiliary (starting) winding displaced $90^\circ$ electrically from the main winding, with a phase difference between the two currents. The methods are:

1. Resistance split-phase: the auxiliary winding has higher resistance and lower reactance than the main winding. The currents differ in phase by about $30^\circ$, giving a weak rotating field and a moderate starting torque. A centrifugal switch cuts the auxiliary winding out at about 75 % speed.
2. Capacitor-start: a capacitor in series with the auxiliary winding gives a phase difference near $90^\circ$ and high starting torque (3 to 4.5 times full load).
3. Capacitor-start, capacitor-run: the capacitor stays in the circuit (a smaller one for running), giving good torque and power factor.
4. Permanent split capacitor: the same capacitor is used for starting and running. Low starting torque, but smooth and quiet.
5. Shaded pole: copper rings around part of each pole delay flux in the shaded part, giving a small rotating field effect. Used for very small motors (fans, toys).

 Main winding ------+
                    +---- supply
 Aux. winding --[C]-+-(centrifugal switch)

Why a single-phase induction motor is not self-starting

When single-phase ac is connected to the stator winding, it produces a pulsating (alternating) magnetic field along a fixed axis, not a rotating field. The field induces currents in the stationary rotor, but the torque produced on one half of the rotor is equal and opposite to the other half, so the net starting torque is zero. The rotor stays at rest, although if it is turned by hand in either direction it will accelerate in that direction.

Double field revolving theory

The single-phase winding carrying alternating current produces a pulsating (alternating) flux along a fixed axis: $\phi = \phi_m\cos\omega t$. According to the double field revolving theory, this pulsating flux is equal to two rotating fluxes, each of constant magnitude $\phi_m/2$, rotating at synchronous speed $N_s = \frac{120f}{P}$ in opposite directions.

$$\phi_m\cos\omega t = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t}$$

• $\phi_f$: forward-rotating component (in the direction of the rotor).
• $\phi_b$: backward-rotating component (opposite).

 phi_f = phi_m/2 -->  (anticlockwise)
          \   /
   phi_b   \ /  resultant = phi_m cos(wt)
 = phi_m/2  X   along fixed axis (pulsating)
 (clockwise)

At every instant the vertical components cancel and the horizontal components add up, giving the pulsating flux $\phi_m\cos\omega t$.

Rotor behaviour

• Forward slip $s_f = s$ and backward slip $s_b = 2 - s$, so torque $T = T_f - T_b$.
• At standstill $T_f = T_b$, so net torque is zero (not self-starting).
• Once the rotor is started in a direction, $T_f > T_b$ in that direction and it accelerates to a speed slightly less than $N_s$.

Capacitor-start motor

It has two stator windings placed $90^\circ$ apart in space: a main (running) winding of thick wire and an auxiliary (starting) winding of thinner wire in series with a capacitor ($C$ about 200 to 600 $\mu$F, electrolytic) and a centrifugal switch.

• The capacitor makes the auxiliary current lead the main current by nearly $90^\circ$. The two windings together give a good two-phase rotating field.
• Starting torque is high: 3 to 4.5 times the full-load torque, with low starting current.
• At about 75 to 80 % of synchronous speed the centrifugal switch opens, removing the auxiliary winding and capacitor. The motor then runs as a single-phase motor on the main winding.
• Applications: compressors, refrigerators, pumps, washing machines, air-conditioners, conveyors.

Capacitor-run motor

Capacitor-run motors keep the auxiliary winding and a capacitor (oil-filled paper, 2 to 20 $\mu$F) permanently in the circuit; there is no centrifugal switch.

• The motor behaves like an unbalanced two-phase motor, so it has smooth running, higher power factor and efficiency, and quieter operation.
• Starting torque is low (50 to 100 % of full load), and the cost is low.
• Applications: ceiling fans, blowers, heaters, small pumps, and office equipment.

A capacitor-start, capacitor-run motor uses both: a large starting capacitor with a switch and a small running capacitor, for high starting torque and good running performance.

 Capacitor start:   Main ---------+
                    Aux--[C]--[S]-+-- supply
 Capacitor run:     Aux--[C]------+ (no switch)

A servo motor is a small, high-performance motor used in a closed-loop control system for accurate control of position, speed or torque. It is fast-responding and its shaft position follows a control signal. Two main types are the AC (two-phase) servo and the DC servo motor.

Construction (two-phase AC servo motor)

• Stator: a laminated stator carries two windings placed $90^\circ$ apart in space: the reference (fixed) winding, fed from a constant ac source (often through a capacitor), and the control winding, fed with the variable ac error signal from the servo amplifier.
• Rotor: a squirrel-cage rotor (or drag-cup rotor) of high resistance ($R/X$ large), with small diameter and long length for low inertia and fast response.
• A tacho-generator and a position sensor (potentiometer or encoder) are fitted on the shaft for feedback.

 Ref. -> [Controller] -> error -> [Amplifier]
 input       ^                       |
 (set)       | feedback              v
        [Sensor] <--- shaft <--- [Servo motor] --> load

Working

1. The reference winding gets a fixed voltage $V_r$. The control winding gets the amplified error voltage $V_c$ (difference between desired and actual position), shifted $90^\circ$ in time from $V_r$.
2. The two currents with $90^\circ$ space and time displacement produce a rotating magnetic field in the air gap, which drags the rotor.
3. The torque and speed depend on the magnitude of the control voltage. A larger error gives a larger torque and speed. The direction of rotation depends on the phase of $V_c$ relative to $V_r$ ($\pm90^\circ$).
4. When the shaft reaches the desired position, the error becomes zero, $V_c = 0$, and the motor stops.

The high rotor resistance gives a negative-slope torque-speed characteristic over the full range, which provides damping and stability and prevents single-phase running when the control signal is removed.

Applications

Robotics, machine tools, radar and antenna positioning, aircraft control surfaces, and printers.

Why a single-phase induction motor is not self-starting

A single-phase supply to the stator winding produces only a pulsating field (equal to two equal rotating fields of $\phi_m/2$ turning in opposite directions). At standstill, the torques due to the forward and backward fields are equal and opposite, so the net starting torque is zero. A rotating field is needed at start, which is obtained by using an auxiliary winding with a phase shift.

Starting methods

1. Split-phase (resistance start) motor

• The stator has a main winding (low resistance, high reactance) and a starting winding (thin wire, high resistance, low reactance), placed $90^\circ$ apart in space.
• The currents $I_m$ and $I_a$ differ in phase by about $30^\circ$ to $40^\circ$, giving an elliptical rotating field and a moderate starting torque (1.5 to 2 times full load).
• A centrifugal switch disconnects the starting winding at about 75 % of synchronous speed.
• Use: fans, blowers, washing machines, grinders.

 Main (R low, X high) ----+
                          +-- supply
 Aux (R high, X low)--[S]-+
 Im lags V by large angle; Ia lags by small angle

2. Capacitor-start motor

• A capacitor is connected in series with the starting winding, giving $I_a$ leading $I_m$ by nearly $90^\circ$.
• A near-circular rotating field is produced, and the starting torque is high (3 to 4.5 times full load) with low starting current.
• The centrifugal switch cuts out the capacitor and the auxiliary winding at about 75 % speed.
• Use: compressors, refrigerators, pumps, air conditioners.

 Main -------------------+
                         +-- supply
 Aux --[Capacitor]--[S]--+

Shaded-pole and permanent-split capacitor motors are other options for small loads.

A single-phase induction motor is not self-starting because a single-phase supply gives only a pulsating magnetic field and no rotating field.

1. The stator winding carries an alternating current, so the flux $\phi = \phi_m\cos\omega t$ only alternates in magnitude along a fixed axis; its axis does not rotate.
2. This flux can be treated as two equal rotating fluxes of $\phi_m/2$, rotating at $N_s$ in opposite directions (double field revolving theory).
3. At standstill the slip is 1 for both fields. The forward and backward fields induce equal rotor currents and produce equal and opposite torques:

$$T_{start} = T_f - T_b = 0$$

4. With zero net torque the rotor cannot start. If it is given a push in either direction, the torque in that direction becomes greater than the opposing torque, and the motor accelerates and keeps running.

So a starting arrangement (auxiliary winding with phase shift, capacitor, or shaded poles) is needed to produce a rotating field at start.

Working principle of a single-phase induction motor

A single-phase induction motor has a stator with a distributed single-phase main winding and a squirrel-cage rotor.

1. The ac supply in the stator winding produces an alternating (pulsating) flux $\phi_m\cos\omega t$ along a fixed axis. By the double field revolving theory this is equal to two rotating fields of $\phi_m/2$, rotating at synchronous speed $N_s = \frac{120f}{P}$ in opposite directions.
2. The field induces emfs and currents in the closed rotor bars (transformer action). At standstill, the forward and backward fields produce equal and opposite torques, so the starting torque is zero. A starting aid (auxiliary winding, capacitor or shaded pole) gives a rotating field at starting.
3. Once the rotor rotates in some direction at slip $s$, the forward field has slip $s$ and the backward field has slip $2-s$. The rotor reactance to the backward field is high and its current is more reactive, and so the backward torque is small. The net torque $T_f - T_b$ is positive in the direction of rotation.
4. The rotor accelerates to near synchronous speed (slip 3 to 8 %). Equivalent explanation (cross-field theory): the rotor emf due to rotation produces a quadrature flux that, with the main flux, gives a rotating field.

 phi (pulsating)  =  phi_f (CCW) + phi_b (CW)
 Rotor at standstill: Tf = Tb -> T = 0
 Rotor running: Tf > Tb -> T > 0

Effect of air gap on the magnetic circuit

• The field in an induction motor is established by the stator alone. The mmf required is mostly spent on the air gap, so the magnetising current increases with air-gap length (mmf $= H_g l_g$ and $H_g = B/\mu_0$).
• A large gap gives a large magnetising current, and so a low power factor, lower efficiency, and greater copper loss in the stator.
• A large gap also reduces leakage reactance but increases the stator current and noise.
• Therefore, the air gap is kept as small as mechanically possible (typically 0.3 to 0.5 mm in small motors) to get a low magnetising current and good power factor. Very small gaps, however, increase the harmonic (tooth) losses, noise and the risk of rubbing.

True.

Justification: In a single-phase induction motor (split-phase type), the auxiliary (starting) winding is constructed differently from the main winding, because a phase difference is required between their currents to create a rotating field at start.

• Main winding: thick wire, more turns, low resistance, high reactance; placed in the bottom of the slots. The current lags the voltage by a large angle.
• Auxiliary winding: thin wire, fewer turns, high resistance and low reactance (or in series with a capacitor); placed in the upper part of the slots, $90^\circ$ electrical away from the main winding.
• Because of the difference in $R/X$ ratio, the auxiliary current $I_a$ lags the supply voltage by a smaller angle than the main current $I_m$. The phase difference (30° to 40°, or nearly 90° with a capacitor) produces a rotating field and the starting torque.
• The auxiliary winding is thin and designed for short-time duty, so it is cut out by a centrifugal switch at about 75 % speed to avoid overheating.

Stepper motor

A stepper motor is a brushless motor that rotates in fixed angular steps for each input pulse. A drive circuit energises the stator phases in sequence, and the rotor turns to the minimum-reluctance (or magnetic alignment) position of each energised phase.

• Phases A, B, C are energised in order. The rotor moves through the step angle at each switching:

$$\beta = \frac{360^\circ}{m\,N_r}$$

($m$ = number of phases, $N_r$ = number of rotor teeth; for $m=3$, $N_r=4$, $\beta = 30^\circ$).

• Total rotation $=$ number of pulses $\times\ \beta$; speed (rpm) $= \frac{\beta f_p}{6}$ for pulse frequency $f_p$ in Hz.
• Reversing the sequence reverses direction. It works in open loop and holds position when a phase is kept energised.

 Pulse train -> [Driver / sequencer] -> A,B,C phases
                                          |
                                      [Stepper] -> step by step

Use: printers, plotters, CNC, robots.

Servo motor

A servo motor is a motor used in a closed-loop system. The shaft position (or speed) is compared with the reference by a sensor; the difference (error) is amplified and fed to the control winding (AC servo: two-phase induction motor with high-resistance rotor, or DC servo: armature or field controlled).

• The torque is proportional to the control voltage. As the error reduces, the control voltage falls and the motor stops when the position matches the reference.
• The direction depends on the polarity or phase of the control voltage.
• Rotor has small diameter and long length, giving low inertia and fast response.

 Ref -> (+) -> [Amplifier] -> [Servo motor] -> load
         ^ -                                   |
         +------- [Sensor / tacho] <-----------+

A servo motor is used in control systems that need fast response to the error signal, such as starting, stopping and reversing in a very short time. This requires a high torque-to-inertia ratio, i.e. a small mechanical time constant.

1. Torque produced is proportional to the volume of the rotor: $T \propto D^2 L$ (D = diameter, L = length). It can be obtained from a long rotor.
2. Moment of inertia of a rotor is $J \propto D^4 L$. The inertia depends on the fourth power of the diameter and only linearly on the length.
3. So, reducing the diameter lowers the inertia drastically, while the torque is kept by increasing the length. The torque-to-inertia ratio $\propto \frac{1}{D^2}$ increases for small $D$.
4. A low inertia rotor accelerates and decelerates quickly, so the motor follows the control signal accurately with little overshoot. The small diameter also gives low rotor energy storage, which makes quick reversal possible.

Hence the servo motor is built long and thin, unlike normal motors, which are designed for efficiency and economy rather than speed of response.