Fractional Kilowatt Motors

Double field revolving theory states that a pulsating (alternating) magnetic field of maximum value $\phi_m$ can be replaced by two fields, each of constant magnitude $\phi_m/2$, rotating in opposite directions at synchronous speed $N_s = 120f/P$.

Pulsating field resolved

The single-phase stator winding produces a flux that only alternates along its axis:

$$\phi = \phi_m \cos\omega t$$

This is equal to the sum of two rotating phasors $\phi_f$ (forward) and $\phi_b$ (backward), each $\phi_m/2$, turning at $\omega$ in opposite directions:

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

 wt = 0      wt = 90 deg     wt = 180 deg
  ^ ^          <-- -->          v v
  | |  phi_f  phi_b               | |
  sum = phi_m  sum = 0       sum = -phi_m
 (both along  (cancel)       (both reversed)
  the axis)
 phi_f turns anticlockwise, phi_b clockwise

At every instant the vertical components add and the horizontal components cancel, so the resultant is the pulsating field.

Torque at standstill - not self-starting

• At standstill each rotating field induces equal rotor currents (slip = 1 for both), so the forward torque $T_f$ and backward torque $T_b$ are equal and opposite.
• Net starting torque $T = T_f - T_b = 0$. Hence a single-phase induction motor is not self-starting.

Torque when running

If the rotor turns at speed $N$ in the forward direction:

$$s_f = \frac{N_s - N}{N_s} = s, \qquad s_b = \frac{N_s + N}{N_s} = 2 - s$$

Using the induction motor torque expression for each field:

$$T_f = \frac{K s R_2}{R_2^2 + (sX_2)^2}, \qquad T_b = -\frac{K (2-s) R_2}{R_2^2 + \left((2-s)X_2\right)^2}$$ $$T = T_f + T_b$$

• Near normal speed $s$ is small, $2 - s \approx 2$: the backward rotor current sees a large frequency $(2-s)f$ and high reactance, so $T_b$ is small and $T_f \gg |T_b|$. The motor keeps running in whichever direction it was started.

        T
        |            _
        |         .-' \   forward run
        |      .-'     |  (Tf > Tb)
 -Ns ---+----.'--------+---> speed
     |  |  .' 0        Ns
      \_|.'
 reverse run      net T = 0 at N = 0
 (mirror image)

The resultant torque-speed curve passes through zero at standstill and is symmetrical: positive in the forward direction, negative in the reverse direction.

Equivalent circuit (based on DFRT)

The rotor is split into two halves, one for each field: forward half with $0.5R_2'/s$ and $0.5X_2'$, backward half with $0.5R_2'/(2-s)$ and $0.5X_2'$, each with $0.5X_m$ in parallel, in series with the stator $R_1 + jX_1$.

 o--R1--X1--+--0.5X2'--0.5R2'/s ------+
 V          |  (0.5Xm parallel)  forward
            +--0.5X2'--0.5R2'/(2-s)---+
            |  (0.5Xm parallel)  backward
 o----------+-------------------------+

Hence, an auxiliary means (split phase, capacitor, shaded pole) is needed only to start the motor; once running, the forward field dominates.

A reluctance motor is a single-phase synchronous motor whose rotor has no DC excitation; it runs at synchronous speed because the rotor tends to align itself in the position of minimum reluctance to the stator field.

Construction

• Stator: the same as a single-phase induction motor - a main winding and an auxiliary winding (split-phase or capacitor type) to produce a rotating field.
• Rotor: a modified squirrel-cage rotor. Some teeth are removed at symmetrical points to form salient poles; the number of salient poles equals the number of stator poles. The cage bars and end rings remain, so it can start as an induction motor.

        ___________
      /  __     __  \     removed teeth (high
     |  |  |   |  |  |    reluctance gaps)
     |  |__|   |__|  |    remaining parts act as
      \_____________/     salient poles
   rotor with cage bars and cut-outs

Principle of operation

1. Starting: the stator rotating field induces currents in the cage bars, so the motor starts and accelerates as an induction motor.
2. Pull-in: near synchronous speed (about 75-80 %), the reluctance torque acts. A piece of magnetic material in a field experiences a force that moves it to the position of minimum reluctance (shortest air path). The salient poles snap into alignment with the stator field poles.
3. Synchronous running: the rotor locks with the rotating field and runs at $N_s = 120f/P$. The auxiliary winding may be disconnected by a centrifugal switch.
4. The reluctance torque is $T \propto \left(\frac{1}{X_q} - \frac{1}{X_d}\right)\sin 2\delta$; it is zero if $X_d = X_q$, which is why saliency is essential.

Features

• Constant speed, no slip rings, brushes or DC supply.
• Low power factor and efficiency; output is about 1/3 of an induction motor of the same frame size.
• Torque-speed curve shows induction motor region up to pull-in, then a vertical line at $N_s$.

Applications

Electric clocks, timers, signalling devices, recording instruments, teleprinters and other constant-speed drives.

An AC servo motor is a two-phase induction motor designed to give torque proportional to a control voltage, used in feedback control systems for accurate position and speed control.

Construction

• Stator: two windings displaced by $90^\circ$ electrical:
  • Reference (fixed) winding, supplied with a constant AC voltage $V_r$.
  • Control winding, supplied from a servo amplifier with variable voltage $V_c$, $90^\circ$ out of phase with $V_r$.
• Rotor: squirrel cage (or drag-cup) with high resistance and small diameter, long length, giving low inertia and fast response.

   Vr (fixed) --> [Reference wdg]
                       |  90 deg
   Vc (from   --> [Control wdg] --> rotor --> load
   amplifier)

Working

• The two quadrature fluxes produce a rotating field; torque is proportional to $V_c$.
• When $V_c = 0$ the field is only pulsating, and due to high rotor resistance the motor stops (no single-phasing run).
• Reversing the phase of $V_c$ reverses rotation.
• High $R_2/X_2$ gives an almost linear torque-speed curve with negative slope, giving positive damping and stability.

 T |\
   | \  Vc = rated
   |\ \
   | \ \  Vc = 0.5 rated
   +--\-\---- speed

Applications

Position control in radar and antenna drives, instrument servos, X-Y recorders, robotics and computer peripherals.

For the two winding currents to be in quadrature, the starting-winding current must lead the main-winding current by $90^\circ$. The capacitor makes the starting-winding branch capacitive.

Given: $Z_m = 4.5 + j3.7\ \Omega$, $Z_s = 9.5 + j3.5\ \Omega$, $f = 50$ Hz. (The rating, 250 W or "250 kW", does not affect the result.)

Angle of main winding current

$$\theta_m = \tan^{-1}\frac{3.7}{4.5} = 39.43^\circ \quad (I_m \text{ lags } V \text{ by } 39.43^\circ)$$

Required angle of starting branch

$I_s$ must lead $I_m$ by $90^\circ$, so it must lead $V$ by $90^\circ - 39.43^\circ = 50.57^\circ$. The impedance angle of the starting branch must be $-50.57^\circ$:

$$\begin{aligned} \tan(-50.57^\circ) &= \frac{X_s - X_C}{R_s} = \frac{3.5 - X_C}{9.5} \\ 3.5 - X_C &= 9.5 \times (-1.2162) = -11.554 \\ X_C &= 3.5 + 11.554 = 15.054\ \Omega \end{aligned}$$

Capacitance

$$C = \frac{1}{2\pi fX_C} = \frac{1}{2\pi \times 50 \times 15.054} = 211.4\ \mu\text{F}$$

           Is (leads V by 50.6 deg)
          /
         /  90 deg between Is and Im
  ------+--------------------> V
         \
          \ Im (lags V by 39.4 deg)

Check: angle between currents $= 50.57^\circ + 39.43^\circ = 90^\circ$.

Answer: $X_C = 15.05\ \Omega$, so a capacitor of about 211 µF must be connected in series with the starting winding.

A single-phase induction motor is not self-starting because its single winding produces a pulsating field, not a rotating one, so the net starting torque is zero.

Why it is not self-starting

By double field revolving theory, the pulsating flux $\phi_m\cos\omega t$ equals two fields of $\phi_m/2$ rotating in opposite directions at $N_s$. At standstill both have slip 1 and produce equal and opposite torques: $T_f = T_b$, so $T_{net} = 0$. If the rotor is pushed in either direction, the torque in that direction becomes larger and the motor keeps running.

Starting methods

To start, a rotating field must be produced, usually by phase splitting:

1. Resistance split-phase motor
2. Capacitor-start motor
3. Capacitor-start capacitor-run (two-value capacitor) motor
4. Permanent split capacitor (PSC) motor
5. Shaded-pole motor
6. Repulsion-start induction-run motor

1. Resistance split-phase method

 AC o---+--------------+
        |              |
   Main wdg (low R,  Aux wdg (high R,
   high X)           low X)
        |              |
        |          centrifugal S
 o------+--------------+

• An auxiliary winding with high resistance (thin wire) and low reactance is placed $90^\circ$ electrical from the main winding.
• Main current $I_m$ lags $V$ by a large angle; auxiliary current $I_a$ lags by a smaller angle, giving a phase difference of about $25-30^\circ$.
• These two currents produce a rotating field and starting torque $T_{st} \propto I_mI_a\sin\alpha$ (moderate, about 1.5-2 times full load).
• A centrifugal switch disconnects the auxiliary winding at about 75 % of $N_s$.
• Used for fans, blowers, washing machines.

2. Capacitor-start method

 AC o---+--------------+
        |              |
     Main wdg       Aux wdg
        |              |
        |          C (electrolytic)
        |              |
        |          centrifugal S
 o------+--------------+

• A capacitor in series with the auxiliary winding makes $I_a$ lead $V$, so the angle between $I_m$ and $I_a$ is close to $90^\circ$.
• Starting torque is high (3-4.5 times full-load torque) with lower starting current.
• The auxiliary winding and capacitor are cut out by the centrifugal switch at 70-80 % of $N_s$.
• Used for compressors, pumps, refrigerators and air-conditioners.

A hysteresis motor is a single-phase synchronous motor whose rotor is a smooth cylinder of hard magnetic material; it develops torque due to hysteresis in the rotor, and runs at synchronous speed silently.

Construction

• Stator: split-phase, capacitor or shaded-pole winding that produces a rotating field.
• Rotor: a smooth cylinder of high-retentivity hard steel (e.g. cobalt or chrome steel) with large hysteresis loop, mounted on a non-magnetic arbour. No teeth, no windings.

   stator (rotating field)
   _____________________
  |  ________________   |
  | | hard steel ring|  |
  | |  (no slots)    |  |
  | |_____arbour_____|  |
  |_____________________|

Working principle

1. The rotating stator field magnetises the rotor. Because of hysteresis, rotor magnetisation lags the stator field by a hysteresis angle $\alpha$; this lag produces a torque (hysteresis torque) that is constant from standstill to synchronous speed.
2. Eddy currents in the rotor also add torque below synchronous speed.
3. At synchronous speed, the rotor becomes permanently magnetised and runs locked to the field as a permanent-magnet synchronous motor.

Features and applications

• Smooth, quiet, vibration-free; constant torque during run-up; can synchronise any load it can accelerate.
• Used in electric clocks, tape recorders, record players and timing devices.

A stepper (stepping) motor is a brushless motor whose shaft rotates in discrete angular steps, one step for each input pulse, so position is controlled without feedback (open loop).

Construction and types

• Stator has salient poles with windings grouped into phases, energised in sequence by a digital driver.
• Rotor types: variable reluctance (soft iron, toothed), permanent magnet, and hybrid (PM + toothed).

Working

• When a phase is energised, the rotor turns to the position of minimum reluctance (or aligns its magnet) with that phase.
• Switching the next phase moves the rotor by one step angle:

$$\beta = \frac{360^\circ}{m N_r} \quad (m = \text{phases},\ N_r = \text{rotor teeth})$$

• Speed is proportional to pulse rate; direction depends on the switching sequence.

 pulses --> [Logic/driver] --> phases A,B,C --> motor
             A -> B -> C -> A  : clockwise steps

Advantages and applications

• Precise positioning, no cumulative error, digital compatibility, holding torque at rest.
• Used in printers, plotters, disk drives, CNC machines, robots and camera lenses.

A single-stack variable-reluctance (VR) stepper motor has one stator and one rotor stack; the rotor moves in fixed steps because it always aligns itself to the position of minimum reluctance with the energised stator phase.

Construction

• Stator: laminated, with salient poles; each pair of opposite poles carries a phase winding (e.g. 3 phases A, B, C on 6 poles).
• Rotor: laminated soft iron with teeth, no winding. The number of rotor teeth $N_r$ differs from stator poles $N_s$ (e.g. 6 stator poles, 4 rotor teeth).

          A
       ___|___
   C' /   __   \ B
     |  [rotor] |      6 stator poles (A,B,C,
   B  \___  ___/  C'   A',B',C'), 4 rotor teeth
          |
          A'

Operating principle (3-phase, 6/4 motor)

1. Phase A energised: flux flows A-A'. The rotor turns so two of its teeth align with poles A and A' (minimum reluctance). Rotor is held there.
2. Phase A off, B on: the nearest rotor teeth are pulled into line with B-B'. The rotor turns by one step.
3. B off, C on: another step in the same direction.
4. Repeating A-B-C-A gives continuous stepping clockwise; the sequence A-C-B-A gives anticlockwise rotation.

Step angle:

$$\beta = \frac{N_s - N_r}{N_s N_r} \times 360^\circ = \frac{6 - 4}{6 \times 4} \times 360^\circ = 30^\circ$$

(equivalently $\beta = 360^\circ/(mN_r) = 360^\circ/(3 \times 4) = 30^\circ$).

Modes and features

• Half-stepping (A, AB, B, BC, ...) halves the step angle to $15^\circ$.
• Speed $= \dfrac{\beta \times f}{360^\circ} \times 60$ rpm, where $f$ = pulses per second.
• No permanent magnet, so there is no detent torque when unenergised; rotor inertia is low, so response is fast.

Applications

Printers, plotters, X-Y tables, machine tools, floppy disk drives and robotics.

A capacitor-start motor is a split-phase single-phase induction motor in which a capacitor is placed in series with the auxiliary (starting) winding so that the two winding currents are nearly $90^\circ$ apart at starting, giving high starting torque.

Construction

 AC o----+-----------------+
         |                 |
      Main wdg          Aux (start) wdg
      (Rm, Xm)             |
         |               C (electrolytic,
         |                 |  70-300 uF)
         |          centrifugal switch S
 o-------+-----------------+
       rotor: squirrel cage

• Main winding and auxiliary winding are placed $90^\circ$ electrical apart in the stator.
• An electrolytic capacitor $C$ and a centrifugal switch S are in series with the auxiliary winding.
• Rotor is a normal squirrel cage.

Operating principle

1. At start, the main winding current $I_m$ lags $V$ by about $40-60^\circ$ (inductive winding).
2. The capacitor makes the auxiliary current $I_a$ lead $V$. Choosing $C$ properly makes the angle $\alpha$ between $I_m$ and $I_a$ about $80-90^\circ$.

       Ia (leads V)
        \
         \  alpha ~ 90 deg
 ---------+--------------> V
           \
            \ Im (lags V)

3. Two windings displaced in space by $90^\circ$ carrying currents displaced in time by about $90^\circ$ produce an almost uniform rotating magnetic field, as in a two-phase motor.
4. The rotating field induces rotor currents and develops starting torque $T_{st} = K I_m I_a \sin\alpha$, which is large (3-4.5 times full-load torque) since $\sin\alpha \approx 1$.
5. At about 70-80 % of synchronous speed, the centrifugal switch opens and disconnects the capacitor and auxiliary winding. The motor then runs on the main winding alone (by double field revolving theory the forward field dominates).

Characteristics and uses

• High starting torque with lower starting current than resistance split-phase.
• Reversible by reversing the auxiliary winding connections.
• Used for compressors, refrigerators, air-conditioners, pumps, conveyors and machine tools.

A capacitor-start capacitor-run (two-value capacitor) motor uses two capacitors with the auxiliary winding: a large one for high starting torque and a small one that remains in circuit while running, so the motor works as a balanced two-phase motor at all times.

Construction

 AC o----+------------------------+
         |                        |
      Main wdg                 Aux wdg
         |                        |
         |                 +------+------+
         |                 |             |
         |               Cs (start,    Cr (run,
         |               electrolytic) oil-paper)
         |                 |             |
         |               switch S        |
         |                 |             |
 o-------+-----------------+-------------+

• Main and auxiliary windings $90^\circ$ apart in space; the auxiliary winding is permanently connected.
• Starting capacitor $C_s$ (large, electrolytic, about 10-15 times $C_r$) in series with a centrifugal switch.
• Running capacitor $C_r$ (small, oil-filled/paper, continuous rating) always in circuit.

Operating principle

1. At start: $C_s$ and $C_r$ are in parallel, giving a large capacitance. The auxiliary current leads $V$ so that $I_a$ and $I_m$ are about $90^\circ$ apart; with equal mmfs, a strong rotating field is produced and starting torque $T = KI_mI_a\sin\alpha$ is high.
2. At about 75 % of $N_s$: the centrifugal switch removes $C_s$.
3. Running: only $C_r$ remains. Its value is chosen so that at full load the two winding currents are still nearly $90^\circ$ apart with balanced mmfs. The backward field is almost eliminated, so the field is nearly uniform and rotating, as in a balanced 2-phase motor.

 T  |\  start (Cs + Cr)
    | \___
    |     \    switch opens
    |      \__________ run (Cr only)
    |                 \
    +------------------\---- speed
                       Ns

Advantages

• Highest starting torque and good running performance of the single-phase motors.
• Better power factor and efficiency, quiet and smooth running (no double-frequency pulsating torque).
• Higher overload capacity.

Applications

Refrigerators, air-conditioners, compressors, pumps, and other loads needing high starting torque and continuous quiet operation.

Both motors are capacitor motors in which a capacitor in series with the auxiliary winding makes its current lead the main winding current, so a rotating field is produced.

Permanent split capacitor (PSC) motor

 AC o----+-------------+
         |             |
      Main wdg      Aux wdg
         |             |
         |           C (oil/paper,
         |             permanently in)
 o-------+-------------+
     no centrifugal switch

Principle:

1. One small capacitor (a few µF, continuous rating) stays in series with the auxiliary winding at all times; there is no centrifugal switch.
2. The capacitor makes $I_a$ lead $I_m$ by nearly $90^\circ$, producing a rotating field and starting torque.
3. The capacitor value is chosen for good running performance, so it is small; starting torque is therefore low (about 50-100 % of full-load torque).
4. While running, it behaves like a two-phase motor: quiet, smooth, with good power factor and efficiency. Speed can be varied by tapping the main winding or changing voltage, and direction is easily reversed.

Applications: ceiling and table fans, blowers, air-circulators, room coolers, office machines, and loads with low starting torque.

Capacitor-start capacitor-run (CSCR) motor

 AC o----+-----------------------+
         |                       |
      Main wdg                Aux wdg
         |                   +---+---+
         |                  Cs       Cr
         |                   |       |
         |                switch S   |
 o-------+-------------------+-------+

Principle:

1. Two capacitors: a large starting capacitor $C_s$ (electrolytic) with a centrifugal switch, and a small running capacitor $C_r$ permanently connected.
2. At start, $C_s \parallel C_r$ gives large capacitance and phase difference near $90^\circ$ with high auxiliary current, so starting torque is high (about 3-4 times full load).
3. At about 75 % of $N_s$ the switch removes $C_s$; $C_r$ keeps the motor working as a balanced two-phase motor for good running performance.

Applications: refrigerators, air-conditioners, compressors, pumps, conveyors.

Comparison

A servo motor is a small motor designed for use in a closed-loop control system, in which its shaft position or speed follows a command signal accurately and quickly.

Requirements

• Torque proportional to control signal, linear torque-speed characteristics.
• Low inertia (small diameter, long rotor) for quick start, stop and reversal.
• Stable operation; no running when the control signal is zero.

Types

Working (closed loop)

 Ref -->(+)--> [Amplifier] --> [Servo motor] --> Load
        (-)                          |
         ^------[Feedback sensor]<---+

• The error between the command and the feedback (from a potentiometer, encoder or tachogenerator) is amplified and applied to the motor.
• The motor turns until the error becomes zero; the direction of the error decides the direction of rotation.

Applications

Robots, CNC machine tools, radar and antenna positioning, aircraft control surfaces, camera auto-focus, printers and instrument servos.

Double field revolving theory states that a pulsating (alternating) magnetic field of maximum value $\phi_m$ can be replaced by two fields, each of constant magnitude $\phi_m/2$, rotating in opposite directions at synchronous speed $N_s = 120f/P$.

Pulsating field resolved

The single-phase stator winding produces a flux that only alternates along its axis:

$$\phi = \phi_m \cos\omega t$$

This is equal to the sum of two rotating phasors $\phi_f$ (forward) and $\phi_b$ (backward), each $\phi_m/2$, turning at $\omega$ in opposite directions:

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

 wt = 0      wt = 90 deg     wt = 180 deg
  ^ ^          <-- -->          v v
  | |  phi_f  phi_b               | |
  sum = phi_m  sum = 0       sum = -phi_m
 (both along  (cancel)       (both reversed)
  the axis)
 phi_f turns anticlockwise, phi_b clockwise

At every instant the vertical components add and the horizontal components cancel, so the resultant is the pulsating field.

Torque at standstill - not self-starting

• At standstill each rotating field induces equal rotor currents (slip = 1 for both), so the forward torque $T_f$ and backward torque $T_b$ are equal and opposite.
• Net starting torque $T = T_f - T_b = 0$. Hence a single-phase induction motor is not self-starting.

Torque when running

If the rotor turns at speed $N$ in the forward direction:

$$s_f = \frac{N_s - N}{N_s} = s, \qquad s_b = \frac{N_s + N}{N_s} = 2 - s$$

Using the induction motor torque expression for each field:

$$T_f = \frac{K s R_2}{R_2^2 + (sX_2)^2}, \qquad T_b = -\frac{K (2-s) R_2}{R_2^2 + \left((2-s)X_2\right)^2}$$ $$T = T_f + T_b$$

• Near normal speed $s$ is small, $2 - s \approx 2$: the backward rotor current sees a large frequency $(2-s)f$ and high reactance, so $T_b$ is small and $T_f \gg |T_b|$. The motor keeps running in whichever direction it was started.

        T
        |            _
        |         .-' \   forward run
        |      .-'     |  (Tf > Tb)
 -Ns ---+----.'--------+---> speed
     |  |  .' 0        Ns
      \_|.'
 reverse run      net T = 0 at N = 0
 (mirror image)

The resultant torque-speed curve passes through zero at standstill and is symmetrical: positive in the forward direction, negative in the reverse direction.

Equivalent circuit (based on DFRT)

The rotor is split into two halves, one for each field: forward half with $0.5R_2'/s$ and $0.5X_2'$, backward half with $0.5R_2'/(2-s)$ and $0.5X_2'$, each with $0.5X_m$ in parallel, in series with the stator $R_1 + jX_1$.

 o--R1--X1--+--0.5X2'--0.5R2'/s ------+
 V          |  (0.5Xm parallel)  forward
            +--0.5X2'--0.5R2'/(2-s)---+
            |  (0.5Xm parallel)  backward
 o----------+-------------------------+

Hence, an auxiliary means (split phase, capacitor, shaded pole) is needed only to start the motor; once running, the forward field dominates.

One starting method: capacitor-start motor

Since net starting torque is zero, a second (auxiliary) winding is placed $90^\circ$ (electrical) from the main winding, with a capacitor $C$ in series and a centrifugal switch S.

 1-ph AC o----+-----------+
              |           |
          Main wdg     Aux wdg
           (Im)           |
              |           C
              |           |
              |        S (opens at ~75% Ns)
 o------------+-----------+

1. The capacitor makes the auxiliary current $I_a$ lead $V$, while the main current $I_m$ lags $V$; the phase difference $\alpha$ approaches $90^\circ$.
2. Two space-displaced windings carrying time-displaced currents produce a rotating field, giving starting torque $T_{st} = K I_m I_a \sin\alpha$, which is high (about 3-4.5 times full-load torque).
3. At about 70-80 % of synchronous speed the centrifugal switch disconnects the auxiliary winding; the motor then runs on the main winding alone, as explained by DFRT.

It is used for compressors, pumps, refrigerators and air-conditioners.

The capacitor must make the starting-winding current lead the main-winding current by $90^\circ$. Starting torque is $T_{st} = K I_m I_s \sin\alpha$, where $\alpha$ is the angle between the two currents. The change in torque is found by comparing with the motor without the capacitor (plain resistance split-phase start), with the same supply voltage $V$.

Capacitor value

$$\theta_m = \tan^{-1}\frac{3}{3} = 45^\circ \ (I_m \text{ lags } V)$$

For $\alpha = 90^\circ$, $I_s$ must lead $V$ by $90^\circ - 45^\circ = 45^\circ$, so the starting branch angle is $-45^\circ$:

$$\begin{aligned} \frac{3 - X_C}{7.5} &= \tan(-45^\circ) = -1 \\ X_C &= 3 + 7.5 = 10.5\ \Omega \\ C &= \frac{1}{2\pi \times 50 \times 10.5} = 303.2\ \mu\text{F} \end{aligned}$$

Starting torque without capacitor

$$\begin{aligned} \theta_s &= \tan^{-1}\frac{3}{7.5} = 21.80^\circ,\quad |Z_s| = 8.0777\ \Omega \\ \alpha_1 &= 45^\circ - 21.80^\circ = 23.20^\circ \\ I_{s1} &= \frac{V}{8.0777} = 0.1238V \\ T_1 &\propto I_m \times 0.1238V \times \sin23.20^\circ = 0.04877\,V I_m \end{aligned}$$

Starting torque with capacitor

$$\begin{aligned} |Z_s'| &= |7.5 - j7.5| = 10.6066\ \Omega,\quad I_{s2} = 0.0943V \\ T_2 &\propto I_m \times 0.0943V \times \sin90^\circ = 0.09428\,V I_m \end{aligned}$$

$I_m$ is the same in both cases, so

$$\frac{T_2}{T_1} = \frac{0.09428}{0.04877} = 1.9333$$

Percentage change $= (1.9333 - 1) \times 100 = 93.33\ \%$ (increase).

        Is (with C, leads V by 45 deg)
       /
      / 90 deg
 ----+-----------------> V
     |\  Is (no C, lags 21.8 deg)
     | \
     |  Im (lags 45 deg)

Answer: $C = 303.2$ µF (series with starting winding); starting torque increases by about 93.3 % compared with the same motor started without the capacitor.

A single-phase induction motor works on electromagnetic induction like a 3-phase induction motor, but its single stator winding produces a pulsating field, which by double field revolving theory is two equal fields rotating in opposite directions.

Working principle

1. AC supply to the stator winding produces a flux $\phi = \phi_m\cos\omega t$ that alternates along one axis.
2. This flux is equivalent to a forward field and a backward field, each $\phi_m/2$, rotating at $N_s$ in opposite directions.
3. At standstill both fields induce equal rotor currents and produce equal and opposite torques, so the motor has no starting torque.
4. If the rotor is started in one direction (by an auxiliary means), the slip for the forward field is $s$ and for the backward field is $2 - s$. The forward torque becomes much larger than the backward torque, and the motor accelerates to a speed slightly below $N_s$ and continues running in that direction.

Torque-slip characteristic

        T
        |            _
        |         .-' \   forward run
        |      .-'     |  (Tf > Tb)
 -Ns ---+----.'--------+---> speed
     |  |  .' 0        Ns
      \_|.'
 reverse run      net T = 0 at N = 0
 (mirror image)

• Net torque is zero at $s = 1$ (standstill).
• Net torque is positive for forward running ($0 < s < 1$) and becomes zero slightly before $s = 0$ because the backward torque is not zero at synchronous speed.
• The curve is symmetrical for reverse rotation ($1 < s < 2$).

One starting method: resistance split-phase

 AC o---+------------+
        |            |
   Main wdg     Aux wdg (high R)
        |            |
        |      centrifugal switch
 o------+------------+

• An auxiliary winding of high resistance and low reactance is placed $90^\circ$ electrical from the main winding.
• Its current lags $V$ less than the main winding current, giving a phase difference of $25-30^\circ$, which creates a rotating field and starting torque $T \propto I_mI_a\sin\alpha$.
• At about 75 % of $N_s$ a centrifugal switch disconnects the auxiliary winding.
• Used in fans, blowers, washing machines and small tools.

A DC servo motor is a small DC motor designed for fast, accurate control of position or speed in a closed-loop (feedback) control system. It is the "actuator" that turns the error signal of a servo system into controlled motion.

Construction

• Similar to an ordinary DC motor, but with a long, small-diameter armature to keep inertia ($J$) low.
• Field is either a permanent magnet or a separately excited winding.
• Low-inertia types: slotless (smooth) armature, disc (printed-circuit) armature and shell/cup armature, which give very small electrical and mechanical time constants.

Types and working

1. Armature-controlled: field current is kept constant and the control voltage is applied to the armature. Torque $T = K_t I_a$, so torque and speed vary linearly with armature voltage. This is the most common type (larger power, good damping from back emf).
2. Field-controlled: armature current is kept constant and the control signal is applied to the field winding. Torque $T \propto \phi \propto I_f$. Used for small powers; the small field current is easy to control, but response is slower (large field inductance) and there is no back-emf damping.

 error   +-----------+  Va   +-------+   shaft
 ------->| Amplifier |------>|  DC   |----------> load
   ^     +-----------+       | servo |     |
   |                         +-------+     |
   +------- feedback (pot / encoder) <-----+

Features

• Linear torque–speed and torque–voltage characteristics
• High starting torque, quick reversal, wide speed range
• High torque-to-inertia ratio, fast response

Applications

Robotics, CNC machine tools, X–Y plotters and printers, disk drives, tracking antennas, aircraft control surfaces and position control systems.

Drawback: brushes and commutator need maintenance and produce sparking, so brushless servo drives are replacing them in many uses.

To send the same current in both windings with a 90° phase shift, the auxiliary branch impedance must have the same magnitude as the main winding impedance but an angle that is 90° less. A resistance alone cannot shift the current by 90°, so both a resistance and a capacitor are added in series with the auxiliary winding.

Given: $f = 50$ Hz, $Z_m = Z_a = 5 + j20\ \Omega$ (before adding anything).

Main winding

$$\begin{aligned} Z_m &= 5 + j20 = 20.616\angle 75.96^\circ\ \Omega \end{aligned}$$

So $I_m$ lags $V$ by $75.96^\circ$.

Required auxiliary impedance

For $|I_a| = |I_m|$ and $I_a$ leading $I_m$ by $90^\circ$:

$$\begin{aligned} \theta_a &= 75.96^\circ - 90^\circ = -14.04^\circ \\ Z_a' &= 20.616\angle -14.04^\circ \\ &= 20.616(\cos 14.04^\circ - j\sin 14.04^\circ) \\ &= 20 - j5\ \Omega \end{aligned}$$

Let $R$ and $X_C$ be added in series: $Z_a' = (5 + R) + j(20 - X_C)$.

a) Resistance to be added

$$\begin{aligned} 5 + R &= 20 \\ R &= 15\ \Omega \end{aligned}$$

b) Capacitance to be added

$$\begin{aligned} 20 - X_C &= -5 \;\Rightarrow\; X_C = 25\ \Omega \\ C &= \frac{1}{2\pi f X_C} = \frac{1}{2\pi \times 50 \times 25} \\ &= 127.3 \times 10^{-6}\ \text{F} \end{aligned}$$

Check

$|Z_a'| = \sqrt{20^2 + 5^2} = 20.616\ \Omega = |Z_m|$, so the currents are equal; angle difference $= 75.96^\circ - (-14.04^\circ) = 90^\circ$.

        I_a (leads V by 14.04 deg)
         ^
         |   /
         |  /
         | /
  -------+-----------------> V
          \
           \ 75.96 deg
            \
             v I_m   (I_a and I_m 90 deg apart)

Answer: add $R = 15\ \Omega$ and $C \approx 127.3\ \mu\text{F}$ ($X_C = 25\ \Omega$) in series with the auxiliary winding.

Double field revolving theory (DFRT)

A single-phase winding produces a pulsating field $B = B_m\cos\omega t$ along a fixed axis, not a rotating one. DFRT states that this pulsating field can be split into two fields of half amplitude ($B_m/2$ each) rotating at synchronous speed in opposite directions:

$$B_m \cos\omega t\,\cos\theta = \frac{B_m}{2}\cos(\theta - \omega t) + \frac{B_m}{2}\cos(\theta + \omega t)$$

• The forward field (FF) gives torque $T_f$ in its direction, the backward field (BF) gives $T_b$ in the opposite direction.
• At standstill both slips are 1, so $T_f = T_b$ and net torque is zero: the motor is not self-starting.
• If the rotor is pushed in either direction, the slip w.r.t. that field becomes small, $T_f > T_b$, and the motor keeps running in that direction.

 T  |   Tf
    |  /\
    | /  \___      net T = Tf - Tb
  --+------------------------------> speed
 -n |\___   /           +n
    |    \ /
    |     \/  Tb

Starting methods

1. Split-phase (resistance start) motor
2. Capacitor-start induction-run motor
3. Capacitor-start capacitor-run (two-value capacitor) motor
4. Permanent split capacitor (PSC) motor
5. Shaded-pole motor

All of them create a second field displaced in space and time so that a rotating field exists at start.

Capacitor-start capacitor-run (CSCR) motor

Construction: a main winding and an auxiliary winding displaced by $90^\circ$ electrical. The auxiliary circuit has two capacitors in parallel: a large electrolytic starting capacitor $C_s$ in series with a centrifugal switch, and a small oil-type running capacitor $C_r$ that stays in circuit.

  o------+-----------+
         |           |
       Main      Auxiliary
         |           |
         |       +---+---+
  V      |       |       |
         |      C_s     C_r
         |       |       |
         |      CS       |
         |       +---+---+
  o------+-----------+

Operation:

• At start $C_s + C_r$ make $I_a$ lead $I_m$ by nearly $90^\circ$, giving high starting torque ($T_{st} \propto I_m I_a \sin\alpha$).
• At about 75% of synchronous speed the centrifugal switch removes $C_s$.
• $C_r$ keeps the motor running as a nearly balanced two-phase motor.

Advantages: high starting torque, high power factor and efficiency, quiet operation. Used in compressors, refrigerators and air-conditioners.

At starting both windings are across the 230 V supply. The capacitor is in series with the auxiliary winding. Line current is the phasor sum of the two winding currents.

Given: $Z_m = 15 + j22.5\ \Omega$, $Z_a = 50 + j120\ \Omega$, $C = 12\ \mu\text{F}$, $V = 230\angle 0^\circ$ V, $f = 50$ Hz.

Capacitive reactance

$$X_C = \frac{1}{2\pi f C} = \frac{1}{2\pi \times 50 \times 12\times10^{-6}} = 265.26\ \Omega$$

Branch impedances

$$\begin{aligned} Z_m &= 15 + j22.5 = 27.04\angle 56.31^\circ\ \Omega \\ Z_a' &= 50 + j(120 - 265.26) = 50 - j145.26 \\ &= 153.62\angle -71.01^\circ\ \Omega \end{aligned}$$

Winding currents

$$\begin{aligned} I_m &= \frac{230\angle 0^\circ}{27.04\angle 56.31^\circ} = 8.505\angle -56.31^\circ\ \text{A} \\ &= 4.718 - j7.077\ \text{A} \\ I_a &= \frac{230\angle 0^\circ}{153.62\angle -71.01^\circ} = 1.497\angle 71.01^\circ\ \text{A} \\ &= 0.487 + j1.416\ \text{A} \end{aligned}$$

Line current

$$\begin{aligned} I_L &= I_m + I_a \\ &= (4.718 + 0.487) + j(-7.077 + 1.416) \\ &= 5.205 - j5.661 \\ &= 7.69\angle -47.40^\circ\ \text{A} \end{aligned}$$

Phase angle between $I_a$ and $I_m$ $= 71.01^\circ + 56.31^\circ = 127.3^\circ$; starting power factor $= \cos 47.40^\circ = 0.677$ lagging.

        I_a (1.50 A, +71.0 deg)
         ^
         |
  -------+--------------> V
         | \
         |  \  I_L (7.69 A, -47.4 deg)
         |   \
         v I_m (8.51 A, -56.3 deg)

Answer: starting line current $I_L \approx 7.69\angle -47.4^\circ$ A, i.e. 7.69 A at 0.677 lagging power factor.

A shaded-pole motor is a simple single-phase induction motor in which the starting rotating field is produced by a short-circuited copper ring (the shading ring) placed on part of each salient pole.

Construction

• Stator: salient poles with a concentrated exciting winding fed from the single-phase supply. Each pole is slotted, and about one-third of the pole face is surrounded by a heavy copper ring (shading coil).
• Rotor: ordinary squirrel-cage rotor.
• No capacitor, no centrifugal switch, no auxiliary winding.

        +----------------------+
        |  unshaded  | shaded  |
        |    part    | [ring]  |  pole face
        +----------------------+
               ( rotor )
   field moves: unshaded ---> shaded

Principle of operation

The flux in the shaded part is made to lag the flux in the unshaded part. Consider one half cycle of the current:

1. Current rising quickly (near zero): the rapidly changing flux induces a large current in the shading ring. By Lenz's law it opposes the change, so most flux passes through the unshaded part. Flux axis is at the unshaded part.
2. Current near peak (little change): almost no emf is induced in the ring, so flux is spread uniformly over the pole. Flux axis is at the centre of the pole.
3. Current falling: the ring current now opposes the decrease, so flux is concentrated in the shaded part. Flux axis moves to the shaded part.

Thus the flux axis sweeps from the unshaded to the shaded part every half cycle. This moving field (two fluxes displaced in space and in time) behaves like a weak rotating field and induces currents in the cage rotor, producing torque in the direction unshaded → shaded.

Characteristics

• Very low starting torque (about 40–50% of full-load torque)
• Low power factor and low efficiency (5–35%) because of copper loss in the ring
• Direction fixed by construction; cannot be reversed electrically
• Speed fairly constant; ratings up to about 1/20 kW (some up to 1/4 HP)

Applications

Table and ceiling fans of small size, exhaust fans, hair dryers, record players, small blowers, toys, advertising displays and electric clocks.

For the currents to be in quadrature, the auxiliary (starting) winding current must lead the main winding current by $90^\circ$. So the angle of the starting branch impedance must be $90^\circ$ less than that of the main winding.

Given: $Z_m = 4.5 + j3.5\ \Omega$, $Z_s = 9.5 + j3.5\ \Omega$, $V = 230$ V, $f = 50$ Hz.

Main winding angle

$$\theta_m = \tan^{-1}\frac{3.5}{4.5} = 37.87^\circ \quad (I_m \text{ lags } V \text{ by } 37.87^\circ)$$

Required angle of starting branch

$$\theta_s = 37.87^\circ - 90^\circ = -52.13^\circ$$

With capacitor: $Z_s' = 9.5 + j(3.5 - X_C)$, so

$$\begin{aligned} \frac{3.5 - X_C}{9.5} &= \tan(-52.13^\circ) = -\frac{4.5}{3.5} = -1.2857 \\ 3.5 - X_C &= -12.214 \\ X_C &= 15.714\ \Omega \end{aligned}$$

Capacitance

$$\begin{aligned} C &= \frac{1}{2\pi f X_C} = \frac{1}{2\pi \times 50 \times 15.714} \\ &= 202.6 \times 10^{-6}\ \text{F} \end{aligned}$$

   o----+-------------+
        |             |
     Z_m=4.5+j3.5   Z_s=9.5+j3.5
        |             |
 230 V  |            [C]  ~203 uF
        |             |
        |            CS (centrifugal switch)
   o----+-------------+

Check: $Z_s' = 9.5 - j12.214 = 15.47\angle -52.13^\circ$; angle between currents $= 37.87^\circ + 52.13^\circ = 90^\circ$.

Answer: starting capacitor $C \approx 202.6\ \mu\text{F}$ ($X_C = 15.71\ \Omega$).

A stepper motor is a brushless motor that converts electrical pulses into discrete angular movements called steps. Each pulse turns the shaft by a fixed angle, so position is controlled without feedback (open loop).

Operating principle

• The stator has several phase windings on salient poles. A drive circuit energises the phases one after another in a fixed sequence.
• The rotor (soft iron, permanent magnet or hybrid) always moves to the position of minimum reluctance (or aligns its magnet poles with the stator poles).
• When the next phase is energised, the field axis jumps by one step and the rotor follows it.

Step angle:

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

where $m$ = number of phases, $N_r$ = rotor teeth, $N_s$ = stator poles.

Example: a 3-phase VR motor with $N_s = 6$, $N_r = 4$ gives $\beta = \frac{6-4}{6\times4}\times360^\circ = 30^\circ$; 12 pulses make one revolution. Speed (rpm) $= \frac{\beta \times f}{6}$ where $f$ = pulses per second.

 pulses  +---------+  A  +--------------+
 ------->|  logic  |---->|   stator     |
 (f pps) | sequenc |  B  |   phases     |--> rotor
 dir --->| + driver|---->|  A,B,C ...   |    steps
         +---------+  C  +--------------+

Types

1. Variable reluctance (VR): soft-iron toothed rotor, works on minimum reluctance.
2. Permanent magnet (PM): magnet rotor, larger step angles, detent torque.
3. Hybrid: magnet plus toothed rotor, very small step angles (e.g. $1.8^\circ$), high torque.

Modes of drive

One-phase-on (full step), two-phase-on (more torque), and half-step (alternating, halves the step angle); microstepping gives even finer steps.

Features

• Rotation angle proportional to number of pulses; speed proportional to pulse rate
• Holding torque at standstill; easy reversal by changing sequence
• No brushes; errors do not accumulate

Applications

• Computer printers, plotters, floppy/CD drives, scanners
• CNC machines, 3D printers, robotics
• Quartz watches, cameras (lens focus), medical equipment
• Valve and process control, X–Y tables, satellite antenna positioning

A capacitor-start capacitor-run (CSCR) motor, also called a two-value capacitor motor, is a single-phase induction motor that uses one large capacitor for starting and a smaller capacitor that stays in circuit while running.

Construction

• Stator with a main winding and an auxiliary winding displaced $90^\circ$ electrical in space.
• Auxiliary branch has two capacitors in parallel:
  • Starting capacitor $C_s$ (large, electrolytic, short-time rated) in series with a centrifugal switch
  • Running capacitor $C_r$ (small, oil-filled paper, continuously rated)
• Squirrel-cage rotor.

  o-------+------------+
          |            |
         Main      Auxiliary
        winding     winding
          |            |
  1-ph    |       +----+----+
  supply  |       |         |
          |      C_s       C_r
          |       |         |
          |      CS         |
          |       +----+----+
  o-------+------------+
          (rotor: squirrel cage)

Operating principle

1. At starting both capacitors are in circuit. The large total capacitance makes the auxiliary current $I_a$ lead $V$, while $I_m$ lags $V$. The angle $\alpha$ between them is close to $90^\circ$.
2. Two currents displaced in time and space produce a rotating magnetic field, so the motor starts. Starting torque $T_{st} = K I_m I_a \sin\alpha$ is high (about 3–4.5 times full-load).
3. At about 70–80% of synchronous speed the centrifugal switch disconnects $C_s$.
4. The running capacitor keeps the two windings working as an almost balanced two-phase motor, so the backward field is small during running.

          I_a (with C_s + C_r)
           ^
           |  alpha ~ 90 deg
  ---------+-------------> V
            \
             \
              v I_m

Speed–torque characteristic

• High starting torque from start to switching speed (upper curve, both capacitors).
• At the switching point the torque drops to the lower curve (only $C_r$).
• Running characteristic is smooth, like a two-phase motor; full-load slip is small.

 Torque
   ^ ___   C_s + C_r
   |    \___
   |        \__  switch opens
   |           |\
   |  C_r only |  \_
   |___________|____\_____> speed
   0          75%    Ns

Advantages and applications

• High starting torque, better power factor and efficiency, quiet running, smooth torque (less 100 Hz pulsation).
• Used in refrigerators, air-conditioners, compressors, pumps and conveyors.

Using double field revolving theory, the rotor and magnetising branches are split into forward and backward halves. Core loss is neglected (no core-loss resistance is given).

Given: $V = 230$ V, $r_1 = 2.51\ \Omega$, $x_1 = 4.62\ \Omega$, $r_2' = 7.81\ \Omega$, $x_2' = 4.62\ \Omega$, $X_m = 150.88\ \Omega$, $s = 0.05$.

  I ->  r1     jx1
 o--/\/\/--mmm--+--------+---------+
                |        |         |
 V           0.5jXm  0.5r2'/s     |  Forward
                |     +j0.5x2'    |  Z_F
                +--------+---------+
                |        |         |
             0.5jXm  0.5r2'/(2-s) |  Backward
                |     +j0.5x2'    |  Z_B
 o--------------+--------+---------+

Forward impedance

$$\begin{aligned} 0.5 X_m &= 75.44\ \Omega,\quad \frac{0.5r_2'}{s} = \frac{3.905}{0.05} = 78.1\ \Omega,\quad 0.5x_2' = 2.31\ \Omega \\ Z_F &= \frac{(j75.44)(78.1 + j2.31)}{78.1 + j77.75} \\ &= 36.60 + j39.01\ \Omega \end{aligned}$$

Backward impedance

$$\begin{aligned} \frac{0.5 r_2'}{2-s} &= \frac{3.905}{1.95} = 2.003\ \Omega \\ Z_B &= \frac{(j75.44)(2.003 + j2.31)}{2.003 + j77.75} \\ &= 1.884 + j2.290\ \Omega \end{aligned}$$

Total input impedance

$$\begin{aligned} Z &= (r_1 + jx_1) + Z_F + Z_B \\ &= (2.51 + 36.60 + 1.884) + j(4.62 + 39.01 + 2.290) \\ &= 40.99 + j45.92 \\ &= 61.55\angle 48.24^\circ\ \Omega \end{aligned}$$

Main winding current and power factor

$$\begin{aligned} I_m &= \frac{V}{Z} = \frac{230\angle 0^\circ}{61.55\angle 48.24^\circ} = 3.737\angle -48.24^\circ\ \text{A} \\ \text{pf} &= \cos 48.24^\circ = 0.666\ \text{lagging} \end{aligned}$$

(For reference: air-gap powers are $P_{gF} = I^2 R_F = 511.0$ W and $P_{gB} = I^2 R_B = 26.3$ W.)

Answer: main winding current $I_m \approx 3.74$ A, power factor $\approx 0.666$ lagging.

Starting torque of a two-winding single-phase motor is proportional to the product of the winding currents and the sine of the angle between them:

$$T_{st} = K I_m I_a \sin\alpha, \qquad I = \frac{V}{|Z|}$$

Given: $V = 120$ V, $f = 50$ Hz, $Z_m = 1.5 + j4\ \Omega$, $Z_a = 3 + j6\ \Omega$, $C = 1000\ \mu\text{F}$.

Main winding (unchanged)

$$Z_m = 1.5 + j4 = 4.272\angle 69.44^\circ\ \Omega,\quad I_m = 28.09\angle -69.44^\circ\ \text{A}$$

Case 1: without capacitor

$$\begin{aligned} Z_a &= 3 + j6 = 6.708\angle 63.43^\circ\ \Omega \\ I_a &= \frac{120}{6.708} = 17.89\angle -63.43^\circ\ \text{A} \\ \alpha_1 &= 69.44^\circ - 63.43^\circ = 6.01^\circ \\ T_1 &= K(28.09)(17.89)\sin 6.01^\circ = 52.62\,K \end{aligned}$$

Case 2: with 1000 μF capacitor

$$\begin{aligned} X_C &= \frac{1}{2\pi \times 50 \times 1000\times10^{-6}} = 3.183\ \Omega \\ Z_a' &= 3 + j(6 - 3.183) = 3 + j2.817 = 4.115\angle 43.20^\circ\ \Omega \\ I_a' &= \frac{120}{4.115} = 29.16\angle -43.20^\circ\ \text{A} \\ \alpha_2 &= 69.44^\circ - 43.20^\circ = 26.25^\circ \\ T_2 &= K(28.09)(29.16)\sin 26.25^\circ = 362.3\,K \end{aligned}$$

Ratio and percentage increase

$$\begin{aligned} \frac{T_2}{T_1} &= \frac{29.16 \sin 26.25^\circ}{17.89 \sin 6.01^\circ} = 6.886 \\ \%\ \text{increase} &= (6.886 - 1)\times 100 = 588.6\% \end{aligned}$$

Answer: starting torque increases about 6.89 times, i.e. an increase of about 589%.

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

A single-phase stator winding produces a pulsating (alternating) field along one fixed axis, not a rotating field. By double field revolving theory, this field equals two fields of half amplitude, $\phi_m/2$, rotating at synchronous speed in opposite directions.

• At standstill the rotor slip with respect to both fields is $s = 1$.
• Both fields induce equal rotor currents and produce equal and opposite torques, $T_f = T_b$.
• Net starting torque $T = T_f - T_b = 0$, so the rotor only hums and does not start.

(Cross-field view: rotor currents at standstill produce a field in line with the stator field, so there is no torque-producing displacement.)

If the rotor is pushed in either direction, $T_f \ne T_b$ and it accelerates in that direction. So an auxiliary means is needed to create a rotating field at start.

Permanent split capacitor (PSC) motor

Construction:

• Main winding and auxiliary winding displaced $90^\circ$ electrical in space.
• One oil-filled paper capacitor of small value (e.g. 2–20 μF) is permanently connected in series with the auxiliary winding. There is no centrifugal switch.
• Both windings are usually identical (same copper), so the motor works like a two-phase motor.

  o-------+---------------+
          |               |
        Main          Auxiliary
       winding         winding
  1-ph    |               |
  supply  |              [C]  (permanent)
          |               |
  o-------+---------------+

Working principle:

1. The capacitor makes $I_a$ lead the supply voltage, while $I_m$ lags it. The angle between them is large (about $90^\circ$ at the design load).
2. Two space-displaced windings carrying time-displaced currents produce a rotating magnetic field (nearly uniform at rated load), and the cage rotor starts and runs like a two-phase induction motor.
3. Because the capacitor is chosen for good running, the starting torque is only moderate (about 50–100% of full-load torque).
4. Direction can be reversed easily by switching the capacitor from one winding to the other.

        I_a
         ^   alpha ~ 90 deg
         |
  -------+--------> V
          \
           v I_m

Features: no switch to fail, high power factor, good efficiency, quiet and smooth running, speed control by tapped windings or voltage.

Applications:

• Ceiling fans, table fans, exhaust fans and blowers
• Air-conditioner and refrigerator fan motors, room coolers
• Oil burners, office machines, motor-operated valves and reversible drives

A stepper (stepping) motor is a brushless, synchronous-type motor that moves in fixed angular steps, one step for each input pulse. It is used for accurate open-loop position control.

Construction

Three common types:

  VR motor: 6 stator poles, 4 rotor teeth
          A
        __|__
   C' /  ___  \ B'
     |  | R |  |     A-A', B-B', C-C' = phases
   B  \ |___| / C    rotor teeth pull into line
        --|--        with excited phase
          A'

Working principle

• Stator phases are energised one at a time (or two at a time) in sequence by a driver circuit controlled by digital pulses.
• VR type: the rotor turns to the position of minimum reluctance, aligning its nearest teeth with the excited poles.
• PM / hybrid type: rotor magnet poles align with the opposite stator poles.
• When the next phase is excited, the field axis shifts by one step, and the rotor follows.

Step angle:

$$\beta = \frac{N_s - N_r}{N_s N_r}\times 360^\circ$$

For $N_s = 6$, $N_r = 4$: $\beta = 30^\circ$, so 12 steps per revolution.

Excitation modes: full step (one phase on), two-phase on (higher torque), half step (step angle halved), and microstepping.

Key relations: shaft angle = $\beta \times$ number of pulses; speed $n = \dfrac{\beta f}{360^\circ}\times 60$ rpm for pulse rate $f$.

Applications

• Printers, plotters, scanners, disk drives
• CNC machine tools, 3D printers, robots, X–Y tables
• Quartz clocks and watches, cameras
• Medical and scientific instruments, valve control
• Satellite and antenna positioning

Double field revolving theory (DFRT) states that a pulsating magnetic field of amplitude $\phi_m$ can be resolved into two rotating fields, each of amplitude $\phi_m/2$, rotating at synchronous speed $N_s$ in opposite directions.

$$\phi_m\cos\omega t \cos\theta = \frac{\phi_m}{2}\cos(\theta - \omega t) + \frac{\phi_m}{2}\cos(\theta + \omega t)$$

Torque from the two fields

• If the rotor runs at speed $N$ in the forward direction, slip w.r.t. forward field: $s_f = s = \dfrac{N_s - N}{N_s}$.
• Slip w.r.t. backward field: $s_b = \dfrac{N_s + N}{N_s} = 2 - s$.
• Forward field gives torque $T_f$, backward field gives $T_b$ (opposite). Net torque $T = T_f - T_b$.
• At standstill $s = 1$, $s_b = 1$: $T_f = T_b$, net torque zero, so the motor is not self-starting. Once rotating, $T_f > T_b$ and it runs on.

 T  |      Tf
    |     /\         Tnet = Tf - Tb
    |    /  \
 ---+---------------------> N
    |  \  /
    |   \/   Tb

Equivalent circuit

Since each rotating field is half the total, the rotor can be represented by two half-rotors: one acted on by the forward field (slip $s$) and one by the backward field (slip $2-s$). Magnetising reactance and rotor impedance are each split into halves.

 I1   R1     X1
o--/\/\--mmm--+----------+
              |  0.5Xm   |  0.5R2'/s
              +--mmm--+--+--/\/\--mmm-- (Forward)
  V           |          |  0.5X2'
              +----------+
              |  0.5Xm   |  0.5R2'/(2-s)
              +--mmm--+--+--/\/\--mmm-- (Backward)
              |          |  0.5X2'
o-------------+----------+

Forward and backward impedances:

$$\begin{aligned} Z_F &= R_F + jX_F = \frac{(j0.5X_m)\left(\frac{0.5R_2'}{s} + j0.5X_2'\right)}{\frac{0.5R_2'}{s} + j0.5(X_m + X_2')} \\ Z_B &= R_B + jX_B = \frac{(j0.5X_m)\left(\frac{0.5R_2'}{2-s} + j0.5X_2'\right)}{\frac{0.5R_2'}{2-s} + j0.5(X_m + X_2')} \end{aligned}$$

Operation through the equivalent circuit

1. Input current: $I_1 = \dfrac{V}{R_1 + jX_1 + Z_F + Z_B}$.
2. Air-gap powers: $P_{gF} = I_1^2 R_F$, $P_{gB} = I_1^2 R_B$.
3. Torques: $T_f = \dfrac{P_{gF}}{\omega_s}$, $T_b = \dfrac{P_{gB}}{\omega_s}$, net $T = \dfrac{P_{gF} - P_{gB}}{\omega_s}$.
4. Mechanical power developed: $P_m = (1-s)(P_{gF} - P_{gB})$.
5. Rotor copper loss: $s P_{gF} + (2-s) P_{gB}$.

Interpretation:

• At $s = 1$: $Z_F = Z_B$, so $P_{gF} = P_{gB}$ and $T = 0$ (no starting torque).
• At normal slip (small $s$): $0.5R_2'/s$ is large, so $Z_F \gg Z_B$. Most of the voltage appears across the forward branch, the forward field is strong and the backward field is weak, giving positive net torque.
• The backward field adds extra rotor copper loss and a double-frequency torque pulsation, so a single-phase motor has lower efficiency and is noisier than a three-phase motor.