1. Introduction
Single-phase induction motors are widely used in domestic and light commercial applications where three-phase supply is not available. These motors are simple, robust, and economical, making them ideal for fans, pumps, refrigerators, and small machine tools.
- Power rating: typically up to 1 kW
- Not self-starting (requires auxiliary mechanism)
- Lower efficiency compared to three-phase motors
- Simple construction and maintenance
2. Double Revolving Field Theory
The fundamental principle explaining single-phase induction motor operation is the Double Revolving Field Theory, proposed by Ferraris. According to this theory, a stationary pulsating magnetic field can be resolved into two rotating magnetic fields of equal magnitude but rotating in opposite directions.
Pulsating MMF Expression
The magnetomotive force produced by a single-phase winding:
$$F = F_m \cos(\omega t) \cos(\theta)$$where \(F_m\) is the maximum MMF, \(\omega\) is the angular frequency, and \(\theta\) is the spatial angle.
Resolution into Forward and Backward Fields
$$F = \frac{F_m}{2}\cos(\omega t - \theta) + \frac{F_m}{2}\cos(\omega t + \theta)$$The first term represents the forward rotating field, and the second represents the backward rotating field.
Forward and Backward Slip
For the forward rotating field:
$$s_f = s = \frac{N_s - N}{N_s}$$For the backward rotating field:
$$s_b = 2 - s = \frac{N_s - (-N)}{N_s} = \frac{N_s + N}{N_s}$$where \(N_s\) is the synchronous speed and \(N\) is the rotor speed.
3. Equivalent Circuit
The equivalent circuit of a single-phase induction motor represents the motor as two separate motors connected in parallel - one representing the forward field and one representing the backward field.
Forward Field Impedance
$$Z_f = r_f + jx_f = \frac{(r_2'/s) + jx_2'}{2}$$Backward Field Impedance
$$Z_b = r_b + jx_b = \frac{(r_2'/(2-s)) + jx_2'}{2}$$Total Input Impedance
$$Z_{in} = r_1 + jx_1 + \frac{Z_f \cdot jx_m}{Z_f + jx_m} + \frac{Z_b \cdot jx_m}{Z_b + jx_m}$$where:
- \(r_1\) = stator resistance
- \(x_1\) = stator leakage reactance
- \(r_2'\) = rotor resistance referred to stator
- \(x_2'\) = rotor leakage reactance referred to stator
- \(x_m\) = magnetizing reactance
4. Torque Analysis
Forward Field Torque
$$T_f = \frac{1}{\omega_s} \cdot \frac{I_f^2 \cdot r_2'}{2s}$$where \(I_f\) is the forward field current and \(\omega_s\) is the synchronous angular velocity.
Backward Field Torque
$$T_b = \frac{1}{\omega_s} \cdot \frac{I_b^2 \cdot r_2'}{2(2-s)}$$where \(I_b\) is the backward field current. This torque opposes the motor rotation.
Net Torque
$$T_{net} = T_f - T_b$$At standstill (s = 1):
$$T_{net} = T_f - T_b = 0$$This explains why single-phase induction motors are not self-starting.
5. Power Relations
Input Power
$$P_{in} = V_1 I_1 \cos\phi_1$$where \(V_1\) is the supply voltage, \(I_1\) is the input current, and \(\phi_1\) is the power factor angle.
Stator Copper Loss
$$P_{cu1} = I_1^2 r_1$$Air Gap Power (Forward and Backward)
$$P_{ag,f} = I_f^2 \cdot \frac{r_2'}{2s}$$ $$P_{ag,b} = I_b^2 \cdot \frac{r_2'}{2(2-s)}$$Rotor Copper Loss
$$P_{cu2} = s \cdot P_{ag,f} + (2-s) \cdot P_{ag,b}$$Mechanical Power Developed
$$P_{mech} = (1-s) P_{ag,f} - (1-s) P_{ag,b}$$Output Power
$$P_{out} = P_{mech} - P_{rot}$$where \(P_{rot}\) represents rotational losses (friction and windage).
Efficiency
$$\eta = \frac{P_{out}}{P_{in}} \times 100\%$$6. Types of Single-Phase Induction Motors
6.1 Split-Phase Motor
Uses two windings: a main winding and an auxiliary (starting) winding. The auxiliary winding has higher resistance and lower inductance, creating a phase difference.
Phase Angle Difference
$$\alpha = \tan^{-1}\left(\frac{X_a}{R_a}\right) - \tan^{-1}\left(\frac{X_m}{R_m}\right)$$where subscripts 'a' and 'm' denote auxiliary and main windings respectively.
Typical phase difference: 25° - 30°
Starting Torque
$$T_{start} = K \cdot V^2 \cdot \sin\alpha$$where K is a constant depending on motor design, V is the supply voltage, and \(\alpha\) is the phase angle between the two winding currents.
6.2 Capacitor-Start Motor
A capacitor is connected in series with the auxiliary winding to achieve a larger phase displacement (approximately 90°).
Optimum Capacitance for Maximum Starting Torque
$$C = \frac{1}{2\pi f X_a}$$For maximum torque per ampere:
$$C = \frac{1}{\omega} \sqrt{\frac{R_m^2 + X_m^2}{R_a^2 + X_a^2}} \cdot \frac{1}{X_a + \sqrt{R_a^2 + X_a^2}}$$where \(f\) is the supply frequency and \(\omega = 2\pi f\).
6.3 Capacitor-Start Capacitor-Run Motor
Uses two capacitors: a large starting capacitor (disconnected after starting) and a smaller running capacitor (permanently connected).
Running Capacitor Value
$$C_{run} = \frac{P_{out}}{2\pi f V^2 \sin\phi}$$where \(P_{out}\) is the output power and \(\phi\) is the desired power factor angle.
6.4 Shaded-Pole Motor
A portion of each pole is surrounded by a short-circuited copper band called a shading coil, creating a time lag in the magnetic field.
Shading Angle
$$\delta = \tan^{-1}\left(\frac{\omega L_{sh}}{R_{sh}}\right)$$where \(L_{sh}\) and \(R_{sh}\) are the inductance and resistance of the shading coil.
Typical values: \(\delta\) = 20° - 40°
7. Performance Characteristics
Synchronous Speed
$$N_s = \frac{120f}{P}$$where \(f\) is the frequency in Hz and \(P\) is the number of poles.
Rotor Speed
$$N = N_s(1-s)$$where \(s\) is the slip.
Torque-Slip Relationship
$$T \propto \frac{s}{r_2'^2 + (sx_2')^2}$$For small values of slip (normal running condition):
$$T \propto s$$Maximum Torque (Breakdown Torque)
$$T_{max} = \frac{3V_1^2}{2\omega_s\left(r_1 + \sqrt{r_1^2 + (x_1+x_2')^2}\right)}$$Slip at maximum torque:
$$s_{max} = \frac{r_2'}{\sqrt{r_1^2 + (x_1+x_2')^2}}$$| Motor Type | Starting Torque (% of FL) | Starting Current (% of FL) | Efficiency |
|---|---|---|---|
| Split-Phase | 100-150% | 600-800% | Low |
| Capacitor-Start | 200-350% | 450-550% | Medium |
| Capacitor-Start Capacitor-Run | 200-350% | 400-500% | High |
| Shaded-Pole | 50-75% | 200-400% | Very Low |
8. Starting Methods and Switching
Centrifugal Switch
Automatically disconnects the starting winding when the motor reaches approximately 75-80% of synchronous speed.
Cut-out Speed
$$N_{cutout} = (0.75 \text{ to } 0.80) \times N_s$$Current Relay
Operates based on the reduction in starting current as the motor accelerates.
Pick-up Current
$$I_{pickup} = (1.2 \text{ to } 1.5) \times I_{rated}$$Drop-out Current
$$I_{dropout} = (0.8 \text{ to } 0.9) \times I_{rated}$$9. Losses and Efficiency
Total Losses
$$P_{loss} = P_{cu1} + P_{cu2} + P_{core} + P_{rot}$$where:
- \(P_{cu1}\) = Stator copper loss = \(I_1^2 r_1\)
- \(P_{cu2}\) = Rotor copper loss = \(s \times P_{ag}\)
- \(P_{core}\) = Core loss (hysteresis + eddy current)
- \(P_{rot}\) = Rotational losses (friction + windage)
Core Loss Components
Hysteresis loss:
$$P_h = \eta B_{max}^{1.6} f V$$Eddy current loss:
$$P_e = K_e B_{max}^2 f^2 t^2 V$$where \(B_{max}\) is maximum flux density, \(t\) is lamination thickness, and \(V\) is volume.
10. Applications and Selection Criteria
Application Guidelines
| Motor Type | Typical Applications |
|---|---|
| Split-Phase | Fans, blowers, small pumps, washing machines |
| Capacitor-Start | Compressors, conveyors, machine tools, refrigerators |
| Capacitor-Start Capacitor-Run | Air conditioners, high-performance fans, pumps |
| Shaded-Pole | Small fans, toys, record players, low-power applications |
- Starting Torque Requirement: High starting load requires capacitor-start or capacitor-start capacitor-run motors
- Running Efficiency: For continuous operation, choose capacitor-run motors
- Cost: Shaded-pole motors are cheapest, capacitor motors are more expensive
- Power Rating: Match motor rating to load requirement with appropriate service factor
- Duty Cycle: Consider continuous vs. intermittent operation
11. Speed Control Methods
1. Voltage Control
Since torque is proportional to the square of voltage:
$$T \propto V^2$$Reducing voltage reduces torque and speed. Common methods include:
- Autotransformer
- Thyristor voltage controller
- Tapped stator winding
2. Frequency Control
Speed is directly proportional to frequency:
$$N \propto f$$For constant V/f ratio (constant flux):
$$\frac{V}{f} = \text{constant}$$This method requires variable frequency drive (VFD).
3. Pole Changing
$$N_s = \frac{120f}{P}$$Provides discrete speed steps (e.g., 2-speed or 4-speed motors).
12. Testing and Performance Evaluation
12.1 No-Load Test
Motor run at rated voltage without mechanical load.
Measurements: \(V_0\), \(I_0\), \(P_0\)
Core Loss and Rotational Loss
$$P_0 = P_{cu1,0} + P_{core} + P_{rot}$$ $$P_{core} + P_{rot} \approx P_0 - I_0^2 r_1$$12.2 Blocked Rotor Test
Rotor is locked, reduced voltage applied to limit current to rated value.
Measurements: \(V_{sc}\), \(I_{sc}\), \(P_{sc}\)
Equivalent Impedance
$$Z_{eq} = \frac{V_{sc}}{I_{sc}}$$Equivalent Resistance
$$r_{eq} = \frac{P_{sc}}{I_{sc}^2} = r_1 + r_2'$$Equivalent Reactance
$$x_{eq} = \sqrt{Z_{eq}^2 - r_{eq}^2} = x_1 + x_2'$$13. Important Design Considerations
Magnetic Circuit Design
Air Gap Flux Density
$$B_g = \frac{\phi}{A_g}$$where \(\phi\) is the flux per pole and \(A_g\) is the air gap area.
Turns per Phase
$$T_{ph} = \frac{V}{4.44 f \phi K_w}$$where \(K_w\) is the winding factor.
Thermal Design
Temperature Rise
$$\Delta T = \frac{P_{loss}}{A \cdot h}$$where \(A\) is the cooling surface area and \(h\) is the heat transfer coefficient.
Current Density
$$J = \frac{I}{A_c}$$Typical values: 3-6 A/mm² for naturally cooled motors
14. Summary of Key Formulas
| Parameter | Formula |
|---|---|
| Synchronous Speed | \(N_s = \frac{120f}{P}\) |
| Slip | \(s = \frac{N_s - N}{N_s}\) |
| Rotor Frequency | \(f_r = s \cdot f\) |
| Forward Slip | \(s_f = s\) |
| Backward Slip | \(s_b = 2 - s\) |
| Efficiency | \(\eta = \frac{P_{out}}{P_{in}} \times 100\%\) |
| Power Factor | \(\cos\phi = \frac{P_{in}}{V_1 I_1}\) |
| Torque | \(T = \frac{P_{mech}}{\omega_m} = \frac{P_{mech}}{2\pi N/60}\) |
| Rotor Copper Loss | \(P_{cu2} = s \cdot P_{ag}\) |
15. Cross-Field Theory (Alternative Analysis)
An alternative approach to analyzing single-phase induction motors, particularly useful for understanding motors with auxiliary windings.
Main Field MMF
$F_m = F_{m,max} \cos(\omega t) \cos(\theta)$Cross Field MMF (in quadrature)
$F_c = F_{c,max} \cos(\omega t) \sin(\theta)$The cross-field is produced by the auxiliary winding and is displaced 90° in space from the main field.
Resultant Rotating Field
$F_{res} = \sqrt{F_m^2 + F_c^2}$For balanced two-phase operation (when \(F_m = F_c\)):
$F_{res} = F_m\sqrt{2}$This creates a true rotating field, similar to a three-phase motor.
16. Capacitor Calculations in Detail
16.1 Starting Capacitor Design
Capacitor Value for 90° Phase Shift
$C_s = \frac{1}{2\pi f \left(X_a + \sqrt{\left(\frac{R_m Z_a}{R_a Z_m}\right)^2 - R_a^2}\right)}$where:
- \(Z_m = \sqrt{R_m^2 + X_m^2}\) - Main winding impedance
- \(Z_a = \sqrt{R_a^2 + X_a^2}\) - Auxiliary winding impedance
Current Relationship
$\frac{I_a}{I_m} = \frac{N_m}{N_a} \cdot \frac{Z_m}{Z_a}$For balanced operation:
$I_a \cdot N_a = I_m \cdot N_m$16.2 Running Capacitor Design
Capacitor for Unity Power Factor
$C_r = \frac{P_{out} \tan\phi}{2\pi f V^2}$For improved power factor from \(\cos\phi_1\) to \(\cos\phi_2\):
$C_r = \frac{P(\tan\phi_1 - \tan\phi_2)}{2\pi f V^2}$Capacitor Voltage Rating
$V_C = I_a \cdot X_C = \frac{I_a}{2\pi f C}$Typical safety factor: Select capacitor with voltage rating 1.5 to 2 times calculated value.
17. Advanced Performance Analysis
17.1 Circle Diagram
The circle diagram is a graphical method to determine motor performance at various loads.
Key Points on Circle Diagram
No-load point:
$I_0 = \frac{V}{Z_0}$Blocked rotor point:
$I_{sc} = \frac{V}{Z_{sc}}$Maximum torque occurs when the line from origin is tangent to the circle.
17.2 Stability Analysis
Stability Criterion
$\frac{dT}{ds} < 0$The motor operates stably when torque decreases with increasing slip (normal operating region).
Pull-out Torque Point
$s_{pullout} = \frac{r_2'}{\sqrt{r_1^2 + (x_1 + x_2')^2}}$Beyond this slip, the motor becomes unstable.
18. Special Operating Conditions
18.1 Plugging (Reverse Braking)
Plugging Slip
$s_{plug} = 2 - s = \frac{N_s + N}{N_s}$For reverse rotation with slip \(s_r\):
$s_r = \frac{N_s - (-N)}{N_s} = \frac{N_s + N}{N_s}$Braking Torque
$T_{brake} = \frac{3V^2 r_2' s_{plug}}{\omega_s(r_1 + r_2'/s_{plug})^2 + (x_1 + x_2')^2}$18.2 Dynamic Braking
DC Injection Braking
DC current injected into stator winding creates stationary field.
$I_{dc} = \frac{V_{dc}}{R_1}$Braking Torque with DC
$T_b = K \cdot I_{dc}^2 \cdot \omega \cdot \frac{r_2'}{r_2'^2 + (\omega x_2')^2}$where \(\omega\) is the rotor angular velocity.
19. Effect of Voltage and Frequency Variations
Voltage Variation Effects
Torque variation:
$T \propto V^2$ $T_2 = T_1 \left(\frac{V_2}{V_1}\right)^2$Current variation (approximately):
$I \propto V$Efficiency change:
$\eta_2 \approx \eta_1 - k\left(1 - \frac{V_2}{V_1}\right)$where \(k\) is a constant (typically 0.01-0.02).
Frequency Variation Effects
Synchronous speed:
$N_{s2} = N_{s1} \cdot \frac{f_2}{f_1}$For constant V/f operation:
$\frac{V_2}{V_1} = \frac{f_2}{f_1}$Flux remains approximately constant:
$\phi_2 \approx \phi_1$Torque capability:
$T_{max} \approx \text{constant}$- Voltage variation: ±10% of rated voltage
- Frequency variation: ±5% of rated frequency
- Combined voltage and frequency variation: ±10%
20. Motor Protection and Safety
20.1 Thermal Protection
Thermal Time Constant
$\tau_{th} = \frac{m \cdot c_p}{A \cdot h}$where:
- \(m\) = mass of motor
- \(c_p\) = specific heat capacity
- \(A\) = cooling surface area
- \(h\) = heat transfer coefficient
Temperature Rise Over Time
$T(t) = T_{\infty}(1 - e^{-t/\tau_{th}})$where \(T_{\infty}\) is the steady-state temperature rise.
20.2 Overcurrent Protection
Full Load Current Estimation
$I_{FL} = \frac{P_{rated} \times 1000}{V \times \eta \times \cos\phi}$Fuse/Breaker Rating
$I_{protection} = (1.15 \text{ to } 1.25) \times I_{FL}$For motors with high starting current:
$I_{protection} = (2.0 \text{ to } 2.5) \times I_{FL}$21. Harmonic Analysis
Space Harmonics in MMF
$F(\theta) = \sum_{n=1,3,5,...}^{\infty} F_n \cos(n\theta)$where the nth harmonic amplitude:
$F_n = \frac{4F_m}{n\pi} \cdot K_{wn}$\(K_{wn}\) is the winding factor for nth harmonic.
Harmonic Torques
For nth space harmonic:
$T_n = \frac{P_n}{\omega_s/n}$Synchronous speed of nth harmonic:
$N_{sn} = \frac{120f}{n \cdot P}$Effects of Harmonics
- Additional heating due to harmonic currents
- Torque pulsations causing noise and vibration
- Reduced average torque and efficiency
- Crawling phenomenon at 1/7th of synchronous speed (7th harmonic)
- Cogging or locking at standstill due to harmonic interaction
22. Worked Example Problem
Example: Capacitor-Start Motor Analysis
Given:
- Power rating: 0.5 HP (373 W)
- Supply: 230 V, 50 Hz
- Main winding: \(R_m = 8\,\Omega\), \(X_m = 10\,\Omega\)
- Auxiliary winding: \(R_a = 12\,\Omega\), \(X_a = 8\,\Omega\)
Solution Steps:
1. Calculate impedances:
$Z_m = \sqrt{R_m^2 + X_m^2} = \sqrt{8^2 + 10^2} = 12.81\,\Omega$ $Z_a = \sqrt{R_a^2 + X_a^2} = \sqrt{12^2 + 8^2} = 14.42\,\Omega$2. Phase angles:
$\theta_m = \tan^{-1}\left(\frac{X_m}{R_m}\right) = \tan^{-1}\left(\frac{10}{8}\right) = 51.34°$ $\theta_a = \tan^{-1}\left(\frac{X_a}{R_a}\right) = \tan^{-1}\left(\frac{8}{12}\right) = 33.69°$3. Phase difference without capacitor:
$\alpha = \theta_m - \theta_a = 51.34° - 33.69° = 17.65°$4. Starting capacitor for 90° phase shift:
$X_C = \sqrt{\left(\frac{R_m Z_a}{R_a Z_m}\right)^2 - R_a^2} - X_a$ $X_C = \sqrt{\left(\frac{8 \times 14.42}{12 \times 12.81}\right)^2 - 12^2} - 8 = 28.5\,\Omega$ $C_s = \frac{1}{2\pi f X_C} = \frac{1}{2\pi \times 50 \times 28.5} = 112\,\mu F$5. Starting currents:
$I_m = \frac{V}{Z_m} = \frac{230}{12.81} = 17.96\,A$ $Z_a' = \sqrt{R_a^2 + (X_a - X_C)^2} = \sqrt{12^2 + (8-28.5)^2} = 23.77\,\Omega$ $I_a = \frac{V}{Z_a'} = \frac{230}{23.77} = 9.68\,A$23. Maintenance and Troubleshooting
Common Problems and Diagnosis
| Symptom | Possible Cause | Diagnostic Test |
|---|---|---|
| Motor won't start | Faulty starting mechanism, open winding | Check continuity, test centrifugal switch |
| Low starting torque | Failed capacitor, wrong capacitor value | Measure capacitance, check ESR |
| Overheating | Overload, poor ventilation, voltage imbalance | Measure current, check voltage, inspect cooling |
| Excessive vibration | Unbalanced rotor, misalignment, worn bearings | Check mechanical alignment, bearing condition |
| Humming without rotation | Locked rotor, single-phasing, high load | Check mechanical freedom, starting circuit |
Insulation Resistance Test
Minimum acceptable insulation resistance:
$R_{ins} = \frac{V_{rated}}{1000} + 1\,\text{M}\Omega$For example, for 230 V motor:
$R_{ins} = \frac{230}{1000} + 1 = 1.23\,\text{M}\Omega$24. Energy Efficiency and Standards
Efficiency Classes (IEC 60034-30)
Standard efficiency (SE): \(\eta_{SE} = \eta_{nominal} - 15\%\)
High efficiency (HE): \(\eta_{HE} = \eta_{nominal} - 10\%\)
Premium efficiency (PE): \(\eta_{PE} = \eta_{nominal} - 5\%\)
Energy Savings Calculation
Annual energy consumption:
$E_{annual} = \frac{P_{rated} \times t_{operation} \times L}{\eta}$Energy savings by upgrading efficiency:
$\Delta E = P_{rated} \times t \times L \times \left(\frac{1}{\eta_1} - \frac{1}{\eta_2}\right)$where:
- \(P_{rated}\) = rated power in kW
- \(t\) = operating hours per year
- \(L\) = load factor (0 to 1)
- \(\eta_1, \eta_2\) = old and new efficiencies
25. Conclusion and Summary
Key Takeaways
- Operating Principle: Single-phase induction motors operate on the principle of double revolving field theory, requiring auxiliary means for starting
- Starting Methods: Various starting methods (split-phase, capacitor-start, shaded-pole) provide different torque-speed characteristics for different applications
- Performance: Performance is analyzed using equivalent circuit, considering both forward and backward rotating fields
- Efficiency: Generally lower than three-phase motors, but adequate for fractional horsepower applications
- Applications: Widely used in residential and light commercial applications up to 1-2 HP
Design Trade-offs
| Parameter | Improvement Method | Trade-off |
|---|---|---|
| Starting Torque | Use capacitor-start design | Increased cost and complexity |
| Efficiency | Use capacitor-run design | Higher initial cost |
| Power Factor | Optimize capacitor value | May reduce torque capability |
| Cost | Use shaded-pole design | Lower efficiency and torque |
26. Quick Reference Formulas
| Category | Formula | Units |
|---|---|---|
| Speed | \(N_s = \frac{120f}{P}\) | RPM |
| Slip | \(s = \frac{N_s - N}{N_s}\) | - |
| Torque | \(T = \frac{9.55 \times P_{mech}}{N}\) | N-m |
| Power | \(P = V I \cos\phi\) | W |
| Efficiency | \(\eta = \frac{P_{out}}{P_{in}}\) | % |
| Capacitance | \(C = \frac{1}{2\pi f X_C}\) | F |