AC Motors (Induction & Special Machines)
Rotating magnetic field derivation, torque-slip dynamics, maximum torque criteria, single-phase double revolving field theory, capacitor-start motors, and stepper/BLDC actuators.
9.1 & 9.2 Rotating Magnetic Field (RMF) & 3-Phase Induction Principle
Mathematical Proof of RMF Production
When three-phase balanced stator windings displaced by \(120^\circ\) in space are excited by balanced three-phase AC currents displaced by \(120^\circ\) in time:
\[\phi_R = \Phi_m \cos(\omega t), \quad \phi_Y = \Phi_m \cos(\omega t - 120^\circ), \quad \phi_B = \Phi_m \cos(\omega t + 120^\circ)\]
The resultant vector sum along any orthogonal axis yields:
\[\Phi_r = \sqrt{\Phi_x^2 + \Phi_y^2} = 1.5 \, \Phi_m\]
(See full derivation in Derivation Sheet Section 1).
This field rotates in the air gap at synchronous speed:
\[N_s = \frac{120 f}{P} \text{ [rpm]}\]
Operating Principle & Slip Definition
1. The rotating magnetic field sweeps across stationary rotor conductors at speed \(N_s\).
2. By Faraday's Law, this relative motion induces an alternating EMF and current in the short-circuited rotor cage.
3. By Lenz's Law, the induced rotor current creates a magnetic field that reacts with the stator RMF, producing a driving torque that drags the rotor in the direction of the rotating field.
An induction motor can never achieve synchronous speed under motoring action. If the rotor were to reach \(N = N_s\), the relative speed between the rotating stator field and rotor conductors would become exactly zero (\(N_s - N = 0\)). Consequently, rate of change of flux linkage would be zero, induced EMF in the rotor would collapse to zero, rotor current would become zero, and driving torque would drop to zero! Mechanical friction and windage would immediately slow the rotor down below \(N_s\).
9.3 & 9.4 Torque Equations, Characteristics & Max Torque Condition
Torque Equation Derivation
Torque is proportional to the product of stator flux \(\Phi\), rotor current \(I_{2r}\), and rotor power factor \(\cos\phi_{2r}\): \[T \propto \Phi I_{2r} \cos\phi_{2r}\] Since \(\Phi \propto E_2\), \(I_{2r} = \frac{s E_2}{\sqrt{R_2^2 + (s X_2)^2}}\), and \(\cos\phi_{2r} = \frac{R_2}{\sqrt{R_2^2 + (s X_2)^2}}\): \[T = \frac{k \, s E_2^2 R_2}{R_2^2 + (s X_2)^2}\]
Differentiating running torque \(T\) with respect to slip \(s\) and setting \(\frac{dT}{ds} = 0\):
\[s_{mT} = \frac{R_2}{X_2}\]
Substituting \(s = s_{mT}\) into the torque expression gives the maximum breakdown torque:
\[T_{\text{max}} = \frac{k E_2^2}{2 X_2}\]
• Peak Maximum Torque is independent of rotor resistance \(R_2\)!
• Increasing rotor resistance \(R_2\) (by adding external resistance in a wound-rotor motor) increases starting torque (\(T_{\text{start}} = \frac{k E_2^2 R_2}{R_2^2 + X_2^2}\)) and shifts the peak torque closer to standstill (\(s = 1\)), but leaves \(T_{\text{max}}\) unchanged.
Master Worked Example: Slip & Frequency Calculation
Part (a): Slip \(s = 5\% = 0.05\).
Part (b): Motor runs at \(N = 600\text{ rpm}\).
9.5 & 9.6 Single-Phase Induction Motors & Starting Methods
Double Revolving Field Theory
A single-phase winding produces an alternating pulsating magnetic field along a stationary axis: \(\Phi(t) = \Phi_m \cos(\omega t)\).
Using Euler's identity, this stationary pulsating flux can be mathematically resolved into two equal oppositely rotating fields:
\[\Phi(t) = \frac{\Phi_m}{2} e^{j\omega t} + \frac{\Phi_m}{2} e^{-j\omega t} = \Phi_f + \Phi_b\]
• Forward Field (\(\Phi_f\)): Rotates forward at synchronous speed \(N_s\) with slip \(s = \frac{N_s - N}{N_s}\).
• Backward Field (\(\Phi_b\)): Rotates backward at \(-N_s\) with slip \(s_b = \frac{N_s - (-N)}{N_s} = \frac{N_s + N}{N_s} = 2 - s\).
At standstill (\(N = 0 \implies s = 1\)):
\[s_f = 1, \quad s_b = 2 - 1 = 1\]
Because both forward and backward torques are identical in magnitude but opposite in direction (\(T_f = T_b\)), the net starting torque is identically zero! Hence, a single-phase induction motor is not self-starting.
Starting Methods for 1-Phase Motors
| Motor Type | Phase-Splitting Mechanism | Starting Torque | Typical Applications |
|---|---|---|---|
| Capacitor-Start Induction Run | High-value electrolytic capacitor in series with auxiliary starting winding. Centrifugal switch disconnects starting winding at ~75% speed. | High (\(300 - 400\%\) of rated) | Refrigerators, air compressors, pumps, conveyor belts. |
| Permanent-Split Capacitor (PSC) | A continuous-rated oil-filled capacitor remains permanently in circuit across auxiliary winding for both starting and running. | Moderate (\(50 - 100\%\)) | Ceiling fans, blower fans, blowers, air conditioners (quiet operation). |
| Shaded-Pole Motor | Salient stator poles with a slot housing a short-circuited copper band (shading ring). The ring delays flux build-up, creating a weak rotating field towards the shaded pole. | Very Low (\(40 - 50\%\)) | Small table fans, hair dryers, record players, vending machines (cheap, rugged). |
9.7 Introduction to Special Purpose Motors
| Motor Type | Operating Principle & Key Construction | Special Characteristic | Applications |
|---|---|---|---|
| Permanent Magnet Brushless DC (BLDC) | Permanent magnet rotor; 3-phase stator driven by electronic inverter switched via Hall-effect sensors. No mechanical commutator/brushes! | Extremely high efficiency (\(>90\%\)), high power density, zero maintenance, long lifespan. | Electric vehicles, drones, computer hard drives, robotics. |
| Stepper Motor | Moves in discrete angular steps per input pulse. Rotor aligns with energized stator teeth (Variable Reluctance or Permanent Magnet). | Precise open-loop position control without encoder feedback. Step angle: \(\beta = \frac{N_s - N_r}{N_s N_r} \times 360^\circ\). | 3D printers, CNC machines, scanner head positioning, robotics. |
| Universal Motor | Series-wound commutator motor operating on either AC or DC. | Very high operating speeds (\(3000 - 15000\text{ rpm}\)), high power-to-weight ratio. | Mixer grinders, vacuum cleaners, portable drills. |
| Hysteresis Motor | Smooth non-magnetic chrome/cobalt steel rotor without slots or windings. Operates on magnetic hysteresis torque. | Zero torque ripple, perfectly constant synchronous speed, absolutely silent. | Sound recording equipment, precision turntables, timing devices. |
| Servo Motor | Low-inertia rotor designed for fast acceleration in closed-loop systems with feedback encoders. | Zero hunting, linear speed-torque curve, high positional accuracy. | Robotic arms, automated machinery, aerospace actuators. |