33.3 Content
33.3.1 Torque on a Current Loop
A rectangular coil carrying current in a magnetic field experiences a torque:
\[\tau = nIAB\sin\theta\]
where: - \(\tau\) = torque (N·m) - \(n\) = number of turns - \(I\) = current (A) - \(A\) = area of coil (m²) - \(B\) = magnetic field strength (T) - \(\theta\) = angle between coil normal and field
- Maximum torque when \(\theta = 90°\) (coil plane parallel to field)
- Zero torque when \(\theta = 0°\) (coil plane perpendicular to field)
33.3.2 Interactive: DC Motor Operation
Visualise torque and rotation in a DC motor:
33.3.3 The Commutator
The commutator is a split-ring device that:
- Reverses current direction every half rotation
- Maintains torque direction so the motor keeps spinning
- Converts DC input to the alternating current needed by the coil
Without a commutator, the motor would oscillate back and forth instead of rotating continuously.
33.3.4 Ways to Increase Motor Torque
From \(\tau = nIAB\sin\theta\), torque can be increased by:
| Method | Effect | Practical Consideration |
|---|---|---|
| More turns (n) | Linear increase | Adds weight and resistance |
| Higher current (I) | Linear increase | More heating (I²R losses) |
| Larger coil area (A) | Linear increase | Larger motor size |
| Stronger magnet (B) | Linear increase | Stronger/heavier magnets |
| Radial field design | Keeps \(\theta = 90°\) | Standard in real motors |
33.3.5 Back EMF in Motors
When a motor spins, the rotating coil acts as a generator, inducing an EMF that opposes the supply voltage:
\[V = IR + \varepsilon_{back}\]
or
\[I = \frac{V - \varepsilon_{back}}{R}\]
When a motor starts, \(\varepsilon_{back} = 0\) (no rotation), so starting current is high: \[I_{start} = \frac{V}{R}\] As the motor speeds up, back EMF increases and current decreases.
33.3.6 Interactive: Back EMF Effects
33.3.7 DC Generator
A DC generator is a motor run in reverse:
- Mechanical energy rotates the coil
- Changing flux through the coil induces an EMF
- The commutator converts AC output to pulsating DC
- Brushes transfer current to the external circuit
The induced EMF varies with angle: \[\varepsilon = nBAω\sin(ωt)\]
33.3.8 AC Generator (Alternator)
An AC generator uses slip rings instead of a commutator:
- Output is sinusoidal AC: \(\varepsilon = \varepsilon_0\sin(ωt)\)
- No sparking at brushes (smoother contact)
- Simpler construction than DC generator
33.3.9 Interactive: Generator Comparison
33.3.10 AC Induction Motor
An AC induction motor converts electrical energy to mechanical rotation without a commutator.
It works because a changing magnetic field induces currents (and therefore forces) in the rotor.
Key idea: rotating magnetic field (stator)
- In an AC motor, the stator produces a magnetic field that changes with time.
- In practical motors (especially in industry), three-phase AC creates a rotating magnetic field.
- This rotating field sweeps past the rotor conductors.
Induction in the rotor (why it’s called “induction”)
- As the stator field moves relative to the rotor, the magnetic flux through the rotor conductors changes.
- By Faraday’s law, this induces an EMF in the rotor.
- That EMF drives a current in the rotor.
- By the motor effect (force on a current-carrying conductor in a magnetic field), the rotor experiences a torque and begins to spin.
Slip (why the rotor can’t match the field exactly)
If the rotor spun at exactly the same speed as the rotating magnetic field, there would be no relative motion, so: - no changing flux in the rotor, - no induced EMF/current, - no torque.
So the rotor must run slightly slower than the field.
Define:
Synchronous speed (speed of rotating field): \[n_s = \frac{120f}{p}\] where (n_s) is in rpm, (f) is supply frequency (Hz), and (p) is the number of poles.
Slip: \[s = \frac{n_s - n_r}{n_s}\] where (n_r) is rotor speed.
Slip is usually a few percent under normal load.
An AC induction motor needs relative motion between the stator field and rotor. That relative motion (slip) is what enables induction, current, and torque.
Where energy goes (losses)
- Rotor currents cause heating losses (like (I^2R)).
- There are also friction and air resistance losses.
- This is why real motors have efficiencies < 100%.
Applications
AC induction motors are widely used because they are: - robust (no commutator), - low-maintenance, - efficient at fixed speeds.
Common uses: fans, pumps, industrial drives, appliances.
Worked example (synchronous speed)
A 50 Hz supply powers a 4-pole induction motor. Find the synchronous speed.
\[ n_s = \frac{120f}{p} = \frac{120\times 50}{4} = 1500\ \text{rpm} \]
If the rotor runs at 1440 rpm, the slip is:
\[ s = \frac{1500 - 1440}{1500} = 0.040 = 4.0\% \]
Practice (HSC-style short answer)
Explain why an induction motor cannot operate with zero slip.
- If rotor speed = field speed → no relative motion (1)
- No changing flux through rotor conductors → no induced EMF/current (1)
- No current → no magnetic force/torque (1)
- Therefore slip is required for continuous torque production (1)
33.3.11 Interactive: AC Induction
33.3.12 Lenz’s Law and Energy Conservation
In a generator, Lenz’s law ensures energy conservation:
- Turning the generator coil requires mechanical work
- This induces a current that creates a magnetic force opposing rotation
- More current drawn → more force to overcome → more mechanical energy needed
- Electrical energy out = Mechanical energy in (minus losses)
If the induced current aided rotation (instead of opposing it): - The generator would accelerate itself - Energy would be created from nothing - This violates conservation of energy