Calculating Induction Motor Speed from Synchronous Generator Input
This problem involves a synchronous generator supplying power to an induction motor. To find the motor's running speed, we first need to determine the frequency of the power generated by the synchronous generator. This frequency then becomes the supply frequency for the induction motor, allowing us to calculate its synchronous speed and finally its actual speed based on the given slip.
Step 1: Calculate the Frequency Generated by the Synchronous Generator
The frequency (f) generated by a synchronous machine is related to its number of poles (P) and speed (N) by the formula:
\[ f = \frac{P \times N}{120} \]
Given:
- Number of poles of synchronous generator, \(P_{gen} = 6\)
- Speed of synchronous generator, \(N_{gen} = 1000\) r.p.m.
Substituting these values into the formula:
\[ f = \frac{6 \times 1000}{120} \]
\[ f = \frac{6000}{120} \]
\[ f = 50 \text{ Hz} \]
So, the synchronous generator produces power at a frequency of 50 Hz. This is the frequency supplied to the induction motor.
Step 2: Calculate the Synchronous Speed of the Induction Motor
The synchronous speed (\(N_s\)) of an induction motor is determined by the supply frequency (f) and the motor's number of poles (P). The formula is:
\[ N_s = \frac{120 \times f}{P} \]
Given:
- Supply frequency, \(f = 50\) Hz (from the generator)
- Number of poles of induction motor, \(P_{motor} = 4\)
Substituting these values into the formula:
\[ N_s = \frac{120 \times 50}{4} \]
\[ N_s = \frac{6000}{4} \]
\[ N_s = 1500 \text{ r.p.m.} \]
The synchronous speed of the 4-pole induction motor with a 50 Hz supply is 1500 r.p.m.
Step 3: Calculate the Actual Speed of the Induction Motor
An induction motor always runs at a speed slightly less than its synchronous speed. The difference is expressed as slip (s). The actual rotor speed (\(N_r\)) is calculated using the formula:
\[ N_r = N_s \times (1 - s) \]
Given:
- Synchronous speed, \(N_s = 1500\) r.p.m.
- Slip, \(s = 4\% = 0.04\)
Substituting these values into the formula:
\[ N_r = 1500 \times (1 - 0.04) \]
\[ N_r = 1500 \times (0.96) \]
\[ N_r = 1440 \text{ r.p.m.} \]
The actual running speed of the induction motor with a slip of 4% is 1440 r.p.m.
Therefore, the motor speed will be 1440 r.p.m.
Summary of Calculations
| Parameter | Value / Formula | Calculation |
|---|---|---|
| Generator Poles (\(P_{gen}\)) | 6 | Given |
| Generator Speed (\(N_{gen}\)) | 1000 r.p.m. | Given |
| Supply Frequency (f) | \(f = \frac{P_{gen} \times N_{gen}}{120}\) | \(f = \frac{6 \times 1000}{120} = 50\) Hz |
| Motor Poles (\(P_{motor}\)) | 4 | Given |
| Motor Synchronous Speed (\(N_s\)) | \(N_s = \frac{120 \times f}{P_{motor}}\) | \(N_s = \frac{120 \times 50}{4} = 1500\) r.p.m. |
| Motor Slip (s) | 4% or 0.04 | Given |
| Motor Actual Speed (\(N_r\)) | \(N_r = N_s \times (1 - s)\) | \(N_r = 1500 \times (1 - 0.04) = 1440\) r.p.m. |
Revision Table: Key Concepts
| Concept | Definition | Relevance in Problem |
|---|---|---|
| Synchronous Generator | An AC generator whose speed is synchronized with the frequency of the generated voltage (\(N = \frac{120f}{P}\)). | Source of power determining the supply frequency for the motor. |
| Induction Motor | An AC motor that runs slightly below synchronous speed, with torque produced by induced currents in the rotor. | The machine whose running speed needs to be calculated. |
| Synchronous Speed (\(N_s\)) | The speed of the rotating magnetic field in an AC motor (\(N_s = \frac{120f}{P}\)). | A reference speed for the induction motor, calculated from supply frequency and poles. |
| Slip (s) | The relative speed difference between the synchronous speed and the rotor speed, expressed as a fraction or percentage (\(s = \frac{N_s - N_r}{N_s}\)). | Quantifies how much slower the rotor runs than the magnetic field, used to find actual speed. |
Additional Information: Synchronous vs. Induction Machines
Synchronous machines and induction machines are both types of AC electrical machines but operate on different principles regarding speed synchronization.
- Synchronous Machines: These machines, whether operating as generators or motors, maintain a constant speed relative to the frequency of the AC power. The rotor speed is directly proportional to the supply frequency and inversely proportional to the number of poles (\(N = \frac{120f}{P}\)). They require a DC excitation current for the rotor field.
- Induction Machines: These machines are the most common type of motor. They operate based on the principle of electromagnetic induction. The rotor currents are induced by the rotating magnetic field of the stator. For induction to occur and torque to be produced, the rotor must rotate at a speed different from (usually slower than) the synchronous speed. This speed difference is called slip. Induction motors do not require a separate DC excitation source for the rotor (for squirrel cage and wound rotor types without external DC).
In this problem, the synchronous generator establishes the system frequency, and the induction motor then operates based on that frequency and its own characteristics (poles and slip).
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