Math Problem Statement
solve and eplain
Solution
To solve this example for a three-phase induction motor, we need to calculate the speed, output torque and power, stator current, power factor, and efficiency when the motor is operated at rated voltage and frequency, given a slip of 1.6%.
The provided motor data are as follows:
- Motor power: 20 kW (rated power)
- Rated voltage: 460 V (line-to-line)
- Frequency: 60 Hz
- Slip: 1.6% ()
- Number of poles: 6
1. Speed Calculation:
The synchronous speed of the motor is given by the formula:
where:
- (frequency)
- (number of poles)
Substituting the values:
The rotor speed is related to the slip by:
where is the slip. Thus:
2. Output Power and Torque:
The output power is given by:
To find the torque, use the formula:
where is the rotor angular velocity in radians per second:
Thus, the torque is:
3. Stator Current:
We can calculate the stator current using the power and voltage. For a three-phase system:
However, we need additional information about the power factor () and the impedance to find the exact stator current.
4. Power Factor:
The power factor is related to the impedance of the motor. Without more information on the phase angle, we can estimate using:
5. Efficiency:
The efficiency of the motor is given by:
With the input power at 20 kW and output power at 19.68 kW:
Summary:
- Speed:
- Torque:
- Efficiency:
- Power Factor and Stator Current would need further impedance calculations.
Would you like further details on calculating the stator current or power factor?
Here are 5 related questions you might be interested in:
- How can the equivalent circuit parameters of an induction motor be used to compute current?
- What factors affect the slip of an induction motor?
- How does the rotor resistance influence motor performance?
- What is the significance of the core loss in motor efficiency?
- How do changes in load affect the slip and speed of an induction motor?
Tip: Always ensure to compute synchronous speed correctly, as it's crucial for determining motor slip and performance.
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Math Problem Analysis
Mathematical Concepts
Electromagnetic Induction
Induction Motors
Power and Energy
Torque Calculation
Formulas
Ns = (120 * f) / P
Nr = Ns * (1 - s)
Pout = Pin - Plosses
T = Pout / ωr
η = Pout / Pin
Theorems
Slip Speed Relation
Electromechanical Torque Equation
Suitable Grade Level
University Level (Electrical Engineering)
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