A 400V, 50Hz slipring type three phase induction motor is star connected and has per phase rotor resistance and standstill reactance of 0.5 and 1.5 ohm respectively. Calculate resistance to be added per phase to achieve maximum torque at starting.
2 Answers
To calculate the resistance that must be added to the rotor circuit of a slip-ring type induction motor to achieve maximum torque at starting, we can use the following formula.
For maximum torque to occur, the total rotor resistance \( R_{\text{total}} \) must equal the rotor reactance at standstill \( X_{2} \), i.e.,
\[
R_{\text{total}} = X_{2}
\]
The total rotor resistance \( R_{\text{total}} \) is the sum of the rotor’s own resistance \( R_{2} \) and the additional resistance \( R_{\text{ext}} \) added externally:
\[
R_{\text{total}} = R_{2} + R_{\text{ext}}
\]
### Step-by-Step Calculation:
**Given:**
- Supply Voltage: 400 V (phase voltage is not needed since we are only working with rotor parameters)
- Frequency: 50 Hz
- Rotor resistance \( R_2 \) = 0.5 Ω
- Rotor reactance \( X_2 \) = 1.5 Ω (at standstill)
- The motor is star-connected (but this does not affect the rotor resistance calculation for achieving maximum torque).
**To achieve maximum torque:**
\[
R_{\text{total}} = X_2 = 1.5 \, \Omega
\]
Now, solving for the external resistance \( R_{\text{ext}} \):
\[
R_{\text{ext}} = R_{\text{total}} - R_2
\]
Substitute the values:
\[
R_{\text{ext}} = 1.5 \, \Omega - 0.5 \, \Omega
\]
\[
R_{\text{ext}} = 1.0 \, \Omega
\]
### Conclusion:
The resistance that needs to be added per phase to achieve maximum torque at starting is **1.0 Ω**.
For maximum torque to occur, the total rotor resistance \( R_{\text{total}} \) must equal the rotor reactance at standstill \( X_{2} \), i.e.,
\[
R_{\text{total}} = X_{2}
\]
The total rotor resistance \( R_{\text{total}} \) is the sum of the rotor’s own resistance \( R_{2} \) and the additional resistance \( R_{\text{ext}} \) added externally:
\[
R_{\text{total}} = R_{2} + R_{\text{ext}}
\]
### Step-by-Step Calculation:
**Given:**
- Supply Voltage: 400 V (phase voltage is not needed since we are only working with rotor parameters)
- Frequency: 50 Hz
- Rotor resistance \( R_2 \) = 0.5 Ω
- Rotor reactance \( X_2 \) = 1.5 Ω (at standstill)
- The motor is star-connected (but this does not affect the rotor resistance calculation for achieving maximum torque).
**To achieve maximum torque:**
\[
R_{\text{total}} = X_2 = 1.5 \, \Omega
\]
Now, solving for the external resistance \( R_{\text{ext}} \):
\[
R_{\text{ext}} = R_{\text{total}} - R_2
\]
Substitute the values:
\[
R_{\text{ext}} = 1.5 \, \Omega - 0.5 \, \Omega
\]
\[
R_{\text{ext}} = 1.0 \, \Omega
\]
### Conclusion:
The resistance that needs to be added per phase to achieve maximum torque at starting is **1.0 Ω**.
To achieve maximum torque at starting for a slipring type three-phase induction motor, you need to add resistance to the rotor circuit. The resistance required can be determined using the formula for maximum torque, which occurs when the rotor resistance is equal to the standstill reactance.
Given:
- Rotor resistance per phase \( R_r = 0.5 \Omega \)
- Standstill reactance per phase \( X_r = 1.5 \Omega \)
The maximum torque at starting is achieved when:
\[ R_{added} = X_r - R_r \]
Substituting the given values:
\[ R_{added} = 1.5 \Omega - 0.5 \Omega = 1.0 \Omega \]
So, to achieve maximum torque at starting, you need to add \( 1.0 \Omega \) of resistance per phase.
Given:
- Rotor resistance per phase \( R_r = 0.5 \Omega \)
- Standstill reactance per phase \( X_r = 1.5 \Omega \)
The maximum torque at starting is achieved when:
\[ R_{added} = X_r - R_r \]
Substituting the given values:
\[ R_{added} = 1.5 \Omega - 0.5 \Omega = 1.0 \Omega \]
So, to achieve maximum torque at starting, you need to add \( 1.0 \Omega \) of resistance per phase.