NAMES: ....................................................... 2025 A \( 15 \mathrm{~kW}, 400 \mathrm{~V}, 50 \mathrm{~Hz} \), 4 poles, three phase delta connected squirrel cage induction motor operates with efficiency of \( 88 \% \) and p.f. of 0.8 . If the machine operates for continuous duty, determine: (i) Starting current if a star/delta starter is used to start the motor (assume starting factor for D.O.L. starter to be 5). (ii) The full load torque, when the motor operates with a slip of \( 4 \% \) (iii) Size of switching devices (iv) Size of cables/wires (v) Size protective devices

To determine the various parameters for the given three-phase squirrel cage induction motor, we'll systematically approach each part of the problem. ### **Given Data:** - **Rated Power (P_rated):** 15 kW - **Line Voltage (V_L):** 400 V - **Frequency (f):** 50 Hz - **Number of Poles (p):** 4 - **Efficiency (η):** 88% (0.88) - **Power Factor (pf):** 0.8 (lagging) - **Slip (s):** 4% (0.04) for part (ii) --- ### **Part (i): Starting Current with Star/Delta Starter** **Approach:** 1. **Calculate Full-Load Current (I_rated):** The apparent power \( S \) is given by: \[ S = \frac{P_{\text{rated}}}{\eta \times pf} = \frac{15,000\,\text{W}}{0.88 \times 0.8} \approx 21,306\,\text{VA} \] For a three-phase system: \[ S = \sqrt{3} \times V_L \times I_{\text{rated}} \] \[ I_{\text{rated}} = \frac{S}{\sqrt{3} \times V_L} = \frac{21,306}{1.732 \times 400} \approx 30.78\,\text{A} \] 2. **Determine Starting Current for Star/Delta Starter:** The starting current for a Direct-On-Line (DOL) starter is given as a multiple of the rated current (starting factor = 5): \[ I_{\text{start,DOL}} = 5 \times I_{\text{rated}} = 5 \times 30.78 \approx 153.9\,\text{A} \] A star/delta starter reduces the starting voltage to \( \frac{V_L}{\sqrt{3}} \), thereby reducing the starting current by the same factor: \[ I_{\text{start,SD}} = \frac{I_{\text{start,DOL}}}{\sqrt{3}} \approx \frac{153.9}{1.732} \approx 88.8\,\text{A} \] **Answer:** The starting current when using a star/delta starter is approximately **89 A**. --- ### **Part (ii): Full Load Torque at 4% Slip** **Approach:** 1. **Determine Synchronous Speed (N_s):** \[ N_s = \frac{120 \times f}{p} = \frac{120 \times 50}{4} = 1500\,\text{RPM} \] 2. **Determine Actual Speed (N):** \[ N = N_s \times (1 - s) = 1500 \times (1 - 0.04) = 1440\,\text{RPM} \] 3. **Calculate Mechanical Power Output (P_mech):** \[ P_{\text{mech}} = P_{\text{rated}} \times \eta = 15,000\,\text{W} \times 0.88 = 13,200\,\text{W} \] 4. **Determine Angular Speed (ω):** \[ \omega = \frac{2\pi N}{60} = \frac{2\pi \times 1440}{60} = 48\pi\,\text{rad/s} \approx 150.8\,\text{rad/s} \] 5. **Calculate Torque (T):** \[ T = \frac{P_{\text{mech}}}{\omega} = \frac{13,200}{150.8} \approx 87.5\,\text{Nm} \] **Answer:** The full load torque at 4% slip is approximately **88 Nm**. --- ### **Part (iii): Size of Switching Devices** **Approach:** - **Switching devices** (like contactors) must handle the maximum current. Since the starting current with a star/delta starter is approximately **89 A**, the contactor should be rated above this value to ensure reliability and safety. - **Common Practice:** Use a safety margin, often **125%** of the maximum expected current. \[ \text{Contactor Rating} \geq 1.25 \times 89\,\text{A} \approx 112\,\text{A} \] However, commercially available contactors typically come in standard sizes such as 100 A or 125 A. Given that 100 A is slightly below, it's prudent to opt for a **125 A** contactor. **Answer:** Use contactors (switching devices) rated for at least **125 A**. --- ### **Part (iv): Size of Cables/Wires** **Approach:** - The conductor size must accommodate the **full-load current** of approximately **31 A**. - **Using Standard Ampacity Tables:** - For copper conductors, a **6 mm²** cross-sectional area is typically rated for up to **36 A**, which provides a comfortable margin for 31 A. **Answer:** Employ **6 mm²** copper conductors for the motor’s cables/wires. --- ### **Part (v): Size of Protective Devices** **Approach:** 1. **Determine Protective Device Rating:** - Protective devices (like circuit breakers) are usually sized at **125%** of the full-load current to allow for normal operation without nuisance tripping. - \[ \text{Breaker Rating} = 1.25 \times 31\,\text{A} \approx 39\,\text{A} \] 2. **Consider Motor Starting:** - Since a star/delta starter reduces the starting current, the protective device must tolerate up to **89 A** momentarily without tripping. - **Solution:** Use a breaker with **thermal-magnetic** characteristics that can handle higher inrush currents temporarily. **Answer:** Use a protective circuit breaker rated at approximately **40 A** to ensure proper protection without tripping during startup. --- ### **Summary of Answers:** 1. **Starting Current with Star/Delta Starter:** ~89 A 2. **Full Load Torque at 4% Slip:** ~88 Nm 3. **Size of Switching Devices:** Contactors rated at least 125 A 4. **Size of Cables/Wires:** 6 mm² copper conductors 5. **Size of Protective Devices:** 40 A circuit breaker