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Calculation No.: Calc-001
Revision: A
Description:
Created By: Engineer Name
Project: Project Name
Project No.: P001
Client: Client Name

2. INTRODUCTION

This calculation performs comprehensive steady-state analysis of a three-phase induction motor under DOL control. The analysis uses per-phase equivalent circuit modeling with Thévenin simplification to calculate torque-speed characteristics, current draw, efficiency, and power output across the full speed range from standstill to synchronous speed. **Motor Type:** Three-Phase **Control Method:** DOL **Rated Voltage:** 400.0 V line-to-line, star-connected equivalent circuit **Rated Frequency:** 50.0 Hz **Number of Poles:** 4 The calculations are based on IEEE Standard 112 for polyphase induction motors and NEMA MG 1 standards.

3. INPUT PARAMETERS

Motor equivalent circuit parameters and operating conditions:
Parameter Symbol Value Notes
Motor Type - Three-Phase Winding configuration
Control Method - DOL Starting and running control method
Rated Voltage V_LL 400 V Rated RMS line-to-line voltage
Equivalent-Circuit Connection - Star Star: V_ph = V_LL/√3; delta: V_ph = V_LL (line current √3 × phase current)
Line Frequency f_line 50 Hz Rated supply frequency
Number of Poles p 4 - Magnetic poles in motor stator
Stator Resistance R₁ 200 mΩ Resistance of stator winding per phase
Stator Reactance X₁ 500 mΩ Leakage reactance of stator winding per phase
Rotor Resistance R₂ 300 mΩ Rotor resistance per phase, referred to stator
Rotor Reactance X₂ 600 mΩ Rotor leakage reactance per phase, referred to stator
Magnetizing Reactance Xₘ 20 Ω Magnetizing reactance per phase
Core Loss P_core 75 W Rotational losses due to hysteresis and eddy currents
Friction & Windage Loss P_fw 25 W Mechanical losses due to friction and windage

5. CALCULATIONS

Per-phase steady-state equivalent circuit analysis:

6. Synchronous Speed

Synchronous speed is the speed of the rotating magnetic field. Where: • f = 50.0 Hz (operating frequency) • p = 4 (number of poles) The rotor rotates slightly slower than synchronous speed, with the difference called slip.

Formula: $$ \begin{aligned} $ n_s &= \frac{120 \cdot f}{p} $ \\ &= \frac{120 \times 50.0}{4} = 1500.0 \; \text{RPM} $ \end{aligned} $$

Angular velocity of the rotating magnetic field in radians per second.

Formula: $ \omega_s = \frac{2\pi \cdot n_s}{60} = \frac{2\pi \times 1500.0}{60} = 157.080 \; \text{rad/s} $

7. Per-Phase Voltage

Star (Y) connected equivalent circuit: each phase sees the line-to-neutral voltage and the line current equals the phase current.

Formula: $ V_{ph} = \frac{V_{LL}}{\sqrt{3}} = \frac{400.0}{\sqrt{3}} = 230.94 \; \text{V} $

8. Thévenin Equivalent Circuit

The stator circuit is simplified using Thévenin theorem as seen from the rotor terminals. This reduces the complex stator impedance network to a simple voltage source (V_th) in series with equivalent impedance (R_th + jX_th).

Formula: $$ \begin{aligned} $ V_{th} &= V_{eff} \times \frac{X_m}{\sqrt{R_1^2 + (X_1 + X_m)^2}} $ \\ &= 230.94 \times \frac{20.000}{\sqrt{0.200^2 + (0.500 + 20.000)^2}} = 230.94 \times \frac{20.000}{20.501} = 225.297 \; \text{V} $ \end{aligned} $$

Equivalent resistance as seen from rotor terminals = 0.190345 Ω

Formula: $ R_{th} = \frac{R_1 \cdot X_m^2}{R_1^2 + (X_1 + X_m)^2} = 0.190345 \; \Omega $

Equivalent reactance as seen from rotor terminals = 0.489662 Ω

Formula: $ X_{th} = \frac{X_1^2 \cdot X_m + X_1 \cdot X_m^2 + R_1^2 \cdot X_m}{R_1^2 + (X_1 + X_m)^2} = 0.489662 \; \Omega $

9. Rotor Current, Stator Current and Torque (Three-Phase)

Rotor current referred to the stator, used for the developed torque. At starting (s=1), rotor current is maximum. Where: • s = slip = (n_s - n_r) / n_s • n_r = rotor speed (RPM) Example shown for starting condition (s=1). I₂ excludes the magnetising current, so it is not the current drawn from the supply — see the stator current below.

Formula: $$ \begin{aligned} $ I_2 &= \frac{V_{th}}{\sqrt{(R_{th} + \frac{R_2}{s})^2 + (X_{th} + X_{2,eff})^2}} $ \\ &= 1} = \frac{225.297}{\sqrt{(0.1903 + \frac{0.300}{1})^2 + (0.4897 + 0.600)^2}} = 188.548 \; \text{A} $ \end{aligned} $$

Stator phase current including the magnetising current (Chapman, Electric Machinery Fundamentals, Example 6-5). Star connection: the line current equals I₁. The current-speed curve and the reported starting currents are line currents. At starting (s=1): I₁ = 194.225 A (DOL locked-rotor line current at rated voltage and frequency: 194.22 A).

Formula: $$ \begin{aligned} $ I_1 &= \frac{V_{ph}}{\left| R_1 + jX_1 + jX_m \parallel \left(\frac{R_2}{s} + jX_2\right) \right|} $ \\ &= 1} = \frac{230.94}{\left| 0.4827 + j1.0866 \right|} = 194.225 \; \text{A} $ \end{aligned} $$

Electromagnetic torque developed by the motor. The factor of 3 accounts for three-phase operation. At starting (s=1): T = 203.69 N·m The complete torque-speed curve is calculated by repeating this calculation for slip values from 1.0 (standstill) to 0.001 (near synchronous speed).

Formula: $$ \begin{aligned} $ T &= \frac{3 \cdot I_2^2 \cdot \frac{R_2}{s}}{\omega_s} $ \\ &= 1} = \frac{3 \times 188.548^2 \times 0.300}{157.080} = 203.69 \; \text{N·m} $ \end{aligned} $$

10. Power Flow Analysis

I²R losses in stator windings. Three phases, each carrying the stator current I₁ of the full equivalent circuit (magnetising current included).

Formula: $$ \begin{aligned} $ I_1 &= \frac{V}{\left| R_1 + jX_1 + jX_m \parallel (R_2/s + jX_2) \right|}, \quad P_{scl} = 3 \times I_1^2 \times R_1 $ \\ &= 1} = 3 \times 194.225^2 \times 0.200 = 22633.9 \; \text{W} $ \end{aligned} $$

Electromagnetic power transferred across air gap from stator to rotor.

Formula: $$ \begin{aligned} $ P_{ag} &= T \times \omega_s $ \\ &= 1} = 203.69 \times 157.080 = 31995.1 \; \text{W} $ \end{aligned} $$

Total electrical power input to motor.

Formula: $$ \begin{aligned} $ P_{in} &= P_{ag} + P_{scl} + P_{core} $ \\ &= 1} = 31995.1 + 22633.9 + 75.0 = 54704.0 \; \text{W} $ \end{aligned} $$

Portion of air gap power converted to mechanical form. At starting (s=1), P_conv = 0 (all power dissipated as heat in rotor).

Formula: $ P_{conv} = P_{ag} \times (1 - s) $

Useful mechanical power available at shaft after subtracting friction/windage (25.0 W). Core loss is already accounted for in P_in.

Formula: $ P_{out} = P_{conv} - P_{fw} $

Ratio of output power to input power. Maximum efficiency achieved at 1478.9 RPM with η_max = 95.25%.

Formula: $ \eta = \frac{P_{out}}{P_{in}} \times 100\% $

11. PERFORMANCE SUMMARY

Starting torque with direct-on-line start at rated frequency (50.0 Hz): T_start = 203.69 N·m; locked-rotor stator (line) current I₁,start = 194.22 A Maximum (pull-out) torque: T_max = 373.86 N·m at 1093.2 RPM This is the maximum torque the motor can develop. Operating beyond this point causes the motor to stall. Pull-out slip = 0.271 (27.1%) Maximum efficiency: η_max = 95.25% at 1478.9 RPM This is the most efficient operating point. Slip at maximum efficiency = 0.0141 (1.41%) **Operating Characteristics:** • Synchronous Speed: 1500.0 RPM • Operating Frequency: 50.0 Hz • Effective Voltage: 230.94 V The complete torque-speed, current-speed, and efficiency-speed curves are calculated across 101 points from standstill (s=1) to near synchronous speed (s=0.001); maximum torque and maximum efficiency are then located between the sweep points by golden-section search.

12. RESULTS SUMMARY

Comprehensive motor performance analysis results:
Parameter Symbol Value Notes
Operating Frequency f_op 50 Hz Actual operating frequency
Per-Phase Voltage V_ph 231 V Rated voltage across one phase of the equivalent circuit (V_LL/√3 star, V_LL delta)
Effective Voltage V_eff 231 V Per-phase voltage after V/Hz scaling (held at rated voltage above base frequency)
Synchronous Speed n_s 1.5 kRPM Speed of rotating magnetic field
Thévenin Voltage V_th 225 V Thévenin equivalent voltage
Thévenin Resistance R_th 190 mΩ Thévenin equivalent resistance
Thévenin Reactance X_th 490 mΩ Thévenin equivalent reactance
DOL Starting Torque T_start 204 N·m Starting torque with direct-on-line start at rated frequency
DOL Starting Current I_1,start 194 A Locked-rotor line current at rated voltage and frequency (stator I₁; × √3 for delta)
Maximum Torque (Pull-out) T_max 374 N·m Maximum torque capacity before motor stalls
Speed at Max Torque n_Tmax 1.09 kRPM Rotor speed at maximum torque
Maximum Efficiency η_max 95.2 % Maximum efficiency operating point
Speed at Max Efficiency n_ηmax 1.48 kRPM Rotor speed at maximum efficiency

14. DESIGN SUMMARY

**Motor Analysis Summary:** This three-phase induction motor has been analyzed under DOL control conditions. Key findings: **Torque Characteristics:** • DOL Starting Torque: 203.69 N·m (at 50.0 Hz) • Maximum Torque: 373.86 N·m (at 1093.2 RPM) • Torque Ratio: T_max / T_start = 1.84 **Efficiency:** • Maximum Efficiency: 95.25% (at 1478.9 RPM) • Efficiency at starting is very low due to high copper losses **Operating Conditions:** • Synchronous Speed: 1500.0 RPM • Operating Frequency: 50.0 Hz **Recommendations:** • Consider higher rotor resistance for improved starting torque (currently T_max/T_start = 1.84) • Consider VFD or soft starter to reduce starting current and mechanical stress • Operate near 1479 RPM for maximum efficiency (95.2%) • Ensure load torque remains below 373.9 N·m to prevent motor stall

15. APPROVAL

This calculation has been prepared in accordance with IEEE Standard 112 and NEMA MG 1.
Created By
Engineer Name
Date: ..........................
Checked
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Date: ..........................
Approved
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Date: ..........................
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