LibraryNumerical Problems on Synchronous Motors

Numerical Problems on Synchronous Motors

Learn about Numerical Problems on Synchronous Motors as part of GATE Electrical Engineering - Power Systems and Machines

Numerical Problems on Synchronous Motors for GATE Electrical Engineering

This module focuses on solving numerical problems related to synchronous motors, a crucial topic for the GATE Electrical Engineering exam, particularly within the Power Systems and Machines syllabus. We will cover key concepts and common problem types to build your problem-solving skills.

Understanding Synchronous Motor Fundamentals for Problem Solving

Before diving into numericals, it's essential to recall the fundamental principles of synchronous motors. These include the concept of rotating magnetic fields, synchronous speed, torque production, and the V-curves.

What is the formula for synchronous speed (Ns) of an AC motor?

Ns = (120 * f) / P, where f is the supply frequency and P is the number of poles.

Synchronous motors operate at a constant speed, known as the synchronous speed, determined by the supply frequency and the number of poles. The torque developed is proportional to the product of the resultant air gap flux and the stator current, and the sine of the load angle (δ).

Key Concepts for Numerical Problems

Several key parameters and equations are central to solving synchronous motor numericals:

Common Numerical Problem Types

Let's explore some typical numerical problem scenarios encountered in GATE exams:

Problem TypeKey Parameters InvolvedCore Concepts/Formulas
Calculating Torque and PowerTerminal Voltage (VtV_t), Back EMF (EfE_f), Synchronous Reactance (XsX_s), Load Angle (δ)Pmech=3VtEfXssin(δ)P_{mech} = \frac{3V_t E_f}{X_s} \sin(\delta), T=PmechωsT = \frac{P_{mech}}{\omega_s}
Determining Power FactorTerminal Voltage (VtV_t), Back EMF (EfE_f), Armature Current (IaI_a), Resistance (RaR_a), Reactance (XsX_s)Phasor diagram analysis: Vt=Ef+Ia(Ra+jXs)V_t = E_f + I_a(R_a + jX_s). Power factor = cos(ϕ)\cos(\phi) where ϕ\phi is the angle between VtV_t and IaI_a.
Finding Minimum Armature Current (Unity PF)Terminal Voltage (VtV_t), Synchronous Reactance (XsX_s), Load Angle (δ) at unity PFThe condition for minimum armature current occurs when the phasor EfE_f is perpendicular to the armature current IaI_a in the phasor diagram. This often involves finding the load angle for unity power factor.
Calculating Excitation Voltage (EfE_f) for a given PFTerminal Voltage (VtV_t), Armature Current (IaI_a), Resistance (RaR_a), Reactance (XsX_s), Desired Power FactorUsing the phasor equation Vt=Ef+Ia(Ra+jXs)V_t = E_f + I_a(R_a + jX_s), solve for EfE_f by considering the phase of IaI_a relative to VtV_t for the given power factor.
Stability LimitsMaximum power output, VtV_t, EfE_f, XsX_sThe maximum power output occurs when δ=90\delta = 90^{\circ}. The pull-out torque is the maximum torque the motor can develop before losing synchronism.

Example Problem Walkthrough

Let's consider a typical problem. A 3-phase, 1000 kW, 11 kV, 50 Hz synchronous motor has a synchronous reactance of 4 Ω\Omega/phase and negligible armature resistance. It is operating at unity power factor. Calculate the back EMF per phase.

To solve this, we first need to determine the armature current (IaI_a). Since the motor is operating at unity power factor and delivering 1000 kW, the output power is Pout=1000 kW=106 WP_{out} = 1000 \text{ kW} = 10^6 \text{ W}. The line voltage is VL=11 kVV_L = 11 \text{ kV}. The phase voltage is Vph=VL/3=11000/3 VV_{ph} = V_L / \sqrt{3} = 11000 / \sqrt{3} \text{ V}. The input power (assuming efficiency close to 1 for simplicity in this context, or if efficiency is given, use that) is Pin=PoutηP_{in} = \frac{P_{out}}{\eta}. For unity power factor, Pin=3VLILcos(ϕ)=3VLILP_{in} = \sqrt{3} V_L I_L \cos(\phi) = \sqrt{3} V_L I_L. Thus, IL=Pin3VLI_L = \frac{P_{in}}{\sqrt{3} V_L}. The armature current per phase is Ia=IL/3I_a = I_L / \sqrt{3}. With Ra=0R_a = 0, the phasor equation is Vph=Ef+jIaXsV_{ph} = E_f + jI_a X_s. Since it's unity power factor, IaI_a is in phase with VphV_{ph} (or we can align VphV_{ph} with the real axis, and IaI_a will also be on the real axis). Therefore, Ef=VphjIaXsE_f = V_{ph} - jI_a X_s. We need to be careful with the phasor representation. A more robust approach is to use the phasor diagram. At unity power factor, the armature current IaI_a is in phase with the terminal voltage VphV_{ph}. The back EMF EfE_f can be found using Ef=VphIa(Ra+jXs)E_f = V_{ph} - I_a(R_a + jX_s). With Ra=0R_a=0, Ef=VphjIaXsE_f = V_{ph} - jI_a X_s. Let's assume VphV_{ph} is along the real axis. Then Vph=1100030V_{ph} = \frac{11000}{\sqrt{3}} \angle 0^{\circ}. The input power is Pin=1000×103ηP_{in} = \frac{1000 \times 10^3}{\eta}. Assuming η1\eta \approx 1, Pin=1000 kWP_{in} = 1000 \text{ kW}. Pin=3VphIacos(ϕ)P_{in} = 3 V_{ph} I_a \cos(\phi). At unity PF, cos(ϕ)=1\cos(\phi) = 1. So, 1000×103=3×110003imesIaimes11000 \times 10^3 = 3 \times \frac{11000}{\sqrt{3}} imes I_a imes 1. Ia=1000imes103imes33imes1100052.4 AI_a = \frac{1000 imes 10^3 imes \sqrt{3}}{3 imes 11000} \approx 52.4 \text{ A}. Now, Ef=VphjIaXs=110003j(52.4)(4)=6350.8j209.6E_f = V_{ph} - jI_a X_s = \frac{11000}{\sqrt{3}} - j(52.4)(4) = 6350.8 - j209.6. The magnitude of EfE_f is Ef=6350.82+(209.6)26354.3 V|E_f| = \sqrt{6350.8^2 + (-209.6)^2} \approx 6354.3 \text{ V}. This is the phase voltage. The line voltage Ef,line=3Ef11006 VE_{f,line} = \sqrt{3} |E_f| \approx 11006 \text{ V}. A more direct method using the power angle: Pin=3VtEfXssin(δ)P_{in} = \frac{3V_t E_f}{X_s} \sin(\delta). At unity PF, the load angle δ\delta can be found from the phasor diagram. The condition for unity power factor is when EfE_f leads VtV_t by an angle α\alpha such that tan(α)=Xs/Ra\tan(\alpha) = X_s/R_a. Since Ra=0R_a=0, this implies α=90\alpha = 90^{\circ}. However, this is for a generator. For a motor, at unity PF, IaI_a is in phase with VtV_t. The phasor equation is Vt=Ef+IaZsV_t = E_f + I_a Z_s. Ef=VtIaZsE_f = V_t - I_a Z_s. Let VtV_t be along the real axis. Vt=1100030V_t = \frac{11000}{\sqrt{3}} \angle 0^{\circ}. IaI_a is also along the real axis for unity PF. Ia=Pin3VL=1000imes1033imes11imes10352.4 AI_a = \frac{P_{in}}{\sqrt{3} V_L} = \frac{1000 imes 10^3}{\sqrt{3} imes 11 imes 10^3} \approx 52.4 \text{ A}. So, Ia=52.40I_a = 52.4 \angle 0^{\circ}. Zs=j4Z_s = j4. Ef=1100030(52.40)(j4)=6350.8j209.6E_f = \frac{11000}{\sqrt{3}} \angle 0^{\circ} - (52.4 \angle 0^{\circ})(j4) = 6350.8 - j209.6. Ef=6350.82+(209.6)26354.3 V|E_f| = \sqrt{6350.8^2 + (-209.6)^2} \approx 6354.3 \text{ V}. This is the phase back EMF. The question asks for back EMF, which usually implies the phase value unless specified otherwise.

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The solution involves calculating the armature current at unity power factor and then using the phasor equation Vph=Ef+IaZsV_{ph} = E_f + I_a Z_s (with Ra=0R_a=0) to find EfE_f. The key is to correctly represent the phasors and their relationships.

Tips for Solving Numerical Problems

Always draw a phasor diagram! It's your best friend for visualizing the relationships between voltages, currents, and impedances in AC circuits.

  1. Identify Given and Required: Clearly list all known parameters and what needs to be calculated.
  2. Choose a Reference Phasor: Usually, the terminal voltage (VtV_t) is taken as the reference (angle 0°).
  3. Determine Armature Current (IaI_a): Calculate IaI_a based on power, voltage, and power factor. Pay attention to the phase angle of IaI_a relative to VtV_t for different power factors.
  4. Apply Phasor Equation: Use Vt=Ef+IaZsV_t = E_f + I_a Z_s (or Ef=VtIaZsE_f = V_t - I_a Z_s) to solve for the unknown.
  5. Check Units and Consistency: Ensure all units are consistent (e.g., Volts, Amperes, Ohms, Watts).

Advanced Concepts and Practice

For higher-level problems, you might encounter concepts like saliency (using d-q axis reactances), hunting, and starting methods. Practice a wide variety of problems from standard textbooks and previous GATE papers to build confidence and speed.

What is the primary advantage of operating a synchronous motor at a leading power factor?

It can supply reactive power to the grid, improving the overall power factor of the system and potentially reducing electricity bills.

Learning Resources

Synchronous Motor Numerical Problems - GATE Electrical Engineering(blog)

This blog post provides solved numerical examples for synchronous motors, focusing on common GATE exam patterns and formulas.

Synchronous Motor Numerical Problems with Solutions(blog)

Offers a collection of solved numerical problems on synchronous motors, covering various aspects like power, torque, and power factor.

GATE Electrical Engineering - Synchronous Machines(tutorial)

A comprehensive course module on synchronous machines, likely including video lectures and practice problems relevant to GATE.

Power System Analysis - Synchronous Machines(video)

NPTEL lectures on power systems, with specific modules dedicated to synchronous machines, often including problem-solving sessions.

Synchronous Motor Numerical Problems - Electrical Engineering(blog)

This resource presents solved numerical problems on synchronous motors, with clear explanations of the steps involved.

Electrical Machines - Synchronous Motor(documentation)

Lecture notes on Electrical Machines II, which typically cover synchronous motors in detail, including theoretical concepts and problem-solving approaches.

GATE Electrical Engineering - Power Systems and Machines(blog)

A forum and resource hub for GATE Electrical Engineering, featuring discussions and solved problems related to power systems and machines.

Synchronous Motor - Working Principle, Types, Applications(blog)

While not solely focused on numericals, this article provides a solid foundation on synchronous motor principles, which is essential for understanding the context of numerical problems.

Electrical Engineering - Synchronous Motor Problems(blog)

A community-driven resource with shared problems and solutions related to synchronous motors, often including GATE-level questions.

Synchronous Machines - GATE Electrical Engineering(tutorial)

This page offers practice questions and answers specifically for Synchronous Machines in GATE Electrical Engineering, which will include numerical problems.