৪৯তম বিসিএস ⎯ তথ্য ও যোগাযোগ প্রযুক্তি (EEE) [ ৮৯২]

When the motor runs at synchronous speed, there is no relative motion between rotor and revolving flux produced by stator currents. Hence e.m.f. in the damper winding placed on the rotor is zero.

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

The resultant flux d in a synchronous motor is fixed in value as the applied stator voltage is constant. Since synchronous motor is a doubly excited machine, both d.c. and a.c, sources may co-operate to produce this fixed resultant flux o. For a given mechanical load, if the entire resultant flux is provided by d.c, excitation alone, the stator will absorb no reactive power (as it is not contributing to the resultant flux). Hence the p.f. of the motor will be unity.

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

Synchronous Reactance (Xs​):The synchronous reactance is the combined reactance of the stator inductance and the magnetic leakage reactance in a synchronous motor. It is a key parameter in the equivalent circuit of the motor, and it represents the opposition to the flow of alternating current (AC) due to the motor’s inductive characteristics.
The synchronous reactance contributes to the impedance seen by the stator, and it significantly impacts the motor's ability to regulate voltage under varying load conditions.


Voltage Regulation:Voltage regulation refers to the ability of the motor to maintain a constant terminal voltage despite changes in the load. When the load on the synchronous motor increases, the terminal voltage can drop due to the drop in the current through the motor’s reactance. The larger the synchronous reactance, the larger the voltage drop under load, resulting in poorer voltage regulation.
The voltage regulation can be calculated as the difference between the no-load voltage and the full-load voltage as a percentage of the full-load voltage. The higher the synchronous reactance, the greater the drop in voltage from no-load to full-load conditions, leading to a larger voltage regulation.


Why the Other Options Are Incorrect:
(ক) The motor’s starting torque:The starting torque of a synchronous motor is primarily influenced by the initial conditions when the motor is started. Synchronous motors typically do not generate a large starting torque, as they need to be synchronized with the supply frequency. The starting torque is more influenced by the synchronous reactance and the field excitation, but not as directly as voltage regulation.


(খ) The motor’s speed:The speed of a synchronous motor is determined by the supply frequency and the number of poles in the motor, and it is fixed at synchronous speed. The synchronous reactance does not directly affect the motor's speed, as it operates at synchronous speed once started.


(ঘ) The motor’s power factor: While the synchronous reactance does influence the power factor (it is related to the phase difference between current and voltage), it is not the primary determinant of the motor’s power factor. The excitation voltage plays a more significant role in determining the power factor of the motor.

Source: Electric Machinery Fundamentals by Stephen J. Chapman

The equation provided below represents the angle of the field (δ) in the context of an alternating current (AC) system, possibly related to synchronous machines or rotating magnetic fields. Let’s break down the formula and the calculation:

The Formula:

δ = (p * α) / 2

Where:

δ = Angle of the field (likely in degrees or radians),

p = Number of poles of the motor (or machine),

α = The electrical angle (possibly related to the phase angle or the electrical angle of the rotor).

Given Values:

p = 40 (this indicates the number of poles),

α = 1 (it seems this represents a unit angle or phase angle, often taken as 1 for simplicity or normalization).

Calculation:

Substitute the given values into the formula:

δ = (40 × 1) / 2 = 20°

δ = 20°: This result represents the angle of the rotor’s magnetic field relative to the stator’s magnetic field. It could be interpreted as the angle between the rotor field and the stator field in an AC system like a synchronous motor. This angle is critical in the operation of machines, as it affects torque production, efficiency, and synchrony between the rotor and stator.


Source: Principles of Electrical Machines by V.K. Mehta, Rohit Mehta

Effect of Reduced Excitation on Power Factor:
When the excitation of a synchronous motor is reduced, it means that the strength of the rotor's magnetic field is decreased. This change affects the relationship between the rotor’s magnetic field and the stator’s rotating magnetic field.

1. Underexcited Condition (Reduced Excitation):Underexcitation occurs when the excitation voltage (DC current supplied to the field windings) is reduced. With reduced excitation, the rotor’s magnetic field becomes weaker, causing the motor to lag more behind the stator’s rotating magnetic field. This results in an increased lagging current. As a result, the power factor of the motor becomes more lagging. In a lagging power factor, the current lags the voltage, meaning that more reactive power is drawn, and less real power is converted into useful mechanical energy.

2. Why Power Factor Becomes Lagging: The synchronous motor operates at a power factor that can be leading or lagging, depending on the level of excitation:

Overexcited (higher excitation): The motor’s power factor is leading, meaning the motor produces a strong rotor field and supplies reactive power to the system.

Underexcited (reduced excitation): The motor’s power factor becomes more lagging, meaning it draws more reactive power from the supply, making the motor act like an inductive load (similar to a lagging power factor).

3. Impact of Reduced Excitation: More Lagging Power Factor: With lower excitation, the rotor’s magnetic field is weaker, and the motor draws more current to maintain torque production. This current is lagging because of the weaker field, increasing the reactive power and making the motor more inductive.
If excitation continues to decrease too much, the motor may become unstable, but typically, in most practical cases, a slight reduction in excitation leads to a more lagging power factor.


Reference Book: Electric Machinery and Transformers by Bhag S. Guru and Huseyin R. Hiziroglu.

 An over-excited loaded synchronous motor supplies reactive power to the 3-phase line as well as absorbs active power (kW). But an unloaded over-excited synchronous motor performs only one function i.e., delivers reactive power to the 3-phase line, like a capacitor. Hence, the name synchronous capacitor.

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

Pull-out Torque:The pull-out torque is the maximum torque that a synchronous motor can develop. It is the point where the motor operates at the maximum load it can handle before it begins to lose synchrony with the stator field and slips out of synchronization. At the pull-out torque, the motor can no longer maintain synchrony with the rotating magnetic field of the stator and will either slow down or stop if the load is increased beyond this point.

Rotor Position During Pull-out Torque:At the point of pull-out torque, the rotor is slipping just enough to cause the poles of the rotor to be positioned mid-way between the North (N) and South (S) poles of the stator. In other words, when the motor is about to slip out of synchronization, the magnetic poles of the rotor are at an angle relative to the magnetic poles of the stator. Specifically, the rotor poles are positioned midway between the stator's North and South poles.
This position corresponds to the maximum torque condition, where the motor can no longer maintain synchronous operation at a constant speed due to excessive load.


Source: Principles of Electrical Machines by V.K. Mehta, Rohit mehta

Synchronous Speed (Ns): The synchronous speed of a synchronous motor is the speed at which the magnetic field of the stator rotates. It is given by the formula:

Ns = (120 × f) / p

Where:

Ns = Synchronous speed (in r.p.m.),

f = Supply frequency (in Hz),

p = Number of poles of the motor.

Rotor Speed (N):The rotor of a synchronous motor rotates at a speed close to the synchronous speed, but the difference between the synchronous speed and the rotor speed is due to **slip** (s).

Slip is defined as:

s = (Ns - N) / Ns

Where:

N = Rotor speed (in r.p.m.),

Ns = Synchronous speed (in r.p.m.),

s = Slip of the motor.

Rearranging the equation to solve for N (Rotor speed), we get:

N = Ns (1 - s)

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

The formula for synchronous speed of an induction motor is:

Ns = (120 × f) / P

Where:
Ns = Synchronous speed (in r.p.m.)
f = Supply frequency (in Hz)
P = Number of poles

Given:
Ns = 1500 r.p.m.
P = 4

Calculation:
f = (Ns × P) / 120

f = (1500 × 4) / 120

f = 6000 / 120

f = 50 Hz

Answer:
The supply frequency is 50 Hz.


(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

So that additional external resistance may be inserted in the rotor circuit at starting to increase the starting torque and decrease the starting cur-rent. As the motor gains speed, these external resistances are cut out of the rotor circuit.

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

 The torque characteristic of a 3-phase induction motor is similar to that of a DC shunt motor. Here’s why:

Torque Characteristics:
3-Phase Induction Motor:

The torque of a 3-phase induction motor increases with the square of the voltage and is proportional to the slip. The torque-speed characteristic of a 3-phase induction motor starts from zero torque at synchronous speed, increases to a maximum value, and then decreases as the motor approaches stall (pull-out torque). This is a characteristic that is somewhat similar to the behavior of DC motors.

DC Shunt Motor:The torque-speed characteristic of a DC shunt motor is relatively flat over a range of speeds. Like the induction motor, the torque in a DC shunt motor increases almost linearly with the armature current. In both types of motors, the torque increases with the increase in load current (or load), and the torque drops at high speeds.

The most important similarity is that both motors exhibit a positive torque-speed characteristic, where the torque increases with increasing load up to a certain point and then begins to decrease as the motor reaches its maximum operating speed or torque. 


Source: Principles of Electrical Machines by V.K. Mehta, Rohit mehta

 Induction Motor on No-Load: When an induction motor is running on no-load, the rotor is rotating at a speed close to the synchronous speed, and the motor is drawing only the magnetizing current (no significant torque is required from the motor). The rotor currents are induced by the rotating magnetic field created by the stator, but there is very little current flowing in the rotor, as the motor is not doing any mechanical work.This situation is similar to a transformer under no-load conditions.

Transformer Under No-Load: When a transformer operates on no-load, the secondary winding is open-circuited, meaning there is no load connected to the secondary side. In this case, the transformer still draws a small no-load current from the supply, primarily to magnetize the core and overcome core losses (hysteresis and eddy currents). This no-load current is similar to the magnetizing current in an induction motor, which is required to create the magnetic field but does not do any useful work.

Source: Principles of Electrical Machines by V.K. Mehta, Rohit mehta.

In an induction motor, only stator winding is fed from a.c. supply. The rotor winding derives voltage and power from the externally energised stator winding through the principle of mutual induction.

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

Rotor Core Loss at No-Load:

Core Loss (Iron Loss):
The core loss in the rotor of an induction motor primarily consists of hysteresis loss and eddy current loss. These losses are caused by the alternating magnetic field in the rotor, which induces currents in the rotor core.

However, at no-load, the rotor is rotating at a speed close to the synchronous speed, and the slip (the difference between synchronous speed and rotor speed) is very small. Because core losses are proportional to the square of the slip, at no-load conditions, the slip is so low that the core losses are negligible.

Rotor Current and Losses:
At no-load, the motor is not delivering torque, and only a small current is induced in the rotor due to the rotating magnetic field of the stator. This small current causes only minimal losses in the rotor core. Essentially, the rotor is not "working" to produce torque, and therefore the core losses are almost zero.

Core Losses and Load:
Rotor core losses increase with the load on the motor because as the load increases, the slip increases, which results in higher induced currents in the rotor. At no-load, the motor is essentially idle in terms of torque production, so the core losses are minimal.

Source: Electric Machinery Fundamentals by Stephen J. Chapman.

As the slip increases in an induction motor, the relative speed between the rotor and the magnetic field increases, which results in higher current in the stator winding. This is because the motor requires more current to produce the additional torque to overcome the increased load at higher slip.

(Electric Machinery Fundamentals by Stephen J. Chapman)

f = NP/120 = 50Hz

Ns​ = 120×f/P​ = (120* 50)/6 =1000

% slip s = (Ns-N)/Ns 
⇒ 3 = (1000-N)/1000
⇒ N = 970

Given Data:

Number of poles (p) = 12

Supply voltage = 600 V

Frequency = 50 Hz

Rotor resistance per phase (R2) = 0.03 Ω/phase

Stand-still reactance per phase (X2) = 0.5 Ω/phase

We are tasked with calculating the slip corresponding to the maximum torque under running conditions.

### Formula for Slip corresponding to Maximum Torque:

The slip corresponding to the maximum torque in a 3-phase induction motor is given by the formula:

s = (R2) / (X2)

Where:

R2 = Rotor resistance per phase

X2 = Stand-still reactance per phase

### Calculation:

Substituting the given values for R2 and X2:

s = 0.03 / 0.5 = 0.06 or 6%

Here, Copper loss=Pcu, 

Mechanical Power develop in Rotor = Pm

Rotor input = P2

P2 : Pcu : Pm = 1 : s : 1-s  

P2/Pcu = 1/s 

(A.C and D.C Machines by B.L.Thereja)

Starting Torque: When the motor starts, it has a high slip (close to 1), which means the relative speed between the rotating magnetic field and the rotor is large. This results in a high induced current in the rotor, producing a high starting torque. However, as the motor speeds up, the slip decreases and the torque reduces.

Full-load Torque: At full load, the motor operates at a lower slip (typically around 2-6% for most motors), which means the rotor is rotating closer to the synchronous speed. The torque at full load is typically the designed torque required to drive the load.
Maximum (Breakdown) Torque: The maximum torque occurs at a certain slip that is neither too high nor too low. At this point, the electromagnetic torque is at its peak, and it is usually much higher than the full-load torque. This breakdown torque is the point at which the motor can handle the highest load without stalling, but if the load exceeds this torque, the motor will stall.

The maximum torque occurs at a slip that is typically greater than the slip at full load but less than the starting slip. This results in a higher torque than at full load, which is one of the distinguishing features of squirrel cage induction motors.

Source: Principles of Electrical Machines by V.K. Mehta, Rohit mehta

Induced e.m.f. between the slip rings at standstill = 72 V

Resistance per phase = 0.5 Ω

Standstill reactance per phase = 3.5 Ω

rotor e.m.f per phase =72/√3 =3.54

Rotor impedance/phase at standstill, = √{(o.5)2 +(3.5)2}  = 3.54 ohm

Rotor current/phase at starting,= 41.5/3.54 =11.76

Starting Methods and Inrush Current:

Direct-On-Line (DOL): In the Direct-On-Line (DOL) starting method, the motor is connected directly to the full supply voltage. This results in a very high inrush current, typically 5-7 times the motor's full-load current, which can cause significant stress on the power supply and electrical components.

Star-Delta Starting: The Star-Delta starting method involves connecting the motor windings in star configuration during startup, which reduces the voltage across each winding to 1/√3 of the line voltage. After the motor reaches a certain speed, the connection is switched to delta to apply the full line voltage. The inrush current is reduced compared to DOL starting, but it is still higher than the autotransformer method.

Autotransformer Starting: The Autotransformer method uses an autotransformer to reduce the voltage applied to the motor during startup. By applying a reduced voltage (typically 50% or 60% of the line voltage), the inrush current is significantly reduced, typically to 2-3 times the motor’s full-load current. After the motor accelerates to a certain speed, the autotransformer is bypassed, and the motor is connected to the full line voltage. This method is the most effective in reducing inrush current because it limits the voltage applied to the motor during startup, which directly reduces the inrush current.

(Reference: Electric Machinery Fundamentals by Stephen J. Chapman)

**Synchronous Speed (N_s):**

The synchronous speed of an induction motor is determined by the supply frequency and the number of poles in the motor. It is given by the formula:

N_s = (120 × f) / P

Where:

N_s = Synchronous speed (in r.p.m.)

f = Supply frequency (in Hz)

P = Number of poles

2. **Rotor Speed (N):**

The actual speed of the rotor is related to the synchronous speed and the **slip (s)**. The relationship is given by:

s = (N_s - N) / N_s

Rearranging the equation to find rotor speed:

N = N_s (1 - s)

3. **Effect of Increasing Slip on Speed:**

As the slip increases, the rotor speed decreases. This is because slip is the difference between the synchronous speed and the actual rotor speed. Higher slip means a larger gap between the synchronous speed and rotor speed, causing the rotor to slow down further.

(Electric Machinery Fundamentals by Stephen J. Chapman)

The **rotor resistance** of an induction motor plays a significant role in determining the slip of the motor. The slip (s) is the difference between the synchronous speed (N_s) and the actual rotor speed (N), and it is given by the formula:

s = (N_s - N) / N_s

Where:

N_s = Synchronous speed of the motor

N = Rotor speed

s = Slip

2. **Effect of Rotor Resistance on Torque and Slip:**

When the rotor resistance increases, the motor generates more current to produce the same torque. This increases the slip for a given load. As slip increases, the rotor speed decreases because the difference between synchronous speed and rotor speed becomes larger.

3. **Rotor Resistance and Motor Efficiency:**

Higher rotor resistance leads to higher losses, which decreases the efficiency of the motor. To compensate for the increased losses, the motor operates at a higher slip, resulting in slower rotor speed.

(Electric Machinery Fundamentals by Stephen J. Chapman)

By the change of number of stator pole, the synchronous speed and hence rotor speed can be changed. Ns=120f/P

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)

The synchronous speed (NsN_sNs​) of an induction motor is directly related to the supply frequency (fff) and the number of poles (PPP) in the motor. The formula for synchronous speed is:

Where:

Ns​ = Synchronous speed (in r.p.m.),
f = Supply frequency (in Hz),
P = Number of poles in the motor.
Ns = 120f/P 

If the supply frequency drops, the synchronous speed will decrease proportionally, because the synchronous speed is directly proportional to the frequency.

If the frequency drops by 20%, the new frequency becomes 80% of the original frequency. This leads to the following relationship:

N_s' = 0.80 × N_s

When the line frequency drops by 20%, the speed of the induction motor(synchronous speed) will drop by 20% as well.

Power Factor of a Single-Phase Induction Motor:

Power Factor Definition:The power factor (PF) is the ratio of real power (active power) to apparent power. It indicates how effectively the motor is converting electrical power into mechanical power. A power factor of 1 indicates perfect efficiency, while a lower value indicates higher losses and inefficiency.

Single-Phase Induction Motor: A single-phase induction motor typically has a power factor ranging from 0.8 to 0.9 under full-load conditions. This means that the motor converts 80% to 90% of the electrical power it draws into useful mechanical power, with the rest being reactive power, which does not perform useful work. The power factor is affected by factors like the motor's load, the type of capacitor (if used), and the motor's design. At full load, the motor's power factor is generally in this range, though it can vary based on the type and quality of the motor.

Why the Other Options Are Incorrect:

(খ) 0.3 to 0.5:
This would represent a very low power factor, which is not typical for a single-phase induction motor under normal operating conditions. Such a low power factor is usually seen in highly inductive loads that are under very light load conditions.


(গ) 1.0:
A power factor of 1.0 (unity) is ideal, but it is rarely achieved in a single-phase induction motor. Typically, the motor operates with a power factor in the range of 0.8 to 0.9. A power factor of 1.0 would imply that there are no reactive power losses, which is unlikely in typical motor operations.

(ঘ) 0.1 to 0.2:
This range represents an extremely low power factor, which would not be typical for a properly operating single-phase induction motor. Such low power factors are usually associated with poorly designed or malfunctioning motors.


Reference Book: Electric Machinery Fundamentals by Stephen J. Chapman

 The synchronous speed (Ns) of any induction motor, including a single-phase induction motor, is determined by the supply frequency (f) and the number of poles (P) in the motor. The formula for synchronous speed is:

Ns = 120 × fP

Where,

Ns = Synchronous speed (in revolutions per minute, r.p.m.),
f= Supply frequency (in Hertz, Hz),
P = Number of poles in the motor.
Reference Book: Electrical Machines, Drives, and Power Systems by Theodore Wildi

Torque Production in a Single-Phase Induction Motor: In a single-phase induction motor, the torque is primarily produced by the interaction between the rotor's magnetic field and the stator's magnetic field. Here’s a breakdown of how this works:
 In a single-phase motor, the stator generates a pulsating magnetic field when supplied with single-phase AC power. This field alternates, but it does not rotate on its own like in a 3-phase motor. To make the motor self-starting, the pulsating field is "split" into two rotating magnetic fields using methods like shaded-pole design or a capacitor. This creates an effective rotating magnetic field that interacts with the rotor.

The rotor is subjected to the rotating magnetic field created by the stator. The rotor becomes magnetized and induces a magnetic field in response to the rotating field. The interaction of the rotor magnetic field with the stator magnetic field produces a force that results in torque.

The rotor experiences a force due to this interaction, which causes the rotor to rotate and produce mechanical output power. This is the fundamental principle of torque generation in induction motors, including single-phase induction motors.

Reference Book: Electric Machinery Fundamentals by Stephen J. Chapman

Single-Phase Induction Motor's Operating Principle: A single-phase induction motor operates by generating a pulsating magnetic field in the stator. This magnetic field induces current in the rotor, creating a magnetic field that interacts with the stator's field to produce torque.
Under no-load conditions, the motor is running, but there is little to no mechanical load on the rotor. This means the rotor is rotating at a speed close to the synchronous speed (with minimal slip).

As a result, the torque required to maintain rotation is minimal, and the current drawn by the motor decreases. The magnetizing current, which is responsible for creating the magnetic field, becomes weaker because the motor doesn't need to supply a significant amount of energy to overcome mechanical load.
Effect on the Magnetic Field:
With the reduced load and lower current, the magnetic field produced by the stator weakens. This is because the stator’s field strength is directly related to the amount of current flowing through the windings, which is reduced at no-load.
The field is still present but weaker compared to the full-load condition, where the motor requires more power to generate torque to overcome the load.

Magnetizing Reactance in a Single-Phase Induction Motor:

1. Magnetizing Reactance:In the equivalent circuit of a single-phase induction motor, the magnetizing reactance represents the reactance associated with the creation of the magnetic field in the motor. This reactance is linked to the stator winding's inductance and is responsible for the motor's ability to produce a magnetic field. The magnetizing reactance can be thought of as the component of the impedance that generates the magnetic flux in the motor, which is essential for inducing current in the rotor and producing torque.

2. Magnetic Field Generation:The magnetizing current (which causes the magnetizing reactance) creates the magnetic field in the stator. This magnetic field is the primary field that induces a current in the rotor, which interacts with the stator's magnetic field to produce torque. In other words, the magnetizing reactance is directly related to the magnetic field generated by the stator, and this magnetic field is essential for the motor's operation.

Why the Other Options Are Incorrect:

(খ) The resistance of the rotor:The rotor resistance in the equivalent circuit represents the resistance of the rotor winding, not the magnetizing reactance. Rotor resistance is associated with energy losses in the rotor due to its resistance.

(গ) The mechanical losses in the motor:Mechanical losses (like friction and windage losses) are not represented by the magnetizing reactance. These losses are typically modeled as separate losses in the motor’s equivalent circuit and do not affect the magnetic field generation directly.

(ঘ) The air gap between the stator and rotor:The air gap represents the physical distance between the stator and rotor and affects the overall motor performance. While the air gap impacts the magnetic flux density, it is not directly represented by the magnetizing reactance. The reactance primarily represents the magnetizing field created by the stator.

Power Factor Definition:The power factor (PF) of a motor is the ratio of real power (active power) to apparent power. It indicates how effectively the motor converts electrical energy into useful mechanical energy. A low power factor means that the motor is drawing more reactive power, which does not contribute to the actual work being done.

Improving Power Factor: A single-phase induction motor typically operates with a lagging power factor, meaning the current lags behind the voltage due to the inductive nature of the motor. This lagging power factor results in inefficient power usage. The power factor can be improved by reducing the lag between the voltage and the current.

Reference Book: Electric Machinery and Transformers by Bhag S. Guru and Huseyin R. Hiziroglu

Split-Phase Induction Motor: In a split-phase induction motor, the motor is designed to create a phase shift in the stator's magnetic field to make it self-starting. Unlike a three-phase motor, which has a naturally rotating magnetic field, a single-phase motor cannot start on its own due to the lack of a rotating magnetic field. The split-phase method overcomes this limitation.


How Phase Shift is Created: In the split-phase method, the stator is equipped with two windings:
1. The main winding, which is responsible for the majority of the magnetic field.
2. A secondary (or auxiliary) winding, which is placed in parallel with the main winding but is designed to have different impedance (higher resistance and inductance) to create a phase difference.
This difference in impedance creates a phase shift between the currents in the two windings. As a result, a rotating magnetic field is created, which allows the motor to start and rotate.

Step angle: 
3 phase = m
24 pole = Nr
α = 360/mNr=360/(3*24) = 5°/step

(Principles of Electrical Machines by V.K. Mehta, Rohit mehta)