Electrical Machines: Transformers

Introduction to Transformers

A transformer is a static electrical device that transfers electrical energy between two or more circuits through electromagnetic induction. It is primarily used to increase (step-up) or decrease (step-down) voltage levels in an alternating current (AC) system. Transformers are fundamental components in power transmission and distribution networks, as well as in electronic devices for voltage regulation.

Principle of Operation

The operation of a transformer is based on Faraday's law of electromagnetic induction and the principle of mutual induction. When an alternating voltage is applied to the primary winding, it creates an alternating current. This current produces a time-varying magnetic flux in the core. This changing flux links with the secondary winding and induces an alternating electromotive force (EMF) across it. The magnitude of the induced EMF in each winding is proportional to the number of turns in that winding.

Let $V_1$ be the voltage applied to the primary winding and $N_1$ be the number of turns in the primary winding. Let $V_2$ be the induced voltage in the secondary winding and $N_2$ be the number of turns in the secondary winding. According to Faraday's law, the induced EMF ($E$) in a winding is given by:

$E = -N \frac{d\Phi}{dt}$

Where $\Phi$ is the magnetic flux. For the primary and secondary windings:

$E_1 = -N_1 \frac{d\Phi}{dt}$

$E_2 = -N_2 \frac{d\Phi}{dt}$

Assuming the flux linking both windings is the same, we can find the ratio of induced EMFs:

$\frac{E_1}{E_2} = \frac{N_1}{N_2}$

In an ideal transformer, the applied voltage $V_1$ is approximately equal to $E_1$, and the terminal voltage $V_2$ is approximately equal to $E_2$. Therefore, the voltage transformation ratio (or turns ratio, $a$) is:

$a = \frac{N_1}{N_2} = \frac{V_1}{V_2} = \frac{I_2}{I_1}$

This shows that the ratio of voltages is equal to the ratio of turns, and the ratio of currents is the inverse of the turns ratio.

Construction of Transformers

Transformers typically consist of two main parts: a magnetic core and windings.

Magnetic Core

The core provides a low-reluctance path for the magnetic flux. It is usually made of thin laminations of high-grade silicon steel to reduce eddy current losses and hysteresis losses. The laminations are insulated from each other by a thin layer of varnish or oxide. There are two main types of core construction:

• Core Type: The windings surround the core. This type has two limbs and two yokes.
• Shell Type: The core surrounds the windings. This type has a central limb and two outer limbs.

Windings

The windings are made of copper or aluminum conductors. They are insulated from the core and from each other. Typically, there are two windings: the primary winding (connected to the input AC supply) and the secondary winding (connected to the load).

Types of Transformers

Transformers can be classified based on their construction, application, and voltage level.

• Power Transformers: Used in transmission and distribution substations to step up or step down voltages.
• Distribution Transformers: Used to step down high voltages to the lower voltages required by consumers.
• Instrument Transformers: Used for measurement purposes.
  • Current Transformers (CTs): Step down high currents to a measurable range for ammeters.
  • Potential Transformers (PTs) / Voltage Transformers (VTs): Step down high voltages to a measurable range for voltmeters.
• Autotransformers: Have a single winding that serves as both primary and secondary. They offer advantages in terms of size and cost for certain applications but lack electrical isolation.

Transformer Losses

No transformer is 100% efficient due to energy losses. The main losses are:

• Core Losses (Iron Losses): These occur in the magnetic core and are constant for a constant applied voltage. They consist of:
  • Hysteresis Loss: Due to the repeated magnetization and demagnetization of the core material.
  • Eddy Current Loss: Due to circulating currents induced in the core by the changing magnetic flux.
• Copper Losses (I²R Losses): These occur in the windings due to the resistance of the conductors. They vary with the square of the load current.

Efficiency of a Transformer

Efficiency ($\eta$) is defined as the ratio of output power ($P_{out}$) to input power ($P_{in}$).

$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + Losses}$

Where $Losses = Core Losses + Copper Losses$.

For a transformer, maximum efficiency occurs when the variable copper losses are equal to the constant core losses.

Let $P_{core}$ be the core loss and $P_{cu}$ be the copper loss at full load. If the load is $x$ times the full load, then the copper loss at this load is $x^2 P_{cu}$.

The efficiency at a load $x$ is:

$\eta(x) = \frac{x \cdot P_{out,fl}}{x \cdot P_{out,fl} + P_{core} + x^2 P_{cu,fl}}$

Where $P_{out,fl}$ is the output power at full load and $P_{cu,fl}$ is the copper loss at full load.

Maximum efficiency occurs when $P_{core} = x^2 P_{cu,fl}$. The value of $x$ for maximum efficiency is $x_{max} = \sqrt{\frac{P_{core}}{P_{cu,fl}}}$.

Equivalent Circuit of a Transformer

The behavior of a transformer can be represented by an equivalent circuit. A simplified exact equivalent circuit includes the winding resistances ($R_1, R_2$) and leakage reactances ($X_1, X_2$), along with the magnetizing reactance ($X_m$) and core loss resistance ($R_c$) referred to the primary side.

The approximate equivalent circuit simplifies this by referring all secondary quantities to the primary side. The series impedance of the primary and referred secondary winding is combined ($R_{eq1} = R_1 + a^2 R_2$, $X_{eq1} = X_1 + a^2 X_2$). The magnetizing branch ($R_c || jX_m$) is placed at the input terminals.

The turns ratio $a = N_1/N_2$.

Voltage Regulation

Voltage regulation is a measure of the change in the secondary terminal voltage from no load to full load, expressed as a percentage of the full-load voltage.

$Voltage Regulation = \frac{V_{2,NL} - V_{2,FL}}{V_{2,FL}} \times 100\%$

Where $V_{2,NL}$ is the secondary voltage at no load and $V_{2,FL}$ is the secondary voltage at full load. A lower voltage regulation indicates a better transformer. It depends on the winding impedances and the power factor of the load.

Electrical Machines: DC Machines

Introduction to DC Machines

A Direct Current (DC) machine is an electrical machine that converts mechanical energy into electrical energy (as a generator) or electrical energy into mechanical energy (as a motor). It operates on the principle of electromagnetic force experienced by a current-carrying conductor placed in a magnetic field. DC machines are versatile and used in various applications, from small toys to large industrial drives.

Construction of DC Machines

A DC machine consists of two main parts: the stator and the rotor.

Stator (Field System)

The stator is the stationary part and provides the magnetic field. It consists of:

• Yoke: The outer frame of the machine, providing mechanical support and acting as a path for the magnetic flux. Usually made of cast iron or rolled steel.
• Poles: Mounted on the inner side of the yoke, they are the source of the magnetic field. Each pole consists of a pole core and a pole shoe. Field windings are wound around the pole cores.
• Field Windings: Coils of insulated wire wound on the poles. When a DC current flows through them, they create the magnetic field.

Rotor (Armature)

The rotor is the rotating part and carries the armature winding.

• Armature Core: A cylindrical structure made of laminated steel to reduce eddy current losses. It has slots on its periphery to accommodate the armature conductors.
• Armature Winding: A system of insulated conductors placed in the slots of the armature core. This is where the main EMF is induced (in a generator) or the torque is produced (in a motor).
• Commutator: A crucial part mounted on the rotor shaft. It is a cylindrical structure made of copper segments insulated from each other and from the shaft. It acts as a mechanical rectifier, converting the AC induced EMF in the armature winding into a unidirectional DC voltage at the brushes (for a generator) or supplying DC current to the armature winding (for a motor).
• Brushes: Stationary conductors (usually carbon) that make contact with the rotating commutator segments. They conduct current between the stationary external circuit and the rotating armature winding.

Principle of Operation

As a Motor: When a DC voltage is applied to the armature winding (while the stator provides a magnetic field), current flows through the armature conductors. According to Lorentz force law, a current-carrying conductor in a magnetic field experiences a force ($F = BIL$). These forces on the armature conductors produce a torque that causes the rotor to rotate.

As a Generator: When the rotor is mechanically driven by an external prime mover, the armature conductors cut the magnetic flux lines produced by the stator poles. According to Faraday's law of electromagnetic induction, an EMF is induced in the armature conductors. The commutator then converts this induced AC EMF into a unidirectional DC voltage at the brushes.

Types of DC Machines (Based on Field Winding Connection)

DC machines can be classified based on how the field winding is connected with respect to the armature winding.

1. Separately Excited DC Machine

The field winding is connected to a separate DC power source, independent of the armature circuit. This allows for independent control of field flux and armature current.

2. Self-Excited DC Machines

The field winding is connected to the armature circuit itself.

• Shunt Machine: The field winding is connected in parallel (shunt) with the armature winding. The field winding has a large number of turns of thin wire, giving it high resistance.
• Series Machine: The field winding is connected in series with the armature winding. The field winding has a few turns of thick wire, giving it low resistance.
• Compound Machine: Has both shunt and series field windings.
  • Cumulative Compound: The magnetic field produced by the series winding aids the field produced by the shunt winding.
  • Differential Compound: The magnetic field produced by the series winding opposes the field produced by the shunt winding. (Rarely used due to instability).

Electromotive Force (EMF) Equation

The EMF induced in the armature winding of a DC generator is given by:

$E_a = \frac{2 P \Phi N Z}{60 A}$

Where:

• $E_a$ = Back EMF (in Volts)
• $P$ = Number of poles
• $\Phi$ = Flux per pole (in Webers)
• $N$ = Speed of the armature in revolutions per minute (RPM)
• $Z$ = Total number of armature conductors
• $A$ = Number of parallel paths in the armature winding

The term $\frac{2 P Z}{60 A}$ is a constant for a given machine and is often denoted by $K$. So, $E_a = K \Phi N$.

For a DC motor, this induced EMF acts as a back EMF, opposing the applied voltage.

Torque Equation

The torque developed by a DC machine (motor or generator) is proportional to the armature current and the flux per pole.

$T = \frac{2 P \Phi Z I_a}{2 \pi A}$ (in N-m)

Or, $T = K \Phi I_a$, where $K = \frac{2 P Z}{2 \pi A}$ is a machine constant.

DC Motor Characteristics

These are graphs that show the relationship between different parameters like speed, torque, and armature current.

1. Shunt Motor Characteristics

a) Torque vs. Armature Current ($T$ vs $I_a$): Approximately a straight line passing through the origin (since $T = K \Phi I_a$ and $\Phi$ is constant). Torque is directly proportional to armature current.

b) Speed vs. Armature Current ($N$ vs $I_a$): Speed decreases slightly as armature current increases. This is because as $I_a$ increases, the back EMF ($E_a = V - I_a R_a$) decreases, and since $N \propto E_a/\Phi$ (with $\Phi$ constant), $N$ decreases. The drop is usually small because $R_a$ is small.

c) Speed vs. Torque ($N$ vs $T$): Speed decreases moderately as torque increases.

2. Series Motor Characteristics

a) Torque vs. Armature Current ($T$ vs $I_a$): In the initial range, the flux $\Phi$ is proportional to $I_a$ (since $I_a$ is the only current, and $\Phi \propto I_{f} \approx I_a$). Therefore, $T = K \Phi I_a \propto I_a^2$. At higher currents, the magnetic field saturates, and $\Phi$ becomes constant. Then, $T \propto I_a$. The curve is initially parabolic and then becomes a straight line.

b) Speed vs. Armature Current ($N$ vs $I_a$): Speed is inversely proportional to torque. At low currents (low torque), $\Phi$ is small, and $N$ is very high ($N \propto 1/(\Phi I_a)$). At high currents (high torque), $\Phi$ increases, and $N$ decreases significantly.

c) Speed vs. Torque ($N$ vs $T$): Speed drops rapidly as torque increases.

3. Compound Motor Characteristics

The characteristics lie between those of shunt and series motors, depending on the relative strengths of the series and shunt fields. Cumulative compound motors have better starting torque than shunt motors and less speed variation. Differential compound motors are unstable and rarely used.

Starting of DC Motors

At standstill ($N=0$), the back EMF ($E_a = K \Phi N$) is zero. If full supply voltage is applied, a very large current ($I_a = (V - E_a) / R_a \approx V/R_a$) will flow through the armature. Since $R_a$ is very small (typically 0.1 to 1 ohm), this current can be several hundred times the rated current, damaging the armature winding and brushes.

Therefore, a starting resistance (or starter) is connected in series with the armature to limit the starting current to a safe value. As the motor speeds up, back EMF increases, and the starting resistance is gradually cut out.

• Two-point Starter: Used for shunt and series motors. Consists of a starting resistance and a holding coil.
• Three-point Starter: Used for shunt motors. Similar to a two-point starter but uses a starting resistance, a no-volt coil (holding coil), and an overload release coil.
• Four-point Starter: Used for shunt motors. Separates the no-volt coil from the line current path, making it more reliable.
• Series Motor Starter: Typically a two-point starter, often combined with a main switch and protective devices.

Speed Control of DC Motors

The speed of a DC motor ($N \propto E_a / \Phi$) can be controlled by varying:

• Armature Voltage ($V$): By reducing the armature voltage (using a variable resistor in series or a variable voltage supply), the back EMF ($E_a = V - I_a R_a$) is reduced, thus reducing the speed. This method is suitable for speed reduction below the rated speed and is common for shunt and compound motors.
• Field Flux ($\Phi$): By reducing the field flux (using a variable resistance called a field rheostat in series with the shunt field winding), the speed increases. This method is only suitable for speed increase above the rated speed and is primarily used for shunt and compound motors. It is more efficient than armature voltage control for speed increase.
• Armature Resistance ($R_a$): By inserting resistance in series with the armature, the back EMF ($E_a = V - I_a (R_a + R_{ext})$) is reduced, leading to reduced speed. This method is inefficient as power is wasted in the external resistance. It is generally used for speed reduction and mainly for series motors.

Electrical Machines: Induction Motors

Introduction to Induction Motors

An induction motor is the most common type of AC electric motor. It is an asynchronous motor because its rotor speed is always less than the synchronous speed of the rotating magnetic field. It works on the principle of electromagnetic induction, where a rotating magnetic field is produced in the stator, which induces currents in the rotor conductors, creating a torque that drives the rotor.

Principle of Operation

When a three-phase AC supply is connected to the stator windings, it produces a rotating magnetic field (RMF) whose speed is called the synchronous speed ($N_s$). This RMF cuts the rotor conductors, inducing EMFs in them according to Faraday's law. Since the rotor conductors form a closed circuit (either through end rings or external resistors), these induced EMFs cause currents to flow in the rotor. The interaction between the rotor currents and the stator's rotating magnetic field produces a torque, causing the rotor to rotate in the same direction as the RMF.

The rotor always rotates at a speed ($N_r$) less than the synchronous speed ($N_s$). The difference between the synchronous speed and the rotor speed is called slip ($s$).

$s = \frac{N_s - N_r}{N_s}$

The induced EMF, current, and torque in the rotor are dependent on this slip. At starting (when $N_r = 0$), the slip is maximum ($s=1$), inducing maximum EMF and current, leading to starting torque. As the motor speeds up, slip decreases, and the induced EMF and current decrease.

Construction of Induction Motors

1. Stator

The stator is the stationary part and consists of a stator core and a stator winding.

• Stator Core: Made of laminated silicon steel to reduce hysteresis and eddy current losses. It has slots on the inner periphery to accommodate the stator winding.
• Stator Winding: Typically a three-phase winding, distributed in the slots. When connected to a three-phase supply, it produces a rotating magnetic field.

2. Rotor

The rotor is the rotating part and is mounted on a shaft. There are two main types of rotors:

• Squirrel Cage Rotor: The most common type. It consists of a cylindrical laminated core with deep slots. In these slots, rotor bars (usually aluminum or copper) are embedded and short-circuited at both ends by end rings. It resembles a squirrel cage. It is simple, robust, and requires no external connection.
• Wound Rotor (Slip Ring Rotor): Consists of a laminated core with slots similar to the squirrel cage rotor. However, it carries a three-phase winding (similar to the stator winding) which is connected to slip rings mounted on the shaft. External resistors can be connected to the slip rings via brushes to control starting torque and speed.

Synchronous Speed

The speed of the rotating magnetic field is called synchronous speed ($N_s$) and is determined by the frequency of the supply ($f$) and the number of poles ($P$) in the stator winding.

$N_s = \frac{120 f}{P}$ (in RPM)

For example, a 4-pole motor operating on a 50 Hz supply has a synchronous speed of $N_s = (120 \times 50) / 4 = 1500$ RPM.

Torque Equation

The torque developed by an induction motor is given by:

$T = \frac{3 I_2^2 R_2'}{s \omega_s}$ (in N-m)

Where:

• $I_2$ = Rotor current per phase
• $R_2'$ = Rotor resistance referred to the stator
• $s$ = Slip
• $\omega_s = 2 \pi N_s / 60$ = Synchronous speed in radians/sec

The rotor current $I_2$ depends on the slip and the rotor impedance. The rotor impedance per phase is $Z_2' = R_2' + j s X_2'$, where $X_2'$ is the rotor leakage reactance at standstill.

So, $I_2 = \frac{s E_2'}{Z_2'} = \frac{s E_2'}{\sqrt{(R_2')^2 + (s X_2')^2}}$, where $E_2'$ is the rotor EMF induced at standstill referred to the stator.

Substituting this into the torque equation gives:

$T = \frac{3 s E_2'^2 R_2'}{R_2'^2 + (s X_2')^2}$

Torque-Slip Characteristics

The torque-slip characteristic of an induction motor is a curve showing the relationship between torque and slip.

• Starting Torque ($s=1$): $T_{start} = \frac{3 E_2'^2 R_2'}{R_2'^2 + (X_2')^2}$.
• Maximum Torque (Breakdown Torque): Occurs at a specific slip $s_{max} = \frac{R_2'}{X_2'}$. The maximum torque is $T_{max} = \frac{3 E_2'^2}{2 (R_2' + X_2')}$. This is the highest torque the motor can produce.
• Full Load Torque: Occurs at the normal operating slip, which is usually small (e.g., 1-5%).

The curve typically starts at $s=1$ (standstill), rises to a maximum torque at $s_{max}$, and then decreases as slip approaches 0 (synchronous speed). The motor operates stably in the region where torque increases with decreasing slip (i.e., $s < s_{max}$).

Speed Control of Induction Motors

Controlling the speed of an induction motor is more complex than DC motors. The main methods include:

• Changing the Number of Poles: By providing multiple stator windings or using special designs, the number of poles ($P$) can be changed, altering the synchronous speed ($N_s = 120f/P$). This provides discrete speed steps.
• Varying the Supply Frequency ($f$): Using Variable Frequency Drives (VFDs), the frequency of the AC supply can be changed. Since $N_s \propto f$, this allows for smooth and wide-range speed control. The voltage is also usually varied proportionally to maintain a constant V/f ratio, which helps in maintaining constant torque capability.
• Varying the Supply Voltage ($V$): Reducing the supply voltage reduces the torque approximately as $T \propto V^2$. This method provides limited speed control and significantly reduces torque, leading to poor performance.
• Varying Rotor Resistance (Wound Rotor Motors): Adding external resistance to the rotor circuit increases the slip at which the motor operates for a given torque, effectively reducing the rotor speed. This method is inefficient as power is dissipated in the external resistors.

Losses and Efficiency

Induction motors have several types of losses:

• Stator Copper Loss ($I_1^2 R_1$): Due to stator current and stator winding resistance.
• Rotor Copper Loss ($I_2^2 R_2$): Due to rotor current and rotor resistance. This loss is directly related to slip: $P_{rotor loss} = s \times P_{airgap}$, where $P_{airgap}$ is the power transferred from stator to rotor across the air gap.
• Core Loss: Hysteresis and eddy current losses in the stator core, assumed constant at constant voltage and frequency.
• Mechanical Losses: Friction and windage losses in bearings and due to air resistance.
• Stray Load Losses: Additional losses that vary with load.

Efficiency ($\eta$) is calculated as:

$\eta = \frac{P_{out}}{P_{in}} = \frac{P_{in} - Losses}{P_{in}} = 1 - \frac{Losses}{P_{in}}$

Modern induction motors, especially larger ones, can achieve efficiencies of over 95%.

Types of Induction Motors

• Squirrel Cage Induction Motor (SCIM): Most common, simple, rugged, low cost, low maintenance.
• Wound Rotor Induction Motor (WRIM) / Slip Ring Induction Motor (SRIM): Used when high starting torque and adjustable speed are required. More expensive and requires more maintenance due to slip rings and brushes.
• Single-Phase Induction Motors: Used in domestic and light commercial applications. Require a starting mechanism (e.g., capacitor start, split-phase) as a single-phase supply does not inherently produce an RMF.

Electrical Machines: Synchronous Machines

Introduction to Synchronous Machines

A synchronous machine is an AC electrical machine in which the rotor rotates at the same speed as the rotating magnetic field produced by the stator. This speed is called the synchronous speed ($N_s$). Synchronous machines can operate as either generators (alternators) or motors. They are known for their constant speed operation, high efficiency, and ability to control power factor.

Construction of Synchronous Machines

Synchronous machines have two main components: the stator and the rotor.

Stator

The stator is similar to that of an induction motor. It consists of a laminated steel core with slots on the inner periphery to accommodate a three-phase winding. This stator winding is connected to the external AC power system. When excited by AC voltage, it produces a rotating magnetic field (RMF) at synchronous speed ($N_s = 120f/P$).

Rotor

The rotor carries the field winding, which is supplied with DC current to produce a magnetic field. The speed of rotation of this rotor magnetic field must be synchronized with the stator RMF for continuous operation. There are two main types of rotors:

• Salient Pole Rotor: Used in low-speed and medium-speed machines (typically below 1000 RPM), such as those driven by hydro turbines. The poles project outwards from the rotor surface. This construction provides a larger pole area and allows for better ventilation. It results in a non-uniform air gap, leading to salient pole effects (difference in magnetic reluctance along different axes).
• Cylindrical Rotor (Non-Salient Pole): Used in high-speed machines (typically 1500 or 3000 RPM) driven by steam turbines. The rotor is a solid cylinder with slots machined into it to accommodate the field winding. The field poles are not distinct but are formed by the distribution of the winding. This design is mechanically strong and suitable for high centrifugal forces. The air gap is uniform.

The DC excitation for the rotor field winding is usually supplied through slip rings and brushes, or via a brushless exciter system (a small AC generator on the same shaft whose output is rectified to DC).

Principle of Operation

As a Generator (Alternator):

The rotor is driven by a prime mover at synchronous speed ($N_s$). DC current is supplied to the rotor field winding, creating a magnetic field. As this rotor field rotates, it cuts the stator conductors, inducing AC EMFs in the stator windings according to Faraday's law. The frequency of the induced EMF is determined by the speed of rotation and the number of poles ($f = P N_s / 120$). The magnitude of the induced EMF depends on the field flux, speed, and winding characteristics.

As a Motor:

A three-phase AC supply is connected to the stator winding, creating an RMF at synchronous speed. DC excitation is applied to the rotor field winding. The rotor's magnetic field 'locks' with the stator's RMF, and the rotor is pulled into synchronism, rotating at the same speed ($N_r = N_s$). Torque is produced due to the interaction between the stator RMF and the rotor magnetic field. If the mechanical load exceeds the maximum torque the motor can produce, the rotor will fall out of synchronism, and the motor will stop.

Synchronous Speed and Frequency

The relationship between synchronous speed ($N_s$ in RPM), supply frequency ($f$ in Hz), and the number of poles ($P$) is fundamental:

$N_s = \frac{120 f}{P}$

This equation applies to both induction motors and synchronous motors.

Synchronous Motor Characteristics

Synchronous motors are unique because their speed is constant regardless of load (up to their pull-out torque). Their most significant characteristic is their ability to operate at leading, lagging, or unity power factor by adjusting the DC field excitation.

• Operation at Unity Power Factor: Occurs when the field excitation is adjusted such that the motor draws minimum line current for a given load.
• Operation at Lagging Power Factor: Occurs when the field excitation is reduced (under-excitation).
• Operation at Leading Power Factor: Occurs when the field excitation is increased (over-excitation).

The relationship between armature current ($I_a$), excitation voltage ($E_f$), and synchronous impedance ($Z_s$) is described by the phasor equation:

$V = E_f - I_a Z_s$ (for a generator)

$V = E_f + I_a Z_s$ (for a motor, where $E_f$ lags $V$)

Or more commonly for a motor, considering phase angle $\delta$:

$V \angle 0 = E_f \angle \delta - I_a (R_a + j X_s)$

Where $V$ is the terminal voltage, $E_f$ is the excitation voltage, $I_a$ is the armature current, $R_a$ is the armature resistance, $X_s$ is the synchronous reactance, and $\delta$ is the load angle (power angle).

Power Factor Control (V-Curves)

The V-curves of a synchronous motor plot armature current ($I_a$) versus field current ($I_f$) for different constant power outputs.

• At no load, increasing $I_f$ from zero causes $I_a$ to decrease until it reaches a minimum at unity power factor, then increases again as $I_f$ is further increased (leading power factor).
• As the mechanical load increases, the minimum $I_a$ point shifts upwards and to the right (requiring more excitation for unity power factor).
• The lowest point on all V-curves corresponds to unity power factor operation.
• The region to the left of the unity power factor (lower $I_f$) represents leading power factor operation.
• The region to the right of the unity power factor (higher $I_f$) represents lagging power factor operation.

Synchronous motors can be used for power factor correction in industrial plants by operating them with over-excitation (leading power factor) even when they are not carrying a significant mechanical load.

Starting of Synchronous Motors

Synchronous motors are not self-starting. They require a starting mechanism to bring the rotor close to synchronous speed before the DC field excitation is applied. Common starting methods include:

• Using a Pony Motor: A small induction motor brings the synchronous motor rotor up to near synchronous speed.
• Damper Windings (Amortisseur Windings): Similar to squirrel cage windings embedded in the rotor pole faces. During starting, the stator RMF induces currents in these windings (like an induction motor), producing starting torque. As the rotor approaches synchronous speed, DC excitation is applied, and the rotor pulls into synchronism, with the damper windings effectively acting as a short-circuited winding during normal operation.
• Using a Variable Frequency Drive (VFD): The VFD starts the motor at a very low frequency and gradually increases it, allowing the rotor to lock with the RMF and start smoothly from standstill.

Applications

• Synchronous Generators (Alternators): The primary source of electrical power in power plants worldwide.
• Synchronous Motors: Used for constant speed drives where high efficiency and power factor correction are important, such as large compressors, pumps, fans, and in rolling mills.
• Synchronous Condensers: Synchronous motors operated with no mechanical load, purely for power factor correction.

Electrical Machines: Characteristics, Starting, Control, Losses, Efficiency

DC Machines: Recap and Deeper Dive

Characteristics Summary

Understanding the performance of DC motors relies on their characteristic curves.

Characteristic	Shunt Motor	Series Motor	Compound Motor (Cumulative)
Torque vs. $I_a$	Linear ($T \propto I_a$)	Non-linear ($T \propto I_a^2$ at low $I_a$, $T \propto I_a$ at high $I_a$)	Between shunt and series
Speed vs. $I_a$	Slight decrease with $I_a$ (constant speed)	High speed at low $I_a$, low speed at high $I_a$ (variable speed)	Less variable than series, more than shunt
Speed vs. Torque	Slight decrease with Torque	Rapid decrease with Torque	Moderate decrease with Torque
Starting Torque	Moderate	Very High	High

Starting and Speed Control Recap

Starting: Essential to limit initial high armature current ($I_a = V/R_a$ at standstill). Starters (2-point, 3-point, 4-point) add resistance in series with the armature, which is progressively removed as speed increases.

Speed Control:

• Shunt/Compound: Speed below rated (reduce armature voltage), Speed above rated (reduce field flux).
• Series: Speed control is limited and often achieved by varying armature voltage or field current (less common). Tapping field winding or adding resistance in series limits speed.

Losses and Efficiency in DC Machines

Total losses in a DC machine are categorized as:

• Constant Losses (Fixed Losses): These are independent of the load.
  • Core Loss: Hysteresis and eddy current losses in the armature core.
  • Mechanical Loss: Friction at bearings and brushes, windage loss due to air resistance.
  • Shunt Field Loss: $I_{sh}^2 R_{sh}$ (for shunt and compound machines), constant if supply voltage is constant.
• Variable Losses (Load Dependent Losses): These vary with the load current.
  • Armature Copper Loss: $I_a^2 R_a$. This is the most significant variable loss.
  • Series Field Loss: $I_s^2 R_s$ (for series and compound machines), varies with load current.

Efficiency ($\eta$) = (Output Power / Input Power) = (Output Power / (Output Power + Total Losses)).

Maximum efficiency occurs when variable losses equal constant losses. For a DC motor, this means $I_a^2 R_a + I_s^2 R_s = P_{constant}$.

Induction Motors: Recap and Deeper Dive

Starting Torque Considerations

The starting torque is critical for applications like cranes, hoists, and traction.

• Squirrel Cage Motors: Starting torque is inherent to the rotor design. High resistance rotors (e.g., deep bars, double cage) provide higher starting torque and lower starting current.
• Wound Rotor Motors: External resistance connected via slip rings can significantly boost starting torque. The resistance required for maximum starting torque is $R_{2,opt}' = s_{max} X_2'$. By adjusting this external resistance, starting torque can be optimized.

Speed Control Methods

VFDs (Variable Frequency Drives) are the most versatile and efficient method for speed control of induction motors, allowing smooth variation from near zero to above synchronous speed while maintaining good torque.

Stator Voltage Control: Simple but inefficient, reduces torque significantly ($T \propto V^2$). Used for light loads or specific applications.

Rotor Resistance Control (Wound Rotor): Inefficient, used for starting torque boost and limited speed reduction.

Pole Changing: Provides discrete speed steps, typically two or three. Used in applications where specific speed ranges are needed (e.g., some fans, machine tools).

Losses and Efficiency in Induction Motors

Losses include: Stator copper loss ($I_1^2 R_1$), Rotor copper loss ($I_2^2 R_2 = s P_{airgap}$), Core loss (constant), Friction & Windage, Stray Load Losses.

Efficiency is generally high for larger motors operating near rated load. Slip is usually small (1-5%) for standard motors, meaning rotor copper losses are minimal.

Synchronous Machines: Recap and Deeper Dive

Power Factor Correction

Synchronous motors are unique in their ability to operate at unity or leading power factors. By over-exciting the field winding (increasing $I_f$), the motor draws leading current, which can offset the lagging reactive power drawn by other inductive loads (like induction motors) in a plant. This improves the overall system power factor, reducing line losses and potentially avoiding penalties from the utility. Such a motor used solely for power factor correction is called a synchronous condenser.

Pull-out Torque

This is the maximum torque a synchronous motor can develop without falling out of synchronism. It depends on the synchronous reactance ($X_s$), armature resistance ($R_a$), field excitation ($E_f$), and terminal voltage ($V$).

For a cylindrical rotor synchronous motor, the pull-out torque is approximately:

$T_{max} \approx \frac{3 V E_f}{\omega_s X_s}$ (ignoring $R_a$)

This occurs at a critical load angle $\delta_c$ (typically around 90 electrical degrees). If the load torque exceeds this value, the rotor loses synchronism.

Starting Challenges

The need for damper windings or VFDs for starting is due to the fact that at standstill, the rotor magnetic field does not align with the stator's RMF, resulting in zero net torque. Damper windings act like a squirrel cage, allowing the motor to start as an induction motor, and then the rotor is pulled into synchronism by the DC field excitation as it approaches synchronous speed.