Direct axis synchronous reactance:

• A direct axis quantity is one whose magnetic effect is along the field pole axis.
• The field pole axis is also known as direct axis.
• Based on this fact, a direct axis synchronous reactance is defined as the reactance offered to the armature flux when the peak of armature mmf is directed along the direct axis.

• Under this condition, the air gap length is minimum and hence the reluctance is also minimum.
• the flux linkage per ampere in the armature winding is called direct axis synchronous inductance Ld.
• The peak of the armature mmf wave coincides with the direct axis.

Direct axis synchronous reactance; $X_d=\frac{V_{max}}{I_{min}}$

Quadrature Axis Synchronous Reactance:

• A quadrature axis is one whose magnetic effect is along the perpendicular to the field pole axis.
• When the peak of armature mmf coincides with the quadrature axis, the reactance offered to the armature mmf is called quadrature axis reactance.

• The peak of armature mmf is along the perpendicular to the field pole axis or direct axis.
• The reactance offered by the two large air gap is maximum.
• For a given armature current, the armature mmf is constant and hence under this case the armature flux is minimum.
• The flux linkage per armature ampere is called quadrature axis or q-axis synchronous inductance Lq.

Quadrature axis synchronous reactance; $X_q=\frac{V_{min}}{I_{max}}$

Concept:

Distribution factor or belt factor or breadth factor or spread factor (Kd) 

• It is a measure of the resultant emf of a distributed winding compared to a concentrated winding.
• Kd = emf induced in distributed winding/emf induced if the winding is concentrated

The mathematical expression of the distribution factor is given as,

$k_d=\frac{\sin \frac{m\beta }{2}}{m\sin \frac{\beta }{2}}$

Where m is the number of slots per pole per phase

β is the slot angle in degrees

Calculation:

3 – phase, 8 – pole synchronous machine.

No. of slots = 240 (stator slots)

$m=\frac{240}{3\times 8}=10$

$\beta =\frac{{180}^{\circ }}{slotsperpole}$

$\beta =\frac{{180}^{\circ }}{240/8}=6^{\circ }$

$K_d=\frac{\sin \frac{10\times 6}{2}}{10\sin \frac{6}{2}}=\frac{\sin {30}^{\circ }}{10\sin 3^{\circ }}$

Winding factor (K_w):

• It is the method of improving the RMS generated voltage so, that the torque and the output voltage do not consist of any harmonics which reduces the efficiency of the machine.
• Winding Factor is defined as the product of Distribution factor (K_d) and the coil span factor (K_p)

mathematically it is represented as

Winding factor (K_w) = K_p × K_d

Pitch Factor or Coil Span Factor or Chording factor (K_p) 

• It is defined as the ratio of emf generated in short pitch coil to the emf generated in full pitch coil.
• It is denoted by Kp and its value is always less than unity.
• It can also be defined as the ratio of the vector sum of induced emf per coil to the arithmetic sum of induced emf per coil.

If α be the angle in electrical degrees, by which the span of the coil is less than a pole pitch in the alternator, then mathematically pitch factor is defined as

K_P = resultant emf of short pitch coil/ resultant emf of full pitch coil = cos α/2

For n^th harmonic

K_p = cos nα/2

Where n is the nth harmonic

Distribution factor or belt factor or breadth factor or spread factor (K_d) 

• It is a measure of the resultant emf of a distributed winding compared to a concentrated winding.
• K_d = emf induced in distributed winding/emf induced if the winding is concentrated

The mathematical expression of the distribution factor is given as,

$k_d=\frac{\sin \frac{m\beta }{2}}{m\sin \frac{\beta }{2}}$

Where m is the number of slots per pole per phase

β is the slot angle in degrees

Armature reaction:

• Armature reaction is the effect of armature reaction (ϕ_a) on main field flux (ϕ_m)
• Armature reaction depends on the magnitude of the load and loads power factor.

At lagging load:

• The armature (ϕ_a) opposes the main filed flux (ϕ_m)
• Therefore, it will oppose and weaken the main field flux. It is said to be demagnetized.

​Phasor diagram of lagging load:

• The EMF will get decreased due to demagnetization.
• Therefore, an alternator supplying lagging power factor loads are generally over-excited to maintain the terminal voltage​ 

Mainly there are two types of Armature windings.

1. Distributed winding.
2. Concentrated winding.

Phase belt:  It is the group of slots belong to each phase under each pole, denoted by m.

m slots = slots/pole/phase (spp).

Distributed winding:

For the distributed winding, the coils are connected in series.

Phase belt (slots/pole/phase) m ≥ 3 slots.

Distribution factor (K_d) = (The emf induced for distributed winding)/(The emf induced for concentrated winding)

$K_d=\frac{\sin \left(\frac{mn\gamma }{2}\right)}{msin\left(\frac{n\gamma }{2}\right)}$

Concentrated winding:

For the concentrated winding

Phase belt, m = 1 slot.

 If spp, m =1 then it is concentrated winding.

The phase difference between E₁, E₂, E₃ is 0.

The resultant of voltage is an arithmetic sum (E_r) = E + E + E = 3E

Advantages of Distributed winding:

• The effect of harmonics is eliminated. So the generated Emf is more sinusoidal.
• With the elimination of harmonics, iron losses are reduced therefore, Efficiency will be increased.
• Armature winding is well balanced since winding is placed in all slots.
• Temperature rise (copper losses) are uniformly distributed. So, the cooling is effective.
• The entire armature core is utilized.
• Less slot depth compared to concentrated winding. Therefore less leakage flux, less leakage reactance, less voltage drop, better regulation.

Potier triangle method:

• This method depends on the separation of the leakage reactance of armature and its effects.
• It is used to obtain the leakage reactance and field current equivalent of armature reaction.
• It is the most accurate method of voltage regulation.
• For calculating the regulation, it requires open circuit characteristics and zero power factor characteristics.​

ΔDEF is Potier triangle which is a right angle triangle 

DE = armature MMF (Fa) or field current which compensates armature reaction  

DF = IaXal = armature leakage reactance

The following assumptions are made in this method.

• The armature resistance is neglected.
• The O.C.C taken on no-load accurately represents the relation between MMF and voltage on load.
• The leakage reactance voltage is independent of excitation.
• The armature reaction MMF is constant.

Synchronous impedance method:

• The synchronous impedance method of calculating voltage regulation of an alternator is otherwise called as the EMF method.
• The synchronous impedance method or the EMF method is based on the concept of replacing the effect of armature reaction by an imaginary reactance.
• It gives a result that is higher than the original value. That's why it is called the pessimistic method.
• For calculating the regulation, the synchronous method requires the armature resistance per phase, the open-circuit characteristic, and the short circuit characteristic.

Armature turn method:

It is also known as the MMF method. It gives a value that is lower than the original value. That's why it is called an optimistic method.

To calculate the voltage regulation by MMF Method, the following information is required. They are as follows:

• The resistance of the stator winding per phase
• Open circuit characteristics at synchronous speed
• Field current at rated short circuit current

Reactive power limitations:

• For a given shaft power, the reactive output of a generator is limited by either field current or armature winding heating.
• The maximum reactive power capability corresponds to the maximum reactive power that a generator may produce when operating with a lagging power factor.
• The minimum reactive power capability corresponds to the maximum reactive power that a generator can absorb when operating with a leading power factor.
• These limitations are a function of the real power output of the generator, that is, as the real power increases, the reactive power limitations move closer to zero.

Reactive power control:

Synchronous condenser:

• A synchronous motor takes a leading current when over-excited and, therefore, behaves as a capacitor
• An over-excited synchronous motor running on no load is known as synchronous condenser
• When such a machine is connected in parallel with the supply, it takes a leading current which partly neutralizes the lagging reactive component of the load
• Thus, the power factor is improved

• The power delivered by the alternator connected to the infinite busbar depends on the speed of the prime mover and torque angle.
• More is the power transferred more becomes the torque angle. To increase the power transfer, more torque angle at a given rpm is required for an alternator.
• The speed (RPM) decides the frequency of the electric power supply which needs to be constant.
• This comes from increasing power to the prime over by steam turbine or hydraulic turbine.
• Thus depending on a load, a known torque angle is required on the shaft to deliver power from prime mover to electric alternator at a given speed to maintain the frequency of the supply.
• Therefore, If the prime mover input of an alternator connected directly to an infinite bus is increased, then its active power output increases and vice versa.

​ 

Under the no-load condition, the power drawn by the prime mover of an alternator is utilized to:

• If the phase angle θ is exactly 90° the synchronous machine must be driven by the prime mover whose function is to supply power losses.
• In this case, a synchronous machine should be considered as a generator with zero power output.
• Since its power losses are covered by the electric power system its power factor can only be close to zero.
• Under the no-load condition, the synchronous machine will only draw a small current (active power) to mainly compensate for friction and windage losses and it can be used to supply reactive power and control the power factor of an external system, by controlling the field current.

Stator frame:

• The stator frame supports the stator and the end brackets.
• It must be strong and accurately machined so that the stator is concentric with the rotor.
• It can be made of cast aluminum, cast iron, or fabricated steel plates.

Stator core:

• The main function of the stator core is to carry the alternating flux
• In order to reduce the eddy current loss, the stator core is laminated
• These laminated types of structure are made up of stamping which is about 0.4 to 0.5 mm thick
• All the stampings are stamped together to form the stator core, which is then placed in the stator frame
• The stamping is made up of silicon steel, which helps to reduce the hysteresis loss occurring in the motor and for high magnetic permeability

Synchronous motor:

V curves for synchronous motor give the relation between armature current and DC field current. The curves are shown below.

A synchronous motor is capable of operating at all types of power factor i.e. either UPF, leading, or lagging power factor.

• Lagging power factor: If field excitation is such that Eb < V the motor is said to be under excited and it has a lagging power factor.
• Leading power factor: If field excitation is such that Eb > V the motor is said to be over-excited and it draws leading current. So that the power factor improves.
• Unity power factor: If field excitation is such that Eb = V the motor is said to be normally excited.

Synchronous generator:

A synchronous generator or alternator is capable of operating at all types of power factor i.e. either UPF, leading or lagging power factor.

• Leading power factor: If field excitation is such that Eb < V the alternator is said to be under excited and it has a leading power factor.
• Lagging power factor: If field excitation is such that Eb > V the alternator is said to be over-excited and it draws lagging current. 
• Unity power factor: If field excitation is such that Eb = V the alternator is said to be normally excited.
• V -curve for synchronous generator or alternator is shown below