Synchronous Generator : Concepts - 4

Synchronous Generator: Parallel-Generator Theorem:

Reference to Figure given above , where 2 generators are connected in parallel. Let the load be I amps at V volts such that V / I = Z.

Then , V = (I1+I2 )Z = [( E1-V)/Zs1+ ( E2-V)/Zs2 ]

= [ E1/Zs1 + E2/Zs2 ] Z – V[1/Zs1 + 2/Zs2]Z

i.e. V [1/Z +1/Zs1 + 1/Zs2]

= E1/Zs1 + E2/Zs2 i.e. V [1/Z0] = Isc

where Isc is the total short circuit current obtained by summing the terms E1/Zs1 and E2/Zs2where

1/Z0=1/Z +1/Zs1 + 1/Zs2

This theorem holds true for any number of generator.

The characteristics of a synchronous generator on infinite bus-bars are quite different from those when it operates on its own local load. In the latter case, a change in the excitation changes the terminal voltage, while the pf is determined by the load. When working on infinite bus-bars, on the other hand, no alternation of the excitation can change the terminal voltage which is fixed by the network, the point however, is affected. In both cases the power developed by a generator (or received by a motor) depends solely upon the mechanical power provided (or load applied to it).

Consider 2 alternators operating in parallel on infinite bus-bars, with identical initial operating conditions, i.e. the active and reactive powers are divided equally. Now suppose the excitation of alternator 1 is increased then as stated earlier, the kW loading of the 2 alternators remains unchanged as the mechanical input remains the same. The change is seen in the KVAR loading due to the changes in the individual load currents and points.

Similarly with change in steam supply of one of the alternator with excitation kept same, the change is observed in the kW loading of the 2 alternators. While the KVAR loading remains unaltered.

SYNCHRONIZING POWER (Ps):

Let for same cause the angle δ changes to δ I δ’.

The synchronous power, Ps = (E+ V) Zs sin (θ + δ) sinδ’

For large generator, Zx =Sx

i.e. θ = 900

Ps = ( E+ V ) Zs cos δ sinδ’

When an unloaded M/c is synchronized to a constant voltage bus bar

δ = 0,

Ps = (E+ V) Zs sin δ/ ph

when δ is small enough

Ps = (E+ V) Zs sin δ’/ ph

=V I_sc δ’ / ph where I_sc = E_f / Z_s

_{← Synchronous Generator > Theory Page 3}

Synchronous Generator : Concepts - 3

Synchronous Generator: BLONDELS TWO REACTION THEORY:

In case of cylindrical pole machines, the direct-axis and the quadrature axis mmfs act on the same magnetic circuits, hence they can be summed up as complexors. However, in a salient-pole machine, the two mmfs do not act on the same magnetic circuit. The direct axis component Fadoperates over a magnetic circuit identical with that of the field system, while the q-axis component Faq is applied across the interpole space, producing a flux distribution different from that of Fad or the Field mmf.

The Blondel's two reaction theory hence considers the results of the cross and direct-reaction components separately and if saturation is neglected, accounts for their different effects by assigning to each an appropriate value for armature-reaction "reactive" respectively Xaq and Xad .

Considering the leakage reactance, the combined reactance values becomes

Xad = X + X ad and X sq = X aq

Xsq < Xsd as a given current component of the q-axis gives rise to a smaller flux due to the higher reluctance of the magnetic path.

Let lq and Id be the q and d-axis components of the current I in the armature reference to the phasor diagram in Figure. We get the following relationships

Iq= I cos (σ+θ) Ia = I cosφ

Id = I sin (σ+ φ) Ir = I sinφ

And I = √(Id2 + Iq2)= = √(Id2 + Ir2)

where Ia and Ir are the active and reactive components of current I.

Voltage Regulation of synchronous generator:

voltage regulation of an alternator is defined as "the rise in voltage when full load is removed (field excitation and speed remaining unaltered) divided by the rated terminal voltage. Thus

% regulation =( E0 – V ) / V x 100

In case of leading load pf the regulation is negative.

Parallel Operation of Synchronous Generators:

A stationary synchronous generator should not be connected to five bus bars because, stator induced e.mf. being zero, a short circuit will result. For proper paralleling of Generators the following three conditions must be satisfied :

1. The terminal voltage of the incoming generator must be same as bus-bar voltage.

2. The speed of the incoming generator must be such that its frequency (PN/120) equal bus-bar frequency.

3. The phase of the synchronous generator voltage must be identical with the phase of the bus voltage.

• ← Synchronous Generator > Theory Page 2
• Synchronous Generator > Theory Page 4 →

Synchronous Generator : Concepts - 2

(3) Load Characteristics of Synchronous Generator:

While the exciting current and the speed remain constant, the terminal voltage changes with the load current in the armature and the relationship between the terminal voltage and load current of an alternator is known as its load characteristics.

When the armature current increases, the terminal voltage drops. This is mainly due to

(a) Resistance and reactance of armature winding, and

(b) Armature reaction.

The load characteristics of an alternator is shown in the figure.

Phasor diagram of synchronous generator under three types of leading conditions :

Simplified equivalent AC circuit (per phase) for synchronous generator:

AT POWER FACTOR LAGGING:

(A) When RA is very small:

α = torque angle

P = 3 VφE0 sin α/ Xs

Torque induced,

T ind = 3 VφE0 sin α/ Xs ωm

where ωm = speed.

(B) General case:

P = 3 E0/Zs [ E0cosθ – V (cosθ + α) ]

where cosθ = Ra / Zs

:. Small Ra implies θ = 90.

For maximum power output:

cosφ = E0/√ (E20 +V2φ)

α = 900

3 P max = (3 Vφ I max E0)/√ (E20 +V2 φ) = 3 Vφ E0 / Xs

• ← Synchronous Generator > Theory Page 1
• Synchronous Generator > Theory Page 3 →

Electric Circuits and Ohm's Law : Concepts - 2

Electric Current and Ohm's Law (Continued):

Resistances in Parallel :

When conductors are joined in parallel, following relations hold good

I = I1 + I2 + I3

1 / R= 1 / R1 + 1 / R2 + 1 / R3

R= ( R1 +R2 + R3 ) / ( R1R2 + R2R3 + R3R1 )

G = G1 + G2 + G3

Effect of Temperature on Resistance :

Resistance of all materials is affected by the variations in temperature. The effect of temperature in general is as follows:

(i) Resistance of most of the metallic conductors increases with rising temperature

(ii) Resistance of non-conductors or insulators usually decreases with rising temperature.

Temperature coefficient of resistance :

It is defined as the increase in resistance per ohm original resistance per

oC rise in temperature. Thus

α = (Rt - Ro )/(Ro . t)

where

Ro = resistance at 0 oC

Rt = resistance at t °C

t = temperature rise in oC

Usually

α is of the order of l0 -4 Ω/ Ω oC for most of the metals.

In case of insulators and electrolytes, α is usually negative.

Temperature coefficient of carbon is negative.

Resistor color coding :

Carbon resistors are physically small in size and color code is used to represent their value in ohms. The scheme is shown in Figure above. Various codes for colors are given in the table below :

Color Code
Color	Value
Black	0
Brown	1
Red	2
Orange	3
Yellow	4
Green	5
Blue	6
Violet	7
Grey	8
White	9

DRIFT VELOCITY :

The drift velocity vd of charge carriers is related to current I by the equation

I = n α e vd

Where

n = density of charge carriers in conductor,

α = area of cross-section of conductor,

e = charge on each carrier.

A large amount of energy has to be supplied to pull an electron from inside to outside of the metal surface. This energy is called work function. This energy is the characteristic of the metal.

SUPER-CONDUCTIVITY :

As temperature of metallic conductor decreases, their resistivity decreases. In certain metallic conductors as temperature decreases, the resistivity falls to zero at a certain temperature called super-conducting temperature. It happens for mercury at 4 K and for tin at 3.72 K. This phenomenon is called super-conductivity.

Resistivity of semiconductors decreases with increase in temperature

ρT = ρo e-(Eg / kT)

where

Eg = band gap energy,

ρT = resistivity at T K,

k = Boltzman constant.

NON LINEAR DEVICES :

The devices for which potential difference V Vs current I curve is not a straight line are called non-linear devices. They do not obey Ohm's law and resistance of these devices is a function of V or I e.g. vacuum tubes, junction diodes, thermistors etc.

The dynamic resistance of such devices is given as

r = Lt∆ t → 0 ∆ V / ∆ I = d V / d I

where

∆V is the change in p.d.

← Electric Circuit > Theory Page 1

Electric Circuits and Ohm's Law : Concepts

Resistances in Parallel :

When conductors are joined in parallel, following relations hold good

I = I1 + I2 + I3

1 / R= 1 / R1 + 1 / R2 + 1 / R3

R= ( R1 +R2 + R3 ) / ( R1R2 + R2R3 + R3R1 )

G = G1 + G2 + G3

Effect of Temperature on Resistance :

Resistance of all materials is affected by the variations in temperature. The effect of temperature in general is as follows:

(i) Resistance of most of the metallic conductors increases with rising temperature

(ii) Resistance of non-conductors or insulators usually decreases with rising temperature.

Temperature coefficient of resistance :

It is defined as the increase in resistance per ohm original resistance per

oC rise in temperature. Thus

α = (Rt - Ro )/(Ro . t)

where

Ro = resistance at 0 oC

Rt = resistance at t °C

t = temperature rise in oC

Usually

α is of the order of l0 -4 Ω/ Ω oC for most of the metals.

In case of insulators and electrolytes, α is usually negative.

Temperature coefficient of carbon is negative.

Resistor color coding :

Carbon resistors are physically small in size and color code is used to represent their value in ohms. The scheme is shown in Figure above. Various codes for colors are given in the table below :

Color Code
Color	Value
Black	0
Brown	1
Red	2
Orange	3
Yellow	4
Green	5
Blue	6
Violet	7
Grey	8
White	9

DRIFT VELOCITY :

The drift velocity vd of charge carriers is related to current I by the equation

I = n α e vd

Where

n = density of charge carriers in conductor,

α = area of cross-section of conductor,

e = charge on each carrier.

A large amount of energy has to be supplied to pull an electron from inside to outside of the metal surface. This energy is called work function. This energy is the characteristic of the metal.

SUPER-CONDUCTIVITY :

As temperature of metallic conductor decreases, their resistivity decreases. In certain metallic conductors as temperature decreases, the resistivity falls to zero at a certain temperature called super-conducting temperature. It happens for mercury at 4 K and for tin at 3.72 K. This phenomenon is called super-conductivity.

Resistivity of semiconductors decreases with increase in temperature

ρT = ρo e-(Eg / kT)

where

Eg = band gap energy,

ρT = resistivity at T K,

k = Boltzman constant.

NON LINEAR DEVICES :

The devices for which potential difference V Vs current I curve is not a straight line are called non-linear devices. They do not obey Ohm's law and resistance of these devices is a function of V or I e.g. vacuum tubes, junction diodes, thermistors etc.

The dynamic resistance of such devices is given as

r = Lt∆ t → 0 ∆ V / ∆ I = d V / d I

where

∆V is the change in p.d.

← Electric Circuit > Theory Page 1