Synchronous Machines notes Part – II for Electrical Engineering

Cylindrical Rotor Synchronous Generator

The alternator is operating on no load i.e., the rotor is rotating and energized and the stator is open-circuited.

• Its circuit diagram is shown in below.

• An equivalent circuit of a synchronous generator shown in below.

• Let X_s = Synchronous reactance, X_ar = Fictitious reactance, X_a = Armature reactance, R_a = Armature resistance, and Z_s = Synchronous impedance.

X_s = X_ar + X_a

Z_s = R_a + j X_s

E_a = V + I_a Z_s

Phasor Diagram

• The phasor diagram for inductive, purely resistive and capacitive loads are shown in the figure below. All these phasor diagrams apply to one phase of a 3-φ machine.

• At lagging, power factor:

• At unity, power factor:

• At leading, power factor:

Power Relationship

• Mechanical power input to the generator P_mechanical = T_sω_s
• DC power input to a wound rotor P_in electrical= I_f
• Total power input: P_in = T_sω_s + I_f
• Real power output:

• Reactive power output:

where, V = Terminal voltage per phase, and E_f = Excitation voltage per phase = Phase angle between E_f and V, and X_s = Synchronous reactance

Salient Pole Synchronous Machine

The component currents I_d and I_q provide component voltage drops jI_d X_d and jI_d X_q as shown in the figure.

E_a = V + I_aR_a + jI_dX_d + jI_qX_q

I = I_d + I_q

If R_a is neglected, E_a = V + jI_dX_d + jI_qX_d

• Phasor Diagram

• For Generating Mode: 

• For Motoring Mode:

 Note: δ = ψ – φ (generating mode), and δ = φ – ψ (motoring mode)

• Power Angle

Use + for synchronous generator, and - for synchronous motor (here R_a is neglected)

• Output Power

P₀ = 3V(I_d sin δ + I_d cos δ)

• Total Power Developed

Cylindrical Rotor Synchronous Motor

where, V = Terminal phase voltage applied to the armature, E_f = Excitation voltage, R_a = Effective armature resistance/phase, X_s = Synchronous reactance/phase, Z_s = Impedance/phase.

E_a = V – I_aR_a – jI_aX_s

• Phasor Diagram

Salient Pole Synchronous Motor

E_a = V – I_aR_a – jI_aX_q – jI_d(X_d – X_q)

• Phasor Diagram:

• Power Developed:

Experimental Determination of Circuit Parameters

In the per phase equivalent circuit model illustrated above first for Generator & Second For Motor, there are three parameters need to be determined: winding resistance Ra, synchronous reactance Xs, and induced emf in the phase winding Ea. The phase winding resistance Ra can be determined by measuring DC resistance of the winding using the volt-ampere method, while the synchronous reactance and the induced emf can be determined by the open circuit and short circuit tests.

Open Circuit Test
Drive the synchronous machine at the synchronous speed using a prime mover when the stator windings are open circuited. Vary the rotor winding current, and measure stator winding terminal voltage. The relationship between the stator winding terminal voltage and the rotor field current obtained by the open circuit test is known as the open circuit characteristic of the synchronous machine.

Short Circuit Test
Reduce the field current to a minimum, by using the field rheostat, and then open the field supply circuit breaker. Short the stator terminals of the machine together through three ammeters; Close the field circuit breaker; and raise the field current to the value noted in the open circuit test at which the open circuit terminal voltage equals the rated voltage while maintaining the synchronous speed. Record the three stator currents. (This test should be executed quickly as the stator currents may be greater than the rated value).

• Under the assumptions that the synchronous reactance Xs and the induced emf Ea have the same values in both the open and short circuit tests,

     and that Xs >> Ra, we have

Effect of Excitation

By controlling the rotor excitation current such that the synchronous condenser draws a line current of leading phase angle, whose imaginary component cancels that of the load current, the total line current would have a minimum imaginary component.

Therefore, the overall power factor of the inductive load and the synchronous condenser would be close to one and the magnitude of the overall line current would be the minimum.

It can also be seen that only when the power factor is a unit or the stator current is aligned with the terminal voltage, the magnitude of the stator current is minimum.

By plotting the magnitude of the stator current against the rotor excitation current, a family of “V” curves can be obtained. It is shown that a larger rotor field current is required for a larger active load to operate at unity power factor.

Voltage Regulation

The variation in the terminal voltage with load is called voltage regulation, hence 

Per-unit voltage regulation = (|V_NL|-|V_FL|)/|V_FL| = |E_f|-|V|/|V|

• (a) zero power factor leading
• (b) 0.8 power factor leading
• (c) 0.9 power factor leading
• (d) unity power factor
• (e) 0.9 power factor lagging
• (f) zero power factor lagging.

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