1 synchronous machines two-pole,3-phase,wye-connected,salient-pole synchronous machine

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Two-pole,3-phase,wye-connected,salient-pole synchronous machineTwo-pole,3-phase,wye-connected,salient-pole synchronous machine

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

In abc reference frame, voltage equations can be written as

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

flux linkage equations

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

flux linkage equations

Referring all rotor variables to the stator windingsReferring all rotor variables to the stator windings

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Referring all rotor variables to the stator windingsReferring all rotor variables to the stator windings

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Referring all rotor variables to the stator windingsReferring all rotor variables to the stator windings

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

TORQUE EQUATION IN MACHINE VARIABLES

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

SWING EQUATION

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Stator Voltage Equations in Arbitrary Reference-frame Variables

SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The rotor voltage equations are expressed only in the rotor The rotor voltage equations are expressed only in the rotor reference frame:reference frame:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The flux linkage equations may be expressed as:The flux linkage equations may be expressed as:

The sinusoidal terms are constant, independent of The sinusoidal terms are constant, independent of and and rr only if only if = = rr

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Therefore, the time-varying inductances are eliminated from the Therefore, the time-varying inductances are eliminated from the voltage equations only ifvoltage equations only if the the reference frame is fixed in the rotorreference frame is fixed in the rotor..

Voltage Equations In Rotor Reference-frame variables: park's Equations

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Park's voltage equations are Park's voltage equations are often written in expanded form:often written in expanded form:

Flux linkages in expanded form:Flux linkages in expanded form:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The Equivalent q-axis Circuits:The Equivalent q-axis Circuits:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The Equivalent d-axis Circuits:The Equivalent d-axis Circuits:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The Equivalent 0-axis Circuits:The Equivalent 0-axis Circuits:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

It is often convenient to express the voltage and flux linkage It is often convenient to express the voltage and flux linkage equations in terms of reactances rather than inductances:equations in terms of reactances rather than inductances:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Also, it is convenient to define:Also, it is convenient to define:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

If we select the currents as independent variables:If we select the currents as independent variables:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

If we select the flux linkages per second as independent If we select the flux linkages per second as independent variables:variables:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Torque Equations in Substitute Variables::

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Rotor Angle :Rotor Angle :

it is convenient to relate the position of the rotor of a synchronous it is convenient to relate the position of the rotor of a synchronous machine to a voltage or to the rotor of another machine.machine to a voltage or to the rotor of another machine.

The electrical angular displacement of the rotor relative to its terminal The electrical angular displacement of the rotor relative to its terminal voltage is defined as the rotor angle,voltage is defined as the rotor angle,

The rotor angle is the displacement of the rotor generally referenced to The rotor angle is the displacement of the rotor generally referenced to the maximum positive value of the fundamental component of the the maximum positive value of the fundamental component of the terminal voltage of phase a:terminal voltage of phase a:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Rotor Angle :Rotor Angle :

It is important to note that the rotor angle is often used as the It is important to note that the rotor angle is often used as the argument in the transformation between the rotor and argument in the transformation between the rotor and synchronously rotating reference framessynchronously rotating reference frames

The rotor angle is often used in The rotor angle is often used in relating torque and rotor speed (if relating torque and rotor speed (if ee is constant): is constant):

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PER UNIT SYSTEMPER UNIT SYSTEM

SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

Base voltage: the rms value ofthe rated phase voltage for the abc variables the peak value for the qd0 variables. Base power: When considering the machine separately, the power base is selected as its volt-ampere rating. When considering power systems, a system power base (system base) is selected Once the base quantities are established, the corresponding base current and base impedance may be calculated.Base torque is the base power divided by the synchronous speed of the rotor:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

The torque expressed in per unit:The torque expressed in per unit:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

ANALYSIS OF STEADY-STATE OPERATION:ANALYSIS OF STEADY-STATE OPERATION:

For balanced conditions:For balanced conditions:the 0s quantities are zero.the 0s quantities are zero.rr is constant and equal to is constant and equal to ee

the rotor windings do not experience a change of flux linkagesthe rotor windings do not experience a change of flux linkagesthe current is not flowing in the short-circuited damper windingsthe current is not flowing in the short-circuited damper windingsthe time rate of change of all flux linkages neglectedthe time rate of change of all flux linkages neglected

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

ANALYSIS OF STEADY-STATE OPERATION:ANALYSIS OF STEADY-STATE OPERATION:

For balanced conditions:For balanced conditions:

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SYNCHRONOUS MACHINESSYNCHRONOUS MACHINES

ANALYSIS OF STEADY-STATE OPERATION:ANALYSIS OF STEADY-STATE OPERATION:

Hence:

and if it is noted that:

Then:

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ANALYSIS OF STEADY-STATE OPERATIONANALYSIS OF STEADY-STATE OPERATION

It is convenient to define the last term on the right-hand side as (excitation voltage):

if rs is neglected, the expression for the balanced steady-state electromagnetic torque in per unit can be written as:

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

Dynamic performance of a hydro turbine generator during a step increase in Dynamic performance of a hydro turbine generator during a step increase in input torque from zero to rated:input torque from zero to rated:

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

Torque versus rotor angle characteristicsTorque versus rotor angle characteristics

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

Dynamic performance of a steam Dynamic performance of a steam turbine generator during a step turbine generator during a step increase in input torque from zero increase in input torque from zero to 50% rated.to 50% rated.

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DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE DYNAMIC PERFORMANCE DURING A SUDDEN CHANGE IN INPUT TORQUEIN INPUT TORQUE

Torque versus rotor angle characteristicsTorque versus rotor angle characteristics

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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS

a hydro turbine generatora hydro turbine generator

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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS

Torque versus rotor angle characteristics:Torque versus rotor angle characteristics:

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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS

a steam turbine generatora steam turbine generator

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DYNAMIC PERFORMANCE DURING A 3 PHASE FAULT AT THE MACHINE TERMINALS

Torque versus rotor angle characteristics:Torque versus rotor angle characteristics:

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COMPUTER SIMULATION

Simulation in Rotor Reference Frame

Where:

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COMPUTER SIMULATION

Simulation in Rotor Reference Frame

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COMPUTER SIMULATION

Simulation of Saturation

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COMPUTER SIMULATION

Simulation of Saturation