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lecture10 - Chapter 8: Transient Analysis of Synchronous...

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FAMU-FSU College of Engineering Chapter 8: Transient Analysis of Synchronous Machines

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(c) Feb 2004 Power Systems I 2 Synchronous Machines Steady state modeling rotor mmf and stator mmf are stationary with respect to each other flux linkage with the rotor are invariant with time no voltages are induced on the rotor circuits Transient modeling flux linkage changes with time differential equations have time-varying coefficients Parks transformation dynamic behavior sub-transient period, transient period, and steady-state period emf X S R A V T I G
(c) Feb 2004 Power Systems I 3 Transient Analysis Transient analysis will be applied in the dynamic study of generators Generators experience dynamic behavior during switching load faults Consider the transient behavior of an RL circuit with a switched voltage source R L t = 0 V(t) i(t)

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(c) Feb 2004 Power Systems I 4 () 2 2 2 1 and , tan , , : where sin sin ) ( ) ( ) ( sin 0 L R Z R L R L Z V I e I t I ti dt tid L R t V m m t m m m ω γ τ α + = = = = + = + = R L t = 0 V(t) i(t) Transient Analysis The voltage source is sinusoidal: The KVL equation are: ( ) + = t V t v m sin ) (
(c) Feb 2004 Power Systems I 5 Example Solve for the time-domain solution of the current for a faulted generator having the following characteristics R = 0.125 Ω L = 10 mH v ( t ) = 151 sin (377 t + α ) which will give (a) zero dc offset current, (b) maximum dc offset current

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(c) Feb 2004 Power Systems I 6 Example ( )( ) () 08 . 0 08 . 0 08 . 0 40 377 cos 40 90 sin 40 90 377 sin 40 ) ( 9.1 90 1. 88 Let (b) 377 sin 40 ) ( 88 Let (a) 88 sin 40 88 377 sin 40 ) ( s 08 . 0 125 .0 01 . 0 A 40 772 .3 151 88 772 77 . 3 125 01 . 0 377 125 t t t m e t e t ti t e t I j j Z + = ° ° = ° = ° ° = = ° = ° ° + = = = = = Ω ° = + = + = α τ
(c) Feb 2004 Power Systems I 7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -40 -20 0 20 40 t, sec i(t) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -40 -20 0 20 40 60 80 t, sec Example (a) zero dc offset current (b) maximum dc offset current

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(c) Feb 2004 Power Systems I 8 Transient Analysis Synchronous Machines Models and analysis were previously developed for steady state behaviors rotor and stator magnetic fields are stationary with respect to each other the flux linkage in the rotor circuit are constant in time the per phase equivalent circuit becomes a constant generated emf in series with a simple impedance Under transient conditions (time varying) the above assumptions are no longer valid changing stator current are reflected in a dynamic flux linkage changing flux linkage induces transient currents in the rotor transient rotor currents in turn react with the stator and the induced voltages
(c) Feb 2004 Power Systems I 9 Synchronous Machine Model The synchronous machine consist of: three ac stator windings mounted on the stator one field winding mounted on the rotor two fictitious windings which model short-circuited paths of the damper windings When modeling, the following are assumed:

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lecture10 - Chapter 8: Transient Analysis of Synchronous...

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