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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 circle6 Steady state modeling rhombus6 rotor mmf and stator mmf are stationary with respect to each other rhombus6 flux linkage with the rotor are invariant with time rhombus6 no voltages are induced on the rotor circuits circle6 Transient modeling rhombus6 flux linkage changes with time rhombus6 differential equations have time-varying coefficients rhombus6 Parks transformation rhombus6 dynamic behavior square6 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 circle6 Transient analysis will be applied in the dynamic study of generators circle6 Generators experience dynamic behavior during rhombus6 switching load rhombus6 faults circle6 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 ( 29 ( 29 ( 29 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 t i dt t i d L t i R t V m m t m m m ϖ ϖ γ τ γ α γ α ϖ α ϖ τ + = = = = - - - + = - - + = - - R L t = 0 V(t) i(t) Transient Analysis circle6 The voltage source is sinusoidal: circle6 The KVL equation are: ( 29 α ϖ + = t V t v m sin ) (
(c) Feb 2004 Power Systems I 5 Example circle6 Solve for the time-domain solution of the current rhombus6 for a faulted generator having the following characteristics R = 0.125 L = 10 mH v ( t ) = 151 sin (377 t + α ) rhombus6 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 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 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 ) ( 1 . 88 Let (a) 1 . 88 sin 40 1 . 88 377 sin 40 ) ( s 08 . 0 125 . 0 01 . 0 A 40 772 . 3 151 1 . 88 772 . 3 77 . 3 125 . 0 01 . 0 377 125 . 0 t t t m e t e t t i t t i e t t i I j j Z - - - + - = ° - - ° - = ° - = ° - ° = = ° = ° - - ° - + = = = = = ° = + = + = α α α 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 i(t) Example (a) zero dc offset current (b) maximum dc offset current

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(c) Feb 2004 Power Systems I 8 Transient Analysis circle6 Synchronous Machines rhombus6 Models and analysis were previously developed for steady state behaviors square6 rotor and stator magnetic fields are stationary with respect to each other square6 the flux linkage in the rotor circuit are constant in time square6 the per phase equivalent circuit becomes a constant generated emf in series with a simple impedance rhombus6 Under transient conditions (time varying) the above assumptions are no longer valid square6 changing stator current are reflected in a dynamic flux linkage square6 changing flux linkage induces transient currents in the rotor square6 transient rotor currents in turn react with the stator and the induced voltages
(c) Feb 2004 Power Systems I 9 Synchronous Machine Model circle6 The synchronous machine consist of: rhombus6 three ac stator windings mounted on the stator rhombus6 one field winding mounted on the rotor rhombus6

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