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Generator Equations Galore

Generator Equations Galore. Summary of the equations for the magnetic and electrical characteristics of a smooth-rotor , 2-pole generator as developed in class to find : Rotor and stator inductances and their mutual inductance The generator equivalent circuit

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Generator Equations Galore

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  1. Generator Equations Galore Summary of the equations for the magnetic and electrical characteristics of a smooth-rotor, 2-pole generator as developed in class to find: Rotor and stator inductances and their mutual inductance The generator equivalent circuit Suppression of harmonics at frequencies: 3n times the fundamental

  2. Rotor and Stator Coupling with Distributed Windings on Both Rotor and Stator • Sinusoidal flux distribution over rotor surface with peak B given by the usual use of Ampere’s Law • Not all flux goes through all turns of the rotor so inductance is slightly smaller than N times the core flux • With cosine distributed stator windings, the mutual inductance has the same form

  3. What’s Going on with Matrix Representation of a Transformer • Every coil current induces a flux in every other coil proportional to that current • The voltage across a coil is the time derivative of the flux through the coil – Faraday’s Law • Multiplying out the stator voltage gives an equation equivalent to a voltage source in series with an inductor. That inductor has the full inductance of the stator winding so the air gap must be chosen to optimize this inductance.

  4. Three-Phase System: two more distributed stator coils

  5. Exciting the Rotor with DC – No Voltage Variation across Its Terminals! • Multiply the matrix row onto the current vector for the rotor voltage – in a single-phase generator, this had a time dependent value. • First term vanishes because the rotor current is DC. Of course, the rotor DC voltage must be enough to satisfy Ohm’s law: • Each squared cosine term has a DC term and a cosine at twice the frequency. The derivative of the DC term is zero and the set of cosines is a phasor sum that is also zero.

  6. Why? What’s Going On? • Exciter produces sinusoidal radial B field in the air gap oriented with the peak along one axis – shown here as a vector • Output current produces similar field rotating in the same direction and speed but directed opposing the exciter field at a different angle • Net is smaller field oriented between them at the angle of the generator output voltage

  7. Why a Wye to Delta Connection at Generator Output? • The phase shift in a three-phase system is actually a time delay rather than a phase shift from circuit reactances because the shift is from a delay as the rotor turns by an amount per phase step of: • Suppose the m’th winding has a harmonic in its voltage at frequency 3n times the fundamental • This means that all three output nodes have the same voltage relative to the center of the star and a delta-wound transformer will have no harmonic at that frequency in its magnetic field!!

  8. Connecting to the Grid: The Power Factor of the Transmission System and Required Rotor Phase Lag • The grid appears to a generator as a “stiff” AC source that is much stronger than any single generator, being the aggregate of hundreds of sources totaling power that is at least 8 to 200 times greater than a typical generator. • The transformers and transmission lines that connect this generator to the grid add a series inductance that makes this appear as a load with power factor < 1. Typically generators are specified to run into a power factor of 0.90 or an angle of degrees. • This equivalent circuit is for one, wye-connected phase of the three phase system.

  9. Phasor Model of the Generator to Grid Connection at Startup and Equilibrium • Equilibrium operation has the generator output current and voltage at nameplate rating. The voltage is higher than the grid because of the grid inductance and power factor. • The exciter current sets the length of E, the magnitude of the emf. • The turbine torque sets the angle to balance the turbine power to the real power delivered to the grid. • At startup there is no grid connection and no current. The generator voltage is E and that must be brought to match frequency, phase and magnitude before the switch is closed. • After the connection is made, the torque and excitation increase together until equilibrium is reached

  10. What Does a Bobolink Look Like?

  11. A Large Three-Phase Transformer!

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