Doubly Excited System in electromachinal energy conversion.pptx

A DC motor is an electrical motor that uses direct current (DC) to produce mechanical force. The most common types rely on magnetic forces produced by currents in the coils. Nearly all types of DC motors have some internal mechanism, either electromechanical or electronic, to periodically change the direction of current in part of the motor.

1. Energy Equation For Doubly Excited Magnetic System Stator coil has resistance R 1 and rotor coil has resistance R 2 . Respective exciting voltages are V 1 and V 2 and currents in coils are I 1 and I 2 . Ø 1 and Ø 2 are respective fluxes by coil 1 and coil 2. λ 1 and λ 2 are total flux linkages of coil 1 and coil 2. L 1 ,L 2 and M are self and mutual inductances of coil 1 and coil 2.

2. Flux linkages with coil 1 and coil2, λ 1 = L 1 *I 1 + M*I 2 λ 2 = L 2 *I 2 + M*I 1 Voltage equations, V 1 = I 1 *R 1 + d λ 1 / dt V 2 = I 2 *R 2 + d λ 2 / dt Putting the value of λ 1 in voltage equation, V 1 = I 1 *R 1 + d (L 1 I 1 + MI 2 )/ dt V 1 = I 1 *R 1 + L 1 dI 1 / dt + I 1 dL 1 / dt + MdI 2 / dt + I 2 dM / dt Multiplying with I 1 on both sides, V 1 *I 1 = I 1 * I 1 *R 1 + L 1 * I 1 dI 1 / dt + I 1 * I 1 dL 1 / dt + M * I 1 dI 2 / dt + I 1 * I 2 dM / dt -------------(1)

3. Putting the value of λ 1 in voltage equation , V 2 = I 2 *R 2 + d ( L 2 *I 2 + M*I 1 )/ dt V 2 = I 2 *R 2 + L 2 * dI 2 / dt + I 2 *dL 2 / dt + M*dI 1 / dt + I 1 dM / dt Multiplying with I 2 on both sides, V 2 *I 2 = I 2 *I 2 *R 2 + L 2 * I 2 dI 2 / dt + I 2 * I 2 *dL 2 / dt + M * I 2 * dI 1 / dt + I 1 * I 2 * dM / dt -----------(2)

4. Adding both equations and multiplying with dt , (V 1 *I 1 + V 2 *I 2 ) dt = (I 1 * I 1 *R 1 + I 2 * I 2 *R 2 ) dt + (L 1 * I 1 dI 1 + L 2 * I 2 dI 2 + M * I 1 dI 2 + M * I 2 dI 1 + 2* I 1 * I 2 dM + I 1 * I 1 dL 1 + I 2 * I 2 dL 2 ) dt (V 1 *I 1 + V 2 *I 2 - I 1 * I 1 *R 1 - I 2 * I 2 *R 2 ) dt = (L 1 * I 1 dI 1 + L 2 * I 2 dI 2 + I 1 * I 1 dL 1 + I 2 * I 2 dL 2 ) + (M * I 1 dI 2 + M * I 2 dI 1 + 2* I 1 * I 2 dM ) Useful electrical energy input = Field energy stored in the system + Electrical to Mechanical energy transfer

5. From above derived relations we can find the stored magnetic energy in doubly excited system considering steady state When the mechanical output is zero the entire electrical energy is stored as magnetic energy in the system. Steady terms in equation dL 1 , dL 2 , dM is zero and after integrating the equation, W fe = ½*L 1 *I 1 *I 1 + ½*L 2 *I 2 *I 2 +M*I 1 *I 2

6. Electromagnetic Torque Equation For Doubly Excited System Stored energy is used to mechanical work. Mechanical work done = Rate of change of stored energy = dW fe / dt dW fe / dt = d (½*L 1 *I 1 *I 1 + ½*L 2 *I 2 *I 2 +M*I 1 *I 2 )/ dt = ½*L 1 *d(I 1 *I 1 )/ dt +½*I 1 *I 1 *dL 1 / dt + ½*L 2 *d(I 2 *I 2 )/ dt +½* I 2 *I 2 *dL 2 / dt + M*I 2 *dI 1 / dt+M *I 1 *dI 2 /dt+ I 1 * I 2 dM / dt = L 1 *I 1 ( dI 1 / dt ) + ½*I 1 *I 1 *dL 1 / dt + L 2 *I 2 ( dI 2 / dt ) + ½*I 2 *I 2 *dL 2 / dt + M*I 2 *dI 1 / dt + M*I 1 *dI 2 / dt + I 1 *I 2 dM/ dt

7. Integrating above equation w.r.t dt , W fe = (L 1 *I 1 dI 1 + L 2 *I 2 dI 2 + M*I 2 *dI 1 + M*I 1 *dI 2 )+( ½*I 1 *I 1 *dL 1 + ½* I 2 *I 2 *dL 2 + I 1 *I 2 * dM ) W fe = W stored + W me W me = ½*I 1 *I 1 *dL 1 + ½*I 2 *I 2 *dL 2 +I 1 *I 2 * dM In rotation while mechanical energy is produced the terms dL 1 , dL 2 , dM are not constant but vary with θm . Differentiating above equation w.r.t θ m , Torque equation = dW me / dθ m =½*I 1 *I 1 *dL 1 /dθ m +½*I 2 *I 2 *dL 2 / dθ m +I 1 *I 2 * dM / dθ m Putting I 2 = 0, we get torque equation for singly excited system, Torque equation = ½*I 1 *I 1 *dL 1 / dθ m

8. Comparison between Singly and Doubly Excited Systems Singly Excited System One coil takes active part in energy conversion process. It has winding on stationary part only. It works on induction or asynchronous principle. Used in constant speed application. Produce useful startup torque. E.g. Magnetic relays, D.C. generator, Induction machine Doubly Excited System Two coils take active part in energy conversion process. It has winding on stationary and rotating part. It works on synchronous principle. Used in variable speed application. Do not produce useful startup torque. E.g. DC motor , Commutator and Synchronous motor