Multiple Excited Magnetic System

Multiple Excited Magnetic System

AU : May-03, 06, 07, 10, 11, 13, 14, 15, 16, Dec.-03, 04, 05, 10, 12, 14, 15, 18, 19, April-04, Oct.-02

• For continuous energy conversion devices like Stator alternators, synchronous motors etc., multiply excited magnetic systems are used. In practice, doubly excited systems are very much in use.

• The Fig. 2.7.1 shows doubly excited magnetic system. This system has two independent sources of excitations. One source is connected to coil on stator while other is connected to coil on rotor.

Let  i₁ = Current due to source 1

i₂ = Current due to source 2

 λ₁ = Flux linkages due to i₁

λ₂= Flux linkages due to i₂

θ = Angular displacement of rotor

T_f = Torque developed

Due to two sources, there are two sets of three independent variables i.e. (λ₁ , λ₂ , θ) or (i₁ , i₂ , θ).

Case 1 : Independent variables λ₁ , λ₂ , θ , i.e. λ₁ , λ₂ , are constants.

From the earlier analysis it is known,

Ex. 2.7.1 Two coupled coils have self and mutual inductance of      

L₁₁ = 3 + 1/3x’;L₂₂ = 1 + 3 1/3x’;L₁₂ = L₂₁ =1/3x

 Over a certain range of linear displacement x. The first coil is excited by a constant current of 10 A and the second by a constant current of - 5 A. Find the mechanical work done if x changes from 0.5 to 1 m and energy supplied by each electrical source for the above case.          AU : April-04, Marks 16      

Sol. : As coils are excited by constant currents, co-energy expressions to be used.

Ex 2.7.2 Two coupled coils have self and mutual inductance of L₁₁ = 3+1/3x^’L₂₂ =1+1/4x’L₁₂ = L₂₁ = 1/2x over a certain range of linear displacement x. Find the expression for the time-average force of field origin at x = 0.6 m if :

 1) Both coils are connected in series across a voltage source of 150 cos 314 t V.

2) Both coils are connected in parallel across a voltage source of 150 cos 314 t V.  

AU : Oct.-02, Marks 8

Sol. :

Ex. 2.7.3 Two coupled coils have self and mutual inductance of L₁₁ = 2 + 1/(2x); L₂₂ =1+1/(2x); L₁₂ = L₂₁ = 1/(2x), over a certain range of linear displacement x. The first coil is excited by a constant current of 20 A and the second by a constant current of - 10 A. Find 

 1) Mechanical work done if x changes from 0.5 to 1 m.     

2) Energy supplied by each electrical source in part (1).

3) Change in field energy in part (1).

 Hence verify that the energy supplied by the sources is equal to the increase in the field energy plus the mechanical work done. AU : Dec.-05, 12, May-10, Marks 8

Sol. :

3) To obtain the change in the field energy, it is necessary to calculate βs

 Ex 2.7.4 In the rectangular elecromagnetic relay, there are two coils. L₁₁=K1/x’, L₁₂=K₂/x and L₂₂ = K₃/x where x is the air gap at each end of the armature, which is free at both the ends. Find the expression for the force on the armature if, i₁ = I₁, sin ???t and i₂ = I₂, sin ????t. Write an expression for the average force. For what relationship between w₁, and w₂ is the average force i) Maximum and ii) Minimum?

 Sol. :

The stored energy is given by,

Ex. 2.7.5 For doubly excited magnetic field system, various inductances are, L₁₁ = (4 + cos2θ) x 10-³H, L₁₂ = 0.15 cos θ H, L₂₂ = (20 + 5 cos 2θ) H       

 Find the torque developed if i₁ = 1 A and i₂ = 0.02 A.

 Sol. : The co-energy for doubly excited magnetic field system is,

The first term is reluctance torque, which appears if self inductances are functions of θ. The second term is torque due to mutual component. The reluctance torque term is a double frequency of the θ as compared to second term. The negative sign indicates that torque is in the direction opposite to θ i.e. acts as restoring torque.

Ex. 2.7.6 Two windings, one mounted in stator and other at rotor have self and mutual inductance of L₁₁ = 4.5 and L₂₂ = 2.5, L₁₂ = 2.8 cosθ H, where θ is the angle between axes of winding. Winding 2 is short circuited and current in winding as a function of time is in = 10 sin ??t A.

i) Determine the expression for numerical value in Newton-meter for the instantaneous value of torque in terms of θ.r[8]

ii) Compute the time average torque in Newton-meter when  θ = 45°[4]

iii) If the rotor is allowed to move, will it continuously rotate or it will come to rest? If later at which value of θ₀.[4]

AU : Dec.-10.15, Marks 16

Sol. :

Ex. 2.7.7 A doubly-excited magnetic field system has coil self and mutual inductance of L₁₁ = L₂₂ = 2 H and L₁₂ = L₂₁ = cos θ. Where θ is the angle between the axes of the coils.

 i) The coils are connected in parallel to a voltage source v = V_m sin ???t. Derive an expression for the instantaneous torque as a function of the angular position θ. Find there from the time-average torque. Evaluate for θ = 30°, v = 100 sin 314 t.           ii) If coil 2 is shorted while coil 1 carries a current of i₁ = I_m sin ?? t, derive expression for the instantaneous and time-average torques. Compute the value of the time-average torque when θ = 45° and i₁ =  √2 sin 314 t.

AU : May-11, Marks 16

 Sol. :

Review Questions

1. Derive expression for co-energy in a multiply excited magnetic field system.   AU : May-03, Marks 8

 2. With an example explain the Multiple - excited magnetic field system.

AU : Dec.-03, 19, May-16, Marks 16

3. Derive the expression for the magnetic torque and force developed for doubly excited magnetic system. AU : Dec.-03, 04, 05, 12, 14, May-10, 13, Marks 10      

4. Show that the torque developed in doubly excited magnetic system is equal to the rate of increase of field energy with respect to displacement at constant current. AU : May-06,14, Dec.-15, Marks 8

5. Derive the field energy, co-energy and force for a doubly excited systems. AU : May-07, Marks 10, Dec.-18, Marks 13

6. Two coils have self and mutual inductances of L₁₁ = L₂₂ = 2/1+2x= 2L₁₂. Calculate the time average force and coil currents at x = 0.5 m if both the coils are connected in parallel across a voltage source of v = 100 cos 314 t.

[Ans. : -0.0338 N, 0.2123 sin 314t A]

 7. Two coupled coils have self and mutual  inductances of L₁₁ =2+1/2x, L₁₂ =L₂₁ =1/2x’  L₂₂ =1+ 1/2x’ over a certain range of linear displacement x. Find the expression for the time average force of field origin at x = 0.5 m if both the coils are connected in series and carry a current of 0.5 cos 314 t A.

[Ans. : - cos2314 N]

8. The self and mututal inductances of the two exciting coils of a multiply excited translatory system are :

L₁₁ = L₂₂ = 36/1+2x’ L₁₂ =L₂₁ = 18/1+2x

Calculate the time, average force and coil currents at x = 0.4 m when both the coils are connected in parallel across a voltage source of 100 cos 314 t.

 [Ans. : 0.1061 sin 314 t, - 0.03752 sin2314 t -0.01876 N]

9. A doubly excited magnetic field system has coil self and mutual inductances of L₁₁ = L₂₂ = 2 and L₂₁ = L₁₂ = cos 2θ where θ is the angle between the axes of the coils. If coil 2 is shorted while coil 1 carries a current of i = I_m sinot, derive the expressions for instantaneous and time average torques. Find the time average torque when θ = π/4 and i₁ = √2 sin 314 t. 

[Ans. : im sin 20 cos20, 0 Nm]

10. With neat sketch explain the multiple excited magnetic field system in eletromechanical energy conversion systems. Also obtain the expression for field energy in the system. AU : May-15, Marks 16