Q. Derive the expression of torque developed in closely excited magnetic system. Clearly explain then assumption made.
Sol. Double - Excited System
A doubly - excited magnetic system has two independent sources of excitations. Examples of such systems are separately excited dc machines synchronous machine, loudspeakers, tachometers etc.Let us consider that both the stator and rotor have silency. Assumptions are as for a singly - excited system.
The flux linkage eq. for the two windings are
Ψ_{1} = L_{1}I_{1} + Mi_{2}
Ψ_{2} = L_{2}I_{2} + Mi_{1}
_{ }
_{ }The instantaneous voltage eq. for the two coils are
V_{1 = }R_{1}i_{1} + d Ψ_{1}/dt
V_{2 = }R_{2}i_{2} + d Ψ_{2}/dt
Substituting the values Ψ_{1} and Ψ_{2}
_{ }
_{ } V_{1 = }R_{1}i_{1 }+ d/dt + (L_{1}i_{1}) + d/dt (Mi_{2})
V_{2 = }R_{2}i_{2 }+ d/dt + (L_{2}i_{2}) + d/dt (Mi_{1})
Now the inductances are independent of currents and depend on the position of the root angle θ_{m} which is a function of time. Similarly, current are time dependent and are not function of inductances. Therfore,
V_{1} = R_{1}i_{1} + L_{1}di_{1}/dt + i_{1}dL_{1}/dt + Mdi_{2}/dt + i_{2}dM/dt
V_{2} = R_{2}i_{2} + L_{2}di_{12}/dt + i_{2}dL_{2}/dt + Mdi_{1}/dt + i_{1}dM/dt
By multiplying we get,
V_{1}i_{1} = R_{1}i_{1}+ L_{1}i_{1}di_{1}/dt + i_{1}^{2}dL/dt + i_{1}Mdi_{2}/dt + i_{1}i_{2}dM/dt
V_{2}i_{2} = R_{2}i_{2}+ L_{2}i_{2}di_{2}/dt + i_{2}^{2}dL/dt + i_{2}Mdi_{1}/dt + i_{1}i_{2}dM/dt
Now we get,
(( v_{1}i_{1 + }v_{2}i_{2 }) dt =(( R_{1}i_{1}^{2} + R_{2}i_{2}^{2} ) dt + (( L_{1}i_{1}di_{1} + L_{2}i_{2}di_{2} + i_{1}Mdi_{2} + 2i_{1}i_{2}dM + i_{1}^{2}dL_{1} + i_{2}^{2}dL_{2} + i_{2}Mdi_{1})
Also, [Useful electrical energy input] = (( v_{1}i_{1 + }v_{2}i_{2 }) dt - (( R_{1}i_{1}^{2} + R_{2}i_{2}^{2} ) dt
[Energy to field storage in the electrical systems] + [Electrical to mechanical energy] = (( L_{1}i_{1}di_{1} + L_{2}i_{2}di_{2} + i_{1}Mdi_{2} +2i_{1}i_{2}dM + i_{1}^{2}dL_{1} + i_{2}^{2}dL_{2} + i_{2}Mdi_{1})
Stored energy in the Magnetic field
The instantaneous value of energy stored in the magnetic field depends on the inductance and current values at the instant considered. This energy may be found by considering the transductor to be stationary and the coils to be energized from zero current to the required instantaneous values of current. There is no mechanical output and W_{em} is zero. The inductance values are constant. Therefore terms dL_{1}, dL_{2} and dM become zero
(dW_{fe} = _{o}^{i}_{1}(L_{1}i_{1}di_{1} + _{o}^{i}_{2}(L_{2}i_{2}di_{2} + _{o}^{i}_{2}^{,i}_{2} ( (i_{2}Mdi_{1} + i_{1} Mdi_{2} )
[Total W_{fe}] = 1/2L_{1}i_{1}^{2} + 1/2L_{2}i_{2}^{2} + Mi_{1}i_{2}
Electromagnetic Torque
If the transductor rotates, the rate of change of field energy with respect to time is given by differentiating.
dW_{fe}/dt = 1/2L_{1} d/dt i_{1}^{2} + 1/2i_{1}^{2} dL_{1}/dt + 1/2L_{2} di^{2}/dt^{2} + 1/2i_{2}^{2} dL_{2}/dt + i_{2}i_{2} dM/dt + i_{1}M di_{2}/dt + i_{2}M di/dt
dW_{fe}/dt = L_{1}i_{1} di_{1}/dt + 1/2i_{1}^{2} dL_{1}/dt + L_{2}i_{2} di^{2}/dt + 1/2i_{2}^{2} dL_{2}/dt + i_{2}i_{2} dM/dt + i_{1}M di_{2}/dt + i_{2}M di/dt
Integrated with respect to time
(dW_{fe} = W_{fe} = ((L_{1}i_{1}di_{1} + 1/2i_{1}^{2}dL_{1} + L_{2}i_{2}di_{2} + 1/2i_{2}^{2}dL_{2}) + i_{1}i_{2}dM + i_{1}Mdi_{1}
This is general eq. for a moving transducer in which L_{1}, L_{2} and M, i_{1} and i_{2} are all varying with position and time. On comparing we get,
W_{em} = [Electrical to mechanical energy] = ((1/2 i_{1}^{2}dL_{1 }+ 1/2i_{2}^{2}dL_{2} + i_{2}i_{2}dM)
Differentiating with respect to θ_{m}
dW_{em}/d θ_{m} = ½ i_{1}^{2} dL_{1}/d θ_{m} = ½ i_{2}^{2} dM/d θ_{m}
as only L_{1}, L_{2} and M are dependent on θ_{m}
_{ }
It includes the case of singly - excited system when one of the two current is equal to zero so that the expression for the torque becomes
Τ_{e = }i^{2}/2 dl/d θ_{m}
_{ }
_{ }The first two terms of the torque are reluctance torques or saliency torques. The last term i_{1}i_{2} dM/dθ is called the co - alignment torque, that is two superimposed fields, that try to align.
For machines having uniform air gaps reluctance torque is not produced.