The rotor input P2, rotor copper loss Pc and gross mechanical power developed Pm are related through the slip s. Let us derive this relationship.
So torque developed by the rotor can be expressed interms of power
input and angular speed at which power is inputted i.e. ?s as,
The torque produced by rotor is gross mechanical torque and due to
mechanical losses entire torque can not be available to drive load. The
load torque is net output torque called shaft torque or useful torque
and is denoted as Tsh. It is related to Pout as,
We have seen earlier that
T = (k s E22 R2)/(R22 +(s X2)2)
and it mentioned that k = 3/(2? ns) . Let us see its proof.
The rotor copper losses can be expressed as,
Pc = 3 x I2r2 x R2
but I2r = (s E2)/?(R22 +(s X2)2), hence substituting above
Now as per P2 : Pc : Pm is 1 : s : 1-s ,
Pc/Pm = s/(1-s)
Now Pm = T x ?
= T x (2?N/60)
Now N = Ns (1-s) from definition of slip, substituting in above,
but Ns/60 = ns in r.p.m.
So substituting in the above equation,