\(N_{s}=N_{r}\Rightarrow\)relative speed = 0
\(\Rightarrow\) No rotor emf/current
\(\Rightarrow\) No Torque

At standstill/blocked rotor \(N_r=0
\Rightarrow\)\(s=1\)

At synchronous speed \(N_s = N_r
\Rightarrow\)\(s=0\)

FREQUENCY OF ROTOR CURRENT

When the rotor is stationary, the frequency of rotor current is
the same as the supply frequency

When rotor starts revolving, then the frequency depends upon the
relative speed or on slip speed

Let any slip-speed, the frequency of the rotor current be \(f_r\). Then \[\begin{aligned}
N_{s}-N & =\dfrac{120f_r}{P}~\mbox{and}~N_{s}=\dfrac{120f}{P}\\
\Longrightarrow\dfrac{f_r}{f} & =\dfrac{N_{s}-N}{N_{s}}=s\\
\Rightarrow f_r & =sf
\end{aligned}\]

Thus rotor currents have a frequency \(f_r=sf\) and when flowing through the
individual phases of rotor winding, give rise to rotor magnetic
fields

These individual rotor magnetic fields produce a combined
rotating magnetic field, whose speed relative to rotor is \[=\dfrac{120f_{r}}{P}=\dfrac{120sf}{P}=sN_{s}\]

However, the rotor itself is running at speed \(N\) w.r.t space

Hence, the speed of rotor field in space = speed of rotor
magnetic field relative to rotor + speed of rotor relative to space
\[=
sN_s+N=sN_s+N_s(1-s)=N_s\]

No matter what the vale of \(s\), rotor currents and stator current each
produce a sinusoidally distributed magnetic field of constant magnitude
and constant space speed \(N_s\)

In other words, both rotor and stator field rotate synchronously,
which means that they are stationary w.r.t each other

These two synchronously rotating magnetic fields, in fact,
superimpose on each other and give rise to the actually existing
rotating field

Relation between Torque and Rotor Power factor

We know, \[\begin{aligned}
T & \propto\phi I_{2}\cos\phi_{2}\\
\Rightarrow T & =K\phi I_{2}\cos\phi_{2}
\end{aligned}\] where, \[\begin{aligned}
I_{2} & =\mbox{rotor current at standstill}\\
\phi_{2} & =\mbox{angle between rotor e.m.f and rotor
current}\\
K & =\mbox{a constant}
\end{aligned}\]

Denoting rotor e.m.f at standstill by \(E_2\), we have \(E_2 \propto \phi\)

Therefore, \(T \propto
E_{2}I_{2}\cos\phi_{2} \quad
\Rightarrow T =K_{1}E_{2}I_{2}\cos\phi_{2}\)