Speed torque characteristics for Induction motor

3 phase, 50 Hz induction motor, represented by equivalent circuit constants X1 = X2 = 0.1 ohm and R1 = R2 = 0.2 ohm is operated at half of rated voltage and frequency. Calculate the ratio of starting torque at half voltage & frequency to rated values.

Answer – Given x1 = x2 =0.1

R1 = R2 = 0.2

Frequency = f= 50 hz

From the above equivalent circuit of Induction motor we see that the Equation of torque is

Now slip can be calculated by

Slip (s) = (ωms- ωm)/ ωms.

Where ωm – Actual speed of motor

ωms – Synchronous Speed

ωms = 120f/ P

f = frequency of supply

P = number of Poles

We want to calculate the starting torque so the ωm would be zero so slip is 1 (maximum). Now the equation becomes

Assumed values

P = number of Poles= 4

Rated voltage = 120 v

Now we substitute the values rated voltage and frequency we get

As ωms = 120*f/ P

ωms ∝ f">ωms ∝ fωms ∝ f

-----------(1)

Similarly we can write the equation for half voltage and frequency

--------(2)

After solving eq 1 and 2 we get 3 and 4 respectively

-------- (3)

------- (4)

Ratio of Starting torque at half voltage and frequencya and rated frequency can be fould by dividing eq 4 by eq 3 we get

T_h/T = 0.5

Case 2 - Speed torque characteristics for Induction motor

Let us consider a 3 phase 4 pole, 50 hz, 230 volts induction motor

also let x1 =0.68

x2=0.672

R1 =0.68

R2 = 0.23

by substuting the values and varying the frequency we can get the speed torque characteristics

For frequency control we change ωs = 120*f/ P

% calculating per phase voltage
V_pp = 230/sqrt(3);
%number of Phases
nph =3;
% number of poles
poles = 4;
% stator resistance & rotor resistance
R1 = 0.680;
R2 = 0.23;
% Stator Inductance 
x1 = 0.680;
x2 = 0.672;
% magnetizing Inductance 
xm = 18.7;
%variable Frequency 
 f = [10,20,30,40,50];
for i = f
    %calculation omega_s wrt f
    omega_s(i) = (4*pi*i)/poles;
    ns(i) = (120*i)/poles;
    for n =1:200  
    S(n) = n/200; %slip
    rpm(n) = ns(i)*(1-S(n)); 
    num(n) = 3*V_pp^2*(R2/S(n));
    den(n) = omega_s(i)*((R1+(R2/S(n)))^2+(x1+x2)^2) ;

    T(n) = num(n) / den(n) ; 
    end
    hold on
    plot (rpm,T)

xlabel('speed rad/s')
ylabel('Torque NM')
legend('10Hz','20Hz','30Hz','40Hz','50Hz');
title('speed torque characteristics for Induction motor at variable frequency')
end

Plotting the torque and speed we get