A460-V. 30-hp. 60-Hz. four-pole. Star-connected induction motor has two possible rotor designs. a single-cage rotor and a double-cage rotor. (The stator is identical for either rotor design.) The motor with the single-cage rotor may be modeled by the following impedances in ohms per phase referred to the stator circuit:
R1= 0.641 Ω X1 = 0.750 Ω R2 = 0.3 Ω X2 = 0.5 Ω Xm= 26.3 Ω
The motor with the double-cage rotor may be modeled as a tightly coupled. High resistance outer cage in parallel with a loosely coupled. Low resistance inner cage The stator and magnetization resistance and reactances will be identical with those in the single-cage design. The resistance and reactance of the rotor outer cage are:
R2o, = 3.2 Ω X2o, = 0.5 Ω The resistance and reactance of the inner cage are R2i = 0.4 Ω X2i = 3.3 Ω
Write a MATLAB script to calculate and plot the two torque-speed characteristics with the two rotor designs. How do they compare?
r1 = 0.641; % Stator
resistance
x1 = 0.750; % Stator
reactance
r2 = 0.300; % Rotor
resistance for single-
% cage motor
r2i = 0.400; % Rotor
resistance for inner
% cage of double-cage motor
r2o = 3.200; % Rotor
resistance for outer
% cage of double-cage motor
x2 = 0.500; % Rotor
reactance for single-
% cage motor
x2i = 3.300; % Rotor
reactance for inner
% cage of double-cage motor
x2o = 0.500; % Rotor
reactance for outer
% cage of double-cage motor
xm = 26.3; %
Magnetization branch reactance
v_phase = 460 / sqrt(3); % Phase
voltage
n_sync = 1800; %
Synchronous speed (r/min)
w_sync = 188.5; %
Synchronous speed (rad/s)
% Calculate the Thevenin
voltage and impedance from Equations
% 7-41a and 7-43.
v_th = v_phase * ( xm / sqrt(r1^2
+ (x1 + xm)^2) );
z_th = ((j*xm) * (r1 + j*x1)) /
(r1 + j*(x1 + xm));
r_th = real(z_th);
x_th = imag(z_th);
% Now calculate the motor
speed for many slips between
% 0 and 1. Note that the
first slip value is set to
% 0.001 instead of exactly
0 to avoid divide-by-zero
% problems.
s = (0:1:50) / 50; % Slip
s(1) = 0.001; % Avoid
division-by-zero
nm = (1 - s) * n_sync; %
Mechanical speed
% Calculate torque for the
single-cage rotor.
for ii = 1:51
t_ind1(ii) = (3 * v_th^2 * r2 /
s(ii)) / ...
(w_sync * ((r_th + r2/s(ii))^2 +
(x_th + x2)^2) );
end
% Calculate resistance and
reactance of the double-cage
% rotor at this slip, and
then use those values to
% calculate the induced
torque.
for ii = 1:51
y_r = 1/(r2i + j*s(ii)*x2i) +
1/(r2o + j*s(ii)*x2o);
z_r = 1/y_r; %
Effective rotor impedance
r2eff = real(z_r); %
Effective rotor resistance
x2eff = imag(z_r); %
Effective rotor reactance
% Calculate induced torque
for double-cage rotor.
t_ind2(ii) = (3 * v_th^2 * r2eff
/ s(ii)) / ...
(w_sync * ((r_th + r2eff/s(ii))^2
+ (x_th + x2eff)^2) );
end
% Plot the torque–speed
curves
% Plot the torque–speed
curves
plot(nm,t_ind1,'Color','k','LineWidth',2.0);
hold on;
plot(nm,t_ind2,'Color','k','LineWidth',2.0,'LineStyle','-.');
xlabel('\itn_{m}','Fontweight','Bold');
ylabel('\tau_{ind}','Fontweight','Bold');
title ('Induction
motor torque–speed characteristics', ...
'Fontweight','Bold');
legend ('Single-Cage
Design','Double-Cage Design');
grid on;
hold off;
