Page 205 - Solutions Manual to accompany Electric Machinery Fundamentals

The resistance is low due to the high cross sectional area, but the reactance is relatively high due to the
quite loose coupling between the rotor and stator.
Calculate the torque-speed characteristic for this induction motor, and compare it to the torque-
speed characteristic for the single-cage design in Problem 6-21. How do the curves differ? Explain the
differences.
SOLUTION The dual-cage rotor has two current paths in parallel, the inner cage and the outer cage. As a
result, the impedance of the rotor is calculated as the parallel combination of these two current paths. For
any given slip, the impedance of the rotor can be calculated as
1
Z  1 1
R
R  i jX i  R  o jX o

where R is the resistance of the inner rotor cage, X is the reactance of the inner rotor cage, and so forth.
i
i
Also, recall that rotor reactance varies with rotor frequency. The rotor reactance is given by the equation
X  sX
o
where s is the slip and X is the rotor reactance at locked-rotor conditions. The rotor impedance and any
o
slip can thus be calculated as
1
Z  1 1
R
R  i jsX oi  R  o jsX oo

where X is the reactance of the inner rotor cage at locked-rotor conditions, and X is the reactance of
oo
oi
the outer rotor cage at locked-rotor conditions. We must apply this equation to calculate the rotor
impedance at any slip, and then divide the resulting reactance by the slip to get to the equivalent
impedance at locked-rotor conditions (the reactance at locked-rotor conditions is the term that goes into
the torque equation).
A MATLAB program to calculate the torque-speed characteristic of this motor is shown below:

% M-file: prob6_32.m
% M-file create a plot of the torque-speed curve of the
% induction motor of Problem 6-32.

% First, initialize the values needed in this program.
r1 = 0.54; % Stator resistance
x1 = 2.093; % Stator reactance

% Resistances and reactances of the dual-cage rotor
r2a = 4.8; % Outer bar rotor resistance
x2a = 3.75; % Outer bar rotor reactance
r2b = 0.573; % Inner bar rotor resistance
x2b = 4.65; % Inner bar rotor reactance

% Resistance and reactance of the single-cage rotor (6-21)
r2 = 0.488; %
x2 = 3.209; %
xm = 51.12; % 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
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