Note: You should first click on "Cell → Run All" in order that the plots get generated.
Import the PyLab namespace (provides set of useful commands and constants like Pi)
In [1]:
%pylab notebook
First, initialize the values needed in this program.
In [2]:
r1 = 0.641 # Stator resistance
x1 = 1.106 # Stator reactance
r2 = 0.332 # Rotor resistance
x2 = 0.464 # Rotor reactance
xm = 26.3 # Magnetization branch reactance
v_phase = 460 / sqrt(3) # Phase voltage
n_sync = 1800 # Synchronous speed (r/min)
w_sync = n_sync * 2*pi/60 # Synchronous speed (rad/s)
Calculate the Thevenin voltage and impedance from Equations 7-41a:
$$ V_{TH} = V_\phi \frac{X_M}{\sqrt{R_1^2 + (X_1 + X_M)^2}} $$and 7-43:
$$ Z_{TH} = \frac{jX_m (R_1 + jX_1)}{R_1 + j(X_1 + X_M)} $$
In [3]:
v_th = v_phase * ( xm / sqrt(r1**2 + (x1 + xm)**2) )
z_th = ((1j*xm) * (r1 + 1j*x1)) / (r1 + 1j*(x1 + xm))
r_th = real(z_th)
x_th = imag(z_th)
Now calculate the torque-speed characteristic for many slips between 0 and 1.
In [4]:
s = linspace(0, 1, 50) # Slip
s[0] = 0.001 # avoid divide-by-zero problems
nm = (1 - s) * n_sync # mechanical speed
Calculate torque for original rotor resistance using:
$$ \tau_\text{ind} = \frac{3 V_{TH}^2 R_2 / s}{\omega_\text{sync}[(R_{TH} + R_2/s)^2 + (X_{TH} + X_2)^2]} $$
In [5]:
t_ind1 = ((3 * v_th**2 * r2/s) /
(w_sync * ((r_th + r2/s)**2 + (x_th + x2)**2)))
Calculate torque for doubled rotor resistance:
In [6]:
t_ind2 = ((3 * v_th**2 * 2*r2/s) /
(w_sync * ((r_th + 2*r2/s)**2 + (x_th + x2)**2)))
Plot the torque-speed curve:
In [7]:
rc('text', usetex=True) # enable LaTeX commands for plot
plot(nm, t_ind2,'k--',
nm, t_ind1,'b',
lw=2)
xlabel(r'$\mathbf{n_{m}}\ [rpm]$')
ylabel(r'$\mathbf{\tau_{ind}}\ [Nm]$')
title ('Induction motor torque-speed characteristic')
legend ((r'Doubled $R_{2}$','Original $R_{2}$'), loc = 3);
grid()