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%pylab notebook
%precision 3
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A 208-V six-pole Y-connected 25-hp design class B induction motor is tested in the laboratory, with the following results:
Based on those measurements:
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Vnl = 208.0 # [V]
Inl = 24.0 # [A]
Pnl = 1400.0 # [W]
fnl = 60.0 # [Hz]
Vlr = 24.6 # [V]
Ilr = 64.5 # [A]
Plr = 2200.0 # [W]
flr = 15.0 # [Hz]
Vdc = 13.5 # [V]
Idc = 64.0 # [A]
From the DC test,
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R1 = Vdc/Idc / 2
print('''
R1 = {:.3f} Ω
============'''.format(R1))
In the no-load test, the line voltage is 208 V, so the phase voltage is:
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Vnl_phase = Vnl / sqrt(3)
print('Vnl_phase = {:.0f} V'.format(Vnl_phase))
Therefore,
$$X_1 + X_M = \frac{V_\phi}{I_\text{A,nl}}$$
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X1_Xm = Vnl_phase / Inl
print('X1 + Xm = {:.2f} Ω @ {:.0f} Hz'.format(X1_Xm, fnl))
In the locked-rotor test, the line voltage is 24.6 V, so the phase voltage is:
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Vlr_phase = Vlr / sqrt(3)
print('Vlr_phase = {:.1f} V'.format(Vlr_phase))
From the locked-rotor test at 15 Hz , $$|Z'_{LR}| = |R_{LR} + jX'_{LR}| = \frac{V_\phi}{I_{A,LR}}$$
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z_lr = Vlr_phase / Ilr
Slr = sqrt(3) * Vlr * Ilr
theta_lr = arccos(Plr / Slr)
print('''
z_lr = {:.3f} Ω
theta_lr = {:.2f}°'''.format(z_lr, theta_lr))
Therefore, $$R_{LR} = |Z'_{LR}| \cos{\theta_{LR}}$$
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Rlr = z_lr * cos(theta_lr)
print('Rlr = {:.3f} Ω'.format(Rlr))
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R2 = Rlr - R1
print('''
R2 = {:.3f} Ω
============'''.format(R2))
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X_lr = z_lr * sin(theta_lr)
print('X_lr = {:.3f} Ω @ {:.0f} Hz'.format(X_lr, flr))
At a frequency of 60 Hz,
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Xlr = (60 / flr) *X_lr
print('Xlr = {:.3f} Ω @ 60 Hz'.format(Xlr))
For a Design Class B motor, the split is $X_1 = 0.4 \cdot X_{LR}$ and $X_2 = 0.6 \cdot X_{LR}$ .
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X1 = 0.4 * Xlr; X1
Out[12]:
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X2 = 0.6 * Xlr; X2
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Therefore,
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Xm = X1_Xm - X1
print('''
Xm = {:.3f} Ω
============'''.format(Xm))
The resulting equivalent circuit is shown below:
Plotting the torque-speed characteristic:
Calculate the Thevenin voltage and impedance:
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V_th = Vnl_phase * ( Xm / sqrt(R1**2 + (X1 + Xm)**2) )
Z_th = ((Xm*1j) * (R1 + X1*1j)) / (R1 + (X1 + Xm)*1j)
R_th = real(Z_th)
X_th = imag(Z_th)
Now calculate the torque-speed characteristic 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.
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s = linspace(0,50,51) / 50 # generate an array with 51 values between 0 and 50
s[0] = 0.001 # avoid division by zero
p = 6 # number of poles
n_sync = 60/(p/2) * fnl # [r/min]
w_sync = 2*pi/60 *n_sync # [rad/s]
nm = (1-s) * n_sync
Calculate torque versus speed
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tau_ind = (3 * V_th**2 * R2 / s) / (w_sync * ((R_th + R2/s)**2 + (X_th + X2)**2) )
Plot the torque-speed curve:
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rc('text', usetex=True) # enable LaTeX commands for plot
title(r'\bf Induction Motor Torque-Speed Characteristic')
xlabel(r'$n_m$ [r/min]')
ylabel(r'$\tau_{ind}$ [Nm]')
plot(nm, tau_ind, linewidth = 2)
grid()