Question 3.AE.7: Terminal voltages of a three-phase induction motor contain t......
Terminal voltages of a three-phase induction motor contain the forward-rotating subharmonic of 6 Hz at an amplitude of V_{0.1}=5%. The fundamental voltage is V_1=240 V and the parameters of a three-phase induction machine are as follows:
240=416V, Δ=Yconnected, f = 60Hz,
n_{rated} = 1738 rpm, p = 4 poles, (E3.7-1)
\begin{aligned} & R_s=7.0 \Omega, \quad \omega_1 L_{s l}=8 \Omega, \quad \omega_1 L_m=110 \Omega, \\ & R_r^{\prime}=5 \Omega, \quad \omega_1 L_{r \ell}^{\prime}=7 \Omega . \end{aligned}Compute and plot the sum of the fundamental and subharmonic torques (T_{e1}+T_{e0.1}) as a function of the fundamental slip s_1.
Parameters of Eqs. 3-47 to 3-54 are listed as a function of the fundamental (h=1) and subharmonic h=0.1 in Table E3.7.1. Fundamental torque values T_{e1} are listed in Table E3.7.2 as a function of the fundamental slip s_1.
The plot of the positively rotating fundamental torque characteristic T_{e1}=f(s_1) is shown in Fig. E3.7.1 and the positively rotating torque Te_{0.1}=f(s_{0.1}) is depicted in Fig. E3.7.2, where the 6 Hz voltage component has an amplitude of 5%, that is, V_{6Hz}=0.05.V_{60Hz}=0.05(240 V)=12 V.
Subharmonic torque values T_{e0.1} are listed in Table E3.7.3 as a function of the subharmonic slip s_{0.1}.
Using Eqs. 3-4d and 3-25 one can compute the fundamental torque T_{e1} and the subharmonic torque T_{e0.1} as a function of the fundamental slip s1 (Table E3.7.4). For example a subharmonic (h=0.1) slip of s_{0.1}=1 corresponds to a fundamental slip of (see Eqs. 3-31b, 3-32, and Table E3.7.3)
s_1=(0.1)\left(s_{0.1}\right)+0.9=1.0 . (E3.7-2)
The plot of fundamental and subharmonic torque values (T_{e1}+T_{e0.1}) as a function of fundamental slip (s_1) is shown in Fig. E3.7.3.
Table E3.7.1 Maximum Torques for Fundamental (h¼1) and Subharmonic (h¼0.1) Voltages, where the Fundamental Voltage is 100% and the Subharmonic Voltage is 5% of the Fundamental Voltage
\begin{array}{l|c|l|l|l|l} \boldsymbol{h} & \boldsymbol{V}_{\text {shTH }}(\mathbf{V}) & \pm \boldsymbol{S}_{\text {hmax }}(-) & \begin{array}{l} \boldsymbol{\omega}_{\text {msh }} \\ (\mathrm{rad} / \mathrm{s}) \end{array} & \begin{array}{l} T_{\text {ehmmax }}^{\text {mot }} \\ (\mathrm{Nm}) \end{array} & \begin{array}{l} T_{\text {Thmmax }}^{\text {gen }} \\ (\mathbf{N m}) \end{array} \\ \hline 1 & 223.34 & 0.312 & 188.49 & 17.98 & -39.89 \\ 0.1 & 9.62 & 0.820 & 18.849 & 0.696 & -4.61 \\ \hline \end{array}
Table E3.7.2 Fundamental Torque T_{e1} as a Function of Fundamental Slip s_1.
\begin{array}{l|c|c|c|c|c} \hline S_1(-) & 2.0 & 1.0 & 0.9 & 0.8 & 0.7 \\ T_{e 1}(\mathrm{Nm}) & 6.78 & 11.61 & 12.44 & 13.37 & 14.40 \\ S_1(-) & 0.6 & 0.5 & 0.4 & 0.3 & 0.2 \\ T_{e 1}(\mathrm{Nm}) & 15.5 & 16.63 & 17.59 & 17.97 & 16.76 \\ S_1(-) & 0.1 & 0.0344 & 0 & -0.1 & -0.2 \\ T_{e 1}(\mathrm{Nm}) & 11.8 & 4.99 & 0 & -18.46 & -34.32 \\ \hline \end{array}
Table E3.7.3 Subharmonic Torque T_{e0.1} as a Function of Subharmonic Slip s_{0.1}.
\begin{array}{l|l|l|l|l|l|c} \hline S_{0.1}(-) & 1.0 & 0.9 & 0.82 & 0.8 & 0.7 & 0.6 \\ S_1(-) & 1.0 & 0.99 & 0.982 & 0.98 & 0.97 & 0.96 \\ T_{e 0.1}(\mathrm{Nm}) & 0.687 & 0.693 & 0.696 & 0.695 & 0.69 & 0.676 \\ S_{0.1}(-) & 0.5 & 0.4 & 0.3 & 0.2 & 0.1 & 0 \\ S_1(-) & 0.95 & 0.94 & 0.93 & 0.92 & 0.91 & 0.9 \\ T_{e 0.1}(\mathrm{Nm}) & 0.648 & 0.602 & 0.528 & 0.415 & 0.247 & 0 \\ S_{0.1}(-) & -0.1 & -0.2 & -0.3 & -0.4 & -0.5 & -0.6 \\ S_1(-) & 0.89 & 0.88 & 0.87 & 0.86 & 0.85 & 0.84 \\ T_{e 0.1}(\mathrm{Nm}) & -0.353 & -0.842 & -1.488 & -2.28 & -3.12 & -3.88 \\ S_{0.1}(-) & -0.7 & -0.8 & -0.82 & -0.9 & -1 & -1.1 \\ S_1(-) & 0.83 & 0.82 & 0.818 & 0.81 & 0.8 & 0.79 \\ T_{e 0.1}(\mathrm{Nm}) & -4.40 & -4.60 & -4.61 & -4.53 & -4.29 & -3.95 \\ S_{0.1}(-) & -1.2 & -1.3 & -1.4 & -1.5 & -1.6 & -1.7 \\ S_1(-) & 0.78 & 0.77 & 0.76 & 0.75 & 0.74 & 0.73 \\ T_{e 0.1}(\mathrm{Nm}) & -3.60 & -3.26 & -2.96 & -2.68 & -2.45 & -2.24 \\ S_{0.1}(-) & -2.0 & -3.0 & -4.0 & -5.0 & -6.0 & -7.0 \\ S_1(-) & 0.70 & 0.60 & 0.50 & 0.40 & 0.30 & 0.20 \\ T_{e 0.1}(\mathrm{Nm}) & -1.76 & -0.98 & -0.67 & -0.51 & -0.40 & -0.34 \\ S_{0.1}(-) & -8.0 & -9.0 & -10.0 & -11.0 & -12.0 & -13.0 \\ S_1(-) & 0.1 & 0 & -0.1 & -0.20 & -0.30 & -0.40 \\ T_{e 0.1}(\mathrm{Nm}) & -0.29 & -0.25 & -0.22 & -0.20 & -0.18 & -0.17 \\ \hline \end{array}
Table E3.7.4 Fundamental and Subharmonic Torque (T_{e1}+T_{e0.1}) as a Function of Fundamental Slip s_1.
\begin{array}{l|c|c|r|r|r|r} \hline S_1(-) & 1.0 & 0.99 & 0.982 & 0.98 & 0.97 & 0.96 \\ T_{e 1}(\mathrm{Nm}) & 11.61 & 11.69 & 11.75 & 11.77 & 11.85 & 11.93 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 12.30 & 12.38 & 12.45 & 12.46 & 12.54 & 12.60 \\ S_1(-) & 0.95 & 0.94 & 0.93 & 0.92 & 0.91 & 0.9 \\ T_{e 1}(\mathrm{Nm}) & 12.01 & 12.10 & 12.18 & 12.27 & 12.35 & 12.44 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 12.66 & 12.70 & 12.71 & 12.68 & 12.60 & 12.44 \\ S_1(-) & 0.89 & 0.88 & 0.87 & 0.86 & 0.85 & 0.83 \\ T_{e 1}(\mathrm{Nm}) & 12.53 & 12.62 & 12.71 & 12.80 & 12.89 & 13.08 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 12.18 & 11.78 & 11.22 & 10.52 & 9.77 & 8.68 \\ S_1(-) & 0.82 & 0.818 & 0.81 & 0.8 & 0.79 & 0.78 \\ T_{e 1}(\mathrm{Nm}) & 13.18 & 13.19 & 13.27 & 13.37 & 13.47 & 13.57 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 8.58 & 8.58 & 8.74 & 9.08 & 9.52 & 9.97 \\ S_1(-) & 0.77 & 0.76 & 0.75 & 0.74 & 0.73 & 0.70 \\ T_{e 1}(\mathrm{Nm}) & 13.67 & 13.77 & 13.87 & 13.97 & 14.08 & 14.40 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 10.41 & 10.81 & 11.19 & 11.52 & 11.84 & 12.64 \\ S_1(-) & 0.60 & 0.50 & 0.40 & 0.30 & 0.20 & 0.10 \\ T_{e 1}(\mathrm{Nm}) & 15.50 & 16.63 & 17.59 & 17.97 & 16.76 & 11.80 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & 14.52 & 15.96 & 17.08 & 17.57 & 16.42 & 11.51 \\ S_1(-) & 0 & -0.10 & -0.20 & -0.30 & -0.40 & 0 \\ T_{e 1}(\mathrm{Nm}) & 0 & -18.46 & -34.32 & -39.84 & -38.01 & 0 \\ T_{e 1}+T_{e 0.1}(\mathrm{Nm}) & -0.25 & -18.68 & -34.52 & -40.02 & -38.19 & 0 \\ \hline \end{array}
