# Online Efficiency Optimization and Speed Sensorless Control of Single-Phase Induction Motors
## 6 Simulation Results  
The presented control scheme is simulated in MATLAB software to verify its performance. A four-pole 0.5 horsepower capacitor-run SPIM is employed, the characteristics of which are listed in Table 1. Other simulation parameters such as PI controller coefficients and motor reference speed are listed in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**Value**} & \multicolumn{1}{c}{**Unit**} \\ \hline \multirow{4}{*}{Nominal Values} & Voltage & 220 & V \\ \cline{2-4}  & Frequency & 50 & Hz \\ \cline{2-4}  & Power & \(0.5\times 746\) & W \\ \cline{2-4}  & Speed & 1440 & rpm \\ \cline{2-4}  & Load Torque & 2.4 & N-m \\ \hline \multirow{4}{*}{Main Winding Stator} & Resistance & 15 & \(\Omega\) \\ \cline{2-4}  & Leakage Inductance & 40 & mH \\ \cline{2-4}  & Resistance & 12.1 & \(\Omega\) \\ \cline{2-4}  & Leakage Inductance & 48.4 & mH \\ \hline \multirow{4}{*}{Auxiliary Winding Stator} & Resistance & 16.5 & \(\Omega\) \\ \cline{2-4}  & Leakage Inductance & 48.4 & mH \\ \cline{2-4}  & Series capacitance & 18 & \(\upmu\)F \\ \hline \multicolumn{4}{c}{Main Mutual Inductance} & 350 & mH \\ \hline \multicolumn{4}{c}{Turns ratio of auxiliary to main winding} & 1.1 & - \\ \hline \multicolumn{4}{c}{Inertia} & 0.01 & Kg-m\({}^{2}\) \\ \hline \multicolumn{4}{c}{Iron Loss Equivalent Resistance} & 1000 & \(\Omega\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Motor parameters. Using a variable-amplitude voltage source, \(K_{\text{s}}\) is calculated for different frequencies varying between 20 to 50 Hz, as described in chapter IV. Figure 5 depicts the diagram of \(K_{\text{s}}\) versus frequency. This diagram is saved as a lookup table and is used as a reference for the controller. The performance of the proposed drive is verified through three simulation scenarios. In the first scenario, the motor is initially at standstill. The reference speed is set to 1440 rpm and the load, which is of fan type, is set to 50% nominal. The motor starts at \(t=0\). Then, at \(t=15\) s, the reference speed is changed from nominal to 1000 rpm. As shown in Figure 5(a)-c, the controller's output voltage and frequency increase with a limited rate to realize a soft start. After the starting period, the speed control loop adjusts the frequency at 51.5 Hz to regulate the motor speed at its set point (1440 rpm) and controls the voltage magnitude at 0.65 pu to ensure the stator current ratio settles at the optimal value of 0.96 (see Figure 5(d).i). The waveforms of the instantaneous and rms stator currents are shown in Figure 5(e)-g. It is seen that during the starting phase, the main winding current is higher than the auxiliary winding current. This behavior is mainly caused by the fact that the voltage to frequency ratio at the starting phase is set to unity. Afterwards, the main winding current drops to below the auxiliary winding current due to the reduction in v/f ratio. It is noteworthy to highlight that the reduction in v/f ratio and hence the flux is the key to enhancing the efficiency at light loading conditions. As shown in Figure 5(h), the electromagnetic torque is higher than the load torque during the starting phase, when the motor accelerates. Since the mechanical and windage losses are omitted, both motor and load torque settle at 1.2 Nm (50% of nominal) at steady-state. The speed estimation error is initially large but reduces to as low as 0.5% in the steady state, providing decent speed regulation.