# A Compact Rectifier Design Method Utilizing Harmonics
## 2 Design and Simulation
### Simulation  
A rectifier is designed to verify the proposed design scenario, which utilizes both the 2 nd and the 3 rd harmonics simultaneously. The operation fundamental frequency is 5.8 GHz. A Schottky diode MA4E1317 is chosen as the rectifying diode, which has a series resistance R\({}_{\text{s}}\) of 4 \(\Omega\), a zero bias junction capacitance C\({}_{\text{j0}}\) of 0.02 pF, and a reverse breakdown voltage V\({}_{\text{b}}\) of 7 V. The material F4B-M with a thickness of 0.8 mm is employed as the printed circuit board (PCB) substrate. The substrate has a relative dielectric constant of 2.65 and a loss tangent of 0.001. The structure of the proposed rectifier is shown in Figure 2 and is simulated under Advanced Designed System (ADS) software. The \(\pi\)-type stub acts as the input impedance matching network. The \(\pi\)-type stub is used because it has more adjustable variables. The DC-pass filter consists of five open stubs. Stubs \(l_{6}\)-\(l_{9}\) correspond to the quarter guided wavelength at fundamental, 2 nd, 4 th, and 6 th harmonic. The length of stub \(l_{6}\) is quarter guided wavelength at fundamental, so the input impedance of stubs \(l_{6}\)-\(l_{11}\) is short circuit at fundamental and the 3 rd harmonic. Since the length of stub \(l_{5}\) is also quarter guided wavelength at fundamental, the input impedance of the DC-pass filter is open circuit at fundamental and the 3 rd harmonic. Stub \(l_{10}\) functions to adjust the input impedance of the DC-pass filter at the 2 nd harmonic. Thus, the input impedance of the DC-pass filter is open circuit at the fundamental and the 3 rd harmonic, and \(-\text{jZ}_{\text{c}}/\tan(4\pi\times l_{10}/\lambda_{\text{g}})\) at the 2 nd harmonic, respectively. \(Z_{\text{c}}\) is the characteristic impedance of the microstrip line, and \(\lambda_{\text{g}}\) is the guided wavelength of the microstrip line at 5.8 GHz. Figure 3 shows the position of the input impedance of the DC-pass filter on the Smith Chart when the length of stub \(l_{10}\) varies from 0 to \(\lambda_{\text{g}}/4\). By jointly optimizing the \(\pi\)-type stub and the DC-pass filter, the design goal can be met. \begin{table}
\begin{tabular}{c c c} \hline \hline
**The Type of Harmonic Utilization** & \(Z_{F,\,2f_{0}}\) **(\(\Omega\))** & \(Z_{F,\,3f_{0}}\) **(\(\Omega\))** \\ \hline
**2 nd and 3 rd** & \(\text{jX}\) or \(+\infty\) & \(\text{jX}\) or \(+\infty\) \\
**2 nd** & \(\text{jX}\) or \(+\infty\) & 0 \\
**3 rd** & 0 & \(\text{jX}\) or \(+\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The input impedances of the DC-pass filter in different harmonic utilization states. Figure 3: For different length of the stub \(l_{10}\): the position of the input impedance of the DC-pass filter in the Smith Chart. Adjusting the length of stub \(l_{10}\) can affect the amplitudes of the 2 nd and 3 rd harmonic simultaneously. Since the harmonics (\(2f_{0}+f_{0}=3f_{0}\), \(3f_{0}-f_{0}=2f_{0}\)) are mixed with the fundamental in the diode, the amplitude of the 2 nd harmonic has a tight relationship with that of the 3 rd harmonic. By adjusting the input impedance of the DC-pass filter (jX) (the length of stub \(l_{10}\)) at the 2 nd harmonic, the amplitude of the 2 nd harmonic will change. Meanwhile, the 3 rd harmonic is also affected by the amplitude of the 2 nd harmonic even if the input impedance of the DC-pass filter maintains open circuit at the 3 rd harmonic. Therefore, the amplitudes of the 2 nd and 3 rd harmonics can be controlled simultaneously by adjusting the length of \(l_{10}\). The harmonic amplitude varying with the length of the stub \(l_{10}\) at 18 dBm input power is shown in Figure 4. When the length of the stub \(l_{10}\) varies between 0-3\(\mathrm{\SIUnitSymbolMicro g}\)/20(0-5 mm), the amplitude of the 2 nd harmonic is lower than that of the 3 rd one. The 2 nd harmonic is reflected back into the diode and mixed with the fundamental, which might excite the new 3 rd harmonic and increases its amplitude.