## 3 Analytical Expressions
### Impedance Matching and Envelope Detector  
An impedance matching network is required to maximize the received RF power of the antenna. To ensure the best possible performance of the diode circuit, the load should be equal to the conjugate of the source impedance. The Schottky diode SMS7630 from Skyworks was chosen as the detector diode due to its sensitivity to the lowest RF power. The most common measure of mismatch is return loss \(S_{11}\). To create an effective impedance matching, passive elements such as capacitors and inductors are used, which reduce the losses due to the reflection of a certain part of the energy of the incident waves. The antenna and the SAW filter are designed for 50 \(\Omega\) source impedance. For the defined transmission frequency of 868 MHz, the load impedance \(Z_{\mathrm{L}}\) is defined as shown in Equation (6).  
\[Z_{\mathrm{L}}=50\,\Omega\cdot\frac{1+S_{11}}{1-S_{11}}. \tag{6}\]  
The reflection coefficient is simulated in ADS Keysight (Figure 6) and the SPICE parameters of the SMS7630 diode. It is found that the diodes are almost entirely capacitive and are matched with two inductors to achieve the best resonant frequency.  
The matching network evaluated by ADS consists of a reversed L network (Figure 7), L1 = 22 nH and a parallel inductance L2 = 3.3 nH. The exact values of the matching network are determined directly on the board using the input reflection coefficient \(S_{11}\) measurement on the network analyzer.  
The maximum diode RF voltage sensitivity at room temperature at 868 MHz, assuming ideal impedance matching, is derived from [42], which is expressed in Equation (7).  
\[\gamma_{\mathrm{RF}}=\beta_{\mathrm{i}}\cdot R_{\mathrm{V}}. \tag{7}\]  
Figure 6: ADS simulation of reflection coefficient S11.  
Figure 7: Reversed L network [41].  
The current sensitivity can be described in Equation (8), which is defined by the angular frequency \(\omega\), junction capacitance \(C_{\mathrm{j0}}\), and video resistance \(R_{\mathrm{V}}\simeq nV_{\mathrm{T}}/I_{\mathrm{s}}\) with saturation current \(I_{\mathrm{s}}\), and the thermal voltage \(V_{\mathrm{T}}\), as well as the series resistance \(R_{\mathrm{s}}\) and the ideality factor \(n\).  
\[\beta_{\mathrm{i}}=\frac{1}{2 nV_{\mathrm{T}}(1+(\omega C_{\mathrm{j0}})^{2}R_{ \mathrm{V}}R_{\mathrm{S}})}. \tag{8}\]  
Furthermore, the envelope detector diodes influence the WuRx sensitivity. For that, the tangential signal sensitivity (TSS), expressed in Equation (9), is used to describe the sensitivity of the detector diodes and is the lowest input signal power level [43] where \(B_{\mathrm{V}}\) is the bandwidth, \(T\) expresses the environmental temperature, and \(k\) is the Boltzmann constant.  
\[P_{\mathrm{TSS}}=\frac{2.5\sqrt{4 kTR_{\mathrm{V}}B_{\mathrm{V}}}}{\gamma_{ \mathrm{RF}}} \tag{9}\]  
The curve of voltage sensitivity and tangential sensitivity are shown in Figure 8. With \(T=298\,\mathrm{K}\) and \(B_{\mathrm{V}}=18.7\,\mathrm{kHz}\) at 868 MHz, the voltage sensitivity is more than \(90\,\mathrm{mV}/\upmu\mathrm{W}\); according to this value, the lowest input power level of the diodes is \(-74.7\,\mathrm{dBm}\).  
The envelope detector circuit is designed using Greinacher voltage multipliers as given in Figure 9. It is used to demodulate the received wake-up signal, as well as the amplification of the input voltage (\(V_{\mathrm{in,ED}}\)). Typically, the output voltage of the voltage multiplier (\(V_{\mathrm{out,ED}}\)) is almost the double of the input voltage, where the used ones affect the voltage level by their reverse voltage.