# A Low Cost BLE Transceiver with RX Matching Network Reusing PA Load Inductor for WSNs Applications
## 2 Proposed Front-End with RX Matching Network Reusing PA Load Inductor
### LNA and LNA Matching Network  
To allow the RX a good coexistence performance with an in-band blocker [9], the LNA and mixer should be designed to be linear to prevent these blockers from degrading the performance. As shown in Figure 2, the LNA adopts a push-pull common-source inductorless topology [17], which inherently achieves higher linearity and power-efficiency when compared with a conventional NMOS-only LNA [18].  
In RX mode, the simplified small signal equivalent circuit diagram of \(Z_{RX}\) is shown in Figure 5, and the output impedance can be written as:  
\[Z_{RX}=Z_{in}||\frac{1}{j\omega C_{ct}}||Z_{TX-off}, \tag{7}\]  
Figure 4: TX impedance analysis. (**a**) TX impedance movement in Smith chart; (**b**) The final \(S_{11}\) of TX impedance from 2.4 GHz to 2.5 GHz.  
Figure 3: TX matching network analysis.  
where C\({}_{\text{ct}}\) represents the sum of all of parasitic capacitances seen from the LNA input, \(Z_{in}\) represents the LNA input impedance and can be expressed as [17]:  
\[Z_{in}=\frac{R_{Fin}}{1+(R_{Fin}C_{\omega}^{gsT}2)}-j\frac{R_{Fin}{}^{2}C_{gsT} \omega}{1+(R_{Fin}C_{gsT}\omega)^{2}}, \tag{8}\]  
\[R_{Fin}=\frac{R_{f}}{1+|G_{V}|}, \tag{9}\]  
\[C_{gsT}=\ C_{3}+\ C_{gsn1}+C_{gsp1}, \tag{10}\]  
where \(C_{gsn1}\) and \(C_{gsp1}\) represent the capacitance between the gate and the source of NM\({}_{1}\) and PM\({}_{1}\), respectively. G\({}_{V}\) represents the LNA voltage gain and can be written as:  
\[G_{V}=\frac{1}{2}\frac{g_{mT}}{R_{s}C_{gsT}\omega_{0}}(r_{ds1}||r_{ds2}||R_{f} ||R_{Mixin}), \tag{11}\]  
\[g_{mT}=\ g_{mn1}+\ g_{mp1} \tag{12}\]  
Here \(g_{mm1}\) and \(g_{mp1}\) represent the transconductance of NM\({}_{1}\) and NM\({}_{2}\) respectively, and \(R_{s}\) is 50 \(\Omega\). Thus, \(Z_{RX}\) can be simplified as a resistor \(R_{r}\) shunted with an inductor \(C_{r}\), and the admittance \(Y_{RX}\) can be expressed as:  
\[Y_{RX}=\frac{1}{R_{r}}+j\omega C_{r}=(\frac{1}{R_{Fin}}+\frac{1}{R_{RL}})+j[ \omega(C_{gsT}+C_{ct}+C_{TL})] \tag{13}\]  
Considering there is no inductive device in the LNA circuit, \(Z_{\text{RX}}\) is in the capacitive region in Smith chart. Therefore, the PA load inductor L\({}_{1}\) is reused in RX matching network. As described above, \(L_{1}\) moves \(Z_{RX}\) to the inductive region in Smith chart. \(Z_{RX1}\) represents the series impedance of \(Z_{RX}\) and \(L_{1}\), which can be written as:  
\[Z_{RX1}=\frac{R_{r}}{1+(\omega R_{r}C_{r})^{2}}-j\frac{\omega R_{r}C_{r}}{1+( \omega R_{r}C_{r})^{2}}+j\omega L_{1}. \tag{14}\]  
Then \(Z_{RX1}\) can be moved to near the center point via the capacitor \(C_{1}\) and trimming the capacitor \(C_{2}\). Figure 6 shows the trajectory of the \(Z_{RX}\) when applying \(L_{1}\), \(C_{1}\) and \(C_{2}\). \(Z_{RX2}\) represents the impedance after adding \(C_{1}\) and \(C_{2}\), and can be expressed as:  
\[Z_{RX2}=(Z_{RX1}+\frac{1}{j\omega C_{1}})||\frac{1}{j\omega(C_{2}+C_{div})}. \tag{15}\]  
As shown in Figure 6b, the simulated LNA NF is around 2.57 dB and the maximum S\({}_{11}\) is about \(-15\) dB.  
Figure 5: RX matching network analysis.