## 2 SPDT Switch Design
The parallel C\({}_{\rm d82,off}\) and L\({}_{1}\) in the RX path are designed to be in resonance and open at a center frequency of 28 GHz. This leads to decent input impedance matching at the TX and antenna nodes (i.e., a decent S\({}_{\rm 11}\) and S\({}_{\rm 33}\)), a low IL, and high TX-to-RX isolation. This can be explained in more detail as follows. In the RX path of the SPDT switch, inductor L\({}_{1}\) is in parallel with DTMOS-R M\({}_{2}\). In the RX mode, DTMOS-R M\({}_{2}\) is in the on-state, equivalent to a small resistance R\({}_{\rm on2}\),to achieve a decent IL. L\({}_{1}\) is negligible since R\({}_{\rm on2}\) dominates the RX path impedance. In the TX mode, DTMOS-R M\({}_{2}\) is in the off-state, equivalent to a large resistance R\({}_{\rm off2}\), in parallel with a capacitance C\({}_{\rm d82,off}\). To achieve decent isolation with the RX path, L\({}_{1}\) and C\({}_{\rm d82,off}\) should be in parallel resonance at an operation frequency of 28 GHz. For SPDT SW1, C\({}_{\rm d82,off}\) is equal to 74.9 fF, so an L\({}_{1}\) of 431.3 pH is chosen. For SPDT SW2, C\({}_{\rm d82,off}\) is equal to 32.7 fF, so an L\({}_{1}\) of 987.5 pH is used. The design and simulation of the \(\lambda\)/8-TL-based equivalent \(\lambda\)/4-TL in the TX path can be explained in more detail as follows. According to the TL theory, the ABCD matrix of a lossless TL with electrical length of \(\theta\) is given by [21]  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}\cos(\theta)&jZ_{\rm C}\sin(\theta)\\ j\sin(\theta)/Z_{\rm C}&\cos(\theta)\end{bmatrix}  \tag{1}\]From (1), \(Z_{C}\) and \(\theta\) can be written as  
\[Z_{C}=\sqrt{B/C} \tag{2}\]  
\[\theta=\cos^{-1}(A)\ (\mathrm{or}\ \cos^{-1}(D)) \tag{3}\]  
That is, the \(Z_{C}\) and \(\theta\) of a TL can be obtained from the simulated ABCD-parameters converted from the simulated S-parameters. For a lossless TL with a \(\theta\) of 90\({}^{\circ}\) (i.e., length of \(\lambda/4\)) and \(Z_{C}\) of R\({}_{0}\) (i.e., 50 \(\Omega\)), (1) can be simplified as  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}0&jR_{0}\\ j/R_{0}&0\end{bmatrix}  \tag{4}\]  
Additionally, the lossless TL can be modeled by a \(\pi\)-network consisting of a series inductor (L\({}_{L}\)) and two parallel end capacitors (C\({}_{L}\)). The ABCD matrix of the \(\pi\)-network is given by  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}1-\omega_{0}^{2}L_{L}C_{L}&j\omega_{0}L_{L} \\ j\omega_{0}C_{L}(2-\omega_{0}^{2}L_{L}C_{L})&1-\omega_{0}^{2}L_{L}C_{L}\end{bmatrix}  \tag{5}\]  
If we let (5) equal (4), we obtain  
\[L_{L}=\frac{R_{0}}{\omega_{0}} \tag{6}\]  
\[C_{L}=\frac{1}{R_{0}\omega_{0}} \tag{7}\]  
In Figure 4a, two additional parallel capacitances C\({}_{P}\) (equal to C\({}_{\mathrm{dSI,off}}\) + C\({}_{\mathrm{gd1,off}}\) or C\({}_{\mathrm{ant}}\)+ C\({}_{\mathrm{gd2,off}}\)) are incorporated to the shorter (electrical length \(\theta_{1}\) of 45\({}^{\circ}\) or length of \(\lambda/8\) in this work) TL with a \(Z_{C}\) of \(Z_{C1}\) to form an equivalent \(\lambda/4\) TL. It has the potential of a small chip area since the TL \(\theta_{1}\) (45\({}^{\circ}\) in this work) can be any value less than or equal to 90\({}^{\circ}\) in theory. For the TL, suppose the corresponding L\({}_{L}\) is L\({}_{L1}\), and C\({}_{L}\) is C\({}_{L1}\). From (1) and (5), L\({}_{L1}\) and C\({}_{L1}\) can be written as  
\[L_{L1}=\frac{Z_{C1}\sin\theta_{1}}{\omega_{0}} \tag{8}\]  
\[C_{L1}=\frac{1-\cos\theta_{1}}{\omega_{0}Z_{T1}\sin\theta_{1}} \tag{9}\]  
The equivalence of the \(\lambda/8\) TL with a \(Z_{C}\) of \(Z_{C1}\) (i.e., C\({}_{L1}\)-L\({}_{L1}\)-C\({}_{L1}\)\(\pi\)-network) and two additional parallel capacitances (C\({}_{P}\)) to a \(\lambda/4\) TL with a \(Z_{C}\) of R\({}_{0}\) (i.e., C\({}_{L}\)-L\({}_{L}\)-C\({}_{L}\)\(\pi\)-network) requires L\({}_{L1}\) = L\({}_{L}\) and C\({}_{L1}\) + C\({}_{P}\) = C\({}_{L}\).