# A Wideband Power Amplifier in 65 nm CMOS Covering 25.8 GHz-36.9 GHz by Staggering Tuned MCRs
## 2 Analysis and Design
### Wideband Output Matching Network Design
To achieve a coupling factor k as high as 0.7, a combination of interleaved and stacked structures is employed [24; 25; 26], as shown in Figure 4b. The primary inductor of the stacked part is metal 8 paralleled with metal 7. The secondary inductor consists of metals 8 and 9 and the AP layer in parallel. These paralleled metals increase the Q factor of the windings. Also with a high k factor, according to Formula (11), the insertion loss of the transformer is decreased, in which \(\eta\), \(Q_{p}\), \(Q_{s}\), and k represent the passive efficiency, the Q factor of primary/secondary windings, and the coupling coefficient of the two windings, respectively [5]. The distance between AP layer and metal 8 is as far as 4.94 \(\upmu\)m, so the parasitic capacitance between \(L_{1}\) and \(L_{2}\) is not high enough to affect the SRF of the output transformer seriously. Figure 5a,b show the finally realized inductance and Q factor of \(L_{1}\) and \(L_{2}\). Figure 6a shows the k between \(L_{1}\) and \(L_{2}\), and Figure 6b shows the 3D view of the whole output matching network, respectively. \[Q_{1}=Q_{2}=Q_{0} \tag{2}\]3  
The power gain of the whole output matching network is presented in Figure 7a. The power gain is defined as [27]:  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]4  
in which \(P_{L}\) represents the power delivered to the load, and \(P_{in}\) represents the power fed into the network. This factor takes into account the situation that the load impedance is not always matching [27]. The minimum loss is \(-1.376\) dB at 28.1 GHz. In general, from  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]5 The lumped model of the output symmetrical MCR4. 25 GHz to 40 GHz, the loss at high frequency is larger than that at low frequency. Figure 7b presents the trans-impedance (Z21) of the MCR4 with Z21 being defined as [28]:  
\[Q_{1}=Q_{2}=Q_{0} \tag{2}\]6  
in which \(v_{2}\), \(i_{1}\), and \(i_{2}\) are defined in \[Q_{1}=Q_{2}=Q_{0} \tag{2}\]7 Compared with an ideal lossless symmetrical MCR, the finite Q parameter of the lossy transformer makes the two amplitude peaks of Z21 not equal anymore [29]. Also, the practical coupling factor k of a lossy transformer is not a constant value compared with an ideal transformer. The test schematic for the Z21 of MCR4 with a 50 \(\Omega\) resistor at the input is shown in \[Q_{1}=Q_{2}=Q_{0} \tag{2}\]8 Although the input-matching condition of the power stage affects the test results, the power stage has good isolation due to the neutralization capacitors. Therefore, directly connecting a 50 \(\Omega\) resistor at the power stage input does not have a significant impact on the test results of the Z21. Also, the test iteration for Z21 is needed when the matching passives are designed. As a result, the initial Z21 test result is used to provide an intuitive perception of the gain shape. With the matching network being placed in the circuit, the \"in-situation\" Z21 test method takes the circuit parasitics into account [20]. \[Q_{1}=Q_{2}=Q_{0} \tag{2}\]9 (**a**) The 3D view of the output transformer (TF4) in the symmetrical MCR4 and (**b**) its layer structure. \[\omega_{1}=\omega_{2}=\omega_{0} \tag{3}\]0 The (**a**) inductance and (**b**) Q factor of \(L_{1}\) and \(L_{2}\) for the output transformer (TF4) in MCR4. Figure 9a shows the power gain (Gp) of the power stage with the whole output matching network. From Figure 9a, we can see that the peak power gain at low frequency is much higher than that at high frequency, which has a similar trend as the Z21 shown in Figure 7b. Figure 9b presents a flattened input impedance of the output symmetrical MCR4 (including \(C_{11}+C_{12}\) of the power stage) over a wide frequency band, especially a wide band near zero imaginary part, which means that the capacitor (\(C_{11}+C_{12}\)) resonates with the right side impedance \(Z_{right}\) shown in Figure 3, so that the drain of the output stage transistors can see a nearly pure real Ropt.