# A Wideband Power Amplifier in 65 nm CMOS Covering 25.8 GHz-36.9 GHz by Staggering Tuned MCRs
## 2 Analysis and Design
### Input-Matching Network Design  
For the input-matching network, an asymmetrical MCR1 satisfying \(\omega_{1}/\omega_{2}=Q_{1}/Q_{2}\) and resulting in two equal Z21 peaks, is used to extend the bandwidth [16]. Figure 20a presents the lumped model of MCR1 based on TF1 for matching. The input impedance of the input stage (ZM1 = 42.94 \(-\) j*119 at 36 GHz) is equivalent to a paralleled tank including a resistor \(R_{2}\) and a capacitor \(C_{2}\). The output impedance of the power splitter (ZM2 = 82.16 \(-\) j*22.12 at 26 GHz) is equivalent to a paralleled tank including a resistor \(R_{1}\) and a capacitor \(C_{11}\). The power splitter is designed by using microstrip transmission-line segments to reactively match the impedance and split the input power to the two-way driver stage 1 in-phase, as shown in Figure 20b. \(C_{12}\) on the left side is the extra added capacitor to satisfy the condition of \(\omega_{1}/\omega_{2}=Q_{1}/Q_{2}\) for the asymmetrical MCR1. In MCR1, we set \(\omega_{1}\) = 26 GHz, \(\omega_{2}\) = 36 GHz. For the input stage, as shown in Figure 1, a pair of paralleled Rg of 500 \(\Omega\), and a pair of series Rs of 10 \(\Omega\) are added at the gate of the input transistors. All these resistors decrease the Q factor of the input impedance of the transistors. Also, Rs improves the common mode stability of the circuit. After adding these resistors, the input impedance of the input stage ZM1 is 42.94 \(-\) j*119 at 36 GHz. Its equivalent paralleled resistance and capacitance are \(R_{2}\) = 372.7 \(\Omega\) and \(C_{2}\) = 32.87 fF, respectively. As \(L_{2}\) resonates with \(C_{2}\) at 36 GHz, we obtain \(L_{2}=\frac{1}{\omega_{2}^{2}C_{2}}\) = 594.6 pH and \(Q_{2}=\omega_{2}*R_{2}*C_{2}\) = 2.771. Based on \(\omega_{1}/\omega_{2}=Q_{1}/Q_{2}\), we obtain \(Q_{1}\) = 2 and \(C_{1}\) = 139 fF. As \(L_{1}\) resonates with \(C_{1}\) at 26 GHz, we acquire \(L_{1}=\frac{1}{\omega_{1}^{2}C_{1}}=269.5\) pH. As mentioned above, \(C_{11}\) is the equivalent output paralleled capacitor of the power splitter, which is 18.7 fF at 26 GHz in this work; thus, the added extra capacitor \(C_{12}\) is 120.33 fF (equals to \((C_{1}-C_{11})\)). So far, the coupling factor k between \(L_{1}\) and \(L_{2}\) has not been determined. Usually, a higher k makes the two Z21 magnitude peaks of an MCR be apart from each other [16; 31; 32; 33]. Figure 21 shows the Z21 of the MCR1 with a k increasing from 0.3 to 0.7, when the inductors and capacitors are all ideal components. With an increasing k, both the bandwidth and in-band ripple are increased. Considering the target frequency, in-band ripple, and the difficulties of an actual transformer implementation, a k of 0.573 at 32 GHz is selected and achieved for this MCR1. The 3D view of the input transformer (TF1) is shown in Figure 20b. The primary inductance (\(L_{1}\)) at 26 GHz is 266.65 pH, and the secondary inductance (\(L_{2}\)) is 596.4 pH at 36 GHz for the finally achieved practical transformer.  
Figure 22a presents the Z21 of MCR1 with the whole input-matching network shown in Figure 20b. Compared with the initial desire with the condition of \(\omega_{1}/\omega_{2}=Q_{1}/Q_{2}\), the two Z21 peaks are not equal anymore. The loss of the transformer affects the Z21 response [29]. Meanwhile, the equivalent \(R_{1}\), \(R_{2}\), \(L_{1}\), \(L_{2}\), and k being non-constant with frequency also affects the Z21 response. In Figure 22a, the Z21 magnitude of MCR1 at the frequency edge (from 25 GHz to 40 GHz) is much higher than that at the frequency center (from 30 GHz to 35 GHz). Therefore, the Z21 drop at the band edge is compensated, like the case in reference [7]. Figure 22b presents the Z21 of each MCR and the Z21 of the total circuit. It indicates that the Z21 amplitudes of the first and second MCRs compensate for that of the third and the fourth. Therefore, a wideband total Z21 of the whole circuit from input to output is achieved.