# A 2-V 1.4-dB NF GaAs MMIC LNA for K-Band Applications
## 2 Circuit Design
### Design Approach  
In this paper, we present a design methodology focused on how to choose the active device geometry and DC bias conditions to obtain the lowest NF possible when source degeneration is applied. A simplified overview of the design procedure is given in Figure 1a. First, to select the device size, the CS amplifier with the source degeneration depicted in Figure 1b is employed. Note that all the ports are matched to 50 \(\Omega\) and the DC biasing circuitry is not shown for simplicity. The device geometry is defined by the total width \((W_{t})\), which is the product of the unitary finger width \((W_{u})\) and the number of fingers \((N_{f})\). \(Z_{opt}\) is the impedance seen looking away from the gate and \(Z_{in}\) is the impedance seen looking into the gate of the active device. The effect of increasing \(W_{t}\), the value of the source inductor \((L_{s})\) and the gate inductor \((L_{g})\) are shown in Figure 1c. In Figure 1b, the source degeneration inductance \(L_{s}\) is swept from 0 to 300 pH. Source degeneration helps reduce the NF at the expense of a gain reduction, improving the amplifier's stability. A value of approximately 150 pH for \(L_{s}\) is enough to bring the \(S_{11}\) and \(S_{opt}\) closer to the unity circle, i.e., to achieve a real part of about 50 \(\Omega\). A gate inductor is then used to bring both \(S_{11}\) and \(S_{opt}\) to the center of the Smith chart, obtaining a maximum gain and minimum NF amplifier. To select the size and biasing of the LNA first stage device, its performance \((NF_{min}\) and \(G_{max})\) against its current density \((J)\) is studied, keeping close attention to the \(S_{11}\) and \(S_{opt}\), as shown in Figure 2, for different device sizes. In these figures, only device geometries that provide \(S_{11}\) and \(S_{opt}\) close to 50 \(\Omega\) are included. Figure 1: Simplified overview of the design procedure (**a**), schematic diagram used for device selection (**b**) and Smith Chart representation of the influence of varying the transistor width (\(W_{t}\)), the source inductor (\(L_{s}\)) and the gate inductor (\(L_{g}\)) on the input impedance (\(Z_{in}\)) and the optimum source impedance (\(Z_{opt}\)) (**c**). Figure 2: Current density and \(S_{11}\), \(S_{opt}\) of the first stage transistor for a finger width of 20 \(\upmu\)m (**a**,**b**), 30 \(\upmu\)m (**c**,**d**) and 40 \(\upmu\)m (**e**,**f**). A finger width of 20 \(\upmu\)m is explored in Figure 2a,b. In Figure 2a, the \(NF_{min}\) and \(G_{max}\) are plotted as a function of the current density. As seen, a device of 4 fingers and 20 \(\upmu\)m biased with a current density of 0.12 mA/\(\upmu\)m results in a \(G_{max}\) of 12.7 dB and 0.79 dB of \(NF_{min}\), which is the lowest \(NF_{min}\) possible. As shown in Figure 2b, a device of 2 fingers \(\times\) 20 \(\upmu\)m would require a very high value of \(L_{g}\) to be matched. It presents a real part higher than 50 \(\Omega\) and would be increased further after applying source degeneration, so this choice is discarded. However, 6 and 4 finger devices present a real part lower than 50 \(\Omega\) and would be suitable for the first stage design. In fact, both options could be matched to the input with a single gate inductor with a very similar value. Nevertheless, since the 4 \(\times\) 20 \(\upmu\)m transistor presents a \(S_{opt}\) closer to the unitary circle, a lower value of \(L_{g}\) can be employed. This situation is highly desirable since a very high \(L_{s}\) decreases the NF, but it also decreases the gain significantly. In Figure 2c,d, a finger width of 30 \(\upmu\)m is explored. Note that a 2 \(\times\) 30 \(\upmu\)m transistor results in an even lower \(NF_{min}\) than the 4 \(\times\) 20 \(\upmu\)m device. Still, this device presents a real part higher than 50 \(\Omega\), so it is discarded for the same reason as the 2 \(\times\) 20 \(\upmu\)m transistor. The 4 \(\times\) 30 \(\upmu\)m transistor would be a better option, but its \(NF_{min}\) is higher than the 4 \(\times\) 20 \(\upmu\)m device.