# A 2-V 1.4-dB NF GaAs MMIC LNA for K-Band Applications
## 2 Circuit Design  
According to the Friis formula [2; 7], the overall noise figure (\(NF_{tot}\)) of a cascaded system with \(n\) stages is given as (1), where \(NF_{n}\) and \(G_{n}\) represent the NF and gain of the \(n\)-th stage, respectively. As seen, the first element contributes the most to \(NF_{tot}\), and the NF of the upcoming stages is attenuated by the gain of the first stage (\(G_{1}\)) and all the preceding stages. Hence, the implementation of the LNA must be thoroughly carried out since its performance is critical for the receiver [2; 7]. \[NF_{tot}=1+(NF_{1}-1)+\frac{NF_{2}-1}{G_{1}}+\ldots+\frac{NF_{n}-1}{G_{1}\cdots G _{n-1}} \tag{1}\]  
The LNA is composed of four cascaded CS stages with source degeneration to achieve a minimum NF design. The Friis formula can be applied to the whole system but it can be applied locally too, as shown in (2). \[NF_{LNA}=1+(NF_{s1}-1)+\frac{NF_{s2}-1}{G_{s1}}+\frac{NF_{s3}-1}{G_{s1}\cdot G _{s2}}+\frac{NF_{s4}-1}{G_{s1}\cdot G_{s2}\cdot G_{s3}} \tag{2}\]  
If the LNA is divided into four CS amplifiers, the first stage (characterized by \(NF_{s1}\), \(G_{s1}\)) is again the main contributor to the overall LNA NF (\(NF_{LNA}\)). Applying (2), one can calculate the requirements of each stage to achieve a certain value of \(NF_{LNA}\). We assume the designer is interested in \(NF_{LNA}\) = 1.4 dB and \(G_{tot}\) = 33 dB. If the first stage presents \(NF_{s1}\) = 1 dB and \(G_{s1}\) = 6.5 dB (which is constrained by the selected process and the active device selection), assuming \(NF_{s2}\) = \(NF_{s3}\) = \(NF_{s4}\) = \(NF_{s234}\) and \(G_{s2}\) = \(G_{s3}\) = \(G_{s4}\) = \(G_{s234}\) for simplicity, then to achieve the desired performance, the \(NF_{s234}\) and \(G_{s234}\) should be:  
\[\left\{ \begin{aligned} & NF_{s234}<1.7\ (\text{dB})\\ & G_{s234}\geq 9\ (\text{dB})\end{aligned} \right., \tag{3}\]  
In this sense, the first stage should introduce the lowest NF possible and a high gain in order to allow a more flexible design of the upcoming stages. Additionally, to achieve \(NF_{s1}\) = 1 dB and \(G_{s1}\) = 6.5 dB, the implementation of the input matching network with a single inductor is desired to minimize the number of elements in the signal path and to minimize the noise contribution of the input matching network. Otherwise, obtaining an LNA with an NF under 1.4 dB is not possible without lowering the NF of stages 2, 3 and 4. However, as explained in the following section, lowering \(NF_{s234}\) is only possible at the expense of a gain reduction, which means an additional stage may be needed to reach \(G_{tot}\) = 33 dB. The performance of the active device used in the CS amplifier is mainly defined by three parameters: the minimum NF it provides (\(NF_{min}\)), the maximum stable gain (\(G_{max}\)) and Rollet's stability factor (\(k\)). The value of these three parameters depends on several factors, namely, the device geometry, physical parameters of the selected process, impedance matching or DC bias conditions, among others. The value of \(NF_{min}\) is given as (4), where \(g_{m}\) is the device transconductance (obtained from the device's DC operating point), \(R_{G}\) and \(R_{S}\) are the gate and source resistances, \(f\) is the frequency of operation and \(f_{T}\) is the transit frequency [1]. Similarly, \(G_{max}\) is defined as (5), and [PERSON]'s stability factor \(k\) is given by (6). \[NF_{min}=1+K\sqrt{g_{m}\cdot(R_{G}+R_{S})}\cdot\frac{f}{f_{t}} \tag{4}\]  
\[G_{max}=\left|\frac{S_{21}}{S_{12}}\right|\cdot\left(k-\sqrt{k^{2}-1}\right) \tag{5}\]  
\[k=\frac{1-\left|S_{11}\right|^{2}-\left|S_{22}\right|^{2}+\left|S_{11}S_{22}- S_{12}S_{21}\right|^{2}}{2\left|S_{21}S_{12}\right|}>1 \tag{6}\]  
Impedance matching selection plays a critical role in the design of each stage of the LNA. To obtain an amplifier that yields the maximum gain, one must match the input of the LNA to \((S_{11})^{*}\), whereas to obtain a minimum NF design, one must match the input to the conjugate of the optimum source impedance \((S_{opt})^{*}\).