## 3 Transformers Implementation  
A transformer is a passive structure composed of primary and secondary inductors (\(L_{p}\) and \(L_{s}\)), with a certain magnetic coupling coefficient \(k\) between them. In a transformer, \(k\) determines the strength of the magnetic coupling between the windings and is given by (2), where \(M\) is the mutual inductance between the primary and the secondary. \[k=\frac{M}{\sqrt{L_{p}\cdot L_{s}}} \tag{2}\]  
Due to the overlapping, the proximity of the windings, and the dielectric constant of the substrate material, designers have to deal with a finite parasitic capacitance (\(C_{par}\)) which can give rise to a low value of the self-resonant frequency (SRF). The SRF is the frequency at which the transformer resonates, determined by (3), and it limits the range of use of the transformer. This is because \(C_{par}\) is proportional to the inductance value (\(L\)). In electronic circuit design, it is desirable for the operating frequency to be a fraction of SRF in order to attain the maximum possible inductance for a given operating frequency. However, as the frequency increases, the utilization of large inductances becomes increasingly challenging due to the corresponding increase in \(C_{par}\) with frequency. Since we are dealing with the design of a K-band LNA, the transformers need to be designed to present an SRF in the order of 50 GHz, which is achieved by means of layout techniques. \[\omega_{sf}=\frac{1}{\sqrt{L\cdot C_{par}}} \tag{3}\]  
To size the transformers defining the values of \(k\), \(L_{p}\), and \(L_{s}\), the designer must obtain the desired source and load impedances (\(Z_{S}\) and \(Z_{L}\), respectively). The impedance  
Figure 3: The 3D representation of the active device layout (**a**), and front view of the same device (**b**). refers to the impedance seen at the input terminal of the transformer (the primary), while impedance \(Z_{L}\) is the impedance at the output node of the transformer (the secondary), which is equivalent to the conjugate value of the desired \(S_{opt}\). This situation is reflected in Figure 4.  
Instead of using a discrete analytical solution as an approximation to the final values of \(k\), \(L_{p}\), and \(L_{s}\), the transformer-based impedance matching tool proposed in [12] is replicated in MATLAB for transformer sizing and inductance equalization. The developed tool receives the working frequency \(f\), the magnetic coupling coefficient \(k\), the source impedance \(Z_{S}\), and the load impedance \(Z_{L}\) as inputs, and it produces the inductance values of a transformer that matches the given impedances at the operating frequency. The MATLAB script generates the values of inductances \(L_{p}\) and \(L_{s}\) to match the two impedances at the given frequency and magnetic coupling coefficient. In addition, an equalization process is implemented, following [12], to adjust the inductors to present the same inductance value, \(L_{eq}\). As a result of the equalization process, an additional passive element has to be added to the source or load of the transformer to perform impedance matching. This is the reason why the capacitors \(C_{in}\), \(C_{m}\), and \(C_{out}\) are added to the schematic in Figure 1.  
Due to the nature of integrated technologies, achieving a \(k\) greater than 0.8-0.9 can be challenging. The EMX full-wave electromagnetic (EM) solver has been used to explore the \(k\) values that can be achieved in the 45 RFSOI technology (-0.7). The MATLAB tool is then used to calculate the inductances needed to match \(Z_{S}\) with \(Z_{L}\) for this \(k\) value. Obtaining the desired inductances depends on the geometry and physical properties of the metals and layers used, so obtaining them is an iterative process, highly dependent on the physical structure used. The final design of the transformers is shown in Figure 5. The layout design has been based on the proposal in [18].