## 3 Design of the Asymmetric Coupled Line Balun  
Figure 3 shows IHP's 130-nm SiGe BiCMOS back-end of the line (BEOL) process cross-section [1; 29; 30]. The BEOL contains six metal layers including two thick low-loss metals (TM2 and TM1), which provide flexibility in choosing different metal layers for the balun design. To design an edged coupled balun, one can use either lower metals (M1-M5) or top metals (TM2, TM1). The TM2 and TM1 have a thickness of 3 \(\upmu\)m and 2 \(\upmu\)m, respectively, while the lower metals have a thickness of roughly 0.49 \(\upmu\)m. In edge-coupling where the thickness of conductors defines the area of  
Figure 1: Symmetric coupled line with ports’ definition (**a**) and the conventional Marchand balun (**b**). Figure 2: Capacitances of coupled lines under odd mode excitation (**a**) and even mode excitation (**b**). the parallel plate, TM2 and TM1, respectively, will give six- and four-times higher coupling than lower metals. This makes the lower metal not suitable to achieve strong mutual capacitive coupling in odd mode. Furthermore, they can increase the insertion loss. Therefore, TM2 and TM1 were utilized in the design of three asymmetric coupled lines, while M3 served as a ground plane. Figure 4 shows the 3D views of the conventional edged coupled balun (a) and the three asymmetric coupled line balun (b). In the three asymmetric coupled line balun, the inner (\(\lambda\)/2) line is surrounded by two (\(\lambda\)/4) lines. The widths (W1, W2) of the lines and separations (S1, S2) among them can be adjusted to increase effective odd mode capacitance as discussed earlier. This helps in equalizing the phase velocities for even and odd mode. In order to match the single-ended input impedance (\(Z_{in}\)) to 50 \(\Omega\) and differential output impedance (\(Z_{out}\)) to 100 \(\Omega\), the parameters (W1, W2, S1, and S2) were determined through the parametric study of bandwidth, return loss, insertion loss and the phase and amplitude imbalance of the balun. For instance, Figure 5 shows the effect of S1 and S2 on the input reflection coefficient (\(S_{11}\)), bandwidth, insertion loss, amplitude imbalance, and phase difference of the asymmetric coupled line balun when all three ports were terminated in 50 \(\Omega\) except for the insertion loss, which was found by placing two baluns in a back-to-back configuration. As can be seen, lower values of S1 and S2 provided a wider bandwidth (Figure 5a,b) and a small amplitude imbalance (Figure 5c,d). However, the matching at the intermediate frequencies was not better than \(-\)15 dB, which caused higher mismatch losses (Figure 5e,f) and a phase difference (Figure 5g,h) less than 170 degrees (phase error \(\geq\)10 degrees). On the other hand, higher values of S1 and S2 enhanced the matching at the intermediate frequencies, which resulted in lower losses and phase error at a narrow bandwidth. The variation of S1 and S2 changed the even/odd capacitances discussed earlier (see Figure 2). Therefore, intermediate values can be selected, which provide a wider bandwidth with lower losses and amplitude and phase imbalance. A similar study can be performed on W1 and W2. The optimal values for the parameters of the balun were W1 = 5 \(\upmu\)m, W2 = 6.5 \(\upmu\)m, S1 = 2.75 \(\upmu\)m, and S2 = 4.75 \(\upmu\)m. The length of the \(\lambda\)/2 and \(\lambda\)/4 sections were adjusted around the center frequency (\(f_{c}\)) of 150 GHz. Furthermore, two adjustment lines made of top metal (TM1) were added in the second arm of the balun where the \(\lambda\)/2 was terminated in open circuit. These lines increased the length of the second arm slightly to decrease the phase imbalance without affecting the amplitude imbalance. A folded layout was used for the balun to reduce its size, which resulted in a small area of 0.0318 mm\({}^{2}\)  
Figure 3: The 130-nm SiGe BEOLprocess cross-section [29]. To see the advantages of three asymmetric coupled lines over the conventional two coupled lines in the balun design, the two baluns were EM-simulated in ADS momentum.