## 3 Design of the Asymmetric Coupled Line Balun
The input and output reflection coefficients were found for the single-ended 50 \(\Omega\) and differential 100 \(\Omega\) loads, respectively. Figure 6 shows a comparison of the input and output reflection coefficients (\(S_{11}\) and \(S_{22}\)) (a) and the amplitude imbalance and phase difference (b) of the two baluns. In the case of Balun 1, \(S_{11}\) and \(S_{22}\) did not matched well to their respective port impedances over a wide bandwidth, which resulted in a large amplitude imbalance, as can be seen in Figure 6b. The main reasons for such performances were the weak coupling between the coupled lines and the lower even-odd mode impedance ratios, resulting in the even/odd mode phase velocity variation over the frequencies. In contrast, the asymmetric coupled line balun not only improved the capacitive coupling, it also adjusted the even/odd mode capacitances over a wider bandwidth, which enhanced the input and output matching and reduced the amplitude imbalance in the frequency band of 88-224 GHz. The phase difference was very close to \(\pm 180^{\circ}\) in both cases. The surface current distribution of the two baluns is shown in Figure 7 at 150 and 200 GHz. At these two frequencies, the magnitude of the surface current was varying significantly for Balun 1, while it was almost similar for Balun 2. The surface current verified the performance of the proposed asymmetric three coupled line balun over the conventional two coupled line balun. Thus, the asymmetric coupled line balun improved the odd mode capacitances, which equalized the even/odd mode phase velocities needed for a high performance balun. Figure 4: 3D views of the conventional edged couple balun (**a**) and a three asymmetric coupled lines balun (**b**), where different important parameters are shown. For the sake of simplicity, the ground plane is not shown. The size of each balun is 215 \(\times\) 148 \(\upmu\)m\({}^{2}\). Balun 1: conventional coupled line balun without the inner lines; Balun 2: the three asymmetric coupled lines balun. W, width; S, separation. Figure 5: Simulated input reflection coefficient (\(S_{11}\)), amplitude imbalance, and phase difference for the two gap (S1 and S2) variations with fixed widths of W1 = 5 \(\upmu\)m and W2 = 6.5 \(\upmu\)m. \(S_{11}\) under S1 (**a**) and S2 (**b**) variation, amplitude imbalance under S1 (**c**) and S2 (**d**) variation, insertion loss of a single balun under S1 (**e**) and S2 (**f**) variation, and phase difference under S1 (**g**) and S2 (**h**) variation. All parameters are in \(\upmu\)m, while absolute values of the phase difference are shown to clearly see the phase imbalance. To validate the design of the asymmetric coupled line balun, it was also simulated in Sonnet EM-simulator. Figure 8 shows a comparison of \(S_{11}\)and \(S_{22}\) (a) and the amplitude imbalance and phase difference (b) of the balun in the two software programs. A good correlation was found between the two simulators. \(S_{11}\), \(S_{22}\) were below \(-10\) dB for the frequency band of 88-224 GHz, which covers  
Figure 6: Simulated input/output reflection coefficients (\(S_{11}\), \(S_{22}\)) (a) and amplitude imbalance and phase difference (b) of the two baluns. Figure 7: Simulated surface current distribution at 150 GHz of Balun 1 (a) and Balun 2 (b) and at 200 GHz of Balun 1 (c) and Balun 2 (d). the entire WR-6 (110-170 GHz) and WR-5 (140-220 GHz) frequency bands. The \(-\)10 dB reflection coefficient (\(S_{11}\)) bandwidth of the balun was 136 GHz. An amplitude imbalance of \(\pm\)1 dB with a phase difference close to \(\pm\)180\({}^{\circ}\) was found in the frequency range of 94-177 GHz. Figure 9 shows the phase imbalance (a) and insertion loss (b) of the balun. The phase imbalance was less than seven degrees. The minimum and maximum insertion losses of two back-to-back baluns including RF-pads in the D-band were 2.45 and 3 dB, respectively, which exhibited less than 1.5 dB of maximum insertion loss for the single balun. Above 200 GHz, the performances of the balun were degraded due to mismatches.