Even / Odd Mode Analysis This is a method of circuit analysis that uses super-positioning to simplify symmetric circuits

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1 NOMNCLATUR ABCD Matrices: These are matrices that can represent the function of simple two-port networks. The use of ABCD matrices is manifested in their ability to be cascaded through simple matrix multiplication. ven / Odd Mode Analysis This is a method of circuit analysis that uses super-positioning to simplify symmetric circuits S-Matrix This is a matrix that defines how, for any circuit, the power of that circuit is distributed when input to each individual port. Thru-Holes These are physically drilled holes that straddle the transmission lines to suppress the propagation of higher order modes that would detract from transmitted power. Return Loss This is a measure of the amount of energy that is reflected back to some port when power is input at that port. Isolation This is a measure of the amount of energy that is transmitted to a port that is designed to have no power transmitted to it. PROCSS The theory used to analyze the Branchline hybrid was adapted from S. Kumar, "A Multisection Broadband Impedance Transforming Branch-Line Hybrid", 1995 I Transactions of Microwave Theory and Techniques, Vol. 43, No. 11, pp A derivation for the design of an arbitrary power division Branchline is detailed below. The resulting equations are then adapted to the specific needs of this hybrid, which is having an equal impedances on each end as well as an equal power division between output ports. Figure 1: Schematic Diagram of a Branchline Hybrid Figure 1 shows the schematic diagram of a generic Branchline Hybrid. ven / Odd Mode Analysis begins the derivation. This type of analysis requires that the schematic diagram be split along an axis of symmetry. The even mode creates open circuits at the split lines while the odd mode creates short circuits. The rest of the derivation of the Branchline coupler is done using these simplified circuits seen below.

2 Figure 2: ven and Odd Modes used for Branchline Analysis (Left and Right, respectively) These simplified circuits can be broken down further by using ABCD matrices. These matrices find their use in their ability to be cascaded simply through matrix multiplication. We see that the simplified even and odd mode circuits are cascaded shunt arms and transmission lines, so we can use the well known ABCD matrices for these individual components and cascade them to get the effect of the entire circuit. This is done for both the even and odd mode circuits and will result in a final ABCD matrix for both the even and odd modes. For the even mode, (1) (2) (3) For the odd mode, (4) (5) (6) ABCD matrices can be transformed into the S matrix using the well known conversions seen below, (7) (8) (9) (10) (11)

3 Continuing to examine the ABCD/S-matrix conversion equations, by design it is known that we want the input power to be transmitted through to the output ports. This in turn means that we do not want any reflected energy at any of the input ports. We can set this restriction by setting the reflections at any of the input ports to zero for both the even and odd modes. Doing so, we obtain the following equations, (12) (13) quations (12) and (13) can be combined to obtain the following relationships, (14) (15) The individual transmissions for the even and odd modes, denoted by in the S-matrix, can be combined to find the total transmission for the whole Branchline. Doing so, the following equations can be written, (16) where, (17) (18) (19) So, the analysis above yields the S-matrix for the Branchline for arbitrary values of the line impedances. Although we have only really found the SX1 parameters (which are the S parameters if power is input at Port 1), because of the symmetry of the circuit, the other S parameters follow the same power distribution pattern. Continuing on, the goal is to find closed form equations for the line impedances for an arbitrary power division ratio and an arbitrary impedance ratio. Both of these restrictions are defined mathematically below, (20) (21) Starting with quation (20), we can substitute in the simpler versions of quations

4 (16) - (19) as follows: (22) Further substitution yields: (23) We can see, from quations (3) and (6), the following relations: (24) Therefore, using quations (14), (15) and (24), quation following: (23) can be simplified to the (25) Squaring both sides, rearranging and noting that quation (3) implies that is a negative quantity yields: (26) To obtain a relationship between a and d, we look to combine quations (14) and (15). This yields the following two equations: (3), (27)

5 (28) Setting quations yields the first relationship between impedances a and d: (27) and (28) equal to each other (29) The second relation is obtained, first, through applying the condition of losslessness: (30) Combining quations (20) and (30) yields: (31) Substituting into quation (31) in terms of the even and odd transmission quations (18) and (19) and using the relations in quations (14), simplification: (15) and (26) will yield, after much algebra, expansion and (32) Using quations (31) and (32) to solve for yields: (33) Substituting for (from quation (3)) into quation (33), and also using quation 2 (27) will yield the second relation between line impedances a and d:

6 (34) Combining quations (29) and = (34) to solve for a and d individually yield the following design equations: (35) (36) Allowing line impedances b and c to be equal and using quations (27), (35) and (36), we find the following design equations: (37) For our specific application, we have equal power division and equal impedances. Therefore, Applying the conditions in quations (38), (39) and (40), our design equations simplify to: (38) (39) (40) (41) (42) (43) Lastly, our characteristic line impedance is given as 50. Therefore, we find our final line impedances: (44) (45) (46)

7 Port4 =90deg F=11G Hz Z=a =90deg F=11G Hz Z=c =90deg F=11G Hz Z=d Port1 Port2 Port3 RSULTS AND DISCUSSION The calculated theoretical values were then simulated, ideally, in Ansoft Designer software. The results, seen below, validated the design. As can be seen, we have a perfect 50/50 power division between output ports 2 and 3 at our center frequency as well as a perfect 90deg difference between the output ports. =90deg Z=50 =90deg Z=b =90deg Z=b =90deg Z=50 =90deg Z=50 =90deg Z=b Figure 3: Theoretical Branchline Model in Ansoft Designer =90deg Z=b =90deg Z=50 Figure 4: Theoretical Transmission Figure 5: Theoretical Output Phase Difference

8 Figure 6: Theoretical Return Loss Figure 7: Theoretical Isolation Figure 8: Theoretical Output Power Division DISCUSSION OF THORTICAL RSULTS As can be seen from Figure 4 to Figure 8, the results of the theoretical simulations were as expected. We see, over the entire bandwidth from 10 to 12GHz, approximately equal power division with an output phase difference of exactly 90deg. These results are, being ideal, the best possible results that can be obtained for the Branchline hybrid. These ideal simulations clearly validate the derived theory. Doing so, the model is now made more complex by introducing losses that would be seen in real world systems. These losses are added into the model by including the substrate, adding bends, tees, discontinuities, etc. This lossy model is referred to as the designer model. These losses cause performance to degrade over all metrics. These include output power division, output phase differences, return loss and isolation. The theoretical values, therefore, must be tweaked to combat these changes. All of this is done in the software program called Ansoft designer. Using this program, the results of these changes can be seen instantly. The goal is to get the designer model to look as much as possible like the theoretical model. The results for the final designer model are seen below:

9 Figure 9: Designer Model Transmission Figure 10: Designer Model Output Phase Difference Figure 11: Designer Model Return Loss Figure 12: Designer Model Isolation We can see from the designer model simulations, which can be seen from Figure 9 to Figure 12, the results are quite close to the ideal model. Note that there is a change in the overall shape of the transmission plot. It should be mentioned that the overall shape of the transmission was changed to this more eye -like pattern to allow for a more constant transmission across the bandwidth. Note that return losses and isolations are very similar and have a maximum of approximately 24dB down. Also note that the output phase differences range from to across the bandwidth. This designer model was then exported to Ansoft HFSS and, through an iterative tuning process, the final HFSS model was generated. Ansoft HFSS is a more robust full-wave solver that can be used for electromagnetic simulations. The results of the final HFSS simulation can be seen from Figure 15 to Figure 18. The tuning process took place between Ansoft HFSS and Ansoft Designer and required several iterations that tuned lengths, widths and radii of different transmission lines based on what could be inferred from the simulation differences. One last thing that should be noted about the HFSS simulations is that thru-holes were added in the simulation to help suppress higher order mode propagation. These are copper plated holes that are drilled through the circuit that straddle the transmission lines. Several views of the HFSS model can be seen in Figure 13 and Figure 14.

10 Y1 Y1 Y1 Y1 [deg] Figure 13: Final HFSS Branchline Model, Top View Figure 14: Final HFSS Branchline Model, Slanted Side View Transmission Tweaked WITH XTRA THRU Phase Differences Tweaked WITH XTRA THRU m6 m (S(2,1))-(S(3,1)) (S(3,4))-(S(2,4)) (S(1,2))-(S(4,2)) (S(4,3))-(S(1,3)) m m m m m m m db(s(1,2)) db(s(1,3)) db(s(2,4)) m4 m6 m db(s(3,4)) db(s(4,2)) db(s(4,3)) Figure 15: Final HFSS Model Transmission Figure 16: Final HFSS Model Output Phase Difference Return Loss Tweaked WITH XTRA THRU Isolation Tweaked WITH XTRA THRU db(s(2,2)) db(s(3,3)) db(s(1,4)) db(s(2,3)) db(s(3,2)) db(s(4,4)) db(s(4,1)) Figure 17: Final HFSS Model Return Loss Figure 18: Final HFSS Model Isolation DISCUSSION OF HFSS RSULTS

11 Y1 Y1 [deg] The HFSS simulation results show, overall, a good match to the designer model results. We see that the average transmission for this model is db ±0.125 db which, when compared with the previous results, is low. This can be accounted for, however, by the fact that HFSS is a more robust solver that takes more losses into account. It was shown, through simulation where the substrate loss tangent was set to zero, all copper traces were replaced with perfect electric conductors, etc, the transmission was seen to be 3.03 db ±0.150 db. We also see that the phase differences at the output ports range from to across the bandwidth, which when compared to the designer model, is slightly lower, but within a quarter of a degree to ninety at all times. Lastly, we see that across the 10-12GHz bandwidth the return loss and isolation actually decreased to 24.7dB down. After these simulations, the boards were sent to be manufactured. This manufacturing had a six week lead time and during that time tolerance analysis was done on the Branchline. This analysis was done to be sure that variation in performance due to different parameters that could change would be tolerable. Several tests were done including: epsilon variation (3.00 +/- 0.04), and top and bottom dielectric thickness variation (60mil +/-5mil). The epsilon variation was taken from the datasheet and the dielectric thickness variation was decided using engineering judgment no value was given and no value was able to be obtained in a timely manner. Because the manufactured thicknesses can sometimes vary by only 5mil, it was decided that if simulations were done at +/-5mil (which is approximately 8.333% of the total thickness of the substrate) it would represent the absolute worst case possible and most likely worse. A sampling of the tolerance analysis results can be seen below from Figure 19 to Figure m m m Transmission Figure 19: Transmission after psilon Variation m4 Tweaked Tolerance Analysis 02 psilon Variation m5 m6 $epsilon='2.96' $epsilon='2.98' $epsilon='3' $epsilon='3.02' $epsilon='3.04' $epsilon='2.96' $epsilon='2.98' $epsilon='3' $epsilon='3.02' $epsilon='3.04' m m m Phase Differences Tweaked Tolerance Analysis 02 psilon Variation Figure 20: Output Phase Differences after psilon Variation m4 m5 m6 $epsilon='2.96' $epsilon='2.98' $epsilon='3' $epsilon='3.02' $epsilon='3.04' $epsilon='2.96' $epsilon='2.98' $epsilon='3' $epsilon='3.02' $epsilon='3.04' $epsilon='2.96' $epsilon='2.98'

12 Y1 [deg] Y1 Y1 Y Return Loss and Isolation Tweaked Tolerance Analysis 02 psilon Variation db(s(1,1 $epsilon='2.96' db(s(1,1 $epsilon='2.98' db(s(1,1 $epsilon='3' db(s(1,1 $epsilon='3.02' db(s(1,1 $epsilon='3.04' db(s(4,1 $epsilon='2.96' db(s(4,1 $epsilon='2.98' Transmission m4 Tweaked Tolerance Analysis 03 Top Dielectric Thickness epsthick='-5mil' epsthick='-4mil' epsthick='-3mil' m5 epsthick='-2mil' epsthick='-1mil' m6 epsthick='0mil' epsthick='1mil' epsthick='2mil' db(s(4,1 $epsilon='3' db(s(4,1 $epsilon='3.02' db(s(4,1 $epsilon='3.04' m m m epsthick='3mil' epsthick='4mil' epsthick='5mil' epsthick='-5mil' Figure 21: Return Loss and Isolation after psilon Variation Figure 22: Transmission after Dielectric Thickness Variation m m m Phase Difference Tweaked Tolerance Analysis 03 Top Dielectric Thickness (S(2,1) epsthick='-5mil' (S(2,1) epsthick='-4mil' (S(2,1) epsthick='-3mil' Ansoft Name Corporation X Y Return Loss and Isolation Tweaked Tolerance Analysis 03 Top Dielectric Thickness epsthick='-5mil' epsthick='-4mil' epsthick='-3mil' (S(2,1) epsthick='-2mil' (S(2,1) epsthick='-1mil' epsthick='-2mil' epsthick='-1mil' (S(2,1) epsthick='0mil' epsthick='0mil' m5 (S(2,1) epsthick='1mil' epsthick='1mil' m4 m6 (S(2,1) epsthick='2mil' (S(2,1) epsthick='3mil' epsthick='2mil' epsthick='3mil' (S(2,1) epsthick='4mil' epsthick='4mil' (S(2,1) epsthick='5mil' (S(3,4) epsthick='-5mil' (S(3,4) Figure 23: Output Phase Differences after Dielectric Thickness Variation Figure 24: Return Loss and isolation after Dielectric Thickness Variation The tolerance analysis, for which only some of the results are shown, yielded no major changes in the overall predicted responses of the systems. The worst change that is seen in shown in the output phase difference responses for which the variation around 90 degrees increases to approximately +/- 0.5deg. Two physical Branchline circuits were manufactured and tested. Once the SMA Launches were characterized, the results of the Branchlines could be captured. The final experimental results for both of the Branchline couplers are seen below from Figure 25 to Figure 29. It is essential to note that these results were captured with the help of the gating feature on the network analyzer. Using the gating feature we were able to take out the negative effects from the SMA Launch and focus on just the results from the component itself. The gating was set to start from just after the SMA Launch ended and end at just before the reflections from the other SMA Launch came back. The specific gate times for both Branchline circuits were a start gate of 0.400ns and an ending gate of 1.250ns. These values were found by visually analyzing the real part of the time domain signal that displayed on the network analyzer

13 Figure 25: xperimental Branchline Transmission, Circuit 1 (left) and Circuit 2 (right) Figure 26: xperimental Branchline Ports 2 and 3 Phase Difference, Circuit 1 (left) and Circuit 2 (right) Figure 27: xperimental Branchline Ports 1 and 4 Phase Difference, Circuit 1 (left) and Circuit 2 (right)

14 Figure 28: xperimental Branchline Return Loss, Circuit 1 (left) and Circuit 2 (right) Figure 29: xperimental Branchline Isolation, Circuit 1 (left) and Circuit 2 (right) Table 1, shown below, contains a comparison between the theoretical, designer, HFSS, tolerance analysis and experimental results/values.

15 Table 1: Theoretical / Designer / HFSS / Tolerance Analysis / xperimental Results Comparison Theoretical Simulation Designer Simulation HFSS Simulation Tolerance: psilon Value Tolerance: Dielectric Thickness xperimental: Circuit 1 xperimental: Circuit 2 TrL Parameters: a, d Line Impedance Ω Ω Ω X X X X lectrical Length 90º º º X X X X TrL Parameters: b, c Line Impedance Ω Ω Ω X X X X lectrical Length 90º º º X X X X Transmission Average db db db db db db db Delta [db] +/ / / / / / / Output Phase Difference Average 90º º º º º º º Delta [ º] +/ / / / / / / Return Loss Maximum db db db db db db db Isolation Maximum db db db db db db db The final experimental results for the Branchline circuits were mixed. The return loss and the isolations matched very well, both in value and in overall trend, with the HFSS simulations. In fact, the only outlier for the return losses and isolations seems to be the trend of S44 for both of the manufactured circuits. They are the only ones that seem to not fit the trend that is set by all the other return losses (shown in Figure 28). Those problems, however, occur only at 11.5GHz and above. Moving on, considering how the isolations and return losses matched so well, the transmission seems a bit low at dB. This suggests power losses occurring somewhere within the circuit. These losses could easily be due to the effects of the SMA launch, which have proved, experimentally, to be problematic. Also, if we compare the transmissions to the simulated HFSS results we see that the eye shape has shifted apart. It is this shifting that is causing the large spread that we see in Table 1. It can be seen to be causing the high transmission value at the low frequency range and the low transmission value at the high frequency range. The last metric, the output phase differences, are, comparatively, the worst result. Considering the tight delta that resulted from simulations, the experimental circuits are both offset DC-wise and have a 10-times worse phase spread. ven the tolerance analysis shows +/-0.5deg and, at the worst, we see +2.25deg. This result is hard to explain considering that that largest phase difference only occurs on one of the phase difference plots. It can be seen, however, that even with these problems the phases are flat across the bandwidth. Comparing the circuits to each other, the repeatability of the design seems to be quite good as both circuits had approximately the exact same responses across all tested parameters. If one wanted to improve the results, a redesign would have to deal with the shifted transmissions and increase in phase difference delta. Further redesign could be done to try and center the return loss and isolation bandwidth at 11GHz. That could result in a 4dB improvement for both parameters.