The goal of the taper is to minimize the reflection coefficient involved with changes in impedance caused by changes in transmission line geometry. In essence we are trying to make the changes so gentle that we don't scare any electrons. The scared electrons reflect at the point of abrupt changes in impedance(flee back in the direction they came from) which weakens the signal delivered to the HDMI cable and, finally, the display. (Forgive the anthropomorphization of the electrons--it is purely for illustrative purposes.)
As the high-frequency differential lines traverse the board from processor pin to connector pin certain parts of the geometry and electromagnetic environment stay fairly constant, other aspects in certain places are not conducive to the transmission environment needed by the HDMI signals we wish to deliver to the connector.
Three principles are at work here: 1. Whatever conditions are apparent over the majority of the conduction path will dominate the transmission characteristics. (You could think of the overall impedance as being similar to a length-weighted average of the local impedance of all the sections along the path.) 2. The abruptness of changes in geometry will determine the abruptness of changes in impedance and thus the reflection coefficient associated with the perturbation. 3. Reflections are more troublesome the further you get from the signal source. Close to the source the reflection arrives at the source during the signal rise time and can be overcome by the line driver.
The trace width stays mostly constant at 5mil except for component pads at processor, ESD chip, and connector and the two through-hole vias used to transition from layer 1 (processor) to layer 6 (room for differential microstrip transmission lines) back to layer 1 (connector).
Both the signal trace copper thickness and the dielectric thickness between signal traces and ground plane change only at the signal vias.
The room available on layer 6 allows us to make a controlled-impedance differential transmission line for a good share of the transmission path. Since the HDMI standard specifies 100 +/-15% Ohm impedance, we have designed the geometry to provide a characteristic impedance close to the upper end of the tolerance of the nominal impedance. We have room to impose this geometry for most of the length of the sojourn. At both ends the space is restricted such that the close quarters will no doubt result in a lower local impedance.
Where we have room, the distance between a differential pair and any other copper (be it another differential pair or ground) is 15mil. At both ends this is restricted by the spacing between lands in the component layouts down to 5-7mil.
From the first principle, we see that the influence of the lower local impedance from the restricted sections will serve to lower the overall impedance. In order to stay within the tolerances of the nominal impedance we attempt to limit the length of the restricted sections (where the inter-pair distance <15mil). In some places this could lead us to maintain 15mil inter-pair distance right up to an obstruction which imposes a 5-7mil inter-pair distance. The second principle leads us to recognize this is an abrupt change and expect that it will cause reflections. The third principle suggests it is more important to deal with abrupt changes at the connector end of the transmission line than the processor end.
Hence, we are exploring the feasibility of tapering the inter-pair distance down from 15mil to 5mil as we get to the connector end, in order to soften the effects of the unavoidable space restrictions at the connector end. The other important point is that since we are dealing with differential signals, we are interested in trying to maintain symmetry in dealing with the two traces of each differential pair, lest we push signal energy into common-mode.
The idea was inspired by my reading of a discussion on "Microwaves101"[*] of an impedance taper first described by R. W. Klopfenstein in a paper titled "A Transmission Line Taper of Improved Design", published in the Proceedings of the IRE, page 31-35, January 1956.
We aren't really doing his work justice as our frequencies are so low that our board is too small to accommodate the length required to get the good low frequency response he demonstrates. Nevertheless, we are interested in making the sequence of small transitions in a somewhat similar fashion.
Transmission Line geometry (widths)
North ground fill keep out Inter-pair Distance = 15mil HDMI TX2P trace = 5mil Intra-pair Spacing = 5mil HDMI TX2N trace = 5mil Inter-pair Distance = 15mil HDMI TX1P trace = 5mil Intra-pair Spacing = 5mil HDMI TX1N trace = 5mil Inter-pair Distance = 15mil HDMI TX0P trace = 5mil Intra-pair Spacing = 5mil HDMI TX0N trace = 5mil Inter-pair Distance = 15mil HDMI TXCP trace = 5mil Intra-pair Spacing = 5mil HDMI TXCN trace = 5mil Inter-pair Distance = 15mil South ground fill keep out
Adding this up yields a total = 135mil
When we scale the Inter-pair Distance = 5mil, the total = 85mil
This drops 50mil in width.
<step> <Inter-pair Distance> <Change> 0 15mil 1 14mil -1mil 2 13mil -1mil 3 12mil -1mil 4 10mil -2mil 5 08mil -2mil 6 07mil -1mil 7 06mil -1mil 8 05mil -1mil
If we use 15mil along the signal conduction path from the onset of one change to the next and 45 degree turns to initiate and complete all the changes, and if we choose a geometry to lengthen TXC (clock) the most and leave unchanged TX2, the length along the signal path of the taper will be 7*15mil + 4*1mil = 109mil after which we have no need of manual keep outs for the ground fill as the board rule of 5mil minimum Cu-Cu spacing will suffice.
Deviations from path in due NorthEast direction (+ signifies change in the NorthWest direction, - signifies change in the SouthEast direction, units in mil) <step> <Northern keepout> <TX1> <TX0> <TXC> <Southern keepout> 0 0 0 0 0 0 1 -1 1 2 3 4 2 -1 1 2 3 4 3 -1 1 2 3 4 4 -2 2 4 6 8 5 -2 2 4 6 8 6 -1 1 2 3 4 7 -1 1 2 3 4 8 -1 1 2 3 4
Diagram attached below.
Good grief that took awhile! I'm now completely sold on the concept of Computer-Aided Design (I've used some awkward implementations before but this was done with pencil, pen, measuring tape, and book spine for straight edge).
Reference: [*] https://www.microwaves101.com/encyclopedias/klopfenstein-taper
