## Deconstructing Gain and Impedance from S11

OK, this topic isn't for everyone, but this year, I have encountered so many people struggling with frequency-domain descriptions of transmission-line behavior that I thought I'd pass along some interesting tricks.

My last column extracted a complete curve of pc-board-trace impedance versus
frequency from TDR (time-domain-reflectometry) or S11-type measurements^{[1]}. This column concentrates on
trace gain using a similar trick. For the purpose of this column, the trace gain
is the gain measured between two points along an infinitely long structure,
where distance x separates the points and the signal travels in only one
direction.

To measure trace gain, hook a network analyzer to any short, unterminated
pc-board trace and capture its S11 data. Now, convert that data to the time
domain and integrate it to produce a picture such as the one in Figure 1. (If you have a TDR-type instrument, it may already produce such a picture;
otherwise, see [1]** **for information
about converting from TDR to S11 format.)

This measurement derives from a simulated, 30-in., unterminated pc-board trace. The trace is 0.006 in. above the reference plane and 0.0083 in. wide. The small initial positive pedestal suggests that the asymptotic high-frequency value of trace impedance must lie slightly higher than the 50Ω source impedance of the network analyzer.

During the first 10 nsec of this waveform, before the big reflection arrives from the far end, observe the gentle upward tilt of the waveform. This tilt hints at a slow modulation of the characteristic impedance with time—therefore, variations with frequency.

Next, look at the rounded shape of the second edge. This edge has twice traversed the structure. It contains a cornucopia of information about the high-frequency gain of the pc-board trace. Unfortunately, the second edge sits superimposed upon the still-rising form of the characteristic-impedance curve. The two curves interfere with each other.

With one S11 measurement you cannot independently deconstruct the gain and impedance information present in this waveform. A second measurement, however, reveals both.

Make the first measurement as usual, with an open-circuit trace at the far end. Make the second measurement with the trace shorted to ground at the far end.

From the two S11 curves you have measured, now calculate the round-trip gain, *H ^{2}(f)*, of your pc-board trace.

The symbol *H ^{2}* should remind you that this gain is the
round-trip (two-way) gain, not just the one-way gain. This calculation works at
extended frequencies corresponding to the full length of the waveform you have
captured, and the reflection time of your TDR test coupon does not limit it:

Note that these expressions return a pure gain function, independent of
loading. The gain function is an "insertion loss" function that assumes perfectly matched source and end
terminations. Perfect terminations are not the same as 50Ω terminations,
especially for transmission lines having a length sufficient to force operation
even partially in the RC mode, such as long backplane traces. Reference [2] discusses the gain function *H(f)*at
length.

This technique uniquely derives the transmission-line gain from measurements you take at only one end of your pc-board trace. Any digitizing TDR instrument captures the raw data you need to apply this technique.

### ReferenceS

**[1]** Johnson, Howard, "See beyond the
edge," *EDN*, Oct 13, 2005.

**[2]** Johnson, Howard, and Martin Graham,*High-Speed Signal Propagation: More Black
Magic* , Prentice-Hall, 2003.