See Beyond the Edge

(Originally published in EDN Magazine, October 13, 2005)
 

To measure the characteristic impedance of a pc-board trace, most engineers use a TDR (time-domain reflectometer). Connect the instrument to a long, unterminated trace, and it blasts into the trace one very quick, precise rising edge. By analyzing the signal that reflects back from the trace, you can deduce the trace impedance. Some instruments provide averaging capabilities to help reduce the noise floor during particularly fast measurements.

In the usual setup, a TDR instrument uses only the first few nanoseconds of your reflected signal. After the initial step edge propagates to the far end of the unterminated trace, it bounces and returns to the instrument, corrupting further readings of characteristic impedance. Figure 1 illustrates the typical result. The top curve (TDR, in red) illustrates the signal you typically observe at the front end of a pc-board trace. This plot displays the initial TDR step (first edge) and the signal that reflects from the far end of the trace (second edge).

You calculate the lower plot (blue, offset below for visual clarity) from the TDR plot. It is the step response of the S-parameter function, S11. In the frequency domain, S11(f) equals (2×(j2πf)×TDR(f)–1). The S11 step response shows only the signal reflected from the trace, in the absence of the outgoing signal. (Subtracting unity accomplishes this reduction.)

The S11 step response displays a pedestal, from whose amplitude you may deduce the effective trace impedance over a scale of time of 1 nsec or so. After the pedestal, this S11 step response also displays a gentle upward tilt. This tilt is the hallmark of a trace marred by significant amounts of skin-effect loss.

Engineers usually consider the second edge the end of usable data in a TDR waveform. Even though the latter stages of the waveform contain a wealth of information about trace loss and impedance, these details are hidden from view—unless you learn to see beyond that second edge.

Here is the catch: You must make two measurements, not one. Make the first measurement as usual, with the trace open-circuited at the far end. Make the second measurement with the trace shorted to ground at the far end. Now, convert both measurements to the frequency domain using an FFT.

If you are unfamiliar with FFT calculations, look up the "FFT windowing functions" in your instrument's help screens. The time-domain window exists to chop off the truly unusable negative edge of your pulse generator in a graceful way without inducing other undesirable side effects, such as wiggles in the frequency-domain output. It sometimes helps to differentiate the waveform before windowing and then patch it up later in frequency-domain form.

Now, convert each of your TDR results to S11 form using S11(f)=(2×(j2πf)×TDR(f)–1).

From the two S11 functions now in your possession and from knowledge of the source impedance, Z_S(f), of your TDR tester (usually 50Ω), you may now calculate the characteristic impedance, Z_C(f), of your pc-board trace. This calculation works at extended frequencies corresponding to the full length of the waveform you capture and is not limited by the reflection time of your TDR-test coupon:

This clever frequency-domain technique derives from procedures that engineers commonly use to calibrate the SMA cables you use with a network analyzer.

All Publications by Dr. Howard Johnson except as noted.
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