Equivalent Circuit Source Impedance

Wong Kin Hong of the Chinese University of Hong Kong has a question about one of the figures in my book "High-Speed Digital Design: A Handbook of Black Magic." We have combined his question with several others and provided a consolidated response below.

I found that your high-speed digital design book is very useful and will recommend it to my students as a reference. One question I want to ask, is what is the source impedance of the equivalent circuit at figure 1.6 (page 13). In the figure, you label it as a "500-ohm lab setup," while in figure 1.8 (page 15) you show a driving point impedance for the same circuit of 503 ohms. Neither seems correct to me.

Here's the figure:

   V(t)==========+---R=1K---+---R=1K--=========to scope---+--V(out)
   50ohm         |          |                             |
   source        |          |                             |
               R=50ohm     DUT                         R=50ohm
                 |          |                             |
                 |          |                             |
                gnd        gnd                           gnd

Figure 1.6—A 500-ohm lab setup for measuring capacitance

(The voltage V(t) comes from a 50-ohm source. A coax cable connects the source to the test area (shown with double lines ======). The first 50ohm resistor terminates the source cable near the DUT. The first 1K resistor then connects the source signal to the DUT. The second 1K resistor connects the DUT to the scope cable. The scope cable is terminated at the right- hand side by the second 50-ohm resistor, which is located internal to the scope.)

To the left of the DUT, I see the first 1K in series with a combined 25-ohm load (the first 50-ohm terminator in parallel with the 50ohm source impedance). This is 1025 ohms.

To the right of the DUT I see the second 1K in series with the 50-ohm input impedance of the scope. (1050 ohms)

The parallel combination of 1025 and 1050 is 519 ohms.

Was the 500 ohms only a simplified approximation based on 10 resistors? If so, where did 503 ohms come from?

Thanks for your interest in High-Speed Digital Design.

Excellent question! I am glad to see you are reading carefully. Other readers may have wondered the same thing, but never asked. Apologies for any confusion.

You are correct. The 500-ohm value is a simplified approximation based on the fact that, with 10resistors, you should get something in the range of 468 to 568 ohms.

In practice, before I use the test circuit, I measure the actual DC resistance from DUT to ground and use the measured value to adjust the test results. (I measure the actual DC resistance with the pulse generator connected, but turned off. On my generator, even when it is turned off its output registers a solid 50-ohm connection to ground.) If you like, it's rather easy to hand-select resistor values. Using values of precisely 975 and 950 ohms, the driving point impedance of the test circuit works out to precisely 500 ohms.

Best Regards,
Dr. Howard Johnson