## Constant-Resistance Termination

In 1945, Hendrik W Bode published *Network Analysis and Feedback Amplifier
Design*, codifying in one classic book the filter and feedback-amplifier
theory upon which much of the electronics industry still relies.

Of the many secrets his seminal work reveals, one of my favorites is the
constant-resistance network (Figure 1). The figure shows only one of many
forms of this circuit. As long as you scale the components such that the time
constant, *Z*_{0}*C*_{IN}, equals the time constant, *L*_{2}/*Z*_{0}, then, in response to a step input,
the rate of decrease in the admittance of the R-C leg precisely matches the rate
of increase in the admittance of the L-R leg. The result is that the impedance, *Z(f)*, of the whole circuit remains constant at all frequencies. At least,
it remains constant until some limit above which the parasitic aspects of the
circuit take over and the C and L components no longer behave like Cs and Ls.

This circuit occasionally sees application in digital systems as a
terminating network. For example, suppose *C*_{IN} represents the
unavoidable input capacitance of a receiver. You may then use components *R*_{1}, *R*_{2}, and *L* to complete the
circuit, forming at the input a compensated termination that returns no echo
regardless of the value of *C*_{IN}.

In that case, the received signal, having passed through *R*_{1} before arriving at the input terminal of the receiver, is delayed by the group
delay, *T*_{group}=*R*_{1}*C*_{IN} of the filter
thus formed. The filter's 10 to 90 percent rise time, *T*_{10-90}=2.2×*R*_{1}*C*_{IN}, degrades the
rise time of the incoming signal. Provided these two artifacts are acceptable,
the termination works in an ideal fashion.

Many other constant-resistance structures are possible. The general theory
says that if you replace *C*_{IN} by any network *a* and *L*_{2} by any network *b* having the impedance relationship *b*=*Z*_{0}^{2}/*a*, the input impedance of the
whole structure will still equal exactly *Z*_{0} at all
frequencies. This remarkable property is provable using ordinary algebra,
although the calculations are hideous and time-consuming. One can only imagine
how many long nights Bode spent in his office at Bell Labs coming up with this
theory. The general theory of constant-resistance networks becomes very
important when you wish to construct an equalizing filter at the end of a long
transmission line. By carefully crafting impedance *a*, you can construct
just about any arbitrary equalization function at the input terminals of the
receiver. You then use impedance *b* to balance the network such that the
input impedance of the whole structure looks like a perfect end termination.

Constant-resistance filters differ significantly from lossless L-C ladder filters, such as the popular Cauer filters you may have encountered. A lossless filter works by either passing power through the network or reflecting it back to its source. A constant-resistance filter works by either passing power through the network to the receiver or shunting it off to the balancing leg of the filter, where it dissipates harmlessly in the form of heat.

### Reference

[1] Hendrik W Bode, *Network Analysis and Feedback Amplifier
Design*, D. Van Nostrand, 1945