## Constant-Resistance Equalizer

My last column converted an imperfect receiving device into an ideal termination using a constant-resistance network (Reference [1]). The schematic in Figure 1 extends that idea, combining a good termination with a useful equalizing function. Such circuits are useful in places where the rewards of improved circuit performance more than make up for the investment in circuit complexity; for example, at the end of a long transmission cable.

The circuit in Figure 1 is specifically tuned for driving a capacitive load. The design
equations in the figure assume that you are operating at some baud rate, *f*_{b}, and wish
to enhance the high-frequency content of your signal by the desired ratio, *G*_{AC}/*G*_{DC}. In this scenario, *G*_{AC} represents the gain at half the baud rate
(*f*_{b}/2), and *G*_{DC} represents the asymptotic
gain at lower frequencies. Within network *a*, capacitor *C*_{IN} represents the unavoidable input capacitance of your
receiver. Components *L*_{1} and *R*_{1} perform the
equalization function. Network *b* (the dummy-load branch) exists solely to
balance the impedance of the structure such that it equals *Z*_{0} at all frequencies. According to the general theory of constant-resistance
circuits, you can use any network *a*, provided that you preserve the
impedance relationship *b*=*Z*_{0}^{2}/*a*, and
the input impedance of the whole structure will still equal exactly *Z*_{0} at all frequencies. Networks *a* and *b* are
known as *dual* circuits.

The example values in Figure 1 correspond to a 50Ω system operating at 2.5 Gbps with a target
value of *G*_{AC}/*G*_{DC}=4. Figure 2 plots the frequency response of the circuit in Figure 1. The top trace (blue) shows the frequency response that results
from removing components *L*_{1} and *R*_{1} and
shorting across components *C*_{2} and *R*_{2},
producing a simple constant-resistance termination with no equalization (Reference [1]). The bottom trace (red) shows
the response of the whole circuit.

If you push *f*_{b}/2 much beyond the natural cutoff,
1/(2π*Z*_{0}*C*_{IN}), the circuit still functions but
at a progressively reduced ratio of *G*_{AC}/*G*_{DC}.

If you ever face a particularly hideous value of *C*_{IN}, it
helps to reduce the line impedance, *Z*_{0}. The lower the
impedance of the line, the more capacitive loading it tolerates. Alternatively,
if you can afford some additional signal attenuation, you can add another
resistor, *R*_{3}, in parallel with *C*_{IN}. The new
resistor reduces the effective impedance driving *C*_{IN}, thus
reducing the time constant associated with its response. You compensate resistor *R*_{3} by raising the dummy-load resistor from *Z*_{0} to a new value of *Z*_{0}+*Z*_{0}^{2}/*R*_{3}.

### Reference

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