9.2.6.3 Resistor Design Equations for the Single-Ended to Differential Configuration of the FDA

The design equations for setting the resistors around an FDA to convert from a single-ended input signal to differential output can be approached from several directions. Here, several critical assumptions are made to simplify the results:

• The feedback resistors are selected first and set equal on the two sides.
• The DC and AC impedances from the summing junctions back to the signal source and ground (or a bias voltage on the nonsignal input side) are set equal to retain feedback divider balance on each side of the FDA.

Both of these assumptions are typical for delivering the best dynamic range through the FDA signal path.

After the feedback resistor values are chosen, the aim is to solve for the R_T (a termination resistor to ground on the signal input side), R_G1 (the input gain resistor for the signal path), and R_G2 (the matching gain resistor on the nonsignal input side); see Figure 96 and Figure 97. The same resistor solutions can be applied to either AC- or DC-coupled paths. Adding blocking capacitors in the input-signal chain is a simple option. Adding these blocking capacitors after the R_T element (as shown in Figure 96) has the advantage of removing any DC currents in the feedback path from the output V_OCM to ground.

Earlier approaches to the solutions for R_T and R_G1 (when the input must be matched to a source impedance, R_S) follow an iterative approach. This complexity arises from the active input impedance at the R_G1 input. When the FDA is used to convert a single-ended signal to differential, the common-mode input voltage at the FDA inputs must move with the input signal to generate the inverted output signal as a current in the R_G2 element. A more recent solution is shown as Equation 8, where a quadratic in R_T can be solved for an exact value. This quadratic emerges from the simultaneous solution for a matched input impedance and target gain. The only inputs required are:

• The selected R_F value.
• The target voltage gain (A_v) from the input of R_T to the differential output voltage.
• The desired input impedance at the junction of R_T and R_G1 to match R_S.

Solving this quadratic for R_T starts the solution sequence, as shown in Equation 8:

Equation 8.

Being a quadratic, there are limits to the range of solutions. Specifically, after R_F and R_S are chosen, there is physically a maximum gain beyond which Equation 8 starts to solve for negative R_T values (if input matching is a requirement). With R_F selected, use Equation 9 to verify that the maximum gain is greater than the desired gain.

Equation 9.

If the achievable A_V(MAX) is less than desired, increase the R_F value. After R_T is derived from Equation 8, the R_G1 element is given by Equation 10:

Equation 10.

Then, the simplest approach is to use a single R_G2 = R_T || R_S + R_G1 on the nonsignal input side. Often, this approach is shown as the separate R_G1 and R_S elements. Using these separate elements provides a better divider match on the two feedback paths, but a single R_G2 is often acceptable. A direct solution for R_G2 is given as Equation 11:

Equation 11.

This design proceeds from a target input impedance matched to R_S, signal gain A_v from the matched input to the differential output voltage, and a selected R_F value. The nominal R_F value chosen for the THS4531A family characterization is 2 kΩ. As discussed previously, going lower improves noise and phase margin, but reduces the total output load impedance possibly degrading harmonic distortion. Going higher increases the output noise, and might reduce the loop-phase margin because of the feedback pole to the input capacitance, but reduces the total loading on the outputs.

Using Equation 9 to Equation 11 to sweep the target gain from 1 to A_V(MAX) < 10 V/V gives Table 9, which shows exact values for R_T, R_G1, and R_G2, where a 50-Ω source must be matched while setting the two feedback resistors to 2 kΩ. One possible solution for 1% standard values is shown, and the resulting actual input impedance and gain with % errors to the targets are also shown in Table 9.

These equations and design flow apply to any FDA. Using the feedback resistor value as a starting point is particularly useful for current-feedback-based FDAs such as the LMH6554, where the value of these feedback resistors determines the frequency response flatness. Similar tables can be built using the equations provided here for other source impedances, R_F values, and gain ranges.

The TINA model correctly shows this actively-set input impedance in the single-ended to differential configuration, and is a good tool to validate the gains, input impedances, response shapes, and noise issues.