How to size/calculate a resistor network where I can change the gain by 1dB with each switched in resistor? [closed]

This is wildly guessing from your description and the really hard to interpret pseudo-schematics you gave us, but making assumptions that don't work well with the self-contradicting parts of your comments (i.e., especially ignoring that you said the amplifier is impedance-matched, and ignoring that you have an idealized amplifier in your schematic, but think that the load matters here, which would indicated that you actually do need impedance matching, but if you need that, your whole approach makes no sense at all and can be discarded without further considerations):

• The voltage gain of your amplifier is defined by the ratio of input resistors to feedback resistors, let's call it \$A = \frac{R_i}{\sum\limits_{n\text{. resistor enabled}}R_n} \$.
• by definition, a gain of 1dB is the ratio defined by \$10^{\frac{1}{20}} =: \delta\$, putting that into a calculator will tell you \$\delta \approx 1.122\$.
• Hence, adding a gain of 1 dB means increasing \$A\$ by a factor of \$\delta\$.
• Assuming you mean to switch the feedback resistors: each switch must reduce the sum of enabled resistors by a factor of \$\delta\$.
• therefore, every resistor must add \$\delta-1\$ in resistance to the sum of all previous resistors

The rest is you picking a first resistor; the second is trivially computed from the first (bold rule in above), the third from the sum of the first and the second, and so on.

Do I think you're on a sensible way to building a programmable gain amplifier? No, I'm afraid not. There's a couple misconceptions your question leaves us guessing at.

And: in your specific case, \$\delta-1\$ is only \$\approx 0.122\$. 12.2%: you will have a non-trivial time finding resistors in that raster, so you need to combine multiple resistors. Because combining resistors adds sums the variance in their tolerances, you pretty quickly end up in a situation where you need to hand-tune your resistive ladder. Then, with resistors of different magnitudes in series, some dissipate much more power than others – and resistors have thermal coefficients. Suddenly you're starting to come up with solutions to thermally couple large numbers of resistors.

Hence, yours is not really a good architecture to realize in discrete electronics. Generally, modern multiplying DACs (hint: look that up on wikipedia!) are not build like that.

There's a host of other ways to achieve adjustable gains. But you will need to ask a new question that actually describes the signal (frequency, amplitudes, powers, fidelity & noise) and the application. Your approach so far is not sufficient to actually help you.

It sounds like you want to create, not a resistor network, but a variable resistor, to use as a component in your fedback amplifier. The variable resistor should be able to be set to several values in a dB ratio.

Given the amplifier topology, any consistency of impedance around the inputs and outputs is moot, the fedback amplifier has a zero output impedance. As long as it has enough current output capability, it will drive the load.

With the topology you have illustrated, the gain is controlled linearly with the value of the feedback resistor. The input resistor sets a current, the feedback resistor gets a voltage developed across itself equal to IR.

All it takes to choose the values is to get your calculator or favourite spreadsheet, and calculate Rn = R0 * 10^(n/20) for n dBs, the inverse of the dB function. Then it's just a question of subtraction to find the incremental values that you switch.

I think 1dB steps would be too small for audio, 2dB is quite fine enough. A good approximation to 2dB steps can be had from the Renard R10 series: 1.0, 1.25, 1.6, 2.0, 2.5, 3.2, 4.0, 5.0, 6.3, 8.0, 10.0.¹ This can be found cropping up everywhere, the most visible to an engineer might be that its alternate terms (the R5 series) are used for the standard voltages for capacitors.

Place this variable resistor in the feedback network of your amplifier wherever it's needed.

^{simulate this circuit – Schematic created using CircuitLab}

I've normalised the values to 100 Ω, of course scale them as required.

I've only drawn 9 steps, 100 to 800 Ω, but I'm sure you can extend the pattern.

You'll notice the resistors I've used are not in a nice progression. This is as a result of taking the small differences between two, larger, approximated values. This nicely illustrates the error magnification feature of taking small differences like this. If you use exact values for the total required resistances instead of the R10 values, then the differences will progress nicely.

If you want more accuracy, then use the formula above rather than the approximate 'nice numbers' R10 series to base the values on. If you do really want 1dB steps, then use the formula above for 1dB steps, or use the R20 series.

If this is simply a volume control for audio, then extreme accuracy on the steps is not really required. Choose the nearest E12 or E24 value resistors for the string of resistors and compute the errors, you'll find they are quite small.

Do bear in mind that (apart from the 1 or 2 dB step size) I've answered the question as you have asked it. I am not endorsing most of the other elements of your question. Your amplifiers are not impedance matched. This is probably not the best way of varying gain in an amplifier. I don't know whether your application is best served by a differential amplifier: note that its input resistance is not symmetrical, and it uses twice as many gain control elements as a single-ended amplifier.

¹ I'm trying to learn to do logs in my head, to at least 1.5 places. Get the first place from internalizing that series, get the second from linear interpolation between them. Handy for approximating roots and powers, as well as multiplication and division, if you're lying awake at night.