# Seemingly Simple Circuits, Part 3: Effect of Resistor Tolerances on Diff-Amp Gains

Resistor tolerances will cause amplification to depart from the ideal as derived in part 1 of this series. This is of significance because, in many seemingly undemanding designs, 1% resistors are not adequate, usually because of excessive CM gain, which ideally is ACM = 0. If a resistor of value R is used in the amplifier and has a tolerance of +/- ε, its range of values is R •(1 +/- &#949)

Resistors in series have a resistance of

For equal-tolerance resistors, &#9491 = &#9492 = &#949. Then, for the worst case, their errors have the same sign, and

The tolerance of resistors in series is the same as the resistor tolerance.

For resistors in parallel,

Again, the combination of resistors, in series or parallel, results in a combined tolerance equal to the resistor tolerance.

For a divider, the result is not as beneficial. A ratio of resistors of equal tolerance has a worst-case error when one has +&#949 and the other has −ε

Thus, the tolerance range for resistor ratios is +/-2∙ε, ε << 1.

These resistor error formulas can be applied to the one-op-amp diff-amp. The worst case (maximum error) occurs for maximum mismatch in the dividers T i + and T i . Defining the ideal diff-amp voltage gain as

and for which ε << 1, Av0 >> 1. Thus, for small ε and Av >> 1, the error with resistor tolerance of ε in the differential gain is about +/-2•ε. The case for +2•ε is derived by exchanging 1 + 2•ε and 1 – 2•ε for the resistance ratios in the above derivation.

The CM gain is

Applying error tolerances,

Applying these derivations to a design example, the requirements are that Av = 1, and that the CM input voltage range ≥ 30 V. The differential input range is ±2.5 V, input resistance is 2 MΩ, and VOS = 2.5 V, half the ADC reference voltage.

The value of Av0 = Rf /Ri results in Rf = Ri . To make input resistance meet spec, let Rf = Ri = 1.00 MΩ +/-1%. Then, for the CM range requirement using a rail-rail-input (RRI) op-amp,

Solving for ROS+ , ROS+ ≤ 111.1 kΩ. Let ROS+ = 100 kΩ, +/-1%. Then the CM range is 32.5 V > 30 V. Resistor mismatch causes a worst-case gain error of about

The maximum output voltage error of the diff-amp is thus about (+/-2.5 V)•(0.01) = +/-25 mV. For the CM gain,

Over a 30 V CM range, the maximum output voltage error is about 30 V/50 = 600 mV. For a 10-bit A/D converter with a 5V input range and 1,000 counts of range (after 24 counts are allotted for calibration), this is 120 counts of error, or about 12% of the input range. The CM error dominates. It usually does unless low-tolerance or closely matched resistors are used.

## 12 comments on “Seemingly Simple Circuits, Part 3: Effect of Resistor Tolerances on Diff-Amp Gains”

1. Teno
October 29, 2014

Very nicely documented post. I didn't go through all the math but it looks correct. CMRR errors are non-linear in nature and a circuit that seems to work well within tolerances at lower common mode input levels rapidy goes out of spec when the common mode voltage increases. Best to make sure the circuit is tested at the highest common mode input voltage.

Precision resistors can be costly and matching can be very time consuming and requires a good stock of resitors. POTs are unreliable and electronic pots do not normally have the required dynamic range for most instrumentation. I posted a circuit on the element 14 Community (search for Low Cost DAC Boosts PSoC Amp Performance by Over 30dB) that uses a very low cost serial DAC and inexpensive .1% resistors to get excellent CMRR.

Another concern is temperature mismatches in the circuit. The main sources are, self heating in the resistors, temperature coefficient differences between the resistors and localized board heating. These issues are best handled by a) placing resitors very close together, b) keep the resistors on a location that has a fairly stable temperater, e.g. away from the current sensing resistor or other hot components and c) using low tempco resistors.

2. JamesBryant
October 30, 2014

I am pleased to see these detailed analyses of precision amplifier circuits, but sorry that, unless I have overlooked something, the obvious conclusion is not made explicit: wherever possible subtractors (one op-amp diff amps) and instrumentation amplifiers (which are in effect subtractors with input buffers) should be monolithic chips with integrated thin-film resistors – these can be matched in resistance to a few ppm, temperature coefficient (TC) to well under a ppm, and actual temperature difference between the various resistors (which are, of course on the same chip) to well under 0.1 degrees C. Monolithic subtractors and instrumentation amplifiers, if available to the required specification, almost always outperform, and are usually cheaper than, ones built with discrete components.

3. D Feucht
October 30, 2014

As for the performance of monolithic over semidiscrete implementations, your point is indisputable. Whether an engineer is designing a semidiscrete circuit or a monlithic integrated circuit, these concepts pertain to both and with them it is possible to more clearly see how (and why) monolithic diff-amps can be superior in performance to semidiscrete diff-amps.

What amazes me about this seemingly simple one-op-amp diff-amp is that it took until August 1991 before the given design formulas were published (with a different derivation of equations) by the circuits-intensive Catalonian, Ramon Pallas-Areny (accent marks omitted); I cite the IEEE Transaction in Part 4 of this series. This simple circuit had by 1991 been around for decades. Why did it take so long for these basic derivations to be published? I know not why; perhaps everyone kept them as trade secrets for a long while.

4. yalanand
October 31, 2014

It usually does unless low-tolerance or closely matched resistors are used.

@Dennis, thanks for the post. I think this is the reason why Analog layout is very challenging task. Matching the resistors in Analog layouts is tricky.

5. yalanand
October 31, 2014

The main sources are, self heating in the resistors, temperature coefficient differences between the resistors and localized board heating.

@Teno, how to make sure that the resistors that is being used are having same temperature coefficient ? How to select best matched resistors ?

6. geek
October 31, 2014

“Another concern is temperature mismatches in the circuit. The main sources are, self heating in the resistors, temperature coefficient differences between the resistors and localized board heating”

@Teno: Shouldn't that depend on the application itself? You may have applications where the circuit is bound to be exposed to high temperatures. What do you do about those? Are there special type of resistors that you should use? Or should you rather look for ways to keep the overall circuit cool despite the heating in the environment?

7. amrutah
October 31, 2014

“Low Cost DAC Boosts PSoC Amp Performance by Over 30dB”

@Teno: Thanks for sharing this, very nice to read.

8. amrutah
October 31, 2014

@Yalanand:  Before Teno replies, here is what I think about the matching

If its discrete design, then better place all the resistors away from the high temperature zones, tightly pack the resistors together and if they are high watt resistors, use heat sinks.

If the resistors are integrated on a chip then,

• best is to split the resistors into fingers and interdigitate the fingers, so that the matching is achieved.
• Use of zero tempco resistors.
• the integrated resistors also suffer nwell, voltage coefficient which needs to be accounted for.
9. vasanjk
October 31, 2014

amrutah

Well said. I would like to add that choosing metal film resistors over carbon film types can help immensely.

10. Teno
October 31, 2014

I was just responding to the post and not intending to write a a complete solution. You are right though, it does depend on the final design implimentation. There are a number of tradeoffs. Ideally the divider network would be close to the inputs of the amplifier but what if you are dealing with high voltage? Do you want to have high voltage bussed all across your board? Can you afford to buffer the divider near the source? Keeping the resisotrs close together usually keeps a low temperature gradient between the resisotrs. My suggestions don't preclude a good thermal and low noise design. I've worked in aerospace and that brings about a whole new set of design challenges. The proof used in the original post can be used to calculate thermal effects too. As suggested, there is no one single solution. Good judgement, awareness of the potential problems and what tradeoffs are available must always be used in a good design. CMRR errors can be a hidden trap if one is not aware of them. All I expected anyone to take away from my comment is some of the issues and solutions related to CMRR errors.

11. D Feucht
October 31, 2014

Teno & others,

Thanks for your comments on the topic. I did not realize that it would push an enginerring hot-button!

Having agreed that monolithic integration is superior to semidiscrete design, let me defend the precision-resistor manufacturers for a moment.

Silicon substrates have high thermal conductivity, and with such short distances, everything on the chip is thermally connected. Precision metal resistors integrated on a ceramic substrate (with interdigitation and all the other applicable techniques applied) are thermally connected through ceramic (which, as aluminum oxide, also has high thermal conductivity) but are isolated from the heat-producing output-stage transistors. If the R array is shielded from air currents and kept close to the active-device circuits, rather impressive precision can be attained, and without the voltage-constant nonlinearity of on-chip diffusion resistors. Of course, nichrome on-chip resistors are the preferred choice for serious analog.

So there is still something to be said for semidiscrete design, especially if through-hole and not surface-mount resistors are used. Through-hole Rs have lower voltage constants.

12. dassa.an
October 31, 2014

@D Fecuht: Well that's some news to me. I was kind of stuck at that point. Let me try as you said here.

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