A designer's guide to op-amp gain error
When selecting the proper operational amplifier for your circuit, you need to know how much gain error your design can tolerate.
By Bonnie Baker -- EDN, September 17, 2009
As you sit down to select the proper operational amplifier for your circuit, the first order of business is to determine the signal bandwidth that your system will send through that amplifier. Once you settle on this parameter, you can start to look for the right amplifier. The high-speed-op-amp gurus warn that you should avoid using analog devices that are too fast for your application. So you try to pick an amplifier with a closed-loop bandwidth just a little higher than the maximum frequency of your signal.
This strategy may sound like a good product-selection recipe, but it will probably bring disaster to your application board. In the lab, you may find that, when you put an input-sine-wave signal at the application's maximum frequency into your system, the output signal from your amplifier does not go across the expected full-scale analog range. The gain on the signal is much less than you would expect. If the slew-rate magnitude of your amplifier is more than adequate and you are not driving the amplifier output into the power-supply rails, then what has gone wrong?
Stop double-checking your resistor values! When designing an amplifier into a gain cell, you must know your signal's maximum bandwidth, the amplifier's closed-loop noise gain, the amplifier's gain-bandwidth product, and how much gain error your design can tolerate. The closed-loop noise gain is the amplifier's gain, as if a small voltage source were in series with the op amp's noninverting input.
You can work this problem through by example. For instance, start with a signal bandwidth of 1 MHz. The amplifier's circuit noise gain in Figure 1 is 10V/V. Figure 1 also shows the open-loop frequency response of an amplifier that has just enough bandwidth for this circuit—or so you think. The amplifier has a 16-MHz gain-bandwidth product. The op amp looks as though it can support a gain of 10V/V, or 20 dB, out to 1 MHz, but look a little closer. The gain of the open-loop gain curve at the signal's bandwidth is
where AOL is the open-loop gain of the amplifier, SBW is the signal bandwidth, and GBWP is the gain-bandwidth product.
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In this case, the amplifier's open-loop gain, AOL-SBW, is 16V/V at 1 MHz. But here's the kicker: The closed-loop gain error in this circuit is NG/(AOL-SBW+NG), where NG is the noise gain. The closed-loop gain error at 1 MHz in this example is 0.385, or a gain error of 38.5%.
For this circuit, if you are willing to tolerate a gain error of 0.05 from your amplifier and you understand that the GBWP of an amplifier can change a maximum of 30% from product to product and over temperature, you need an amplifier that has a GBWP greater than 246 MHz. The guiding formula is
where GBWPOPA is the op amp's gain-bandwidth product.
Use this formula during your first pass when you choose an amplifier for your circuit. After you determine the amplifier's bandwidth, you can start to delve into the other important amplifier characteristics for your application.
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Looking at the data sheets, there is one more parameter which needs to be looked into while selecting op-amps for high frequencies. This parameter is shown as a graph as the Peak-to-peak undistorted output [Vo(pp)] vs. frequency.
As an example, TLC072 is a 10MHz wide bandwidth op-amp, whose differential votage gain at 3MHz is shown as +10dB.
However, Vo(pp) at 3MHz is shown as 2.0V (Vdd of 5V and RL of 10K).
Combining the two information above, whatever be the input signal (above 4nV), at 3MHz, maximum output available from the op-amp will be limited to 2.0V(pp), of course, undistorted.
Another example, LT1797, a 10MHz Rail-to-Rail Output Op-amp; voltage gain at 3MHz is shown as +5dB. However, Vo(pp) at 1MHz is shown as 0.5V and no information is given for higher frequencies. Voltage gain at 1MHz is shown as +15dB.
Kunal Ghosh - 2009-12-10 21:57:00 PDT -
Hi Bonnie:
I think your treatment of this topic is accurate for DC, but not for AC. The fact is that the AC gain error of opamps is dominated not by the magnitude of their gain, but by the phase of their gain.
The gain error is thrown out of phase with the signal. That is why opamps are more accurate in magnitude than their gain implies. That is also why at higher frequencies still, opamps can actually peak before they roll off.
Any equations relating AC open loop gain to closed loop gain must include the phase(s). The phase is not included in the GBW product spec.
Glen Brisebois - 2009-29-9 13:47:00 PDT -
Suppose, from knowing the transfer function well, you basically know the schematic of an active anti-alias filter. Use can use TI FilterPro for this. It works well. As an example, suppose one wants a 5-pole, 100kHz Butterworth LPF. In this more difficult case, is there a way to determine the Aol of the three op-amps required in order to make the theoretical cutoff spec, say at 1 MHz, that is not trial and error? I believe quite a fast opamp is required, even for this moderate cutoff frequency.
Alexander Jacobson - 2009-29-9 13:11:00 PDT -
Unfortunately, the conclusion in the column is wrong! Ms. Baker's analysis assumes there is no phase shift of the open loop gain. In fact, the open loop gain at 1MHz is close to 90 degrees. Taking this phase shift into account, the actual closed loop gain is 8.48 for an error of 15.2%, not the 38.5% error as claimed.
Stuart R. Michaels - 2009-26-9 11:50:00 PDT -
Bonnie
Interesting article and useful guide.
However my experience is that over the past 10 years or so, Op amps from quality suppliers such as TI have become so accurate and repeatable that figures like 30% are way out of line, more like 2% drift between units.
If you use a single source then you can usually deign from the spec sheets and be confident that 95% of ubits will work as designed.
However 'belt and braces' design is always good for critical apps.
Ian Proffitt - 2009-18-9 00:51:00 PDT
