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A designer's guide to op-amp gain error

Selecting an op amp for an amplification circuit is a task that is familiar to analogue designers. For any given application, you first have to determine the amplifier's closedloop gain and 3-dB frequency. Op amps that are too fast for the application tend to introduce drawbacks in terms of current consumption, offset error, component cost or stability issues, which can lead to layout restrictions or even an additional development loop. Consequently, we always try to select an op amp with a gainbandwidth product appropriate to our needs.

Conventionally we calculate the gain-bandwidth product GBP required for the given circuit using the 3-dB frequency f_3dB and the feedback factor k_f, representing the portion of the op amp output voltage returned to the differential input of the op amp.

The two resistors of the feedback circuitry determine the feedback factor.

The feedback factor is negative due to the connection of the resistors to the negative op amp input. Its value is equal for the inverting and the non-inverting amplification circuit in Figure 1.

We also calculate the GBP using the closed-loop gain A_ideal- and A_ideal+ in these two examples.

The commonly used simplified expression GBP ≅ f_3dB·A is exact for a non-inverting amplifier but it is also a valid approximation for high-gain inverting amplifier circuits.

But we might still not be satisfied with the result. In fact, from a gainaccuracy point of view, the gain error at the cutoff frequency f_3dB reaches almost 30%, which is an unacceptably high value in most applications. Instead, you must design an op amp circuit based on an effective bandwidth BW_α, where the gain error is less than, or equal to, a specified value α that you set by reference to eac h specific application.

Designers often use the following simple equation (Reference 1) to determine the gain error [α] at a given frequency BW_α]

The previous equation can be used as a rough worst-case estimation of the gain error. But the error calculation at BW= 0.1 f_3dB yields a reduction of 9% instead of 0.5%; thus it is far from being an accurate method of error calculation. Instead, you also need to take into account the op-amp's complex frequency- dependent gain including the its dc open-loop gain A_0DC:

Equation 6 gives the closed-loop gain (Reference 2).

The following equation gives the relative gain error α.

As a consequence of high-frequency operation, not only is the gain of the signal reduced but the signal is also affected by a phase-shift. However, we will concentrate on the gain error only. Substitution of (5) and (6) in (7) results in a relative gain error of

For small gain errors, equation 8 can be simplified to

A graphical representation of the exact value and of the approximations of the gain error is shown in Figure 2. The calculation has been done for an absolute value of the gain error at dc of 1% and a cut-off frequency f_3dB of 1 MHz. The rule of thumb from equation 4 is acceptable only if errors of 30% are tolerable; for most applications this is not the case, and that rule of thumb is not a suitable approximation. The approximation from equation 9 delivers good values for small errors but yields values that are too high for frequencies close to f_3dB. We will look at a solution using the latter approximation first, before taking into account the influence of higher error values.

There exist several combinations of GBP and A_0DC values that lead to the same error α. To choose an appropriate op amp we choose the (GBP₀, A_{0DC 0}) combination, where at the given frequency BW_α the dc-error and the frequency- dependent part of equation 9 add the same portion, α/2 to the total gain error α.

There is of course the possibility of increasing GBP₀ obtain a lower value of A_{0DC 0}, or vice versa. But even if one of the values is greatly increased, the error that the other introduces may never exceed the total gain error α. The minimum values of A_0DC for an infinite GBP and of a GBP for infinite A_0DC are

We can use equations 10 and 11 of A_{0DC 0} and GBP₀ in equation 8 to obtain an expression for the accurate gain error as a function of A_0DC/A_{0DC 0} and GBP/GBP₀. Solving this equation for GBP yields

where |α| is the total gain error and A_0DC/A_{0DC 0} is the factor by which the dc open-loop gain increases or decreases. The resulting value of the GBP includes the change due to the increased or decreased A_0DC but also eliminates the calculation error that the approximation used in equation 9 yields.

Equation 13 is depicted in Figure 3. Because of the approximation that lead to the calculation of GBP₀ and A_{0DC 0}, the (0dB, 1)-point is only part of the |α|≤1% curve. For higher values of gain error |α|, you can adjust the values of GBP or A_0DC/ using Figure 3, or directly from equation 13. As a practical example of how to use these results, we will go through the calculation of an example step by step:

Let us assume an op-amp circuit with a gain of +500 or -500 where the total gain error should not exceed |α|≤10% for frequencies up to BW_α =100 kHz. To select an appropriate op amp fulfilling these requirements we have to calculate A_{0DC 0} and GBP₀, first using equations 10 and 11 with kf ≈ -|gain|^-1 =-0.002:

This is already a good approximation. But the -10% curve of Figure 3 shows that we could still reduce the GBP by 10%. Therefore, we have achieved the performance that we require using an op amp with a dc openloop gain of 80 dB and a gain-bandwidth product greater than 140 MHz.

Finding an op amp with an openloop gain of 80 dB might be easy whereas the gain-bandwidth product could be the more critical parameter. If we increase the op amp's open-loop gain by a factor of three (+10 dB) to decrease the value of the bandwidth product, we end up with another set of values using, once more, the -10% curve in Figure 3:

Of course the calculated values represent the minimum that you can use for this particular application. Keep in mind that the GBP changes by up to 30% from product to product and over temperature, and that the open-loop-gain variation from the typical value is in the order of ±3 dB for precise op amps but may reach up to ±20 dB. We should search for op amps with a typical GBP of at least 160 MHz.

Once these values are stable, you can start fixing other important opamp characteristics. Keep in mind, in particular, that the amplifiers' slew rate may also affect the gain error.

REFERENCES

Bonnie Baker: "A designer's guide to op-amp gain error", EDN Europe, October 2009, pg 35, www.edn-europe.com/article.asp?articleid=3461. Sergio Franco: "Design with operational amplifiers and analog integrated circuits", McGraw-Hill, 1998.

Author Information

Jochen Zwick has been working as developmentengineer at automotive supplierContinental in Germany since 2002. Heholds master of science degrees in electricalengineering from the Universität Karlsruhe(TH), Germany and the École NationaleSupérieure d'Électronique et de Radioélectricitéde Grenoble (ENSERG), France.Since 2008 he lectures in electronics as associateprofessor at the Duale HochschuleBaden-Württemberg (DHBW, formerBerufsakademie) Ravensburg, Germany.