# How much bandwidth does your op amp need?

When selecting an operational amplifier (op amp), you have many parameters to consider. DC parameters such as input offset voltage and input bias current are easier to account for because they do not vary with frequency.

When determining accuracy, however, or how close the output is to the expected value, it’s important to consider the analog input signal frequency. You must select an op amp with enough bandwidth to introduce minimal gain error across the range of frequencies it needs to amplify. And while you might be tempted to match an op amp’s bandwidth to the product of the frequency range of the analog signal source and the closed-loop gain, this method can lead to significant gain error at higher frequencies. So how do you select an op amp with sufficient gain bandwidth product to provide accurate amplification at your frequency of interest?

**How much bandwidth do you need?**

Let’s say that you have an application where the input is an 8-kHz 50-mVpp signal. The noninverting closed-loop gain is 90 V/V and your gain error requirement is 1%. Applying the 1% specification gives you an acceptable gain of 89.1 V/V. *Fig. 1* shows an example schematic of an op amp configured in a noninverting gain of 90 V/V.

Equation 1 calculates the minimum bandwidth required to amplify the input signal without exceeding the predetermined accuracy loss:

where f_{signal} is the input signal frequency (8 kHz in the example), G_{DC} is the noise gain of the circuit (90 V/V in the example) and G_{maximum frequency} is the noise gain of the circuit with the allowable error (89.1 V/V in the example).

Because Equation 1 is complex, Texas Instruments developed a calculator tool called the Gain Error vs. Bandwidth calculator, housed within the Analog Engineer’s Calculator. Download the tool, launch it, click the Plus button next to Amplifier and Comparators, and click Gain Error vs. Bandwidth.

Within the first tab of the calculator, which is called Find Gain Bandwidth, plug in these values:

- 8k (Hz) for signal frequency
- 90 (V/V) for closed loop gain
- 1 (%) for acceptable gain error

Clicking “Find Gain Bandwidth” will give you a minimum gain bandwidth product of about 5 MHz. If you select an op amp with about 5 MHz of bandwidth, you can expect to see 90 V/V at DC and 89.1 V/V at 8 kHz as shown in the frequency response plot in *Fig. 2*.

Remember, the equation and calculator provides the minimum bandwidth required to achieve the desired gain with maximum error (89.1 V/V) at your frequency of interest (8 kHz). It is a worst-case requirement for the circuit. A slightly higher-bandwidth op amp would certainly decrease the gain error, but a bandwidth that is too wide may result in higher power consumption and sometimes integrated noise.

**Does the ****A _{OL}**

**matter?**

A more thorough analysis of the example requires an understanding of loop gain and feedback.

When learning about open-loop gain (A_{OL}), it’s often easiest to break the parameter down into two subcategories: infinite and finite A_{OL}. In an ideal op amp, A_{OL} is infinite, meaning that there is no gain error at the output. In real-world op amps, A_{OL} is finite, which will introduce gain error when applying feedback. Going back to the example, you may notice that Equation 1 did not consider A_{OL}. That is because Equation 1 assumes that the A_{OL} is infinite, which is not possible.

Considering finite A_{OL} requires another calculation. To set the closed-loop gain, apply negative feedback – in this example, Ri and Rf, as shown in *Fig. 3*. You now have what is called a closed-loop gain (A_{CL}), expressed by Equation 2:

Again, this equation is complex, so the Analog Engineer’s Calculator has a second tab within the Gain Error vs. Bandwidth calculator called A_{OL} Error Considerations. The desired A_{CL} is 90 V/V, but you will see with finite A_{OL} that this value shifts.

Assume that you have selected a 5-MHz op amp with an A_{OL} of 90 dB. Plugging 90 dB into the second tab of the calculator yields a gain of 89.745 V/V at DC rather than the expected 90 V/V, a discrepancy that contributes to even more accuracy loss at the frequency of interest, as shown in *Fig. 4*.

The intended error of 1% increases to 1.28096%. If you had chosen an op amp like TI’s OPA2182, a 5-MHz high-precision op amp with a minimum A_{OL} of 140 dB, then the gain error remains more consistent with the predetermined acceptable error, as shown in *Fig. 5*. (You cannot see the red curve with finite A_{OL} because it’s so close in value to the white curve with infinite A_{OL}.)

**What about your resistors and capacitors?**

These analyses assume that the resistors and any capacitors in the circuit are perfectly matched. Unfortunately, that’s impossible in the real world. In production, resistors and capacitors are labeled according to their tolerance ratings. For example, a 1% 1-kΩ resistor can have up to a ±10-Ω error and still be within tolerance. This tolerance doesn’t include any temperature drift of the resistor or capacitor – and these errors can lead to significant gain-error degradation.

The third tab in the Gain Error vs. Bandwidth calculator, called Resistor Tolerance, takes resistor tolerance into account and displays a curve with worst-case gain error bands. Assuming 1% resistors, the worst-case performance would be that each resistor shifts 1% in the opposite direction, making them 2% off from each other.

The expected error of ±1% can be as low as -2.98% to 0.98%, depending on how these 1% resistors shift in production, as shown in *Fig. 6*.

**Conclusion**

What seems like a simple question – “how much bandwidth do you need?” – requires several iterations of calculations and an understanding of A_{OL}, feedback and component selection. Finding an op amp with the appropriate amount of bandwidth for accurate amplification can be challenging, so the Analog Engineer’s Calculator can help you determine the appropriate bandwidth for your application.

*The author would like to acknowledge Arthur Kay, the original creator of the Analog Engineer’s Calculator.*

**Additional resources:**

- Download the Analog Engineer’s Calculator to find the Gain Error vs. Bandwidth calculator discussed in this article.
- Learn more about op-amp bandwidth theory in the TI Precision Labs – Op amps: Bandwidth video series.

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