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.

Fig. 1: Noninverting op amp with gain of 90 V/V (Image: Texas Instruments)

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.

Fig. 2: Gain versus frequency plot with minimum gain bandwidth (Image: Texas Instruments)

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:

Fig. 3: Op amp with a closed-loop noninverting gain of 90 V/V (Image: Texas Instruments)

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.

Fig. 4: Gain versus frequency plot with finite A_OL (Image: Texas Instruments)

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.)

Fig. 5: Gain versus frequency plot with 140 dB of A_OL (Image: Texas Instruments)

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.

Fig. 6: Gain versus frequency plot with 1% resistors (Image: Texas Instruments)

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.