How op amps work and why you should use them: part 3

Operational amplifiers (op amps) are almost the perfect amplifier. As long as you keep a few important details in mind, their imperfections will seem invisible.

In part 1, Equations 2 and 3 used the term A_V for the voltage gain of the simple noninverting amplifier shown in Figure 1. The equations use α for the attenuation factor of the feedback network. The feedback network is a simple voltage divider, and α is commonly used for the attenuation factor of a voltage divider network. Here is that figure duplicated with the feedback network redrawn to make the voltage divider aspect more obvious.

Non-inverting op amp — Figure 1. This simple non-inverting op amp amplifier is powered from bipolar supplies.

We are moving more into the servo amplifier analysis that we started in part 2, so let’s change some terms to avoid any confusion. I will use AVOL for open-loop gain and AVCL for closed-loop gain. Also, I will use β for the feedback factor rather than α. Using β implies a feedback network that may be far more complex than a simple two-resistor network.

In the simple two-resistor feedback network, the feedback network’s factor (now called out as β instead of α) is expressed as:

The right side of this equation should look familiar as the voltage divider formula.

A_VCL for a noninverting amplifier expressed as:

Combining these two equations gives us this equation:

This says that the closed-loop gain is the reciprocal of the feedback factor. Incidentally, in some literature, you will see the 1/β term called noise gain. Don’t be concerned that we are suddenly expecting the amplifier circuitry to be noisy. It’s just a shorthand way to describe one of the terms commonly used. If you want to calculate the output voltage (V_OUT) with respect to the input voltage (V_IN), you simply multiply V_IN by A_VCL. Alternatively, you could write a simple transfer function as:

In part 2, Figure 9 (Equation 2), the output voltage (V_OUT) equation in terms of the input voltage (V_IN), the feedback factor (β), and the open-loop gain (A_VOL) is repeated here using the revised term for open-loop gain:

Here I mathematically combined the feedback and input resistors as a black box that simply attenuated the output voltage with a factor of unity or smaller and labeled it β. You can do some algebraic manipulation of Equation 4 in three steps to get the often-seen gain equation (output voltage divided by input voltage) as follows:

Now this looks more complicated than Equation 4. But fear not. I wrote the term in the square brackets the way I did to help clarify what happens as A_VOL degrades. If you do a little mental math and assume that A_VOL is a very large number, the right side of the equation becomes [a very large number] divided by [that same very large number plus one more] times the reciprocal of β. A very large number divided by that same very large number plus one more is almost exactly one; 1 times the reciprocal of β is the reciprocal of β. Thus, if A_VOL is very large, Equation 6c is pretty much the same as the combination of Equations 3 and 4. Put some numbers in and see for yourself. If you used A_VOL of 10⁶, the multiplier to the β term would be 0.999999 instead of 1. For most engineering work, that’s close enough.

Perhaps now you can see where this is going — we’re getting to the heart of the problem. At the end of part 2, I left you with this concern: What happens at higher frequencies? Why is the output no longer simply a gain-up version of the input? The answer is that at higher frequencies, A_VOL is no longer a very large number. If A_VOL is only 100 V/V instead of one million, you get 0.990099 for the multiplier to β. If A_VOL were 10 V/V, you would get 0.9090909 for the multiplier to β.

That means at higher frequencies, the 1/β term gets smaller and by extension, the op amp’s output gets smaller. This results in the high-frequency content being rolled off and the transient response being degraded. As a practical example, if you want to design a pre-amp for a microphone used for music, make sure the chosen op amp has plenty of open-loop gain and bandwidth.

While we are on the topic of microphone pre-amps and similar circuits, let’s consider a few more important details to keep in mind when designing op amp circuits for low-level signals:

- For mic pre-amps, make sure your selected device is classified as a low-noise op amp.
- For sensor pre-amps used with (for example) pH sensors, thermocouples, and photodetectors, you need low noise, low-drift op amps. Use devices with very low to ultra-low bias current and offset voltage specs. In addition, the photodetector circuits usually need high bandwidth op amps. Photodetectors are used in high bandwidth communications apps and rapid rise time pulse amplifiers/wave shapers. More on these apps another time. Study the datasheet carefully.
- The input bias current and input offset voltage specs are not especially important in audio circuits – they are usually AC coupled, so a little DC offset at the output won’t have any bad effects.

Related to the frequency response described above, you also need to consider the phase response of the op amp being used. As an example, consider the OPAx863A from Texas Instruments. If we look at the datasheet Figure 7-50 (Figure 2), we see the open-loop frequency response (somewhat similar to what we saw for the LF444 in part 2 of this series) and an additional curve for the phase response (in red).

op amp open-loop — Figure 2. As frequency increases, an op amp’s open-loop bandwidth vs. frequency drops, and the phase shifts. Image: Texas Instruments

Compared to the LF444, this op amp can amplify signals accurately at much higher frequencies. At those higher frequencies, the phase relationship (output signal compared to input signal) changes significantly. Measured from the op amp’s inverting input to the output, at very low frequencies (e.g., 1 Hz), the output shows about a 180° phase shift, just like you’d expect. As the frequency of the applied signal increases, the phase lag increases. It stabilizes for a few decades of frequency at 90° and then lags some more. At 100 MHz, it shows a 0° phase shift — the op amp’s inverting input will now act as a noninverting input. Without proper circuit elements added around the op amp (output to input and/or across the inputs), this op amp would become a high-frequency oscillator.

For a more detailed analysis, take a look at the Analog Devices MT-033 tutorial titled Voltage Feedback Op Amp Gain and Bandwidth or the Texas Instruments app-note sboa15 titled Feedback Plots Define Op Amp AC Performance.

Additional things to keep in mind

When op amp circuits were first implemented, they were often powered from ±15 VDC supplies. Output voltage ranges could typically swing within a few volts of the positive and negative supply rail before clipping occurred. The Input voltage range was usually similar. Exceeding those limits would cause clipping or input phase reversal. As above, this is where the op amp’s noninverting input acts like the inverting input and the inverting input acts like the noninverting input. As you might guess, wild oscillation could occur, or the output might just latch high or low.

Now, it’s much more common to see op amps powered from +5 VDC, +3.3 VDC, or even +1.8 VDC. When your circuits are powered from such low voltages, the op amps need to accept input voltages and produce output voltages that are within millivolts of ground and the positive supply rail with very low distortion (typically manifested as clipping). These op amps will be marketed as rail-to-rail, input/output, abbreviated as RRIO.

Study the datasheet carefully to see exactly how close the input and output can get to the supply rails before soft or hard clipping (distortion) occurs. Overlooking this detail will result in circuits that perform poorly or not at all.