How op amps work and why you should use them, part 2

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

In part 1, I gave a simplified explanation of how an op amp, functioning as a servo amplifier, amplifies a small signal by comparing it to the attenuated signal from the amplifier’s output. I said that the amplifier acted to make the feedback equal to the input. The op-amp, when not configured with negative feedback, has extremely high voltage gain – perhaps 100 k-V/V or 1 M-V/V – so practically infinite. If the gain were infinite, the servo action (Figure 1) would act to make the voltages at the negative and positive inputs the same. The differential input stage would amplify the input voltage differentially (which we just defined as zero) and multiply that by an infinite gain, which would produce an unknown output because it’s mathematically undefined.

Figure 1. This is a simple single-supply circuit featuring AC coupling at both the input and output.

Voltage gain

Looking again at Figure 1, when you provide a small signal to the op amp’s noninverting input (R13-C6 node), amplify it by a very large number, then send the output (at +C7) through the attenuation network (R9 and R10) to the op amp’s inverting input (R9-R10 node), the servo action makes the voltage differential vanishingly small — but not zero. The closeness to zero is directly related to how close the voltage gain of a large V/V ratio approaches an infinite voltage gain. The distortion level is also related to this. A perfect amplifier would, of course, have no distortion.

I’ll push into this at the end of this article and subject you to the mathematical analysis. Still, I hope that this simplified and intuitive approach will establish a framework for understanding.

Input structure characteristics

Any op amp, fabricated with bipolar transistors, JFETs, MOSFETs, or even vacuum tubes, will exhibit less-than-ideal characteristics. Predominant among the shortcomings of the input stage are input offset voltage and input bias current. In Figure 2, this is a stripped-down version of the previous amplifier circuit. To simplify the input biasing circuitry, it’s powered from bipolar supplies. The output coupling capacitor and supply bypass capacitors are omitted for clarity.

Figure 2. To understand a real op amp, think of a perfect op amp with tiny amounts of input offset voltage at the differential inputs.

To symbolically illustrate what input offset voltage looks like, we can imagine tiny batteries connected in series with the inputs of a perfect op amp, as shown in Figure 2. Their terminal voltages might be on the order of a few microvolts to a few millivolts. Polarity could be as shown or could be reversed depending on the specifics of the op amp’s input structure.

For amplifiers used with AC signals, such as audio pre-amps, inductive tachometer pickups, vibration sensors, and RF, the offset can often be ignored. For amplifiers used with DC signals, such as thermocouples, photodetectors, electrometers, and electrochemical cells, use a low-offset-voltage op amp or add circuitry to adjust and nullify any offset voltage.

Figure 3. Think of a real op amp as a perfect op amp with tiny amounts of input bias current at the differential inputs.

To symbolically show what input bias current looks like, we can add tiny current sources at the op amp’s inputs, as shown in Figure 3. These current sources range from femtoamps to microamps. Their polarity may be as shown or reversed, depending on the input structure of the op amp.

This bias current will flow through the resistance present at the input of the op amp (so R6 for the noninverting input and R4//R5 for the inverting input). To the extent these are not matched, the bias current flowing and the subsequent I-R drop will effectively create an additional offset voltage. As mentioned above, it may be necessary to nullify this.

Bandwidth or frequency response

Having addressed the primary issues surrounding the op amp’s input, we now turn our attention to the bandwidth specification. Bandwidth of the op amp is generally expressed as its open-loop voltage gain – i.e., with no feedback added – up to the frequency where that gain drops to unity or 0.0 dB. The gain drops with increasing frequency due to the low-pass characteristic of the internal circuitry. There is usually one dominant pole created by an internal capacitor and associated drive resistance (that’s the low-pass part) that rolls off the frequency. As an example, have a look at a portion of the Texas Instruments quad op amp LF444 data sheet, reproduced here in Table 1. I’ve drawn a blue box around the spec for the typical open-loop gain of one section. The gain bandwidth product (GBW) is shown as 1 MHz (typical).

Table 1. The gain-bandwidth product for one section of the LF444. (Image: Texas Instruments)

Figure 4. This graph shows the open-loop gain vs. frequency for one section of the LF444. (Image: Texas Instruments)

Now, referring to Figure 4 from the same data sheet, we see that the gain vs. frequency shows 100 dB from a very low frequency (probably DC) up to 10 Hz. It then rolls off as a straight line and hits the 0 dB point at around 1 MHz. Therefore, the information shown in the typical graph aligns with the data in the table. Keep in mind that both the x-axis (frequency) and the y-axis (voltage gain) are logarithmic scales, so each tick mark on the x-axis is one decade of frequency (10, 100, 1 kHz, etc.) and each tick mark on the y-axis is 20 dB (20, 40, 60, etc.). The log scales make mathematical manipulation easier for some of the evaluations we need to perform: you can add or subtract gain values in decibels rather than multiplying gain values in V/V.

Figure 5. The op amp is configured for a gain of –R1/R2 or –10 V/V in this example. The minus sign indicates an inverting amplifier, where VOUT is measured with respect to VIN.

Note, just because the bandwidth of the op amp extends to 1 MHz doesn’t mean you can use this op amp to amplify signals up to 1 MHz. Well, you could, but you wouldn’t be happy with the results. To see why, we’ll dive back into the theory of operation of the op amp, considering closed-loop response compared with open-loop response.

If you want an audio pre-amp circuit using an op amp with a gain of 10 V/V (equivalent to +20 dB), it’s a very simple circuit. Referring to Figure 5, pick R1 = 10.0 kΩ and R2 = 1.00 kΩ.

Figure 6. The blue line indicates the op-amp circuit’s closed-loop gain.

To see what this circuit will do, we can superimpose our closed-loop gain over the open-loop gain graph. I’ve done this as shown in Figure 6 with the added blue line.

From V_IN to V_OUT, we get a gain of 20 dB out to 100 kHz. Then the closed-loop gain tracks the open-loop gain down to 1 MHz, where the gain becomes 0 dB (or a gain of 1 or unity). Looks good enough for audio, right? Nope. In actuality, this is quite undesirable. The gain margin (mathematical difference between open and closed-loop gains) at 10 kHz is only 20 dB, unlike at 10 Hz, where it is 80 dB. This means you will get distortion.

Figure 7. An open-loop system uses no feedback.

We’ll revisit the beginning of servo amp design to gain a better understanding. Rather than using the op amp symbol and specific resistors, as we did in Figures 1 and 6, we’ll draw blocks in our diagram and add the necessary values to represent gains, attenuations, and the point where signals are added together (comparable to what the op amp’s differential input structure does). So, the open-loop amplifier and the relevant gain equation (Equation 1) — called the transfer function — is shown in Figure 7.

The transfer function is simply another way to describe the relationship between the output and the input.

Figure 8. A closed-loop system feeds back a portion of its output signal to its inputs.

We aim for a closed-loop system, similar to the one depicted in Figures 1 and 6. We can use a more generic representation of the feedback resistors by just drawing a block and calling it β. We can represent the differential input as a circle with an X inside it; this is the generic symbol for a summing junction. Referring to Figure 6, the summing junction is the node where R1 and R2 are connected to the op amp’s negative input. The closed-loop servo system is shown in Figure 8.

Some of the terms used are self-explanatory, while others are not. Here are all the terms and their meanings:

V_IN is the input voltage.
V_OUT is the output voltage.
A_OL is the open-loop gain of the amplifier before any feedback is added.
β is the attenuation factor of the β box (the feedback network).
V_OUT β is the output from the β box.
V_IN – V_OUT β is the output from the summing junction.

So far, the attenuation (β) has been expressed as the ratio of two resistors. Keep in mind that the feedback network can be more complex and complicated than just two resistors: Diodes, capacitors, and inductors are used, and even a second op amp may be present in the feedback loop. In some published articles, the term V_IN – V_OUT • β is referred to as the error voltage (E) or the summation voltage (Σ).

This op amp has a finite (but very large) open-loop gain, and the Σ voltage is very small. Thus, the very small Σ multiplied (gained up) by the very large A_OL gives us the V_OUT at precisely the voltage we expected, mostly. With higher frequency signals (e.g., 10 kHz), A_OL is significantly smaller than at DC or low frequencies. This means the output no longer follows the input (albeit gained up) but instead won’t be quite what we expected (i.e., distorted).

To understand exactly why this happens, we’ll need to examine the mathematical analysis further. We’ll save that for my next article on op amps.