FEEDBACK AND OPERATIONAL AMPLIFIERS (Some selected section from CHAPTER 3 of the book "The Art of Electronics", written by Paul Horowitz and Winfield Hill, Cambridge University Press , 1985.)

	BASIC OP-AMP CIRCUITS
	The golden rules for working out op-amp behavior
	Departures from ideal op-amp performance
	Integrators
	COMPARATORS AND SCHMITT TRIGGER
	FEEDBACK WITH AMPLIFIERS
	FEEDBACK AMPLIFIER FREQUENCY COMPENSATION

INTRODUCTION TO FEEDBACK AND OPERATIONAL AMPLIFIERS

Feedback has become such a well-known concept that the word has entered the general vocabulary. In control systems, feedback consists in comparing the output of the system with the desired output and making a correction accordingly. The "system" can be almost anything: for instance, the process of driving a car down the road, in which the output (the position and velocity of the car) is sensed by the driver, who compares it with expectations and makes corrections to the input (steering wheel, throttle, brake). In amplifier circuits the output should be a multiple of the input, so in a feedback amplifier the input is compared with an attenuated version of the output.
 
 

3.01 Introduction to feedback

Negative feedback is the process of coupling the output back in such a way as to cancel some of the input. You might think that this would only have the effect of reducing the amplifier's gain and would be a pretty stupid thing to do. True, it does lower the gain, but in exchange it also improves other characteristics, most notably freedom from distortion and nonlinearity, flatness of response (or conformity to some desired frequency response), and predictability. In fact, as more negative feedback is used, the resultant amplifier characteristics become less dependent on the characteristics of the open-loop (no-feedback) amplifier and finally depend only on the properties of the feedback network itself. Operational amplifiers are typically used in this high-loop-gain limit, with open-loop voltage gain (no feedback) of a million or so.

A feedback network can be frequency dependent, to produce an equalization amplifier (with specific gain-versus-frequency characteristics,), or it can be amplitude-dependent, producing a nonlinear amplifier (a popular example is a logarithmic amplifier). It can be arranged to produce a current source (near-infinite output impedance) or a voltage source (nearzero output impedance), and it can be connected to generate very high or very low input impedance. Speaking in general terms, the property that is sampled to produce feedback is the property that is improved. Thus, if you feed back a signal proportional to the output current, you will generate a good current source.

Feedback can also be positive; that's how you make an oscillator, for instance. As much fun as that may sound, it simply isn't as important as negative feedback. More often it's a nuisance, since a negative feedback circuit may have large enough phase shifts at some high frequency to produce positive feedback and oscillations. It is surprisingly easy to have this happen, and the prevention of unwanted oscillations is the object of what is called compensation.

Having made these general comments, we will now look at a few feedback examples with operational amplifiers.

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3.02 Operational amplifiers

Most of our work with feedback will involve operational amplifiers, very high gain dc-coupled differential amplifiers with singleended outputs. [You can think of the classic differential amplifiers with its two inputs and single output as a prototype, although real op-amps have much higher gain (typically 105 to 106) and lower output impedance and allow the output to swing through most of the supply range (you usually use a split supply, most often ± 15V)]. Operational amplifiers are now available in literally hundreds of types, with the universal symbol shown in Figure 3.1, where the (+) and (-) inputs do as expected: The output goes positive when the noninverting input (+) goes more positive than the inverting input (-), and vice versa. The (+) and (-) symbols don't mean that you have to keep one positive with respect to the other, or anything like that; they just tell you the relative phase of the output (which is important to keep negative feedback negative). Using the words "noninverting" and "inverting," rather than "plus" and "minus," will help avoid confusion. Power-supply connections are frequently not displayed, and there is no ground terminal. Operational amplifiers have enormous voltage gain, and they are never used without feedback. Think of an op-amp as fodder for feedback. The open-loop gain is so high that, for any reasonable closed-loop gain, the characteristics depend only on the feedback network.
 

Fig. 3-1. Universal symbol	Fig. 3-2 . Mini-DIP integrated circuit.	Fig. 3-3.

The most common op-amp is the mA741 C, or just 741 for short. It is a wee beastie packaged in the so-called mini-DIP (dual in-line package), and it looks as shown in Figure 3.2. Inside is a piece of silicon containing 20 transistors and 11 resistors. The pin connections are shown in Figure 3.3. The dot in the corner, or notch at the end of the package, identifies the end from which to begin counting the pin numbers. As with most electronic packages, you count pins counterclockwise, viewing from the top. The "offset null" terminals have to do with correcting (externally) the small asymmetries that are unavoidable when making the op-amp.
 

3.03 The golden rules

Here are the simple rules for working out op-amp behavior with external feedback. They're good enough for almost everything you'll ever do.

1. First, the op-amp voltage gain is so high that a fraction of a millivolt between the input terminals will swing the output over its full range, so we ignore that small voltage => The output attempts to do whatever is necessary to make the voltage difference between the inputs zero.

 
2. Second, op-amps draw very little input current (0.08~A for the 741; picoamps for FET-input types); we round this off => The inputs draw no current.

. Figure 3-4

BASIC OP-AMP CIRCUITS

3.04 Inverting amplifier

Let's begin with the circuit shown in Figure 3-4. The analysis is simple, if you remember your golden rules:

1. Point B is at ground, so rule 1 implies that point A is also.
2. This means that (a) the voltage across R₂ is V_out, and (b) The voltage across R₁, is V_in,
3. So, using rule 2I, we have

Later you will see that it's often better not to ground B directly, but through a resistor. However, don't worry about that now.

To understand how feedback works, just imagine some input level, say + 1 volt. For concreteness, imagine that R₁, is 10k and R₂ is 100k. Now, suppose the output decides to be uncooperative, and sits at zero volts. What happens? R₁ andR₂form a voltage divider, holding the inverting input at +0.91 volt. The op-amp sees an enormous input unbalance, forcing the output to go negative. This action continues until the output is at the required - 10.0 volts, at which point both op-amp inputs are at the same voltage, namely ground. Similarly, any tendency for the output to go more negative than - 10.0 volts will pull the inverting input below ground, forcing the output voltage to rise.

What is the input impedance? Simple. Point A is always at zero volts (it's called a virtualground). So Z_in = R₁,. At this point you don't yet know how to figure the output impedance; for this circuit, it's a fraction of an ohm.

Note that this analysis is true even for dc - it's a dc amplifier. So if you have a signal source offset from ground (collector of a previous stage, for instance), you may want to use a coupling capacitor (sometimes called a blocking capacitor, since it blocks dc but couples the signal). For reasons you will see later (having to do with departures of op-amp behavior from the ideal), it is usually a good idea to use a blocking capacitor if you're only interested in ac signals anyway.

This circuit is known as an inverting amplifier. Its one undesirable feature is the low input impedance, particularly for amplifiers with large (closed-loop) voltage gain, where R₁, tends to be rather small. That is remedied in the next circuit  (Fig. 3-5).

. Figure 3-5

3.05 Noninverting amplifier

Consider Figure 3.5. Again, the analysis is simplicity itself:
V_A = V_in
But V_A comes from a voltage divider:
V_A = V_outR₁/(R₁ + R₂)
Set V_A = V_in, and you get
gain = V_out/ V_in = 1 + R₂/R₁

This is a noninverting amplifier.In the approximation we are using, the input impedance is infinite (with a 741 it would be hundreds of megohms, at least; a FET opamp might have Z_in ~ 10¹²W or more). The output impedance is still a fraction of an ohm. As with the inverting amplifier, a detailed look at the voltages at the inputs will persuade you that it works as advertized.
 

. Figure 3-6

Once again we have a dc amplifier. If the signal source is ac-coupled, you must provide a return to ground for the (very small) input current, as in Figure 3-6. The component values shown give a voltage gain of 10 and a low-frequency 3dB point of 16Hz.
 

ac Amplifier

Again, if only ac signals are being amplified, it is often a good idea to "roll off" the gain to unity at dc, especially if the amplifier has large voltage gain, in order to reduce the effects of finite "input offset voltage." The circuit in Figure 3.7 has a low-frequency 3dB point of 17Hz, the frequency at which the impedance of the capacitor equals 2kW. Note the large capacitor value required. For noninverting amplifiers with high gain, the capacitor in this ac amplifier configuration may be undesirably large. In that case it may be preferable to omit the capacitor and trim the offset voltage to zero, as we will discuss later (Section 3.12). An alternative is to raise R₁ and R₂ perhaps using a T network for the latter (Section 3.18).
 

. Figure 3-7

3.06 Follower

Figure 3.8 shows the op-amp version of an emitter follower. It is simply a noninverting amplifier with R₁ infinite and R₂ zero (gain = 1). There are special op-amps, usable only as followers, with improved characteristics (mainly higher speed), e.g., the LM310.

An amplifier of unity gain is sometimes called a bufferbecause of its isolating properties (high input impedance, low output impedance).
 

. Figure 3-8

3.08 Basic cautions for op-amp circuits

In all op-amp circuits golden rules 1 and 2 (Section 3.03) will be obeyed only if the op-amp is in the active region, i.e., inputs and outputs not saturated at one of the supply voltages.

For instance, overdriving one of the amplifier configurations will cause output clipping at output swings near V_ccor V_ee. During clipping the inputs will no longer be maintained at the same voltage. The op-amp output cannot swing beyond the supply voltages (typically it can swing only to within 2V of the supplies). Likewise, the output compliance of an op-amp current source is set by the same limitation.

2. The feedback must be arranged so that it is negative. This means (among other things) that you must not mix up the inverting and noninverting inputs.

3. There must always be feedback at dc in an op-amp circuit. Otherwise the op-amp is guaranteed to go into saturation.

 

 

For instance , we werw able to put a capacitor from the feeddback network to ground in the noninverting amplifier (to reduce gain at dc to 1), but we could not similarly put a capacitor in series between the output and the inverting input.

4. Many op-amps have a relatively small maximum differential input voltage limit. The maximum voltage difference between the inverting and noninverting inputs might be limited to as little as 5 volts in either polarity. Breaking this rule will cause large input currents to flow, with degradation or destruction of the op-amp.



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AN OP-AMP SMORGASBORD

In the following examples we will skip the detailed analysis, leaving that fun for you, the reader.


3.09 Linear circuits

Optional inverter

The circuits in Figure 3.13 let you invert, or amplify without inversion, by flipping a switch. The voltage gain is either + 1 or - 1, depending on the switch position.
 


. Figure 3-13


 

Follower with bootstrap

As with transistor amplifiers, the bias path can compromise the high input impedance you would otherwise get with an op-amp, particularly with ac-coupled inputs, where a resistor to ground is mandatory. If that is a problem, the bootstrap circuit shown in Figure 3.14 is a possible solution. As in the transistor bootstrap circuit, the 1mF capacitor makes the upper 100k resistor look like a high-impedance current source to input signals. The low-frequency rolloíf for this circuit will begin at about 10Hz, and the response will drop at 12dB per octave tor frequencies somewhat below this, where both capacitors contribute to the rolloff.
 


. Figure 3-14


 

Ideal current-to-voltage converter

Remember that the humble resistor is the simplest l-to-V converter. However, it has the disadvantage of presenting a nonzero impedance to the source of input current; this can be fatal if the device providing the input current has very little compliance or does not produce a constant current as the output voltage changes. A good example is a photovoltaic cell, a fancy name for a sun battery. Even the garden-variety signal diodes you use in circuits have a small photovoltaic effect (there are amusing stories of bizarre circuit behavior finally traced to this efifect). Figure 3.15 shows the good way to convert current to voltage while holding the input strictly at ground. The inverting input is a virtual ground; this is fortunate, since a photovoltaic diode can generate only a few tenths of a volt. This particular circuit has an output of 1 volt per microamp of input current. The resistor from the noninverting input to ground could be omitted, but it is highly desirable; its function will be explained shortly in connection with op-amp shortcomings.
 


Figure 3-15

Of course, this transresistance configuration can be used equally well for devices that source their current via some positive excitation voltage, such as V_cc. Photomultiplier tubes and phototransistors (both devices that source current from a positive supply when exposed to light) are often used this way (Fig. 3.16).
 


. Figure 3-16


 

Differential amplifier

The circuit in Figure 3.17 is a differential amplifier with gain R₂/R₁,. As with the current source that used matched resistor ratios, this circuit requires precise resistor matching to achieve high common-mode rejection ratios. The best procedure is to stock up on a bunch of 100k 0.01% resistors next time you have a chance. All your differential amplifiers will have unity gain, but that's easily remedied with further (single-ended) stages of gain.
 


. Figure 3-17


 

Summing amplifier

The circuit shown in Figure 3.18 is just a variation of the inverting amplifier. Point X is a virtual ground, so the input current is V₁/R + V₂/R + V₃/R.That gives V_out = - (V₁ + V₂ + V₃). Note that the inputs can be positive or negative. Also, the input resistors need not be equal; if they're unequal, you get a weighted sum. For instance, you could have four inputs, each of which is + 1 volt or zero, representing binary values 1, 2, 4, and 8. By using input resistors of 10k, 5k, 2,5k, and 1,25k you will get an output in volts equal to the binary count input. This scheme can be easily expanded to several digits. It is the basis of digital-to-analog conversion, although a different input circuit (an R-2R ladder) is usually used.
 


. Figure 3-18


 

RIAA preamp

The RIAA preamp is an example of an amplifier with a specifically tailored frequency response. Phonograph records are cut with approximately flat amplitude characteristics; magnetic pickups, on the other hand, respond to velocity, so a playback amplifier with rising bass response is required. The circuit shown in Figure 3.19 produces the required response. The RIAA playback amplifier frequency response (relative to 0dB at 1 kHz) is shown in the graph, with the breakpoints given in terms of time constants. The 47mF capacitor to ground rolls off the gain to unity at dc, where it would otherwise be about 1000; as we have hinted earlier, the reason is to avoid amplification of dc input "offsets". The 739 is a low-noise dual op-amp intended for audio applications.
 


	. Figure 3-19  Op-amp RIAA phono playback amplifier.


 


3.18 Integrators

Op-amps allow you to make nearly perfect integrators, without the restriction that V_out<<V_in. Figure 3.43 shows how it's done. Input current V_in / R flows through C. Since the inverting input is a virtual ground, the output voltage is given by
 


<--Figure 3-43

The input can, of course, be a current, in which case R will be omitted. One problem with this circuit as drawn is that the output tends to wander off, even with the input grounded, due to op-amp offsets and bias current. This problem can be minimized by using an FET op-amp for low input current and offset, trimming the op-amp input offset voltage, and using large R and C values. In addition, in many applications the integrator is zeroed periodically by closing a switch placed across the capacitor (usually an FET), so only the drift over short time scales matters. As an example, an inexpensive FET op-amp like the LF355 (30pA bias current) trimmed to a voltage offset of 0.2mV and used in an integrator with R = 10MW and C = 10mF will produce an output drift of less than 0.003 volt in 1000 seconds.
 


Figure 3-44 Op-amp integrators with reset switches

If the residual drift of the integrator is stil too large for a given application, it may be necessary to put a large resistor R₂ across C to provide dc feedback for stable biasing. The effect is to roll off the integrator action at very low frequencies, f < 1 /R₂C. Figure 3.44 shows integrators with FET zeroing switch and with resistor bias stabilization.
 


. Figure 3-45

The feedback resistor may become rather large in this sort of application. Figure 3.45 shows a trick for producing the effect of a large feedback resistor using smaller values. In this case the feedback network behaves like a single 10MW resistor in the standard inverting amplifier circuit giving a voltage gain of - 100. This technique has the advantage of using resistors of convenient values without the problems of stray capacitance, etc., that occur with very large resistor values.


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F - Použitie integrátora v jadrovej elektronike - nábojový zosilňovač 


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3.19 Differentiators

Differentiators are similar to integrators, but with R and C reversed (Fig. 3.46). Since the inverting input is at ground, the rate of change of input voltage produces a current / = C(dV_in / dt) and hence an output voltage
 


Figure 3-46 ->

Differentiators are bias-stable, but they generally have problems with noise and instabilities at high frequencies because of the op-amp's high gain and internal phase shifts. For this reason it is necessary to roll off the differentiator action at some maximum frequency. The usual method is shown in Figure 3.47. The choice of the rolloff components R₁ and C₂ depends on the noise level of the signal and the bandwidth of the op-amp. At high frequencies this circuit becomes an integrator, due to R₁ and C₂.
 


. Figure 3-47

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