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

We mentioned in Section 3.12 that the finite open-loop gain of an op-amp limits its ; performance in a feedback circuit. Specifically, the closed-loop gain can never exceed the open-loop gain, and as the open-loop gain approaches the closed loop gain, the amplifier begins to depart from the ideal behavior we have come to expect. In this section we ; will quantify these statements so that you will be able to predict the performance of a feedback amplifier constructed with real (less than ideal) components. This is important also for feedback amplifiers constructed entirely with discrete components (transistors), where the open-loop gain is usually much less than with op-amps. In these cases the output impedance, for instance, will not be zero. Nonetheless, with a good understanding of feedback principles you will be able to achieve the performance required in any given circuit.
 

. Figure 3-58

 

3.24 Gain equation

Let's begin by considering an amplifier of finite voltage gain, connected with feedback (Fig. 3.58). The amplifier has open-loop voltage gain A, and the feedback network subtracts a fraction B of the output voltage from the input. (Later we will generalize things so that inputs and outputs can be currents or voltages.) The input to the gain block is then V_in - BV_out.But the output is just the input times A:
 

A (Vin - B Vout) = Vout

In other words,
and the closed-loop voltage gain, V_out/V_in is just

Some terminology: The standard designations for these quantities are as follows:
G = closed-loop gain,
A = open-loop gain,
AB = loop gain,
1 + AB = return difference, or desensitivity.
The feedback network i sometimes called the beta network (no relation to transistor beta, h_fe,b_F).

3.25 Effects of feedback on amplifier circuits

Let's look at the important effects of feedback. The most significant are predictability of gain (and reduction of distortion changed input impedance, and change output impedance.
 

Predictability of gain

The voltage gain is A/(1 + AB). In the limit of infinite open-loop gain A, G = 1 /B. We saw this result in the noninverting amplifier configuration, where a voltage divider on tl output provided the signal to the inverting input. The closed-loop voltage gain was just the inverse of the division ratio of the voltage divider. For finite gain A, feedback still acts to reduce the effects of variations of A (with frequency, temperature,amplitude, etc.). For instance, suppose A depends frequency as in Figure 3.59.

. Figure 3-59

This will surely satisfy anyone's definition of a poor amplifier (the gain varies over a factor of 10 with frequency). Now imagine we introduce feed back, with B = 0.1 (a simple voltage divider will do). The closed-loop voltage gain now varies from 1000/[ 1 + (1000 x 0.1)], or 9.90, to 10,000/[ 1 + (10,000 x 0.1)], or 9.99, a variation of just 1 % over the same range of frequency! To put it in audio terms, the original amplifier is flat to ± 10dB, whereas the feedback amplifier is flat to ±0.04dB. We can now recover the original gain of 1000 with nearly this linearity by just cascading three such stages. It was for just this reason (namely, the need for extremely flat telephone repeater amplifiers) that negative feedback was invented.

It is easy to show that relative variations in the open-loop gain are reduced by the desensitivity:
Thus for good performance the loop gain AB should be much larger than 1. That's equivalent to saying that the open-loop gain should be much larger than the closed-loop gain.

A very important consequence of this is that nonlinearities, which are simply gain variations that depend on signal level, are reduced in exactly the same way.

Input impedance

Feedback can be arranged to subtract a voltage or a current from the input (these are sometimes called series feedback and shunt feedback, respectively). The noninverting op-amp configuration, for instance, subtracts a sample of the output voltage from the differential voltage appearing at the input, whereas in the inverting configuration a current is subtracted from the input. The effects on input impedance are opposite in the two cases: Voltage feedback multiplies the open-loop input impedance by 1 + AB, whereas current feedback reduces it by the same factor. In the limit of infinite loop gain the input impedance (at the amplifier's input terminal) goes to infinity or zero, respectively. This is easy to understand, since voltage feedback tends to subtract signal from the input, resulting in a smaller change (by the factor AB)across the amplifier's input resistance; it's a form of bootstrapping. Current feedback reduces the input signal by bucking it with an equal current.

. Figure 3-60

Let's see explicitly how the effective input impedance is changed by feedback. We will illustrate the case of voltage feedback only, since the derivations are similar for the two cases. We begin with an op-amp model with (finite) input resistance as shown in Figure 3.60. An input V_in is reduced by BV_out , putting a voltage V_diff=V₁-BV_out across the inputs of the amplifier. The input current is therefore

The classic op-amp noninverting amplifier is exactly this feedback configuration, as shown in Figure 3.61. In this circuit, B = R₁ /(R₁ + R₂), giving the usual voltage gain expression G_v= 1 + R₂ /R₁ and an infinite input impedance for the ideal case of infinite open-loop voltage gain A. For finite loop gain the equations as previously derived apply.

. Figure 3-61

In the case of current (shunt) feedback, the input to the amplifier is at the "summing junction'" (the inverting input of the amplifier), where the currents from feedback and input signals are combined (this amplifier connection is really a "transresistance" configuration; it converts a current input to a voltage output). Feedback reduces the impedance looking into the summing junction (including the resistance of the feedback resistor) by the factor 1 + AB. In cases of very high loop gain (e.g., an op-amp) the input impedance is reduced to a fraction of an ohm, a good characteristic for a current-input amplifier. Some good examples are the photometer amplifier in Section 3.21.

Figure 3-62  Input impedance of the inverting amplifier

The classic op-amp inverting amplifier connection is a combination of a shunt feedback amplifier and a series input resistor, as shown in Figure 3.62. As a result, the input impedance equals the sum of R₁, and the impedance looking into the summing junction. For the ideal case of infinite open-loop voltage gain, the input impedance of the entire amplifier circuit is just R₁, and the gain is -R₂/R₁.

. Figure 3-63

Output impedance

Again, feedback can extract a sample of the output voltage or the output current. In the first case the open-loop output impedance will be reduced by the factor 1 + AB, whereas in the second case it will be increased by the same factor. We will illustrate this effect for the case of voltage sampling. We begin with the model shown in Figure 3.63. This time we have shown the output impedance explicitly. The calculation is simplified by a trick: Short the input, and apply a voltage V to the output; by calculating the output current I, we get the output impedance R'_o = V/I. Voltage V at the output puts a voltage -BV across the amplifier's input, producing a voltage -ABVin the amplifier's internal generator. The output current is therefore

If feedback is connected instead to sample the output current, the expression becomes
 

R'o = Ro(1 + AB)

It is possible to have multiple feedback paths, sampling both voltage and current. In the general case the output impedance is given by Blackman's impedance relation
where (AB)_SC is the loop gain with the output shorted to ground and (AB)_OC is the loop gain with no load attached. Thus feedback can be used to generate a desired output impedance. This equation reduces to the previous results for the usual situation in which feedback is derived from either thé output voltage or the output current.
 

l  Loading by the feedback network

In feedback computations, you usually assume that the beta network doesn't load the amplifier's output. If it does, that must be taken into account in computing the open-loop gain. Likewise, if the connection of the beta network at the amplifier's input affects the open-loop gain (feedback removed, but network still connected), you must use the modified open-loop gain. Finally, the preceding expressions assume that the beta network is unidirectional, i.e., it does not couple signal from the input to the output.

13.26 Two examples of transistor amplifiers with negative feedbacks

Figure 3.64 shows a transisotr amplifier with negative feedback.

. Figure 3-64  Transistor power amplifier with negative feedback

 

l Circuit description

It may look complicated, but it is extremely straightforward in design and is relatively easy to analyze. Q₁ and Q₂ form a differential pair, with common-emitter amplifier Q₃ amplifying its output. R₆ is Q₃'s collector load resistor, and push-pull pair Q₄ and Q₅ form the output emitter follower. The output voltage is sampled by the feedback network consisting of voltage divider R₄ and R₅, with C₂ included to reduce the gain to unity at dc for stable biasing. R₃ sets the quiescent current in the differential pair, and since overall feedback guarantees that the quiescent output voltage is at ground, Q₃'s quiescent current is easily seen to be 10mA (V_EEacross R₆, approximately). The diodes bias the push-pull pair into conduction, leaving one diode drop across the series pair R₇, and R₈, i.e., 60mA quiescent current. That's class AB operation, good for minimizing crossover distortion, at the cost of 1 watt standby dissipation in each output transistor.

From the point of view of our earlier circuits, the only unusual feature is Q₁'s quiescent collector voltage, one diode drop below V_CC. That is where it must sit in order to hold Q₃ in conduction, and the feedback path ensures that it will. (For instance, if Q₁ were to pull its collector closer to ground, Q₃ would conduct heavily, raising the output voltage, which in turn would force Q₂ to conduct more heavily, reducing Q₁'s collector current and hence restoring the status quo.) R₂ was chosen to give a diode drop at Q₁'s quiescent current in order to keep the collector currents in the differential pair approximately equal at the quiescent point. In this transistor circuit the input bias current is not negligible (4mA), resulting in a 0,4 volt drop across the 100k input resistors. In transistor amplifier circuits like this, in which the input currents are considerably larger than in op-amps, it is particularly important to make sure that the dc resistances seen from the inputs are equal, as shown (a Darlington input stage would probably be better here).
 

l  Analysis

Let's analyze this circuit in detail, determining the gain, input and output impedances, and distortion. To illustrate the utility of feedback, we will find these parameters for both the open-loop and closed-loop situations (recognizing that biasing would be hopeless in the open-loop case). To get a feeling for the linearizing effect of the feedback, the gain will be calculated at +10 volts and -10 volts output, as well as the quiescent point (zero volts).

l Open loop

Input impedance: We cut the feedback at point X and ground the right side of R4. The input signal sees 100k in parallel with the impedance looking into the base. The latter is h_fe times twice the intrinsic emitter resistance plus the impedance seen at Q₂'s emitter due to the feedback network at Q₂'s base. For h_fe ~ 250, Z_in~ 250 X[(2 X 25) + (3.3k/250)]; i.e., Z_in~ 16k.

Output impedance: Since the impedance looking back into Q₃'s collector is high, the output transistors are driven by a 1.5k source (R₆). The output impedance is about 15 ohms (h_fe ~ 100) plus the 5 ohm emitter resistance, or 20 ohms. The intrinsic emitter resistance of 0.4 ohm is negligible.

Gain: The differential input stage sees a load of R₂ paralleled by Q₃'s base resistance. Since Q₃ is running 10mA quiescent current, its intrinsic emitter resistance is 2.5 ohms, giving a base impedance of about 250 ohms (again, h_fe ~ 100). The differential pair thus has a gain of

250 || 620  or         3.5 2 x 25W

The second stage, Q₃, has a voltage gain of 1.5k/2.5 ohms, or 600. The overall voltage gain at the quiescent point is 3.5 x 600, or 2100. Since Q₃'s gain depends on its collector current, there is substantial change of gain with signal swing, i.e., nonlinearity. The gain is tabulated in the following section for three values of output voltage.
 

l Closed Ioop

Input impedance: This circuit uses series feedback, so the input impedance is raised by (1 + loop gain). The feedback network is a voltage divider with B= 1 /30 at signal frequencies, so the loop gain AB is 70. The input impedance is therefore 70 x 16k, still paralleled by the 100k bias resistor, i.e., about 92k. The bias resistor now dominates the input impedance.

Output impedance: Since the output voliage is sampled, the output impedance is reduced by (1 + loop gain). The output impedance is therefore 0.3 ohm. Note that this is a small-signal impedance and does not mean that a 1 ohm load could be driven to nearly full swing, for instance. The 5 ohm emitter resistors in the output stage limit the large signal swing. For instance, a 4 ohm load could be driven only to 10 volts pp, approximately.

Gain: The gain is A/(1 + AB). At the quiescent point that equals 30.84, using the exact value for B. In order to illustrate the gain stability achieved with negative feedback, the overall voltage gain of the circuit with and without feedback is tabulated at three values of output level at the end of this paragraph. It should be obvious that negative feedback has brought about considerable improvement in the amplifier's characteristics, although in fairness it should be pointed out that the amplifier could have been designed for better open-loop performance, e.g., by using a current source for Q₃'s collector load and degenerating its emitter; by using a current source for the differential pair emitter circuit, etc. Even so, feedback would still make a large improvement.

 

l Series feedback pair

Figure 3.65 shows another transistor amplifier with feedback. Thinking of Q₁ as an amplifier of its base-emitter voltage drop (thinking in the Ebers-Moll sense), the feedback samples the output voltage and subtracts a fraction of it from the input signal. This circuit is a bit tricky because Q₂'s collector resistor doubles as the feedback network. Applying the techniques we used earlier, you should be able to show that G (open loop) ~ 200, loop gain ~ 20, Z_out (open loop) ~ 10k, Z_out (closed loop) ~ 500 ohms, and G (closed loop) ~ 9.5.

. Figure 3-65

 

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