Download Op-Amp Basics - Analog Electronics - Lecture Notes and more Study notes Computer Science in PDF only on Docsity!
431/531 Class Notes 9
6 Op-Amp Basics
The op erational ampli er is one of the most useful and imp ortant comp onents of analog elec- tronics. They are widely used in p opular electronics. Their primary limitation is that they are not esp ecially fast: The typical p erformance degrades rapidly for frequencies greater than ab out 1 MHz, although some mo dels are designed sp eci cally to handle higher frequencies. The primary use of op-amps in ampli er and related circuits is closely connected to the concept of negative feedback. Feedback represents a vast and interesting topic in itself. We will discuss it in rudimentary terms a bit later. However, it is p ossible to get a feeling for the two primary typ es of ampli er circuits, inverting and non-inverting, by simply p ostulating a few simple rules (the \golden rules"). We will start in this way, and then go back to understand their origin in terms of feedback.
6.1 The Golden Rules
The op-amp is in essence a di erential amplifer of the typ e we discussed in Section 5.7 with the re nements we discussed (current source load, follower output stage), plus more, all nicely debugged, characterized, and packaged for use. Examples are the 741 and 411 mo dels which we use in lab. These two di er most signi cantly in that the 411 uses JFET transistors at the inputs in order to achieve a very large input imp edance (Zin 109 ), whereas the 741 is an all-bip olar design (Zin 106 ). The other imp ortant fact ab out op-amps is that their open-loop gain is huge. This is the gain that would b e measured from a con guration like Fig. 29, in which there is no feedback lo op from output back to input. A typical op en-lo op voltage gain is 104 {10^5. By using negative feedback, we throw most of that away! We will so on discuss why, however, this might actually b e a smart thing to do.
in 1
in 2
out
Figure 29: Op erational ampli er.
The golden rules are idealizations of op-amp b ehavior, but are nevertheless very useful for describing overall p erformance. They are applicable whenever op-amps are con gured with negative feedback, as in the two ampli er circuits discussed b elow. These rules consist of the following two statements:
- The voltage di erence b etween the inputs, V+ V , is zero. (Negative feedback will ensure that this is the case.)
- The inputs draw no current. ( This is true in the approximation that the Zin of the op-amp is much larger than any other current path available to the inputs.)
When we assume ideal op-amp b ehavior, it means that we consider the golden rules to b e exact. We now use these rules to analyze the two most common op-amp con gurations.
6.2 Inverting Ampli er
The inverting ampli er con guration is shown in Fig. 30. It is \inverting" b ecause our signal input comes to the \ " input, and therefore has the opp osite sign to the output. The negative feedback is provided by the resistor R 2 connecting output to input.
R 1
R 2
VIN VOUT
Figure 30: Inverting ampli er con guration.
We can use our rules to analyze this circuit. Since input + is connected to ground, then by rule 1, input is also at ground. For this reason, the input is said to b e at virtual ground. Therefore, the voltage drop across R 1 is vin v = vin , and the voltage drop across R 2 is vout v = vout. So, applying Kircho 's rst law to the no de at input , we have, using golden rule 2: i = 0 = iin + iout = vin =R 1 + vout =R 2
or G = vout=vin = R 2 =R 1 (34) The input imp edance, as always, is the imp edance to ground for an input signal. Since the input is at (virtual) ground, then the input imp edance is simply R 1 :
Zin = R 1 (35)
The output imp edance is very small (< 1 ), and we will discuss this again so on.
6.3 Non-inverting Ampli er
This con guration is given in Fig. 31. Again, its basic prop erties are easy to analyze in terms of the golden rules.
vin = v+ = v = vout
R
1 R 1 + R 2
Perhaps the b est way to b eat these efects, if they are a problem for a particular appli- cation, is to cho ose op-amps which have go o d sp eci cations. For example, IOS can b e a problem for bi-p olar designs, in which case cho osing a design with FET inputs will usually solve the problem. However, if one has to deal with this, it is go o d to know what to do. Fig- ure 32 shows how this might b e accomplished. Without the 10 k resistors, this represents a non-inverting ampli er with voltage gain of 1 + (10^5 = 102 ) 1000. The mo di ed design in the gure gives a DC path from ground to the op-amp inputs which are aproximately equal in resistance (10 k ), while maintaining the same gain.
OUT
IN
10k
10k
100
100k
Figure 32: Non-inverting ampli er designed to minimize e ect of IOS.
Similarly, the inverting ampli er con guration can b e mo di ed to mitigate o set currents. In this case one would put a resistance from the input to ground which is balanced by the R 1 and R 2 in parallel (see Fig. 30). It is imp ortant to note that, just as we found for transistor circuits, one shp ould always provide a DC path to ground for op-amp inputs. Otherwise, charge will build up on the e ective capacitance of the inputs and the large gain will convert this voltage (= Q=C ) into a large and uncontrolled output voltage o set. However, our mo di ed designs to ght IOS have made our op-amp designs worse in a general sense. For the non-inverting design, we have turned the very large input imp edance into a not very sp ectacular 10 k. In the inverting case, we have made the virtual ground into an approximation. One way around this, if one is concerned only with AC signals, is to place a capacitor in the feedback lo op. For the non-inverting ampli er, this would go in series with the resistor R 1 to ground. Therefore, as stated b efore, it is b est, where imp ortant, to simply cho ose b etter op-amps!
6.5 Frequency-dep endent Feedback
Below are examples of simple integrator and di erentiator circuits which result from making the feedback path have frequency dep endence, in these cases single-capacitor RC lters. It is also p ossible to mo dify non-inverting con gurations in a similar way. For example, problem (3) on page 251 of the text asks ab out adding a \rollo " capacitor in this way. Again, one would simply mo dify our derivations of the basic inverting and non-inverting gain formulae by the replacements R! Z , as necessary.
6.5.1 Integrator
Using the golden rules for the circuit of Fig. 33, we have
vin v R
vin R
= iin = iout = C
d(vout v ) dt
= C
dvout dt So, solving for the output gives
vout =
RC
Z
vin dt (38)
And for a single Fourier comp onent! , this gives for the gain
G(! ) =
! RC
Therefore, to the extent that the golden rules hold, this circuit represents an ideal inte- grator and a low-pass lter. Because of the presence of the op-amp, this is an example of an active lter. In practice, one may need to supply a resistor in parallel with the capacitor to give a DC path for the feedback.
IN OUT
R
C
Figure 33: Op-amp integrator or low-pass lter.
6.5.2 Di erentiator
The circuit of Fig. 34 can b e analyzed in analogy to the integrator. We nd the following:
vout = RC
dvin dt
G(! ) = ! RC (41)
So this ideally represents a p erfect di erentiator and an active high-pass lter. In practice, one may need to provide a capacitor in parallel with the feedback resistor. (The gain cannot really increase with frequency inde nitely!)
6.6 Negative Feedback
As we mentioned ab ove, the rst of our Golden Rules for op-amps required the use of negative feedback. We illustrated this with the two basic negative feedback con gurations: the inverting and the non-inverting con gurations. In this section we will discuss negative feedback in a very general way, followed by some examples illustrating how negative feedback can b e used to improve p erformance.
6.6.2 Input and Output Imp edance
We can now also calculate the e ect that the closed-lo op con guration has on the input and output imp edance. The gure b elow is meant to clearly show the relationship b etween the de nitions of input and output imp edances and the other quantities of the circuit. The quantity Ri represents the op en-lo op input imp edance of the op-amp, that is, the imp edance the hardware had in the absence of any negative feedback lo op. Similarly, Ro represents the Thevenin source (output) imp edance of the op en-lo op device.
B
v out
vin R i
Ro
b
Figure 36: Schematic to illustrate the input and output imp edance of a negative feedback con guration.
We start the calculation of Zin with the de nition Zin = vin =iin. Let us calculate the current passing through Ri :
iin =
vin vb Ri
vin B vout Ri
Substituting the result of Eqn. 42 gives
iin =
Ri
vin B vin
A
1 + AB
Rearanging allows one to obtain
Zin = vin =iin = Ri [ 1 + AB ] (43)
A similar pro cedure allows the calculation of Zout vop en =ishort. We have vop en = vout and the shorted current is what gets when the load has zero input imp edance. This means that all of the current from the ampli er go es into the load, leaving none for the feedback lo op. Hence, B = 0 and
ishort = A (vin B vout) =Ro = Avin =Ro =
Avout Ro G
Avout Ro
1 + AB
A
vout Ro
(1 + AB )
This gives our result
Zout = vop en =ishort =
Ro 1 + AB
Therefore, the efect of the closed lo op circuit is to improve b oth input and output imp edances by the identical lo op-gain factor 1 + AB AB. So for a typical op-amp like a 741 with A = 103 , Ri = 1 M , and Ro = 100 , then if we have a lo op with B = 0 : 1 we get Zin = 100 M and Zout = 1.
6.6.3 Examples of Negative Feedback Bene ts
We just demonstrated that the input and output imp edance of a device employing negative feedback are b oth improved by a factor 1 + AB AB , the device lo op gain. Now we give a simple example of the gain equation Eqn. 42 in action. An op-amp may typically have an op en-lo op gain A which varies by at least an order of magnitude over a useful range of frequency. Let Amax = 104 and Amin = 103 , and let B = 0 :1. We then calculate for the corresp onding closed-lo op gain extremes:
Gmax =
10(1 10 ^3 )
Gmin =
10(1 10 ^2 )
Hence, the factor of 10 op en-lo op gain variation has b een reduced to a 1% variation. This is typical of negative feedback. It attenuates errors which app ear within the feedback lo op, either internal or external to the op-amp prop er. In general, the b ene ts of negative feedback go as the lo op gain factor AB. For most op-amps, A is very large, starting at > 105 for f < 100 Hz. A large gain G can b e achieved with large A and relatively small B , at the exp ense of somewhat p o orer p erformance relative to a smaller gain, large B choice, which will tend to very go o d stability and error comp en- sation prop erties. An extreme example of the latter choice is the \op-amp follower" circuit, consisting of a non-inverting ampli er (see Fig. 31) with R 2 = 0 and R 1 removed. In this case, B = 1, giving G = A=(1 + A) 1. Another interesting feature of negative feedback is one we discussed brie y in class. The qualitative statement is that any signal irregularity which is put into the feedback lo op will, in the limit B! 1, b e taken out of the output. This reasoning is as follows. Imagine a small, steady signal vs which is added within the feedback lo op. This is returned to the output with the opp osite sign after passing through the feedback lo op. In the limit B = 1 the output and feedback are identical (G = 1) and the cancellation of vs is complete. An example of this is that of placing a \push-pull" output stage to the op-amp output in order to b o ost output current. (See text Section 2 :15.) The push-pull circuits, while b o osting current, also exhibit \cross-over distortion", as we discussed in class and in the text. However, when the stage is placed within the op-amp negative feedback lo op, this distortion can essentially b e removed, at least when the lo op gain AB is large.
6.7 Comp ensation in Op-amps
Recall that an RC lter intro duces a phase shift b etween 0 and =2. If one cascades these lters, the phase shifts can accumulate, pro ducing at some frequency ! the p ossibility of a phase shift of . This is dangerous for op-amp circuits employing negative feedback, as a phase shift of converts negative feedback to positive feedback. This in turn tends to
