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Physics 3330 Experiment #4 Fall 2013
Operational Amplifiers and Negative Feedback
Purpose
This experiment shows how an operational amplifier (op-amp) with negative feedback can be used to make an amplifier with many desirable properties, such as stable gain, high linearity, and low output impedance. You will build both non-inverting and inverting voltage amplifiers using an LF356 op-amp.
Introduction
The purpose of an amplifier is to increase the voltage level of a signal while preserving as accurately as possible the original waveform. In the physical sciences, transducers are used to convert basic physical quantities into electric signals, as shown in Figure 4.1. An amplifier is usually needed to raise the small transducer voltage to a useful level.
A small voltage or current change at the input of the amplifier controls a much larger signal at the input of whatever circuit or instrument the amplifier’s output is connected to. Measuring and recording equipment typically requires input signals of 1 to 10 V. To meet such needs, a typical laboratory amplifier might have the following characteristics:
- Predictable and stable gain. The magnitude of the gain is equal to the ratio of the output signal amplitude to the input signal amplitude.
- Linear amplitude response, so that the output signal is directly proportional to the input signal.
Transducer
Oscilloscope
Data Analysis System
Amplifier
Temperature, Magnetic Field, Radiation, Acceleration, etc.
DC Power
Volts to mVolts
Figure 4.1 Typical Laboratory Measurement
Volts
- According to the application, the frequency dependence of the gain might be a constant independent of frequency up to the highest frequency component in the input signal (wideband amplifier), or a sharply tuned resonance response if a particular frequency must be picked out.
- High input impedance and low output impedance are usually desirable. These characteristics minimize changes of gain when the amplifier is connected to the input transducer and to other instruments at the output.
- Low noise is usually important. Every amplifier adds some random noise to the signals it processes, and this noise often limits the sensitivity of an experiment. We will learn about amplifier noise in Experiment 8.
Commercial laboratory amplifiers are readily available, but a general-purpose amplifier is expensive (>$1000), and most of its features might be unneeded in a given application. Often, it is preferable to design your own circuit using a cheap (<$1) op-amp chip.
Op-amps have many other circuit applications. They can be used to make filters, limiters, rectifiers, oscillators, integrators, and other devices (see FC 12.9 – 12.15). To get an idea of the variety of op-amps available, have a look at the National Semiconductor web site (www.national.com). They currently make over 300 types, offering trade-offs between speed, cost, power consumption, precision, etc. The op-amp is the most important building block of analog electronics.
Suggested Readings
- FC Sections 12.2 – 12.15. The basic rules of op-amp behavior and the most important op-amp circuits. [Note while FC discussed basic characteristics of op amps in 12.2 it does not have the I, II “Golden Rule” analysis that we will discuss in class.] Do not worry right now about the transistor guts of op-amps. We will learn about transistors in Experiments 7-8.
- Horowitz and Hill, Chapter 4.
- The IC Op-Amp Cookbook, by Walter Jung, 3rd^ edition (Prentice Hall, 1997).
- Diefenderfer Chapter 9
- D.V. Bugg, Chapter 7, particularly Sections 7.1–7.4, 7.7, and 7.12.
If we assume that A → ∞, that the op-amp input impedance is infinite, and that the output impedance is zero, then the behavior of these circuits can be understood using the simple “Golden Rules”:
I. The voltage difference between the inputs is zero. (“voltage rule”)
II. No current flows into (or out of) each input. (“current rule”)
The “Golden Rule” analysis is very important and is always the first step is designing op-amp circuits, so be sure you understand it before you read the material below, where we consider the effect of finite values of A, mainly so that the frequency dependence of the closed loop gain G can be understood.
GAIN EQUATION – NON-INVERTING CASE The basic formula for the gain of feedback amplifiers is derived in FC, Section 12.5. From Fig. 4.2 we can see that: Vout A V^ in BVout
Solving this equation yields:
AB
A
V
G V
in
out
With B=R/(R+RF) the above equation for the closed loop gain then gives
which is the Golden Rule result. If we don't care about corrections due to the finite value of A, then Equation (1) is not needed and the analysis can be done using just the Golden Rules.
INPUT AND OUTPUT IMPEDANCE – NON-INVERTING CASE
Formulas for the input and output impedance are derived in H&H Section 4.26. The results are
where Ri and Ro are the input and output impedances of the op-amp alone, while the primed symbols refer to whole the amplifier with feedback. These impedances will be improved from the values for the bare op-amp if the loop gain A·B is large.
Ri Ri (1 AB ) Ro Ro / (1 AB )
G (^) 1
R (^) f R if^ A^ ^ ^1
FREQUENCY DEPENDENCE – NON-INVERTING CASE
The above formulas are still correct when A and/or B depend on frequency. B will be frequency independent if we have a resistive feedback network, but A always varies with frequency. For most op-amps, including the LF356 (and others with dominant pole compensation, see H&H Section 4.34), the open loop gain varies with frequency like an RC low-pass filter:
A
A 0
1 i f
f 0
The 3dB frequency f 0 is usually very low, around 10 Hz. Data sheets do not usually give f 0 directly; instead they give the dc gain A 0 and the unity gain frequency fT, which is the frequency where the magnitude of the open loop gain A is equal to one. The relation between A 0 , f 0 , and fT is
The frequency dependence of the closed loop gain G can be found by substituting Equation (2) into Equation (1). You will find the result
The frequency response of the amplifier with feedback is therefore also the same as for an RC low-pass filter. The 3dB bandwidth fB with feedback is given by
At frequencies well below fB the gain is
GAIN-BANDWIDTH PRODUCT – NON-INVERTING CASE
We can now derive an example of a very important general rule connecting the gain and bandwidth of feedback amplifiers. Multiplying the low frequency gain G 0 by the 3 dB bandwidth fB gives
In words, this very important formula says that the gain-bandwidth product G 0 fB equals the unity
fT A 0 f 0.
G
A 0 1 A 0 B 1 i f f (^) 0 1 A 0 B
G^0
1 i f f (^) B
f (^) B f (^) 0 1 A 0 B .
G 0
A 0
1 A 0 B
G 0 f B
A 0
1 A 0 B
f (^) 0 1 A 0 B A 0 f (^) 0 ,
G 0 fB fT.
impedance of an inverting amplifier is not usually greater than about 100 k, while the input impedance of a non-inverting amplifier can easily be as large as 10^12 . When A is not large the formula for the input impedance is
The formula for the output impedance is the same as for a non-inverting amplifier:
FREQUENCY DEPENDENCE – INVERTING CASE
As for the non-inverting case, all of the above formulas are still correct when A and/or B depend on frequency. (To make B frequency dependent, R and/or RF may be replaced by networks
containing capacitors or inductors.) We again assume that the op-amp has dominant pole compensation:
(4)
so that the relation between the open loop 3 dB frequency f 0 , the dc open loop gain A 0 and the unity gain frequency fT is still
The frequency dependence of the closed loop gain G for the feedback amplifier can be found by substituting Equation (4) into Equation (3). The result is
When B is frequency independent, the frequency response of the amplifier with feedback is again the same as an RC low-pass filter. The 3dB bandwidth fB with feedback is the same as for
the non-inverting amplifier
At frequencies well below fB the gain is
The gain-bandwidth product relation is
This is the same (except for the sign) as the non-inverting result when the closed loop gain is
Zin R 1 R F A.
Ro Ro / (1 AB ).
A A^0
1 i f f (^) 0
f (^) T A 0 f 0.
G
A 0 1 B
^1 ^ A 0 B
1 i (^) f f
0 ^1 ^ A 0 B
G^0 1 i ff B
.
f (^) B f (^) 0 1 A 0 B .
G 0 A 10 ^ ^1 A^ ^ B 0 B^
.
G 0 f (^) B A 10 ^ ^1 A ^ B 0 B^
f 0 1 A 0 B A 0 f 0 1 B ,
G 0 f (^) B fT (1 B ).
large (B << 1, G 0 >> 1), but at unity closed loop gain (B = 1/2, G 0 = -1) the inverting
amplifier has only half as much bandwidth as a non-inverting follower.
Apparatus
This experiment will use both +15V and -15V to power the LF356 op-amp. Look back at Experiment 2 to remember how to connect the power supply to your prototyping board.
Turn off the power while wiring your op-amp circuits. Figure 4.4 shows pin-out data and a layout for the gain=100 amplifier.
0 V
Figure 4.4 Op-amp pin diagram and possible layout for gain 100 amplifier
+15 V
V n in
V ut
out
RF
R
356
local ground panel ground
356
+15 V
Input (^) V ut
o
2 (^34)
(^76)
7
8
6 4 5
3
1 2
bal. 1 V V
NC +15 V V bal. 2
1
8 4
5
view from above
0 V
+15 V 0 V
R
R
F
V ut
out
V n
in bypass cap
Everyone makes mistakes in wiring-up circuits. Thus, it’s a good idea to check your circuit before applying power. You may well save a transistor or chip from burnout and save yourself a lot of frustration.
The following procedure will help you wire up a circuit accurately:
- Draw a complete schematic in your lab book, including all ground and power connections, and all IC pin numbers. Try to layout your prototype so the parts are arranged in the same way as on the schematic, as far as possible.
- Know the color or number code for resistors and capacitors so you can avoid putting the wrong part into the circuit. When in doubt about the value of a part, measure it with an Ohm meter or impedance bridge.
- Adhere to a color code for wires. Red for positive power and black for ground is a very widely honored standard. Beyond that, just try to be consistent. For example:
0 V (ground) Black +15 V Red -15 V Blue analog signals Yellow digital signals Purple
The op-amp chip sits across a groove in the circuit board. Before inserting a chip, gently straighten the pins. After insertion, check visually that no pin is broken or bent under the chip. To remove the chip, use a small screwdriver in the groove to pry it out.
You will have less trouble with spontaneous oscillations if the circuit layout is neat and compact, especially the feedback path should be as short as possible to reduced unwanted capacitive coupling and lead inductance.
To help prevent spontaneous oscillations due to unintended coupling via the power supplies, use bypass capacitors to filter the supply lines. A bypass capacitor between each power supply lead and ground, will provide a miniature current “reservoir” that can quickly supply current when needed. This capacitor is normally in the range 1 F – 10 F. Compact capacitors in this range are usually electrolytic tantalum or aluminum and are polarized, meaning that one terminal (marked +) must always be positive relative to the other. If you put a polarized capacitor in backwards, it will burn out. It is often helpful to put a smaller (0.01-0.1 μF) non-polar ceramic capacitor between power and ground as well. Ceramic capacitors offer less capacitance than electrolytic ones, but work better at high frequencies. Bypass capacitors should be placed as close as possible to the op-amp supply pins.
Experiment
GETTING STARTED: TESTING THE OP-AMP
You can save yourself some frustration by testing your op-amp chips to make sure they are not burned out. Connect the op-amp as a voltage follower with the input grounded (see Fig. 4.5). Since Vout = Vin for a follower, you expect Vout = 0 for this circuit. A "bad" chip will often have Vout = ±Vsat because of a burned-out output
circuit.
Measure the DC voltages with a multimeter. Be careful not to connect pins 6 & 7, since this will burn out the op-amp. You should find Pin 4 = – 15 V, Pin 7 = +15 V, Pin 2 = 0 V, Pin 3 = 0 V, Pin 6 = 0 V. The small voltage on pin 2 is equal to the input offset voltage VOS, typically less than 3 mV. If Vout = ± 15 V, then the chip is bad.
Throw it in the trash to save another person from having to deal with it. (In case you are wondering, the LF356 costs $0.50 when purchased in quantities of 100. Check out a distributor like www.digikey.com if you are curious about the cost of any part we use.)
0 V
Figure 4.5 Test circuit for op-amp
+15 V
V uto
356
local ground panel ground
2 6
7
3 4
deviation from the performance of an ideal follower. The measurement at high frequency will depend on many details of your setup and you are unlikely to find a simple RC filter type falloff. Using the 10X probe, measure the gain at every decade in frequency from 10 MHz down to 10 Hz. Do you find any deviation from unity gain? Be sure that the output amplitude is below the level affected by the slew rate! Plot the low and high frequency data on a Bode diagram. Do you find a simple fall-off as suggested by the theory for the ideal follower (fT=fB)? If so, find the 3dB frequency. How does your measured 3dB frequency compare to what is expected from the op-amp datasheet? Even if the fall-off doesn't have a simple behavior, at what frequency do you observe more than a 10% deviation from the ideal gain?
If you observed "ideal" behavior, you're lucky! At frequencies above a few MHz, the simple model of the frequency response of the op-amp (see theory section) is not accurate. Furthermore, once you are in this frequency range, many physical details of your circuit and breadboard can have large effects in the circuit. Building reliable circuits at these frequencies typically requires careful attention to grounding and minimization of capacitive and inductive coupling between circuit elements and to ground. At lower frequencies, our op-amp model will work much better.
2. GAIN = 100 NON-INVERTING AMPLIFIER
Change the negative feedback loop to the one in Figure 4.8, with RF = 10 k and R = 100 . Use 1%, 1/2 Watt, precision resistors for the feedback loop. Measure R and RF with the multimeter before inserting them into the circuit board. Recalculate G 0 and fB from these measured values and the op-amp's value of fT measured in the last part.
2A. Measure the low frequency gain G0 for 1 kHz sine waves. Vary the input amplitude until you observe saturation in the output. What are the output saturation levels, +Vsat and -Vsat? Now, reduce the input amplitude so that the output voltage is less than half the saturated value.
2B. Determine the 3dB bandwidth fB. Then vary the frequency to obtain data at decade intervals for checking your Bode diagram.
2C. Using the gain-bandwidth relation G 0 fB=fT, determine the fT for your op-amp. How does this compare with the value from the op-amp datasheet?
2D. The input impedance of the amplifier should be exceptionally high. Can you see any change in output when you connect a 1 M resistor in series
0 V
Figure 4.8 Gain of 100 Amplifier
– 15 V
+15 V
Vin
Vout
R R F
with the input? Is this consistent with your homework result?
2E. The output impedance should be very low, provided that the output current is less than the maximum available (25 mA). What happens when you connect a load of 220 from output to ground? Explore as a function of output amplitude. Be sure to pull out this 220 before proceeding to the next section.
3. INVERTING AMPLIFIER
Build the inverting amplifier that you designed in Problem 5.
3A. Measure the gain versus frequency and record the result on a Bode plot, comparing with your prediction.
3B. Find the gain-bandwidth product and see how it compares to the value your observed for the gain 100 non-inverting amplifier.
3C. Measure the input impedance and see if it agrees with your expectations.
