Physics 219 Fall, 1995



Amplifiers

Table of Contents

List of amplifier related lab exercises to do from The Student Manual 3

Transistor Amplifiers 5

Part I (Lab 4, continued) 5

Amplification and Gain 5

Emitter Follower 7

Input Impedance of the Emitter Follower 9

Biasing the Emitter Follower 12

Output Impedance of the Emitter Follower 13

Transistor Current Source 14

Common Emitter Amplifier 16

Output Impedance of The Common Emitter 18

Summary of results for emitter follower and common emitter amplifiers 18

Crossover Distortion in a Push-Pull Amplifier 19

Op Amps I (Lab 8) 22

Differential Amplifiers 22

Introduction to Op amps 23

Input bias current 25

Feedback 25

Some Op Amp Circuits 27

Inverting Amplifier 27

Non-Inverting Amplifier 29

Op Amp Follower 30

Op Amp Current Source 30

Current-to-Voltage Converter 31

Summing Amplifier 33

Summing Amplifier as a Digital to Analog Converter 34

Op Amp Buffer for Push-Pull Amplifier 35

Op Amps II (Lab #9) 38

Op Amp Limitations 39

Op Amp Integrator 41

Active Rectifier 43

Op Amps III (Lab #10) 44

Comparator 44

Application: Digital Voltmeter (DVM) 45

Schmitt Trigger 47

RC Relaxation Oscillator 49

A real op amp circuit 52

Stabilizing the length of a diode laser cavity to a part in 108 using a 40 cent op amp 52

List of amplifier related lab exercises to do from The Student Manual

Lab 4 – Transistors Part I

4-2 Emitter Follower Amplifier

4-3 Input and Output Impedances of the Emitter Follower Amplifier

4-4 Single Supply Follower

4-7 Common Emitter Amplifier

Lab 5 – Transistors Part II

5-6 Push-Pull Amplifier

Lab 8 – Op Amps, Part I

8-1 Open-Loop Test Circuit

8-2 Inverting Amplifier

8-3 Non-Inverting Amplifier

8-4 Follower

8-6 Current to Voltage Converter

8-7 Summing Amplifier

8-8 Push Pull Buffer

Lab 9 – Op Amps, Part II

9.1a Op Amp Limitations: Slew Rate

9.2 Op Amp Integrator

9.4 AC amplifier: microphone amplifier

Clap Sensor - Did sound exceed a certain threshold?

9.5 Active rectifier/ Active peak detector (peak detector is not in Student Manual)

Lab 10 – Op Amps, Part III

10.1 Comparator and Schmitt trigger

10.2 RC relaxation oscillator

Transistor Amplifiers

Part I (Lab 4, continued)

Amplification and Gain

So far we have seen the value of a transistor in acting as an electronic switch. Another important capability of a transistor is to provide a means for amplifying electronic signals.

Consider the following example, which is representative of a very broad and important class of problem. Suppose we want to build a “public address system” where the goal is to use a microphone to "detect" a sound and then ultimately have a louder version of the sound emanate from a loudspeaker.

[pic]

Your first impulse might be to simply to connect the microphone directly to the speaker. This approach won't work at all for two reasons: 1) The amplitude of the voltage signal produced by the microphone is usually quite small to begin with. and 2) Worse still, the output impedance of the microphone is typically quite high (many kΩ at least). If one were to connect the microphone directly to the speaker, which typically has an input impedance of only 8 Ω, then the already small voltage produced by the microphone would undergo further severe attenuation.

As I said this problem can be viewed as emblematic of a near universal problem: there’s a small electronic signal, and you want to make it bigger!

We will solve these difficulties in two stages. First we'll show how a transistor configured as an “emitter follower” can help solve the "impedance mismatch" problem:

[pic]

An emitter follower configuration

Later we'll show how a different transistor circuit, the common emitter, can amplify the signal and allow you to really rattle the windows.

Before we formally analyze the input and output impedances of the emitter follower, let us first develop a more intuitive understanding of what is going on. When we say "the microphone has a large output impedance”, what does that really mean?

The key point is that sources with large output impedances cannot supply a large current to a load. (Remember, the maximum current that can be supplied is the "short circuit" current given by

[pic]

where[pic] and [pic] are the output impedances and the Thevenin voltages of the device.) Because of the transistor's current gain, the emitter follower boosts the amount of current that can be delivered from the source (e.g. microphone) to the load (e.g. speaker).

The output impedance of a device can be thought of as a measure of its ability to provide current to a load.

The fundamental criterion that we must keep in mind when connecting a “load” to a source:

[pic]

[pic]

In practice a useful, somewhat arbitrary, rule of thumb is to require

[pic]

Emitter Follower

(Student Manual Lab 4-2)

[pic]

First let us recall:

The Four Golden Rules of Bipolar Transistors

Here are the rules for npn transistors:

Rule 1. [pic] by at least a few tenths of a volt. The collector voltage must be more positive than the emitter.

Rule 2. In "normal operation" the base emitter junction behaves like a forward-biased diode, so that there is approximately a 0.6 V drop from base to emitter. Thus [pic] or [pic].

Rule 3. There are limits on [pic], etc. which, if exceeded, will destroy the transistor.

Rule 4. When rules I through 3 are obeyed, then [pic] where [pic] is a constant with a typical value of about 100.

With these rules in mind, let us analyze the above circuit:

1) Golden Rule #2 implies

[pic]

Since very little base current flows across the small base resistor

[pic]

so that

[pic]

Thus, in the emitter follower circuit , the output voltage [pic], "follows” the input voltage [pic].

This will work provided [pic]; if [pic] then [pic] won't follow [pic] since current can only flow in the direction of the arrow through the 3.3 kΩ resistor to ground. Thus [pic] must always be [pic].

Thus if we want the output to follow an ac signal we need to connect the load resistor to a negative power supply, as shown below.

[pic]

What's the point of this?

Input Impedance of the Emitter Follower

(Student Manual Lab 4-3)

[pic]

For ac signals we define the input impedance by

[pic]

where the "Δ" denotes that the quantities are undergoing small changes associated with a small amplitude ac signal.

For the emitter follower

[pic]

Ohm's Law implies:

[pic]

or

[pic]

Golden Rule #4 implies

[pic]

These last two results, taken together imply:

[pic]

so, finally,

[pic]

[pic]

Therefore, the emitter follower increases the input impedance of the load by a factor [pic].

Thus the emitter follower acts as a buffer between the source and the load, making it easier for the source to drive the load without being attenuated.

To measure the input impedance of the emitter follower experimentally, build the following circuit:

[pic]

Now,

[pic]

You can measure amplitude of the input voltage,[pic], directly with the oscilloscope. You can then deduce [pic]by looking at the voltage drop across the resistor:

[pic]

Biasing the Emitter Follower

(section 2.05 of text, p.86 of Student Manual)

Sometimes one only has access to a power supply with only one polarity (in a battery operated device this is often the case.) We can use a single polarity power supply to operate the emitter follower by pulling the transistor’s quiescent voltages (the voltages that the terminals are out when there is zero input voltage) off-center , “biasing” it away from zero volts:

[pic]

The biasing divider must be “stiff enough” to hold the base of the transistor where we want it (about midway between the positive power supply voltage (often referred to as [pic]) and ground. We thus have two requirements:

[pic]

and

[pic]

where as we know, the impedance of the divider is given in this case by

[pic]

In words: We require that the impedance of the divider small compared to the input impedance of the emitter follower, but large compared to the impedance of the source.

Output Impedance of the Emitter Follower

Above we showed that for an emitter follower

[pic]

Therefore, the emitter follower increases the input impedance of the load by a factor [pic].

Alternatively we can view the emitter follower as a circuit as something that lowers the impedance of a source by a factor of [pic]

[pic]

Proof:

Recall [pic]

Because of the transistor’s current gain, when an amount of current [pic] is drawn from the output the amount of current that flows through the source impedance is only [pic]. Thus [pic] is factor of [pic] less than it would be without the transistor.

Therefore, the emitter follower decreases the impedance of the source by a factor [pic].

Transistor Current Source

(See Student Manual, Lab section 4-6)

[pic]

An “Ideal Voltage Source” supplies a constant voltage regardless of the value of the load resistor.

Similarly: An “Ideal Current Source” supplies a constant current regardless of the value of the load resistor.

[pic]

The simplest way in practice to get a not so bad constant current source is to simply use a battery and a big resistor:

[pic]

If [pic]then [pic] independent of the value of [pic].

Downside:

1) Since R must be large, you need a big V in order to appreciable I.

2) A lot of power is dissipated (and hence wasted) in the resistor R. The power “lost” in R is given by

[pic]

while the power actually “used” by the load is given by

[pic]

so if [pic]then [pic]

We can do much better using just one transistor. For example, consider the following circuit:

[pic]

The analysis of this circuit is easy:

1) The base of the transistor is held at 5 V so, from golden rule #2 [pic].

2) From Ohm’s Law [pic]

3) [pic] independent of [pic].

Thus the current through the load is constant regardless of the value of the load resistor.

Some Jargon:

Stiffness – This term is used to describe the ability of a voltage source to supply a constant voltage as the resistance of the load varies.

Compliance – This term is used to describe the ability of a current source to supply a constant current as the resistance of the load varies.

Limits on the compliance of the transistor current source:

1) When [pic] becomes large enough so that [pic] then the transistor saturates and the current will begin to drop.

2) We assumed above that [pic]always. But actually [pic] varies somewhat as [pic] changes. (This is known as the Early Effect)

3) See p. 61 of Horowitz and Hill for more.

Common Emitter Amplifier

(See Student Manual, Lab section 4-7)

[pic]

The circuit is similar to the current source except now [pic] changes.

Therefore [pic] changes.

Therefore[pic] changes.

Specifically:

1) Golden Rule #2 => [pic]

2) Ohm’s Law => [pic]

3) Rule #4 => [pic]

4) Ohm’s law => [pic]

Thus

[pic]

[pic]

(The minus sign means phase is shifted by 180 degrees.)

The Common Emitter amplifier is a voltage amplifier with a gain equal to [pic].

The input impedance of the common emitter is the same as for the emitter follower (analysis is the same):

[pic]

Output Impedance of The Common Emitter

[pic]

Recall [pic]

Easy to see that, since all output current must come through the collector resistor [pic]. Thus for the common emitter amplifier [pic].

Summary of results for emitter follower and common emitter amplifiers

| |Input Impedance |Output Impedance |Gain |

|Emitter Follower |[pic] |[pic] |1 |

|Common Emitter |[pic] |[pic] |[pic] |

These days it is becoming increasingly rare to build or use circuits that rely on a few discrete transistors. More commonly one makes use of integrated circuits (ICs), which contain anywhere from a few to millions of transistors on a single piece of silicon. The time we have spent studying "simple" circuits with one or two transistors has been useful in large part because of the insights provided into what is going on inside the ICs that will occupy much of our attention for the remainder of the course. Much of the material in labs 5 and 6 is interesting and useful, but we will skip most of it in the interest of spending our time on even more important topics. In fact, the only lab sections that you are asked to do are 5-6, which covers the push-pull amplifier.

Crossover Distortion in a Push-Pull Amplifier

(See Student Manual, Lab section 5-6)

Recall the emitter follower:

[pic]

The npn emitter follower can only "source" current into the load resistor.

Similarly, a pnp emitter follower can only "sink" current out of the load resistor.

Previously, in order to be able to follow both polarities of an ac input signal we resorted either to a dc bias scheme:

[pic]

or we used a "split" power supply that provided both positive and negative bias voltages:

Both of these schemes suffer from the drawback of requiring large “quiescent currents”. That is lots of current flow even when there is no ac signal. In addition, the voltage divider employed in the first scheme often serves to lower the input impedance of the amplifier.

[pic]

One simple alternative is the push-pull amplifier:

[pic]

When [pic] is positive the npn transistor “turns on” and “sources” current to the load while when [pic]is negative the pnp transistor “turns on” and “sinks” current from the load.

This circuit has many advantages. Unlike the "split supply follower" (Lab section 46) this circuit can drive a load that has one side grounded. Also there is no dc offset and no quiescent current. Furthermore, no voltage divider is required at the input.

There is however one potentially serious problem: The crossover distortion that results from the fact that for input voltages between -0.6 V and +0.6 V neither transistor is “on”.

The result It is that the output voltage looks like:

[pic]

See if you can hear the crossover distortion on your speakers. Shortly, we will see a beautiful way to virtually eliminate this problem with the clever use of an op amp.

Op Amps I (Lab 8)

(Do all sections Lab #8 in the Student Manual except 8-5)

Differential Amplifiers

Amplifiers such as the common emitter have a single input terminal into which one feeds a voltage [pic] measured with respect to ground. The output voltage, [pic] also measured with respect to ground is given by

[pic]

Now, ve[pic]y often we’d like to measure the difference between two input signals. Here’s one important reason why:

[pic]

Ordinary Amplifier - subject to "pick-up" of unwanted signals

Differential Amplifier – minimizes unwanted “pick up”

In the case of the differential amplifier, both the "signal lead" and the "ground lead" are subject to the same pick up; the differential amplifier "rejects this common mode signal."

(Section 6-4 of the Student Manual shows how to build a simple differential amplifier using two transistors. You can build this circuit as an optional exercise.)

Introduction to Op amps

An "ideal" differential amplifier would have the following characteristics:

• very large (nearly infinite) differential mode gain

• zero common mode gain

• infinite input impedance

• zero output impedance

So-called operational amplifiers (“op amps”) are a class of amplifiers that come amazingly close to satisfying this ideal. They are available as integrated circuits (ICs) typically consisting a 10 -20 transistors and related resistors, capacitors and diodes, all etched into a single slice of silicon. They are cheap, selling for as little as 20 cents.

The schematic symbol for an op amp looks like:

[pic]

The (+) and (-) symbols on the inputs do not mean that one in put is necessarily more positive than the other. Rather they denote the "non-inverting" and "inverting" inputs respectively. The output voltage of the op amp is given by

[pic]

where [pic] is the differential mode gain and [pic] and [pic] are the voltages applied to the non-inverting and inverting inputs respectively.

The op amps we will use come packaged in Dual In-line Packages (DIPs). The pinout diagram for the (now obsolete) 741 and the 411 op amp is shown below. By convention, the "notch" in the package serves to identify pin #1.

[pic]

(The major difference between these op amps is that the input stage of the 411 uses field effect transistors (FETs) whereas the 741 uses bipolar transistors. This results in the 411 having a substantially high input impedance than the 741.)

Note that these op amps require both a positive and a negative power supply.

Like all op amps, the 741 and 411 both have an extremely large differential mode gain (also called "open loop gain", for reasons that will become clear). You can get a feel for this fact by building the following circuit: (See lab section 8-1 of the Student Manual.)

[pic]

• Use the potentiometer on your breadboards for the variable resistor.

• Adjust the voltage divider so that the input voltage is ±1 mV.

• Due to the very large differential gain ([pic] for the 411 is > 2 x 105), you should see the output switch between + 15 V and -15 V just as the input voltage changes sign.

Input bias current

The input impedance of a real op amp is not infinite. In fact, the inputs draw a small amount of current, called the input bias current ([pic]) even when they are connected together and then shorted to ground.

To measure [pic] build the following circuit:

[pic]

Note the sign of [pic]. Is the op amp "sinking" or "sourcing" current?

If you try the same measurement with a 411 op amp, you should find a much smaller value for IB, owing to the extremely high input impedance of this op amp.

Feedback

Feedback can roughly be defined as when one "feeds" some portion of an output back into the input of a system. For example:

[pic]

The above is an example of positive feedback, meaning that the output is fed back in such a manner as to add to, or reinforce the input. The result is that the sound gets louder and louder. On the other hand, it is possible to have negative feedback, where the output is fed back so as to partially "cancel" some of the input.

[pic]

Negative feedback tends to act as a "correcting" influence, stabilizing the system at the expense of lowering the gain of the system. A good everyday example of negative feedback is the act of driving a car on the highway. If we view the "output signal" of this system as how far your eyes tell you that you've deviated from the center of your lane, then the driver can be thought to be feeding this output back into the "input" (angular position of the steering wheel) in order to achieve stable, straight driving.

[pic]

Real op amp circuits (almost) always use some form of feedback:

At first we'll use mainly negative feedback (Labs 8 and 9) and then later (Lab 10) we'll see some of the uses of positive feedback.

Assume negative feedback is in place as shown above. Now, suppose that V-, the voltage at the inverting (-) input, begins to drift slightly below ground. The large differential gain of the op amp will generate a much larger positive voltage at the input. This will cause current to flow through the feedback resistor RFB back to the inverting input, thus raising V- back toward ground. A similar argument holds for the case where V- starts drifting above ground. Thus we are lead to the first of two Golden Rules for op amps:

Op Amp Golden Rule I

With negative feedback in place, the output of the op amp will try to do whatever is necessary to keep the voltage difference between the inputs equal to zero.

The second Golden Rule follows as an immediate consequence of the very high input impedance of op amps:

Op Amp Golden Rule II

Due to their very high input impedance, the inputs of an op amp will neither source nor sink appreciable currents.

Some Op Amp Circuits

Inverting Amplifier

(See Student Manual, Lab section 8-2)

[pic]

The analysis is simplicity itself:

• Golden Rule I implies:

[pic]

• Ohm's Law implies:

[pic]

• Golden Rule II implies:

[pic]

• Ohm's Law implies:

[pic]

or

[pic]

This is closely analogous to the result we obtained for the common emitter amplifier. The gain in this case is given by:

[pic]

independent of the differential gain of the amp.

• The minus sign appearing in the gain indicates that the output is inverted.

• As is usually the case in amplifiers, the "output swing" is limited by the supply voltages.

• The input impedance of this configuration is given by:

[pic]

Note that the input impedance in this case is not necessarily high. This is a drawback of this design, but in most other ways the amplifier performs wonderfully, so if you don't need a very high input impedance this circuit is highly recommended.

• The output impedance can be found experimentally to be very low (fractions of an ohm!!) for small signals. This is a big advantage in using this op amp-based design as opposed to a common emitter amplifier. However, op amps are usually limited in the amount of load current they can supply, so the output impedance effectively increases for larger signals.

Non-Inverting Amplifier

(See Student Manual, Lab section 8-3)

[pic]

The analysis is once again pretty simple:

• Golden Rule I implies:

[pic]

• Ohm's Law implies:

[pic]

Golden Rule II implies:

[pic]

Ohm's Law implies:

[pic]

or

[pic]

Thus the gain of this configuration is given by:

[pic]

Like the inverting amplifier this configuration has a very low output impedance (for small signals). However the input impedance of this configuration is very high, which is a very attractive feature of this design. (It tends not to be quite a stable as the inverting amplifier shown above.)

Op Amp Follower

(See Student Manual, Lab section 8-4)

If we consider the non-inverting amplifier in the limit that [pic] and [pic] we see that we now have an amplifier with a gain of one, very high input impedance, and very low output impedance. It is a nearly ideal follower! (save for its inability to supply large currents. This is not an insignificant drawback, since we often want followers to supply large currents)

[pic]

Op Amp Current Source

(See Student Manual, Lab section 8-5)

If you calculate the current in the variable resistor [pic] you will see that it is independent of the value of [pic] over a wide range of resistances. Thus we have an excellent current source, better in fact than the transistor-based one that you built in Lab #4.

[pic]

Current-to-Voltage Converter

(See Student Manual, Lab section 8-6)

Photodiodes - You are probably familiar with light emitting diodes (LEDs): When current flows through a diode made of certain types of semiconductors, gallium arsenide for example, light is emitted. (A hand-held remote control unit typically uses infra-red light generated by a gallium arsenide LED.)

The reverse process is also possible: When light is absorbed at the p-n junction of a diode, a "photo-current" is generated. Since the size of the photo-current is approximately linearly proportional to the intensity of the illuminating light, a diode can serve as a light detector. In this case it is called a photodiode. (The infra-red beam emitted by your remote control unit is detected by a silicon photodiode located inside your TV set.) Surprisingly, the direction of the photo-current is in the direction opposite to the normal direction of current flow in a diode. Why this is so requires a reasonable amount of solid state physics to explain. I’ll resist the temptation and ask you to simply accept this for now as a rule handed down in the spirit of our other golden rules.

In order to "detect" the induced photo-current we would like to "convert" the current into a voltage. The simplest I-to-V converter is the humble resistor. However in this case it has two major disadvantages: 1) the photodiode is not a very compliant current source, the biggest output voltage it can sustain is about 0.5 V and 2) As a voltage is allowed to develop across the photodiode, the "response" varies, thus diminishing the linearity of the device.

[pic]

The following op amp circuit makes for a simple, but vastly improved I-to-V converter.

[pic]

An ordinary light emitting diode can be used as a photodiode (The light detection process described above is exactly the inverse of the way light is generated in an LED, where an “injected” current of electrons and holes meet up at the p-n junction and “recombine”, resulting in the emission of photons.) (If you want the best performance you wouldn’t use an LED as a light detector, the materials used to make efficient LEDs, such as gallium arsenide, don’t work as well as silicon-based photodiodes.)

If the photodiode is replaced by a phototransistor, the current gain of the phototransistor makes for a much more sensitive light detector. (We need to now use a smaller feedback resistor; otherwise the circuit is too sensitive.)

[pic]

The figure above shows how to wire an npn phototransistor with an op amp configured as a current to voltage converter. Note that in this case, in contrast to the situation with a photodiode, the current flow is in the direction of the arrow.

When you get this working, make sure to check out the suggestion on page 182 of the Student Manual where you make an oscilloscope “part of the feedback loop”. By simply aiming the phototransistor at an oscilloscope trace that is monitoring the output of the above circuit you should see the trace move to “avoid”, or deflect around, the phototransistor. This trick, which shows off some of the power (and magic) of the feedback principle, will work best in a darkened room.

Summing Amplifier

[pic]

The currents through the input resistor add at the summing junction and flow through the feedback resistor:

[pic]

The summing junction is a virtual ground so

[pic]

[pic]

If [pic] then

[pic]

so the output voltage is proportional to the sum of the input voltages.

Summing Amplifier as a Digital to Analog Converter

(See Student Manual, Lab section 8-7)

Here's a preview of digital electronics. Suppose we let 5 V represent a binary "1" and 0 V represent a binary "0". The following "weighted summing amplifier" produces an analog output voltage that is proportional to a 4-bit input number:

[pic]

[pic]

[pic]

[pic]

[pic]

Note that:

[pic]

so that

[pic]

[pic]

[pic]

Thus the circuit acts as a digital to analog converter.

Op Amp Buffer for Push-Pull Amplifier

(See Student Manual, Lab section 8-8)

Recall the push-pull amplifier:

[pic]

Advantages:

• useful for supplying large currents to low impedance loads

• can follow both positive and negative input voltages

• no large quiescent currents

Disadvantages:

• crossover distortion

• provides no gain; must rely on another "stage" to provide gain

Goal: Use relatively high impedance small signal source (e.g. a microphone) to provide a lot of current to a low impedance load (e.g. a loud speaker). Specifically, suppose we try to design a circuit that can take a 50 mV signal produced by microphone (source impedance = 2.2 kΩ ) and produce an 2 volt signal into a 8Ω load. Obviously we need a circuit with an overall gain of 40.

First try:

[pic]

This works except you can see (and hear?) the crossover distortion:

[pic]

Second try:

[pic]

Look at [pic] with your oscilloscope. The crossover distortion disappears! What happened!?

Analysis:

• From Golden Rule I:

[pic]

• From Ohm's Law and Golden Rule II:

[pic]

• From Ohm's Law:

[pic]

[pic]

We see from this analysis that the output voltage should always equal -20 times the input voltage, whatever the input voltage is! Thus this result confirms that there should be no crossover distortion, yet one can't help feeling that a swindle has just taken place. We still haven't really answered the question "Where did the crossover distortion go?"

The answer is that, according to Golden Rule I, the output of the op amp (labeled [pic] in the drawing above) will try to do whatever it takes to keep the inputs at the same voltage. In the present case this apparently includes compensating for the crossover distortion:

[pic]

• Whenever [pic] the circuit is "sourcing" current into the load resistor. (Note that very little current flows into the load via the 400 kΩ resistor.) This means the npn transistor in the push-pull amplifier in "on" and [pic].

• Whenever [pic] the circuit is "sinking" current from the load resistor. This means the pnp transistor in the push-pull amplifier in "on" and [pic].

Therefore, whenever [pic] crosses through 0 V, the op amp must abruptly change [pic] between +0.6 V and - 0.6 V. Real op amps cannot change their output voltages instantly; they have a limited "slew rate", as we shall soon see. This limits the ability of a real op amp to fully compensate for the crossover distortion.

Op Amps II (Lab #9)

(Do sections 9-1 through 9-5 in Student Manual .)

Op Amp Limitations

Real op amps do not quite behave in the idealized manner that we have assumed. For example, there are mild violations of the two Golden Rules. Corrected versions of the Golden Rules would look like:

Modified Golden Rule I

With negative feedback in place, the output of the op amp will try to do whatever is necessary to keep the voltage difference between the inputs equal to a very small voltage difference, called the offset voltage [pic].

Typical offset voltages for the op amps that we use in the lab are ~ a few mV.

Modified Op Amp Golden Rule II

Due to their very high input impedance, the inputs of an op amp will sink a very small current, called the input bias current [pic].

Because of their FET input stage, the typical input bias currents for a 411 op amp are extremely low, ~ 3 pA (That's picoamps. 1 pA =10-12 A.) The 741 op amps have input bias currents of a few nanoamps.

Another op amp limitation is relatively small amount of output current that they can supply. The 411 can supply at most 25 mA to a load.

(There exist specialized op amps for delivering more substantial currents. We have for example in our parts shelf the “386” op amp which is designed for audio applications and is capable of directly driving a small 8 ohm speaker.)

Perhaps the most noteworthy op amp limitation is the fact that the open loop gain of the op amp "rolls off" at high frequencies. The gain curve for a 411 op amp is shown below:

[pic]

This drop in gain limits the op amps ability to respond to high frequency signals, resulting in a limited slew rate for the op amp. The slew rate is defined as the maximum rate at which the output of the op amp's output can change. The 411 has a slew rate of about 15 V/_s while the 741 has a slew rate of about 0.5 V/_s.

(This roll off in gain is actually intentional. In effect the output of the op amp is fed through a low pass filter with a [pic] of about 100 Hz. The reason for doing this has to do with op amp stability. Without this roll-off the op amp would be subject to spontaneous high frequency oscillations. These high frequency oscillations result when stray capacitance causes phase shifts in the feedback loop, so that the feedback unintentionally becomes positive for high frequency signals. See Horowitz and Hill section 4.33 for more on this.)

The effect of the slew rate can easily be seen by observing how an op amp follower responds to a square wave input:

[pic]

(The 10 k resistor serves to limit the input current in the event that a clumsy user allows [pic] to exceed the supply voltages.)

Note that as the amplitude of the square wave increases, the distortion will become more apparent. As noted previously, the slew rate limits the ability of the op amp to compensate for the crossover distortion in the push-pull buffer circuit.

Op Amp Integrator

Recall the simple op amp integrator. It worked well only when [pic] For low frequency signals (ω 0 negative feedback is in place and the circuit acts as a follower. For [pic], there is no feedback and the op-amp output saturates at the negative supply voltage while [pic] remains at ground. Try using this circuit, along with the digital scope to look at your rectified claps. For the ac coupled microphone, it is necessary to tie the non-inverting input of the op amp to ground through a 300 kΩ resistor. (Need to have dc bias.)

Actually, if you think about it, what you'd really like for the clap sensor is a peak detector. Try replacing the 10 kΩ resistor with a 0.1 μF capacitor. What happens to the signal produced on the digital scope. Can you figure out what is going on?

Op Amps III (Lab #10)

(Do sections 10-1 through 10-2 in Student Manual .)

Comparator

A comparator is a device which (not surprisingly) compares two different voltages and indicates which of the two voltages is larger. A simple op amp without any feedback can serve as a comparator:

[pic]

The output of the op amp will swing to one of the two supply voltages, depending on which of the inputs is larger. (In other words, the output "changes state", depending on which of the two input voltages is larger.)

While any garden variety op amp (e.g., the 411) can serve as a comparator, chips that are especially designed to act as comparators (e.g., the 311) offer improved performance. For example, for the above application, as well as many others, it is desirable for the comparator to have as fast a slew rate as possible. To achieve a fast slew rate, comparators such as the 311 make use of an open collector output, as shown below. With an external, user-supplied pull-up resistor connected to +5 V, the comparator output will switch between near ground and 5 V, depending on whether the output transistor is "on" or "off":

[pic]

When [pic], the output transistor is "off" and [pic].

When [pic], the output transistor is "on", the transistor saturates, and [pic].

Thus the two output states of the comparator correspond to digital "1" and "0" states.

Application: Digital Voltmeter (DVM)

You’ve used a comparator before. It was part of scheme that you used to do analog to digital conversions with the LogoChip using your a simple digital to analog converter:

[pic]

A comparator can also be used in a different approach to analog to digital conversion that is typically employed when high precision conversions are required, for example in your digital voltmeters.

[pic]

The voltage that is to be measured is applied to one input of an op amp while the DVM supplies an internally generated voltage ramp to the other op amp input. The DVM measures the time it takes for the op amp to switch its output state. This approach tends to give very accurate voltage measurements, largely because it is quite easy to measure time intervals to a very high precision with digital circuits.

A down side to this approach is that it takes a relatively long time to make a typical measurement as you wait for the voltage ramp to reach the target voltage, but in a device like a digital voltmeter this presents no problem; you’re generally quite happy to have the measurement take a few milliseconds longer if it results in a more accurate measurement.

(Aside: How do you generate a voltage ramp? One simple way is to use an op amp integrator circuit It would be interesting to try making high precision voltage measurements with a LogoChip based on this scheme.

[pic]

.)

Schmitt Trigger

A weakness in the above comparator circuits is that noise in the input signal can lead to multiple transitions at the output. If you think about an application like the clap sensor you can see why this might be a problem:

[pic]

An elegant solution is the Schmitt trigger, which makes use of positive feedback to largely eliminate the multiple transitions:

[pic]

The voltage at the non-inverting input, V+, determines the "threshold" voltage. The key point is that because of the positive feedback provided by the 100 k resistor in the above circuit, the value of the threshold voltage V+ will depend on the output state [pic].

When [pic] then [pic]

When [pic] then [pic].

[pic]

This means that there are two different thresholds in the circuit, the one which determines when the output will switch from "high" to "low" is equal to +0.5 V while the one which determines when the output will switch from "low" to "high" is equal to 0 V.

As you can see from the drawing above, this scheme avoids the multiple transitions problem.

Design Challenge on ps#4: Design a comparator circuit to be used with the peak rectifier you built last time that enables you to set a "threshold level" for detecting claps. The output of the comparator should give one digital "pulse" per clap. See the worked example on pages 227-231 of the Student Manual for help. (It will then be fairly easy to use a LogoChip to count these claps.)

RC Relaxation Oscillator

Ever wonder how your function generators can “generate” a periodically varying voltage signal? Here is a simple way to generate a square wave with a user-determined frequency:

[pic]

Note that in this RC relaxation oscillator circuit both positive and negative feedback are used and there is no input signal. Assume that when the power to the circuit is first turned on the op amp output goes into positive saturation (it's actually a toss-up which way it will go, but as you'll see, it doesn't matter.) The voltage at the non-inverting input [pic] will thus be at about +1.3 volts due to the voltage divider formed by the 100k and 10 k series resistors.

[pic]

The capacitor (and hence [pic]) begins charging up toward +15 V, with time constant RC. When it reaches the threshold voltage of about +1.3 V, the op amp's output will quickly swing towards the negative power supply voltage (-15 V in this case), changing the voltage at the non-inverting input [pic] to about +1.3 volts. The capacitor (and hence [pic]) begins discharging down toward -15 V, again with a time constant RC. Switching will now occur at the new threshold voltage of -1.3 V. By varying the RC time constant the frequency of the square wave can be adjusted.

Oscillators based on this principle are known as relaxation oscillators. They are inexpensive and simple, and with careful design they can be made quite stable in frequency. They can be used for example to provide a simple very tiny “clock” for a microcontroller. Microcontrollers these days are becoming quite small and inexpensive (much smaller than a single 3904 transistor and costing about 80 cents, which is great if you want to start building computation into everyday objects.) A “crystal-based clock” circuit can end up being larger (and more expensive) than the microcontroller itself, so simple RC based relaxation oscillators, which can be microscopic in both size and cost when built directly into the microcontroller IC, can be quite useful at times.

A real op amp circuit

Stabilizing the length of a diode laser cavity to a part in 108 using a 40 cent op amp

The next page shows the circuit used to stabilize the length of a diode laser cavity to a part in 108 as part of the laser cooling experiment that we are working on down in the laser lab. It features 3 garden variety op amps housed inside a 40 cent op amp package. You should be able to recognize a lot of familiar features in this circuit.

[pic]

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