Session 9 - Friday, February 27

An AC microphone amplifier

Microphone + amplifier + speaker = fun! :D
In this lab we used all the knowledge we've gained thus far in the semester to build a circuit that takes in a signal from a microphone, amplifies that signal, and then feeds it to a speaker.

Setting the Gain

To get us started we were given the diagram above. We were then tasked with designing CKT #1 so that there would be a reasonable voltage gain and reasonable was defined as having the output voltage be in the range 0V to 3.5V. 

Given that the input voltage coming from the microphone will be approximately 15mV we need to create CKT #1 so that the op amp will have a gain of roughly 100 in order to put our output voltage at 1.5V which is within the reasonable range described in the previous paragraph. To accomplish this we chose to make CKT #1 a resistor with a value of 10kΩ. With a 10kΩ resistor the ratio of CKT #1 to the 1MΩ resistor in the formula Vout = (1+ R1/R2)*Vin gives us a gain of 101 which is approximately 100.

We then built this circuit with a 10kΩ resistor in the place of CKT #1 and drove it with a 1kHz sine wave at Vpp = 20mV and no DC offset in order to test that it works as expected. As expected from the lab write-up the output is clipped on the low-voltage (negative voltage) side since the input is symmetric about zero due to being a sine wave and the output cannot go below the value of ground which is 0V. Also we found that the ration of input to output amplitudes in the unclipped regions was approximately 100 as we expected as can be seen in the photo below:

 The scale of the yellow is 500mV while the scale of the blue is 5mV showing that there is in fact a 100x increase.

Next we increased the input amplitude gradually in steps of 5mV and watched what happened to the output. When we reached input values above 4V the output was clipped at roughly 4V. This makes sense because the maximum output voltage is approximately 1.5V below the positive supply which is +5V. You can see this clipping in the image below:

Volume Control

The next step in our mission to build an awesome microphone to speaker circuit is to have a volume control knob. First, we needed some sound output so we connected a buzzer as our speaker. The buzzer had one pin grounded and the other connected to the amplifier output. The presence of the buzzer in the circuit changed the output signal as can be seen in the photo below: 

Now that we had a volume output, we modified our CKT #1 to give us control over the volume. We did this by adding a potentiometer in series with the resistor that was previously all of CKT#1 with the potentiometer's other side being connected to ground. Since the potentiometer has a resistance of its own ranging from 0Ω to 10kΩ, we swapped the 10kΩ resistor that was previously in CKT#1 with a 5kΩ resistor. This will give our amplifier a gain between 200x when potentiometer is set to 0Ω and 67x when potentiometer is at 10kΩ. Gains were calculated from the ratio of 1MΩ and the resistance of CKT #1.

Avoiding Clipping
Now we must change the circuit so that the signal doesn't get clipped below 0V in order to "have room" for the voltage swings of the audio signal. To do this we will make the signal sit on a DC offset. The lab write-up tells us this offset will be 1.75V, however when we drive the circuit with a 20mVpp 1kHz sine wave from the function generator and gradually raise the DC offset by 10mV at a time we find that the output saturates and hits a rail at only a DC offset of approximately 70mV. This is because the DC offset is being routed through the op amp and also receiving a gain of 100.

To prevent the DC offset portion of the signal from being amplified we want the op amp to act as a follower at DC, but an amplifier at AC. This is what a high-pass filter does, so we accomplished this by further modifying CKT#1 to include a high-pass filter. We added a capacitor between the 5kΩ resistor and the rest of the circuit. To select the value of the capacitor we used the formula 1/(w*C) = Zcap < 5kΩ and the fact that we need 20Hz to pass through the amplifier to calculate the capacitance from the following: 1/(2*pi*20*C) = 5000Ω. We end up with C = 1.5 microFarads and since we didn't have a 1.6 microFarad capacitor we used one 1.2 microFarad capacitor and three 0.1 microFarad capacitors in parallel to achieve that value. You can see the resulting circuit (and the several capacitors) in the photo below:

We confirmed that the DC offset at the output is equal to the DC offset at the input while the AC signal is amplified and you can see this in the photo below: 

However, the amplifier does not have the same gain at all frequencies and thus the amplifier has distortion. The change in gain between 20Hz and 20kHz was 2.16V to 780mV and thus we know that the amplifier is distorted by a factor of 3.

Adding a Microphone

Now that we have a circuit that will take in a signal and output a volume controlled audio output through the speaker that is not clipped and has "space" to swing as needed, we need to change the circuit so that the input signal comes from the microphone instead of the function generator. We added the microphone to a separate section of the breadboard and set it up as shown in the photo below:

We confirmed that the microphone worked by connecting the Vout of the circuit shown above to the oscilloscope. We could see amplitude changes on the oscilloscope when we spoke into the microphone, but as we have no clue what the signal pattern of our voice should look like we also tested it with a tuning fork. Tuning forks produce a sine wave signal and you can see that test in the video below:

It's important to note that even though the signal in the video is a sine wave it did not come from the signal generator, but from the microphone. We just wanted to make sure everything worked as expected and thus used a tuning fork so that we would know what the output should look like (a sine wave). We tried several different tuning forks and we also raised and lowered our voices in pitch while talking in to the microphone and the higher the pitch the faster/higher the frequency of the signal output. 

Connecting the Microphone to Amplifier

Now that we have confirmed that the microphone is working as expected, we need to connected to the amplifier and speaker. However, we cannot just directly them because the microphone's output is a +5V DC signal with a very small AC signal of roughly 20mV on top. This DC signal is too much for our amplifier so we need to remove it from the microphone signal before it reaches the amplifier. However, as we know from the "Avoiding Clipping" section of this lab, we do need some DC offset, specifically 1.75V. To accomplish this we need to design another circuit, CKT#2, to connect the microphone to the rest of the circuit via the (+) input of the amplifier.

First, in order to cut the +5V DC offset off of the microphone signal and isolate the AC signal we need to use a high-pass filter. Then to add back in a 1.75V portion of a DC voltage we need to use a voltage divider made of two resistors. We can combine the high-pass filter and the voltage divider so that the the bottom resistor of both is the same. A sketch of this circuit can be seen below:

Now we just need to determine the values of the components in the sketch above. We will use a +5V voltage source since we have the option of 5 or 12V. The two resistors need to have a ratio of 0.65 in order to pass 1.75V from the 5V voltage source and we want them to be very high in order to have lower power dissipation (P = V^2/R). Thus we chose the values of 1MΩ and 560kΩ (closest to 538kΩ which is the calculated value) for the resistors. Now for the capacitor, we know we want our high-pass filter to pass 20Hz so we use the formula 1/(2*pi*R*20) = C to get a capacitance value of 0.113 microFarads which we rounded to 0.1 microFarads.

Now that all of our components have values, we build the circuit! We lengthened the distance between the microphone and speaker using more of the cables to connect the pins of the microphone and the speaker to the correct spots on the bread board. The circuit worked!!! And it was very fun to play with :).

Unfortunately I don't have any good videos of the microphone and speaker working because there was so much ambient noise in the room and the microphone on my phone isn't very good.

During my playing with the speaker though, I noticed that high frequencies came through more clearly and more loudly than low frequencies. This was especially noticeable when I used the child's xylophone that was in the blue cabinet to produce an octave of frequencies. I think the clarity of the high tones versus the low tones has to do with the properties of our high pass-filter, but it might also be because of the properties of the buzzer itself or even human hearing which is logarithmic. It would be very interesting to explore this further and see what is the cause and how best to compensate. I wonder if this is why speakers for phones and smaller devices often sound "tinny" and don't produce bass notes well.

Session 8 - Tuesday, February 24

Op Amp Current Source

If you have a variable resistor as R2 in the configuration shown below:

We can apply the Op Amp Golden Rules and Ohm's Law to show the following:

1. V- = V+ = Vin
2. IR1 = V- / R1 = Vin / R1
3. IR2 = IR1 = Vin / R1

Thus we know that the current flowing through R2 is independent of the value of R2 over a wide range of resistances and therefore this circuit is an excellent current source.

Op Amp Current Source, experimentally

Student Manual Section 8-5

In this lab we create an op amp current source by connecting a 15kΩ resistor from a +12V voltage source to a 1kΩ resistor which is then connected to ground. This is a voltage divider which we will connect the middle of to the (+) input pin of the op amp. The (-) input pin is connected to a 180Ω resistor in series with ground. The output of the op amp is connected to a 10k "load" potentiometer and digital multimeter in series and then connected between the (-) input and the 180Ω resistor.

As we vary the current source with the potentiometer we measure currents between 1.08mA and 4.16mA. Because of our 15kΩ and 1kΩ voltage divider we know that the voltage input is +0.75V or 1/16th of the +12V supply. Since we know from the golden rules that V- = V+, V- is also 0.75V. This 0.75V is connected to ground by a 180Ω resistor and using the law V=IR we know that the current should be at most 4.2mA and this is very near our measurement of 4.16mA showing that the results are as expected.

This current source is much more precise and stable than the transistor current source, but it unfortunately needs a "floating load" meaning that the load is not connected to ground on either side. Another disadvantage of this current source is that its speed limitations are not very good with either output current of load impedance varying at microsecond speeds.

Photodiodes
We are already familiar with LEDs from a previous lab and we know that one photon of light is emitted for every electron of current through the LED. A photodiode is a reverse of this process with an electron of current produced for every photon of light absorbed. Since the current produced is approximately linearly proportional to the intensity of light, photodiodes serve as light detectors.

For us to measure the induced photo-current from a photodiode we need to convert the photocurrent into a voltage. Usually a resistor is the standard current to voltage converter, but this will not work well for a photodiode because it is not a very good current source and thus the biggest output voltage is can sustain is about 0.5V and also as voltage develops across the photodiode the linearity of the device is diminished because of the varying response.

Op Amp Current-to-Voltage Converter
In order to have a better current source than our photodiode, we will be using a phototransistor which is essentially a photodiode with gain, making it a much more sensitive light detector.

It is important to note that the current for a photodiode flows towards the voltage source which is backwards from what we would expect, but for phototransistors the current flows in the more logical direction towards the next circuit element.

Current-to-Voltage Converter, experimentally
Lab 8-6 in the Student Manual
For this lab we constructed the circuit shown below with the +15V source changed to +12V and the 100kΩ resistor changed to a 10kΩ.

We measured Vout to be 0.7V and thus the average input photocurrent in this circuit is I = V/R = (0.7V)/10kΩ = 70 micro-amps. The output as seen on the oscilloscope can be seen in the photo below:

We then found the percentage modulation of this circuit by taking the change in Vpp which was 620mV and the change in Vbase which was 800mV. Putting these values into the formula (Vpp/Vbase)*100 = percent-modulation gave us a percent modulation value of 77.5%

Summing Amplifier
Op amps get there name from being "operational amplifiers" meaning that they can be configured to do mathematical operations. To do the mathematical operation of addition you can configure the op amp to take in two voltages sources by getting a current from applying that voltage to a resistor and adding the resulting currents together at a "summing junction" so that the sum of the two currents is fed into the feedback resistor. This summing junction is a virtual ground so we know that if the two resistors between the voltage sources and the summing junction are equal then Vout = (-Rfeedback / Rsource) * (V1 + V2)

Question: Can you design a circuit using op amps that subtracts two voltages?
To create an op amp that would subtract two voltage one would just need to invert one of the voltages using an inverting follower op amp and then feed the inverted voltage and the other voltage into a summing amplifier.

Summing Amplifier as a Digital to Analog Converter
Another use of the summing amplifier is as a digital to analog converter. If we say 6V represents a binary  "1" and 0V represents a binary "0" then by proportionally increasing the resistance by a factor of two from each power source for each additional digit we can create a digital to analog converter. To explain further the most significant digit will have the smallest resistor , let's say a 1kΩ, and the next digit will have a 2kΩ resistor, and the next a 4kΩ, and so on. Using the law V = IR this means that the smaller the resistor the larger the current at the summing junction leading to a weighted sum. Since the number coming in is in binary and the resistors are bases of two, the weighted sum is equivalent to the analog base ten current and thus the circuit acts as a digital to analog converter.

Summing Amplifier, experimentally
Student Manual Section 8-7
In this lab we constructed the circuit shown in the diagram below, but with the + and - 15V sources changed to + and - 12V and also the 15kΩ resistor changed to a 12kΩ.

This circuit sums a DC level with the input signal allowing you to add a DC offset to the signal. We applied a 5Vpp sine wave at 1kHz to the input and then adjusted the offset voltage. As we approached the end points of our offset range we noticed that the voltage peaks were being clipped. We tried this again with smaller input signal amplitudes and found that with a small enough Vpp we could avoid clipping all together because the Vout never got near the 12V limit. We also experimented with the AC coupling option and found that if the oscilloscope was AC coupled the signal would "bounce" briefly but then settled back to 0V. 

Op Amp Limitations
As has been mentioned in prior blog posts, real op amps are not ideal. To deal with the reality of op amps instead of the ideal version we can use the following two modified golden rules:

• Modified Golden Rule IWith 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, Vos.
• Modified Golden Rule IIDue to their very high input impedance, the inputs of an op amp will sink a very small current, called the input bias current, Ibias.

Typical input bias currents for a 411 op amp are on the order of 3 pico-amps and a 411 op amp can supply at most 25mA. Another very important limitation is that at high frequencies the open loop gain of the op amp will "roll off"and decrease as frequency increases because of the slew rate. The slew rate for the 411 op amp is about 15V per micro-second. We will explore the effects of the slew rate in lab 9-1.
Op Amp Limitations, experimentally
Lab 9-1 in the Student Manual
For this lab we constructed a circuit with the (-) input of the op amp connected directly to Vout and the (+) input connected via a 10kΩ to the input signal. Driving this circuit with a square input wave of Vpp = 10V we found the falling slope to be 12.8mV per nanosecond and the rising slope to be 10.4mV per nanosecond. These measurements give us a slew rate for slewing down to be 12.8V per microsecond and the slew rate for slewing up to be 10.4V per microsecond.

We then changed the input to be a sine wave with an amplitude of 10V and measured the frequency at which the output waveform begins to distort. We found that frequency to be approximately 500kHz and the output appeared to look like a sawtooth wave. Using the equation 2*pi*f*A = 2*pi*500kHz*10 = 31.4V per microsecond. This did not match the expected value of  approximately 10 to 13V per microsecond we had just measured with the square wave.

Because sine waves were difficult for us to notice distortions on until the distortions were fairly significant, we changed back to a square wave and again measured the frequency at which distortions began. With a square wave we found this frequency to be 10kHz and the new calculated slew rate to be 6.3 V per microsecond, much closer than 31.4 to our expected values of 10-13V. I expect that if we had a more accurate method of measuring than just estimating with our eyes that we could get an even closer value.

Op Amp Integrator
As mentioned before, op amps can perform several mathematical operations. We have explored summation and now we will consider integration. An op amp integrator consists of a capacitor with the (+) input grounded, the (-) input connected via a resistor to Vin and a feedback loop containing a capacitor connecting (-) input to Vout.

Using an op amp to integrate, instead of the RC integrator circuit we explored earlier in the semester, allows us to remove the restriction of Vout << Vin. Using the Golden Rules and the definition of capacitance we can derive the expression Vout(t) = -(1/(R*C))*INTEGRAL(Vin(t))dt and this result will work for a wide range of frequencies.

This op amp integrator works very well which can cause a problem if there is even a small DC offset on the input voltage. Over time the offset is accumulated on the capacitor until the Vout "drifts to one of the supply rails".

We can correct this drifting by adding a large by-pass resistor to the feedback loop in parallel to the capacitor. This will give the DC current another path through the resistor and thus prevent it from accumulating on the capacitor.

Op Amp Integrator, experimentally
Lab 9-2 int he Student Manual
In this lab we constructed the circuit shown in the image below:

We then drove this circuit with a 1kHz square wave. As the lab write-up tells us, this circuit is sensitive to small DC offsets which causes the output to go into saturation near the 12V supplies or to not be symmetric about 0V. We used the function generator's offset controls to prevent these problems.

Next we used the component values to predict the triangle wave amplitude produced from an input of 2Vpp, 500Hz square wave. A 2Vpp means a value of +-1V for Vin and thus a current of (1V/100kΩ) = 1*10^(-5)A which we can use in the formula deltaV = (I/C)*delta(t) with the values t = 1ms and C = 0.01 microFarad. This gives us a change in voltage of -1V as the gain is negative.

The 10MΩ resistor serves to keep the output signal from railing. You can see the results of removing the 10MΩ resistor in the video below:

As you can see, without the resistor the DC current doesn't have another path and is forced to build up on the capacitor causing the output to drift to the rail.

Session 7 - Friday, February 20

Operational Amplifiers
Amplification and Gain

We have previously explored the use of transistors as electronic switches, but in this lab we explore their uses for amplifying electronic signals.

If you want to connect a microphone "source" to a speaker "load", it won't work well at all if you just directly connect them. This is for two main reasons. The first reason is that microphones don't produce very large voltage signals and there wouldn't be enough voltage to power the speaker sufficiently. The second reason is that microphone output impedance is usually very high (~1kΩ), while the input impedance of speakers is usually very low (~8Ω). This means that the signal would be very strongly attenuated as it moved from the microphone to the speaker and since microphones already produce very small voltages, this would leave almost no voltage for the speaker.

This problem is a common one in electronics (needing to amplify a small electronic signal), and we can learn to solve it with the help of the following two statements:

1. The output impedance of a device can be thought of as a measure of its ability to provide current to a load. (smaller Zout = better current supply to load)
2. When connecting a "load" to a "source" the output impedance of the source must be small compared to the input impedance of the load. (Zout << Zin)

Differential Amplifiers

Measuring the difference between two input signals is important in order to minimize unwanted pick-up.

For differential amplifiers, the two leads (ground and signal) are affected by the same pick-up, but in a differential amplifier this common pick-up mode is rejected.

Ideal differential amplifiers (which reject the pick-up) have the following the following characteristics:

• nearly infinite differential mode gain (Gdiff = Vout / ((V+) - (V-)))
• zero common mode gain (zero Vout when the two inputs are given identical non-zero voltages)
• infinite input impedance
• zero output impedance

As is usually the case, ideal things don't exist in the real world. However the operational amplifier, also known as "op amp" is amazingly close to this ideal. Op amps have an inverting input symbolized with (-) and a non-inverting input symbolized by (+). Op amps also have other pins, but in this lab we are only concerned with the non-inverting and inverting inputs, the positive and negative voltage sources and the output pin. We get our +12V and -12V voltage sources from a 5-pin DIN which we use to modify the wall's power supply and plug it into the bread board.

Open Loop Gain of an Op Amp, Experimentally
Lab 8-1 in the Student Manual

We began this lab by carefully setting up the circuit with 0.01 microFarad capacitors connecting the positive and negative voltage sources to ground in order to "decouple" the power supplies. Decoupling stabilizes the power supply and thus minimizes "fuzz" or noise on the oscilloscopes which will make for better results later.

We then connected our circuit to a potentiometer and when we spun the potentiometer the output reading on the oscilloscope alternated between the two rails of ~+12V and -12V. Some groups were able to delicately spin the pot so that it just lined up so that the output was ~0V, but we didn't have a very good pot (nor hand stability). The behavior of jumping between rails is as expected though because the gain is 200V/mV and thus even 1mV is multiplied so that it is 200V, this means that you would have to adjust the pot so that the voltage is between -0.06mV and +0.06mV in order to see anything other than a railed value of 12, which is very difficult to do, as you can see in the video below:

Feedback
Feedback is what we call taking some of the output and feeding it back into the input. Positive feedback is when we have a feedback designed to reinforce the input. This reinforcement of the input causes the sound to get louder and louder and the growing outputs keep getting resubmitted to the input causing the next output to be even greater. This kind of feedback is what causes the loud screeching sounds caused when you sometimes get the microphone too close to the speaker in common audio systems.

This screeching is not very useful, so let's look at the other type of feedback called negative feedback. Negative feedback pulls the output into the input in such a way that the output partially "cancels" some of the input. This stabilizes the system and corrects it when it varies too far from which is good, but unfortunately it also lowers the gain of the system, which is not so good, but is usually worth the benefits of the stabilization.

Almost all op amp circuits use some form of feedback and we will be exploring this later in this lab.

Op Amp Golden Rules

  1. 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.

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

Inverting Amplifier
By the application of the Op Amp Golden Rules and Ohm's law to a circuit with a negative feedback, we know the following:

1.  V- = V+ = 0
2. IR1 = Vin/R1
3. IR1 = IR2
4. Vout = 0 - IR2 * R2 = -(Vin/R1) * R2
5. Vout = (-R2/R1) * Vin (simplification of 4)

It is important to note that if R2>R1 the signal will be amplified and that for the circuit to work as an inverter the differential gain needs to be large, but the gain is independent of the differential gain of the amp. Also note that the minus sign in 4. and 5. above show that the output is 180degrees out of phase relative to the input and is thus inverted, which is why it is called an inverting amplifier.

Two other quick things to note before doing the lab 8-2; the input impedance of this circuit is Zin = Vin/Iin = R1, and the maximum output voltage will be ~1 volt less than the positive supply voltage while the minimum output voltage will be ~1 volt greater than the negative supply voltage. 

Inverting Amplifier, Experimentally
Lab 8-2 in the Student Manual
For this lab we constructed an inverting amplifier with a 1k resistor between the input voltage and the (-) input and a 10k resistor in the negative feedback loop. Also (+) input was grounded. We powered the op amp with +-12V and drove the circuit with a 1kHz sine wave. The output is shown in the photo below:

The gain is 10x as you can you see from the photo above by the fact that the scale for CH1 (output) is 5V while the scale for CH2(input) is 500mV. The maximum output swing is 1V for the input and 10V for the output and we can see that the output is 180 degrees out of phase with the input. We repeated this with a triangle wave as you can see in the photo below:

We then repeated it again with sine waves of various frequencies. As you can see in the video below, the amplifier stops working well as the frequencies get higher.

This has to do with the slew rate, but for the rest of this session we will treat the op amp as ideal with good slewing (we're on our honeymoon with the op amp :] )

Going back to driving the circuit with a 1kHz sine wave we measured the input impedance of the amplifier circuit by adding a 1kΩ resistor in series with the input. We found Vin to be 1V and Vout to be 520mV, telling us that the input impedance is 1kΩ since there is a voltage drop of roughly 50%.

We then attempted to measure the output impedance. After a long time fiddling with various small resistors and a very cool resistor box and lowering our voltage as much as possible, we could only conclude that the Zout is significantly smaller than 4Ω. This difficulty in measuring the output impedance has to do with the effects of limit on op amp output current. We hit a limit with our current output and thus couldn't conclude anything from results past this limit. You can see evidence of this limit in the clipping of the signal in the photo below:

Non-Inverting Amplifier
For this part, we explored the properties of a non-inverting amplifier which has Vin connected to the (+) input and has the (-) input connected to ground with a resistor between it and ground. It also has a feedback loop from Vout to (-) input with a resistor.

Applying the Op Amp Golden Rules and Ohm's Law to this circuit reveals the following properties:

1. V- = V+ = Vin
2. IR1 = (V-)/R1 = Vin/R1
3. IR1 = IR2 
4. Vout = V- + IR2 * R2 = Vin + (Vin/R1) * R2
5. Vout = (1 + R2/R1) * Vin (simplification of 4)

The gain for this setup is G = (1 + R2/R1). The output will be in phase with the input, which is why it is called a non-inverting amplifier. Similar to the inverting, this amplifier has a very small output impedance for small signals. Also, because the input signal is connected directly to the input of the op amp, the input impedance is very high. However, this circuit is not as stable as the inverting amplifier at high gains.

Non-Inverting Amplifier, Experimentally
Lab 8-3 in the Student Manual
For this lab we built a non-inverting amplifier with a 1kΩ resistor between (-) input and ground and a 10kΩ in the feedback loop and drove it with a 1kHz sine wave signal. We found Vin to be 1.06V and Vout to be 11.3V, giving a gain of 10.66 or ~11x. We then tried to measure the circuit's input impedance by putting a 1MΩ resistor in series with the input, and since the Vout was approximately the same we concluded that the input impedance must be >> 1MΩ.

This non-inverting amplifier maintains the low output impedance we measured in the inverting amplifier because it is essential the same circuit with the only difference being whether (+) or (-) input is connected to Vin. Since every other element is the same, we know that we maintain the low output impedance.

Op Amp Follower 
An Op Amp Follower is a special case of the non-inverting amplifier. When a non-inverting amplifier approaches the limit of R2 = 0 and R1 = infinity, the gain becomes one. Such a circuit is called a follower because the output signal "follows" the input signal.

Followers are useful because they maintain the high input impedance and very low output impedance of a non-inverting amplifier and thus can act as a buffer between a source and load without changing the output signal (because it has a gain of 1).

Op Amps are almost ideal followers, with their drawback being the inability to supply large currents.

Op Amp Follower, Experimentally
Lab 8-4 in the Student Manual
For this lab, we built the very simple circuit of a follower by connecting the Vin directly to the (+) input and connected the (-) input to Vout with no resistors. We confirmed that this follower circuit has the same Zin (very large) and Zout (very small) by briefly trying to take this measurements by using resistors in series.

Rather than having us measure the follower's input impedance, we added a 1kΩ resistor in series with the output in order to show that the feedback is producing the low output impedance. By looking at Vout with and without a 1kΩ load attached and with the feedback loop connected either before or after the 1kΩ resistor (not the load resistor). When the feedback loop is connected before the 1kΩ resistor which is in series with the 1kΩ load we see a ~50% attenuation in the signal which is expected because half the voltage drop is occurring across each 1kΩ resistor. However, when the feedback loop is connected in between the two 1kΩ, that is the 1kΩ resistor is treated as a part of the follower, we see no attenuation of the signal.

This results are not at all surprising since a follower is simply a special case of the non-inverting amplifier. As the lab writeup said, "No surprises here: Rout had better be 1kΩ."