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:
- 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)
- 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:
- V- = V+ = 0
- IR1 = Vin/R1
- IR1 = IR2
- Vout = 0 - IR2 * R2 = -(Vin/R1) * R2
- 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:
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:
- V- = V+ = Vin
- IR1 = (V-)/R1 = Vin/R1
- IR1 = IR2
- Vout = V- + IR2 * R2 = Vin + (Vin/R1) * R2
- 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."
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