Module 4: The Operational Amplifier Operational Amplifiers: General Introduction In the laboratory, analog signals (that is to say continuously variable, not discrete signals) often require amplification. Amplification of a signal is the process of increasing a signal s power. For an ideal amplifier, the information contained in the signal is not altered, only its power level is. In addition, we often need to process the signal. For example, it might be necessary to subtract two signals, add two signals, or even carry out more sophisticated operations such as integration, differentiation, or taking the logarithm. A standard, packaged device which allows us to easily perform these operations is the operational amplifier, or for short, the op amp. An op amp is a device which contains many transistors and complicated circuitry. We will concentrate here only on the behavior of the amplifier, not the details of its construction. The generic representation of an op amp is shown below: v Vnon The op amp is powered via the supply rails, and. The output of the device is given by. has a range which is usually close to the range given by the supply rails. The output cannot exceed the supply rail voltage. There are two input terminals to the op amp, the input labeled with a is known as the non-inverting input and the input labeled with a - is the inverting input. The voltages applied to these terminals are written as Vnon and v, respectively. To first approximation, the output voltage is controlled by the difference voltage to the inputs. The op amp is, therefore, a difference amplifier with effective input voltage vdif = vnon vinv. There are, however, restrictions on the allowable values of the input. Again, these values are usually close to the range spanned by the supply rails. To begin our analysis, we will consider an ideal device. In the real world, though, nothing behaves ideally! Since the op amp looks at the input voltage and delivers an output voltage, an ideal amplifier would have infinite input impedance (so any device could be attached to it) and the output would have zero source impedance (so any load can be driven). The relationship between the effective input vdif and the output, vout is described by the transfer characteristic shown in the figure below. Notice that because the amplification is high, it only takes a small vdif to drive vout over its active range. Outside of the active range, the amplifier saturates. The gain G of the amplifier is given by the slope of the active portion of the curve: dvout G = dv For typical op amps, G is in the range of 10 4 10 7. dif
sat active range Vdif sat Vdif active range Operational Amplifiers with Feedback Up to now, we have used the op amp simply as a differential amplifier. In most applications, the op amp is not used this way, since the output is very sensitive to imperfections of the device and not particularly stable. Instead, we apply negative feedback to the input as shown below. The feedback loop is a way of assuring that any errors in the output are self-corrected. The boxed g in the figure represents a control on the amount of feedback we apply, known as the transfer function. g g Assuming infinite gain of the amplifier and that it is operating in its active region, we can see that initially there will be a difference voltage between the two inputs which gets amplified and shows up as a non-zero voltage on the output. But because of the feedback, a voltage of g now appears on one of the inputs, thereby reducing the voltage difference and, therefore, changing the output to reflect the new input difference. Very rapidly, the configuration approaches the state where the input difference is very close to zero. In that limit, g = 0, and we see that the output voltage is simply given by:
= g Naturally, real op amps do not have infinite gain, but we can show that for a wide range of operating values, the infinite gain approximation actually works well. In analyzing op amp circuits, we begin by neglecting the non-ideal characteristics of an op amp such as: voltage offset bias current offset current open-loop gain rolloff slewing rate limitations output current limit We approximate the op amp as having the following two principal characteristics (Golden Rules): 1) The inputs have infinite impedance, and therefore, draw no current. 2) The op amp will deliver whatever output voltage it needs to keep the difference of the input at zero. [Exercise 1: Small Signal Amplification] Using the Proto-Board and a 741 op amp, construct the circuit shown schematically below: 15 100K -15 1M 150 2 3 7 4 6 Signal Generator 47K 100 We wish to study the op amp within its active range in order to measure G, the small signal gain. To do this we will pass the voltage from our signal generator through a voltage divider as shown in the schematic. This allows us to conveniently monitor the signal going to the op amp, while keeping the actual signal small.
(a) Calculate the relationship between Vnon and. In an ideal op amp, the output is zero if vdif is zero. In a real op amp, there are various internal voltages which unbalance the input. These are known collectively as the input offset voltage vio. Typically vio will be on the order of a few millivolts. We will compensate for vio by applying a small voltage to the inverting input. (b) Set up the function generator to deliver a sine wave at 1 khz, symmetric about ground. Reduce and rebalance the control voltage to the inverting input until the output remains in the active range and vout is symmetric about ground. Measure the amplification G of the amplifier. Repeat the measurement for frequency above and below 1 khz spanning from 100 Hz to 30 khz using intervals of factors of ~3 (i.e. 100 Hz, 300 Hz, 1kHz, 3kHz...). For each frequency note the relative phase between the input and output. (c) Make a log-log plot of your results for G vs. frequency. [End Exercise 1] You should have found that the amplification G of the op amp decreases at higher frequencies. The op amp is specifically designed to have G ~ 1/f above a corner frequency of ~ 10 Hz. For a frequency of 1 Hz, G is almost constant. This is an example of amplification rolloff. It is done to keep the op amp stable when used in a feedback configuration (next week s lab). When an op amp amplification automatically rolls off, it is said to be internally compensated. The Comparator: An Example of an Overdriven Amplifier In Exercise 1, we used the op amp to provide small signal amplification. In practice, we would usually not use the op amp in this configuration because 1) the gain is not necessarily linear, 2) the frequency response is designed for a feedback configuration, 3) this setup is sensitive to drifts in vio. There are times, however when we might use an op amp without feedback, an example being the comparator. A comparator is a device which lets us know when an input signal crosses a certain threshold value. We can overdrive the amplifier, i.e. swing the output from negative to positive saturation, to signal that the threshold crossing has just occurred. Provided we are working with large voltages, we can ignore vio.
[Exercise 2: The Comparator] Setup the circuit shown below with vnon initially connected to ground. Use the triangle wave setting of the function generator. Signal Generator 100K 15 100K Vref 2 3 7 4 6-15 47K 0.05 uf (a) Using both traces of the oscilloscope, observe and to determine at what level of the output switching occurs. (b) Measure the positive and negative output saturation voltages. (c) Now connect vnon to the adjustable voltage section of the circuit. Lets call this voltage Vref. Vary Vref and the level of. Observe where the op amp switching now occurs. (d) You should have noticed that the output moves in a direction opposite to that of the signal. Trade inputs to the amplifier to obtain a comparator whose output moves in the same direction as the input signal. (e) The accuracy of the comparator depends on how fast the input signal changes over the active region and on the intrinsic speed of the op amp. The maximum rate that the op amp can respond is known as the slewing rate. Vary the amplitude and frequency of to see how the rate of change of is affected. (f) Measure the slewing rate of the 741 op amp by changing the input signal to a square wave. [End Exercise 2]
Applications of Feedback Circuits A) The Voltage Follower: Consider the following circuit This circuit corresponds to negative feedback with g = 1. As a result, =, and the configuration is known as a voltage follower. Why build a voltage follower? A follower allows one to input a signal from a source with high output impedance (i.e. a source that cannot deliver much power) to a load with low impedance (i.e. a source which will draw significant current). The voltage follower is a unity gain circuit for the voltage, clearly, but it can deliver a gain in the power. In this application, the voltage follower is used as a power amplifier. Another name for this circuit is a buffer. [Exercise 3: The Voltage Follower] (a) Build a voltage follower using a 741 op amp. With a frequency of 1 khz, and a 5V p-p amplitude, apply a sine and square wave and observe the output. (b) Look closely at the output of the follower with a square wave input. Measure the risetime of the output voltage. Does the output follow the input exactly? Explain. (c) Now substitute a 3140 op amp for the 741 and repeat part (b). Is there a difference in behavior? Explain. [End Exercise 3] For the exercises that follow, you will need to choose resistors appropriate for the circuit gain you are trying to achieve. As a guide, choose resistor values in the 1K-100K range to avoid invoking some of the non-ideal behavior of the op amp and continue to use the 3140 op amp. B) The Non-inverting Amplifier Now, instead of feeding the output directly back to the input, lets decrease the voltage with a resistive voltage divider network as shown in the figure below:
R1 R2 [Exercise 4: The Non-inverting Amplifier] (a) Show that the output of the non-inverting amplifier is given by: R1 R2 = R1 (b) Build a non-inverting amplifier with a gain of 10 and investigate its performance over a range of amplitudes and frequencies for a sine wave input. [End Exercise 4] C) The Inverting Amplifier For the non-inverting amplifier, both inputs are usually maintained at a non-zero voltage and this can lead to lower performance of the amplifier when the non-ideal characteristics of the op amp are considered. A way to increase performance is to operate the amplifier in an inverting amplifier configuration. R1 R2 [Exercise 5: The Inverting Amplifier] (a) Show that the output of the inverting amplifier is given by:
R2 = R1 (b) Build an inverting amplifier with a gain of 10 and investigate its performance over a range of amplitudes and frequencies for a sine wave input. [End Exercise 5] D) The Integrator To construct an integrator we simply replace R2 in the inverting amplifier with a capacitor: R C Let s see how the integrator works. If the non-inverting input is grounded, then the inverting input is kept at virtual ground and the current which flows through R is simply i = / R The current through the capacitor is related to the voltage across it by the standard formula V i = C t Combining these equations, we get t 1 = dt V0 RC 0 where V 0 is any voltage on the capacitor at time t = 0. Of course, we must not forget that this for an ideal op amp. You will find that the non-ideal bias and offset currents which exist will cause the capacitor to charge even when = 0! [Exercise 6: The Integrator] (a) Design and build an integrator to produce a triangle wave output with 10V p-p output at 5 khz using a 5V p-p square wave input centered about ground. Be sure to list and discuss your design considerations in your lab book. Note any non-ideal behavior you observe.
(b) Add a large resistor across the capacitor. Does this help alleviate some of the non ideal behavior? Explain. Hint: You will want to install a switch in the circuit to periodically reset the integrator to zero. [End Exercise 6]