MAS.967 Problem Set #1: Op Amp Circuits

This problem set is designed to step the student through the basics of op amp circuit analysis and design. The analysis questions will present a number of basic op amp circuits and will ask the reader to calculate the transfer function (ratio of output voltage to input voltage(s)). The student will first be asked to calculate a number of intermediate voltages and currents necessary to achieve the final result, to give them an idea of the methods by which such systems can be analyzed. Analysis Questions The following laws/rules are necessary to complete the following problems: Kirchoff’s First Law: The voltages in any closed loop sum to zero. Kirchoff’s Second Law: The currents into and out of any node sum to zero. Ohm’s Law: The voltage drop across a resistor is equal to the current flowing through it times the resistance. First Ideal Op Amp Rule: No current flows into the inputs of an ideal op amp. More rules will be added as we continue. 1. We want to calculate the transfer function (Vo/Vi) of the circuit shown in Figure 1.

Figure 1: Inverting Op Amp Configuration

a) Calculate the current I1 running through resistor R1. b) How does that current relate to the current (I2) running through R2? c) Using I2, find another expression for Vo in terms of V- and Vi. d) Express Vo in terms of the voltage at the inverting terminal (V-) using the via the intrinsic

op amp gain (Av). e) Using c) and d), find the transfer function in terms of Av. Then take the limit as Av goes

to infinity to find the ideal transfer function for the inverting op amp configuration. f) Given the ideal transfer function, find the ideal value of V-.

Figure 2: Another Inverting Op Amp Configuration

g) Now consider the circuit in Figure 2. Using the technique of parts a)-c), and assuming V-

=0, find Vo in terms of V1 and V2. For R0 = R1, what mathematical operation does the input stage implement?

2. We will now calculate the ideal transfer function of an instrumentation amplifier, as shown in Figure 2. To do so, we will use the Second Ideal Op Amp Rule: an ideal op amp with negative feedback works to make the voltage at the two inputs terminals equal.

Figure 3: Instrumentation Amplifier

a) What are the voltage at the inverting terminals of op amps 1 and 2. b) Use those voltages to find the voltages at the output of op amps 1 and 2, Vo1 and Vo2

respectively. We will calculate the gain of the third stage using the superposition principle. c) For op amp 3, find Vo in terms of Vo1 for Vo2 grounded. What op amp configuration is

this? d) For op amp 3, find Vo in terms of Vo2 for Vo1 grounded. What op amp configuration does

this resemble? e) By superposition, the total output Vo of op amp 3 is the sum of the above two results.

Find the complete input/output relationship of op amp 3. Using that relationship and the values of Vo1 and Vo2 found above, find Vo in terms of V1 and V2.

3. We now consider a non-inverting configuration op amp, but with a small change. We replace the ground in the feedback path with an arbitrary voltage reference Vref. Find the new transfer function by calculating the voltage difference between the output and the voltage reference.

Figure 4: Biased Non-Inverting Op Amp Configuration

This technique, known as biasing an amplifier, is essential for use with single supply op amps, those for which there is only a positive saturation voltage. It is also useful for amplifying sensor signals which are produced relative to a voltage reference, as described in class. The transfer function will bear this out.

4. Our final circuit is very similar to the one above, but with one difference: the output feeds back to the non-inverting terminal instead of the inverting terminal. This system therefore has positive feedback instead of negative, meaning that it is unstable, and the output will rapidly increase until it saturates (reaches the maximum voltage provided to the op amp). Further, the second ideal op amp rule no longer applies, as we now have positive feedback. We will analyze the operation of this circuit.

Figure 5: Comparator with Hysteresis

a) What is the voltage at the non-inverting input (V+) if the output is in positive saturation (Vo = Vsat)?

b) What is the voltage at the non-inverting input (V-) if the output is in negative saturation (Vo = -Vsat)?

c) Assuming that the op amp is currently in positive saturation, at what input voltage (Vi) will it switch to negative saturation?

d) This circuit has hysteresis, meaning that the current transfer function is based on the previous voltage inputs. For Vref=0, draw a graph of Vo versus Vi, showing how to achieve the various output voltage states.

Design Question Perhaps the most important thing to learn about analog signal conditioning for sensors (after straight DC gain and offset adjustments, as covered in the analysis problems) is filtering. This problem is structured to take you through simple design of elementary passive and active filters of the kind that you should know about in doing this work, enabling you to make a frequency "window" around the signal of interest and remove any unwanted noise or background at other frequencies. Note that "active" here means containing a powered, current-or-voltage-generating gain component (like an op amp or transistor), and "passive" means not having such components (e.g., only having resistors and capacitors and/or inductors). This is a very practical problem, so please specify actual component values for the resistors, capacitors, and inductors (where used). No need to specify an op amp when called for - assume an ideal op amp, as before. 1. Filter Design a) Design a passive, first-order RC highpass filter, with 3 dB rolloff at 1 kHz, and an impedance (at infinite frequency) of 10 kΩ (i.e., 10,000 Ω). Assume a zero impedance source and infinite impedance load (the ideal case).

b) Design a passive, first-order RC lowpass filter, with 3 dB rolloff at 10 kHz, and an impedance of 10 kΩ (at zero frequency). Again, assume a zero impedance source and infinite impedance load. c) Design an active filter around a single operational amplifier (inverting configuration), with:

• A maximum (e.g., passband) gain of -8 • A minimum input impedance (at infinite frequency) of 10 kΩ • A first-order highpass response of rolling off below 1 kHz and a first-order lowpass

response rolling off beyond 10 kHz (again, these are the 3 dB rolloff points; this filter gives roughly uniform gain between 1 and 10 KHz, and rolls off at 6 dB/Octave at either end).

• No inductors (of course). d) Assume that the op amp picked above was a rail-to-rail type, running between 0 and 5 V to feed a microprocessor's ADC input. We'll thus need to bias the zero-point of this op amp to be ideally midway between the rails for maximum dynamic range (e.g., at 2.5 V). Assuming you have access to only 5 V and ground, add a simple circuit, using only resistors, to bias this op amp up so that its quiescent point (e.g., output level given no input) is at 2.5 V. e) Design a passive RLC (resistor-inductor-capacitor) bandpass filter, with centre frequency at 10 kHz and a Q of 5. Pick reasonable values for the components (hint: inductors shouldn't be greater than 10 mH and capacitors shouldn't be bigger than 100 nF (or in this case less than 100 pF either). Assume a zero impedance source and an infinite impedance load. f) Although LC filters are quite useful tools to have in your toolkit (as in part e) above), they are rarely used a lower frequencies, mainly because of non-idealities and other problems with the inductors. To get a steeper filter rolloff (e.g., if you need more isolation from other extraneous frequency components), you'll occasionally need to design higher-order filters, and/or design bandpass filters only with resistors and capacitors (using capacitors in feedback around the op amp to simulate the action of an inductor). It is therefore very useful to know about the minimal such designs - e.g., single-op amp second-order filters, namely VCVS or Sallen-Key filters. These were covered in class, and are discussed in several texts, such as the Active Filter Cookbook, the Art of Electronics, etc. As an exercise, design a second-order Sallen-Key highpass Butterworth (what Lancaster calls "Flattest Amplitude") filter, with a cutoff at 5 kHz. Gain isn't critical (e.g., between a factor of 1 and 3 is fine) Draw the schematic, and give reasonable values for the components (again, consult the references for guideline). g) As in part d) above, show how this circuit can be biased up so its output sits quiescently at 2.5 V when you have a single 5 V supply and a rail-to-rail op amp.