EK 307 Homework #5 Fall 2015

Problem 1) Below are two circuit diagrams for op-amps.

a) Calculate the gain for each circuit.

b) A practical voltage source is connected to each circuit as shown below. Does this change the gain? Determine the gain for each circuit.

Problem 2: Design a circuit with one op-amp that provides a gain of 5.5. Assume you have a resistor Ri = 10kΩ, what value would you choose for Rf?

Problem 3: Determine the current io in the circuit below

Problem 4: Determine vo in the circuit below.

Problem 5: For the circuit below, find V0 in terms of VS.

Problem 6:

a) For the circuit below, find the output voltage of v0 in terms of the input voltages v1 and v2. b) If v1= 1V, what is the range of values v2 can have without saturating the op-amp?

Problem 7:

Calculate the value for the input voltage v2 to obtain an output voltage of v0 = -16.5 V?

Boston University EK307 – Fall 2017

Problem 8. [Revision 2]. (Part 2; continued from Problem Set 4 – Problem 10) Design problem: toward a digital-to-analog converter (DAC). In the last problem set, you looked at the Thevenin equivalent of the 3-bit R-2R DAC with respect to each of the three bit input voltages (v0, v1, v2), and you showed the output voltage for an arbitrary analog set of input voltages and for all 3-bit digital input combinations at (v0, v1, v2) using 1V sources. This week, we want to understand how this DAC design behaves and whether it works (1) when you connect a resistive load to it, and (2) to develop a short Matlab code that can let you quickly analyze the circuit with arbitrary choice of resistance values for the 6 resistors, and show that it works. Then, test how the DAC performs if your resistors are off by 5% from their target values (fabrication tolerance) in a few different ways you choose.

(a) Give a single equation that gives the output voltage, vout, as a function of the three (v0, v1, v2) input voltages. Use the superposition principle to do this. (b) Find vout when you connect an arbitrary load resistor RL to the DAC output. Give the single equation, just like in part (a). Hint: you can look at the Thevenin equivalents you found in the last homework, with the load connected to it, and use superposition. (c) If the six resistors in the DAC are R and 2R as shown in the diagram, plot the voltage vs. analog value of all 8 3-bit digital inputs (0 to 7), for RL = infinity (last problem set, ie open circuit), RL = R, and RL = R/2. You should have three curves on one plot. What happens to the voltage for smaller loads? Does the DAC still work? (d) Now, add an op-amp based unity-gain buffer between the 3-bit DAC and the load resistor, and repeat the plot (3 curves) in part (c). How does the buffer help?

Boston University EK307 – Fall 2017

(e) Now, using the node voltage method (or your favorite method), we will solve the same circuit, but assuming arbitrary resistors R1 to R6 (see Figure 2 below). Assume no connected load again. Please show a matrix node voltage equation, where the unknown vector has the node voltages, including the one we’re interested in, vout. Give the matrix equation.

(f) Put your matrix equation into Matlab, and write a function that returns vout given inputs (v0,v1,v2). Then, write a script that sets R1 to R6 to the circuit from Homework 4 Prob 10, with R = 1kΩ, (i.e. 1kΩ and 2kΩ resistors) and run it through the 8 bit combinations to produce the same plot you got by hand in Homework 4. Show the plot from Matlab, and the code. Keep the code as short and compact as possible. (g) Set the resistors (R1 to R6) using resistor values that are 5% off the target value, i.e. use R = 0.95kΩ or 1.05kΩ instead of 1kΩ. For each of the 6 resistors pick randomly whether the resistor is 5% too high or low. Use your Matlab code to plot the output voltage vs. 0 to 7 digital input for three cases (ie choose the 6 resistors randomly – or it doesn’t have to be random – choose them in some combination you wish – three times). Show those 3 curves on top of the line for the ideal case from part (d) with RL = infinity. Keep your code. We will use it in next week’s problem set.

Boston University EK307 – Fall 2017

ERRATUM SUMMARY (CHANGES FROM ORIGINAL): In Problem 8(e), the figure was slightly modified to relabel the resistors, so as to make the pattern in the resulting node-voltages matrix more obvious. The figure was changed from the following in the original document:

to this revised figure in the current document:

The order in which the resistors are labeled was just reversed. Note that the addition of the second, equivalent drawing is just for clarity and does not change the problem – the left and right drawings are equivalent. Please be sure to use the new resistor labels in your solution. The solutions to the problem set will use this new labeling.