Lab 6: Logic Circuits

Score: 50 pts.

Objectives:

§  Design Combinational Circuits

§  Implement and simulate using logisim software. It is an easy to use and easy to understand digital logic tool.

§  Download software from www.cburch.com/logisim, and install on your machine. From the menu, click on Help then Tutorial to familiarize yourself with the program. You can do so as you implement your systems.

I.      Full Adder (This was adapted from David M. and Sarah L. Harris, 2012)

In this section, you will design a simple digital circuit called a full adder. You will

use the logisim tool to enter a schematic and simulate your design.

Background: Adders

An adder, not surprisingly, is a circuit whose output is the binary sum of its inputs. Since adders are needed to perform arithmetic, they are an essential part of any computer. The full adder will be an integral part of the microprocessor that you design in later labs.

A full adder has three inputs (A, B, Cin) and two outputs (S, Cout), as shown in Figure 1 below. Inputs A and B each represent 1-bit binary numbers that are being added, and S represents a bit of the resulting sum.

The Cin (carry in) and Cout (carry out) signals are used when adding numbers that are more than one bit long. To understand how these signals are used, consider how you would add the binary numbers 101 and 001 by hand:

As with decimal addition, you first add the two least significant bits. Since 1+1=10 (in binary), you place a zero in the least significant bit of the sum and carry the 1. Then you add the next two bits with the carry, and place a 1 in the second bit of the sum. Finally, you add the most significant bits (with no carry) and get a 1 in the most significant bit of the sum.

When a sum is performed using full adders, each adder handles a single column of the sum. Figure 2 below shows how to build a circuit that adds two 3-digit binary numbers using three full adders. The Cout for each bit is connected to the Cin of the next most significant bit. Each bit of the 3-bit numbers being added is connected to the appropriate adder’s inputs and the three sum outputs (S2:0) make up the full 3-bit sum result.

Note that the rightmost Cin input is unnecessary, since there can never be a carry into the first column of the sum. This would allow us to use a half adder for the first bit of the sum. A half adder (shown in Figure 3 below) is similar to a full adder, except that it lacks a Cin and is thus simpler to implement. To save you design time, however, you will only build a full adder in this lab.

1.    Design

A partially completed truth table for a full adder is given in Table 1 below. The table indicates the values of the outputs for every possible input, and thus completely specifies the operation of a full adder. As is common, the inputs are shown in binary numeric order. The values for S (sum) are given, but the Cout (carry out) column is left blank. Complete the table by filling in the correct values for Cout so that adders connected as in Figure 2 will perform valid addition.

From the truth table, we now want to implement our design using logic gates. The sum output (S) can be produced from the inputs by connecting two 2-input XOR gates as shown in Figure 4.

You should convince yourself that this circuit produces the outputs for S as given in the table.

Using only two-input logic gates (AND, OR, XOR) and inverters (NOT), design a circuit that takes A, B, and Cin as its inputs and produces the Cout output. Try to use the fewest number of gates possible. Sketch your schematic.

2.    Schematic

Now that you know how to produce both the sum (S) and carry out (Cout) outputs using simple logic gates, you will now construct a working full adder circuit: Enter the schematic representation of your logic into the software package; then simulate the circuit and test that it works the way you expect it to.

II.    Simulations

One motivation for drawing your schematics in Logisim is that you can now use the software to simulate the operation of the circuit.

From the menu, check Simulation Enabled.

Use the poke tool to switch through different combinations of your truth tables. If they all match, the circuit works.

III.   Submissions

Please submit your lab report through Blackboard.

Your report must include:

1. A few sentences describing the purpose of this lab
2. Your completed truth table
3. Your completed schematics, including the logic gate from both S and Cout. You must show the equations derived from the truth table, optimizations with KMap, and build the schematics from the optimized equations
4. Your simulation of the full adder, including all inputs and outputs

The simulation file can be produced using File à Export Image

Be sure to uncheck the “Printer View” checkbox.