Lab 04: Logic Functions

Combinational logic circuits implement logic functions using a combination of logic gates. Recall that all logic functions can be implemented with AND, OR, and NOT functions.

Consider a 2-input XOR function. Recall that when inputs are different, the output is 1; when inputs are the same, the output is 0. Assume that the input variables are A and B. If A is true (1) and B is false (0) or if A is false (0) and B is true (1) the function will be true. Putting that into a logic equation:

\[A \oplus B = A\overline B + \overline A B\]

One way to prove that two logic equations implement the same function is to show that they have the same truth table. The truth table below shows that \(A \oplus B\) and \(A\overline B + \overline A B\) have the same truth table, and, thus, they must implement the same function.

The logic diagrams for these two functions are:

If we implement both circuits they should have the same behavior. A logic diagram does not have enough information to implement a function. A wiring diagram is similar to a logic diagram but also has information about how to wire the chips such as chip type, unit numbers, and pin numbers. Below are the wiring diagrams for both logic functions.

Wiring diagram for logic equation A ⊕ B:

Here is the wiring diagram for the logic function AB + AB:

This circuit has three chips, U1 is a 7404 (Inverter), U2 is a 7408 (AND), and U3 is a 7432 (OR). Line up these chips along the same section of the breadboard, all facing the same way. Leave some space between them in case one of them is faulty and has to be extracted.

For each chip connect the power pin (pin 14) to V_CC (+5V) and the ground pin (pin 7) to GND (0V). Next, connect all of the wires as shown in the wiring diagram. For example, connect pin 2 of U1 to pin 1 of U2. To keep track of which connections have been made, after you connect two pins with a wire, place a small checkmark on that wire in the wiring diagram. It is also a good idea to place a checkmark next to pin 14 and pin 7 in the power and ground table after you connect each to V_CC and GND, respectively.

Note: there is a problem drawing wiring diagrams as discussed above. You must constantly refer to the KB03: TTL IC Specifications or else go online to find each chip's pin numbers and then to place them in your drawing. Instead, you can use a circuit-drawing program called KiCad. (SCH stands for "schematic", meaning a circuit diagram). It allows you to place and connect symbols for gates and chips in your diagram that contain all pin numbers plus Vcc and ground (so a Vcc-ground table is no longer needed).

Once you draw a circuit with this tool, it is all right there. (You may want to print it out so you can check off connecting wires as you install them.) The program is available on the lab PC's plus you can download it from the web for free. Refer to the KB04: KiCad Tutorial for all instructions on how to download it to your computer and how to use it. It is of real benefit when wiring up your circuit to have your diagram on the PC screen, complete with pin and U (unit) numbers.

Rarely will a circuit work the first time it is built? Inevitably there will be mistakes, also known as "bugs", either due to incorrect design or a mistake during wiring. In order to find the mistake, you need to troubleshoot or "debug" your circuit. For a combinational logic circuit here are some steps to debugging a circuit:

Steps for troubleshooting (debugging) a combinational logic circuit:

1. Determine whether or not the circuit generates the correct outputs for all possible input combinations. If so, the circuit works properly. If not, record the input combinations that generate incorrect outputs. (The number of inputs can not be too high, else too many combinations to test.)
2. For one of the input combinations that physically produced an incorrect output, shown on the diagram what the correct input and output of each gate should be (internal gates as well as the
   final output).
3. Now, set the inputs to the circuit equal to that input combination. Note that while debugging a circuit it is best to manually control the inputs by connecting them to switches, or digital output pins from the ADALM1000 device, or by driving the 7493 counter with a manually clock signal.
4. Using the logic probe, start from the output of the circuit and trace the values backward through the various components until you identify an incorrect value (a value that differs from the one you determined in step 2).
5. Identify and fix the bug.
6. Repeat steps 1 through 5 until there are no bugs remaining in the circuit.

There are two keys to being successful in troubleshooting your design. First, you must understand how your circuit is supposed to behave. Also, you need to have a good understanding of the behavior of each logic gate, hence the importance of Lab 3.

The circuits shown in the following debugging session have been drawn using KiCad. Notice how gates from different chips show different U numbers and how pin numbers are included automatically. Once you are familiar with KiCad, you'll see how much more efficient it is to draw your circuits.

Debugging steps:

Step 1: with the bug, the circuit will generate the output values shown in the following truth table under Actual Value, instead of the correct values shown under A ⊕ B. There are two input combinations that generate the incorrect results shown in bold. From the values in the Actual Value column, it shows that the expression for the output of the circuit is now just A.

Step 2: for input combination, 00, the correct output value should be 0, but a check with your logic probe at the output of U3 (OR) will show it is 1. We start troubleshooting (debugging) by working backward from the output gate using the logic probe. The triangular symbols (red=1, green=0) represent the LED indicators on the logic probe. As you proceed, record the 1 and 0 values on your diagram.

Figure 4.9: Circuit with Error, Output Becomes just A

1. We first check the input pins of the OR. If the OR output is 0, then its inputs, pins 1 and 2, should both be 0's. The input at pin 1 turns out to be 0, as expected, so it's OK. But the input at pin 2 is a 1, which explains why the output of the OR is also 1 (i.e. 0 + 1 = 1). Since pin 1 is OK, we will ignore the upper half of the circuit and focus instead on the bottom half to see why the input to pin 2 is wrong.
2. Now, if pin 2 of the OR is properly connected to pin 6, as the diagram shows, then the AND output will also be a 1. But an AND output of 1 implies that both its inputs, pins 4 and 5, must be 1's (output = 1 • 1 = 1), and the logic probe should show that they are.
3. That pin 5 is a 1 is expected since B is the complement of B (0). On the other hand, we find pin 4 to be 1 even though it is supposed to connect directly to A (0). If we follow the wire from pin 4, we find that it ends, not at A but at A, the output of the upper inverter. We have located the bug. The circuit output is, therefore, AB (from lower AND) + AB (from upper AND) = A (B + B) = A, as the above truth table indicates.

Alternative debugging method

Assuming there are no electrical (as opposed to logic) problems with your circuit, problems like a bad chip, or one with no connection to Vcc or ground, or a broken wire, the following approach may save time since it doesn't involve poking around inside the circuit with a logic probe as in the previous example. It just requires adding a column to the original truth table with output values found when testing the actual circuit. We'll use the table for the XOR circuit shown at the top of the previous page to illustrate this.

The 1's in the Actual Values column lie in the first two rows where A = 0. So we could say that the circuit output is simply A. But this implies a circuit with no AND and OR gates — very different from the XOR circuit. In order to compare the two, we want the same structure for each (i.e. two inverters, two ANDs, and an OR). So, instead, we write separate AND expressions (product terms) for the first and second rows in the Actual Values column and combine them as shown:

Actual output = A B + A B

XOR output = A B + A B

We can see right away that the problem is the underlined A which should be A. In other words, the wire into pin 4 of the lower AND should come directly from the A and not from the output A of the upper Inverter. Problem solved.

A majority voter is a circuit with three inputs that we will refer to as A, B, and C. It has one output which we will refer to as V. The value of V will agree with the majority value on the inputs. Thus, if the majority (two or more) of the values are 0, V will be 0. If the majority of the values are 1, V will be 1. Given this definition of the majority voter, complete the truth table below for V, and record it in your report.

Let's think about this circuit. If two or three inputs are 1, output V will be 1. Otherwise, it will be 0. Verify that your values for V in the truth table agree with this definition. If so, then we can determine a logical expression for V; i.e. V = AB + BC + AC. We can verify that this is correct by showing that the truth table for V and for AB + BC + AC are the same. Complete the truth table below to prove that these two functions are the same, and record it in your report.

Question

3. Build this circuit and verify that it produces the same results as given in the truth table for V:
   1. Draw the logic diagram for V = AB + BC + AC. Mark the chip numbers and pin numbers on all logic gates in the diagram. (Check the diagrams in Exp #4.2)
   2. Implement the circuit and verify that it behaves correctly. If it does not, use the debugging steps in Exp #4.2 to troubleshoot your design. Take a photo of your circuit connection on the breadboard, and paste it into the report.

Assuming that you were lucky and wired your circuit properly, you haven't had a chance to practice your debugging skills. In this section, you will each take turns inserting a bug for your partner after which your partner will troubleshoot the circuit. Follow the steps shown in Exp #4.2 to debug your circuit. Document your process as shown in Exp #4.2 in your lab journal. Demonstrate your debugging technique to your instructor.