Project 3 Combinational Logic Circuits

An introduction to logic minimization techniques



This project presents several circuit requirements that are described with higher-level, natural-worded descriptions. Your job is to create circuits that behave according to the descriptions.

When starting circuit designs from higher-level, purely behavioral descriptions like those presented below, it’s a good idea to identify and follow a set of good design practices. In general, good design practices follow several steps: first, be sure you clearly understand the design intent, and exactly what is being asked; second, cast the worded description into formalisms like a block diagram and truth table that shows all inputs & outputs and their functional relationships; third, capture a circuit based on the formalisms (that is, write a Verilog description); and fourth, implement the circuit and verify its performance. Each of the requirements below present problems that are involved enough that all of these steps should be followed.

Before you begin, you should:

  • Have a working knowledge of Vivado and a functioning Boolean board;
  • Be comfortable working with logic equations and logic circuits;
  • Be able to describe logic circuits in Verilog;
  • Know how to implement logic circuits on the Bolean board.

After you’re done, you should:

  • Be able to write more complex Verilog descriptions;
  • Be able to analyze more complex descriptions, and design circuits to implement the behavior;
  • Know how to find a minimum expression for any given logic requirement.


Each of the problems in this project describe a two-state output (either on or off, true or false, 1 or 0, etc.) that is a logic function of some number of two-state (or binary) inputs. It follows that each of these problems can be represented using a truth table, and the truth table forms the specification for a circuit. The truth table exactly specifies the behavior of a circuit, but not the structure. Before a physical circuit can be constructed, its structure must be defined.

For any given behavior description, any number of physical/structural circuits could be built to implement the same behavior. Of all the possible circuits that could be constructed, our goal is to find the most efficient one. Note that “efficiency” could imply several end goals – the most efficient circuit could use the fewest number of transistors, or it could use the least amount of power, or it could run the fastest. All of these end-points could result in different circuits. For the most part, we will look for the circuit that uses the fewest number of transistors. Several methods have evolved to find the most efficient circuit, and they are discussed in the topic documents.

To solve these problems, you should cast the described behavior in a truth table, then analyze/process that truth table to find a minimal circuit (using the methods described in the topic documents), then capture the circuit in the Vivado design tool, and then implement and verify your design.


When Verilog source files are created for more complex problems like the ones in this project, it is quite possible (likely?) they contain some errors. Before implementing the design in hardware, or using it as a component in some other design, it makes sense to use a logic simulator to check the circuit’s performance. The logic simulator creates an executable model of a circuit from a Verilog design source file, applies user-defined input signals to the circuit’s input ports, and then simulates the circuit’s behavior. The simulated circuit outputs are shown in a waveform viewer so the designer can verify that expected outputs are generated for all combinations of inputs.

There are many benefits to simulating a circuit early in the design process, before pressing on with further design work. In addition to detecting and correcting design flaws, signal timings can be checked and verified, misnamed signals can be identified, proper operations in corner-cases can be examined, and so on.

For all but the simplest designs, you should adopt the habit of simulating your code and verifying your circuit’s performance as soon as the source code is complete. If you start using the simulator now, with simpler circuits, you will develop some amount of proficiency. Then, as your designs become more complex, running the simulator will not feel burdensome. Like thousands of engineers before you, you will find that if you simulate and verify your code early in the design process, you will decrease your work and increase your productivity.

A background topic document presents a guided tutorial that will walk you through creating a test bench for the majority of five circuit in the first requirement below.


1. Design a “majority of five” circuit

Design a majority-of-five circuit that outputs a 1 when any three or more of its five inputs are asserted. Implement the circuit on the Boolean board, using five slide switches as inputs and an LED as the output.

The first step in the design process, understanding the objective and intent, is straight forward for this design.

The next step is to cast the requirement into an engineering formalism, which means creating a truth table to capture the required behavior. Then the truth table can be analyzed using K-maps, and a minimal circuit defined. In the figure below, the truth table is shown in an uncompressed “super K-map” and in an entered-variable K-map. Optimal equations can be looped from either the super map or EV map – an initial set of loops is shown. After the equations have been obtained, they can be captured in a Verilog design file.

Figure 1. Majority of five block diagram
Figure 1. Majority of five block diagram
Figure 2. Representations for the Majority of Five circuit
Figure 2. Representations for the Majority of Five circuit

You might entertain an alternative way of looking at the problem, both as a way to gain more insight, and as a check on the formal design procedure using truth tables and K-maps. In fact following a methodical design procedure is often the best way to tackle any given design, but thinking about the problem from a different angle is often advantageous as well.

In this case, with a little thinking, you might realize that all the terms in the final equation will have three logic variables, and all unique three-variable terms will be included in the equation (why?). A “5 choose 3” calculation shows there are 10 ways to choose three variables out of five; it follows that ten 3-input terms can be defined from a pool of five input variables.

You can find the loigc terms needed in the final equation through looping, or perhaps you can discern them just by thinking through the combinations. Or better yet you can do both, and make sure both approaches arrive at the same solution (they do). Either way, the Verilog code below captures the requirement. Pause a moment to think about this, and make sure you agree and understand.

module majority_of_five(input [4:0] sw, output led);

assign led =	(sw[0] & sw[1] & sw[2]) | //ABC
		(sw[0] & sw[1] & sw[3]) | //ABD
		(sw[0] & sw[1] & sw[4]) | //ABE
		(sw[0] & sw[2] & sw[3]) | //ACD
		(sw[0] & sw[2] & sw[4]) | //ACE
		(sw[0] & sw[3] & sw[4]) | //ADE
		(sw[1] & sw[2] & sw[3]) | //BCD
		(sw[1] & sw[2] & sw[4]) | //BCE
		(sw[1] & sw[3] & sw[4]) | //BDE
		(sw[2] & sw[3] & sw[4]);  //CDE

2. Design a five-way light switch

A room has five doors with a light switch next to each door, and one light in the center of the room. Design a circuit that allows any light switch to change the state of the light (that is, if the light is currently off, any switch can turn it on, and vice-versa). Use five switches and one LED on your Boolean board to build and demonstrate your circuit.

Each of the five inputs can be on or off (a ‘1’ or a ‘0’), so you can represent all possible combinations of inputs in a truth table. The first row (all 0’s) represents a state where the light is OFF. From there, if any input toggles to a “1”, the light toggles ON. When the output is on, any new input toggle will set the light OFF again, and then any additional toggle will turn the light back ON, etc. You must cast this behavior into a truth table, and then use that truth table to capture a circuit.

The videos in this project show circuits implmented on the Blackboard, but the performance on the Boolean board is entirely similar.

3. Design a temperature indicator

A digital thermometer produces a continuously varying voltage signal between 0V and 5V, where 0V represents 0 degrees and 5V represents 100 degrees. This signal is digitized using an Analog-to-Digital converter that produces an 8-bit binary number proportional to temperature, where 00000000 represents 0 degrees and 11111111 represents 100 degrees (so each binary number represents a multiple of 100/256 degrees).

Design a logic circuit that outputs a logic high signal whenever the temperature is greater than 62.5 degrees but less than 72.5 degrees. Use K-maps to define the circuit, then create and simulate a Verilog description (you will need to create a test bench), and then program the Boolean board with your circuit. Use the eight slide switches to emulate the thermometer output, and an LED to indicate when the temperature is within the desired range.

Hint: You can make an excel spreadsheet with 0-255 in one column, scaled numbers in the next column (by multiplying the first column by 100/254, and rounding to 1 digit after the decimal point), and binary numbers in the third column (by using excel’s DEC2BIN function on the first column of decimal numbers). Then you can identify the numbers in the required range, and devise a circuit to detect only those numbers.

Example: Binary 10110000 would convert to 68.75 degrees (68.8 degrees after rounding), which is within the desired range, and so would cause the output signal to run high.


1. Design an odd number detector (and an even number detector)

Implement a circuit that illuminates an LED when an odd number of the eight slide switches are set to “1”, and illuminates a second LED when an even number of slide switches are set to “1”.

2. Enhance the odd number detector with pushbutton inputs

Illuminate a second LED when the odd LED in the circuit above is illuminated, and 0, 2, or all 4 of the pushbuttons are pressed.