Hey folks! This is the beginning of Project 3 where we explore a variety of combinational circuits and how to analyze and design them.
What are Combinational Circuits?
There are two main types of circuits: combinational and sequential. Combinational circuits are circuits whose outputs are determined only by the present combination of inputs entered into the circuit. The circuit performs operations that it is made to perform by a set of Boolean functions. Several of these combinational circuits are employed when designing digital systems. They are called standard components and they perform specific digital functions commonly needed in the design of the circuit. We will explore common standard components in this project such as adders, decoders, encoders, and multiplexers.
Here is a block diagram showing n inputs and m outputs through a combinational circuit:
It is important to note that for n inputs, there are 2^n combinations of binary inputs and for each combination, there is only one possible value for each output variable. Therefore, we can represent a combinational circuit as a truth table that lists the output values for each combination of input variables. The circuit can also be represented as m Boolean functions, one for each output variable where each output function is represented in terms of the n input variables.
Analysis Procedure
The purpose of analyzing any circuit is to determine the functions that make up that circuit. Given a logic diagram of a specific circuit, we are meant to produce a truth table and Boolean functions. Sometimes analyzing the circuit allows us to discover that the circuit produces useful values, in that case, we should also explain what the circuit can do.
To better illustrate the procedure, we will be analyzing the following circuit:
Finding the Boolean Functions
Step One: Ensure the circuit is actually a combinational circuit
The diagram of a combinational circuit has logic gates with no feedback paths or memory elements. This means that electrical signals flow in one direction from the inputs to the outputs with no connection from the output of one gate to the input of a second gate. If there are feedback paths or memory elements, the circuit would be sequential. Here is an example of a sequential circuit:
See how the output of the first NOR gate is connected to the input of the second NOR gate and the output of the second NOR gate is connected to the input of the first NOR gate. This feedback loop makes this circuit a sequential circuit. You can see that the circuit we are working with in this example does not have any connections that look like that and therefore, it is a combinational circuit and we can move to step two.
Step Two: Labelling intermediate gates
We can see from the circuit that there are two output functions: F1 and F2. We can also see that these functions are made of intermediate products of terms: T1, T2, and T3.
We created these labels arbitrarily but they do hold meaning for us. Step two is labelling and expressing the intermediate products in terms of a combination of input variables so that we are one step closer to identifying the output functions as Boolean functions in terms of the input variables.
T1, T2, and T3 can be expressed as the following:
• T1 = (ABC)’
• T2 = (A+B+C)’
• T3 = F2’ + T2
Now that we have all the intermediate gates expressed as functions of the input variables, we can move to step three.
Step Three: Labelling output gates
Now, we label the output gates as F1 and F2 and determine the functions that represent these gates.
F1 and F2 can expressed as follows:
• F1 = (T1)(T3)
• F2 = AB + AC
Now, through repeated substitution and Boolean algebra, we find the Boolean functions that express F1 and F2 in terms of the input variables:
Now that we’ve found our Boolean functions, preparing the truth tables should be easy.
Preparing the Truth Table
Step One: Determine the number of input variables for the circuit
For n inputs, we will have a 2^n combinations of inputs that produce a unique output. Our first step is so to determine how many inputs there are and begin preparing the truth table.
In our case, we have 3 inputs: A, B, and C. Therefore, we will have 2^3 = 8 combinations.
Step Two: Determine the values for each output function
The simplest way I have found for making error-proof truth tables is to take the function and break it down into one term at a time. For example, to find F1 = A’ + B’C’, we will have a column for A’, B’, and C’ and then a column for B’C’, and then a column for A’ + B’C’.
1) Finding values of F1 = A’ + B’C:
2) Finding values of F2 = AB + AC:
Step Three: Summarize the truth tables by merging the previous truth table results into one table
Now, to summarize the results of our analysis:
The Boolean functions:
• F1 = A’ + B’C’
• F2 = AB + AC
The truth table:
The point of analyzing circuits is to figure out what the circuit does. This particular example does not generate anything useful but understanding what we’ve done here will allow you to understand the examples we will be looking at in the upcoming weeks. Those examples will produce useful circuits that are often used in digital design.
I created a video to go along with this article - feel free to check it out!
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