Combinational Logic uses a combination of basic logic gates AND, OR and NOT to create complex functions. For each output, the design procedure is:
- Derive the truth table.
- Simplify the boolean expression using Karnaugh Map (K map).
- Draw a logic diagram that represents the simplified Boolean expression. Verify by analysing or simulating the circuit.
Your objective is increase your score in the 3 areas: Truth Table, K Map, and Logic Gates while designing different digital circuits
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Black Box Continue
A black box shows the relationship between its inputs and outputs without knowing its implementation.
Click on the inputs and observe the output(s) to understand the specification.
- When the input, output or wire is grey, its value is 0 (false).
- When the input, output or wire is green, its value is 1 (true).
Truth Table Check Skip
Your task is replace all the ? in the output column by clicking on the cell. When you are done click Check.
- When you click on the inputs of the blackbox, the corresponding row is highlighted in the truth table.
- On the truth table, note the input variables and their values.
- Recall that when the input, output in the blackbox is grey, its value is 0. When it is green, its value is 1.
- Use the output color to determine if the output cell should be 0 (grey) or 1 (green).
- If the blackbox has more than one output, we will solve one output at a time.
The Truth Table shows the values of the circuit output for all input values. The number of rows is 2n where n is the number of inputs. So for 2 inputs, there are 4 rows and for 3 inputs there are 8 rows and so on.
You can also fill up the truth table using the specification.
Your task is to select the minterm(s) that represent ? in the Sum of Product. When you are done, click Check
- Hover or tap on ? to find the respective output cell.
- Study the other minterms to understand how they are formed.
There are 2 steps to obtain the Sum of Products from its truth table.
- A Minterm is a product • (AND) term containing all input variables. A variable X appears in complemented form X if it is a 0 in the row of the truth-table, and as a true form X if it appears as a 1 in the row.
- The Sum of Products is obtained by taking the sum + (OR) of the minterm of the rows where a 1 appears in the output.
K Map Check Skip
Your task is to replace all the ? in the K Map by clicking on that cell. When you are done, click Check
- When you hover or tap on an output cell in the truth table, the corresponding cell in the K Map is highlighted.
- Understand the relationship by studying the variables in the headers of both the truth table and the K map.
The Karnaugh map (K map) provides a simple and straight-forward method of minimising boolean expressions.
A K map is a two-dimensional truth-table. Note that the squares are numbered so that any two adjacent cells differ by a change in only one variable.
Your task is to memorise and group adjacent cells containing 1's.
- When you hover or tap on the product terms in the Boolean expression, you will see groups of adjacent 1's. Memorise all of them and click Hide.
- Now click on the cells containing 1's to group them as memorised earlier. When you have selected a group, click Group.
- Repeat the previous step until the button changes to Check.
Rules for Grouping together adjacent cells containing 1's
- Groups must contain 1, 2, 4, 8, 16 (2n) cells.
- Groups must contain 1's only.
- Groups may be horizontal or vertical, but not diagonal.
- Groups should be as large as possible.
- Each cell containing a 1 must be in at least one group.
- Groups may overlap.
- Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.
- There should be as few groups as possible.
Your task is to select the product term(s) that represent ? in the Boolean Expression. When you are done, click Check
- Hover or tap on ? to find the respective group.
- Study the other product terms to understand how they are formed.
Obtaining Product Terms
- If X is a variable that has value 0 in all of the squares in the grouping, then the complemented form X is in the product term.
- If X is a variable that has value 1 in all of the squares in the grouping, then the true form X is in the product term.
- If X is a variable that has value 0 for some squares in the grouping and value 1 for others, then it is not in the product term.
Logic Gates Check Skip
Your task is to connect up the logic gates to realise the Boolean Expression obtained from the K Map. When you are done, click Check
- Hover or tap on the gate to find the boolean expression.
- Study the connected gate to understand its function.
- To connect, click from the output of an element and drag to the input of the element that needs to be connected.
- If you are using a mobile device, select zoom/pan if you need to zoom or pan the logic diagram and select wire if you want connect the circuit.
Connecting the logic gates
- An AND gate performs the same function as product • operator in boolean algebra. It has only 1 output but can have 2 or more inputs.
- An OR gate performs the same function as sum + operator in boolean algebra. It has only 1 output but can have 2 or more inputs.
- A NOT gate inverts its input. It has only 1 output and only 1 input. If the input is X, then its output is X
Design Complete Continue
There are three ways to describe a Boolean function: logic circuit, truth table, and Boolean expression.
To convert a truth table to its circuit
- Convert the truth table to a Boolean expression using the sum of products technique.
- Minimise the Boolean expression using Karnaugh map (or Boolean algebra).
- Convert the minimised Boolean expression into a circuit.