To analyze, simplify, and optimize a given logic circuit using Boolean algebra, implement the optimized circuit using NAND gates, and verify its correctness through simulation and truth table evaluation.
Breadboard
Logic gate IC (AND:74HC08, OR:74HC32, NOT:74HC04)
Digital trainer kit
Connecting wires
Multimeter
Simulation software (Multisim)
The process of converting a given logic circuit to an equivalent Boolean expression is straightforward. Starting from the input side, the output of every logic gate is written in terms of its inputs. At the final output, we obtain the Boolean algebraic expression for the circuit.
The obtained algebraic expression may be simplified by employing the following postulates and theorems of Boolean algebra:
Identity Law: x + 0 = x; x . 1 = x
Null Law: x + 1 = x; x . 0 = 0
Idempotent Law: x + x = x; x . x = x
Complement Law: x + x' = 1; x . x' = 0
Associative Law: (x + y) + z = x + (y + z); (x . y) . z = x . (y . z)
Distributive Law: x(y + z) = xy + xz; x + yz = (x + y)(x + z)
De Morgan’s Theorems: (x + y)' = x' . y'; (xy)' = x' + y'
The simplified Boolean expression can be realized in another logic circuit with generally a lesser number of logic gates.
Writing Boolean expression from a logic diagram involves translating the gates into a mathematical expression. Follow these steps for this translation:
Locate the input variables (e.g. A, B, C) and the final output (e.g. F) of the circuit.
Start from the inputs and trace the path through each gate towards the output.
Write intermediate Boolean expressions for each gate. Use Boolean identities to simplify if possile.
Substitute the intermediate expressions into the final output expression.
Write truth table for the logic diagram from above Boolean expression.
A-----------| \
|AND)----\ \ F
+--| / )OR)--
|\ | +--/ /
B-+--| O-+ |
| |/ |
| |
+------| \ |
|AND)----+
C--------| /
Converting an AND-OR-NOT logic diagram to a NAND-only logic diagram involves replacing all AND, OR, and NOT gates with equivalent NAND gate configurations. This process is based on the fact that NAND gates are universal and can be used to implement any logic function. Below are stepwise instructions to achieve this conversion:
Analyze the given AND-OR-NOT logic diagram.
Identify all the AND, OR, and NOT gates in the diagram.
Label the inputs and outputs of each gate for clarity.
A NOT gate can be implemented using a NAND gate by connecting both inputs of the NAND gate together.
If the input to the NOT gate is A, the output is NOT A.
Replace the NOT gate with a NAND gate where both inputs are A. The output will be (A NAND A) = NOT A.
A --+--| \
| |NAND)O--
+--| /
NOT
A NAND gate is an AND gate followed by a NOT gate.
Convert and AND into a NAND by adding a bubble (O) at the output of the AND gate.
Add another bubble on the same line at some distant so that the signal remains same before reaching next gate.
A---| \
|AND)O---O-
B---| /
An OR gate can be converted to NAND by adding two NOT gates at its inputs.
According to De Morgan's Law (A'+B') = (A+B)', which is NAND.
Add a bubble at every input to the OR gate. If there is already an extra bubble, use it.
If there is no extra bubble, add a pair of bubbles, one at input to the OR gate and other at a distant to make the signal same.
Replace an OR gate with all inverted inputs to a NAND gate.
Redraw the entire logic diagram using only the NAND gates.
Ensure all connections are correctly labeled and bubbles (inversions) are clearly marked.
Compare the truth table of the original AND-OR-NOT logic diagram with the new NAND-only logic diagram.
Ensure both diagrams produce the same output for all possible input combinations.
By adding bubbles to above figure, we get following logic diagram:
A-----------| \
|AND)O--O\ \ F
+--| / )OR)--
|\ | +-O/ /
B-+--| O-+ |
| |/ |
| |
+------| \ |
|AND)O---+
C--------| /
Replacing AND-NOT and NOT-OR with NAND gates, we get:
A--------------| \
|NAND)O--| \
+--| / |NAND)O--
B-+-| \ | +--| /
| |NAND)O-+ |
+-| / |
| |
+------| \ |
|NAND)O-----+
C--------| /
Implement the above circuit on breadboard.
Express the Boolean expression of the given circuit.
Simplify the obtained Boolean expression using postulates of Boolean algebra.
Construct a logic circuit using the NAND implementation on the breadboard by following the datasheet of 74LS00.
Verify the design by evaluating truth tables for both realizations (original and simplified).
Pin# +--u--+ Pin#
A1 ( 1)--¦o ¦--(14) A3
B1 ( 2)--¦ 7 ¦--(13) B3
Y1 ( 3)--¦ 4 ¦--(12) Y3
A2 ( 4)--¦ L ¦--(11) A4
B2 ( 5)--¦ S ¦--(10) B4
Y2 ( 6)--¦ 0 ¦--( 9) Y4
GND ( 7)--¦ 0 ¦--( 8) GND
+-----+
Design the original circuit in Multisim and simulate it.
Evaluate the truth table and verify its correctness.
Implement the simplified circuit and run the simulation.
Compare the outputs of both circuits.
Redesign the circuit using NOR universal gate and repeat steps 1 to 4.
+---+---+---+---+
| A | B | C | F |
+---+---+---+---+
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
+---+---+---+---+
Ensure all connections are secure to avoid errors in circuit implementation.
Verify the functionality of logic gates before using them.
Double-check the Boolean simplifications before implementation.
Handle the digital trainer kit and multimeter with care.
Cross-check the truth tables to confirm the correctness of the circuit.
This experiment demonstrated the optimization of logic circuits. By implementing the logic circuit using NAND universal gates, we achieved a more efficient circuit design with minimized hardware requirements. The truth tables confirmed that both implementations yielded the same outputs, validating the optimization process.
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Circuit and Logic Diagram (3 points)
Simulation, Implementation, and Working (3 points)
Experimental Results (3 points)
Critical Analysis and Conclusion (3 points)
Viva (3 points)