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Realization of Boolean Expressions using only Universal Gates

Here, we are going to learn about the Realization of Boolean Expressions using only Universal Gates in Digital Electronics.
Submitted by Saurabh Gupta, on November 23, 2019

We already know that NAND and NOR are recognized as the universal gates using which we can perform the functioning of any other logic gate. Thus, any Boolean Expressions can also be realized using NAND/NOR Gate only.

Generally, in Digital Electronics, the minimal Boolean expression are obtained either in SOP (Sum of Products) form or POS (Product of Sums) form. Sometimes, it can also be found in a hybrid of both forms. If the Boolean expression is in SOP form then NAND gates should be used for realization, on the contrary, if POS form is available we use NOR Gate for the realization of the logic circuit.

Once, we perform realization using hybrid AOI logic, then there may arise a case of multilevel logic which results in non-uniform propagation delay between input and output which results in a logic race. However, in case of realization using the universal gates, they give rise to two-level logic. Two-level logic provides a uniform delay between the input and output because each input signal has to pass through two gates to reach the output. So, there are no chances of logic race in realization using universal gates.

Any AOI logic can be converted into NAND/NOR logic following some steps which are listed below.

Step 1: Draw the circuit in AOI logic.

Step 2: If the circuit is to be drawn only using NOR Gates, we have to add a circle at the output of each OR Gate and the input of each AND Gate.

Step 3: If the circuit is to be drawn only using NAND Gates, we have to add a circle at the output of each NAND Gate and the input of each OR Gate.

Step 4: Now, add or subtract an inverter on each line where we have drawn a circle in steps 2 or 3 so that the polarity of signals on those lines remains unchanged from those of the original diagram.

Example 1: Realize the Boolean expression

Y = A + AB.(C + D), Using only (a) NAND Gate and (b) NOR Gate

Solution: Our first step is to draw the circuit using AOI logic which can be drawn as:

Realization of boolean expressions example 1

(a) For realization using NAND logic, we will follow step 3, and add a circle to the output of each AND gate and also at the inputs of each OR gate which can be shown as,

Realization of boolean expressions example 2

Now, moving on to step 4, we have to add or remove an inverter on every line where we have drawn a circle in the previous step which can be shown as

Realization of boolean expressions example 3

We have added inverters at line x, y, z, p and q as a circle was drawn here, also in the line r addition of an inverter is not required since two circles were drawn and adding two inverters will cancel out each other.

Therefore, the final circuit using only NAND can be represented as,

Realization of boolean expressions example 4

(b) For realization using NOR logic, we will follow step 2, and add a circle to the output of each OR gate and also at the inputs of each AND gate which can be shown as,

Realization of boolean expressions example 5

Now, moving on to step 4, we have to add or remove an inverter on every line where we have drawn a circle in the previous step which can be shown as,

Realization of boolean expressions example 6

We have added inverters at line a, b, c and d as a circle was drawn here, also in the line e addition of an inverter is not required since two circles were drawn and adding two inverters will cancel out each other. Therefore, final circuit using only NOR can be represented as,

Realization of boolean expressions example 7





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