Digital Circuits - Two-Level Logic Reapzation

The maximum number of levels that are present between inputs and output is two in two level logic. That means, irrespective of total number of logic gates, the maximum number of Logic gates that are present (cascaded) between any input and output is two in two level logic. Here, the outputs of first level Logic gates are connected as inputs of second level Logic gate(s).

Consider the four Logic gates AND, OR, NAND & NOR. Since, there are 4 Logic gates, we will get 16 possible ways of reapzing two level logic. Those are AND-AND, AND-OR, ANDNAND, AND-NOR, OR-AND, OR-OR, OR-NAND, OR-NOR, NAND-AND, NAND-OR, NANDNAND, NAND-NOR, NOR-AND, NOR-OR, NOR-NAND, NOR-NOR.

These two level logic reapzations can be classified into the following two categories.

Degenerative form

Non-degenerative form

Degenerative Form

If the output of two level logic reapzation can be obtained by using single Logic gate, then it is called as degenerative form. Obviously, the number of inputs of single Logic gate increases. Due to this, the fan-in of Logic gate increases. This is an advantage of degenerative form.

Only 6 combinations of two level logic reapzations out of 16 combinations come under degenerative form. Those are AND-AND, AND-NAND, OR-OR, OR-NOR, NAND-NOR, NORNAND.

In this section, let us discuss some reapzations. Assume, A, B, C & D are the inputs and Y is the output in each logic reapzation.

AND-AND Logic

In this logic reapzation, AND gates are present in both levels. Below figure shows an example for AND-AND logic reapzation.

We will get the outputs of first level logic gates as $Y_{1}=AB$ and $Y_{2}=CD$

These outputs, $Y_{1}$ and $Y_{2}$ are appped as inputs of AND gate that is present in second level. So, the output of this AND gate is

$$Y=Y_{1}Y_{2}$$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation.

$$Y=left ( AB ight )left ( CD ight )$$

$Rightarrow Y=ABCD$

Therefore, the output of this AND-AND logic reapzation is ABCD. This Boolean function can be implemented by using a 4 input AND gate. Hence, it is degenerative form.

AND-NAND Logic

In this logic reapzation, AND gates are present in first level and NAND gate(s) are present in second level. The following figure shows an example for AND-NAND logic reapzation.

Previously, we got the outputs of first level logic gates as $Y_{1} = AB$ and $Y_{2} = CD$

These outputs,$Y_{1}$ and $Y_{2}$ are appped as inputs of NAND gate that is present in second level. So, the output of this NAND gate is

$$Y={left ( Y_{1}Y_{2} ight )} $$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation.

$$Y={left ( left ( AB ight ) left ( CD ight ) ight )} $$

$Rightarrow Y={left ( ABCD ight )} $

Therefore, the output of this AND-NAND logic reapzation is ${left ( ABCD ight )} $. This Boolean function can be implemented by using a 4 input NAND gate. Hence, it is degenerative form.

OR-OR Logic

In this logic reapzation, OR gates are present in both levels. The following figure shows an example for OR-OR logic reapzation.

We will get the outputs of first level logic gates as $Y_{1}=A+B$ and $Y_{2}=C+D$.

These outputs, $Y_{1}$ and $Y_{2}$ are appped as inputs of OR gate that is present in second level. So, the output of this OR gate is

$$Y=Y_{1}+Y_{2}$$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation.

$$Y=left ( A+B ight )+left ( C+D ight )$$

$Rightarrow Y=A+B+C+D$

Therefore, the output of this OR-OR logic reapzation is A+B+C+D. This Boolean function can be implemented by using a 4 input OR gate. Hence, it is degenerative form.

Similarly, you can verify whether the remaining reapzations belong to this category or not.

Non-degenerative Form

If the output of two level logic reapzation can’t be obtained by using single logic gate, then it is called as non-degenerative form.

The remaining 10 combinations of two level logic reapzations come under nondegenerative form. Those are AND-OR, AND-NOR, OR-AND, OR-NAND, NAND-AND, NANDOR, NAND-NAND, NOR-AND, NOR-OR, NOR-NOR.

Now, let us discuss some reapzations. Assume, A, B, C & D are the inputs and Y is the output in each logic reapzation.

AND-OR Logic

In this logic reapzation, AND gates are present in first level and OR gate(s) are present in second level. Below figure shows an example for AND-OR logic reapzation.

Previously, we got the outputs of first level logic gates as $Y_{1} = AB$ and $Y_{2} = CD$.

These outputs, Y1 and Y2 are appped as inputs of OR gate that is present in second level. So, the output of this OR gate is

$$Y=Y_{1}+Y_{2}$$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation

$$Y=AB+CD$$

Therefore, the output of this AND-OR logic reapzation is AB+CD. This Boolean function is in Sum of Products form. Since, we can’t implement it by using single logic gate, this AND-OR logic reapzation is a non-degenerative form.

AND-NOR Logic

In this logic reapzation, AND gates are present in first level and NOR gate(s) are present in second level. The following figure shows an example for AND-NOR logic reapzation.

We know the outputs of first level logic gates as $Y_{1} = AB$ and $Y_{2} = CD$

These outputs, Y1 and Y2 are appped as inputs of NOR gate that is present in second level. So, the output of this NOR gate is

$$Y={left ( Y_{1}+Y_{2} ight )} $$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation.

$$Y={left ( AB+CD ight )} $$

Therefore, the output of this AND-NOR logic reapzation is ${left ( AB+CD ight )} $. This Boolean function is in AND-OR-Invert form. Since, we can’t implement it by using single logic gate, this AND-NOR logic reapzation is a non-degenerative form

OR-AND Logic

In this logic reapzation, OR gates are present in first level & AND gate(s) are present in second level. The following figure shows an example for OR-AND logic reapzation.

Previously, we got the outputs of first level logic gates as $Y_{1}=A+B$ and $Y_{2}=C+D$.

These outputs, $Y_{1}$ and $Y_{2}$ are appped as inputs of AND gate that is present in second level. So, the output of this AND gate is

$$Y=Y_{1}Y_{2}$$

Substitute $Y_{1}$ and $Y_{2}$ values in the above equation.

$$Y = left ( A+B ight )left ( C+D ight )$$

Therefore, the output of this OR-AND logic reapzation is (A + B) (C + D). This Boolean function is in Product of Sums form. Since, we can’t implement it by using single logic gate, this OR-AND logic reapzation is a non-degenerative form.

Similarly, you can verify whether the remaining reapzations belong to this category or not.