# Digital circuits - Logical realization at two levels

Tutorial on digital circuits 2020-11-18 02:31:17# Digital circuits - Two-level logic implementation

The maximum number of levels present between the inputs and the output is two in the two-level logic ** levels **. This means that, regardless of the total number of logic gates, the maximum number of logic gates present (in cascade) between any input and output is two in two-level logic. Here, the outputs of the first level logic gates are connected as the second level logic gates inputs.

Consider the four logic gates AND, OR, NAND & NOR. Since there are 4 logic gates, we will have 16 possible ways to achieve two-level logic. These 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 logical realizations can be classified into the following two categories.

- Degenerative form
- Non-degenerative form

## Degenerative form

If the output of the two-level logical realization can be achieved by using a single logical gate, then it's called a ** degenerative form **. Obviously, the number of inputs of a single logic gate increases. Because of this, the logic gate fan-in increases. It is an advantage of degenerative form.

Only ** 6 combinations ** of two-level logical realizations out of 16 combinations are in degenerative form. These are AND-AND, AND-NAND, OR-OR, OR-NOR, NAND-NOR, NORNAND.

In this section, let's talk about some achievements. Suppose that A, B, C and D are the inputs and Y is the output in each logical realization.

### AND-AND logic

In this logical realization, AND gates are present in both levels. The figure below shows an example of ** logical AND-AND ** realization.

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

These outputs, $ Y_ {1} $ and $ Y_ {2} $ are applied as inputs to the AND gate present at the second level, so the output of this AND gate is

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

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

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

$ Rightarrow Y = ABCD $

Therefore, the output of this logical AND-AND realization is ** ABCD **. This Boolean function can be implemented using a 4-input AND gate. Therefore, it is ** degenerative form **.

### AND-NAND logic

In this logical realization, the E gatesT are present at the first level and NAND gates are present at the second level. The following figure shows an example of realization of ** AND-NAND logic **.

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

These outputs, $ Y_ {1} $ and $ Y_ {2 } $ are applied as inputs to NANDgate which is present at level 2. So the output of this NAND gate is

$$ Y = {left (Y_ {1} Y_ {2} right)} '$$

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

$$ Y = {left (left ( AB right) left (CD right) right)} '$$

$ Rightarrow Y = {left (ABCD right)} ' $

Therefore, the output of this sheavelogical AND-NAND is $ {left (ABCD right)} '$. This Boolean function can be implemented using a 4 input NAND gate. It is therefore ** degenerative form **.

### OR-OR logic

In this logical realization, OR gates are present in both levels. The following figure shows an example of realization of ** logic OR-OR **.

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

These outputs, $ Y_ {1} $ and $ Y_ {2} $ are applied as inputs to the OR gate that is present at the second level. Thus, the exit from this OR gate is

$$Y=Y_ Icône1 Cobra+Y_BUILDERS$$

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

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

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

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

Likewise, you can check whether the remaining achievements belong to this category or not.

## Non-degenerative form

If the output of the two-level logical realization cannot be obtained using a single logical gate, then it is called as ** non-degenerative form **.

The rest ** 10 combinations ** of two-level logical realizations come in non-degenerative form. These 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 achievements. Suppose, A, B, C & D shave the inputs and Y is the output in each logical realization.

### AND-OR logic

In this logical realization, the AND gates are present at the first level and the OR gate (s) are present at the second level. The figure below shows an example of realization of ** AND-OR logic **.

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

These outputs, Y1 and Y2 are applied as inputs of the OR gate which is present at the second level. So the output of this OR gate is

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

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

$$ Y = AB + CD $$

Therefore, the output of this logical realization AND-OR is ** AB + CD **. This Boolean function is in the form ** Sum of products **. Since, we cannot implement it using a single logic gate, this AND-OR logic realization is a ** non-degenerative form **.

### AND-NOLogique R

In this logical realization, AND gates are present at the first level and NOR gates are present at the second level. The following figure shows an example of realization of ** AND-NOR logic **.

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

These outputs, Y1 and Y2 are applied as inputs of the NOR gate which is present at the second level. So the output of this NOR gate is

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

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

$$ Y = {left (AB + CD right)} '$$

Therefore, the output of this logical AND-NOR realization is $ {left (AB + CD right)} '$. This Boolean function is in the form ** AND-OR-Invert **. Since we cannot implement it using a single logic gate, this AND-NOR logic realization is a ** non-degenerative form **

### OR-AND Logic

In this logical embodiment, the OR gates are present at the first level and the AND gate (s) are present at the second level. The following figure shows an example of realization of ** logic OR-AND **.

Previously we were getting the outputs offirst level logic gates like $ Y_ {1} = A + B $ and $ Y_ {2} = C + D $.

These outputs, $ Y_ {1} $ and $ Y_ {2} $ are applied as inputs to the AND gate which is present at the second level. So the output of this AND gate is

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

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

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

Therefore, the output of this logical OR-AND realization is ** (A + B) (C + D) **. This Boolean function has the form ** Product of sums **. Since we cannot implement it using a single logic gate, this OR-AND logic realization is a ** non-degenerative form **.

Likewise, you can check whether or not the resulting achievements belong to this category.