Write Short notes on (Any two)

  1. RTL
  2. State Reduction
  3. POS

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a) RTL

  1. The basic RTL device is a NOR gate.
  2. The inputs represent either logic level HIGH (1) or LOW (0).
  3. The logic level LOW is the voltage that drives corresponding transistor in cut-off region, while logic level HIGH drives it into saturation region.
  4. If both the inputs are LOW, then both the transistors are in cut-off i.e. they are turned-off. Thus, voltage Vcc appears at output I.e. HIGH.
  5. If either transistor or both of them are applied HIGH input, the voltage Vcc drops across Rc and output is LOW.

Advantages of RTL Logic circuit:

The primary advantage of RTL technology was that it involved a minimum number of transistors, which was an important consideration before integrated circuit technology, as transistors were the most expensive component to produce

b) State Reduction

Any design process must consider the problem of minimizing the cost of the final circuit. The two most obvious cost reductions are reductions in the number of flip-flops and the number of gates.

The number of states in a sequential circuit is closely related to the complexity of the resulting circuit. It is therefore desirable to know when two or more states are equivalent in all aspects. The process of eliminating the equivalent or redundant states from a state table/diagram is known as state reduction.

Example: Let us consider the state table of a sequential circuit shown in below table.

Present State Next State Output
x = 0 x = 1 x = 0 x = 1
A B C 1 0
B F D 0 0
C D E 1 1
D F E 0 1
E A D 0 0
F B C 1 0

It can be seen from the table that the present state A and F both have the same next states, B (when x=0) and C (when x=1). They also produce the same output 1 (when x=0) and 0 (when x=1). Therefore states A and F are equivalent. Thus one of the states, A or F can be removed from the state table. For example, if we remove row F from the table and replace all F’s by A’s in the columns, the state table is modified as shown in below

Present State Next State Output
x = 0 x = 1 x = 0 x = 1
A B C 1 0
B A D 0 0
C D E 1 1
D A E 0 1
E A D 0 0

It is apparent that states B and E are equivalent. Removing E and replacing E’s by B’s results in the reduce table shown in Table 8.

Present State Next State Output
x = 0 x = 1 x = 0 x = 1
A B C 1 0
B A D 0 0
C D B 1 1
D A B 0 1

The removal of equivalent states has reduced the number of states in the circuit from six to four. Two states are considered to be equivalent if and only if for every input sequence the circuit produces the same output sequence irrespective of which one of the two states is the starting state.

c) POS

Product of sum (POS) is a form in which products of the dissimilar sum of inputs are taken, which are not arithmetic result & sum although they are logical Boolean AND & OR correspondingly. Before going to understand the concept of the product of the sum, we have to know the concept of the max term.

The maxterm can be defined as a term that is true for the highest number of input combinations otherwise that is false for single input combinations. Because OR gate also provides false for just one input combination. Thus Max term is OR of any complemented otherwise non-complemented inputs.

X Y Z Max term (M)
0 0 0 X+Y+Z = M0
0 0 1 X+Y+Z’ = M1
0 1 0 X+Y’+ Z = M2
0 1 1 X+Y’+Z’ = M3
1 0 0 X’+Y+Z= M4
1 0 1 X’+Y+Z’ = M5
1 1 0 X’+Y’+Z = M6
1 1 1 X’+Y’+Z’ = M7

In the above table, there are three inputs namely X, Y, Z and the combinations of these inputs are 8. Every combination has a max term that is specified with M.

For example, the following Boolean function is a typical product-of-sum expression:

Product of Sum Expressions

Q = (A + B).(B + C).(A + 1)

and also

(A + B + C).(A + C).(B + C)

However, Boolean functions can also be expressed in nonstandard product of sum forms like that shown below but they can be converted to a standard POS form by using the distributive law to expand the expression with respect to the sum. Therefore:

Q = A + (BC)

Becomes in expanded product-of-sum terms:

Q = (A + B)(A + C)

Another nonstandard example is:

Q = (A + B) + (A.C)

Becomes as an expanded product-of-sum expression:

Q = (A + B + A)(A + B + C)

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