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awzyCombinational CircuitsCombinational Circuits

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Dr. Sahar FawzyDr. Sahar Fawzy

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Combinational CircuitsCombinational Circuits

A combinational circuit consists of logic A combinational circuit consists of logic gates whose outputs at any time are gates whose outputs at any time are determined by the inputs using logic determined by the inputs using logic operations.operations.

For n input variables, there are 2n possible binary input combinations.

For each binary combination of the input variables, there is one possible binary value on each output.

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Combinational CircuitsCombinational Circuits

A combinational circuit can be described by:

A truth table that lists the output values for each combination of the input variables, OR

m Boolean functions, one for each output variable.

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Analysis of a combinational circuit is the determination of the Boolean function that the circuit implements.

Combinational Circuit Analysis

The analysis starts with a given logic circuit diagram and ends with a set of Boolean functions OR a truth table

Proceed from inputs to outputs (systematically). Label intermediate functions. Simplify as possible.

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Combinational Circuit Analysis

T1 = B’C T2 = A’B

T3 = A+T1=A+B’C

T4 = T2 D =(A’B) D = A’BD’ + AD + B’D

T5 = T2 +D = A’B + D

F1 = T3 + T4 = A + B’C +A’BD’+AD+B’D

=A +B’C + BD’ + B’D

F2 = T5 = A’B + D

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Combination Circuit DesignCombination Circuit Design Design of a combinational circuit is the

development of a circuit from a description of its function

Procedure:

- Problem statement

- Truth table and describing Boolean Algebra

- Simplification

- Implementation

- Verification

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Code Converter ExampleCode Converter Example

Design a circuit that converts a binary-coded-decimal (BCD) to the seven signals required to drive a seven-segment light-emitting diode (LED) display.

Assuming the signal 1 illuminates the segment and a logic-0 signal turns off the segment

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Code Converter Example Derive the Boolean function for each output

- e.g., using the following K-map to derive the

Boolean function for output a

ABCD

00 11 1001

00

11

10

01

1

0 1

0 0

1 1 0

0 1 1

1 1

0 0

0

a = A’C + A’BD + A’B’D’+

AB’C’AC

BD

C’

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Adder Design Example

By cascading four 1-bit full adders so that the carry out from one becomes the carry in to the next higher bit position, we can construct a 4-bit adder

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Signed Binary Addition and Signed Binary Addition and SubtractionSubtraction

Representing negative numbers in their 2’s complement representation, the addition and subtraction of two signed numbers are the same as that of two unsigned numbers.

4-bit Adder-subtractor circuit: when s=0, it performs A+B; when s=1, C0=1 and it performs A-B (A adds the 2’s complement of B)

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DecoderDecoderA decoder is a combinational circuit that converts binary information from the n coded inputs to a maximum of 2n unique outputs

• Three inputs A0-A3

are decoded into eight outputs D0-D7

• Each output Di represents one of minterms of the 3 inputs.

• The output variables are mutually exclusive

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Decoder: Decoders are a handy source of minterms. Any

combinational circuit with n inputs and m outputs can be implemented with a n-to-2n-line decoder and m OR gates

Recall full adder equations, and let AiBiCi be the minterm index: Si(Ai, Bi, Ci)= m ( 1, 2, 4, 7 ) and

Ci+1(Ai, Bi, Ci) = m ( 3, 5, 6, 7 )

Since there are 3 inputs and a total of 8 minterms, we need a 3-to-8 decoder

Si

Ci+1

Ai

Bi

Ci

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Decoder: Large decoder construction:

• Example: a 3-to-8 decoder can be built from two 2-to-4 decoders with the use of enable

• Enable =0 forces all outputs to 0

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Multiplexer

A digital multiplexer is a combinational circuit that selects one input from many input lines and directs it to a single output line.

For a 2n-to-1 multiplexer, there are 2n input lines and n selection lines whose bit combination determines which input is selected.

4-to-1 MUX:

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awzyThank youThank you

see you in see you in Sequential CircuitsSequential Circuits