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Combinational Circuits and Sequential Circuits

Combinational Circuits

Combinational Circuits

 A combinational circuit is consist of input variables,logic gates and output variables.The logic gates accept signal from input and generate signal to the outputs.The block diagram of combinational circuits can be drawn as :

Both input and output data are represented by binary signals therefore for N input variables there are total  2possible combinations of binary input values.  

 

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OR Gate | Truth Table for OR gate

OR gate :-  OR gate generates high output signal if any of it’s input signals is high.It has n input and 1 output.
                    For a 2-input OR gate,if A and B are the input signal then output signal Y can be represented as Y= A+B .
This gate is represented as :
                
Truth Table for OR gate :   

A B Y
0 0 0
0 1 0
1 0 0
1 1 1

Tips- You can see in the table that  we get 1 if any one of A or B or both are 1.

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AND Gate | Truth Table for AND gate

AND gate :-  AND gate generates high output signal iff all it’s input signal are high. It has n   input and 1 output.
                      For a 2-input AND gate,if A and B are the input signal then output signal Y can be represented as Y=A.B .
This gate is represented as :
    
Truth Table for AND gate :             

 A

B

 Y

0

0

 0

0

1

 0

1

0

 0

1

1

 1

Tips- You can see in the table that  we get 1 only if both A and B are 1.

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NOR gate | Truth Table for NOR gate

NOR gate :- It is complement of OR gate. It produces high signal output iff all the input signals are low. It has n input and 1 output.
For a 2-input NAND gate,if A and B are the input signal then output signal Y can be represented as  Y=  
This gate is represented as 

Truth Table for NOR gate :

 A

B

 Y

0

0

 1

0

1

 0

1

0

 0

1

1

 0

 Tips-You can see in the table that  we get 1 only if both A and B are 0.

 

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NAND gate

NAND gate :- It is complement of AND gate. It produces low signal output iff all the input signals are high. It has n input and 1 output.
For a 2-input NAND gate,if A and B are the input signal then output signal Y can be represented as  Y=\(\overline{A.B}\)​ ​​.
This gate is represented as
             
 

Truth Table for NAND gate :
                 

 A

B

 Y

0

0

 1

0

1

 1

1

0

 1

1

1

 0

 

TipsYou can see in the table that  we get 0 only if both A and B are 1.

        
 

 

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Half Adder

Half Adder :
A combinational circuit that performs the addition of two bits is called half adder. It needs two input bits that are to be added and produces two output bits that are sum and carry.
The truth table for half adder is:

 x y C S
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0

Tips:- Here carry output is 0 unless both input bits are 1.
We can derive simplified boolean function for half adder with the above truth table as :
S = x'y+xy' = x⊕y
C = xy
The half adder can be implemented with an exclusive-OR gate and an AND gate.The logic diagram for this implementation is as below:

   

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Half Subtractor

Half Subtractor :
A combinational circuit that performs the subtraction of two bits is called half subtractor. It needs two input bits that are to be subtracted and produces two output bits that are difference and borrow.
The truth table for half subtrator is:

 x y B D
0 0 0 0
0 1 1 1
1 0 0 1
1 1 0 0

Tips:- Here borrow bit is 0 unless x<y.
We can derive simplified boolean function for half subtractor with the above truth table as :
D = x'y+xy' = x⊕y
B = x'y
The logic diagram for this implementation is as below:

 

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Full Adder

Full Adder: 
A combinational circuit that performs the arithmetic sum of three input bits is called full adder. It consist of three input bits that are to be added and two output bits that are sum and carry.
The truth table for half adder is:

x y z C S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1

Here x and y are the two significant bits to be added and z is the carry from the previous lower significant position.
We can derive simplified boolean function for full adder with the above truth table as :
S = x'y'z+x'yz'+xy'z'+xyz = (x⊕y)⊕z 
C = xy+xz+yz
The full adder can be implemented using two half adder and one OR gate.The logic diagram for this implementation is as below:

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Full Subtractor

Full Subtractor: 
A combinational circuit that performs subtraction between two input bits is called full subtractor. It consist of three input bits and two output bits .
The truth table for full subtractor is:

x y z B D
0 0 0 0 0
0 0 1 1 1
0 1 0 1 1
0 1 1 1 0
1 0 0 0 1
1 0 1 0 0
1 1 0 0 0
1 1 1 1 1

Here x ,y and z are minuend, subtrahend, and previous borrow, respectively. The two output B and D represents borrow and difference , respectively.
We can derive simplified boolean function for full subtractor with the above truth table as :
D = x'y'z+x'yz'+xy'z'+xyz = (x⊕y)⊕z 
B = x'y+x'z+yz
The logic diagram for this implementation is as below:

 

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Magnitude Comparator

Magnitude Comparator :
A combinational circuit that compares two input bits is called magnitude comparator.It consists of two input bits and three output bits.
If the input bits are X and Y then the output bits are corresponding to one of the case, X>Y, X=Y or X<Y.
We can write a table for magnitude comparator as :

X Y    F1
(X=Y)
  F2
(X<Y)
 F3
(X>Y)
0 0 1 0 0
0 1 0 1 0
1 0 0 0 1
1 1 1 0 0

F1 =X'Y'+XY = X (Ex-NOR)Y
F2=X'Y
F3=XY'
The logic diagram for this implementation is as below:
    
 

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Decoder

Decoder :
It is a combinational circuit that convert binary information from N input lines to a maximum of 2N unique output lines.
The block diagram of deocder can be drawn as:

For example, consider 3x8 decoder circuit as below:

Truth table for above 3x8 decoder:

x y z D0 D1 D2 D3 D4 D5 D6 D7
0 0 0 1 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 0 0
0 1 1 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0
1 0 1 0 0 0 0 0 1 0 0
1 1 0 0 0 0 0 0 0 1 0
1 1 1 0 0 0 0 0 0 0 1

Tips:-  A decoder provides the 2N minterm of n variables. Since any boolean function can be expressed in sum of minterms canonical form,one can use a decoder to generate the minterms and an external OR gate to form the sum.
 

 

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Encoder | Non Priority Encoder | Priority Encoder

Encoder:
It is a combinational circuit that performs inverse operation of decoder.It performs lossless compression.
An encoder has 2N or fewer input lines and N output lines.
These are of two types :
1. Non priority encoder - These encoders do not support simultaneous input activation.
For example, consider 8x3 encoder circuit as below:

Truth table for 8X3 encoder:
 

D0 D1 D2 D3 D4 D5 D6 D7 X Y Z
1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 0 1 1
0 0 0 0 1 0 0 0 1 0 0
0 0 0 0 0 1 0 0 1 0 1
0 0 0 0 0 0 1 0 1 1 0
0 0 0 0 0 0 0 1 1 1 1

2. Priority encoder- These encoders support simultaneous input activation and used for interrupt servicing.
The operation of priority encoder is such that if two or more than two inputs are equal to 1 at the same time then, the input having the highest priority will take precedence.
For example, consider the truth table of a 4x2 priority encoder

D0 D1 D2 D3 X Y
1 1 1
1 0 1 0
1 0 0 0 1
1 0 0 0 0 0

X = D3+D2
Y = D3+D2'D1 

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Lecture on Digital Electronics
Content covered: 

In this video we have covered previously asked questions on multiplexers, particularly based on mux application of implementing a boolean function.

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GATE Question Solutions 2015 EC SET2 Q14
Content covered: 

Solved question on topic of logic gates asked in GATE 2015 EC paper.

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GATE Question Solutions 2016 EC SET1 Q17
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Solved question on topic of logic gates asked in GATE 2016 EC paper.

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Introduction to SR Flip Flop
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Introduction to SR Flip Flop

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Introduction to JK flip flop
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Introduction to JK flip flop

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Introduction to Counters
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Introduction to Counters

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GATE Question Solutions 2015 EC SET2 Q14
Content covered: 

Solved question on topic of logic gates asked in GATE 2015 EC paper.

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GATE Question Solutions 2016 EC SET1 Q17
Content covered: 

Solved question on topic of logic gates asked in GATE 2016 EC paper.

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  • This quiz contains 10 questions on the topic Combinational Circuits, Sequential Circuits
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  • This quiz contains 10 questions on the topic Topic Name
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Difficulty Level:  intermediate