Combinational Circuits (Adder, Subtractor, Encoder, Decoder, and Multiplexer

Combinational Circuits

Definition: In simple terms a circuit in which output depends only on the inputs but not on the state of that circuit is known as combinational circuits.

We have different types of combinational circuits like Adder, Subtractor, Encoders, Decoders, Multiplexers and De-Multiplexers.There are lots of books you can follow to study these circuits, so there is no need of going  in-depth study of these circuits here, at this post I will give you some tricks or ideas how to use knowledge of these circuit in solving GATE problems, and how to create a circuit for a given function for these Adder, Subtractor, Encoders, Decoders, Multiplexers.Here is an applet which I have used in my Digital Logic post and again here I am presenting this applet again for all of you. Just use radio buttons which is here as "Gates1, Gates2, Full Adder ..." and get more practical experience about these circuits

NOTE-1: Any n-variable can be realized or implemented  using 2^(n-1)*1 MUX(multiplexer).

NOTE-2: Number of m*1 multiplexer required to form n*1 multiplexer is

here for fractional value of this log term we have to use the ceiling value of that fraction.

this formula is very helpful in solving such questions where you encounter such a question where you have to find number of small multiplexers need to create a larger multiplexers.

If no applet appears, you have to install the JRE of http://java.sun.com.

here it is a java applet for half adder if this is not being displayed in your browser install java in your browser.
for more digital circuit view this applets page for digital circuits 

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Finding r's complement and (r-1)'s complement of a number (1s complement and 2s complement)

Finding the r's and (r-1)'s complement

Here we are going to learn how to convert a number to its r's and (r-1)'s complement. 

Method:

Let 'N' is a number and r is its base where r>1 and in N,  'n' is the number of digits before its decimal point then  we can write 

r's complement of number = r^n-N

EX.

                       N = (23)10

                              here r = 10

                              n = 2 and N = 23 

hence we can write the 10's complement of this number as 10^2 - 23 = 77.

hence we can say that 10's comp of 23 is 77. 

Although this method is good enough to solve any problem regarding to this concept, but we will follow different method for finding r's and r-1's complement.

Easy Method:

Let we have to find again the 10's comp of 23 then this method tells us to divide 3 from 10 and 2 from 9 (i.e 10-9). which gives us a result of 77.

                               9  10

                           -  2    3

                                       

                               7    7

i.e the generalized form of writing a r's comp of a number 'abc' which is in r base, we can write.

                             (r-1)   (r-1)   r

                        -     a            b    c

                                                       

this difference gives us the r's comp of that number.

i.e we can find r's complement of a number by subtracting its right most digit by r and all digits by r-1.

Finding (r-1)'s complement:

We can do this easily by subtracting all the digits of that number from (r-1) where r is the base of that number.

EXAMPLES:  

Find the 10's and 9's complement of (348)10. 

ans:                   9   9   10

                       - 3   4     8

                                       

                        6    5    2    here 652 is 10's comp of 348 

9's comp          9    9    9 

                     -  3   4    8

                                      

                        6    5    1    here 651 is 9's comp of 348

from this method you can find the r's and (r-1)'s complement of any number with base r.

if any questions DO COMMENTS..

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Digital Logic gates (AND gate, OR gate, NOT gate)

Digital Logic Gates

Logic gates are the fundamental building blocks of digital systems.

There are three basic gates

1.  AND

2.  OR

3.  NOT

here in this picture you can find that for AND gates out-put becomes 0 if any of the input is Zero(0).

similarly for OR gate out-put is 1 if any of the input is 1.

NOT gate simply reverse the input at its out-put.

Here we are not going to the details of hardware realization of logic gates.(Since it is not in the syllabus of gate exam.)

Although we can realize any circuit using AOI(AND, OR, NOT) gates, there are also two different gates which can also realize any circuit using these gates. So we call these gates as universal gates. Name of these gates are NAND and NOR. from this figure we can also find their truth tables.
Here I am giving you an overview how these gate works we will come later here for more sophisticated   operation of logic gates.

here it is a java applet use this to understand the how logic gate works.

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Digital Electronics and Gates(And gate, OR gate, Not gate)

Digital Electronics

Introduction: When we listen the term digital logic the first thing which comes in our mind is digital values i.e 0 and 1. In digital electronics generation of zero volt input is almost impossible to achieve hence we think values of  ~ 0.5, 0.6, 0.7 V~ as 0V input and  ~ 4.5, 4.6, 4.7V ~ as 5V.

So here question is that why we use digital signal if they are actually not discrete, ( as we know we take ~ 0.5, 0.6, 0.7 V~ as 0V input and  ~ 4.5, 4.6, 4.7V ~ as 5V. etc...)

So lets starts with some different concepts.

Let us take an analog signal

Analog to Digital

we will quantize this signal using some discrete quantum values, and the portion which comes in between two quantum is known as quantization error. So if we take small quantum then this error will be less.
And after getting this discrete values we use comparator to give them a some level for a particular range, and when we get a discrete value we use it with logic gates.

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