Hi everybody, this is Atik. I welcome all of you to my YouTube channel Engineering Drive. Today I am going to discuss the concept of design of fast adders. So, before I am going to speak out on this particular concept, we should be having somewhat clear idea on what is a binary number and what is a BCD number.
Okay. So, let me just give a small picture on this. Then I am going to proceed with my concept.
So, my dear students, you know that our computer understands only one language that is machine language or we can call it as binary language in the form of zeros and ones. So, along with the binary language. our computer makes use of one more representation to store the set of numbers that is binary coded decimal representation or simply bcd representation okay now in this particular concept of fast adders i am going to explain to you how we can convert a set of binary numbers to a corresponding bcd set of numbers okay so to explain this concept let us take first of all one sample set of decimal numbers so here i have taken numbers from 0 to 19 okay now this 0 to 19 how we can represent represent this set of decimal numbers into corresponding bcd representation binary representation we are going to see here okay to represent the set of bcd numbers we have taken the this set of bits s1 s2 s4 s8 and c you know that the Our computer's language of instruction will start from right side. That is why I have took from S1, from right side S1, S2, S4, S8 and C. Similarly, the binary representation, we have taken the name of the bits as Z1, Z2, Z4, Z8 and K.
So, C and K are the carry bits. That is why we have taken with separate letters C and K. C is a carry bit of DCD representation.
K is a carry bit of binary representation. Now... We can see one thing which is very clear to us from decimal numbers 0 to 9, the BCD representation and binary representation is same.
Okay, you can see here decimal representation of 0 for BCD 0s as well as for binary is also 0. If you take the decimal number as 1 or that the representation of BCD is same as well as the representation of binary sum is also same. Okay, let me keep one more line between this. Thank you.
Now, when the change will happen here? The change will come during the numbers from 10 to 19 which means when you get two digits in a decimal number, the BCD representation and binary representation will differ. Why it will differ? You know if you want to convert this number 10 in binary form, what you will do?
divide this 10 by 2 you will get this binary representation directly okay Whereas to represent this 10 in BCD what we need to do means what our computer makes use of. So, there you know that in this 10 we are having two bits. First bit is 0, second first bit is 1, from left side second bit is 0. If you take from right side the first bit is 0, the second bit is 1. Now to represent this 0 our computer makes use of this 4 bits. To represent this one. it will be it will be stored in the carry bit c ok.
Similarly, when we take the numbers second number as 11 ok, how this 11 will be stored ok. This first one this left side one will be stored in carry bit as it is whereas, the remaining 4 bits are used to store this second one you can see as this one will be stored ok. So, you got one clear picture that in when the decimal number contains 2 digits, so you First digit is stored separately in binary representation and second bit is stored in the form of carry. Which means the main difference between the binary representation and BCD representation is binary and BCD representation is same when the decimal number contains only one bit. When our decimal number contains two bits or two digits at that time the binary representation of two digits also will become same whereas the BCD representation of two digits is different.
The reason behind this is our BCD makes use of separate representation for every digit. For example, let us say we are having three numbers 125. Then For one it will be can it will stored separately in binary form, two will be stored separately in binary form, five will be stored separately in binary form. This is nothing but our BCD representation.
Ultimately our BCD representation makes use of more bits to store the values ok. So now the concept has been clear to you how BCD sum will be represented and binary sum will be represented. Now in our next slide what we are going to do?
we are going to add what we are going to add we are going to add binary sum and we are going to do input and finally we are going to produce the bcdc okay that's what i am going to explain in the next slide okay now here we have this formula let us take this formula first c is equal to k plus z8 into z4 plus z8 into z2 so here in order to find out the value of c what we are going to do We are going to add three terms. one is k another one is zx into z4 another one is zx into z4 so what is this one sir and what about these diagrams are whether it is difficult to understand no my dear students it is very easy to understand just to concentrate what i am going to say you know this is very very important how this fast what is the purpose of fast adders means fast adders will take binary sum as the input whatever fast adder will do they will take binary sum so in order to have sum in order to perform some What we need to do? We need to take at least two numbers.
So let us take, I have taken two numbers, two and three. So two is nothing but addend, three is nothing but augment. And in between, what I am doing?
Addition. Okay. The difference in this diagram is here, we are not taking decimal numbers here.
Here we are taking binary numbers. Z1, Z2, Z4, Z8 and K, they are the bits of binary representation. So this is, that is why we are calling this up box as binary sum. This is a 4 bit adder and the output is also a 4 bit adder.
The main difference between this is the first 4 bit adder is an input for binary, the second 4 bit adder is output for BCD. That is what we need to see ok. Now plus indicates we are going to perform addition because it is a adder ok. It has been cleared to you now. Now what we are going to do means finally what is the output we need to get from here?
We need to get the output binary So, what we are going to do? First of all we need to find out the carry bit. How the carry bit is represented means with the help of C.
Now, we are going to make use of two gates here. One is the power gate and two more gates we are going to make use of. They are known as AND gates.
So, how many AND gates are there? Two AND gates. How many OR gates are there?
One OR gate. So OR gate, what is the use of OR gate addition? What is the use of AND gate? The main purpose of AND gate is multiplication.
So both addition and multiplication we are using in our formula here. Find out the value of C. Now what we are doing?
First we are adding k. So here k. This is the k.
This k we are passing to the OR gate. That's why we are making use of plus symbol here. Next what we are doing?
We are multiplying the bits of binary sum. Z gate into Z gate. who's that girl? So here Z8, here Z4. These two bits are getting multiplied to AND gate and the result we are passing to OR gate.
Again what we are doing Z8 into Z2. So this is again Z8 and here where we are having Z2, here we are having Z2. This Z8 and Z2 we are passing again to the second AND gate and again we are passing this one to the OR gate.
That is why we got here 1 more plus. Now, finally once they are added the result will be transferred to the OR gate C and this OR gate value will be the result of this C value will be passed to the 4 bit binary adder. And along with this the along with the value of C ok the bits of binary sum that is Z 8, Z 4, Z 2 and Z 1 will be passed to the 4 bit binary adder.
And what is the final outcome we will get? The final outcome is S1, S2, S3, S4, S8. These are nothing but the bits of the binary coded decimal. And what about the carry bit?
Ultimately, we got the carry bit also here. It is C. Okay.
So, in this way, we can convert our binary sum into a corresponding BCD sum. Hopefully, the concept has been clear to you. With this, let me close my today's session of video.
See you soon, everybody. Take care. Allah Hafiz.