## Decimal or BCD AdderThe BCD-Adder is used in the computers and the calculators that perform arithmetic operation directly in the decimal number system. The BCD-Adder accepts the binary-coded form of decimal numbers. The Decimal-Adder requires a minimum of nine inputs and five outputs. ## Note: The decimal number requires 4 bits to represent in the BCD code, and the circuit must have an input carry and an output carry.There is the following table used in designing of BCD-Adder. From the above table, it is clear that if the produced sum is between 1 to 9, the Binary and the BCD code is the same. But for 10 to 19 decimal numbers, both the codes are different. In the above table, the binary sum combinations from 10 to 19 give invalid BCD. There are the following points that help the circuit to identify the invalid BCD. - It is obvious from the table that a correction is needed when the
**'Binary Sum'**has an output carry K=1. - The other six combinations from 10 to 15 need correction in which the bit on the Z
_{8}position is 1. - In the Binary sum of 8 and 9, the bit on the Z
_{8}position is also 1. So, the second step fails, and we need to modify it. - To distinguish these two numbers, we specify that the bit on the Z
_{4}or Z_{2}position also needs to be 1 with the bit of Z_{8} - The condition for a correction and an output carry can be expressed by the Boolean function:
Once the circuit found the invalid BCD, the circuit adds the binary number of 6 into the invalid BCD code to make it valid. In the above diagram, - We take a 4-bit Binary-Adder, which takes addend and augend bits as an input with an input carry
**'Carry in'**. - The Binary-Adder produces five outputs, i.e., Z8, Z4, Z2, Z1, and an output carry K.
- With the help of the output carry K and Z8, Z4, Z2, Z1 outputs, the logical circuit is designed to identify the C
_{out} - The Z8, Z4, Z2, and Z1 outputs of the binary adder are passed into the 2
^{nd}4-bit binary adder as an Augend. - The addend bit of the 2
^{nd}4-bit binary adder is designed in such a way that the 1^{st}and the 4^{th}bit of the addend number are 0 and the 2^{nd}and the 3^{rd}bit are the same as C_{out}. When the value of C_{out}is 0, the addend number will be 0000, which produce the same result as the 1^{st}4-bit binary number. But when the value of the C_{out}is 1, the addend bit will be 0110, i.e., 6, which adds with the augent to get the valid BCD number.
- First, add both the numbers using a 4-bit binary adder and pass the input carry to 0.
- The binary adder produced the result 0001 and carried output 'K' 1.
- Then, find the C
_{out}value to identify that the produced BCD is invalid or valid using the expression C_{out}=K+Z_{8}.Z_{4}+Z_{8}.Z_{2}. K = 1 Z_{8}= 0 Z_{4}= 0 Z_{2}= 0 C_{out}= 1+0*0+0*0 C_{out}= 1+0+0 C_{out}= 1 - The value of C
_{out}is 1, which expresses that the produced BCD code is invalid. Then, add the output of the 1^{st}4-bit binary adder with 0110. = 0001+0110 = 0111 - The BCD is represented by the carry output as:
BCD=C_{out}Z_{8}Z_{4}Z_{2}Z_{1}=1 0 1 1 1
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