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4 bit binary Adder introduction:-Binary adders are implemented to add two binary numbers. So in order to add two 4 bit binary numbers we need to use 4 full-adders. The connection of full-adders to create binary adder circuit is discussed in block diagram below.
In this implementation, carry of each full-adder is connected to previous carry.  Detailed discussion on full-adder is covered on this link. Example

Lets discuss one example for 4 bit binary Adder.  In this example we will use some terms from Register Transfer Level (RTL) implementations.
Question:
Add two binary numbers 7 and 15 with previous carry = 0.
Solution:
Load the values in two registers R1 and R2.
So, R1 = 7 (decimal) = 0111 (in binary A3A2A1A0)
&   R2 = 15 (decimal) = 1111 (in binary B3B2B1B0)

So from the above implementation we have - Stage
Previous carry
Augends bits A
Sum
Next carry
0
C0=0
A0=1
B0=1
S0=0
C1=1
1
C1=1
A1=1
B1=1
S1=1
C2=1
2
C2=1
A2=1
B2=1
S2=1
C3=1
3
C3=1
A3=0
B3=1
S3=0
C4=1
Sum of two binary numbers 7 and 15 from above table
Is C4S3S2S1S0 = 10110 (In Binary) = 16 (decimal)
In the results we have appended C4 in front of the sum digits to accommodate overflow bit from the binary addition flow. Overflow is discussed in this section.
Clock Crossing Async FIFO Half Adder Full Adder Binary Adder Overflow Overflow Det Adder-Subtractor Multiplier Parity check RTL guidelines NAND to INVERTER VHDL RTL Arith Micro-ops Stack Org Parallel proc. Pipeline proc CMOS Intro
Resources
Clock Domain Crossing rtl & testbench.
Rate change (asynchronous) FIFO design and fifo depth calculation.
RTL coding guidelines. ICG cell, \$assertkill Digital design Interview questions.
FPGA Interview. FPGA flow.
Guide to Graduate studies in US
Pipeline vs. Parallel processing.
Lets start with the expressions for the FULL ADDER :-
Sum_out = [(in_x) XOR (in_y)] XOR [(carry_in)]
Carry_out = [(in_x) AND (in_y)] OR [ (in_x XOR in_y) AND carry_in ]