CS355 Sylabus

The Full Adder

The "staged adder" circuit is called a "Full Adder" circuit
The full adder adds 2 bits together, along with a carry in from the previous bit-addition.
The full adder generate a sum bit and a carry out bit for the next addition stage.
Thus, the full adder has 3 inputs:
- a, one of the bits to be added
- b, the other bit to be added
- c_in, the carry in bit from the previous bit addition
and it has 2 outputs:
- s, the sum bit
- c_out, the carry out bit for the next staged bit addition
Schematically:

The figure on the right shows the operation of the full adder circuit when it adds the bits 1 and 0, while the previous stage generates a carry (c_in = 1).
We can design the full adder circuit using the technique learn a couple of weeks ago (if you have forgotten it, then: click here)
Here is the boolean function in table form for the adder circuit:

c_in a b c_out 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

You can see that:
- c_out = majority(a, b, c_in)
- s = a XOR b XOR c

c_in	a	b		c_out	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

The adder circuit is a very common and important circuit and electrical engineers have determine the optimal digital circuit that implements the full adder.

The optimal full adder circuit is as follows:

The EDiSim macro definition for the Full_Adder is as follows:

Define Full_Adder CarryIn a b | CarryOut Sum; Xor aa a b x; Xor aa x CarryIn Sum; And aa a b y; And aa CarryIn x z; Or aa y z CarryOut; Endef;
/home/cs355001/bin/cs355sim /home/cs355001/demo/circuits/1-full-adder.m

This diagram will help you see the connection (signal names) used in EDiSim macro:

Note:

I used aa as coordinate for every circuit in the Full_Adder macro
That's OK, because I will only see a rectangle box when the circuit is run...

To make a multi-bit adder, we concatenate or cascade a number of full adder circuits.
For example, the following circuit will implement a 4-bit adder circuit that can be used to add two 4-bit numbers:

The EDiSim macro for this circuit is:

Define FourBit_Adder a3 a2 a1 a0 b3 b2 b1 b0 | CarryOut s3 s2 s1 s0; Full_Adder ch ZERO a0 b0 | c1 s0; Full_Adder cf c1 a1 b1 | c2 s1; Full_Adder cd c2 a2 b2 | c3 s2; Full_Adder cb c3 a3 b3 | CarryOut s3; Endef;
/home/cs355001/bin/cs355sim /home/cs355001/demo/circuits/4-bit-adder.m