## Chapter 9 - Combinational Logic Functions

The half-adder is extremely useful until you want to add more that one binary digit quantities. The slow way to develop a two binary digit adders would be to make a truth table and reduce it. Then when you decide to make a three binary digit adder, do it again. Then when you decide to make a four digit adder, do it again. Then when ... The circuits would be fast, but development time would be slow. Looking at a two binary digit sum shows what we need to extend addition to multiple binary digits.
           11
11
11
---
110

Look at how many inputs the middle column uses. Our adder needs three inputs; a, b, and the carry from the previous sum, and we can use our two-input adder to build a three input adder. Σ is the easy part. Normal arithmetic tells us that if Σ = a + b + Cin and Σ1 = a + b, then Σ = Σ1 + Cin. What do we do with C1 and C2? Let’s look at three input sums and quickly calculate:
Cin + a + b = ?
0 + 0 + 0 = 0       0 + 0 + 1 =  1       0 + 1 + 0 =  1      0 + 1 + 1 = 10
1 + 0 + 0 = 1       1 + 0 + 1 = 10       1 + 1 + 0 = 10      1 + 1 + 1 = 11

If you have any concern about the low order bit, please confirm that the circuit and ladder calculate it correctly. In order to calculate the high order bit, notice that it is 1 in both cases when a + b produces a C1. Also, the high order bit is 1 when a + b produces a Σ1 and Cin is a 1. So We will have a carry when C1 OR (Σ1 AND Cin). Our complete three input adder is: For some designs, being able to eliminate one or more types of gates can be important, and you can replace the final OR gate with an XOR gate without changing the results. We can now connect two adders to add 2 bit quantities. A0 is the low order bit of A, A1 is the high order bit of A, B0 is the low order bit of B, B1 is the high order bit of B, Σ0is the low order bit of the sum, Σ1 is the high order bit of the sum, and Cout is the Carry. A two binary digit adder would never be made this way. Instead the lowest order bits would also go through a full adder. There are several reasons for this, one being that we can then allow a circuit to determine whether the lowest order carry should be included in the sum. This allows for the chaining of even larger sums. Consider two different ways to look at a four bit sum.
           111            1<-+  11<+-
0110             |  01  |  10
1011             |  10  |  11
-----          -  | ---- | ---
10001          1  +-100  +-101