Combinatorial Circuits III

1
Combinatorial Circuits III

• The Half Adder, the Full Adder,
• the DeMux, the Encoder

2
How do you add 2 bits?

• A BS C
• 0 00 0 S
• 0 11 0
• 1 01 0 C
• 1 10 1
• Why Half?

S
A
H.A
B
C
3
The Half Adder
4
The Full Adder

• Truth table
• Sum
• C_out

A
S
HA
HA
Cin A B S C_out
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
C
B
5
Full Adder
6
The n-bit Ripple Carry Adder
a an-1.a2a1a0 b bn-1.b2b1b0
an-1
bn-1
a2
b2
a0
b0
a1
b1
c0
c1
FA
FA
FA
FA
??
.
cout ?
sn-1
s2
s1
s0
7
RippleCarry Adders (contd.)

• Ripple-carry adders can be slow
• Delay proportional to number of bits added
• Carry lookahead adders
• Eliminate the delay of ripple-carry adders -
• Impl. carry in 2 parts Gi Ai.Bi Pi Ai xor
  Bi
• then Ci1 Gi Pi.Ci etc.
• Carry-ins are generated independently
• C0 A0 B0
• C1 A0 B0 A1 A0 B0 B1 A1 B1
• . . .
• Requires complex circuits
• Usually, a combination carry lookahead and
  ripple-carry techniques are used

8
Practical Devices Programmable Logic Arrays

• PLAs (also fPGA field P Gate A)
• Implement sum-of-product expressions
• No need to simplify the logical expressions
• Take N inputs and produce M outputs
• Each input represents a logical variable
• Each output represents a logical function output
• Internally uses
• An AND array
• Each AND gate receives 2N inputs
• N inputs and their complements
• An OR array

9
A 2 i/p 2 o/p PLA
10
Implementation of Boolean Functions using PLAs
11
The Demux

• The inverse of the MUX it has
• 2n outputs, 1 input and n selectors
  (addressing) terminals
• a2 a1 a0 d x7 x6 x5 x4 x3 x2 x1 x0
• -------------------------------------
• 0 0 0 c 0 0 0 0 0 0 0 c
• 0 0 1 c 0 0 0 0 0 0 c 0
• 0 1 0 c 0 0 0 0 0 c 0 0
• 0 1 1 c 0 0 0 0 c 0 0 0
• 1 0 0 c 0 0 0 c 0 0 0 0
• 1 0 1 c 0 0 c 0 0 0 0 0
• 1 1 0 c 0 c 0 0 0 0 0 0
• 1 1 1 c c 0 0 0 0 0 0 0
• Used to send data from a single source to one of
  a number of destinations. Is also a universal
  gate
• (assign 2).

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Encoder/ Priority encoder
Table for a priority encoder

• A 2n to n device that delivers the binary index
  of a bit pattern.
• Trouble with straight encoder.
• Hence
• The MSB has highest priority etc.
• Used when multiple devices use the same resource.
  Each device is granted a priority that is used to
  resolve any conflicts.
• (circuit in class)