„
ECE 274 - Digital Logic
Digital Design
Lecture 9
Datapath Components: Adders: 2-bit adder
„
Lecture 9 - Adders
„
Half-adders
„
Full-adders
„
Carry-ripple Adder
Functional Requirements:
„
Design a circuit that will add two 2-bit binary numbers
„
„
Input: A1A0, B1B0
Output: S1S0: sum of inputs
C: carry bit
1
2
Digital Design
Digital Design
Datapath Components: Adders: 2-bit Adder: Truth Table
Datapath Components: Adders: 16-bit Adder: Truth Table
a15a14a13a12a11a10a9a8a7a6a5a4a3a2a1a0b15b14b13b12b11b10b9b8b7b6b5b4b3b2b1b0
000000000000000000000000000000
….
s15s14s13s12s11s10s9s8s7s6s5s4s3s2s1s0 c
0000000000000000
….
16-bit Adder: 16-bits * 2-operands = 32 inputs
Over 4,000,000,000 rows
3
4
Digital Design
Digital Design
Datapath Components: Adders: Exponential Growth
Datapath Components: Adders: Carry-Ripple
Exponential Growth for Two-Level
Adder Implementation
„
Functional Requirements:
„
Design a circuit that will add two bits
„
„
5
Input: A, B
Output: S: sum of inputs
C: carry bit
6
„
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple
Datapath Components: Adders: Carry-Ripple
Functional Requirements:
„
Input: A, B
Output: S: sum of inputs
C: carry bit
Design a circuit that will add two bits
„
„
Input: A, B
Output: S: sum of inputs
C: carry bit
C2
C1
C0
A3
A2
A1
A0
B3
B2
B1
B0
S3
S2
S1
S0
7
8
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple: Half-Adder
Datapath Components: Adders: Carry-Ripple: ½ Adder
2) Convert to Equations
1) Capture the Function
co = ab
s = a’b + ab’ = a xor b
9
10
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple: ½ Adder
Datapath Components: Adders: Carry-Ripple: Full Adder
„
Functional Requirements:
„
Design a circuit that will add three bits
„
3) Create the Circuit
„
Input: A, B, Cin
Output: S: sum of inputs
C: carry bit
co = ab
s = a’b + ab’ = a xor b
11
12
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple: Full Adder
Datapath Components: Adders: Carry-Ripple: Full Adder
Input: A, B, Cin
Output: S: sum of inputs
C: carry bit
2) Convert to Equations
1) Capture the Function
co = bc + ac + ab
s = a xor b xor c
13
14
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple: Full Adder
Datapath Components: Adders: Carry-Ripple: Full Adder
4-bit Adder:
Inputs (8-bits): a3a2a1a0, b3b2b1b0
Outputs (5-bits): c0, s3s2s1s0
3) Create the Circuit
co = bc + ac + ab
s = a xor b xor c
.3ns .3ns .3ns
.8ns
.6ns
For a 4-bit adder composed of full adders, what
would be the soonest time you could expect a
result? A) .9 ns B) 2.3 ns C) 3.6 ns
15
16
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple: Full Adder
Datapath Components: Adders: Carry-Ripple: Full Adder
4-bit Adder:
Inputs (9-bits): a3a2a1a0, b3b2b1b0, cin
Outputs (5-bits): c0, s3s2s1s0
NOTE: Adder will exhibit temporarily incorrect (spurious) results
until the carry bit from the rightmost bit has had a chance to
propagate (ripple) all the way through to the leftmost bit.
17
18
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple
Datapath Components: Adders: Compensating Scale
19
„
20
Digital Design
Digital Design
Datapath Components: Adders: Compensating Scale
Datapath Components: Adders: Compensating Scale
Functional Requirements:
„
Design a circuit that will weigh an object, and add an
adjustment from a user
„
„
Adjustment
Knob
Input: 8-bit Weight Reported by Sensor
3-bit User Adjustment
Output: S7..0: sum of inputs
21
22
Digital Design
Digital Design
Datapath Components: Adders: Carry-Ripple
Datapath Components: Adders
„
8-bit Carry-Ripple Adder:
Inputs (16-bits): a7a6a5a4a3a2a1a0, b7b6b5b4b3b2b1b0
Outputs (5-bits): c0, s3s2s1s0
8-bit Calculator Functional Requirements:
„
Design a circuit that will add two eight-bit inputs controlled
by DIP switches and output the result using eight LEDs
„
„
23
Input: A (8-bits), B (8-bits)
Output: S (8-bits): sum of inputs
24
Digital Design
Digital Design
Datapath Components: Adders
Datapath Components: Adders
„
8-bit Calculator Functional Requirements:
„
Design a circuit that will add two eight-bit inputs controlled
by DIP switches and output the result using eight LEDs. The
output should only be updated with the user presses a
calculate button
„
„
A (8-bits), B (8-bits),
e: input from calculate button
Output: S (8-bits): sum of inputs
Input:
8-bit DIP switch based adding calculator. The addition 2+3=5 is shown.
25
Digital Design
Datapath Components: Adders
8-bit DIP switch-based adding calculator, using a register to block spurious LED outputs.
The LEDs only get updated after the button is pressed, which loads the output register.
27
26