1. ECE 301 – Digital Electronics
Multiple-bit Adder Circuits
(Lecture #13)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.

2. Multiple-bit Adder Circuits
How do you design a combinational logic circuit to add two 4-bit binary numbers?
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ECE Digital Electronics

3. ECE 301 - Digital Electronics
A 4-bit Adder Circuit
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ECE Digital Electronics

4. ECE 301 - Digital Electronics
A 4-bit Adder Circuit
Design a two-level logic circuit
Construct a truth table
9 inputs (A3..A0, B3..B0, Cin)
5 outputs (S3..S0, Cout)
Derive minimized Boolean expressions
What is the problem with this design approach?
What happens when n gets large?
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ECE Digital Electronics

5. ECE 301 - Digital Electronics
A 4-bit Adder Circuit
Use a hierarchical design approach.
Design a logic circuit (i.e. module) to add two 1-bit numbers and a carry-in.
3 inputs (A, B, Cin)
2 outputs (S, Cout)
Connect 4 modules to form a 4-bit adder.
This design approach can easily be extended to n bits.
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ECE Digital Electronics

6. Multiple-bit Adder Circuits
Two designs for multiple-bit adders:
1. Ripple Carry Adder
2. Carry Lookahead Adder
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7. ECE 301 - Digital Electronics
Ripple Carry Adder
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8. ECE 301 - Digital Electronics
Ripple Carry Adder 1 +
Carry-in
Carry-out
Carry ripples from one column to the next
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ECE Digital Electronics

9. ECE 301 - Digital Electronics
Ripple Carry Adder
An n-bit RCA consists of n Full Adders.
The carry-out from bit i is connected to the carry-in of bit (i+1).
Simple design
Relatively slow
Each sum bit can be calculated only after the previous carry-out bit has been calculated.
Delay ~ (n) * (delay of FA)
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ECE Digital Electronics

10. ECE 301 - Digital Electronics
Ripple Carry Adder C0 C1 C2 … C3
Cn-1 Cn S0 A0 B0
Carry-out
Carry ripples from one stage to the next
Carry-in
LSB position
MSB position A1 B1 A2 B2
An-1
Bn-1 S1 S2
Sn-1
FAn-1
FA2
FA1
FA0
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ECE Digital Electronics

11. Multiple-bit Adder Circuits
The Ripple Carry Adder (RCA) may become prohibitively slow as the number of bits to add becomes large.
The Carry Lookahead Adder (CLA) provides a significant increase in speed at the cost of additional hardware (i.e. logic gates).
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ECE Digital Electronics

12. ECE 301 - Digital Electronics
Carry Lookahead Adder
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13. ECE 301 - Digital Electronics
Carry Lookahead Adder 1 +
Carry Generate
Carry End
Carry Propagate A B
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14. ECE 301 - Digital Electronics
Carry Lookahead Adder
A CLA uses the carry generate and carry propagate concepts to produce the carry bits.
A carry is generated iff both A and B are 1.
Generate: G(A,B) = A.B
A carry is propagated if either A or B is 1.
If Cin = 1 and (A or B) = 1 then Cout = 1
Propagate: P(A,B) = A + B
Alternate Propagate: P*(A,B) = A xor B
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15. ECE 301 - Digital Electronics
The Full Adder Circuit
A xor B = P*(A,B)
A.B = G(A,B)
Source: Wikipedia – Adder (Electronics) (
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16. ECE 301 - Digital Electronics
Carry Lookahead Adder
Source: Wikipedia – Adder (Electronics) (
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ECE Digital Electronics

17. ECE 301 - Digital Electronics
Carry Lookahead Adder
For each bit (or stage) of the multiple-bit adder, the carry-out can be defined in terms of the generate and propagate functions, and the carry-in:
Ci+1 = Gi + (Pi . Ci)
Ai+Bi
Ai.Bi
carry-in
carry-out
Pi* can also be used.
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ECE Digital Electronics

18. ECE 301 - Digital Electronics
Carry Lookahead Adder
For bit 0 (LSB):
C1 = G0 + (P0 . C0)
C1 = (A0 . B0) + ((A0 + B0) . C0)
C1 = (A0 . B0) + ((A0 xor B0) . C0)
C1 is a function of primary inputs
Three-level circuit, therefore 3-gate delay
Not a function of previous carries (except C0), therefore no ripple carry.
using Pi*
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ECE Digital Electronics

19. ECE 301 - Digital Electronics
Carry Lookahead Adder
For bit 1:
C2 = G1 + (P1 . C1)
C2 = (A1 . B1) + ((A1 + B1) . C1)
C2 = (A1 . B1) + ((A1 + B1) . ((A0 . B0) + ((A0 + B0) . C0))
C2 is a function of primary inputs
Three-level circuit, therefore 3-gate delay
Not a function of previous carries (except C0), therefore no ripple carry.
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ECE Digital Electronics

20. ECE 301 - Digital Electronics
Carry Lookahead Adder
For bit 2:
C3 = G2 + (P2 . C2)
C3 = G2 + (P2 . (G1 + (P1 . C1))
C3 = G2 + (P2 . (G1 + (P1 . (G0 + (P0 . C0)))
C3 is a function of primary inputs
Three-level circuit, therefore 3-gate delay
Not a function of previous carries (except C0), therefore no ripple carry.
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ECE Digital Electronics

21. ECE 301 - Digital Electronics
Carry Lookahead Adder
For bit i:
Ci+1 = F(G0..Gi, P0..Pi, C0)
For i > 4, the silicon area required for the carry circuits becomes prohibitively large.
Tradeoff: speed vs. area.
How, then, do you build a bigger adder?
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22. ECE 301 - Digital Electronics
A 16-bit Adder Circuit C0 C4 C8
C12
C16
S3-0
A3-0
B3-0
A7-4
B7-4
S7-4
A11-8
B11-8
S11-8
A15-12
B15-12
S15-12
Ripple carry (between CLAs)
CLA3
CLA2
CLA1
CLA0
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23. ECE 301 - Digital Electronics
A 4-bit CLA
(Standard Component)
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24. Multiple-bit Adder/Subtractor Circuit
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25. Multiple-bit Adder/Subtractor
Build separate binary adder and subtractor
Not common.
Use 2's Complement representation
Addition uses binary adder
Subtraction uses binary adder with 2's Complement representation for subtrahend
Issues
Cannot represent a positive number with the same magnitude as the most negative n-bit number
Must detect overflow
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ECE Digital Electronics

26. ECE 301 - Digital Electronics
A 4-bit Subtractor
A – B = A + (-B)
represent with 2's complement
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27. Multiple-bit Adder/Subtractor1 n – x c
-bit adder y
Add 
Sub
control
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28. ECE 301 - Digital Electronics
Detecting Overflow
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29. Detecting Overflow for Addition
Overflow occurs if the result is out of range.
Overflow cannot occur when adding a positive number and a negative number.
Overflow occurs when adding two numbers with the same sign.
Two positive numbers → negative number
Two negative numbers → positive number
Can you write a Boolean expression to detect overflow?
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ECE Digital Electronics

30. Detecting Overflow for Subtraction
Overflow occurs if the result is out of range.
Overflow cannot occur when subtracting two numbers with the same sign.
Overflow occurs when subtracting a positive number from a negative number or a negative number from a positive number.
positive # - negative # → negative #
negative # - positive # → positive #
Can you write a Boolean expression to detect overflow?
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ECE Digital Electronics

31. ECE 301 - Digital Electronics
Questions?
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ECE Digital Electronics