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.

Multiple-bit Adder Circuits How do you design a combinational logic circuit to add two 4-bit binary numbers? Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics A 4-bit Adder Circuit Spring 2011 ECE 301 - Digital Electronics

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? Spring 2011 ECE 301 - Digital Electronics

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. Spring 2011 ECE 301 - Digital Electronics

Multiple-bit Adder Circuits Two designs for multiple-bit adders: 1. Ripple Carry Adder 2. Carry Lookahead Adder Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Ripple Carry Adder Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Ripple Carry Adder 1 + Carry-in Carry-out Carry ripples from one column to the next Spring 2011 ECE 301 - Digital Electronics

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) Spring 2011 ECE 301 - Digital Electronics

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 Spring 2011 ECE 301 - Digital Electronics

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). Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Carry Lookahead Adder Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Carry Lookahead Adder 1 + Carry Generate Carry End Carry Propagate A B Spring 2011 ECE 301 - Digital Electronics

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 Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics The Full Adder Circuit A xor B = P*(A,B) A.B = G(A,B) Source: Wikipedia – Adder (Electronics) (http://en.wikipedia.org/wiki/Adder_electronics) Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Carry Lookahead Adder Source: Wikipedia – Adder (Electronics) (http://en.wikipedia.org/wiki/Adder_electronics) Spring 2011 ECE 301 - Digital Electronics

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. Spring 2011 ECE 301 - Digital Electronics

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* Spring 2011 ECE 301 - Digital Electronics

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. Spring 2011 ECE 301 - Digital Electronics

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. Spring 2011 ECE 301 - Digital Electronics

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? Spring 2011 ECE 301 - Digital Electronics

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 Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics A 4-bit CLA (Standard Component) Spring 2011 ECE 301 - Digital Electronics

Multiple-bit Adder/Subtractor Circuit Spring 2011 ECE 301 - Digital Electronics

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 Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics A 4-bit Subtractor A – B = A + (-B) represent with 2's complement Spring 2011 ECE 301 - Digital Electronics

Multiple-bit Adder/Subtractor 1 n – x c -bit adder y Add  Sub control Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Detecting Overflow Spring 2011 ECE 301 - Digital Electronics

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? Spring 2011 ECE 301 - Digital Electronics

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? Spring 2011 ECE 301 - Digital Electronics

ECE 301 - Digital Electronics Questions? Spring 2011 ECE 301 - Digital Electronics