Lecture 9

1ECE 274 - Digital LogicLecture 9

Lecture 9 - Adders Half-adders Full-adders Carry-ripple Adder

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Digital DesignDatapath Components: Adders: 2-bit 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

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Digital DesignDatapath Components: Adders: 2-bit Adder: Truth Table

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Digital DesignDatapath Components: Adders: 16-bit Adder: Truth Table

..

0000000000000000000000000000000000000000000000

s15s14s13s12s11s10s9s8s7s6s5s4s3s2s1s0 c a15a14a13a12a11a10a9a8a7a6a5a4a3a2a1a0b15b14b13b12b11b10b9b8b7b6b5b4b3b2b1b0

16-bit Adder: 16-bits * 2-operands = 32 inputs

Over 4,000,000,000 rows

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Digital DesignDatapath Components: Adders: Exponential Growth

Exponential Growth for Two-Level Adder Implementation

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Digital DesignDatapath Components: Adders: Carry-Ripple

Functional Requirements: Design a circuit that will add two bits

Input: A, B Output: S: sum of inputs

C: carry bit

7Digital DesignDatapath Components: Adders: Carry-Ripple

Functional Requirements: Design a circuit that will add two bits

Input: A, B Output: S: sum of inputs

C: carry bit

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Input: A, BOutput: S: sum of inputs

C: carry bit

C2

A3

B3

S3

C1

A2

B2

S2

C0

A1

B1

S1

A0

B0

S0

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Digital DesignDatapath Components: Adders: Carry-Ripple: Half-Adder

1) Capture the Function

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Digital DesignDatapath Components: Adders: Carry-Ripple: Adder

2) Convert to Equations

co = abs = ab + ab = a xor b

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Digital DesignDatapath Components: Adders: Carry-Ripple: Adder

3) Create the Circuit

co = abs = ab + ab = a xor b

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Digital DesignDatapath Components: Adders: Carry-Ripple: Full Adder

Functional Requirements: Design a circuit that will add three bits

Input: A, B, Cin Output: S: sum of inputs

C: carry bit

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1) Capture the FunctionInput: A, B, CinOutput: S: sum of inputs

C: carry bit

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2) Convert to Equations

co = bc + ac + abs = a xor b xor c

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co = bc + ac + abs = a xor b xor c

3) Create the Circuit

.3ns .3ns .3ns

.6ns

.8ns

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

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4-bit Adder:Inputs (8-bits): a3a2a1a0, b3b2b1b0Outputs (5-bits): c0, s3s2s1s0

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4-bit Adder:Inputs (9-bits): a3a2a1a0, b3b2b1b0, cinOutputs (5-bits): c0, s3s2s1s0

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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.

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Digital DesignDatapath Components: Adders: Compensating Scale

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Functional Requirements: Design a circuit that will weigh an object, and add an

adjustment from a user Input: 8-bit Weight Reported by Sensor

3-bit User Adjustment Output: S7..0: sum of inputs

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AdjustmentKnob

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8-bit Carry-Ripple Adder:Inputs (16-bits): a7a6a5a4a3a2a1a0, b7b6b5b4b3b2b1b0Outputs (5-bits): c0, s3s2s1s0

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Digital DesignDatapath 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 Input: A (8-bits), B (8-bits) Output: S (8-bits): sum of inputs

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8-bit DIP switch based adding calculator. The addition 2+3=5 is shown.

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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 Input: A (8-bits), B (8-bits),

e: input from calculate button Output: S (8-bits): sum of inputs

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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.

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Lecture 9