Gates and Circuits Monday/Wednesday Week 7. Electronic Circuits  Two types of circuit diagrams See pp. 238 in The Analytical Engine by Decker and Hirshfield.

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Presentation transcript:

Gates and Circuits Monday/Wednesday Week 7

Electronic Circuits  Two types of circuit diagrams See pp. 238 in The Analytical Engine by Decker and Hirshfield for the "mechanical switch" type of diagram. Second style (which we will call "symbolic diagrams") uses pictures for the boolean logic operators. These pictures are called gates.

Three Main Gates  AND  OR  NOT

Gate Diagrams  Example 1: (MR) + S

Gate Diagrams  Example: What does it represent?

Gate Diagrams  Example: ((MR) + S) (RS)’

Truth Table to Gates  First, build the Boolean algebra expression that gives Z Z = AB + A’B’ Z = (A AND B) OR (NOT A AND NOT B) ABZ TTT TFF FTF FFT

Truth Table to Gates  Z = AB + A’B’  Next, build the circuit that goes with the Boolean algebra expression Z ABZ TTT TFF FTF FFT

Z = AB + A’B’

Binary Arithmetic  We can add binary numbers just like decimal numbers only using base two arithmetic.  For example:

Binary Addition  Notice in addition: FalseTrue False True Sum Carry ABSum (1) T (0) F (1) T(0) F(1) T (0) F(1) T (0) F

Sum and Carry ABCarry ABSum

Sum Circuit ABSum Sum = AB’ + A’B

Carry Circuit ABCarry Carry = AB

Half Adder - Sum and Carry

Half Adder  Handles the case where we add two binary digits with no inward carry.

Full Adder  Takes a carry in and produces the result and carry out.  So, we have 3 inputs and two outputs.  Combine two half-adders together with an or gate to get a full adder for each binary digit.  How many half adders would we need to add two 8-digit binary numbers? How many gates?

Full Adder

Binary Subtraction  We do binary subtraction like decimal subtraction only the borrowing is done in 2’s instead of 10’s

Subtraction ABSub ABBorrow

Binary Multiplication  Again, just like decimal except we add and multiply in binary. * x 7x

Exercises  1 - From the book, p 266, Construct circuits with the following properties (using AND, OR or NOT gates):

Exercises A)B) ABZ ABCZ

NAND Gates and NOT  This gate represents (A NAND NOT B).

NAND Truth Table ABA NAND B TTF TFT FTT FFT

Exercises  2 - Fill in a truth table and give a Boolean expression for the following circuits.

Exercises  3 - How would you create a one binary digit multiplier? A two-digit by one-digit multiplier? A two-digit by two-digit multiplier? *