Logic Gates

Introduction to Logic Gates

Logic gates are the physical implementation of Boolean functions. They are the fundamental building blocks of digital circuits, from simple devices to complex computers.

A logic gate is an electronic device that:

• Accepts one or more binary inputs
• Produces a single binary output
• Implements a Boolean function

Basic Logic Gates

1. NOT Gate (Inverter)

Symbol:

    x ──┬─>○── y
        │

Function: Inverts the input

Boolean Expression: y = x̄ or y = NOT x

Truth Table:

Properties:

• 1 input, 1 output
• Simplest gate
• Also called inverter

Example:

• Input: 1 → Output: 0
• Input: 0 → Output: 1

2. AND Gate

Symbol:

    x ──┐
        ├──── y
    z ──┘

Function: Output is 1 only when ALL inputs are 1

Boolean Expression: y = x · z or y = x ∧ z

Truth Table:

Properties:

• 2 or more inputs, 1 output
• Output = 1 only if ALL inputs = 1
• Commutative: x·z = z·x

Applications:

• Masking operations
• Enabling circuits
• Control logic

3. OR Gate

Symbol:

    x ──┐
        )──── y
    z ──┘

Function: Output is 1 when AT LEAST ONE input is 1

Boolean Expression: y = x + z or y = x ∨ z

Truth Table:

Properties:

• 2 or more inputs, 1 output
• Output = 1 if ANY input = 1
• Commutative: x+z = z+x

Applications:

• Combining signals
• Decision logic
• Alert systems

4. NAND Gate (NOT-AND)

Symbol:

    x ──┐
        ├─>○── y
    z ──┘

Function: Opposite of AND (inverted AND)

Boolean Expression: y = (x · z)’ or y = x̄ + z̄

Truth Table:

Special Properties:

• Functionally complete - can build ANY logic circuit using only NAND gates!
• Universal gate
• Cheaper and faster than other gates in CMOS

Building Other Gates from NAND:

NOT: x NAND x = x̄

AND: (x NAND y) NAND (x NAND y) = xy

OR: (x NAND x) NAND (y NAND y) = x + y

5. NOR Gate (NOT-OR)

Symbol:

    x ──┐
        )─>○── y
    z ──┘

Function: Opposite of OR (inverted OR)

Boolean Expression: y = (x + z)’ or y = x̄ · z̄

Truth Table:

Special Properties:

• Also functionally complete!
• Universal gate
• Can build any circuit using only NOR

Building Other Gates from NOR:

NOT: x NOR x = x̄

OR: (x NOR y) NOR (x NOR y) = x + y

AND: (x NOR x) NOR (y NOR y) = xy

6. XOR Gate (Exclusive OR)

Symbol:

    x ──┐
        )═══ y
    z ──┘

Function: Output is 1 when inputs are DIFFERENT

Boolean Expression: y = x ⊕ z = x̄z + xz̄

Truth Table:

Properties:

• Output = 1 when inputs differ
• Output = 0 when inputs are same
• Commutative: x ⊕ z = z ⊕ x
• Associative: x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z

Applications:

• Parity checking
• Addition circuits (half adder)
• Error detection
• Comparison

Special Properties:

• x ⊕ 0 = x
• x ⊕ 1 = x̄
• x ⊕ x = 0
• x ⊕ x̄ = 1

7. XNOR Gate (Exclusive NOR)

Symbol:

    x ──┐
        )══>○── y
    z ──┘

Function: Output is 1 when inputs are SAME (equality detector)

Boolean Expression: y = (x ⊕ z)’ = xz + x̄z̄

Truth Table:

Properties:

• Output = 1 when inputs are equal
• Also called equivalence gate
• Complement of XOR

Applications:

• Equality checking
• Error detection
• Digital comparators

Gate Characteristics

Fan-in

Definition: Number of inputs a gate can accept

Examples:

• NOT gate: fan-in = 1
• 2-input AND: fan-in = 2
• 4-input OR: fan-in = 4

Fan-out

Definition: Number of gates that can be driven by a single output

Example: One AND gate output feeding 5 other gates = fan-out of 5

Propagation Delay

Definition: Time taken for output to change after input change

Typical values: 1-10 nanoseconds for modern gates

Power Consumption

Energy used by the gate during operation.

Combinational Circuits

Definition

A combinational circuit is a circuit where:

• Output depends ONLY on current inputs
• No memory elements
• No feedback loops

Examples: Adders, multiplexers, encoders, decoders

Example 1: Half Adder

Purpose: Add two 1-bit numbers

Inputs: A, B Outputs: Sum (S), Carry (C)

Truth Table:

Boolean Expressions:

• Sum = A ⊕ B
• Carry = A · B

Circuit:

• XOR gate for Sum
• AND gate for Carry

Example 2: Full Adder

Purpose: Add two 1-bit numbers plus carry-in

Inputs: A, B, Cin Outputs: Sum (S), Carry out (Cout)

Boolean Expressions:

• Sum = A ⊕ B ⊕ Cin
• Cout = AB + ACin + BCin = AB + (A⊕B)Cin

Circuit: Two half adders + OR gate

Example 3: 2-to-1 Multiplexer

Purpose: Select one of two inputs based on control signal

Inputs: I0, I1, S (select) Output: Y

Truth Table:

Boolean Expression: Y = S̄I0 + SI1

Circuit:

• 2 AND gates
• 1 OR gate
• 1 NOT gate

Example 4: 2-to-4 Decoder

Purpose: Convert 2-bit input to 4 output lines (one active)

Inputs: A, B Outputs: Y0, Y1, Y2, Y3

Truth Table:

Expressions:

• Y0 = Ā·B̄
• Y1 = Ā·B
• Y2 = A·B̄
• Y3 = A·B

Implementing Boolean Functions with Gates

From SOP Expression

Steps:

1. Each product term → AND gate
2. OR all AND outputs

Example: f = AB + ĀC

Circuit:

1. AND gate: A, B → AB
2. AND gate: Ā, C → ĀC
3. OR gate: AB, ĀC → f

Gates needed: 2 AND, 1 OR, 1 NOT

From POS Expression

Steps:

1. Each sum term → OR gate
2. AND all OR outputs

Example: f = (A+B)·(Ā+C)

Circuit:

1. OR gate: A, B → (A+B)
2. OR gate: Ā, C → (Ā+C)
3. AND gate: (A+B), (Ā+C) → f

Gates needed: 2 OR, 1 AND, 1 NOT

Two-Level Implementation

SOP: AND-OR (2 levels) POS: OR-AND (2 levels)

Advantages:

• Predictable delay
• Easier to design
• Standard approach

Multi-Level Implementation

Example: f = (AB + CD)(E + F)

Advantages:

• Fewer total gates
• Less chip area
• Lower power

Disadvantages:

• More delay (longer path)
• More complex design

Universal Gates: NAND and NOR

Why Universal?

Because you can build:

• NOT gates
• AND gates
• OR gates
• Any other gates!

Using ONLY NAND (or ONLY NOR)

Practical Advantages

1. Manufacturing: Easier to produce one type of gate
2. Cost: Bulk production reduces cost
3. Speed: NAND/NOR typically fastest in CMOS
4. Flexibility: Can build anything with one gate type

Example: Implementing f = AB + CD using NAND

Original: 2 AND gates + 1 OR gate

NAND version:

1. AB using NAND: (A NAND B) NAND (A NAND B)
2. CD using NAND: (C NAND D) NAND (C NAND D)
3. OR using NAND: (AB NAND AB) NAND (CD NAND CD)

Gate Symbols Summary

Practical Considerations

1. Gate Delays

Longer paths = more delay = slower circuit

Example:

• Single gate: 5 ns
• 3-level circuit: 15 ns
• 5-level circuit: 25 ns

2. Power Consumption

More gates = more power = more heat

3. Cost

Fewer gates = cheaper circuit = lower manufacturing cost

4. Reliability

Simpler circuits = fewer components = more reliable

Standard Logic Families

TTL (Transistor-Transistor Logic)

• Older technology
• Fast
• Higher power consumption

CMOS (Complementary Metal-Oxide-Semiconductor)

• Modern standard
• Very low power when static
• High integration density
• Most common today

ECL (Emitter-Coupled Logic)

• Fastest
• Very high power
• Expensive
• Used in high-speed applications

Key Points for Exams

1. 7 basic gates: NOT, AND, OR, NAND, NOR, XOR, XNOR
2. NAND and NOR are universal - can build any circuit
3. XOR output = 1 when inputs differ
4. XNOR output = 1 when inputs are same
5. AND: all inputs 1 → output 1
6. OR: any input 1 → output 1
7. Two-level implementation: SOP = AND-OR, POS = OR-AND
8. Know gate symbols for all 7 gates
9. Half adder: Sum = A⊕B, Carry = AB
10. Propagation delay increases with circuit depth

Practice Problems

1. Draw the logic circuit for f = AB + ĀC

2. Implement 3-input OR using only 2-input NOR gates

3. Design a half adder using XOR and AND gates

4. Convert f = (A+B)(C+D) to NAND-only circuit

5. What is the output of XOR gate when both inputs are 1?

6. How many 2-input NAND gates needed to implement NOT?

7. Design a circuit to detect when 3-bit input is 101

8. Implement f = ABC + ĀB̄C̄ using minimum gates

9. What gate gives output 1 only when inputs are equal?

10. Draw a 2-to-1 multiplexer using basic gates