Second Semester_Digital Logic_Question Answer

 * Define combination circuit. Explain the operation of octal to binary encoder with circuit diagram and truth table

A combinational circuit is a type of digital logic circuit in which the output depends only on the present inputs and not on any past history (no memory).

• It consists of logic gates (AND, OR, NOT, etc.).

• The relationship between inputs and outputs can be expressed with Boolean functions or truth tables.

• Examples: adders, subtractors, encoders, decoders, multiplexers, demultiplexers.
  
  

  Octal to Binary Encoder

  Concept

  • An encoder converts information from one form (like decimal or octal) into a coded form (like binary).

  • An octal-to-binary encoder has 8 input lines (D0–D7) and 3 output lines (Y2, Y1, Y0).

  • At any given time, only one input is active (logic 1), and the output generates the corresponding binary code for that input.

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  Truth Table: Octal-to-Binary Encoder

  Input	Output (Binary)
  D0=1	Y2Y1Y0 = 000
  D1=1	001
  D2=1	010
  D3=1	011
  D4=1	100
  D5=1	101
  D6=1	110
  D7=1	111

  👉 Note: Only one input should be high at a time (otherwise it creates ambiguity unless a priority encoder is used).

  ---

  Boolean Expressions

  From the truth table:

  $$Y0 = D1 + D3 + D5 + D7$$ $$Y1 = D2 + D3 + D6 + D7$$ $$Y2 = D4 + D5 + D6 + D7$$

  (where + means OR operation).

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  Circuit Diagram

  • 8 inputs (D0–D7) connected to OR gates.

  • Each OR gate combines appropriate inputs to produce one of the output bits Y0, Y1, Y2.

  D0 ────────┐
             │
  D1 ────┐   │
         │   │
         └──► OR ─── Y0 (LSB)
  D3 ────┘
  D5 ────┐
  D7 ────┘
  
  D2 ────┐
  D3 ────┤
  D6 ────┤──► OR ─── Y1
  D7 ────┘
  
  D4 ────┐
  D5 ────┤
  D6 ────┤──► OR ─── Y2 (MSB)
  D7 ────┘

* Define register. Explain the 4-bit binary ripple counter with circuit diagram

A register is a group of flip-flops connected together and used to store binary data.

• Each flip-flop stores one bit (0 or 1).

• An n-bit register can store n bits of information.

• Registers are widely used in digital systems for data storage, transfer, and manipulation.

Types include:

• Shift registers

• Parallel registers

• Counters (special type of register)

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2. 4-bit Binary Ripple Counter

Concept

• A counter is a sequential circuit that goes through a predetermined sequence of states (binary numbers) when clock pulses are applied.

• A ripple counter is an asynchronous counter in which the flip-flops are triggered one after another, so the clock “ripples” through them.

• A 4-bit ripple counter counts from 0000 (0) to 1111 (15) in binary, then resets to 0000.

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Operation

• Consists of 4 T flip-flops (or JK flip-flops connected as T).

• T is tied to logic 1, so each flip-flop toggles on the active clock edge.

• The first flip-flop (FF0) receives the external clock.

• Each subsequent flip-flop receives the output of the previous FF as its clock.

• Output sequence:

  0000 → 0001 → 0010 → 0011 → … → 1111 → 0000
  

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Truth Table (Count Sequence)

Clock Pulse	Q3 Q2 Q1 Q0 (Output)	Decimal Equivalent
0	0000	0
1	0001	1
2	0010	2
3	0011	3
...	...	...
14	1110	14
15	1111	15
16	0000 (repeats)	0

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Circuit Diagram (Textual Representation)

        CLK ───►|> FF0 (Q0) ───► Q0 (LSB)
                   │
                   ▼
                |CLK
                 FF1 (Q1) ───► Q1
                   │
                   ▼
                |CLK
                 FF2 (Q2) ───► Q2
                   │
                   ▼
                |CLK
                 FF3 (Q3) ───► Q3 (MSB)

• All flip-flops are T flip-flops with T = 1.

• Q0 toggles with each clock pulse.

• Q1 toggles when Q0 goes from 1 → 0.

• Q2 toggles when Q1 goes from 1 → 0.

• Q3 toggles when Q2 goes from 1 → 0.

* Define Sequential Circuit. Draw a logic diagram, graphic symbol, characteristic table and characteristic equation of clocked RS flip-flop

A sequential circuit is a digital circuit whose output depends not only on the present inputs but also on the past history of inputs (i.e., it has memory).

• Built using logic gates + storage elements (flip-flops/latches).

• Two types:

  1. Synchronous (clock controlled)

  2. Asynchronous (no global clock)

Examples: Counters, Registers, Flip-flops, State machines.

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2. Clocked RS Flip-Flop

A flip-flop is the basic memory element used in sequential circuits.
The RS (Reset-Set) flip-flop stores 1 bit of data.

Inputs: R (Reset), S (Set), CLK (Clock)
Outputs: Q, Q̅

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(a) Logic Diagram

• Two NAND gates cross-coupled.

• Inputs S and R are fed through NAND gates with the Clock signal.

• The outputs form a feedback loop.

          ┌───┐
   S ──┬──►   │
       │  NAND┐
       │   ┌──┘
   CLK─┘   │
           │
           ▼
          ┌───┐
          │   │───► Q
          │NAND│
          │   │
          └───┘
           ▲
           │
           │   ┌───┐
   CLK─┐   │   │   │
       │   └──►NAND│
   R ──┴──────►   │
                  └───┘
                    │
                    └────► Q̅

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(b) Graphic Symbol

   _________
  |         |
  |   RS    |
  | Flip-   |
  |  Flop   |
  |_________|
   |   |   |
   S   R   CLK
          ----
           |
          Q, Q̅

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(c) Characteristic Table

This shows next state (Qⁿ⁺¹) in terms of present state (Qⁿ), inputs S, R, and Clock.

CLK	S	R	Qⁿ (Present)	Qⁿ⁺¹ (Next)	Operation
1	0	0	Qⁿ	Qⁿ	No change (hold)
1	0	1	X	0	Reset
1	1	0	X	1	Set
1	1	1	X	Invalid	Not allowed

👉 Note: Inputs are only effective when CLK = 1 (active high).

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(d) Characteristic Equation

From the table:

$$Q^{n+1} = S + \overline{R} \cdot Q^n$$

This means:

• If S = 1 → Qⁿ⁺¹ = 1 (Set)

• If R = 1 → Qⁿ⁺¹ = 0 (Reset)

• If S = R = 0 → Qⁿ⁺¹ = Qⁿ (Hold)

* What are universal gates Construct the basic gates using NAND Gate.

• A universal gate is a logic gate that can be used to construct any other basic gate (AND, OR, NOT, NOR, XOR, etc.).

• The two universal gates are:

  1. NAND gate

  2. NOR gate

👉 Because using only NAND or only NOR, we can implement all other logic gates.

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2. Constructing Basic Gates Using NAND Gate

(a) NOT Gate using NAND

• Connect both inputs of a NAND gate together.

$$Y = \overline{A \cdot A} = \overline{A}$$

Diagram (text form):

 A ─┐
    │
    └──►|NAND|──► Y = A̅

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(b) AND Gate using NAND

• Step 1: NAND gives output $\overline{A \cdot B}$.

• Step 2: Pass that output through a NOT (made from NAND).

$$Y = \overline{(\overline{A \cdot B})} = A \cdot B$$

Diagram (text form):

 A ───┐
      │
      ├──►|NAND|───►|NAND as NOT|──► Y = A·B
 B ───┘

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(c) OR Gate using NAND

• Use DeMorgan’s theorem:

$$A + B = \overline{(\overline{A} \cdot \overline{B})}$$

• Implementation:

  1. First make NOT gates (from NAND) for A and B.

  2. Then feed them into another NAND.

Diagram (text form):

 A ─►|NAND as NOT|──┐
                    │
                    ├──►|NAND|──► Y = A + B
 B ─►|NAND as NOT|──┘

* Write the features of Digital system.

Discrete Signal Representation

• Works with binary digits (0 and 1) instead of continuous values.

High Accuracy and Reliability

• Less affected by noise and signal distortion compared to analog systems.

• Error detection and correction methods (like parity, Hamming code) are possible.

Programmability and Flexibility

• Easy to program, modify, and upgrade using software or logic design changes.Data Storage Capability

• Digital systems can store large amounts of data using memory units (RAM, ROM, registers, etc.).

Reproducibility

• Signals and operations can be reproduced exactly without degradation.

Complex Operations

• Can perform arithmetic, logical, and decision-making operations very fast.

Integration with Computers

• Directly compatible with computers and microprocessors, making automation easier.

Scalability and Miniaturization

• Easy to integrate millions of components into a small chip (VLSI/ULSI).Low Power Consumption (in modern technologies)

• Especially with CMOS technology, digital circuits consume very little power.

Wide Applications

• Used in computers, mobile devices, communication, control systems, signal processing, robotics, etc.

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Second Semester_Digital Logic_Introduction to Digital System

 

1.1 Introduction to Analog and Digital Systems

• Analog Systems: Continuous signals, infinite possible values.

  • Examples: Temperature sensors, microphones.

• Digital Systems: Discrete signals, limited set of values (usually 0 and 1).

  • Examples: Computers, digital clocks.

• Key Difference: Accuracy, noise tolerance, and ease of storage/processing.

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1.2 Features of Digital Systems

• Reliability: Less prone to noise.

• Precision: Exact values represented in binary.

• Storage and Processing: Easier with binary data.

• Programmability: Digital systems can be reprogrammed for different tasks.

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1.3 Number Systems

• Decimal (Base 10): 0–9

• Binary (Base 2): 0–1

• Octal (Base 8): 0–7

• Hexadecimal (Base 16): 0–9, A–F

• Conversions:

  • Binary ↔ Decimal

  • Binary ↔ Octal

  • Binary ↔ Hexadecimal

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1.4 Binary Arithmetic and Complement Systems

• Binary Arithmetic: Addition, subtraction, multiplication, division in base 2.

• Complement Systems: Used for subtraction.

  • 1’s Complement: Invert all bits.

  • 2’s Complement: 1’s complement + 1

  • 9’s Complement: For decimal subtraction

  • 10’s Complement: 9’s complement + 1

• Advantage: Simplifies subtraction to addition.

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1.5 Codes

• BCD (Binary Coded Decimal): Each decimal digit represented in 4-bit binary.

• XS-3 (Excess-3): BCD + 3, used in decimal arithmetic.

• Gray Code: Only one bit changes between consecutive numbers.

• Hamming Code: Error detection and correction.

• Alphanumeric Codes: Represent characters digitally.

  • ASCII (7 or 8 bits), EBCDIC (8 bits), UNICODE (16 or 32 bits).

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1.6 Error Detecting and Error Correcting Codes

• Parity Bits: Single-bit detection (even/odd parity).

• Checksums & CRC: Detect burst errors in data transmission.

• Hamming Code: Detects and corrects single-bit errors.

• Importance: Ensures reliable data communication and storage.

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Second Semester_Digital Logic_Logic Gates and Boolean Algebra

 

2.1 Basic Definition of Boolean Algebra

• Boolean Algebra: A mathematical system for analyzing and simplifying logic circuits.

• Values are binary: 0 (False) and 1 (True).

• Operations follow specific rules and laws.

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2.2 Basic Theory of Boolean Algebra, Boolean Functions, Logical Operations

• Boolean Functions: Functions that take binary inputs and produce a binary output.

  • Example: F(A, B) = A·B + A'·B

• Logical Operations:

  1. AND (·): Output 1 if all inputs are 1.

  2. OR (+): Output 1 if at least one input is 1.

  3. NOT (’): Output is the complement of input.

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2.3 Logic Gates, IC Digital Logic Families. Basic Gates

• Logic Gates: Hardware implementation of Boolean operations.

  • AND, OR, NOT gates

• Digital Logic Families: Types of IC technologies:

  • TTL (Transistor-Transistor Logic)

  • CMOS (Complementary MOS)

  • ECL (Emitter-Coupled Logic)

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2.4 Universal Gates (NAND and NOR), Other Gates (XOR, XNOR)

• Universal Gates: Can implement any Boolean function.

  • NAND gate

  • NOR gate

• Other Gates:

  • XOR (Exclusive OR): Output 1 if inputs are different.

  • XNOR (Exclusive NOR): Output 1 if inputs are the same.

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2.5 Boolean Identities, De Morgan’s Laws

• Boolean Identities: Rules to simplify expressions. Examples:

  • A + 0 = A, A·1 = A, A + A = A, A·A = A

• De Morgan’s Laws: Convert AND ↔ OR with complements:

1. (A·B)’ = A’ + B’

2. (A + B)’ = A’·B’

🔹 Basic Logic Gates

1. NOT Gate (Inverter)

• Equation: $Y = \overline{A}$

• Truth Table:

  • $A=0 \Rightarrow Y=1$

  • $A=1 \Rightarrow Y=0$

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2. AND Gate

• Equation: $Y = A \cdot B$ (sometimes written as $AB$)

• Truth Table:

  • $0 \cdot 0 = 0$

  • $0 \cdot 1 = 0$

  • $1 \cdot 0 = 0$

  • $1 \cdot 1 = 1$

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3. OR Gate

• Equation: $Y = A + B$

• Truth Table:

  • $0+0 = 0$

  • $0+1 = 1$

  • $1+0 = 1$

  • $1+1 = 1$

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🔹 Universal Gates

4. NAND Gate (NOT-AND)

• Equation: $Y = \overline{A \cdot B}$

• Truth Table:

  • $0 \cdot 0 = 0 \Rightarrow 1$

  • $0 \cdot 1 = 0 \Rightarrow 1$

  • $1 \cdot 0 = 0 \Rightarrow 1$

  • $1 \cdot 1 = 1 \Rightarrow 0$

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5. NOR Gate (NOT-OR)

• Equation: $Y = \overline{A + B}$

• Truth Table:

  • $0+0 = 0 \Rightarrow 1$

  • $0+1 = 1 \Rightarrow 0$

  • $1+0 = 1 \Rightarrow 0$

  • $1+1 = 1 \Rightarrow 0$

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🔹 Exclusive Gates

6. XOR (Exclusive-OR)

• Equation: $Y = A \oplus B = A\overline{B} + \overline{A}B$

• Truth Table:

  • $0 \oplus 0 = 0$

  • $0 \oplus 1 = 1$

  • $1 \oplus 0 = 1$

  • $1 \oplus 1 = 0$

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7. XNOR (Exclusive-NOR / Equivalence)

• Equation: $Y = \overline{A \oplus B} = AB + \overline{A}\,\overline{B}$

• Truth Table:

  • $0 \text{ XNOR } 0 = 1$

  • $0 \text{ XNOR } 1 = 0$

  • $1 \text{ XNOR } 0 = 0$

  • $1 \text{ XNOR } 1 = 1$

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✅ Summary Table (Equations Only)

Gate	Symbol	Equation
NOT	$\overline{A}$	Inverts input
AND	$A \cdot B$	Outputs 1 if both are 1
OR	$A + B$	Outputs 1 if any is 1
NAND	$\overline{A \cdot B}$	Inverse of AND
NOR	$\overline{A + B}$	Inverse of OR
XOR	$A \oplus B = A\overline{B} + \overline{A}B$	Outputs 1 if inputs differ
XNOR	$\overline{A \oplus B} = AB + \overline{A}\,\overline{B}$	Outputs 1 if inputs are same

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