Digital Electronics — Complete Notes IT Sem 1

Number Systems

| System | Base | Digits | Example | |---|---|---|---| | Binary | 2 | 0,1 | 1010₂ | | Octal | 8 | 0-7 | 12₈ | | Decimal | 10 | 0-9 | 10₁₀ | | Hexadecimal | 16 | 0-9, A-F | A₁₆ |

Conversions

Decimal → Binary: divide by 2, collect remainders (LSB first)
  10 ÷ 2 = 5 r0
   5 ÷ 2 = 2 r1
   2 ÷ 2 = 1 r0
   1 ÷ 2 = 0 r1  → 1010₂

Binary → Decimal: multiply by powers of 2
  1010₂ = 1×8 + 0×4 + 1×2 + 0×1 = 10₁₀

Hex → Binary: each hex digit = 4 bits
  A3F₁₆ = 1010 0011 1111₂

Complement Arithmetic

1's complement: flip all bits   1010 → 0101
2's complement: 1's comp + 1   1010 → 0110

-5 in 8-bit 2's complement:
  5 = 00000101
  1's comp = 11111010
  2's comp = 11111011 = -5

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Logic Gates

Universal Gates:

• NAND and NOR are universal — any logic function implementable

Implementing NOT using NAND:

A ──┬── NAND ── Y (= A')
    └──/

Implementing AND using NAND:

A ── NAND₁ ── NAND₂ ── Y (= A·B)
B ──/

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Boolean Algebra

Basic Laws:

| Law | AND form | OR form | |---|---|---| | Identity | A·1 = A | A+0 = A | | Null | A·0 = 0 | A+1 = 1 | | Idempotent | A·A = A | A+A = A | | Complement | A·A' = 0 | A+A' = 1 | | Double complement | A'' = A | — |

De Morgan's Theorems:

(A·B)' = A' + B'   ← NAND = NOT-OR
(A+B)' = A'·B'     ← NOR = NOT-AND

Simplification Example:

F = AB + AB' + A'B
  = A(B+B') + A'B
  = A·1 + A'B
  = A + A'B
  = A + B        (absorption: A + A'B = A + B)

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Karnaugh Map (K-Map)

Graphical method to simplify Boolean expressions.

2-variable K-Map:

     B'   B
A'  [ 0 | 1 ]
A   [ 2 | 3 ]

Rules for grouping:

• Group 1s in powers of 2 (1, 2, 4, 8...)
• Groups can overlap and wrap around
• Largest groups = maximum simplification

Example:

F(A,B,C) = Σm(0,1,2,3,4,5)

K-Map:
      C'D'  C'D   CD   CD'
AB'  [  1  |  1  |  1  |  1 ]
AB   [  1  |  1  |  0  |  0 ]

Groups: top row (4) = B', left two cols (4) = C'
F = B' + C'

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Combinational Circuits

Half Adder

Sum  = A ⊕ B
Carry = A·B

Full Adder

Sum   = A ⊕ B ⊕ Cin
Carry = A·B + Cin(A ⊕ B)

Multiplexer (MUX)

• 2ⁿ data inputs, n select lines, 1 output
• Acts as data selector
• Can implement any Boolean function

Demultiplexer (DEMUX)

• 1 input, 2ⁿ outputs, n select lines
• Opposite of MUX

Encoder / Decoder

• Encoder — 2ⁿ inputs → n-bit output (e.g., 8→3)
• Decoder — n-bit input → 2ⁿ outputs (e.g., 3→8)

Comparator

• Compares two n-bit numbers: A>B, A=B, A<B outputs

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Sequential Circuits — Flip-Flops

SR Flip-Flop

S=0, R=0 → No change
S=1, R=0 → Q=1 (Set)
S=0, R=1 → Q=0 (Reset)
S=1, R=1 → Invalid!

D Flip-Flop (Data/Delay)

On clock edge: Q(next) = D
Use: Registers, memory, shift registers

JK Flip-Flop (no invalid state)

J=0, K=0 → No change
J=1, K=0 → Q=1 (Set)
J=0, K=1 → Q=0 (Reset)
J=1, K=1 → Q' (Toggle!)

T Flip-Flop (Toggle)

T=0 → No change
T=1 → Toggle (Q = Q')
Use: Counters

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Counters and Registers

Asynchronous (Ripple) Counter:

• Each FF triggered by output of previous
• Simple but slow (propagation delay adds up)
• Mod-N counter: N flip-flops count 0 to 2ᴺ-1

Synchronous Counter:

• All FFs triggered by same clock
• Faster, no ripple delay
• Used in digital systems

Shift Register:

• SISO — Serial In Serial Out
• SIPO — Serial In Parallel Out
• PISO — Parallel In Serial Out
• PIPO — Parallel In Parallel Out

Applications: Data conversion, delay lines, digital filters

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Memory Types

| Type | Volatile? | Writable? | Use | |---|---|---|---| | SRAM | Yes | Yes | Cache memory | | DRAM | Yes | Yes | Main RAM | | ROM | No | No | BIOS | | PROM | No | Once | Programmable ROM | | EPROM | No | UV erase | Erasable PROM | | EEPROM | No | Electrically | Flash drives | | Flash | No | Block erase | SSDs, USB drives |