⭐ 3.1 ADDERS
1️⃣ Half Adder
Adds two single-bit numbers (A and B).
Outputs:
2️⃣ Full Adder
Adds three bits (A, B, Cin).
Outputs:
-
Sum = A ⊕ B ⊕ Cin
-
Cout = AB + BCin + ACin
3️⃣ 4-bit & 8-bit Parallel Adder
Uses four full adders connected so that carry passes from one stage to next (ripple carry).
⭐ 3.2 SUBTRACTORS
Half Subtractor
Inputs: A, B
Outputs:
-
Difference = A ⊕ B
-
Borrow = A' • B
Full Subtractor
Inputs: A, B, Bin
Outputs:
-
Difference = A ⊕ B ⊕ Bin
-
Borrow = A'B + B Bin + A' Bin
⭐ 3.3 ADDER–SUBTRACTOR COMBINED CIRCUIT
A single circuit performs both add & subtract using XOR-based control.
Control input M:
-
M = 0 → Add
-
M = 1 → Subtract
⭐ 3.4 MAGNITUDE COMPARATOR
Compares two numbers A and B and generates:
-
A > B
-
A = B
-
A < B
Used in CPU for decision-making (conditional branches).
⭐ 3.5 MULTIPLEXER (MUX) & DEMULTIPLEXER (DEMUX)
Multiplexer (MUX)
Selects one input from many and sends to output.
Example: 4-to-1 MUX
Select lines: S1, S0
Output:
Y = S1'S0'I0 + S1'S0I1 + S1S0'I2 + S1S0I3
Demultiplexer (DEMUX)
Opposite of MUX — one input → many outputs.
Example: 1-to-4 DEMUX.
⭐ 3.6 DECODER & ENCODER
Decoder
Converts n inputs → 2ⁿ outputs.
Example:
2-to-4 decoder
3-to-8 decoder
Used in memory address decoding.
Encoder
Converts 2ⁿ inputs → n outputs.
Example:
8-to-3 encoder
Priority encoder (ignores lower priority inputs)
⭐ 3.7 LATCHES & FLIP-FLOPS
1️⃣ LATCH (Level Triggered)
SR Latch
| S | R | Q(next) |
|---|---|---|
| 0 | 0 | Q (no change) |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | Invalid |
2️⃣ Flip-Flop (Edge Triggered)
D Flip-Flop
Stores 1-bit of data.
Q(next) = D
JK Flip-Flop
| J | K | Qnext |
|---|---|---|
| 0 | 0 | Q |
| 0 | 1 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | Toggle |
T Flip-Flop
T = 1 → Toggle
T = 0 → Hold
⭐ 3.8 COUNTERS
1️⃣ Ripple Counter (Asynchronous)
Every flip-flop triggers the next one.
Example: 4-bit ripple counter (counts 0–15).
2️⃣ Ring Counter
A single 1 rotates through flip-flops.
3️⃣ Johnson Counter
Complement of last flip-flop output is fed to first.
⭐ 3.9 SEQUENTIAL CIRCUITS
A sequential circuit =
combinational circuit + memory (flip-flops).
Used in:
-
Registers
-
Counters
-
Control units of CPUs
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