Counters; Logic Families; Detailed K-Map Walkthroughs

Counters

17.1 Modulus

Modulus-N counter: Counts 0 to N-1 then resets.

Mod-10 (decade): 0–9 → needs 4 FFs (2⁴=16 ≥ 10)
Terminal count for mod-10: 1001 (9 in binary)

17.2 Truncated Modulus

Counters that don't use full 2ⁿ count — e.g., mod-9, mod-11, mod-15.

17.3 Synchronous vs Asynchronous

Synchronous: All FFs share same clock — no ripple delay
Asynchronous (ripple): Output of one FF clocks the next

Logic Families

18.1 Key Parameters

Parameter	Definition
Fan-in	Max inputs a gate can accept
Fan-out	Max gate inputs one output can drive
Noise margin	Voltage tolerance before logic level misread
Power dissipation	Heat generated (mW)
Propagation delay	tPLH (0→1), tPHL (1→0)

18.2 RTL (Resistor-Transistor Logic)

Resistor input, transistor switching
Obsolete

18.3 DTL (Diode-Transistor Logic)

Diode logic + transistor amplifier
Predecessor to TTL

18.4 TTL (Transistor-Transistor Logic)

Advantage	Disadvantage
Strong drive	High power
Less noise than ECL	Slower than ECL
Fast saturation	More power than CMOS

18.5 ECL (Emitter-Coupled Logic)

Advantage	Disadvantage
Fastest logic family	Highest power
High fan-out	Worst noise immunity
Stable with temperature	Complex design

18.6 CMOS (Complementary MOS)

PMOS + NMOS transistors
Highest fan-out
Best noise immunity
Lowest static power
Slowest propagation (historically; modern CMOS is dominant)

CMOS inverter: PMOS pull-up + NMOS pull-down — output inverts input.

Detailed K-Map Walkthroughs

21.1 Three-Variable Map Layout (Gray Code Order)

 C' C
 A'B' 00 01
 A'B 01 11
 AB' 11 10
 AB 10 00

Rows: A'B', A'B, AB', AB Columns: C', C

21.2 Example: F(A,B,C) = Σ(1,5,6,7)

Place 1s at minterms m1, m5, m6, m7:

Minterm	Binary ABC	Cell
1	001	A'B'C
5	101	AB'C
6	110	ABC'
7	111	ABC

Grouping: The four cells form a quad covering A=1 region.

Result: F = A (only A varies; B and C cancel)

21.3 Example: F = x'y'z + x'yz + xy'

Minterms: m1 (001), m3 (011), m4 (100)

After grouping adjacent 1s:

Group x'z (cells with z=1 in x'=0 rows)
Group xy' (AB' row, C don't care)

Minimal SOP: F = x'z + xy'

21.4 Four-Variable K-Map

16 cells in 4×4 grid. Variables typically A,B on rows and C,D on columns (Gray order).

Example: F(A,B,C,D) = Σ(0,2,4,6,8,9,10)

Plot 1s, form largest groups of 8, 4, 2 or 1.

Don't care example: F = Σ(0,4,5) + d(2,3,6)

Use X for 2,3,6 — treat as 1 when grouping to minimize literals.