6.05 Chapter 6 Master PYQ Solutions & Counter Puzzles
Dedicated problem-solving supplement for Chapter 6 (Registers & Counters). Contains step-by-step worked solutions for the 12-bit register decoding puzzle and complete K-maps/schematics for 4-bit irregular sequence counters asked in past university exams.
1. 12-Bit Register Content Decoding Master Puzzle [PYQ: 2022]
PYQ Master Problem (2022): 12-Bit Register Decoding
Question: The state of a 12-bit register is 100010010111. What is its content if it represents [12 Marks]:
(i) Three decimal digits in BCD?
(ii) Three decimal digits in the Excess-3 code?
(iii) Three decimal digits in the 8, 4, -2, -1 code?
(iv) A binary number?
1.1 Step-by-Step Solution
Divide the 12-bit string into three 4-bit nibbles: 1000, 1001, 0111.
(i) As Three Decimal Digits in BCD (8421 Code)
Evaluate each 4-bit nibble directly into decimal:
• Nibble 1: 1000 = 8
• Nibble 2: 1001 = 9
• Nibble 3: 0111 = 7
Result: 897
(ii) As Three Decimal Digits in Excess-3 Code
Excess-3 is obtained by adding 3 (0011) to each decimal digit. To decode, subtract 3 from each nibble:
• Nibble 1: 1000 (8) - 3 = 5
• Nibble 2: 1001 (9) - 3 = 6
• Nibble 3: 0111 (7) - 3 = 4
Result: 564
(iii) As Three Decimal Digits in 8, 4, -2, -1 Code
Apply weights 8 x B3 + 4 x B2 - 2 x B1 - 1 x B0:
• Nibble 1: 1000 → 8(1) + 4(0) - 2(0) - 1(0) = 8
• Nibble 2: 1001 → 8(1) + 4(0) - 2(0) - 1(1) = 8 - 1 = 7
• Nibble 3: 0111 → 8(0) + 4(1) - 2(1) - 1(1) = 4 - 2 - 1 = 1
Result: 871
(iv) As a Pure Binary Number
Evaluate the 12-bit string 100010010111 as a single base-2 integer:
100010010111 = 2048 + 128 + 16 + 4 + 2 + 1 = 2199
2. Irregular Sequence Counter 1: 15 → 10 → 9 → 8 → 7 → 6 [PYQ: 2015, 2020]
PYQ Master Problem (2015, 2020): Sequence Counter Design
Question: Design a synchronous counter that will count 15 → 10 → 9 → 8 → 7 → 6 and repeat using JK flip-flops [13 Marks].
2.1 State & Excitation Table
Flip-flops D, C, B, A (where D is MSB). Unused states (0-5, 11-14) are Don’t Cares (X).
| Present State | Next State | JD, KD | JC, KC | JB, KB | JA, KA |
|---|---|---|---|---|---|
| 0110 (6) | 1111 (15) | 1 X | X 0 | X 0 | 1 X |
| 0111 (7) | 0110 (6) | 0 X | X 0 | X 0 | X 1 |
| 1000 (8) | 0111 (7) | X 1 | 1 X | 1 X | 1 X |
| 1001 (9) | 1000 (8) | X 0 | 0 X | 0 X | X 1 |
| 1010 (10) | 1001 (9) | X 0 | 0 X | X 0 | 1 X |
| 1111 (15) | 1010 (10) | X 0 | X 1 | X 0 | X 1 |
2.2 K-Map Drive Equations
• JD = A’, KD = C
• JC = A’, KC = A
• JB = D’, KB = 0
• JA = C’ + B’, KA = 1
3. Irregular Sequence Counter 2: 15 → 11 → 9 → 8 → 4 → 6 → 1 [PYQ: 2024]
PYQ Master Problem (2024): 7-State T Flip-Flop Counter
Question: Design a counter that will follow the sequence 15 → 11 → 9 → 8 → 4 → 6 → 1 and repeat using T flip-flops [12 Marks].
3.1 State & Excitation Table (T FF Excitation: T = Qn ⊕ Qn+1)
| Present State | Next State | TD | TC | TB | TA |
|---|---|---|---|---|---|
| 0001 (1) | 1111 (15) | 1 | 1 | 1 | 0 |
| 0100 (4) | 0110 (6) | 0 | 0 | 1 | 0 |
| 0110 (6) | 0001 (1) | 0 | 1 | 1 | 1 |
| 1000 (8) | 0100 (4) | 1 | 1 | 0 | 0 |
| 1001 (9) | 1000 (8) | 0 | 0 | 0 | 1 |
| 1011 (11) | 1001 (9) | 0 | 0 | 1 | 0 |
| 1111 (15) | 1011 (11) | 0 | 1 | 1 | 0 |
3.2 Drive Logic Equations
• TD = D’ C’ A + D C’ B’ A’
• TC = D’ C’ A + D B’ A’ + C B A’ + C B A
• TB = D’ C’ A + D C B A + D C’ B A + D’ C A’
• TA = D C’ B’ A + D’ C B A’