Comparators — Hinglish notes

3.3.9 · Hardware › Combinational Circuits

1. Hum actually kya bana rahe hain?

Sabse simple kya hai? Ek 1-bit comparator. Ek single bit $A$ ko ek single bit $B$ se compare karo.

Hum shuru kaise karein? Truth table likho — 2 inputs → 4 rows.

$A$	$B$	$A > B$	$A = B$	$A < B$
0	0	0	1	0
0	1	0	0	1
1	0	1	0	0
1	1	0	1	0

2. 1-bit logic scratch se derive karna

YEH equations kyun? Truth table ko column by column padho aur har "1" ko ek product term mein translate karo.

$A > B$ : sirf ek row hai jahan yeh 1 hai — $A = 1, B = 0$ . Woh product term hai $A \overset{ˉ}{B}$ . $G = A \overset{ˉ}{B}$

$A < B$ : sirf ek 1 hai $A = 0, B = 1$ par, yaani $\overset{ˉ}{A} B$ . $L = \overset{ˉ}{A} B$

$A = B$ : 1 jab dono 0 hon ya dono 1 hon. Yeh exactly XNOR hai: $E = \overset{ˉ}{A} \overset{ˉ}{B} + A B = \overline{A \oplus B}$

3. Multi-bit comparator banana (asli trick)

Giant truth table kyun nahi? $n$ -bit numbers ke liye $2^{2 n}$ rows chahiye honge — 4-bit ke liye woh 256 rows hain. Haath se design karna impossible hai. Humein ek structured, first-principles method chahiye: insaan ki tarah numbers compare karo.

Bits ke liye formalise kaise karein. Maano $A = A_{n - 1} \dots A_{0}$ , $B = B_{n - 1} \dots B_{0}$ .

Har bit ki equality define karo: $x_{i} = \overline{A_{i} \oplus B_{i}}$ (1 jab bit $i$ match kare).

$A > B$ tab hota hai jab upar se scan karte waqt sabse upar wala differing bit $A_{i} = 1, B_{i} = 0$ ho, aur uske upar ke saare bits equal rahe hon:

G = A_{n - 1} \overset{ˉ}{B}_{n - 1} + x_{n - 1} A_{n - 2} \overset{ˉ}{B}_{n - 2} + x_{n - 1} x_{n - 2} A_{n - 3} \overset{ˉ}{B}_{n - 3} + \dots

$x$ chain kyun? Term $x_{n - 1} x_{n - 2}$ ka matlab hai "position $n - 3$ se upar ke saare bits equal the, isliye yeh pehli jagah hai jahan fark hai" — bilkul wahi insaan wala rule.

Symmetrically:

L = \overset{ˉ}{A}_{n - 1} B_{n - 1} + x_{n - 1} \overset{ˉ}{A}_{n - 2} B_{n - 2} + \dots

Aur equality matlab "har bit match kare": $E = x_{n - 1} x_{n - 2} \dots x_{1} x_{0}$

4. Worked example — 2-bit comparator scratch se

$A = A_{1} A_{0}$ ko $B = B_{1} B_{0}$ se compare karo.

Step 1 — per-bit equals. $x_{1} = \overline{A_{1} \oplus B_{1}}$ , $x_{0} = \overline{A_{0} \oplus B_{0}}$ . Yeh step kyun? Yeh reusable "match" signals hain jo insaan wale method ko chahiye.

Step 2 — Equal. Dono bits match karein: $E = x_{1} x_{0}$ Kyun? Numbers equal tabhi hain jab MSB aur LSB dono agree karein.

Step 3 — Greater. MSB pehle decide karta hai; agar MSBs tie karein toh LSB decide karta hai: $G = A_{1} \overset{ˉ}{B}_{1} + x_{1} A_{0} \overset{ˉ}{B}_{0}$ Kyun? Pehla term: $A$ ka MSB directly jeet jaata hai. Doosra: MSBs equal hain ( $x_{1}$ ), isliye LSB $A_{0} \overset{ˉ}{B}_{0}$ tie tod ta hai.

Step 4 — Less. Mirror image: $L = \overset{ˉ}{A}_{1} B_{1} + x_{1} \overset{ˉ}{A}_{0} B_{0}$

Check $A = 1 0_{2} = 2$ , $B = 0 1_{2} = 1$ ke saath: $A_{1} \overset{ˉ}{B}_{1} = 1 \cdot 1 = 1$ → $G = 1$ . ✔ (2 > 1)

Check $A = 01$ , $B = 01$ ke saath: $x_{1} = 1, x_{0} = 1$ → $E = 1$ . ✔

5. Worked example — IC 7485 (4-bit) cascade

Kyun care karein? Real chips jaise 7485 cascade inputs $(A > B)_{in}, (A = B)_{in}, (A < B)_{in}$ lete hain taaki tum unhe wider numbers ke liye chain kar sako.

Rule jab internally tie ho: agar 4 local bits sab equal hon, toh chip cascade input ko pass through karta hai.

(A > B)_{o u t} = G_{l oc a l} + E_{l oc a l} \cdot (A > B)_{in}

Yeh step kyun? $E_{l oc a l}$ matlab "in 4 bits ne kuch decide nahi kiya, isliye neeche wali less-significant chip par bharosa karo" — wahi insaan wala first-differing-digit rule, ab poore chips ke across.

8-bit ke liye cascade wiring: low nibble ke outputs ko high nibble ke cascade inputs mein feed karo. Low chip ke ground-level inputs tied hain $(A = B)_{in} = 1$ ke saath (taaki agar sab kuch equal ho, toh output equal aaye).

6. Common mistakes

7. Active recall

Recall Khud test karo (answers chupaao)

Konsa gate 1-bit equality implement karta hai? → XNOR
Magnitude pehle konsa bit decide karta hai? → MSB
Lower-bit terms ko $x_{i}$ se guard kyun karein? → taaki woh tabhi act karein jab higher bits ne tie kiya ho
Cascaded 7485 mein $E_{l oc a l}$ kya karta hai? → cascade input ko pass through karta hai
Sum identity jo hamesha hold karni chahiye? → $G + E + L = 1$

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho do bacche number cards dikhaa rahe hain, sabse bada pehle. Woh front card palte hain: agar kisi ka bada ho, toh wahi baccha poora game jeet jaata hai — aage dekhne ki zaroorat nahi. Agar front cards match karein, toh woh agla card dekhte hain, aur aise hi chalta rehta hai. Agar har card match kare, toh tie hai. Ek comparator ek chhoti electric machine hai jo exactly yahi card game 1s aur 0s ke saath, instantly, khelti hai.

8. Flashcards

Ek comparator kaunse teen outputs produce karta hai?

A > B

(G),

A = B

(E),

A < B

(L), mutually exclusive.

1-bit equality equation?

E = \overline{A \oplus B}

(XNOR).

1-bit greater-than equation?

G = A \overset{ˉ}{B}

1-bit less-than equation?

L = \overset{ˉ}{A} B

MSB pehle compare kyun karein?

Magnitude most significant bit se dominate hota hai; pehla differing bit decide karta hai.

x_{i} = \overline{A_{i} \oplus B_{i}}

signals ka purpose?

Per-bit equality flags jo lower-bit decision terms ko guard karne ke liye use hote hain.

2-bit greater-than expression?

G = A_{1} \overset{ˉ}{B}_{1} + x_{1} A_{0} \overset{ˉ}{B}_{0}

n-bit equality expression?

E = x_{n - 1} x_{n - 2} \dots x_{0}

(saare bits match karein).

Comparator outputs ke beech konsi identity hamesha hold karti hai?

G + E + L = 1

7485 IC mein cascade inputs ka role?

Multiple comparators chain karo; use hota hai jab local bits sab equal hon taaki less-significant stage par decision liya ja sake.

7485 cascaded output formula for A>B?

(A > B)_{o u t} = G_{l oc a l} + E_{l oc a l} (A > B)_{in}

Common XOR-vs-XNOR trap?

XOR difference detect karta hai (inequality); equality ke liye XNOR chahiye.

9. Connections

XOR and XNOR Gates — equality primitive
Truth Tables and Minterms — $G, E, L$ kaise read out karte hain
Binary Subtraction — subtract karke sign bit check karna ek alternate comparison method hai
Multiplexers — ek aur decision-making combinational block
Ripple Carry Adder — contrast: adders LSB-first jaate hain, comparators MSB-first
7485 IC — standard cascadable comparator chip