Dynamic vs static power consumption: Visual walkthrough

6.4.1 · D2 · Hardware › Power, Thermal & Reliability › Dynamic vs static power consumption

Yeh page parent note ka headline formula rebuild karta hai

$P_{switching} = α C V_{dd}^{2} f$

bilkul scratch se — koi calculus nahi, koi physics assumed nahi. Har letter ko hum ek picture ke saath earn karenge, pehle woh appear ho. End tak tumhe sirf formula yaad nahi rahega; tum actually dekhoge ki har symbol kahan se aata hai aur woh wahan kyun baitha hai.

Agar tum switch se pehle mil chuke ho, woh CMOS Inverter Design mein rehta hai; "bucket" Capacitance in VLSI mein rehta hai. Is page ko follow karne ke liye tumhe koi bhi nahi chahiye — hum dono ko zero se build karte hain.

Step 1 — Switch aur bucket

KYA. Hum woh object draw karte hain jo hum study kar rahe hain: ek CMOS gate. Transistor internals ko ignore karo. Jo matter karta hai woh yeh hai ki ek gate ek wire aur agli gate ki input drive karta hai, aur yeh sab aise behave karta hai jaise ek bucket jo electric charge hold karta hai.

YEH PICTURE KYUN. Power energy per second hai. Digital gate mein energy almost poori ki poori is charge-bucket ko bharne aur khaali karne mein jaati hai. Toh kisi bhi formula se pehle, humein agree karna hai ki kya bharta hai aur woh kisme bharta hai.

PICTURE. Figure mein, daayein taraf ka tank bucket hai. Upar ka tap supply rail $V_{dd}$ se connect hai (ek fixed "high water pressure"). Neeche ka drain ground se connect hai (pressure zero). Switch decide karta hai kaunsa pipe open hai.

Figure — Dynamic vs static power consumption

Teen primitive words jo unhe aapas mein baandhti hain — bucket ki definition — yeh hai:

$q = C \cdot V$

$q$ — charge jo abhi bucket mein baitha hai (coulombs)
$C$ — bucket ki size (farads); wire aur gate ki ek fixed property
$V$ — abhi bucket ke across pressure (volts)

Toh bucket ka pressure supply pressure $V_{dd}$ tak bilkul upar uthane ke liye, tumhe total charge daalna padega:

$Q = C \cdot V_{dd}$

Woh number $Q = C V_{dd}$ yaad rakho. Yeh "full bucket" amount hai.

Step 2 — Ek flip supply ko kya kharcha deta hai

KYA. Hum switch flip karte hain toh output 0 se 1 ho jaata hai. Tap khulta hai, charge supply se bucket mein flow karta hai jab tak bucket pressure $V_{dd}$ ke barabar nahi ho jaata. Ab hum poochhte hain: supply ne kitni energy kharchi?

YEH SAWAAL KYUN. Supply woh cheez hai jo tumhari battery drain kar rahi hai. Usse fark nahi padta energy kahan jaati hai — woh sirf jaanti hai ki usne kitna charge push kiya aur kis pressure par. Constant-pressure source se energy simply pressure × pushed charge hai.

PICTURE. Baayein bar supply hai, hamesha full pressure $V_{dd}$ par push karta hai (woh kabhi sag nahi karta). Har coulomb jo woh deta hai woh usi full pressure par diya jaata hai — yeh shaded rectangle hai.

$E_{supply} = V_{dd} \cdot Q = V_{dd} \cdot (C V_{dd}) = C V_{dd}^{2}$

$V_{dd}$ — supply pressure, constant poore time, isliye seedha product se bahar aata hai
$Q = C V_{dd}$ — Step 1 se full-bucket charge
$C V_{dd}^{2}$ — do $V_{dd}$ s multiply hote hain: ek pressure hai, ek $Q$ ke andar chhupa hai

Woh squared voltage hamara famous $V_{dd}^{2}$ ka pehla darshan hai. Uska origin yaad rakho: ek factor pressure hai, ek factor kitna charge us pressure ko move karna pada hai. Wahi knob do baar ghuma.

Step 3 — Aadhi energy kahan gaayab hoti hai

KYA. Supply ne $C V_{dd}^{2}$ kharcha kiya. Lekin abhi bucket mein actually kitni energy store hai, reuse ke liye ready? Hum "kharcha" aur "rakha" compare karte hain.

YEH KYUN MATTER KARTA HAI. Agar supply ne jo kharcha kiya woh sab store ho jaata, toh kuch bhi heat mein nahi badalti aur chips kabhi garam nahi hote. Kharche aur stored ke beech ka gap exactly heat hai. Humein woh gap dhundna hai.

PICTURE. Do rectangles overlaid hain. Lamba wala (area $C V_{dd}^{2}$ ) woh hai jo supply ne pay kiya — charge full pressure par poore time deliver kiya. Neeche ka triangle woh hai jo bucket ne rakha — kyunki bucket ka apna pressure 0 se start hota hai aur sirf $V_{dd}$ tak climb karta hai jab woh bharta hai. Bucket ne sirf $V_{dd} /2$ ka average pressure feel kiya.

$E_{stored} = \frac{1}{2} Q V_{dd} = \frac{1}{2} C V_{dd}^{2}$

$Q$ — wahi full-bucket charge
$\frac{1}{2} V_{dd}$ — average pressure jo bucket ne fill hote waqt feel kiya (full $V_{dd}$ nahi)
$\frac{1}{2} C V_{dd}^{2}$ — exactly half jo supply ne pay kiya

Missing half switch ki resistance mein heat ban ke jalti hai jab charge us se guzarta hai:

$E_{heat, charge} = E_{supply} - E_{stored} = C V_{dd}^{2} - \frac{1}{2} C V_{dd}^{2} = \frac{1}{2} C V_{dd}^{2}$

Step 4 — Khaali karne ka stroke utna hi kharcha karta hai

KYA. Ab wapas flip karo: output 1 → 0. Tap band hota hai, drain khulta hai, aur bucket apna stored $\frac{1}{2} C V_{dd}^{2}$ ground mein dump kar deta hai. Hum track karte hain ki woh energy kahan jaati hai.

YEH INCLUDE KYUN KAREIN. Ek real gate sirf ek baar fill nahi hoti — woh fill aur empty hoti hai, baar baar. Agar hum Step 3 par rukk jaate toh hum sirf aadha cycle count kar rahe hote aur heat undercount ho jaati.

PICTURE. Bucket lower switch se drain hota hai. Stored energy ka koi bhi hissa supply ko wapas nahi jaata — sab kuch ground ke raaste pull-down switch ki resistance mein heat ban ke jal jaata hai.

$E_{cycle} = charging heat \frac{1}{2} C V_{dd}^{2} + discharging heat \frac{1}{2} C V_{dd}^{2} = C V_{dd}^{2}$

pehla term — fill ke dauran pull-up switch mein heat
doosra term — empty ke dauran pull-down switch mein heat
$C V_{dd}^{2}$ — dono halves milke ek saaf whole banate hain

Toh: ek gate ke ek full flip ki keemat $C V_{dd}^{2}$ joules of heat hai. Yeh dynamic power ka atom hai.

Step 5 — "Per flip" se "per second" tak

KYA. Energy per flip achhi baat hai, lekin power energy per second hai. Hum convert karte hain yeh poochhkar: har second kitne flips hote hain?

KYUN. Tumhare cooling system aur tumhari battery ko heat ki rate, watts (joules per second), ki chinta hai, na ki ek single event ki keemat ki.

PICTURE. Ek clock jo $f$ baar per second tick karta hai. Har tick flip karne ka ek mauka hai. Uske neeche, wahi clock lekin gate sirf kuch ticks par actually flip karta hai — busy ticks marked hain.

Agar gate har tick par flip karta, toh power energy-per-flip times ticks-per-second hoti:

$P = E_{cycle} \cdot f = C V_{dd}^{2} f$

$E_{cycle} = C V_{dd}^{2}$ — joules per flip, Step 4 se
$f$ — flips per second (agar har tick par flip karta)
product — joules per second = watts

Step 6 — Honesty knob: activity factor

KYA. Real gates har tick par flip nahi karti. Ek gate jo steady value hold kar raha hai woh idle baitha rehta hai jabki clock tick karta rehta hai. Hum count ko sirf unhi ticks tak shrink karte hain jahan actually flip hota hai.

KYUN. Is correction ke bina hum power massively overestimate karte. Ek chip mein zyaadatar gates zyaadatar time quiet rehte hain.

PICTURE. Ek row mein das clock ticks; unme se sirf do flip dikhate hain. Jitna fraction lit up hai woh activity factor hai.

Flip-count ko $α$ se scale karne par full switching-power law milta hai:

Dependence ki shapes notice karo, seedhi derivation se:

$V_{dd}^{2}$ quadratic hai — voltage half karo, power quarter ho jaati hai. Isliye Dynamic Voltage Frequency Scaling (DVFS) pehle voltage par rely karta hai.
$f$ linear hai — clock double karo, yeh power double hoti hai.
$α$ ek discount hai — quiet chips free mein thande rehte hain.

Step 7 — Edge cases: knobs ko unke limits par push karna

KYA. Jis formula par tum trust karo woh apne extremes mein survive karna chahiye. Hum har knob ko zero ya uski ceiling tak drive karte hain aur check karte hain ki picture abhi bhi sense banaati hai.

KYUN. Agar equation $α = 0$ ya $f = 0$ par nonsense deta, toh hum jaante ki koi step galat tha. Yeh tests pass karna hi result mein belief earn karna hai.

PICTURE. Chaar mini-panels, har ek ek degenerate setting hai, surviving power value ke saath.

Step 8 — Do worked numbers

KYA. Hum real values ko law mein dalte hain taaki units concrete ho jaayein.

KYUN. Symbols tabhi click karte hain jab tum dekho pico aur giga milli mein cancel hote hain.

Ek-picture summary

Har step ek single chain mein fold hota hai: bucket → fill cost → half lost → both strokes → per second → real flips.

Recall Feynman retelling — ise zor se bolo

Har gate charge ka ek chota bucket drive karta hai. Bucket ko supply pressure tak bharne ke liye tumhe charge ka ek fixed dhera move karna padta hai, aur kyunki supply hamesha full pressure par push karti hai jabki bucket dheere dheere build up karta hai, jo energy tum kharcha karte ho uski exactly aadhi rakhi jaati hai aur aadhi heat ban jaati hai. Bucket khaali karna rakhi hui aadhi bhi jala deta hai. Toh ek complete flip — upar aur neeche — $C V_{dd}^{2}$ pure heat ka kharcha karta hai. Ab flips gino: clock $f$ chances per second offer karta hai, lekin gate sirf unka ek fraction $α$ leta hai. Cost per flip ko flips per second se multiply karo aur tumhare paas watts aa jaate hain: $P = α C V_{dd}^{2} f$ . Voltage squared appear hota hai kyunki tumne wahi pressure knob do baar ghuma — ek baar pressure ke roop mein, ek baar charge ke roop mein jo us pressure ne move kiya — isliye voltage kam karna sabse strong lever hai jo hamare paas hai.

Khud test karo

V_{dd}^{2}

kahan se aata hai?

Ek factor supply pressure hai, ek factor charge

Q = C V_{dd}

hai jo woh pressure move karta hai — wahi voltage knob do baar appear hota hai.

Supplied energy ka sirf aadha capacitor mein kyun store hota hai?

Bucket ka pressure linearly

0 \to V_{dd}

badhta hai, isliye woh

V_{dd} /2

ka average pressure feel karta hai; stored energy charge times woh average hai = rectangle ka aadha.

α

se multiply kyun karte hain?

Real gates har clock tick par flip nahi karti;

α

woh fraction hai ticks ka jo genuine flips hain, isliye woh count discount karta hai.

Jab clock gated off ho (

f = 0

) toh switching power ka kya hota hai?

Woh zero ho jaati hai — koi ticks nahi matlab koi flips nahi matlab koi switching heat nahi.

Related vault stops: Thermal Design Power (TDP) (jahan yeh watts heat ke roop mein dump hone chahiye), Subthreshold Slope (power story ka leakage wala aadha hissa), aur Amdahl's Law (kyun raw $f$ apni power ke liye ek maatra speed lever nahi hai).