Link budget — path loss, EIRP, G - T, Eb - N0: Visual walkthrough

3.6.25 · D2 · Physics › Spacecraft Structures & Systems Engineering › Link budget — path loss, EIRP, G - T, Eb - N0

Yeh poora link budget hai jo pictures mein banaya gaya hai, shuru hota hai sirf ek transmitter se jo empty space mein switch on hua ho. Last figure tak aap poori chain draw kar paoge — power out, power spreading, power collected, noise fighting back, aur final "kya bits survive kiye?" number — ek napkin par.

Har symbol ko hum wahan define karte hain jab woh pehli baar aata hai. Agar aapne kabhi decibel ya logarithm nahi dekha, tab bhi aap line one follow kar paoge.

Step 1 — Space mein akela transmitter: "power spreading" ka matlab kya hai

KYA HAI. Hum ek radio transmitter switch on karte hain. Woh utne watts pump out karta hai jise hum $P_{t}$ kehte hain — transmit power. Ek watt simply "joules of energy per second" hai; ise socho ki energy antenna se kitni fast nikalti hai.

KYUN. Kisi bhi gain, loss, ya antenna se pehle, hume radio ki ek raw fact picture karni chahiye: energy ek point se nikalti hai aur badhte hue sphere ki tarah bahar failti hai. Budget mein baaki sab kuch is ek picture ka correction hai.

PICTURE. Figure mein, red dot transmitter hai. Is second jo energy woh bhejta hai woh ek expanding ball ki surface par hoti hai. Distance $r$ par (transmitter se metres mein) us ball ki surface ka area hota hai

area of a sphere of radius r 4 π r^{2}

Figure — Link budget — path loss, EIRP, G - T, Eb - N0

Intensity — har square metre par landing wali power — total power ko us area se divide karne par milti hai:

I_{iso} = \frac{P _{t} watts sent out}{m ^{2} of sphere 4 π r ^{2}} [\frac{W}{m ^{2}}]

"iso" subscript ka matlab hai isotropic — sabhi directions mein equally radiate karna, jaise ek bare light bulb. Real antennas isse better karte hain, jo Step 2 mein hai.

Step 2 — Power ko aim karna: gain aur EIRP

KYA HAI. Hum bare bulb ko ek directional antenna se replace karte hain — ek dish jo same watts ko ek narrow beam mein focus karti hai, jaise ek flashlight reflector same bulb ko focus karta hai.

KYUN. Hum aur watts nahi bana sakte (spacecraft par power precious hai), toh iske bajaye hum jo watts hamare paas hain unhe aim karte hain. Hume ek number chahiye jo capture kare "beam bare bulb se kitna zyada bright hai."

PICTURE. Figure mein same $P_{t}$ se do intensity patterns dikhaye gaye hain: ek round isotropic glow (thin black) aur ek focused red beam. Red beam apni axis ke along kaafi zyada intense hai.

Hum focusing factor ko transmit gain $G_{t}$ kehte hain — ek plain dimensionless ratio (koi units nahi):

G_{t} = \frac{I _{beam} intensity on axis}{bare-bulb intensity I _{iso}}

$300$ ka gain matlab hai "beam same power ke bare bulb se on-axis $300 \times$ zyada bright hai." $G_{t}$ physically kahan se aata hai? Bada dish → tighter beam → higher gain — dekho Antenna Gain and Effective Aperture.

Ab clever bookkeeping trick. $P_{t}$ aur $G_{t}$ ko alag track karne ki bajaye, ek number define karo, Effective Isotropic Radiated Power (EIRP) — woh wattage jo ek bare bulb ko chahiye hogi hamari beam match karne ke liye:

EIRP = real watts P_{t} \times focusing factor G_{t} [W]

Toh on-axis intensity simply isotropic formula hai jisme $P_{t}$ ki jagah EIRP hai:

S = \frac{EIRP}{4 π r ^{2}}

Step 3 — Decibels: multiply ko add mein badalna (taaki hum Step 6 survive kar sakein)

KYA HAI. Hum apni units raw ratios se decibels (dB) mein switch karte hain.

KYUN. Agle steps mein numbers monstrous ho jaate hain — signals $1 0^{27}$ ke factors se weak ho jaate hain. Aisi numbers ko haath se multiply karna hopeless hai. Ek logarithm multiplication ko addition mein badal deta hai, toh poora budget ek tidy sum ban jaata hai: gains add karo, losses subtract karo.

PICTURE. Figure ek number line hai. Upar, linear scale bade numbers ko ek unusable smear mein cram kar deta hai. Neeche, same values ek dB (log) scale par evenly, readably spaced hain.

Rule. Kisi bhi power ratio $x$ ke liye:

x_{dB} = 10 lo g_{10} (x)

Yahan $lo g_{10} (x)$ sawaal ka jawab deta hai "10 ko kaunsi power deni ho ki $x$ mile?" Humne base $10$ choose kiya (base $e$ nahi) purely kyunki engineers powers of ten mein count karte hain. Do facts jo aap baar baar reuse karoge:

lo g_{10} (a \cdot b) = lo g_{10} a + lo g_{10} b lo g_{10} (a^{2}) = 2 lo g_{10} a

Pehla wahi reason hai kyun multiply add ban jaata hai. Doosra reason hai kyun $r^{2}$ path loss mein $20$ ka factor aata hai (woh hai $10 \times 2$ ).

Do named reference units:

dBW = dB relative to $1$ watt → $P_{dBW} = 10 lo g_{10} (P /1 W)$ .
dBi = dB relative to isotropic → $G_{dBi} = 10 lo g_{10} (G_{t})$ .

Toh Step 2 ka boxed result ek addition ban jaata hai:

EIRP (dBW) = power in dB P_{t} (dBW) + gain in dB G_{t} (dBi)

Step 4 — Receiver ek raindrop pakadta hai: effective aperture

KYA HAI. Door ek receiving dish expanding wavefront ka ek patch intercept karke use electronics mein funnel karti hai.

KYUN. Transmitter ki power ab lakhon km wide sphere par smear ho chuki hai. Hamari dish sirf woh sliver pakadti hai jo uske munh par girta hai. Hume jaanna hai ki dish kitna bada bite leta hai.

PICTURE. Figure mein door wavefront (ab nearly flat) sweep karte hue dikhaya gaya hai, aur red dish ek chhota sa circular area $A_{e}$ label karke scoop out kar rahi hai.

Dish ki effective aperture $A_{e}$ (m² mein) woh wavefront ka area hai jo woh effectively collect karti hai. Pakdi hui power = intensity $\times$ area:

P_{r} = W/m^{2} arriving S \times m^{2} collected A_{e} = \frac{EIRP \cdot A _{e}}{4 π r ^{2}}

Antenna theory aperture ko receive gain $G_{r}$ aur wavelength $λ$ (ek wave ki length, metres mein) se link karta hai:

A_{e} = \frac{G _{r} receive gain λ ^{2} wave size squared}{4 π}

$λ$ kyun aata hai? Ek dish wave ko tab achhi tarah focus karti hai jab dish kai wavelengths cross ho; chhoti wavelengths (higher frequency) same physical dish ko zyada grab karne deti hain — poori kahani Antenna Gain and Effective Aperture mein hai. Substitute karne par:

P_{r} = EIRP \cdot G_{r} \cdot the geometry factor \frac{λ ^{2}}{( 4 π r ) ^{2}}

Step 5 — Villain ko isolate karna: free-space path loss $L_{p}$

KYA HAI. Hum pure distance-and-frequency penalty ko antenna gains se alag karte hain, aur use path loss $L_{p}$ naam dete hain.

KYUN. Gains ( $G_{t}$ , $G_{r}$ ) woh cheezein hain jo hum design karte hain. Distance penalty physics aur geometry se fixed hai — Mars wahan hai jahan Mars hai. Ise isolate karne se hum clearly dekh sakte hain ki hum kya fight kar rahe hain.

PICTURE. Figure Step 1 waala hi expanding sphere hai, lekin ab ek log distance axis ke against "surviving power" ki shrinking bar annotate ki gayi hai — red curve $r$ badhne ke saath plunge karti hai.

$L_{p}$ ko us geometry factor ka reciprocal define karo (toh yeh ek number $> 1$ hai jisse hum divide karte hain):

L_{p} = (\frac{4 π r}{λ})^{2}

Ise term by term padho: $4 π r$ sphere ki spreading hai, denominator mein $λ$ wavelength ki helping hand hai, aur outer square inverse-square law hai jo show ho raha hai. dB mein — aur yahan $lo g (a^{2}) = 2 lo g a$ rule square ko $20$ ke factor mein badal deta hai:

L_{p} (dB) = 20 lo g_{10} (\frac{4 π r}{λ}) = farther \to worse 20 lo g_{10} r + higher freq \to worse 20 lo g_{10} f + constant 20 lo g_{10} \frac{4 π}{c}

$λ = c / f$ use karte hue jahan $c = 3 \times 1 0^{8}$ m/s. Constant fold karke aur convenient units mein convert karke ( $r$ km mein, $f$ MHz mein) woh working form milta hai jo poori Friis duniya mein use hoti hai:

L_{p} (dB) = 32.45 + 20 lo g_{10} (r_{km}) + 20 lo g_{10} (f_{MHz})

Step 6 — Noise floor: G/T aur $C / N_{0}$ ka janam

KYA HAI. Hum poori chain ko ek saath rakh kar carrier-to-noise-density ratio $C / N_{0}$ nikalte hain — signal versus noise.

KYUN. Ek faint signal useless hai agar noise loud ho. Har warm object (electronics, sky, ground) random RF noise radiate karta hai. Message detect karne ka matlab hai ki carrier power $C$ is hiss ke upar khada rehna chahiye.

PICTURE. Figure mein do bars hain: tall received carrier $C$ saare adds aur subtracts ke baad, aur ek fuzzy red band — noise density $N_{0}$ . Unke beech ka gap hi budget hai.

Received carrier, dB mein, ek running total hai — loud start karo, focusing add karo, distance subtract karo, collecting add karo:

C = loud start EIRP - distance L_{p} + dish grabs G_{r}

Noise. Kisi bhi receiver ki noise ko ek system noise temperature $T_{sys}$ (kelvin mein) describe karta hai — ek made-up temperature jiska thermal hiss real noise ke barabar hai (dekho Noise Temperature and Noise Figure). Bandwidth ke har hertz mein noise power hai

N_{0} = k T_{sys}, k = 1.38 \times 1 0^{- 23} J/K = - 228.6 dBW/K/Hz

jahan $k$ Boltzmann's constant hai — temperature ko power-per-hertz mein convert karne ka nature ka formula. $C$ ko $N_{0}$ se divide karo, aur receiver ke do knobs — gain $G_{r}$ (bada chahiye) aur $T_{sys}$ (chhota chahiye) — ek figure of merit mein merge ho jaate hain, G/T ratio:

\frac{G}{T} (dB/K) = G_{r} (dBi) - 10 lo g_{10} (T_{sys})

Sab kuch ek saath rakh kar (negative $k$ subtract karna = $228.6$ add karna):

\frac{C}{N _{0}} (dB-Hz) = EIRP - L_{p} + \frac{G}{T} - 10 lo g_{10} (k)

Step 7 — Per-bit accounting: $E_{b} / N_{0}$ , woh number jo success decide karta hai

KYA HAI. Hum raw signal-to-noise ko energy per bit over noise density, $E_{b} / N_{0}$ mein convert karte hain — woh metric jo actually errors predict karta hai.

KYUN. $C / N_{0}$ ko nahi pata ki hum kitni fast bhej rahe hain. Same power ko double bits per second mein cram karo toh har bit ko half energy milti hai. Bit errors energy per bit par depend karte hain, isliye hume data rate divide out karna hoga.

PICTURE. Figure mein fixed carrier power dikhaya gaya hai jo $R_{b}$ bit-boxes mein slice ho raha hai: kuch bade boxes (slow, robust) vs. bahut saare thin boxes (fast, fragile). Red slice "ek bit ki energy" hai.

Data rate $R_{b}$ (bits per second) par carrier mein se har bit ka share:

E_{b} = \frac{C}{R _{b}} ⟹ \frac{E _{b}}{N _{0}} = \frac{C}{N _{0} R _{b}}

dB mein — ek clean subtraction:

\frac{E _{b}}{N _{0}} (dB) = \frac{C}{N _{0}} (dB-Hz) - 10 lo g_{10} (R_{b})

Yeh payoff number hai. Ise apni modulation ke liye required $E_{b} / N_{0}$ se compare karo (Forward Error Correction ki madad se); difference hi tumhara link margin hai. Iska ultimate floor Shannon-Hartley Theorem se set hota hai.

Ek picture mein summary

Har step ek single left-to-right ledger mein ek arrow hai: loud start karo, focus karo, void cross karo, collect karo, noise se louder raho, bits mein split karo, threshold se compare karo.

Steps 1-3 EIRP - Steps 4-5 L_{p} + Step 6 \frac{G}{T} - 10 lo g k - Step 7 10 lo g R_{b} = \frac{E _{b}}{N _{0}}

Recall Feynman retelling — poora walkthrough plain words mein

Ek transmitter switch on hota hai aur uski energy ek growing bubble ki tarah bahar daudti hai; bubble ki skin par spread hokar, woh one-over-distance-squared ki tarah thin hoti jaati hai (Step 1). Hum ise ek dish se aim karte hain taaki beam bare bulb se kai guna brighter ho — woh focus times real watts hi EIRP hai (Step 2). Aane wale giant numbers mein multiply ki jagah add karne ke liye, hum decibels mein switch karte hain, jahan multiplying, adding ban jaati hai aur squares, twenty ke factors ban jaate hain (Step 3). Door, hamari apni dish thinned bubble ka ek chhota patch scoop karti hai — uska effective aperture — aur woh patch, plus wavelength, set karta hai ki hum kitna pakad sakte hain (Step 4). Antennas strip out karne par pure distance-aur-frequency penalty bachti hai, path loss, Mars ke liye ek monstrous ~279 dB (Step 5). Faint survivor ke saamne receiver ka noise khada hai, jo G/T se summarised hai; signal ko noise-per-hertz se divide karna C/N₀ deta hai, voice aur hiss ke beech ka gap (Step 6). Finally hum puchte hain ki hum kitni fast baat kar rahe hain: carrier ko bits per second mein split karo energy per bit paane ke liye, aur E_b/N_0 batata hai ki har bit sahi padha jaayega ya nahi (Step 7). Gains add karo, losses subtract karo, rate subtract karo — ek honest sum decide karta hai ki signal journey survive karega ya nahi.

Quick self-check

Path loss

20 lo g

of range hai,

10 lo g

nahi — kyun?

Kyunki power

r^{2}

se fall karta hai, aur

lo g (r^{2}) = 2 lo g r

, isliye dB factor

10 \times 2 = 20

hota hai.

Agar aap data rate double kar do, toh

E_{b} / N_{0}

ka kya hoga?

Woh

10 lo g_{10} (2) \approx 3

dB drop karega — har bit ko half energy milti hai.

G_{r}

aur

T_{sys}

ko G/T mein kyun merge karte hain?

Kyunki signal-to-noise gain over noise par depend karta hai; ek figure of merit se aap poore receivers ko ek nazar mein compare kar sakte ho.