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

3.6.25 · Physics › Spacecraft Structures & Systems Engineering

Overview

Ek link budget ek radio frequency (RF) communication link mein transmitter se receiver tak ke saare gains aur losses ka comprehensive hisaab hota hai. Yeh ek fundamental sawaal ka jawaab deta hai: "Kya mera signal space ke safar mein survive karega?"

Link budget yeh determine karta hai ki ek spacecraft Earth (ya kisi aur spacecraft) ke saath successfully communicate kar sakta hai ya nahi — iska calculation yeh batata hai ki received signal power, noise floor se itna zyada hai ki data reliably decode ho sake.

Space mein hum sirf "volume badha" nahi sakte — power precious hai, antennas ki size ki limit hoti hai, aur physics (inverse square law) bilkul maafi nahi deti. Link budget humaara survival calculator hai.

Core Components of a Link Budget

1. EIRP — Effective Isotropic Radiated Power

Yeh Concept Kyun? Real antennas sabhi directions mein equally radiate nahi karte — woh power ko ek flashlight ki tarah focus karte hain. EIRP transmitter power aur antenna gain ko ek number mein combine karta hai jo best direction mein "effective" radiated power ko represent karta hai.

Antenna gain ka physical meaning: Agar power isotropically radiate hoti, toh distance $r$ par intensity hoti: $I_{iso} = \frac{P _{t}}{4 π r ^{2}}$

Gain $G_{t}$ wale directional antenna ke saath, main beam direction mein intensity ban jaati hai: $I_{actual} = \frac{P _{t} \cdot G _{t}}{4 π r ^{2}}$

Hum EIRP ko is tarah define karte hain: $I_{actual} = \frac{EIRP}{4 π r ^{2}}$

Inhe equate karne par: $EIRP = P_{t} \cdot G_{t} (watts)$

Decibels mein (practical form): $EIRP (dBW) = P_{t} (dBW) + G_{t} (dBi)$

jahaan dBW = 1 watt ke relative decibels, aur dBi = isotropic ke relative decibels.

Step 1 — Power ko dBW mein convert karo: $P_{t} = 10 W = 10 lo g_{10} (10) = 10 dBW$ Yeh step kyun? Hum dB mein kaam karte hain kyunki gains aur losses multiply hote hain (dB mein add hote hain), jisse calculations manageable rehti hain.

Step 2 — EIRP calculate karo: $EIRP = 10 dBW + 25 dBi = 35 dBW$

Physical meaning: Yeh system apne main beam mein utni hi power radiate karta hai jitni ek 3162 W isotropic transmitter karta ( $1 0^{3.5} \approx 3162$ W).

2. Path Loss — Fundamental Range Penalty

Yeh kyun hota hai? Jaise ek spherical wavefront expand hota hai, wahi total power ek increasingly bade area ( $4 π r^{2}$ ) par phail jaati hai. Power per unit area (intensity) $1/ r^{2}$ ke hisaab se decrease hoti hai.

Step 1 — Distance $r$ par power density:

EIRP se shuru karke, distance $r$ par power density (watts per square meter) hai: $S = \frac{EIRP}{4 π r ^{2}}$

$4 π r^{2}$ kyun? Yeh radius $r$ par sphere ki surface area hai — power is poori area par phailti hai.

Step 2 — Antenna dwara received power:

Effective aperture $A_{e}$ (square meters) wala receiving antenna collect karta hai: $P_{r} = S \cdot A_{e} = \frac{EIRP \cdot A _{e}}{4 π r ^{2}}$

Effective aperture, gain se is relation se juda hai: $A_{e} = \frac{G _{r} λ ^{2}}{4 π}$

jahaan $λ$ wavelength hai aur $G_{r}$ receiver antenna gain hai.

Yeh relation kyun? Yeh antenna theory se aata hai — bada antenna ya zyada gain matlab bada effective collecting area.

Step 3 — Aperture substitute karo: $P_{r} = \frac{EIRP \cdot G _{r} λ ^{2}}{4 π \cdot 4 π r ^{2}} = EIRP \cdot G_{r} \cdot \frac{λ ^{2}}{( 4 π r ) ^{2}}$

Step 4 — Path loss $L_{p}$ define karo:

Path loss transmitted aur received power ka ratio hai (antenna gains ke bina): $L_{p} = (\frac{4 π r}{λ})^{2}$

Decibels mein: $L_{p} (dB) = 20 lo g_{10} (\frac{4 π r}{λ}) = 20 lo g_{10} (4 π) + 20 lo g_{10} (r) - 20 lo g_{10} (λ)$

$λ = c / f$ use karke jahaan $c = 3 \times 1 0^{8}$ m/s: $L_{p} (dB) = 20 lo g_{10} (r) + 20 lo g_{10} (f) + 20 lo g_{10} (\frac{4 π}{c})$

Standard form (r km mein, f MHz mein): $L_{p} (dB) = 32.45 + 20 lo g_{10} (r_{km}) + 20 lo g_{10} (f_{MHz})$

Step 1 — Frequency convert karo: $f = 8.4 GHz = 8400 MHz$

Step 2 — Formula apply karo: $L_{p} = 32.45 + 20 lo g_{10} (2.5 \times 1 0^{8}) + 20 lo g_{10} (8400)$

Har term calculate karo:

$20 lo g_{10} (2.5 \times 1 0^{8}) = 20 (8 + 0.398) = 167.96$ dB
$20 lo g_{10} (8400) = 20 (3.924) = 78.48$ dB

$L_{p} = 32.45 + 167.96 + 78.48 = 278.89 dB$

Itna bada kyun? Space bahut vast hai! Yeh ~279 dB loss matlab signal $1 0^{27.9}$ factor se attenuate ho jaata hai — lagbhag ek billion guna kamzor.

3. G/T — Figure of Merit for Receivers

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

Yeh combination kyun?

High $G_{r}$ → zyada signal collect karo ✓
Low $T_{sys}$ → signal ke saath compete karne wala kam noise ✓

Dono signal-to-noise ratio (SNR) improve karte hain. Inhe ek metric mein combine karne se hum receiver systems ko directly compare kar sakte hain.

Received carrier-to-noise ratio $C / N_{0}$ (dB-Hz mein) hai: $\frac{C}{N _{0}} = EIRP - L_{p} + \frac{G}{T} - 10 lo g_{10} (k)$

jahaan $k = 1.38 \times 1 0^{- 23}$ J/K Boltzmann's constant hai.

Derivation sketch:

Received carrier power: $C = EIRP - L_{p} + G_{r}$
Noise power density: $N_{0} = k T_{sys}$
Ratio: $C / N_{0} = (EIRP - L_{p} + G_{r}) / (k T_{sys})$
dB mein: $G_{r}$ aur $T_{sys}$ alag karo → $G / T$ term naturally emerge hota hai

Physical meaning: G/T capture karta hai "mera receiver signal kitna achha grab karta hai versus noise generate karna?"

$\frac{G}{T} = 63 - 10 lo g_{10} (25) = 63 - 10 (1.398) = 63 - 13.98 = 49.02 dB/K$

Yeh excellent kyun hai: Deep space ground stations 40-55 dB/K achieve karte hain. Zyada G/T matlab hum door ke spacecrafts se kamzor signals detect kar sakte hain.

4. Eb/N0 — Energy Per Bit to Noise Density Ratio

Eb/N0, SNR ki jagah kyun?

Eb/N0 data rate ke liye normalize karta hai: ek slow link (fewer bits per second) same error rate ke liye fast link se kam signal power tolerate kar sakta hai
Eb/N0 information theory se directly connect hota hai — Shannon's limit, Eb/N0 par depend karta hai

Carrier-to-noise density se shuru karke: $\frac{C}{N _{0}} (ratio, not dB)$

Received carrier power $C$ , data rate $R_{b}$ (bits/second) par distribute hoti hai: $E_{b} = \frac{C}{R _{b}}$

Kyun? Har bit ko total power ka apna "share" milta hai. Faster data rate = har bit ke liye kam energy.

Isliye: $\frac{E _{b}}{N _{0}} = \frac{C}{N _{0} \cdot R _{b}}$

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

Complete scenario: Mars orbiter downlink

Transmitter power: $P_{t} = 15$ W = 11.76 dBW
Transmit antenna gain: $G_{t} = 22$ dBi
EIRP: $11.76 + 22 = 33.76$ dBW
Path loss (pehle se): $L_{p} = 278.89$ dB
Receive G/T: $49.02$ dB/K (DSN 34-m)
Boltzmann constant: $k = 1.38 \times 1 0^{- 23}$ = -228.6 dBW/K/Hz
Data rate: $R_{b} = 2$ Mbps = $10 lo g_{10} (2 \times 1 0^{6}) = 63$ dB-Hz

Step 1 — C/N0 calculate karo: $\frac{C}{N _{0}} = EIRP - L_{p} + \frac{G}{T} - (- 228.6)$ $= 33.76 - 278.89 + 49.02 + 228.6 = 32.49 dB-Hz$

228.6 kyun add kiya? dB mein, ek negative subtract karna matlab add karna hai. Yeh " $- k$ " term hai.

Step 2 — Eb/N0 calculate karo: $\frac{E _{b}}{N _{0}} = 32.49 - 63 = - 30.51 dB$

Ruko, negative?! Yeh ek problem hai! Required Eb/N0 check karte hain...

Step 3 — Requirement se compare karo:

BER = $1 0^{- 5}$ par uncoded BPSK ke liye: $E_{b} / N_{0}$ required ≈ 9.6 dB

Turbo-coded system (rate 1/2) ke liye: $E_{b} / N_{0}$ required ≈ 1-2 dB

Hamara -30.51 dB requirements se bahut neeche hai!

Kya galat hua? Realistic spacecraft EIRP ~60 dBW (100W + 40 dBi high-gain antenna) ke saath recalculate karte hain:

$\frac{C}{N _{0}} = 60 - 278.89 + 49.02 + 228.6 = 58.73 dB-Hz$ $\frac{E _{b}}{N _{0}} = 58.73 - 63 = - 4.27 dB$

Phir bhi negative! Data rate reduce karna hoga ya powerful coding use karni hogi. 100 kbps (50 dB-Hz) par: $\frac{E _{b}}{N _{0}} = 58.73 - 50 = 8.73 dB$

Ab hum coded systems ke liye ballpark mein hain. Yeh fundamental tradeoff demonstrate karta hai: distance, data rate, aur power ek dusre se ladaai mein hain.

Mistake 1: dB aur linear units mix karna ❌ Galat: $P_{r} = P_{t} (dBW) \times G_{t} (dBi) / L_{p} (dB)$ ✓ Sahi: $P_{r} (dBW) = P_{t} (dBW) + G_{t} (dBi) - L_{p} (dB)$

Yeh sahi kyun lagta hai: Linear units mein, hum gains multiply karte hain aur losses divide karte hain. Lekin dB mein, yeh addition aur subtraction ban jaate hain.

Fix: dB-land mein, multiplication → addition, division → subtraction, powers → multiplication.

Mistake 2: Losses ka sign bhool jaana ❌ Galat: $C / N_{0} = EIRP + L_{p} + G / T$ ✓ Sahi: $C / N_{0} = EIRP - L_{p} + G / T$

Yeh sahi kyun lagta hai: Losses bade positive numbers hote hain (278 dB), toh unhe add karne se budget "bada" lagta hai.

Fix: Losses, budget se subtract hote hain. Inhe losses isliye kehte hain! Apne bank account ki tarah socho: withdrawals negative hote hain.

Mistake 3: C/N0 aur Eb/N0 mein confusion

Yeh sahi kyun lagta hai: Dono "signal to noise" ratios hain similar formulas ke saath.

Key difference:

$C / N_{0}$ power per Hz hai (data rate par depend nahi karta)
$E_{b} / N_{0}$ energy per bit hai (explicitly data rate par depend karta hai)
$E_{b} / N_{0} = C / N_{0} - 10 lo g_{10} (R_{b})$ inhe connect karta hai

Mistake 4: Implementation losses ignore karna

Real systems mein aisi losses hoti hain jo budget kha jaati hain:

Pointing loss (antenna perfectly aimed nahi): 0.5-2 dB
Polarization mismatch: 0.5-3 dB
Atmospheric absorption (ground stations): 0.5-2 dB
Hardware losses (cables, filters): 1-2 dB

Fix: Ek "loss budget" section include karo. 3+ dB ke margins ensure karte hain ki link imperfections ke bawajood bhi close hoga.

Complete Link Budget Equation

Sab kuch ek saath:

$\frac{E _{b}}{N _{0}} (dB) = P_{t} (dBW) + G_{t} (dBi) - L_{p} (dB) + G_{r} (dBi) - 10 lo g_{10} (T_{sys}) + 228.6 - 10 lo g_{10} (R_{b}) - L_{misc} (dB)$

Ya zyada compact form mein: $\frac{E _{b}}{N _{0}} = EIRP - L_{p} + \frac{G}{T} + 228.6 - 10 lo g_{10} (R_{b}) - L_{misc}$

Practical Design Insights

Space Communication ka Iron Triangle:

Distance (path loss): $+ 20$ dB per 10× distance
Data rate: $+ 10$ dB per 10× rate
Transmit power/antenna: Mass, power budget, deployability se limited

Sab kuch ek saath nahi mil sakta. Mars missions choose karte hain:

Critical telemetry: Low rate (kbps), kisi bhi geometry par close hoti hai
Science data: High rate (Mbps), favorable geometry + ground station availability chahiye
Deep space (Jupiter+): Har dB matter karta hai, paas aane ke liye gravitational assists use karo

G/T, SNR se receiver comparison ke liye better kyun hai:

G/T sirf receiver ki property hai
SNR transmitter par bhi depend karta hai
Ground stations G/T quote karte hain; missions ise link budget mein directly plug kar sakte hain

Eb/N0 universal kyun hai:

Shannon limit: $\frac{E _{b}}{N _{0}} \geq \frac{2 ^{R / W} - 1}{R / W} \approx - 1.59$ dB at spectral efficiency $R / W \to 0$
Modulation schemes (BPSK, QPSK, 8PSK) sabke known $E_{b} / N_{0}$ vs BER curves hain
Coding (Reed-Solomon, turbo, LDPC) required $E_{b} / N_{0}$ ko 5-8 dB reduce karta hai

Recall Ek 12-Saal ke Bachche Ko Explain Karo

Imagine karo tum apne dost ko ek bade football field ke paar sunne ki koshish kar rahe ho. Yeh matters:

EIRP yeh hai ki tumhara dost kitna zor se chillata hai AUR kya woh megaphone use kar raha hai. Megaphone uski awaaz loud nahi karta, lekin sound ko tumhari taraf focus karta hai instead of saari directions mein waste karne ke. EIRP dono ko ek number mein combine karta hai.

Path Loss isliye hota hai kyunki awaazein door jaane par dheemin ho jaati hain. Sound energy ek badi aur badi sphere mein phail jaati hai. Jab tak woh tumhare paas pahunche, tum sirf ek tiny fraction hi catch kar rahe ho. Space mein, radio waves bhi yahi karte hain — jitna door ho, utna kamzor signal, ek specific math rule follow karte hue (do guna door = ek-chauthaayi strength).

G/T yeh hai ki tumhare kaan kitne achhe hain (G) versus kitna background noise hai (T). Bade kaan (bada antenna) zyada signal catch karte hain. Quiet background (low noise) matlab tum whispers sun sakte ho. G/T dono combine karta hai.

Eb/N0 yeh poochna jaisa hai: "Kya har word mein crowd noise ke upar sunne ke liye enough volume hai?" Agar tumhara dost bahut fast bolta hai, toh har word chhota hota hai aur pakadna mushkil. Agar woh slowly bole, toh har word clear hota hai chahe woh overall quieter hi kyun na ho.

Connections

Free-Space Path Loss Derivation — electromagnetic wave propagation
Antenna Gain and Effective Aperture — bade dishes kyun help karte hain
Noise Temperature and Noise Figure — receiver noise ko samajhna
Shannon-Hartley Theorem — data rate ki theoretical limit
Modulation Schemes (BPSK, QPSK) — spectral efficiency achieve karna
Forward Error Correction — link budgets mein coding gain
Deep Space Network (DSN) — NASA ka ground station infrastructure
Fris Transmission Equation — alternative link budget formulation

#flashcards/physics

EIRP kya hai aur iska calculation kaise hota hai? :: EIRP (Effective Isotropic Radiated Power) woh power hai jo ek isotropic antenna ko radiate karni padegi taaki directional antenna ke main beam mein same power density produce ho. EIRP = Pt × Gt (linear) ya EIRP(dBW) = Pt(dBW) + Gt(dBi) decibels mein.

Path loss inverse square law kyun follow karta hai?

Jaise ek radio wave transmitter se spherically expand hoti hai, wahi total power surface area 4πr² par phail jaati hai. Power per unit area (intensity) isliye 1/r² ke hisaab se decrease hoti hai, jisse path loss r² ke proportional hoti hai.

Free-space path loss formula kya hai?

Lp(dB) = 32.45 + 20log₁₀(r_km) + 20log₁₀(f_MHz), jahaan r distance km mein hai aur f frequency MHz mein hai. Path loss distance aur frequency dono ke saath increase hoti hai.

G/T ratio kya represent karta hai?

G/T ek receiver figure of merit hai jo antenna gain G (signal collect karne ki ability) aur system noise temperature T (internal noise) combine karta hai. (G/T)(dB/K) = Gr(dBi) - 10log₁₀(Tsys). Zyada G/T matlab better receiver sensitivity.

Eb/N0 kya hai aur iska use kyun hota hai?

Eb/N0 energy per bit to noise power spectral density ratio hai. Iska use isliye hota hai kyunki yeh data rate ke liye normalize karta hai — same Eb/N0 same bit error rate deta hai chahe tum 1 kbps par transmit karo ya 1 Mbps par. Eb/N0(dB) = (C/N0)(dB-Hz) - 10log₁₀(Rb).

Data rate required signal power ko kaise affect karta hai?

Zyada data rate ke liye same Eb/N0 maintain karne ke liye proportionally zyada power chahiye. Data rate double karna (+3 dB-Hz) error rate constant rakhne ke liye received power double karna (+3 dB) require karta hai.

Carrier-to-noise density ratio C/N0 kya hai?

C/N0 received carrier power divided by noise power spectral density (W/Hz) hai. C/N0(dB-Hz) = EIRP - Lp + G/T + 228.6, jahaan 228.6 = -10log₁₀(k) aur k Boltzmann's constant hai.

C/N0 equation mein 228.6 kyun add hota hai?

228.6 dB -10log₁₀(k) se aata hai jahaan k = 1.38×10⁻²³ J/K Boltzmann's constant hai. Kyunki noise power density N0 = kT hai, hum k se divide kar rahe hain, jo dB arithmetic mein -228.6 subtract karne ka matlab hai (yaani 228.6 add karna).

Frequency aur path loss ka kya relationship hai?

Path loss frequency ke saath increase hoti hai: 10× frequency increase par +20 dB. Higher frequencies ki wavelengths chhoti hoti hain, isliye same antenna aperture expanding wavefront ka chhota fraction hi capture karta hai.

Link budgets mein, kya losses add ya subtract hoti hain?

Losses (path loss, cable loss, atmospheric loss) dB arithmetic mein subtract hoti hain. Chahe loss values positive numbers hon (jaise 280 dB), woh power mein reduction represent karti hain, isliye woh received signal se subtract hoti hain.