Power systems — solar arrays (I-V curve, power tracking), batteries (DoD, cycles), RTG

3.6.22 · Physics › Spacecraft Structures & Systems Engineering

Overview

Spacecraft power systems energy (solar, nuclear, chemical) ko electrical power mein convert karte hain — spacecraft ke sabhi subsystems ke liye. Teen primary architectures hain: solar photovoltaic arrays (LEO/GEO ke liye sabse common), rechargeable batteries (eclipse periods ke liye), aur radioisotope thermoelectric generators (RTG, deep-space missions ke liye). I-V curves of solar cells, maximum power point tracking (MPPT), depth-of-discharge (DoD) constraints, aur RTG decay kinetics ko samajhna mission power budget design ke liye critical hai.

Solar Photovoltaic Arrays

The I-V Characteristic Curve

YEH SHAPE KYUN? Ek solar cell ek diode plus ek current source hai. Photocurrent $I_{L}$ (light-generated) constant hoti hai, lekin diode ka ek exponential I-V relation hota hai. Dono ko combine karke:

First principles se derivation:

Photon absorption → carrier generation: $E > E_{g}$ (bandgap) wale photons electron-hole pairs create karte hain. Absorbed photons ka flux current $I_{L} \propto$ light intensity deta hai.
P-N junction diode behavior: Light ke bina, ek diode $I = I_{0} (e^{V / n V_{T}} - 1)$ conduct karta hai. Yeh depletion region ke across minority carrier diffusion se Shockley equation hai.
Superposition: Solar cell ek current source $I_{L}$ hai jo ek diode ke parallel mein hai. Kirchoff use karke: $I_{o u t} = I_{L} - I_{d i o d e}$ .
Non-idealities: Real cells mein series resistance $R_{s}$ (wire/contact resistance) aur shunt resistance $R_{s h}$ (edge defects) hoti hai. Simplicity ke liye upar, ideal case mein $R_{s} \approx 0$ aur $R_{s h} \to \infty$ hai.

Power output:

P (V) = V \cdot I (V) = V [I_{L} - I_{0} (e^{\frac{V}{n V _{T}}} - 1)]

Short circuit par ( $V = 0$ ): $I = I_{L} = I_{sc}$ .
Open circuit par ( $I = 0$ ): $V = V_{oc} \approx n V_{T} ln (\frac{I _{L}}{I _{0}} + 1)$ .

Fill factor $F F$ quantify karta hai ki curve kitni "square" hai:

F F = \frac{V _{m p} \cdot I _{m p}}{V _{oc} \cdot I _{sc}} = \frac{P _{ma x}}{V _{oc} \cdot I _{sc}}

Typical silicon cells: $F F \approx 0.7-0.85$ . Triple-junction GaAs cells (space mein use hoti hain): $F F \approx 0.85-0.88$ .

Figure — Power systems — solar arrays (I-V curve, power tracking), batteries (DoD, cycles), RTG

Diya hua: $I_{L} = 5 A$ , $I_{0} = 1 0^{- 12} A$ , $n = 1.5$ , $T = 300 K$ (toh $V_{T} = 0.026 V$ ), ideal cell ( $R_{s} = 0, R_{s h} = \infty$ ).

Find karein: $V_{m p}$ , $I_{m p}$ , $P_{ma x}$ .

Solution:

Power hai $P = V I = V [I_{L} - I_{0} (e^{V / n V_{T}} - 1)]$ .
Maximize karne ke liye, set karein $\frac{d P}{d V} = 0$ :

\frac{d P}{d V} = I + V \frac{d I}{d V} = 0

Compute karein $\frac{d I}{d V} = - \frac{I _{0}}{n V _{T}} e^{V / n V_{T}}$ .
Substitute karein:

I_{L} - I_{0} (e^{V / n V_{T}} - 1) - V \cdot \frac{I _{0}}{n V _{T}} e^{V / n V_{T}} = 0

Yeh transcendental hai; numerically solve karein ya approximate karein. Is example ke liye, $V_{oc} \approx 1.5 \times 0.026 ln (5 \times 1 0^{12}) \approx 1.15 V$ .
Numerical solution deta hai $V_{m p} \approx 1.0 V$ , $I_{m p} \approx 4.9 A$ , $P_{ma x} \approx 4.9 W$ .
$F F = \frac{4.9}{1.15 \times 5} \approx 0.85$ .

Yeh step kyun? Derivative condition $d P / d V = 0$ woh voltage find karta hai jahan incremental voltage increase, current decrease ke saath power ke liye optimally trade-off hota hai.

Maximum Power Point Tracking (MPPT)

ZARURAT KYUN? MPP shift hota hai:

Temperature ke saath: Zyada $T$ se $V_{oc}$ kam hota hai (kyunki $V_{oc} \propto ln (intensity / I_{0} (T))$ aur $I_{0}$ , $T$ ke saath exponentially badhta hai). Typical coefficient: per cell $- 2.2 mV/°C$ .
Illumination ke saath: Kam intensity se $I_{sc}$ proportionally ghatta hai lekin $V_{oc}$ logarithmically.
Degradation ke saath: Space mein radiation damage saalon mein $I_{L}$ reduce karta hai aur $I_{0}$ badhata hai.

MPPT KAISE KAAM KARTA HAI (Perturb-and-Observe algorithm):

Current $P_{k} = V_{k} I_{k}$ measure karo.
Voltage perturb karo: $V_{k + 1} = V_{k} + Δ V$ (chhota step, e.g., 0.5 V).
Naya power $P_{k + 1}$ measure karo.
Agar $P_{k + 1} > P_{k}$ , usi direction mein chalte raho ( $Δ V$ same sign).
Agar $P_{k + 1} < P_{k}$ , direction reverse karo ( $Δ V \to - Δ V$ ).
~10–100 Hz par repeat karo.

Alternative: Incremental conductance ( $\frac{d I}{d V} = - \frac{I}{V}$ at MPP). Zyada efficient lekin zyada complex.

I-V par effect:

$I_{sc}$ ghat ke $I_{sc, 0} \times 0.866$ ho jaata hai.
$V_{oc}$ $n V_{T} ln (0.866) \approx - 1.5 \times 26 \times 0.144 \approx - 5.6 mV$ ghat jaata hai (chhota).
$P_{ma x}$ $\approx 13%$ ghat jaata hai.

MPPT response: Algorithm power drop detect karta hai, naya $V_{m p}$ search karta hai (thoda kam voltage, bahut kam current), lock on karta hai.

Yeh sahi kyun lagta hai: Batteries ~constant voltage loads hoti hain (e.g., Li-ion ~3.7 V/cell). Agar battery voltage $V_{m p}$ se match kare, toh max power milega.

Fix: Reality mein, $V_{m p}$ temperature/angle/age ke saath drift karta hai, aur battery voltage state-of-charge ke saath change hoti hai (Li-ion ke liye 3.0–4.2 V). Ek DC-DC converter (buck-boost) with MPPT, array aur battery ke beech mein hota hai, array ko $V_{m p}$ par operate karne ke liye force karta hai jabki battery ko jo bhi voltage chahiye woh supply karta hai.

Batteries for Eclipse Power

Depth of Discharge (DoD)

KYUN MATTER KARTA HAI: Zyada DoD → kam lifetime cycles. Yeh relationship empirical hai (accelerated life testing se):

Derivation logic: Har charge/discharge cycle cause karta hai:

Solid-electrolyte interphase (SEI) growth anode par (irreversible Li consumption).
Cathode structure degradation (transition metal dissolution).
Electrolyte decomposition (especially high voltages par, yaani near full charge).

Zyada deep discharge → zyada voltage swing → zyada side reactions → faster capacity fade.

Agar hum 80% DoD allow karein:

Usable energy: $560 \times 0.8 = 448 Wh$ (kaafi hai).
Cycle life: ~2500 cycles.
Mission duration: $2500/ (365.25 \times 365.25/90 min \times 1440/90) \approx 2500/5840 \approx 0.43$ years (6 months).

Agar hum 30% DoD tak limit karein:

Usable: $560 \times 0.3 = 168 Wh$ (enough nahi!).
Badi battery chahiye: $290/0.3 = 967 Wh$ , ya 28 V par 34.5 Ah.
Cycle life: ~15,000 cycles → ~2.6 years.

Yeh step kyun? Hum battery mass vs. lifetime trade karte hain. 5-year mission ke liye, aap 30–40% DoD ke liye design karenge (aur battery accordingly size karenge), mass penalty accept karte hue.

Battery Technologies in Space

Type	Energy Density	Cycle Life (30% DoD)	Notes
NiH₂	60 Wh/kg	40,000+	Legacy (Hubble, ISS); low DoD tolerance lekin bahut robust
Li-ion	120–180 Wh/kg	10,000–15,000	Modern standard; thermal manage karna zaroori (runaway risk)
Solid-state Li	200+ Wh/kg (future)	TBD	Development mein; safer, higher density

Thermal management: Li-ion 0–40°C par operate karta hai. Space extreme hai (±150°C swing). Heaters + radiators 15–25°C maintain karte hain.

Recall

Ek 12-saal ke bacche ko explain karo: Socho solar panel ek nadi mein water wheel ki tarah hai. Agar paani freely flow karne do (short circuit), wheel fast ghoomti hai lekin zyada energy nahi milti kyunki koi resistance nahi hai. Agar puri tarah block kar do (open circuit), pressure build up hota hai lekin kuch nahi hilta — phir bhi, koi energy nahi. Trick yeh hai ki bilkul sahi amount ka load lagao (jaise wheel se attached ek grain mill) taaki sabse zyada grinding ho sake. Yahi "maximum power point" hai. Space mein, ek computer continuously "mill size" tweak karta rehta hai taaki sun angle aur temperature change hone par bhi sabse zyada power extract hoti rahe. Batteries ek backup bucket ki tarah hain jo sun shine karte waqt bharta hai aur Earth sun block karne par (eclipse) khali hota hai. Lekin agar har baar bucket zyada drain karo, toh bottom crack ho jaata hai aur aap isse kuch sau baar hi refill kar sakte ho pehle ki woh toot jaaye. Toh engineers battery ka sirf kuch hissa (30–40%) use karte hain taaki woh saalon tak chale.

Radioisotope Thermoelectric Generators (RTG)

Power Output Over Time

Electrical power:

P_{e l ec} (t) = η (t) \cdot P_{t h} (t)

jahan $η$ thermoelectric conversion efficiency hai (typically beginning-of-life par 6–7%).

EFFICIENCY KYUN GIRTI HAI: Thermoelectric materials degrade hoti hain (Te ka sublimation, radiation damage). Empirical model:

η (t) = η_{0} e^{- t / τ}

jahan $τ \approx 15-20$ years. Combine karke:

P_{e l ec} (t) = η_{0} P_{0} \cdot 2^{- t /87.7} \cdot e^{- t / τ}

Yeh step kyun? Do decay processes compound hote hain: isotope half-life (slow) aur TE degradation (faster). Mission design ensure karna chahiye ki end-of-life par minimum power load se zyada rahe.

Seebeck Effect (Thermoelectric Conversion)

Carrier statistics se derivation:

Hot side mein carriers ki energy zyada hoti hai → cold side ki taraf zyada diffusion.
Charge imbalance electric field create karta hai.
Equilibrium par, $e V = Δ μ$ (electrochemical potential difference).
Metals/semiconductors ke liye, $μ \propto k_{B} T ln (n)$ , toh $V \propto T$ (simplified).

Efficiency (Carnot-limited):

η_{ma x} = \frac{T _{h} - T _{c}}{T _{h}} \cdot \frac{1 + Z T - 1}{1 + Z T + T _{c} / T _{h}}

jahan $Z T$ thermoelectric figure of merit hai (dimensionless):

Z T = \frac{S ^{2} σ T}{κ}

$σ$ = electrical conductivity.
$κ$ = thermal conductivity.

EFFICIENCY KAM KYUN? High $σ$ chahiye (good electrical conductor) lekin low $κ$ (poor thermal conductor) — yeh contradictory hai. Best materials (Bi₂Te₃, SiGe, skutterudites): $Z T \approx 1-1.5$ → $η \approx 6-10%$ .

Yeh sahi kyun lagta hai: 87.7-year half-life zyaadatar missions se lambi hai. Batteries ki tarah cycles se koi degradation nahi.

Fix:

Cost: $^{238} Pu$ scarce hai (reactors mein produce hota hai, globally ~10 kg/yr). Ek MMRTG ki cost ~$100M hai. Sirf unhi missions ke liye jahan solar impossible ho.
Mass: 110 W ke liye 45 kg (0.4 kg/W). Solar arrays: ~0.02 kg/W (1 AU par 20× halka per watt).
Safety: Launch approval ke liye extensive containment analysis chahiye (though $^{238} Pu$ alpha-only hai, gamma nahi, toh shielding manageable hai).

Power Budget and System Design

Sunlight charge balance:

P_{solar} \geq P_{load,sun} + \frac{E _{battery}}{t _{charge}}

Sizing example (GEO satellite, 5-year mission):

Eclipse: 72 min, saal mein do baar (equinox).
Load: 3 kW average.
Battery: Li-ion, 30% DoD target.
Energy per eclipse: $3 \times 1.2 = 3.6 kWh$ .
Battery size: $3.6/0.3 = 12 kWh$ (28 V par 430 Ah).
Solar array (end-of-life par, 20% degradation ke saath): $(3 + 3.6/ (23.8/60)) \times 1.2/0.8 \approx 16.4 kW$ BOL.
Array area (30% efficiency, 1367 W/m²): $16.4/ (0.3 \times 1.367) \approx 40 m^{2}$ .

Connections

Spacecraft Thermal Control — batteries aur power electronics heat generate karte hain; solar arrays ko back side par radiators chahiye.
Orbital Mechanics (Keplerian) — eclipse duration orbit altitude aur inclination par depend karta hai; $β$ -angle seasonal solar flux determine karta hai.
Semiconductor Physics — p-n junctions, Fermi levels, carrier diffusion solar cells ki neenv hain.
Thermodynamics & Heat Transfer — Carnot efficiency, Seebeck effect, blackbody radiation (RTG radiator sizing).
Radiation Effects on Materials — solar cells mein displacement damage (coverglass thickness vs. mass trade), TE material degradation.
Power Electronics — DC-DC converters, MPPT algorithms, battery charge controllers.

#flashcards/physics

Solar cell ka maximum power point (MPP) kya hota hai? :: Woh voltage aur current pair $(V_{m p}, I_{m p})$ jahan I-V curve par product $P = V \cdot I$ maximize hota hai; typically jahan $d P / d V = 0$ hota hai.

Solar cell ka open-circuit voltage

V_{oc}

temperature ke saath kyun kam hota hai?

Zyada temperature dark saturation current

I_{0}

ko exponentially badhata hai. Kyunki

V_{oc} \propto n V_{T} ln (I_{L} / I_{0})

I_{0}

mein increase se

V_{oc}

reduce hota hai (typical coefficient: per cell

- 2.2

mV/°C).

MPPT (Maximum Power Point Tracking) kaise kaam karta hai? :: MPPT algorithms (e.g., Perturb-and-Observe) operating voltage ko continuously adjust karte hain — power measure karke, voltage perturb karke, aur increasing power ki direction mein move karke — taaki conditions change hone par bhi MPP par bana rahe.

Depth of discharge (DoD) kya hai aur spacecraft batteries ke liye yeh kyun matter karta hai?

DoD discharged total battery capacity ka percentage hai. Zyada DoD se cycle life reduce hoti hai — increased electrode stress aur side reactions ki wajah se; typical space design 10,000+ cycles achieve karne ke liye 30–40% DoD use karta hai mission life ke dauran.

Li-ion batteries ke liye DoD aur cycle life ke beech empirical relationship kya hai?

N_{cycles} \approx N_{0} (D oD / D o D_{0})^{- k}

jahan

k \approx 1.5-2

; DoD ko 40% se 80% par double karne se cycle life roughly aadhi ho jaati hai.

RTG power output time ke saath kaise change hota hai?

RTG power do processes se decay hoti hai: isotope decay

P_{t h} (t) = P_{0} \cdot 2^{- t / t_{1/2}}

(half-life 87.7 yr for

^{238} Pu

) aur thermoelectric efficiency degradation

η (t) = η_{0} e^{- t / τ}

(τ ≈ 15–20 yr), combined as

P_{e l ec} (t) = η_{0} P_{0} \cdot 2^{- t /87.7} \cdot e^{- t / τ}

Seebeck effect kya hai?

Seebeck effect ek conductor ya semiconductor ke across voltage

V = S Δ T

ka generate hona hai jab temperature gradient

Δ T

apply hota hai — hot se cold side ki taraf charge carriers ke diffusion ki wajah se.

RTGs heat ko electricity mein convert karne mein sirf ~6–7% efficient kyun hain?

Thermoelectric conversion ke liye high electrical conductivity

σ

lekin low thermal conductivity

κ

chahiye, jo contradictory hain. Best materials

Z T \approx 1

achieve karte hain, jo Carnot-adjusted efficiency ko 6–10% tak limit karta hai.

Solar cell ka fill factor (FF) kya hai? :: $F F = \frac{V _{m p} I _{m p}}{V _{oc} I _{sc}} = \frac{P _{ma x}}{V _{oc} I _{sc}}$ , yeh measure karta hai ki I-V curve kitni "square" hai; space-grade cells ke liye typical values 0.7–0.88 hain.

Spacecraft batteries ko thermally manage kyun karna padta hai?

Li-ion batteries sirf 0–40°C range mein safely operate karti hain; is range se bahar thermal runaway ka risk hai. Space environment ±150°C swing karta hai, jiske liye 15–25°C maintain karne ke liye heaters aur radiators chahiye.