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

3.6.22 · D2 · Physics › Spacecraft Structures & Systems Engineering › Power systems — solar arrays (I-V curve, power tracking), ba

Ye page parent note ka central result — I-V curve aur uska maximum power point — ek-ek picture ke through rebuild karta hai. Hum assume karte hain ki tumhe diodes ya solar cells ke baare mein kuch bhi nahi pata. Har symbol ko pehle samjhaya jayega, phir use kiya jayega. End tak tum dekh paoge ki curve ki shape aisi kyun hai aur power ke liye exactly ek hi sweet spot kyun hota hai.

Parent: 3.6.22-Power-systems-—-solar-arrays-(I-V-curve,-power-tracking),-batteries-(DoD,-cycles),-

Step 1 — "Current" aur "voltage" ka matlab yahan kya hai

KYA HAI. Ek solar cell ke do terminals hote hain — inhe do metal tabs samjho. Do numbers batate hain ki un tabs par kisi bhi instant mein kya ho raha hai:

Current $I$ — har second kitne charged particles baahir nikalte hain, amps (A) mein measure hota hai. Ek nadi socho: current matlab kitna paani ek point se har second guzarta hai.
Voltage $V$ — do tabs ke beech electrical "push," volts (V) mein measure hota hai. Ek waterfall ki unchaai socho: jitna bada girna, utna zyada push.

Power $P = V \cdot I$ woh hai jo humein ultimately chahiye (measured in watts, W) — yeh push times flow hai, spacecraft ko har second mein actually deliver hone wali energy.

YE DO KYUN. Ek solar array real hardware ko feed karta hai. Hardware ko power chahiye, aur power in exactly inhi do quantities ka product hai. Toh poora game yeh hai: diye gaye sunlight ke liye, kaun sa pair $(V, I)$ sabse bada product $V \cdot I$ deta hai?

PICTURE. Cell ek black box ki tarah hai jiske do tabs hain, ek ammeter $I$ read kar raha hai, ek voltmeter $V$ read kar raha hai.

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

Step 2 — Andheron mein cell sirf ek diode hai

KYA HAI. Cell ko dhak do taaki koi light na pade. Ab yeh ek plain semiconductor diode ki tarah behave karta hai: current ke liye ek one-way valve. Iska current-voltage law Shockley equation hai:

I_{diode} = tiny leak I_{0} (e^{V / (n V_{T})} - 1)

Har symbol samjhte hain:

$I_{0}$ — dark saturation current. Ek tiny trickle (yahaan $\sim 1 0^{- 12} A$ ) jo bina kisi push ke bhi backward leak hoti hai. Ek thoda leaky valve socho.
$V_{T} = \frac{k _{B} T}{q}$ — thermal voltage. Yeh "scale" set karta hai ki diode voltage ke baare mein kitna care karta hai. $k_{B}$ Boltzmann's constant hai (temperature ko energy se link karta hai), $T$ kelvin mein temperature hai, $q$ electron charge hai. Room temperature par ( $T = 300 K$ ) yeh lagbhag $26 mV$ hota hai.
$n$ — ideality factor, 1 aur 2 ke beech ek fudge number jo real (imperfect) junctions ke liye voltage scale ko stretch karta hai.
$e^{(\cdot)}$ — exponential function. Exponential kyun, straight line kyun nahi?

1 kyun subtract karte hain. $V = 0$ par (koi push nahi) net current zero honi chahiye. $V = 0$ plug karo: $e^{0} - 1 = 1 - 1 = 0$ . " $- 1$ " exactly isliye hai taaki push off hone par current zero ho jaye.

PICTURE. Classic diode curve: negative/small $V$ ke liye flat aur near-zero, phir jaise hi $V$ ek threshold ke paas pahunchta hai steeply upar ki taraf.

Step 3 — Light on karo: ek constant current source appear hoti hai

KYA HAI. Ab cell par sunlight daalo. Energy-wise material ke bandgap $E_{g}$ se zyada energy wale photons (light ke particles) electrons ko loose kar dete hain, freed charges ka ek steady flow create karte hain. Yeh hai photocurrent $I_{L}$ .

I_{L} \propto (light intensity)

$I_{L}$ — light-generated current. Sunlight double karo, $I_{L}$ double ho jayega. Yeh voltage ki zyada parwah nahi karta — yeh ek near-constant current source hai, jaise ek pump jo paani ki uthaan ki unchaai se befikar hamesha utni hi water push karta hai.

CONSTANT KYUN. Har second ek fixed number of photons aate hain aur har ek (zyada se zyada) ek electron-hole pair free karta hai. Woh freeing rate light se set hoti hai, terminal voltage se nahi. Isliye $I_{L}$ poore $V$ axis par flat hai.

PICTURE. Height $I_{L}$ par ek flat horizontal line — photocurrent — Step 2 ke (ab ulte-upar-kiye gaye) diode curve ke upar baithi hui.

Step 4 — Superposition: source se diode subtract karo

KYA HAI. Real solar cell current source $I_{L}$ hai jo diode ke parallel mein hai. Kirchhoff's current law se (charge conserved hoti hai — jo andar aata hai woh baahir bhi nikalna chahiye), terminals par jo current actually milती है woh hai:

I = light gives I_{L} - diode eats I_{0} (e^{V / (n V_{T})} - 1)

Term by term padhte hain:

$I_{L}$ current baahir load ki taraf push karta hai — achha wala.
Diode term internally current chura leta hai, aur yeh $V$ ke saath exponentially badhta hai. Kam $V$ par yeh almost kuch nahi churaata; high $V$ par yeh sab kuch khaa jaata hai.

SUBTRACT KYUN. Light source aur diode same do terminals share karte hain lekin opposite ways mein point karte hain. Charge conservation force karta hai ki output difference ho. Yeh ideal single-diode model hai — parent note ki central equation (shunt/series resistance idealized away ke saath, $R_{s h} \to \infty$ , $R_{s} \to 0$ ).

PICTURE. Flat $I_{L}$ line lo aur uske right end ko exploding diode term se neeche bend karo — jaana-pehchana solar-cell curve saamne aa jaata hai.

Step 5 — Do anchor points padhna: $I_{sc}$ aur $V_{oc}$

KYA HAI. Do special operating points curve ke ends pin down karte hain.

Short circuit — tabs ko wire se connect karo taaki $V = 0$ :

I_{sc} = I_{L} - I_{0} (e^{0} - 1) = I_{L} - 0 = I_{L}

$I_{sc}$ — short-circuit current, sabse zyada current. Koi push nahi, diode kuch nahi churaata, toh saari photocurrent baahir aa jaati hai.

Open circuit — tabs ko disconnect chhod do taaki $I = 0$ . Poore expression ko zero set karo aur $V$ ke liye solve karo:

0 = I_{L} - I_{0} (e^{V_{oc} / (n V_{T})} - 1) ⟹ V_{oc} = n V_{T} ln (\frac{I _{L}}{I _{0}} + 1)

$V_{oc}$ — open-circuit voltage, sabse zyada voltage. Logarithm $ln$ Step 2 ke exponential ka exact inverse hai — yeh answer deta hai "kaun sa voltage diode ko saari photocurrent khilaa deta hai?"

YE DO KYUN IMPORTANT HAIN. $I_{sc}$ top-left corner hai, $V_{oc}$ bottom-right corner hai. Har real operating point inke beech curve par rehta hai. Dhyaan do: $V = 0$ par power zero hai (koi push nahi), aur $I = 0$ par power zero hai (koi flow nahi). Dono extremes zero watts dete hain — toh best point beech mein kaheen hona chahiye. Yeh Step 6 setup karta hai.

PICTURE. Poora curve jisme current axis par $I_{sc}$ mark hai, voltage axis par $V_{oc}$ mark hai, aur dono endpoints " $P = 0$ " label kiye gaye hain.

Recall

Short circuit par, output current kitni hoti hai? ::: $I_{sc} = I_{L}$ (diode $V = 0$ par kuch nahi churaata). $V_{oc}$ ka formula? ::: $V_{oc} = n V_{T} ln (I_{L} / I_{0} + 1)$ .

Step 6 — Power curve aur uska single peak (MPP)

KYA HAI. Har point par height times width multiply karo: $P (V) = V \cdot I (V)$ . $P$ ko $V$ ke against plot karo. Yeh $0$ se start hota hai (at $V = 0$ ), ek single hump tak badhta hai, phir wapas $0$ par crash karta hai (at $V = V_{oc}$ ).

Hump ka top maximum power point (MPP) hai, coordinates $(V_{m p}, I_{m p})$ par.

DERIVATIVE KYUN. Ek hill ka exactly top dhundhne ke liye hum poochhte hain: slope flat kahan hai? Derivative $\frac{d P}{d V}$ hi power curve ka slope hai. Peak par slope zero hota hai:

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

Yeh tool kyun, guess kyun nahi? Kyunki "slope $= 0$ " "aur upar nahi ja sakte" ka precise mathematical statement hai. Product rule se expand karo ( $P = V \cdot I$ ):

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

$I$ — current jo abhi bhi hai.
$V \frac{d I}{d V}$ — jaise voltage badhate hain current kitni tezi se kum hoti hai ( $\frac{d I}{d V} < 0$ falling shoulder par).
Inhe equal aur opposite set karna matlab: zyada voltage se gain exactly kam current se hone wale loss ko cancel karta hai. Woh balance point hi MPP hai.

PICTURE. I-V curve aur P-V hump same voltage axis par; ek vertical dashed line power peak se gir kar $V_{m p}$ mark karti hai, aur across $I_{m p}$ tak. I-V curve ke neeche sabse bada inscribed rectangle area $P_{ma x}$ hai.

Step 7 — Fill factor: curve kitna "square" hai?

KYA HAI. Real best rectangle $V_{m p} I_{m p}$ ko ideal (unreachable) rectangle $V_{oc} I_{sc}$ se compare karo:

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

Numerator — actual best power (real rectangle).
Denominator — "dream" power agar curve perfect right angle hoti.
$F F$ — 0 aur 1 ke beech ek number. 1 ke jitna paas, utna squarer curve ka shoulder aur MPP corner ke utne paas.

USEFUL KYUN. Ek number curve quality capture karta hai. Space-grade triple-junction GaAs cells $F F \approx 0.85$ – $0.88$ tak pahunche hain; ordinary silicon $0.7$ – $0.85$ par baithi hai.

PICTURE. Do nested rectangles — chota solid wala ( $V_{m p} I_{m p}$ ) bade dashed wale ( $V_{oc} I_{sc}$ ) ke andar — unka area ratio hi $F F$ hai.

Step 8 — Edge aur degenerate cases (reader kabhi curve se neeche na gire)

Har corner case, apne reasoning ke saath:

Poora andhera / eclipse ( $I_{L} = 0$ ). Source line axis par gir jaati hai; poora curve Step 2 ke dark diode par collapse ho jaata hai. Output power zero hai — exactly yahi woh waqt hai jab batteries ko over lena padhta hai. Eclipse timing ke liye Orbital Mechanics (Keplerian) dekho.
Hotter cell (temperature badhti hai). $I_{0}$ $T$ ke saath exponentially badhta hai (zyada electrons thermally gap jump karte hain). Kyunki $V_{oc} \propto ln (I_{L} / I_{0})$ , bada $I_{0}$ $V_{oc}$ ko shrink karta hai (lagbhag $- 2.2 mV/^{\circ} C$ per cell). Poora curve baayein slide karta hai, aur $P_{ma x}$ gir jaata hai — ek Spacecraft Thermal Control problem. Meanwhile $I_{sc}$ thoda sa upar jaata hai.
Dimmer sun (off-angle, intensity $\times cos β$ ). $I_{sc} = I_{L}$ intensity ke saath linearly girta hai, lekin $V_{oc}$ sirf logarithmically girta hai (intensity ratio ka ek $ln$ ) — toh voltage almost unchanged rehta hai jabki current sag jaata hai. MPP neeche aur thoda baayein move karta hai.
Reverse bias ( $V < 0$ ). Wrong way push karo aur diode term $e^{V / (n V_{T})} \to 0$ , toh $I \to I_{L} + I_{0} \approx I_{L}$ : cell abhi bhi $\approx I_{L}$ deliver karta hai lekin ab power absorb karta hai ( $P = V I < 0$ ). Yeh danger hai jo ek shaded cell series string mein face karta hai — yeh ek heat-dumping load ban jaata hai (Radiation Effects on Materials aur hotspot concerns).
Radiation ageing. Saalon ki space radiation $I_{L}$ ko cut karti hai (kam collectable carriers) aur $I_{0}$ badhaa deti hai (zyada leakage). Dono curve ko shrink karte hain — MPP wander karta hai, yahi precisely woh reason hai ki array ko active MPPT tracking chahiye na ki ek fixed operating point.

PICTURE. Overlaid curves ka ek panel: baseline (magenta), hot (orange, left shifted), dim (violet, shorter), aur aged (navy, shrunk) — dikhata hai ki MPP har case mein kaise migrate karta hai.

Ek-picture summary

Sab kuch ek single figure par: flat photocurrent line ( $I_{L}$ ) minus exploding diode term solar-cell curve produce karta hai; uske endpoints $I_{sc}$ aur $V_{oc}$ hain; uske neeche sabse bada rectangle curve ko MPP $(V_{m p}, I_{m p})$ par touch karta hai; aur us rectangle ka $V_{oc} \times I_{sc}$ box se ratio fill factor hai.

Recall Feynman retelling (seedha bolte hain)

Andheron mein ek solar cell ek one-way valve hai — ek diode — jo current tabhi jaane deta hai jab tum kaafi zor se push karo, aur yeh explosively karta hai (ek exponential, kyunki charges ko ek hill chadhni padti hai aur sirf energetic wale hi kar paate hain). Uspe light daalo aur tum ek steady pump add karte ho, photocurrent $I_{L}$ , jo ek fixed amount of current push karta hai chahe kuch bhi ho. Jo current actually milti hai woh pump minus woh hai jo valve secretly peeta hai: $I = I_{L} - I_{0} (e^{V / n V_{T}} - 1)$ . Terminals short karo aur valve kuch nahi peeta, toh saara pump milta hai: $I_{sc} = I_{L}$ . Terminals open karo aur voltage tab tak chadhta hai jab tak valve poora pump nahi pee leta, giving $V_{oc} = n V_{T} ln (I_{L} / I_{0})$ . Power push times flow hai, aur dono ends par zero hai — toh best spot beech mein ek hump hai. Us hump ka flat top dhundho ( $d P / d V = 0$ ) aur tumne maximum power point dhundh liya. Curve kitna square hai — fill factor — batata hai ki woh best rectangle dream rectangle ke kitna paas pahunchti hai. Heat, shade, wrong angle, aur age sab hump ko idhar-udhar dhakelti hain, isliye ek spacecraft kabhi MPP ek baar nahi nailta aur bhool jaata: woh hamesha iske liye hunt karta rehta hai.

Prerequisites & connections

Exponential/diode physics: Semiconductor Physics
Boltzmann factor & thermal voltage: Thermodynamics & Heat Transfer
$V_{oc}$ temperature drop: Spacecraft Thermal Control
Eclipse timing (jab $I_{L} \to 0$ ): Orbital Mechanics (Keplerian)
MPPT converters jo moving MPP chase karte hain: Power Electronics
Long-term curve shrink: Radiation Effects on Materials