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

Overview

Spacecraft power systems convert energy (solar, nuclear, chemical) into electrical power for all spacecraft subsystems. The three primary architectures are: solar photovoltaic arrays (most common for LEO/GEO), rechargeable batteries (for eclipse periods), and radioisotope thermoelectric generators (RTG, for deep-space missions). Understanding the I-V curves of solar cells, maximum power point tracking (MPPT), depth-of-discharge (DoD) constraints, and RTG decay kinetics is critical for mission power budget design.

Solar Photovoltaic Arrays

The I-V Characteristic Curve

WHY this shape? A solar cell is a diode plus a current source. The photocurrent $I_L$ (light-generated) is constant, but the diode has an exponential I-V relation. Combining:

Derivation from first principles:

1. Photon absorption → carrier generation: Photons with $E > E_g$ (bandgap) create electron-hole pairs. The flux of absorbed photons givesocurrent $I_L \propto$ light intensity.
2. P-N junction diode behavior: Without light, a diode conducts $I = I_0 (e^{V/nV_T} - 1)$. This is the Shockley equation from minority carrier diffusion across the depletion region.
3. Superposition: The solar cell is a current source $I_L$ in parallel with a diode. Using Kirchoff: $I_{out} = I_L - I_{diode}$.
4. Non-idealities: Real cells have series resistance $R_s$ (wire/contact resistance) and shunt resistance $R_{sh}$ (edge defects). For simplicity above, $R_s \approx 0$ and $R_{sh} \to \infty$ in the ideal case.

Power output:

At short circuit ($V=0$): $I = I_L = I_{sc}$.
At open circuit ($I=0$): $V = V_{oc} \approx nV_T \ln\left(\frac{I_L}{I_0} + 1\right)$.

The fill factor $FF$ quantifies how "square" the curve is:

Typical silicon cells: $FF \approx 0.7–0.85$. Triple-junction GaAs cells (used in space): $FF \approx 0.85–0.88$.

Given: $I_L = 5 \text{ A}$, $I_0 = 10^{-12} \text{ A}$, $n=1.5$, $T=300 \text{ K}$ (so $V_T = 0.026 \text{ V}$), ideal cell ($R_s=0, R_{sh}=\infty$).

Find: $V_{mp}$, $I_{mp}$, $P_{max}$.

Solution:

1. Power is $P = V I = V [I_L - I_0(e^{V/nV_T}-1)]$.
2. To maximize, set $\frac{dP}{dV} = 0$:

3. Compute $\frac{dI}{dV} = -\frac{I_0}{nV_T} e^{V/nV_T}$.
4. Substitute:

5. This is transcendental; solve numerically or approximate. For this example, $V_{oc} \approx 1.5 \times 0.026 \ln(5 \times 10^{12}) \approx 1.15 \text{ V}$.
6. Numerical solution gives $V_{mp} \approx 1.0 \text{ V}$, $I_{mp} \approx 4.9 \text{ A}$, $P_{max} \approx 4.9 \text{ W}$.
7. $FF = \frac{4.9}{1.15 \times 5} \approx 0.85$.

Why this step? The derivative condition $dP/dV = 0$ finds the voltage where incremental voltage increase trades off current decrease optimally for power.

Maximum Power Point Tracking (MPPT)

WHY needed? The MPP shifts with:

• Temperature: Higher $T$ lowers $V_{oc}$ (because $V_{oc} \propto \ln(\text{intensity}/I_0(T))$ and $I_0$ increases exponentially with $T$). Typical coefficient: $-2.2 \text{ mV/°C}$ per cell.
• Illumination: Lower intensity reduces $I_{sc}$ proportionally but $V_{oc}$ logarithmically.
• Degradation: Radiation damage in space reduces $I_L$ and increases $I_0$ over years.

HOW MPPT works (Perturb-and-Observe algorithm):

1. Measure current $P_k = V_k I_k$.
2. Perturb voltage: $V_{k+1} = V_k + \Delta V$ (small step, e.g., 0.5 V).
3. Measure new power $P_{k+1}$.
4. If $P_{k+1} > P_k$, keep moving in same direction ($\Delta V$ same sign).
5. If $P_{k+1} < P_k$, reverse direction ($\Delta V \to -\Delta V$).
6. Repeat at ~10–100 Hz.

Alternative: Incremental conductance ($\frac{dI}{dV} = -\frac{I}{V}$ at MPP). More efficient but more complex.

Effect on I-V:

• $I_{sc}$ drops to $I_{sc,0} \times 0.866$.
• $V_{oc}$ drops by $nV_T \ln(0.866) \approx -1.5 \times 26 \times 0.144 \approx -5.6 \text{ mV}$ (small).
• $P_{max}$ drops by $\approx 13\%$.

MPPT response: Algorithm detects power drop, searches for new $V_{mp}$ (slightly lower voltage, much lower current), locks on.

Why it feels right: Batteries are ~constant voltage loads (e.g., Li-ion ~3.7 V/cell). If battery voltage happens to match $V_{mp}$, you'd get max power.

The fix: In reality, $V_{mp}$ drifts with temperature/angle/age, and battery voltage changes with state-of-charge (3.0–4.2 V for Li-ion). A DC-DC converter (buck-boost) with MPPT sits between array and battery, forcing the array to operate at $V_{mp}$ while supplying whatever voltage the battery needs.

Batteries for Eclipse Power

Depth of Discharge (DoD)

WHY it matters: Higher DoD → fewer lifetime cycles. The relationship is empirical (from accelerated life testing):

Derivation logic: Each charge/discharge cycle causes:

1. Solid-electrolyte interphase (SEI) growth on the anode (irreversible Li consumption).
2. Cathode structure degradation (transition metal dissolution).
3. Electrolyte decomposition (especially at high voltages, i.e., near full charge).

Deeper discharge → more voltage swing → more side reactions → faster capacity fade.

If we allow 80% DoD:

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

If we limit to 30% DoD:

• Usable: $560 \times 0.3 = 168 \text{ Wh}$ (not enough!).
• Need bigger battery: $290 / 0.3 = 967 \text{ Wh}$, or 34.5 Ah at 28 V.
• Cycle life: ~15,000 cycles → ~2.6 years.

Why this step? We trade battery mass vs. lifetime For a 5-year mission, you'd design for 30–40% DoD (and size battery accordingly), accepting the mass penalty.

Battery Technologies in Space

Thermal management: Li-ion operates 0–40°C. Space is extreme (±150°C swing). Heaters + radiators maintain 15–25°C.

Radioisotope Thermoelectric Generators (RTG)

Power Output Over Time

Electrical power:

where $\eta$ is thermoelectric conversion efficiency (typically 6–7% at beginning-of-life).

WHY efficiency drops: Thermoelectric materials degrade (sublimation of Te, radiation damage). Empirical model:

with $\tau \approx 15–20$ years. Combined:

Why this step? Two decay processes compound: isotope half-life (slow) and TE degradation (faster). Mission design must ensure minimum power at end-of-life exceds load.

Sebeck Effect (Thermoelectric Conversion)

Derivation from carrier statistics:

1. Hot side has higher carrier energy → more diffusion to cold side.
2. Charge imbalance creates electric field.
3. At equilibrium, $eV = \Delta \mu$ (electrochemical potential difference).
4. For metals/semiconductors, $\mu \propto k_B T \ln(n)$, so $V \propto T$ (simplified).

Efficiency (Carnot-limited):

where $ZT$ is the thermoelectric figure of merit (dimensionless):

• $\sigma$ = electrical conductivity.
• $\kappa$ = thermal conductivity.

WHY low efficiency? Need high $\sigma$ (good electrical conductor) but low $\kappa$ (poor thermal conductor)—contradictory. Best materials (Bi₂Te₃, SiGe, skuterudites): $ZT \approx 1–1.5$ → $\eta \approx 6–10\%$.

Why it feels right: 87.7-year half-life is longer than most missions. No degradation from cycles like batteries.

The fix:

1. Cost: $^{238}\text{Pu}$ is scarce (produced in reactors, $\sim$10 kg/yr globally). One MMRTG costs ~$100M. Only for missions where solar is impossible.
2. Mass: 45 kg for 110 W (0.4 kg/W). Solar arrays: ~0.02 kg/W (20× lighter per watt at 1 AU).
3. Safety: Launch approval requires extensive containment analysis (though $^{238}\text{Pu}$ is alpha-only, not gamma, so shielding is manageable).

Power Budget and System Design

Sunlight charge balance:

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

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

Connections

• Spacecraft Thermal Control — batteries and power electronics generate heat; solar arrays need radiators on back side.
• Orbital Mechanics (Keplerian) — eclipse duration depends on orbit altitude and inclination; $\beta$-angle determines seasonal solar flux.
• Semiconductor Physics — p-n junctions, Fermi levels, carrier diffusion underpin solar cells.
• Thermodynamics & Heat Transfer — Carnot efficiency, Sebeck effect, blackbody radiation (RTG radiator sizing).
• Radiation Effects on Materials — displacement damage in solar cells (coverglass thickness vs. mass trade), TE material degradation.
• Power Electronics — DC-DC converters, MPPT algorithms, battery charge controllers.

#flashcards/physics

What is the maximum power point (MPP) of a solar cell? :: The voltage and current pair $(V_{mp}, I_{mp})$ at which the product $P = V \cdot I$ is maximized on the I-V curve; typically occurs where $dP/dV = 0$.

How does MPPT (Maximum Power Point Tracking) work? :: MPPT algorithms (e.g., Perturb-and-Observe) continuously adjust the operating voltage by measuring power, perturbing voltage, and moving in the direction of increasing power to stay at the MPP as conditions change.

What is the fill factor (FF) of a solar cell? :: $FF = \frac{V_{mp} I_{mp}}{V_{oc} I_{sc}} = \frac{P_{max}}{V_{oc} I_{sc}}$, a measure of how "square" the I-V curve is; typical values are 0.7–0.88 for space-grade cells.

Concept Map

Hinglish (regional understanding)