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

False — largest current is the short-circuit current $I_{sc}$ at $V=0$, where power is zero. MPP is the interior point where the product $P=VI$ peaks, always at a voltage below $V_{oc}$ and a current below $I_{sc}$.

False — $I_{sc}$ scales nearly linearly with intensity, but $V_{oc}=n{V}_T\ln (I_L/I_0+1)$ grows only as the logarithm of intensity, so doubling light adds only $\sim n{V}_T\ln 2\approx 27$ mV per cell.

False — heat raises the dark saturation current $I_0$ (it climbs exponentially with $T$), which pulls $V_{oc}$ down by roughly $-2.2$ mV/°C per cell; the tiny gain in $I_{sc}$ never compensates, so hot cells lose power. See Spacecraft Thermal Control.

False — $FF=P_{max}/(V_{oc}{I}_{sc})$ is the ratio of the actual MPP power to the rectangle $V_{oc}\times I_{sc}$ that bounds it, so $FF<1$ always; space-grade triple-junction cells reach $\sim 0.88$ but never 1.

False — the battery pins the array to its terminal voltage (state-of-charge dependent), which almost never equals the drifting $V_{mp}$; you need a DC-DC converter with MPPT in between. See Power Electronics.

False — cycle life falls faster than linearly with DoD ($N\propto {\text{DoD}}^{-k}$, $k\approx 1.5$–2), so two shallow cycles are gentler than one deep cycle even though the total charge moved is the same.

False — the thermal power decays with the isotope's half-life, but electrical output falls faster because the thermocouples themselves degrade (sublimation, dopant diffusion), so end-of-life electrical power is lower than half-life prediction alone.

True — differentiating $P=VI$ and setting $dP/dV=0$ gives $I+VdI/dV=0$, i.e. $dI/dV=-I/V$; this exact equality is what the incremental-conductance MPPT algorithm tests.

Wrong — at short circuit $V=0$ so $P=VI=0$; the array delivers zero useful power there. Efficiency (power out per photon in) is maximized at the MPP, not at maximum current.

The exponential term is $e^0-1=0$, so $I=I_L$ exactly at $V=0$, not $I_L-I_0$. The $-1$ inside the parenthesis exists precisely so the diode term vanishes at zero bias.

52% DoD is not lifetime-free — every cycle still grows the SEI layer and fades capacity; "fine for eclipse" only means energy fits, while cycle count over the mission (thousands of orbits in LEO) still sets the real limit.

Only the intensity — and hence $I_{sc}$ — scales by $\cos \beta$; $V_{oc}$ drops merely as $n{V}_T\ln (\cos \beta )$, a much smaller change. Treating both as $\cos \beta$ badly overestimates the voltage loss. See Orbital Mechanics (Keplerian).

It hunts around the MPP, oscillating by one step $\Delta V$ each cycle; it never rests exactly on the peak, and a too-large $\Delta V$ wastes power in the wobble while a too-small one tracks moving conditions too slowly.

Radiation damage also raises $I_0$ (more recombination centres), which drops $V_{oc}$ and rounds off the knee of the curve, lowering $FF$ too. Both current and voltage sides degrade. See Radiation Effects on Materials.

Because the diode term $I_0{e}^{V/n{V}_T}$ is negligible at low voltage (flat, current-source region) but explodes near $V_{oc}$ (steep, diode region); the crossover between these two regimes bends the curve, and a sharper knee means a higher fill factor. See Semiconductor Physics.

A resistor can only dissipate the mismatch as heat, wasting it; a buck-boost converter transforms the array's $(V_{mp},I_{mp})$ into whatever $(V,I)$ the battery needs while conserving power minus small switching losses. See Power Electronics.

Deeper discharge means a larger voltage swing across the electrodes, driving more electrolyte decomposition and SEI growth per cycle; these irreversible side reactions consume lithium and fade capacity, so wear grows super-linearly with DoD.

An exponent of 1 would mean wear is purely proportional to charge moved; the mechanical and chemical damage (electrode particle cracking, SEI regrowth on fresh surface, cathode metal dissolution) scale faster than the discharge depth because deeper swings expose more fresh material and larger strain per cycle, pushing $k$ above 1.

Solar intensity falls as $1/r^2$ from the Sun, so beyond Jupiter the flux is too weak for practical arrays; an RTG's thermal output depends only on radioactive decay, independent of distance from the Sun.

It sets the natural voltage scale for how thermally-agitated carriers surmount the junction barrier; at temperature $T$ the Boltzmann factor $e^{qV/k_{B}T}=e^{V/V_T}$ governs how many carriers have enough energy, which is exactly the exponential in the Shockley equation. See Thermodynamics & Heat Transfer.

The array degrades over the mission (radiation, thermal cycling, micrometeoroids), so sizing to BOL power alone leaves the spacecraft short of watts years later; you size the array so that the degraded EOL power still meets demand.

Exactly zero — no photons means $I_L=0$, so the source term vanishes and the cell can only behave as a passive (slightly leaky) diode; all load power must come from the battery.

Zero — by definition $I=0$ at open circuit, so $P=VI=0$ even though the voltage is at its maximum; useful power lives strictly between the two axes.

$V_T=k_{B}T/q\to 0$ and $I_0\to 0$, so $V_{oc}=n{V}_T\ln (I_L/I_0+1)$ tends toward the bandgap-limited maximum; colder cells give higher voltage, which is why deep-space panels can run "hot on volts" when cold.

$N\propto {\text{DoD}}^{-k}$ predicts the fewest possible cycles, and near 100% real cells risk over-discharge below their safe cutoff, causing copper dissolution and permanent damage — so designers cap DoD well below 100%.

Tracking becomes pointless — the array cannot even power its own control electronics, so the system falls back to battery power and the MPPT may idle or shut down until light returns (e.g., exiting eclipse).

The shaded cell becomes a reverse-biased load that blocks the whole series current and can overheat (hot-spot); bypass diodes are added precisely to route current around a dead cell so one shadow does not kill the string.

The string current is capped by the weakest cell, so the curve develops steps and multiple local power humps — one per shading level; a naive MPPT can lock onto a lower local peak and miss the true global maximum.

Parallel strings share voltage but add currents, so a shaded string simply contributes less current at the common voltage — no reverse-bias hot-spot forces the issue, which is why large arrays mix series (for voltage) and parallel (for shade-tolerant current).