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

This page assumes you have seen nothing. We build every letter the parent note uses, one at a time, so that when you meet $I = I_{L} - I_{0} (e^{V / n V_{T}} - 1)$ you already own every symbol in it.

Part 1 — The four quantities that flow

Before any solar cell, we need four everyday physical ideas. Picture a water system: this analogy carries us the whole way.

Why the topic needs these: Every box on a spacecraft — radios, heaters, computers — asks for a certain number of watts. Since $P = V I$ , the power system must deliver the right voltage and current together. The entire "I-V curve" story is about which $(V, I)$ pairs a solar cell can actually offer.

Part 2 — Energy vs. power, and the "-hour" units

Power is a rate. Energy is the total amount delivered — power multiplied by time.

Part 3 — Percentages, ratios, and logarithms

Two mathematical tools appear all over the parent note. We earn them now.

Why the topic needs $ln$ and $e$ : The diode inside a solar cell has an exponential current-voltage law, $e^{V / n V_{T}}$ . Its inverse — the logarithm — is exactly what lets us solve for the open-circuit voltage $V_{oc} \approx n V_{T} ln (I_{L} / I_{0} + 1)$ . The exponential says "current explodes as voltage rises"; the logarithm "undoes" it to recover the voltage. See Semiconductor Physics for why a diode obeys this law.

Part 4 — The symbols inside the solar-cell equation

Now every ingredient is defined. Here is the parent note's core formula with each symbol named in plain words:

Symbol	Plain words	Picture
$I$	current the cell delivers to the load	flow rate out of the cell
$I_{L}$	photocurrent — charge freed by sunlight	a steady pump set by brightness
$I_{0}$	dark saturation current — tiny leak even with no light	a slow drip backwards
$V$	terminal voltage across the cell	the pressure at the cell's outputs
$n$	ideality factor ( $1$ – $2$ ), how "non-ideal" the diode is	a fudge dial on the exponential
$V_{T}$	thermal voltage $= k_{B} T / q \approx 26 mV$	how hard heat jiggles the charges
$R_{s h}$	shunt resistance — unwanted leakage path	a small crack letting current bypass the load

Reading the equation as a story: sunlight pumps $I_{L}$ out (positive), the diode drinks some back as it turns on exponentially (the middle term), and a leaky crack $R_{s h}$ steals a little more (last term). What remains is $I$ .

The rate at which charge flows past a point, measured in amperes (coulombs per second).

Write the master power relation.

P = V \cdot I

(watts = volts × amps).

How do you get energy from power?

Multiply power by time:

E = P \cdot t

(e.g. watts × hours = watt-hours).

Convert 28 V and 20 Ah into stored energy.

28 \times 20 = 560

Wh.

What question does

ln (x)

answer?

"e raised to what power gives

x

?" — it undoes an exponential.

Why does a diode's current grow exponentially with voltage?

The fraction of charges with enough thermal energy to cross the junction follows the Boltzmann factor

e^{V / n V_{T}}

What is the thermal voltage

V_{T}

and its room-temperature value?

V_{T} = k_{B} T / q \approx 26

mV at 300 K.

In the cell equation, what does

I_{L}

represent?

The photocurrent — charge freed by absorbed sunlight, proportional to brightness.

Why set

d P / d V = 0

to find the maximum power point?

At the top of the power hill the slope is zero, so zero slope locates the peak.

What does 80% DoD physically mean?

80% of the battery's total charge capacity has been discharged.

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

Part 1 — The four quantities that flow

Part 2 — Energy vs. power, and the "-hour" units

Part 3 — Percentages, ratios, and logarithms

Part 4 — The symbols inside the solar-cell equation

Part 5 — Why a derivative appears

Prerequisite map

Equipment checklist