Stage 3 of 6 · The numbers · one page, the whole semester

The semester's equations

The course's mathematics fits on one page, and that is a feature: each formula earned its place by governing a real bench decision, so each comes with the decision it governed. Review them as tools, not strings: for each, know what it computes, what each symbol is, and the bench number it once produced. An equation you can apply with the kit's values is yours; one you can only recite is the exam's.

How to use this page

Three passes, in the retrieval spirit of the next page. First pass: cover the right columns, and for each formula state meaning and a worked number from memory. Second pass: cover the formula column, and reconstruct each from its meaning. Third pass, the engineer's: for each, answer "what goes wrong on the bench if I ignore this?", because that is the form exam applications take. Derivations live in the home pages linked; this page is deliberately the card, not the lecture.

Where each equation lives

Equations have addresses. Pin each formula to its component and recall gets a route: a question about a frozen dashboard walks you to the page's staleness bound; a question about a lingering offset walks you to the rule's e_ss. The card below is this figure, expanded.

The card: all of them

Equation	Says, in one line	Bench numbers, once	Home
I = (V_s − V_f) / R	The resistor sets the LED's current from what is left of the supply after the diode's drop	(3.3 − 2.0) / 330 ≈ 4 mA, comfortably inside the pin budget; 220 Ω ≈ 6 mA, still safe	W4
V_out = V_in · R₂ / (R₁ + R₂)	The divider turns the LDR's changing resistance into a voltage the ADC can read	As the LDR's share of the total shrinks in bright light, V_out slides across the ADC's range	W8
12-bit ADC: 4096 levels	A continuous voltage lands on one of 2¹² discrete counts; resolution is range/4096	GPIO34, ADC1 (the Wi-Fi-safe bank); full range ↦ counts 0 to 4095	W8
σ_mean = σ / √n	Averaging n samples cuts random noise by √n, the cheapest filter there is	read_avg with n = 16: noise ÷ 4, and the D-term downstream stops chattering	W8, W10
silence > 1.5 × keepalive ⇒ client presumed dead	The broker's death test; what makes the LWT fire and status topics honest	keepalive = 30 s ⇒ a vanished node is declared within ~45 s	W9
e = SP − PV	Error is want minus have; the sign tells the controller which way to push	SP 60 %, PV 42 % ⇒ e = +18: push brightness up	W10
u = K_p·e + K_i·∫e dt + K_d·de/dt	The PID law: react to the error, to its accumulation, and to its rate	The A2 loop; W10's chosen gains K_p 2.0, K_i 3.0, K_d 0.18 cut overshoot and settling together	W10
I_k = I_k−1 + K_i·e·Δt · · D_k = K_d·(e − e_k−1)/Δt	The integral and derivative a loop can actually compute, one step at a time	With the node loop's Δt ≈ 0.1 s (PERIOD_MS = 100), each step adds a sliver and compares neighbours	W10
e_ss = (SP − d) / (1 + K·K_p)	P-only control settles short by this much: gain shrinks the offset, only integral removes it	W10's derivation with bench numbers; octupling K_p shrank the offset and bought oscillation	W10
derivative noise ∝ 2πf · amplitude	Differentiation amplifies fast wiggles in proportion to their frequency: the D-term's tax	Why raw ADC into K_d chatters, and read_avg's ÷4 is the first cure	W10
staleness ≤ T_pub + T_poll	The dashboard's number lags by at most one publish interval plus one poll interval	~1 s publish + 2 s fetch ⇒ the "at most 3 s old" requirement, with the 3 defensible	W11
contracts ≤ n(n − 1)/2	Pairwise interfaces grow quadratically; each used one can fail on either side	Your chain uses about six; the skeleton exercises all of them in one afternoon	W12
probes = ⌈log₂ n⌉	Half-splitting finds one faulty segment among n in logarithmically few looks	n = 8 ⇒ 3 probes, guaranteed; one-by-one averages 4 and can cost 7	W12

Symbols are half the marks

For every formula, be able to say what each symbol is and its unit, because application questions are usually symbol questions in disguise: handed a kit with V_f = 2.1 V and asked for a safe resistor, the work is identifying which symbol the 2.1 is. The Ziegler-Nichols table (0.6 K_u, 1.2 K_u/T_u, 0.075 K_uT_u) is the one table-shaped result; it lives with its honesty notes in Week 10.

Checklist for this stage

Pass one done: meanings and worked numbers from memory, formulas visible
Pass two done: formulas reconstructed from meanings
Pass three done: for each equation, I can say what ignoring it breaks on the bench
Every symbol in the PID law and in e_ss named, with units

Check yourself

A kit arrives with 3.3 V pins, LEDs with V_f = 2.1 V, and a bag of 150 Ω resistors. Safe for a pin budgeted at 8 mA?

I = (3.3 − 2.1) / 150 = 1.2 / 150 = 8 mA: exactly at the budget, no margin for a brighter diode or a warm day. The engineering answer is therefore no, not with these; pick the next standard value up (220 Ω gives 1.2/220 ≈ 5.5 mA, comfortable). The marks here are for the symbol work, 2.1 is V_f, not V_s, and for treating "at the limit" as a design smell rather than a pass.

Averaging already costs time. Why does the card still call σ/√n the cheapest filter, and what is the price of raising n from 16 to 64 on this node?

Cheapest in hardware and code: zero components, three lines, no new theory, against which any analog filter or digital IIR is expensive to a beginner bench. The price of n is time and diminishing returns: noise falls as √n, so 16 → 64 buys ÷8 instead of ÷4 (one extra halving) while quadrupling sampling time per reading; on the node's 100 ms loop the reading burst must still fit the period, and the PV grows more sluggish, which the D-term, ironically the term you were protecting, then differentiates as lag. The card's n = 16 is the knee of that curve for this bench.

Use two different card rows to explain why the LED answers a shadow instantly while the dashboard takes seconds, with numbers.

The LED rides the fast loop: the node reads every PERIOD_MS = 100 ms step, the rule's u follows e = SP − PV within a step or two, so the action lands in tenths of a second. The pixel rides the human loop and is bounded by staleness ≤ T_pub + T_poll ≈ 1 s + 2 s = 3 s worst case. Same shadow, two loops, two timescales, and both numbers were design choices, not accidents: the fast loop as fast as the node comfortably samples, the slow loop as slow as human attention allows.