Significantly update explanations

EXPLANATIONS.md

@@ -176,6 +176,43 @@ band-plan lookup), say that plainly and cite the band plan in
 `source` — confidence stays low (3–4) until someone finds a deeper
 hook.
 
+### The Hilfsmittel note
+
+Candidates may use the official exam aid (`Hilfsmittel_12062024.pdf`)
+during the exam. Its complete contents — every formula and table, with
+**printed page numbers** — are catalogued in
+[`references/Hilfsmittel.md`](references/Hilfsmittel.md). When a
+question's answer is a lookup or a direct application of something in
+that sheet, append a short note to the end of the `explanation` body:
+
+```
+... <u>Hilfsmittel:</u> <pointer>
+```
+
+Rules — this is the part that was historically done badly (generic
+boilerplate stamped on everything), so be strict:
+
+- **Only cite what is actually in the sheet.** Verify against
+  `references/Hilfsmittel.md`. If the fact the question turns on is a
+  memory item — diode forward voltages (~0.6 V / ~0.3 V), `tan δ = 1/Q`,
+  S-meter step = 6 dB, harmonics = n × fundamental, ppm/percent
+  arithmetic, semiconductor behaviour, definitions, antenna
+  length/shortening factors — **do not add a Hilfsmittel note at all.**
+- **Name the specific formula(s) or table, and the page.** Not "the
+  formula is in the Formelsammlung" but e.g. `P = U²/R (Leistung,
+  S.12)` or `the Widerstands-Farbcode table (S.11)`.
+- **For multi-step calculations, give the order:** "first
+  `U_eff = Û/√2` (Wechselspannung, S.12), then `P = U_eff²/R`
+  (Leistung, S.12)". State plainly when a value comes from outside the
+  sheet (e.g. a diode drop) versus from a sheet formula.
+- **For a pure table lookup**, say it is a lookup and cite the table +
+  page (e.g. "a table lookup, not a memory item — the band limits …
+  are in the Frequenzbereichszuweisung (Anlage 1, Tabellarische
+  Übersicht, S. 2–3)").
+- **Page numbering:** the printed page number = PDF page − 2 (the cover
+  and the "Hinweis" page are unnumbered). Always cite the printed
+  number, as `references/Hilfsmittel.md` does.
+
 ---
 
 ## 5. Confidence scale

references/Hilfsmittel.md

@@ -0,0 +1,212 @@
+# Hilfsmittel — official exam aid (Formelsammlung + tables)
+
+Catalog of everything in the BNetzA exam aid sheet that candidates are
+**allowed to use during the exam**, with the **printed page numbers** as
+they appear on each page. This file is the source of truth for the
+`<u>Hilfsmittel:</u>` notes appended to explanations (see
+`EXPLANATIONS.md`): a note may only cite a formula/table that is listed
+here, and should name it and its page.
+
+- Source PDF: `Hilfsmittel_12062024.pdf`
+  (https://www.bundesnetzagentur.de/SharedDocs/Downloads/DE/Sachgebiete/Telekommunikation/Unternehmen_Institutionen/Frequenzen/Amateurfunk/AntraegeFormulare/Hilfsmittel_12062024.pdf?__blob=publicationFile&v=3)
+- **Page numbering:** the printed page number = PDF page − 2 (the cover
+  and the "Hinweis" page are unnumbered). All "S. N" below are the
+  printed numbers.
+- Errata (per the PDF "Hinweis" page): on S. 15 the parabolic-dish gain
+  formula, on S. 17 the SWR formula, and on S. 21 the specific-resistance
+  table (Zink → Zinn) were corrected against the 3rd-edition catalog.
+
+If the fact a question needs is **not** in this list (e.g. diode forward
+voltages ~0.6 V / ~0.3 V, tan δ = 1/Q, semiconductor behaviour,
+definitions), it is a memory item — **do not** add a Hilfsmittel note.
+
+---
+
+## S. 1–3 — Frequenzbereichszuweisung (Anlage 1, Amateurfunkverordnung)
+
+- **S. 1 — Nutzungsbedingungen (¶1–4):** general usage rules (incl. the
+  **50 W ERP** cap for remote/automatic terrestrial stations above 30 MHz,
+  with a possible **1000 W ERP** exception for links) and, in ¶3, the
+  **definitions** of a *primärer* and *sekundärer Funkdienst* and their
+  protection/priority rights. This is a *text* page, not a table. (No
+  750 W figure appears here — 750 W PEP is a class-A limit in the S. 2
+  table.)
+- **S. 2 — Tabellarische Übersicht:** per frequency band: status
+  (primary/secondary), the allowed frequency ranges, max. **power per
+  licence class (A/E/N) — in PEP or ERP exactly as printed** (not EIRP),
+  and additional-usage references.
+  Note the units: class N on **10 m is 10 W ERP**, but on **2 m/70 cm it
+  is 6,1 W ERP** (= ≈10 W EIRP after ×1,64, see S. 15) — the table does
+  *not* print "10 W EIRP". Power limits are in PEP/ERP, never EIRP.
+- **S. 3 — Zusätzliche Nutzungsbestimmungen (numbered conditions):**
+  max. occupied **bandwidth** per band (800 Hz, 2,7 kHz, 7 kHz, 40 kHz,
+  2 MHz …), and special power limits such as the 1247–1263 MHz
+  **3,05 W ERP** sub-band cap (= ≈5 W EIRP). The bandwidth values live in
+  these numbered conditions, not in the S. 2 table.
+
+Not in Anlage 1: the general HF/VHF/UHF (KW/UKW) range *terminology*,
+the ISM-band *definition*, and foreign-country prefixes (Landeskenner) —
+those are memory items.
+
+## S. 4–7 — Rufzeichenplan für den Amateurfunkdienst in Deutschland
+
+- **S. 4** — intro: structure of German call signs (prefix DA–DR excl.
+  DE/DI, digit, 2–3-char suffix).
+- **S. 5 — Tabelle "Rufzeichen mit 2- oder 3-buchstabigen Suffixen":**
+  the main **call-sign-series → class/use** table. DA0 = Klubstation;
+  DL1–DL9 = person-bound class A; DO1–DO9 = class E; DN9 = class N;
+  DP0–DP1 = exterritorial class A; etc.
+- **S. 6** — club-station suffixes (1-char) table.
+- **S. 7** — club-station suffixes (4–7-digit); **§ 5** Kurzzeitzulassungen
+  ausländischer Funkamateure (visitors in Germany sign **DL/** class A,
+  **DO/** class E); **§ 6** Kennungen zum Betrieb leistungsschwacher Sender
+  (beacon/peil identifiers **MO, MOE, MOI, MOS, MOH, MO5**); §§ 7–8 rules.
+
+## S. 8 — International gebräuchliche Rufzeichenzusätze / Ausbildung / Remote
+
+- **§ 9** operating suffixes appended with `/`: `/m` (mobile, incl.
+  inland-waterway), `/mm` (maritime mobile), `/am` (aeronautical mobile),
+  `/p` (portable/temporary fixed; optional in Germany), etc.
+- **§ 10** Ausbildungsfunkbetrieb (`/Trainee` speech, `/T` CW/digital).
+- **§ 11** Remotebetrieb (`Remote` speech, `/R` CW/digital).
+
+Foreign-country prefixes (e.g. Swiss HB3/HB9 for a German operating
+*abroad*) are **not** in this plan — memory item.
+
+## S. 9–10 — IARU-Bandplan
+
+- **S. 9 — 2 m** and **S. 10 — 70 cm**: per segment, max bandwidth,
+  preferred mode and usage (CW, SSB, FM, digital, beacons, satellite…).
+
+## S. 11 — Formelsammlung: Grundlagen / Widerstände
+
+- Zehnerpotenzen & SI-Präfixe (p…T) — table.
+- Zweierpotenzen / Bit (2^0…2^12) — table.
+- **Ohmsches Gesetz:** `U = R·I`, `R = U/I`, `I = U/R`.
+- **Innenwiderstand:** `R_i = ΔU/ΔI`.
+- **Widerstand von Drähten:** `R = ρ·l/A_Dr`, `A_Dr = d²·π/4 = r²·π`.
+- **Widerstands-Farbcode** — table (Farbe / Wert / Multiplikator / Toleranz).
+
+## S. 12 — Widerstandsnetzwerke / Leistung / Wechselspannung
+
+- **Reihenschaltung:** `R_G = R1+R2+…+R_N`; 2 R: `R_G = R1+R2`.
+- **Parallelschaltung:** `1/R_G = 1/R1+1/R2+…`; 2 R: `R_G = R1·R2/(R1+R2)`.
+- **Spannungsteiler (unbelastet):** `U_G = U1+U2`, `U1/U2 = R1/R2`,
+  `U2/U_G = R2/(R1+R2)`.
+- **Stromteiler:** `I_G = I1+I2`, `I2/I1 = R1/R2`.
+- **Vorzugsreihen** E6/E12/E24 — table.
+- **Leistung:** `P = U·I = U²/R = I²·R`; `U = P/I = √(P·R)`;
+  `I = P/U = √(P/R)`.
+- **Arbeit/Energie:** `W = P·t`.
+- **Wirkungsgrad:** `η = P_ab/P_zu (·100 %)`; `P_ab = P_zu − P_V`.
+- **Wechselspannung:** `Û = U_eff·√2`, `U_SS = 2·Û`, `ω = 2·π·f`,
+  `Z = √(R²+X²)`, `T = 1/f`, `f = 1/T`.
+
+## S. 13 — Induktivität / Transformator / Kapazität
+
+- **Induktiver Blindwiderstand:** `X_L = ω·L`.
+- **L Reihe:** `L_G = L1+…+L_N`; **L Parallel:** `1/L_G = 1/L1+…`.
+- **Ringspule / lange Zylinderspule:** `L = μ0·μr·N²·A_S / l(_m)`.
+- **Ringkernspulen (auch mehrlagig):** `L = N²·A_L`.
+- **Magnetische Feldstärke (Ringspule):** `H = I·N/l_m`.
+- **Magnetische Flussdichte:** `B_m = μr·μ0·H`.
+- **Transformator-Übersetzungsverhältnis:**
+  `ü = N_P/N_S = U_P/U_S = I_S/I_P = √(Z_P/Z_S)`.
+- **Belastbarkeit von Wicklungen:** `I = S·A_Dr` mit `S ≈ 2,5 A/mm²`.
+- **Kapazitiver Blindwiderstand:** `X_C = 1/(ω·C)`.
+- **C Reihe:** `1/C_G = 1/C1+…`; **C Parallel:** `C_G = C1+…`.
+- **E-Feld im homogenen Feld:** `E = U/d`.
+- **Kapazität Kondensator:** `C = ε0·εr·A/d`.
+
+## S. 14 — Filter / Schwingkreis / Transistor / ZF & Spiegelfrequenz
+
+- **RC TP/HP:** `f_g = 1/(2π·R·C)`; **RL TP/HP:** `f_g = R/(2π·L)`
+  (`f_g` = −3 dB-Grenzfrequenz).
+- **Schwingkreis:** `f0 = 1/(2π·√(L·C))`, Resonanz `X_C = X_L`.
+- **Reihenschwingkreis:** `B = R_s/(2π·L)`, `Q = f0/B = X_L/R_s`.
+- **Parallelschwingkreis:** `B = 1/(2π·R_p·C)`, `Q = f0/B = R_p/X_L`.
+- **Transistor (DC):** `B = I_C/I_B`, `I_E = I_C+I_B`.
+- **Transistor (AC):** `v_I = β = ΔI_C/ΔI_B`, `v_U = β = ΔU_CE/ΔU_BE`,
+  `v_P = β² = v_U·v_I`. *(Transcribed verbatim. The printed
+  `v_U = β` is physically dubious — voltage gain is not generally the
+  current gain — but the aid prints it this way; do not silently
+  "correct" it.)*
+- **Zwischenfrequenz:** `f_ZF = |f_E − f_OSZ|` (the sheet notes that the
+  `f_ZF = f_E + f_OSZ` case is not considered).
+- **Spiegelfrequenz:** `f_S = 2·f_OSZ − f_E`.
+
+## S. 15 — Pegel / Antennen
+
+- **Pegel:** `p = 10·log10(P/1mW)` dBm; `p = 10·log10(P/1W)` dBW;
+  `u = 20·log10(U/0,775V)` dBu.
+- **Verstärkung/Gewinn:** `g = 10·log10(P2/P1)` dB;
+  `g = 20·log10(U2/U1)` dB.
+- **Dämpfung/Verluste:** `a = 10·log10(P1/P2)`; `a = 20·log10(U1/U2)`.
+- **dB ↔ Verhältnis-Tabelle** — exactly **11 rows** (Leistungs- /
+  Spannungsverhältnis): −20 (0,01/0,1), −10 (0,1/0,32), −6 (0,25/0,5),
+  −3 (0,5/0,71), −1 (0,79/0,89), 0 (1/1), +1 (1,26/1,12), +3 (2/1,41),
+  +6 (4/2), +10 (10/3,16), +20 (100/10). **There is no 16 dB row** (or
+  any value off this list): combine rows, e.g. 16 dB = +10 dB (×10) +
+  +6 dB (×4) → ×40.
+- **ERP:** `p_ERP = p_S − a + g_d`; `P_ERP = P_S·10^((g_d−a)/10dB)`.
+- **EIRP:** `p_EIRP = p_ERP + 2,15 dB`; `P_EIRP = P_ERP·1,64`.
+- **Feldstärke Fernfeld:** `E = √(30Ω·P_A·G_i)/d = √(30Ω·P_EIRP)/d`
+  (ab `d > λ/(2π)`).
+- **Gewinn:** `G_i = G_d·1,64`, `g_i = g_d + 2,15 dB`, `G = 10^(g/10dB)`.
+- Halbwellendipol `G_i=1,64 / g_i=2,15 dB`; λ/4-Vertikal mit
+  Bodenreflexion `G_i=3,28 / g_i=5,15 dB`; Parabolspiegel
+  `g_i = 10·log10[(π·d/λ)²·η] dB`.
+
+## S. 16 — Rauschen / Amplitudenmodulation / Frequenzmodulation
+
+- **Thermisches Rauschen:** `P_R = k·T_K·B`; `Δp_R = 10·log10(B1/B2)`;
+  `U_R = 2·√(P_R·R)`.
+- **Rauschzahl:** `F = (P_S/P_N)_Eingang/(P_S/P_N)_Ausgang`;
+  `a_F = 10·log10(F)`; `a_F = SNR_Eingang − SNR_Ausgang`.
+- **SNR:** `SNR = 10·log10(P_S/P_N) = 20·log10(U_S/U_N)`.
+- **Shannon-Hartley:** `C = B/1Hz · log2(1 + P_S/P_N)` bit/s.
+- **log2:** `log2(x) = log10(x)/log10(2)`.
+- **AM Modulationsgrad:** `m = Û_mod/Û_T`; **AM Bandbreite:**
+  `B = 2·f_mod,max`.
+- **FM Modulationsindex:** `m = Δf_T/f_mod`; **Carson-Bandbreite:**
+  `B ≈ 2·(Δf_T + f_mod,max)`.
+
+## S. 17 — Wellenlänge & Frequenz / Reflexion / Wellenwiderstand
+
+- **c = f·λ**, `f = c/λ`, `λ = c/f`; `c0 ≈ 3·10⁸ m/s`;
+  `f[MHz] ≈ 300/λ[m]`, `λ[m] ≈ 300/f[MHz]`.
+- **Verkürzungsfaktor:** `k_v = l_G/l_E = 1/√εr = c/c0`.
+- **Stehwellenverhältnis (SWR):**
+  `s = U_max/U_min = (U_v+U_r)/(U_v−U_r) = (√P_v+√P_r)/(√P_v−√P_r) = (1+|r|)/(1−|r|)`;
+  `s = R2/Z` (R2>Z) bzw. `s = Z/R2` (R2<Z).
+- **Reflexionsfaktor:** `|r| = (s−1)/(s+1) = |(R2−Z)/(R2+Z)| = |U_r/U_v| = √(P_r/P_v)`.
+- **Rücklaufende Leistung:** `P_r = P_v·|r|²`.
+- **An R2 abgegebene Leistung:** `P_ab = P_v·(1−|r|²)`.
+- **Wellenwiderstand:** HF-Leitung `Z = √(L'/C')`; Koax
+  `Z = 60Ω/√εr·ln(D/d)`; symm. Zweidraht (a/d>2,5)
+  `Z = 120Ω/√εr·ln(2a/d)`; Viertelwellentransformator `Z = √(Z_E·Z_A)`.
+
+## S. 18 — Weitere Formeln
+
+- **MUF:** `MUF ≈ f_c/sin(α)`; `f_opt = MUF·0,85`.
+- **Empfindlichkeit von Messsystemen:** `E_MESS = R_i/U_i = 1/I_i`.
+- **Relativer maximaler Fehler:** `F_W = ±(G/100)·(W_E/W_M)`.
+- **Abtasttheorem:** `f_abtast > 2·f_max`; für Nicht-Basisband-Signale
+  `f_abtast > 2·(f_max − f_min)` **wenn `f_abtast < f_min` oder
+  `f_abtast > f_max`**.
+- **Datenübertragungs-/Symbolrate:** `C = R_S·n`.
+
+## S. 19–22 — Konstanten und Tabellen
+
+- **S. 19–20** — symbol/quantity glossary and physical constants
+  (k, c0, μ0, ε0, e, …).
+- **S. 21** — material tables: **relative permittivity ε_r** and
+  **specific resistance ρ** (per the S. 21 erratum, Zink → Zinn).
+- **S. 22 — Kabeldämpfungsdiagramm Koaxialkabel:** a chart of cable
+  attenuation in **dB per 100 m vs. frequency** for common coax types
+  (RG58 ≈ 4,95 mm, RG174 ≈ 2,8 mm, PE-foam 10,3/12,7 mm, …). Read the
+  per-100 m loss at the operating frequency, then **scale linearly with
+  cable length**. This chart *is* part of the aid — cable-loss questions
+  are Hilfsmittel lookups.
+</content>
+</invoke>