# 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
`Hilfsmittel:` 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;
DP is intended for exterritorial-location call signs; DP0–DP1 =
class A with use categories KS, RL/FB and SZ; DP2 = class E with the
same use categories; DP8 = class N with the same use categories.
- **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` (R22,5)
`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.