RF Engineering Cheat Sheet

01 — Noise & Sensitivity

Noise & Sensitivity

Thermal Noise Power

P = kTB (watts)
= −174 dBm/Hz at T=290 K
Noise floor (dBm) = −174 + 10·log(B[Hz])

k=1.381×10⁻²³ J/K · T=290 K · B=bandwidth in Hz

Noise Figure & Noise Temperature

NF = 10·log(F) dB · F = SNR_in/SNR_out
T_e = 290·(F−1) K
F = 1 + T_e/290

NF=3 dB → F=2 → T_e=290 K · NF=0.5 dB → T_e=35 K

Friis Cascaded NF

F_total = F₁ + (F₂−1)/G₁ + (F₃−1)/G₁G₂ + …
All linear (not dB) · First stage dominates

Loss L before LNA: F_sys = L·F_LNA (first stage = attenuator+LNA)

Receiver Sensitivity

S_min = −174 + NF + 10·log(B) + SNR_min
All in dB/dBm · B in Hz

WiFi 20 MHz, NF=5, SNR=10: −174+5+73+10 = −86 dBm

P1dB & IIP3

IIP3 = P_in + ΔIM3/2 (dBm)
OIP3 = IIP3 + G
P1dB_in ≈ IIP3 − 9.6 dB (theoretical)

ΔIM3 = P_fund − P_IM3 (output, dBc) · measured in 2-tone test

Cascaded IIP3 & SFDR

1/IIP3_tot = 1/IIP3₁ + G₁/IIP3₂ + …
SFDR = (2/3)·(IIP3 − P_noise) dB

Last stage dominates IIP3 (opposite of NF) · high gain hurts IIP3

Standard	BW	NF (typ)	SNR_min	Sensitivity
GSM 900	200 kHz	8 dB	10 dB	−103 dBm
LTE 10 MHz	9 MHz	7 dB	−1 dB	−98 dBm
WiFi 802.11n 20 MHz	20 MHz	8 dB	4 dB	−89 dBm
5G NR 100 MHz	100 MHz	7 dB	1 dB	−83 dBm
GPS L1	2 MHz	3 dB	−27 dB	−135 dBm
Bluetooth LE	2 MHz	10 dB	−3 dB	−104 dBm

02 — Path Loss & Link Budget

Path Loss & Link Budget

Free-Space Path Loss (FSPL)

FSPL = 20·log(4πd/λ) dB
= 32.44 + 20·log(f[MHz]) + 20·log(d[km])
= 92.44 + 20·log(f[GHz]) + 20·log(d[km])

×2 distance → +6 dB · ×2 frequency → +6 dB

Friis Link Budget

P_RX = P_TX + G_TX + G_RX − FSPL − L
EIRP = P_TX + G_TX (dBm + dBi)
Link margin = P_RX − S_min (dB)

L = all other losses (cable, atmosphere, polarisation mismatch)

Log-Distance Path Loss

PL(d) = PL(d₀) + 10n·log(d/d₀) + X_σ
n = path loss exponent
X_σ = Gaussian shadowing (std dev σ dB)

Free space n=2 · Urban NLOS n=3–4 · Indoor n=1.6–4

3GPP 5G NR UMi LOS

PL = 32.4 + 21·log(d[m]) + 20·log(f[GHz])
UMa NLOS: 13.54 + 39.08·log(d) + 20·log(f)
InH LOS: 32.4 + 17.3·log(d) + 20·log(f)

3GPP TR 38.901 · f in GHz · d in metres

Rain Attenuation (ITU-R P.838)

3.5 GHz: ~0.03 dB/km at 25 mm/hr
28 GHz: ~4.5 dB/km at 25 mm/hr
60 GHz: ~6 dB/km rain + 15 dB/km O₂

O₂ peak at 60 GHz naturally limits range to ~200 m — used for secure links

Radar Range Equation

R_max = ⁴√(P_T·G²·λ²·σ / ((4π)³·S_min))
P_T=TX power · σ=RCS (m²)
SNR = P_RX / kTBNF

Radar: signal travels 2×range → path loss ∝ R⁴ not R²

Scenario	Freq	Distance	FSPL
WiFi 2.4 GHz	2.4 GHz	50 m	80 dB
WiFi 5 GHz	5 GHz	20 m	82 dB
5G NR n78	3.5 GHz	500 m	117 dB
5G mmWave	28 GHz	200 m	127 dB
GPS L1	1.575 GHz	20,200 km	182 dB
GEO Satellite Ku	12 GHz	35,786 km	205 dB

03 — Transmission Lines

Transmission Lines

Characteristic Impedance

Z₀ = √(L/C) = √((R+jωL)/(G+jωC))
Lossless: Z₀ = √(L/C) · 50 Ω standard
Coax: Z₀ = (60/√εr)·ln(D/d)

Microstrip: use Hammerstad-Jensen · Stripline: use Wadell

Reflection & VSWR

Γ = (Z_L − Z₀)/(Z_L + Z₀)
VSWR = (1+|Γ|)/(1−|Γ|)
RL = −20·log|Γ| dB · Mismatch loss = −10·log(1−|Γ|²)

VSWR 2:1 → |Γ|=0.33 → RL=9.5 dB · mismatch loss=0.51 dB

Propagation

v_p = c/√εeff = 1/√(LC)
λ = v_p/f = c/(f·√εeff)
β = 2π/λ · γ = α + jβ

Microstrip FR4: εeff≈3.1 → v_p≈0.57c → λ at 2.4 GHz ≈ 71 mm

Input Impedance

Z_in = Z₀·(Z_L + jZ₀tanβl)/(Z₀ + jZ_Ltanβl)
λ/4 transformer: Z_in = Z₀²/Z_L
λ/2: Z_in = Z_L (repeats)

Open stub: Z_in=−jZ₀cotβl · Short stub: Z_in=jZ₀tanβl

Skin Depth & Conductor Loss

δ_s = 1/√(πfμ₀σ) metres
Copper at 1 GHz: δ_s ≈ 2.1 μm
R_s = 1/(σ·δ_s) = √(πfμ₀/σ) Ω/sq

δ_s ∝ 1/√f · At 10 GHz: 0.66 μm · Trace must be >> δ_s thick

Microstrip Quick Formulas

W/h > 2: Z₀ ≈ 120π/[√εeff·(W/h+1.393+…)]
εeff ≈ (εr+1)/2 + (εr−1)/2·(1+12h/W)^−½
50 Ω on FR4 (εr=4.4, h=1.6mm): W ≈ 3.0 mm

Thinner substrate → narrower trace → higher loss · use Rogers for mm-wave

Substrate	εr	tan δ	50Ω W (h=1.6mm)	Best use
FR4	4.2–4.6	0.018–0.025	~3.0 mm	≤3 GHz general
Rogers 4350B	3.48	0.0037	~3.5 mm	3–20 GHz RF
Rogers 5880	2.20	0.0009	~5.0 mm	mmWave, <0.5 dB/cm
Alumina 96%	9.8	0.0003	~0.8 mm	Microwave ICs, hybrids
PTFE (Duroid)	2.1–10.5	0.0002	varies	Microwave, radar

04 — Antennas

Antennas & Arrays

Gain, Directivity & Efficiency

G = η·D · G(dBi) = 10·log(η·D)
EIRP = P_TX·G_TX (linear) = P_TX + G_TX (dBm+dBi)
Effective area: A_eff = G·λ²/(4π)

Isotropic = 0 dBi · λ/2 dipole = 2.15 dBi · Patch ≈ 5–8 dBi

Aperture Antennas

Parabolic dish: G = η·(πD/λ)² (η≈0.55–0.65)
Horn: G ≈ 10·log(10·A/λ²) dBi
HPBW ≈ 70·λ/D degrees

1m dish at 12 GHz: G ≈ 44 dBi · 3dB BW ≈ 1.75°

Patch Antenna

Resonant length: L ≈ λ/(2√εeff)
Width: W ≈ c/(2f)·√(2/(εr+1))
Gain ≈ 5–9 dBi · BW ≈ 2–5% (FR4)

2.4 GHz on FR4 (εr=4.4): L ≈ 29 mm · W ≈ 38 mm

Array Gain & Beamwidth

N elements: G_array = G_element + 10·log(N) dB
HPBW ≈ 0.886·λ/(N·d) rad
Grating lobe when d > λ/(1+|sin θ_s|)

64 elements: +18 dB · 128 elements: +21 dB · d=λ/2 is standard

λ/2 Dipole

Z_in ≈ 73 + j42.5 Ω at resonance
Gain = 2.15 dBi
Resonant length: L ≈ 0.475·λ (with end effect)

2.4 GHz dipole: L ≈ 59 mm · match to 50 Ω with balun + L-network

Yagi-Uda

Director: 0.4λ shorter than driven
Reflector: 5% longer than driven
Gain ≈ 10·log(0.5·N) dBi (N elements)

10-element: ≈7 dBi · 20-element: ≈10 dBi · spacing≈0.3λ

05 — S-Parameters

S-Parameters & Network Conversions

S-Parameter Definitions

S11 = b1/a1|a2=0 (input reflection, Z_out=Z₀)
S21 = b2/a1|a2=0 (forward transmission gain)
S12 = b1/a2|a1=0 (reverse isolation)
S22 = b2/a2|a1=0 (output reflection)

a = incident wave · b = reflected wave · all ports terminated in Z₀

Key Relationships

Gain (dB) = 20·log|S21|
RL (dB) = −20·log|S11|
Insertion loss = −20·log|S21|
Isolation = −20·log|S12|

S21=0.707 (−3dB) → 50% power · S21=0.5 → −6 dB · S21=2 → +6 dB

S → ABCD (Z₀=50 Ω)

A = ((1+S11)(1−S22)+S12S21)/(2S21)
B = Z₀·((1+S11)(1+S22)−S12S21)/(2S21)
C = (1/Z₀)·((1−S11)(1−S22)−S12S21)/(2S21)
D = ((1−S11)(1+S22)+S12S21)/(2S21)

S → Z-Parameters

Z11 = Z₀·(1+S11)(1−S22)+S12S21 / Δ
Z12 = Z₀·2S12 / Δ
Z21 = Z₀·2S21 / Δ
Z22 = Z₀·(1−S11)(1+S22)+S12S21 / Δ
Δ = (1−S11)(1−S22)−S12S21

S → Y-Parameters

Y11 = (1/Z₀)·(1−S11)(1+S22)+S12S21 / Δ
Y12 = (1/Z₀)·−2S12 / Δ
Y21 = (1/Z₀)·−2S21 / Δ
Y22 = (1/Z₀)·(1+S11)(1−S22)+S12S21 / Δ
Δ = (1+S11)(1+S22)−S12S21

Stability (Rollett)

K = (1−|S11|²−|S22|²+|Δ|²)/(2|S12S21|)
|Δ| = |S11S22−S12S21|
Unconditionally stable: K>1 AND |Δ|<1
μ = (1−|S11|²)/(|S22−ΔS11*|+|S12S21|) >1

μ > 1 is single-condition stability criterion (preferred)

S11 (dB)	\|Γ\|	VSWR	Mismatch Loss	% Power reflected
−6 dB	0.50	3.0:1	1.25 dB	25%
−10 dB	0.316	1.93:1	0.46 dB	10%
−15 dB	0.178	1.43:1	0.14 dB	3.2%
−20 dB	0.100	1.22:1	0.044 dB	1%
−30 dB	0.032	1.065:1	0.004 dB	0.1%

06 — Filters

RF Filters

Filter Selectivity Comparison

Butterworth: Maximally flat · −3 dB at ωc
Attenuation (dB) = 10·log(1+(ω/ωc)^2n)
Chebyshev: Equiripple · steeper rolloff
Elliptic: Steepest rolloff · has notches

Butterworth: −20n dB/decade · Chebyshev: better by ~6 dB at octave

LC Ladder — Prototype to Scaled

L_scaled = L_proto · Z₀/ωc
C_scaled = C_proto / (Z₀·ωc)
BPF: L_series→L+C · C_shunt→L||C
BW ratio = ω₂/ω₁ = FBW

Element values from tables (Zverev, Matthaei) · Z₀=50 Ω standard

Resonator Q & Insertion Loss

IL_min ≈ 4.34·n·(ωc/BW)·(1/Q_u) dB
Q_loaded = f₀/BW_3dB
Q_external = 1/coupling factor
1/Q_L = 1/Q_u + 1/Q_ext

Higher Q_u → lower IL · Cavity Q≈10000 · LC Q≈50–200 · SAW Q≈1000+

Group Delay

τ_g = −dφ/dω (seconds)
Flat GD → linear phase → no pulse distortion
Bessel filter: maximally flat GD
GD variation: ΔGD = 1/BW (rough estimate)

1 ns GD variation over 1 GHz BW causes ISI · critical for wideband data

Filter Type	Rolloff	In-Band Ripple	Group Delay	Best For
Butterworth	−20n dB/dec	0 dB (flat)	Moderate variation	General purpose
Chebyshev I	Steeper than BW	ε dB ripple	Worse near band edge	Sharp rejection needed
Elliptic	Steepest	ε dB ripple	Worst near band edge	Adjacent channel rejection
Bessel	Gentlest	0 dB (flat)	Maximally flat	Pulse / data fidelity
SAW/BAW	Very steep	<1 dB	Moderate	Mobile, small form factor

07 — Amplifiers & Mixers

Amplifiers & Mixers

Amplifier Gain Types

G_T = |S21|²·(1−|Γ_S|²)/(|1−S11Γ_S|²)·(1−|Γ_L|²)/|1−S22Γ_L|²
G_A = available gain (Γ_L=Γ_out*)
G_P = operating power gain

Maximum available gain: MAG = |S21/S12|·(K−√(K²−1)) when K>1

LNA Design Key Numbers

NF_min ≈ 1+2·(R_n/G_a)·(G_opt²+B_opt²)
Noise match ≠ power match
F = F_min + (R_n/G_s)·|Y_s−Y_opt|²

Match to Γ_opt not Γ_MS for minimum NF · typical LNA NF=0.5–2 dB

Mixer Fundamentals

Products: f_out = |m·f_RF ± n·f_LO|
Conversion loss ≈ 6–8 dB (passive)
SSB NF ≈ Conversion loss (if no 1/f)
Image: f_im = f_LO − f_IF (low-side LO)

Half-IF spur at f_RF ± f_IF/2 — design IF > signal BW × 5 to avoid

Power Amplifier Efficiency

PAE = (P_out − P_in)/P_dc × 100%
Class A: η_max=50% · Class B: 78.5%
Class D/E/F: >85% (switching)
PAPR penalty: back-off = PAPR (dB)

OFDM PAPR≈10dB → PA at 10% average efficiency → DPD restores 3–5 dB

08 — Unit Conversions

Unit Conversions

Power

dBm = 10·log(P[mW])
P[mW] = 10^(dBm/10)
P[W] = 10^((dBm−30)/10)
0 dBm = 1 mW = 224 mV into 50Ω

+10 dBm = 10 mW · +30 dBm = 1 W · +43 dBm = 20 W

Voltage (50 Ω)

V_rms = √(P·R) = √(P·50)
V_pk = V_rms·√2
dBm = 20·log(V_rms) + 13.01
0 dBm → 223.6 mV_rms → 316 mV_pk

dB Ratios

+3 dB = ×2 power = ×1.414 voltage
+6 dB = ×4 power = ×2 voltage
+10 dB = ×10 power = ×3.16 voltage
+20 dB = ×100 power = ×10 voltage

Nepers ↔ dB

1 Np = 8.686 dB
1 dB = 0.1151 Np
α[dB/m] = α[Np/m] × 8.686
Used in attenuation formulas

Noise Temperature ↔ NF

T_e = 290·(10^(NF/10)−1)
NF = 10·log(1+T_e/290)
T_e=0 → NF=0 dB · T_e=290→NF=3 dB
T_e=75 K → NF=1 dB (good LNA)

Frequency ↔ Wavelength

λ[mm] = 300/f[GHz] (free space)
λ_g[mm] = 300/(f[GHz]·√εeff)
1 GHz: λ=300 mm · 2.4 GHz: 125 mm
28 GHz: 10.7 mm · 77 GHz: 3.9 mm

dBm	mW	V_rms (50Ω)	Common reference
−174 dBm/Hz	4×10⁻²¹ mW	0.45 fV/√Hz	Thermal noise floor (290 K)
−100 dBm	0.01 pW	707 nV	Typical LTE sensitivity
−60 dBm	1 nW	7.07 μV	Strong indoor WiFi
0 dBm	1 mW	224 mV	Test signal reference
+10 dBm	10 mW	707 mV	WiFi TX power
+30 dBm	1 W	7.07 V	Femtocell TX
+43 dBm	20 W	31.6 V	LTE macro base station TX

09 — Constants & Key Numbers

RF Constants

k

1.381×10⁻²³ J/K

Boltzmann constant

T₀

290 K (16.85°C)

Standard noise temperature

kT₀

−174 dBm/Hz

Thermal noise floor @ T₀

c

3×10⁸ m/s

Speed of light (free space)

η₀

377 Ω

Free-space wave impedance

μ₀

4π×10⁻⁷ H/m

Permeability of free space

ε₀

8.854×10⁻¹² F/m

Permittivity of free space

σ_Cu

5.8×10⁷ S/m

Copper conductivity

δ_Cu@1GHz

2.09 μm

Cu skin depth at 1 GHz

Z₀

50 Ω

Standard RF impedance

Z₀_video

75 Ω

Broadcast/cable TV impedance

kT₀B

−101 dBm @ 20 MHz

WiFi noise floor at 290 K

Frequency Band	Range	λ (free space)	Key Applications
HF	3–30 MHz	10–100 m	Shortwave, amateur radio, OTHR radar
VHF	30–300 MHz	1–10 m	FM radio, TV, ATC, TETRA
UHF	300 MHz–3 GHz	10 cm–1 m	GSM, LTE, WiFi, GPS, Bluetooth
SHF / Microwave	3–30 GHz	1–10 cm	5G NR, radar, satellite, microwave links
EHF / mmWave	30–300 GHz	1–10 mm	5G FR2, 77 GHz radar, imaging, WiGig
Sub-THz	300 GHz–3 THz	0.1–1 mm	Security imaging, spectroscopy

10 — Master Thumb Rules

Master Thumb Rules

These rules encode decades of RF engineering experience into instantly-usable heuristics. Each one is a mental shortcut that lets you estimate the right answer in seconds — before you even open a calculator. They are not approximations to memorise blindly: the explanation tells you exactly when each rule holds and when it breaks.

01

−174 dBm/Hz — the floor of the universe

Thermal noise at 290 K in 1 Hz. Every receiver sensitivity calculation starts here. Add NF, add 10·log(BW), add SNR_min. If your sensitivity is worse than −174+NF+10·log(BW)+SNR, you have a problem upstream.

02

Every 1 dB before the LNA costs 1 dB of system NF

A lossy cable, filter, or switch before the LNA adds its loss directly to system NF — because a passive attenuator L at T₀ has F=L. This is why LNAs go at the antenna and switches go after them.

03

×2 distance or frequency → +6 dB path loss

FSPL = 20·log(d) + 20·log(f) + constant. Doubling either adds 20·log(2) = 6.02 dB. Every engineer should be able to estimate path loss in their head for any link.

04

28 GHz is 18 dB worse than 3.5 GHz (same distance)

20·log(28/3.5) = 18 dB. This is exactly the gain a 64-element phased array provides. 5G mmWave works because beamforming gain cancels the frequency penalty — but only when the beam is pointing correctly.

05

P1dB ≈ IIP3 − 9.6 dB

Theoretical relationship from Taylor expansion of nonlinearity. In practice P1dB ≈ IIP3 − 8 to −12 dB. If you measure P1dB you can estimate IIP3 immediately — useful when you don't have a second tone source.

06

IM3 power grows at 3:1 slope — IIP3 grows at 2:1

Raise the input 1 dB: fundamental rises 1 dB, IM3 rises 3 dB. The difference ΔIM3 shrinks 2 dB per dB of drive. IIP3 is where the extrapolated lines cross — so IIP3 = P_in + ΔIM3/2.

07

N elements in a phased array → +10·log(N) dB gain

Coherent combining gives N× power gain over a single element. 64 elements = 18 dB. This is the mechanism that allows 5G mmWave to overcome 18 dB of excess path loss vs 3.5 GHz.

08

Skin depth ∝ 1/√f — Cu skin depth at 1 GHz = 2.1 μm

δ = 1/√(πfμ₀σ). At 10 GHz: 0.66 μm. At 100 GHz: 0.21 μm. A 35 μm copper trace is 33 skin depths thick at 1 GHz (full current use), but only 0.21 μm/0.66 μm at 10 GHz matters. Thinner plating means higher conductor loss at higher frequencies.

09

50 Ω on FR4 (h=1.6mm) → trace width ≈ 3.0 mm

The most common microstrip design point. εr=4.4 → εeff≈3.1 → Z₀=50Ω at W/h≈1.88 → W≈3.0mm. Thinner board or higher εr → narrower trace → more conductor loss. Rogers 4350B (εr=3.48) gives wider trace and lower loss at same impedance.

10

VSWR 2:1 → 0.5 dB mismatch loss and 11% reflected power

|Γ|=1/3 → mismatch loss = −10·log(1−1/9) = 0.51 dB. Often ignored but real. For a receive chain with VSWR 2:1 at the LNA input, 0.5 dB of the noise penalty is mismatch before the LNA even operates.

11

Q_L doubles → phase noise drops 6 dB (Leeson)

Leeson's equation: L(f) ∝ 1/Q_L². The single biggest lever in oscillator design. Switching from LC (Q=50) to DRO (Q=5000) improves phase noise 40 dB. Crystal oscillators (Q=10⁵–10⁶) achieve the lowest phase noise for this reason.

12

PLL in-band noise = ref noise + 20·log(N)

Dividing the VCO by N in the feedback loop multiplies the reference phase noise by N at the output. For N=500 (e.g. 5 GHz from 10 MHz reference) that's +54 dB — the main reason fractional-N PLLs exist (smaller effective N).

13

10× wider BW → 10 dB higher noise floor

Noise floor = −174 + 10·log(B). 5G NR at 100 MHz is 10 dB noisier than LTE at 10 MHz. The only way to compensate without better NF is beamforming gain or denser cell deployment — not better LNA, since NF is already near limits.

14

λ/4 line transforms impedance: Z_in = Z₀²/Z_L

Quarter-wave transformer. Open → short. Short → open. 200 Ω load + 100 Ω line → 50 Ω. The most used single-frequency matching technique in microwave design. Works exactly at one frequency and degrades with bandwidth.

15

60 GHz: O₂ absorption = 15 dB/km — 200 m maximum range

Oxygen absorption peak at 60 GHz limits links to ~100–200 m regardless of TX power. This is intentionally exploited for WiGig and secure short-range links. At 77 GHz (automotive radar) absorption drops to 0.4 dB/km — effectively transparent.

16

OFDM PAPR ≈ 10–12 dB — PA back-off costs efficiency

When N subcarriers align in phase, peak power = N × average. Statistical PAPR for LTE/5G NR is 10–12 dB. A 40 W PA running 10 dB back-off delivers 4 W average — 10% efficiency. DPD linearises the PA, allowing 3–5 dB less back-off.

17

Rayleigh fading: 99% coverage needs 20 dB fade margin

For Rayleigh fading (dense urban, no LOS), 1% outage probability requires 20 dB margin above the mean received power. With 2-antenna MRC diversity, the same coverage needs only 10 dB — diversity is far more power-efficient than brute-force margin.

18

Concrete wall: +12–15 dB penetration loss at 3.5 GHz

Building penetration loss is often the biggest single term in indoor coverage planning. One concrete wall = 12–15 dB at 3.5 GHz (worse at mmWave: 20–40 dB). Glass = 2–3 dB. Plasterboard = 3–5 dB. Critical for indoor 5G small cell placement.