RF Engineering Formulas

AntennasIntermediate

Antenna Effective Aperture

Physical area from which an antenna captures power

A_e = \frac{G \lambda^2}{4\pi}

Alternative Forms

Gain from Aperture:Calculate gain given physical aperture

With Efficiency:Effective aperture accounting for aperture efficiency

Variables

G– Antenna Gain

f– Frequency

apertureeffective areaantenna gaincapture areadish antenna

Transmission LinesIntermediate

Coaxial Cable Characteristic Impedance

Calculate impedance from coax dimensions and materials

Z_0 = \frac{138}{\sqrt{\varepsilon_r}} \log_{10}\left(\frac{D}{d}\right)

Alternative Forms

Natural Log Form:Using natural logarithm

Air Dielectric:For air-insulated coax (εᵣ = 1)

Variables

D– Outer Conductor ID

d– Inner Conductor OD

ε_r– Relative Permittivity

coaximpedancetransmission linecharacteristic impedancecable design

System DesignBasic

Decibel Power Conversion

Convert between watts and dBm/dBW

P_{dBm} = 10\log_{10}\left(\frac{P_{W}}{0.001}\right) = 10\log_{10}(P_{mW})

Alternative Forms

dBW Conversion:Convert to/from dBW (referenced to 1 watt)

Watts from dBm:Convert dBm back to watts

Variables

P_W– Power (Watts)

decibeldBmdBWpowerconversion+1

PropagationBasic

Free Space Path Loss (FSPL)

Signal attenuation in free space due to geometric spreading

FSPL = 20\log_{10}(d) + 20\log_{10}(f) + 20\log_{10}\left(\frac{4\pi}{c}\right)

Alternative Forms

Simplified (dB):Practical form with frequency in MHz and distance in km

Linear Form:Path loss as a power ratio (not in dB)

Variables

d– Distance

f– Frequency

path lossFSPLpropagationfriisfree space+1

System DesignIntermediate

Friis Cascade Noise Formula

Calculate total noise figure of cascaded amplifiers

F_{total} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots

Alternative Forms

Two-Stage:Simplified form for two stages

In dB (Two-Stage):Approximate dB calculation

Variables

NF_1– Stage 1 Noise Figure

G_1– Stage 1 Gain

NF_2– Stage 2 Noise Figure

G_2– Stage 2 Gain

+1 more variables

cascadenoise figurefriisLNAreceiver design+1

PropagationIntermediate

Friis Transmission Equation

Calculate received power given transmit power and antenna gains

P_r = P_t + G_t + G_r - FSPL

Alternative Forms

Linear Form:Power in watts, gains as ratios

Full dB Form:All terms in dB/dBm/dBi

Variables

P_t– Transmit Power

G_t– Transmit Antenna Gain

G_r– Receive Antenna Gain

f– Frequency

+1 more variables

friislink budgetreceived powerantenna gainpath loss

AntennasBasic

Half-Wave Dipole Length

Physical length of a resonant half-wave dipole antenna

L = \frac{\lambda}{2} \times k = \frac{c}{2f} \times k

Alternative Forms

Practical Formula (meters):Includes 95% velocity factor for typical wire antennas

Practical Formula (inches):Length in inches for the US market

Variables

f– Frequency

k– Velocity Factor

dipoleantenna lengthhalf-waveresonanceham radio

System DesignAdvanced

Link Budget Equation

Complete power budget for a wireless communication link

P_r = P_t + G_t - L_t - L_{fs} - L_m + G_r - L_r

Alternative Forms

Link Margin:Margin above minimum required signal level

EIRP Form:Using Effective Isotropic Radiated Power

Variables

P_t– Transmit Power

G_t– Transmit Antenna Gain

L_t– Transmit Losses

f– Frequency

+5 more variables

link budgetsystem designEIRPpath lossmargin+1

NoiseIntermediate

Noise Figure and Noise Temperature

Convert between noise figure and equivalent noise temperature

T_e = T_0(F - 1)

Alternative Forms

Noise Figure from Temperature:Calculate noise factor from equivalent temperature

Noise Figure in dB:Express noise figure in decibels

Variables

NF– Noise Figure

noise figurenoise temperatureLNAreceiversensitivity

Signal ProcessingBasic

Nyquist Sampling Theorem

Minimum sampling rate to avoid aliasing

f_s \geq 2 f_{max}

Alternative Forms

Maximum Frequency:Highest frequency that can be represented (Nyquist frequency)

With Guard Band:Practical rule including filter roll-off margin

Variables

f_{max}– Maximum Signal Frequency

samplingnyquistaliasingADCdigital signal processing

MatchingIntermediate

Quarter-Wave Transformer

Match two impedances using a quarter-wave transmission line

Z_T = \sqrt{Z_0 \cdot Z_L}

Alternative Forms

Length Formula:Physical length of the transformer section

Bandwidth Approximation:Bandwidth for maximum acceptable reflection Γm

Variables

Z_0– Source Impedance

Z_L– Load Impedance

f– Frequency

matchingquarter wavetransformerimpedance transformation

MatchingBasic

Reflection Coefficient from Impedance

Calculate reflection coefficient from load and line impedance

\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}

Alternative Forms

Normalized Form:Using normalized impedance z = Z_L/Z_0

From Admittance:Using admittance instead of impedance

Variables

Z_L– Load Impedance

Z_0– Characteristic Impedance

reflection coefficientimpedance mismatchsmith chartmatching

Transmission LinesIntermediate

Skin Depth

Penetration depth of electromagnetic waves in conductors

\delta = \sqrt{\frac{2\rho}{\omega\mu}} = \sqrt{\frac{\rho}{\pi f \mu}}

Alternative Forms

For Copper:Simplified formula for copper at room temperature

Surface Resistance:RF surface resistance per square

Variables

f– Frequency

ρ– Resistivity

skin depthconductorlosscoppershielding

Signal ProcessingIntermediate

SNR and Eb/N0 Relationship

Convert between SNR and energy-per-bit to noise density

\frac{E_b}{N_0} = \frac{S}{N} \cdot \frac{B}{R_b}

Alternative Forms

In dB:Decibel form for practical calculations

With Spectral Efficiency:Using spectral efficiency η = Rb/B

Variables

SNR– Signal-to-Noise Ratio

B– Bandwidth

R_b– Bit Rate

SNREb/N0digital communicationsBERmodulation

NoiseBasic

Thermal Noise Power

Johnson-Nyquist noise power in a bandwidth

P_n = kTB

Alternative Forms

In dBm at Room Temperature:Practical form for 290K reference temperature

Noise Voltage:RMS noise voltage across resistance R

Variables

T– Temperature

B– Bandwidth

noisethermal noisejohnson noisenoise floorsensitivity

Transmission LinesBasic

VSWR and Reflection Coefficient

Relationship between VSWR and reflection coefficient

VSWR = \frac{1 + |\Gamma|}{1 - |\Gamma|}

Alternative Forms

Reflection Coefficient from VSWR:Calculate reflection coefficient given VSWR

Return Loss:Return loss in dB from reflection coefficient

Variables

Γ– Reflection Coefficient

VSWRreflectionmismatchreturn lossimpedance matching

PropagationBasic

Wavelength-Frequency Relationship

Convert between wavelength and frequency using speed of light

\lambda = \frac{c}{f}

Alternative Forms

Frequency Form:Solving for frequency given wavelength

Practical Units:Wavelength in meters, frequency in MHz

Variables

f– Frequency

wavelengthfrequencyspeed of lightfundamental

RF Formulas

Antenna Effective Aperture

Coaxial Cable Characteristic Impedance

Decibel Power Conversion

Free Space Path Loss (FSPL)

Friis Cascade Noise Formula

Friis Transmission Equation

Half-Wave Dipole Length

Link Budget Equation

Noise Figure and Noise Temperature

Nyquist Sampling Theorem

Quarter-Wave Transformer

Reflection Coefficient from Impedance

Skin Depth

SNR and Eb/N0 Relationship

Thermal Noise Power

VSWR and Reflection Coefficient

Wavelength-Frequency Relationship

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