Radio Equations

This is a quick reference for various equations useful in Amateur Radio, especially ones on the various exams. Many of these are useful in general in Engineering, but this is not meant to be a reference guide for all engineering equations.

NOTE: some equations are missing here and need to be filled in!

Basics

Wavelength and frequency, band conversions, etc.

<math>\lambda = \frac{c}{f}</math>

frequency (mhz) to quarter wavelength (ft)

246 / MHz = feet (useful for antenna calculations)

frequency (mhz) to wavelength (m)

300 / MHz = m

ohm's law, power, impedance

• <math>V = IR</math>
• <math>P = IV = I^2 R</math>
• <math>Z_C = \frac{-1}{\omega C}</math>
• <math>Z_L = \omega L</math>
• <math>\omega = 2 \pi f </math>

• series resistance adds
• two parallel resistances: <math>R_1||R_2 = \frac{R_1R_2}{R_1+R_2}</math>
• multiple parallel resistances: <math>R_1||...||R_n = \frac{1}{\frac{1}{R_1}+...+\frac{1}{R_n}}</math>
• parallel combinations of reactances
• time constant
• current / voltage lead/lag with reactance

resonance

See also ARRL Handbook 1996 pg 6.35

• <math>f = \frac{1}{2 \pi \sqrt{LC}}</math>
• <math>\omega = \frac{1}{\sqrt{LC}}</math>

Quality:

• <math>Q = \frac{f}{BW} = \frac{V_Z}{V_R}</math>
• <math>BW = f_{high} - f_{low}</math>
• <math>V = V_i e^{T/\tau}</math>

Transformers, inductors, and toroids

• transformer equation
  • impedance transformer examples

powdered iron toroidal inductor

<math>L = \frac{ A_L N^2}{10000}</math>

<math>L = \text{inductance in }\mu H</math>

<math>A_L = \text{inductance index in } \mu H / 100 \text{ turns}</math>

<math>N = \text{number of turns}</math>

ferrite toroidal inductors

same as above except for units: <math>mH/1000</math> instead of <math>\mu H/100</math>, and use <math>10^6</math> instead of <math>10^4</math>

See also ARRL Handbook 1996, pg 6.25

SWR, RMS, PEP

• <math>V_{rms} = \frac{V_{peak}}{ \sqrt 2} \approx 0.707\ V_{peak}</math>

• relation between SWR and impedance mismatch
• PEP ?

Decibels

<math>dB = 10 \log \frac{A_1}{A_2} </math>

Propagation

Warning: these equations are approximations for the radio horizon. There are no exact equations due to localized variations in terrain geometry and conductivity. (There are software models that can estimate it closer than these equations though.)

VHF propagation via groundwave propagation is limited by line of sight with some refraction towards the earth. Thus, your maximum distance is limited by the curvature of the earth. According to the Antenna Book (19th ed, p23-5) the radio horizon is

<math>D = C \sqrt{H}</math>

where D is distance, H is height, and C is

C	D units	H units
1.415	miles	feet
4.124	km	meters

This equation is an approximation for VHF that combines the exact geometric relation with the offset for refraction. The amount of refraction changes with frequency making this equation less accurate outside of vhf.

An alternate equation[1] calculates the horizon vs. frequency based on where diffraction effects take over and weaken your signal:

<math> D_{km} = \frac{80}{\sqrt[3]{f_{MHz}}} </math>