# Formulas on the Extra Exam

This page contains all the formulas needed for the Extra class ham radio license exam.  You might want to print out these formulas and review them just before entering the exam room, but leave this sheet in the car!  Do not bring it into the exam room with you!

 Prefix Prefix Value name symbol International System of Units (SI) giga- G 109 1,000,000,000 one billion mega- M 106 1,000,000 one million kilo- k 103 1,000 one thousand (none) (none) 100 1 one centi- c 10−2 .01 one one-hundredth milli- m 10−3 .001 one one-thousandth micro- µ 10−6 .000001 one one-millionth nano- n 10−9 .000000001 one one-billionth pico- p 10−12 .000000000001 one one-trillionth

Antenna gain in dBd vs dBi:

$$gain\text-of\text-antenna\text-in\text-dBd = gain\text-of\text-antenna\text-in\text-dBi - 2.15~ \text{dB}$$

$$ERP = transmitter \text{-}power \times 10^\left(\frac{gain~in~dB}{10}\right)$$

Length of transmission line:

$$\lambda = \frac{c~ \times~ velocity \text{-}\!factor}{f}$$

Forward and reflected power:

$$power\text-to\text-load = forward\text-power - reflected\text-power$$

Third-order intermodulation products:

Formula  Solve for ƒ2
$$f_i = 2f_1 + f_2$$ $$f_2 = f_i - 2f_1$$
$$f_i = 2f_1 - f_2$$ $$f_2 = 2f_1 - f_i$$
$$f_i = 2f_2 + f_1$$ $$f_2 = \frac{f_i - f_1}{2}$$
$$f_i = 2f_2 - f_1$$ $$f_2 = \frac{f_i + f_1}{2}$$ Operational amplifiers:

$$V_{OUT} = -V_{IN} × \frac{R_F}{R1}$$

$$A_V = \frac{R_F}{R1}$$

Image response frequencies:

$$f_{img1} = f_{RF} - 2 \times f_{IF}$$

$$f_{img2} = f_{RF} + 2 \times f_{IF}$$

Noise floor:

$$BNF = NF + 10 \times \text{log}(BW)$$

where:

$$BNF$$ is the bandwidth noise floor (the noise for the entire received bandwidth) (in dBm)

$$NF$$ is the 1-Hz noise floor (in dBm/Hz)

$$BW$$ is the receive filter bandwidth (in Hz)

Time constant (all components in parallel):

$$R_t = \frac{R_i}{n}$$

$$C_t = C_1 + C_2$$

$$T = R \times C$$

Parts per million:

$$maximum \text{-} error = measurement \times accuracy$$

Resonant frequency:

$$f_R = \frac{1000}{2\pi \sqrt{LC}}$$

where:

$$L$$ in $$\mu\text{H}$$

$$C$$ in $$\text{pF}$$

$$f_R$$ in $$\text{MHz}$$

Half-power bandwidth:

$$hal\!f \text{-} power \text{-} bandwidth = \frac {f_R}{Q}$$

Transformer turns #1:

$$N = 100 \times \sqrt{\frac{L}{A_L}}$$

where:

L in $$\mu\text{H}$$

AL in $$\mu\text{H} / 100 \text{ turns}$$

Transformer turns #2:

$$N = 1000 \times \sqrt{\frac{L}{A_L}}$$

where:

L in $$\text{mH}$$

AL in $$\text{mH} / 1000 \text{ turns}$$

Frequency modulation:

$$deviation \text{-}ratio = \frac{D_{MAX}}{M_{MAX}}$$

$$modulation \text{-}index = \frac{frequency\text-deviation}{modulating\text-frequency}$$

Inductive and capacitive reactances:

$$X_L = 2\pi fL$$

$$X_C = \frac{1}{2\pi fC}$$

$$X = X_L - X_C$$

Phase angle:

$$\theta = \text {arctan} \left(\frac {X}{R}\right)$$

Power factor:

$$power \text{-}\!factor = \text{cos} \left(\theta \right)$$

$$apparent \text{-}power = V_{RMS} \times I$$

$$true \text{-}power = apparent \text{-}power \times power \text{-}\!factor$$

$$P = I^2 \times R$$