# AERO230 Formulae

## Terms

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Basic Electromagnetism
Wavevector
k = 2*pi / lambda
= 2*pi * f / c

lambda = wavelength
f = frequency
c = speed of light

Decibels
two formats!
Power(dB) = 10 log10 (Power)
Power dBm = 10 log10 (Power(dBm))
Area dBsm = 10 log10 (Power(dBsm))

E(dBmicroV/meter) = 20 log10 (E(microV/meter)

Antenna Gain
Gr = Gt = ?

Gr = 4pi Ae / lambda^2 = Gt

Ae = Antenna effective area
lambda = wavelength

Gain transmitted
Gt = 4pi / (dTHETA dPHI)

dTHETA = angular beamwidth vertical
dPHI = angular beamwidth horizontal

Vertical Beamwidth
general / circular?

general:
dTHETA = lambda / 2a
circular:
dTHETA = 1.02 * lambda/2a

lambda = wavelength

Pr = (Pt Gt Gr lambda^2 sigma Ls) / ((4pi)^3 R^4))

Pt = power transmitted
Gt = transmitter gain
lambda = wavelength
sigma = RCS radar cross section
Ls = Loss factor (0.3 - 0.5)
R = range

N = k T0 B F
k = Boltzmann's Constant
= 1.38 * 10 ^(-23)J/K
T0 = temperature
B = bandwidth
F = "noise figure"

Signal-to-Noise Ratio
SNR = Pr/N =
(Pt Gt Gr lambda^2 Ls) / ((4pi)^3 R^4 N)

N = Noise
Pt = Power transmitted
Gt = Gain transmitted
lambda = wavelength
Ls = loss factor (0.3 - 0.5)
R = range

Range Resolution
(rough)

dR = 0.8*c*t0

c = speed of light
t0 = pulse duration

Range accuracy
dR = c / (2B sqrt(2SNR))

c = speed of light
B = bandwidth
SNR = signal-to-noise ratio

Doppler Shift
fd = 2 vr / lambda

lambda = wavelength

frequency (bandwidth)
fBW = 1 / t0

t0 = pulse length

Range and Max Range
R = c * tr / 2
Rmax = c / 2fPRF

c = speed of light
tr = time to recieve
fPRF = pulse repetition frequency

Two dopplers
Integration time
Frequency error

tint = 1/dfd
df = 1/(tint sqrt(2SNR))
dvr = lambda / (2tint sqrt(2SNR))

tint = integration time
dfd = difference between doppler shifts
df = frequency error
SNR = signal-to-noise ratio
lambda = wavelength

Summation of many scatterers

sigma = sum(sqrt(sigma-m) e^(j phi-m)

sigma = RCS
sigma-m = RCS of individual scatterer
phi-m = 2-way phase difference

phi / 2 pi = (d sin(theta)) / lambda

phi = phase difference between elements
d = distance between elements
lambda = wavelength

Gain GaTHETA

GaTHETA = abs(EaTHETA)^2 / N^2
= (sin^2(N pi (d/lambda) sin(theta)) / (N^2 sin^2(pi (d/lambda) sin(theta)))

EaTHETA = field intensity pattern
N = number of antennae in array
d = distance between antennae
lambda = wavelength

cross-range resolution

dcr = R lambda / 2a

R = range to target
lambda = wavelength

cross-range resolution to a

dcr = lambda / 2thetaB = 2a/2 = a

lambda = wavelength
thetaB = beamwidth

Ring Laser Gyro
df = 4 A omega / (L lambda)

A = path area
omega = rate of rotation
L = path length
lambda = wavelength (laser average, in this case)

Fibre Optical Gyro
dPHI = (8pi A N omega)/(c lambda)

A = path area
N = number of turns
omega = rate of rotation
c = speed of light
lambda = wavelength

Schuler Frequency
theta..
and T

theta.. = (a - g sin(theta)) / Re = (a - gtheta) / Re

gsin(theta) = horizontal error due to earth's rotation
Re = radius of the earth
gtheta = horizontal error again

T = 4pi sqrt(Re/g) = 84 minutes

LORAN
D = sqrt((x-x1)^2 + (y-y1)^2) - sqrt((x-x2)^2 + (y-y2)^2) = c*td

D = distance
x1,y1 / x2,y2 = fixed points
x, y = observer point
c = speed of light
td = time delay

Imaging Systems
angular resolution
dTHETA = theta / N

dTHETA = angular resolution
theta = field of view (rads)
N = number of pixels

Imaging Systems
Rayleigh Criterion
dTHETA = 1.22lambda / D

lambda = wavelength
D = aperture diameter

Imaging Systems
Johnson Criteria (3)
Rd = (dX/3)/dTHETA
Rr = (dX/7)/dTHETA
Ri = (dX/11)/dTHETA

Rd = Range of Detection
Rr = Range of Recognition
Ri = Range of Identification
dX = Critical Dimension (smallest of height/width/length)
dTHETA = angular resolution

Air Data Sensors
Pitot tube Airspeed
Indicated versus true?

Q = .5 rho V^2 = Pt - Ps

Q = dynamic pressure
rho = air density
V = airspeed
Pt = total pressure
Ps = static pressure