AERO230 Formulae
Terms
copy deck

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)

Radar Equations
Antenna Gain
Gr = Gt = ?

Gr = 4pi Ae / lambda^2 = Gt
Ae = Antenna effective area
lambda = wavelength

Radar Equations
Gain transmitted

Gt = 4pi / (dTHETA dPHI)
dTHETA = angular beamwidth vertical
dPHI = angular beamwidth horizontal

Radar Equations
Vertical Beamwidth
general / circular?

general:
dTHETA = lambda / 2a
circular:
dTHETA = 1.02 * lambda/2a
lambda = wavelength
a = aperture radius

Radar Equations
Power Received

Pr = (Pt Gt Gr lambda^2 sigma Ls) / ((4pi)^3 R^4))
Pt = power transmitted
Gt = transmitter gain
Gr = receiver gain
lambda = wavelength
sigma = RCS radar cross section
Ls = Loss factor (0.3  0.5)
R = range

Radar Equations
Radar Noise

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

Radar Equations
SignaltoNoise Ratio

SNR = Pr/N =
(Pt Gt Gr lambda^2 Ls) / ((4pi)^3 R^4 N)
Pr = Power received
N = Noise
Pt = Power transmitted
Gt = Gain transmitted
Gr = Gain received
lambda = wavelength
Ls = loss factor (0.3  0.5)
R = range

Radar Equations
Range Resolution
(rough)

dR = 0.8*c*t0
c = speed of light
t0 = pulse duration

Radar Equations
Range accuracy

dR = c / (2B sqrt(2SNR))
c = speed of light
B = bandwidth
SNR = signaltonoise ratio

Radar Equations
Doppler Shift

fd = 2 vr / lambda
vr = radial velocity
lambda = wavelength

Radar Equation
frequency (bandwidth)

fBW = 1 / t0
t0 = pulse length

Radar Equations
Range and Max Range

R = c * tr / 2
Rmax = c / 2fPRF
c = speed of light
tr = time to recieve
fPRF = pulse repetition frequency

Radar Equations
Two dopplers
Integration time
Frequency error
Radial Velocity 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 = signaltonoise ratio
dvr = radial velocity error
lambda = wavelength

Radar Equations
Radar Cross Section (RCS)
Summation of many scatterers

sigma = sum(sqrt(sigmam) e^(j phim)
sigma = RCS
sigmam = RCS of individual scatterer
phim = 2way phase difference

Radar Equations
Phased Array Radar

phi / 2 pi = (d sin(theta)) / lambda
phi = phase difference between elements
d = distance between elements
lambda = wavelength

Radar Equations
Phased Array Radar
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

Radar Equations
Synthetic Aperture Radar
crossrange resolution

dcr = R lambda / 2a
R = range to target
lambda = wavelength
a = antenna radius

Radar Equations
Synthetic Aperture Radar
crossrange resolution to a

dcr = lambda / 2thetaB = 2a/2 = a
lambda = wavelength
thetaB = beamwidth

Inertial Navigation Systems
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)

Inertial Navigation Systems
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

Intertial Navigation Systems
Schuler Frequency
theta..
and T

theta.. = (a  g sin(theta)) / Re = (a  gtheta) / Re
a = acceleration reading
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

Radio Navigation Systems
LORAN

D = sqrt((xx1)^2 + (yy1)^2)  sqrt((xx2)^2 + (yy2)^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
Indicated is without temperature reading