Engineering Reference - Equations
VSWR to RSLdB
Voltage Reflection Coefficient
Z -Z
Γ = Z r + Zo
r
o
where:
RSLdB = 20Log
( )
VSWR-1
VSWR+1
Γ = reflection coefficient
Zr = impedance at reflection
Zo = characteristic impedance (typically 50Ω)
1
Impedance Z = R±jX = = Z0 1 + Γ
Y
1-Γ
Admittance Y = G±jX =
( )
( )
Voltage Standing Wave Ratio (VSWR)
1
= Y0 1 - Γ
Z
1+Γ
VSWR = r =
1 + Γ
1 - Γ
( )
2
Pr
=Γ = r - 1
r+1
Pi
2
Pt
=1-Γ = 4r 2
Pi
(r + 1)
RSLdB to VSWR
(( ) )
(( ) )
10
VSWR =
2
10
RSL
dB
20
RSL
dB
20
where: r = VSWR
Γ = reflection coefficient
Pr = reflected power
Pi = incident power
Pt = transmitted power
+1
-1
Material Parameters at 20°C Table 1.1
Nonmetals
ε″ at frequency
ε′ ,
εr, at frequency
Material
µr
60
10 6
1010
Nylon
Plexiglas
Polyethylene
Teflon (22°C)
1
1
1
1
3.60
3.45
2.26
2.10
3.14
2.76
2.26
2.10
2.80
2.50
2.26
2.10
60
106
0.018
0.022
0.064
0.104
(<0.0002)
(<0.005)
1010
(V/inch)
0.0110
0.0050
0.005
0.004
400
990
1200
1500
Metals
Material
µr
εr
Silver
Copper
Aluminum
Brass
1
1
1
1
1
1
1
1
σ( /m)
6.17
5.8
3.72
1.6
x
x
x
x
10 7
10 7
10 7
10 7
Depth of penetration δ for plane waves (m)
0.064/√ f
0.066/√ f
0.083/√ f
0.013/√ f
All specifications are subject to change without notice
Kathrein Inc., Scala Division Post Office Box 4580 Medford, OR 97501 (USA) Phone:(541) 779-6500 Fax:(541) 779-3991
Engineering Reference - Equations
Characteristic Impedance of Free Space
√
ηo =
where:
Effective Aperature Related to Gain of Antenna
µo
εo = 120πΩ = 377Ω
µo
Aem =
= free space permeability
= 4.0π x 10-7(H/m)
= free space permitivity
εo
G=
−9
=
( )
10 (F/m)
36π
λ2 G
4π
4πA em
λ2
Watts to dBm
1
c = √µoεo = propagation velocity = 2.997925 x 108 m/s
(≅ 3 x 108 m/s)
In free space the wavelength is:
c
λ= f
dBw = 10 log10(Power Watts)
dBm = (10 log10(Power Watts))+30
dBw to Watts
For a nonmagnetic dielectric:
λ
c
λd = f ε = o
√ r √ εr
where: εr is relative dielectric from Table 1.1
Watts = 10
(dBw
10 )
( dBw+30
10 )
Milliwatts = 10
dBm to Watts
Characteristic Impedance of Coaxial Line
a
( ) √ µε
b
Zo = 138 log10 a
(dBm
10 )
Milliwatts = 10
r
b
Friis Transmission Equation
2
( )
where: Pr = received power
Pt = transmitted power
R = separation distance
λ
Voltage Gain/Loss to dB
dB(gain/loss) = 20Log10(Gain or Loss)
dBm to Volts/µVolts
( )
( )
Volts = Log10-1 dBm-13
20
λ GG
t r
4πR
µVolts = Log10-1 dBm-107
20
dBw to Volts/µVolts
2
( )
4πR
( dBm-30
10 )
r
where: a = inner diameter
b = outer diameter
µr = relative permeability (usually = 1)
εr = permitivity (dielectric constant)
as given in Table 1.1
Pr
=
Pt
Watts = 10
= free space loss
Using effective areas
A A
Pr
= et er
Pt
λ 2R
( )
( )
Volts = Log10-1 dBw-17
20
µVolts = Log10-1 dBw-137
20
All specifications are subject to change without notice
Kathrein Inc., Scala Division Post Office Box 4580 Medford, OR 97501 (USA) Phone:(541) 779-6500 Fax:(541) 779-3991
Engineering Reference - Equations
Quarter Wave Matching
Radio Horizon (in miles)
Z = √ZoZL
H = √2 (Tx)½ (Rx)½
where: Z = line impedance
Zo = desired input impedance
ZL = given load impedance
where: H = horizon
Tx = transmit height in feet
Rx = receive height in feet
Noise Factor
F=
Gain of Parabolic Antenna
Pno
SNRIN
=
=
GAPni
SNROUT
()
Te
-1
To
πD
G = 10 logK
λ
2
( )
where: G = parabolic antenna gain
K = eff L 55%
D = diameter in feet
λ = wavelength in feet
Noise Figure
NF = 10LogF
Cascade Noise Factor
Beamwidth of Parabolic Antenna
F -1
F3-1
F = F1 + 2 +
+…
GA1
GA1GA1
Ψ=
Freespace Path Loss
70 λ
D
where:
L = 96.6 + 20 log(d) + 20 log(f)
Ψ = beamwidth
D = diameter in feet
λ = feet
where: L = freespace path loss
d = distance in miles
f = frequency in GHz
Directive Antenna Gain
G = 41253
.
θ φ
theoretical
G = 32400
.
θ φ
corrected for efficiencies
where: G = directive antenna gain
θ = horizontal beamwidth
φ = vertical beamwidth
All specifications are subject to change without notice
Kathrein Inc., Scala Division Post Office Box 4580 Medford, OR 97501 (USA) Phone:(541) 779-6500 Fax:(541) 779-3991
Engineering Reference - Equations
0
2πL = 10
λ
Normalized Directivity, dB
-2
-4
2π L = 100
λ
-6
Cos θ
-8
-10
0
10
20
30
40
50
60
70
80
90
Scan Angle From Broadside, θ
Reprinted from “Microwave Scanning Antennas”, edited by R. C. Hansen, Vol. 1, p. 20, published by Peninsula Publishing, Los Altos, California.
Courtesy of R. C. Hansen
All specifications are subject to change without notice
Kathrein Inc., Scala Division
Post Office Box 4580
Medford, OR 97501 (USA)
Phone:(541) 779-6500
Fax:(541) 779-3991
Engineering Reference - Equations
Beam Broadening Versus Sidelobe Ratio
1.7
1.6
Taylor one-parameter distribution
Beam Broadening
1.5
1.4
1.3
1.2
1.1
1.0
10
20
30
40
50
Sidelobe Ratio, dB
Courtesy of Dr. R. C. Hansen
All specifications are subject to change without notice
Kathrein Inc., Scala Division
Post Office Box 4580
Medford, OR 97501 (USA)
Phone:(541) 779-6500
Fax:(541) 779-3991