RF Calculators

Impedance & Matching
Propagation
Antenna Design
RF Components
Signal & Noise
Fundamentals
Microstrip Patch Antenna
Designs a rectangular microstrip patch antenna. Outputs dimensions, effective ε r , and inset feed position for a 50Ω match.
Design Parameters
Resonant Frequency GHz
Substrate
Substrate ε r
Substrate Height (h) mm
Thicker substrates give wider bandwidth but larger patch size.
Conductor Thickness (t) mm
Standard PCB copper: 1 oz = 0.035 mm, 2 oz = 0.070 mm.
Patch Diagrams
Cross-Section
Patch ε r Ground Plane W h Cross-section (not to scale)
Top View — Inset Feed
W L y₀ 50Ω feed line
y₀ = inset distance for 50Ω impedance match
Patch Dimensions & Feed
Patch Width W
-
mm
Patch Length L
-
mm
Effective ε r
-
dimensionless
Fringe Extension ΔL
-
mm each end
Edge Impedance Z in
-
Ω (radiating edge)
Inset Feed y₀
-
mm for 50 Ω
Free-Space λ
-
mm
Guide λ /2
-
mm
Bandwidth (est.)
-
% (VSWR ≤ 2)
Equations: Balanis, Antenna Theory Ch. 14, transmission line model. Width from half-wave resonance in effective medium; length shortened by fringe extension ΔL at each radiating edge. Edge impedance Z in = 1/(2(G 1 +G 12 )) — inset feed y₀ = (L/π) × arccos(√(50/Z in )). Bandwidth estimate: BW ≈ 3.77 × ((ε r −1)/ε r ²) × (W/L) × (h/λ). Results are a starting point; final dimensions should be verified by simulation (e.g. HFSS) and measurement.
Equations
Patch Width:    \( W = \dfrac{c}{2f}\sqrt{\dfrac{2}{\varepsilon_r+1}} \)
Effective permittivity:    \( \varepsilon_{r,\text{eff}} = \dfrac{\varepsilon_r+1}{2} + \dfrac{\varepsilon_r-1}{2}\left(1+\dfrac{12h}{W}\right)^{-1/2} \)
Fringe extension:    \( \Delta L = 0.412\,h\,\dfrac{(\varepsilon_{r,\text{eff}}+0.3)(W/h+0.264)}{(\varepsilon_{r,\text{eff}}-0.258)(W/h+0.8)} \)
Patch Length:    \( L = \dfrac{c}{2f\sqrt{\varepsilon_{r,\text{eff}}}} - 2\Delta L \)
Radiation conductance:    \( G_1 = \dfrac{W/\lambda_0}{120\pi^2}\!\left[1 - \tfrac{1}{24}(k_0 h)^2\right] \)
Mutual conductance:    \( G_{12} \approx G_1\,\operatorname{sinc}(k_0 L/2) \)
Edge impedance:    \( Z_{\text{in}} = \dfrac{1}{2(G_1+G_{12})} \)
Inset feed distance:    \( y_0 = \dfrac{L}{\pi}\arccos\!\sqrt{\dfrac{50}{Z_{\text{in}}}} \)
Bandwidth (est.):    \( \text{BW} \approx 3.77\,\dfrac{\varepsilon_r-1}{\varepsilon_r^2}\,\dfrac{W}{L}\,\dfrac{h}{\lambda_0} \)
References
[1] C. A. Balanis, Antenna Theory: Analysis and Design , 4th ed., Wiley, 2016. Ch. 14.
[2] D. M. Pozar, “Microstrip antennas,” Proc. IEEE , vol. 80, no. 1, pp. 79–91, Jan. 1992.
[3] E. O. Hammerstad, “Equations for microstrip circuit design,” 5th Eur. Microw. Conf. , 1975.
Interactive Smith Chart
Interactive Smith Chart. Click or drag to plot impedance. All RF parameters update in real time.
Input
Reference Impedance Z 0
Impedance R + j X Ω
Or click anywhere on the chart to place a point
Parameters
|Γ|
-
angle: - deg
VSWR
-
:1
Return Loss
-
dB
Mismatch Loss
-
dB
Impedance Z
-
Ω
Admittance Y
-
mS
Transmission Line Rotation (toward generator)
Frequency GHz
Length mm
Velocity Factor
Rotation
-
degrees
Z at Source
-
Ω
VSWR
-
same as load
Line Length
-
wavelengths
Equations
Reflection coefficient:    \( \Gamma = \dfrac{Z - Z_0}{Z + Z_0} \)
Impedance from \(\Gamma\):    \( Z = Z_0\,\dfrac{1+\Gamma}{1-\Gamma} \)
VSWR:    \( \text{VSWR} = \dfrac{1+|\Gamma|}{1-|\Gamma|} \)
Return Loss:    \( \text{RL} = -20\log_{10}|\Gamma| \quad \text{dB} \)
Mismatch Loss:    \( \text{ML} = -10\log_{10}(1-|\Gamma|^2) \quad \text{dB} \)
TL rotation (to generator):    \( \Gamma_\text{in} = \Gamma_L\,e^{-j2\beta\ell}, \quad \beta = \dfrac{2\pi f \cdot \text{VF}}{c} \)
References
[1] P. H. Smith, “Transmission line calculator,” Electronics , vol. 12, pp. 29–31, Jan. 1939.
[2] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 2.
[3] G. Gonzalez, Microwave Transistor Amplifiers , 2nd ed., Prentice Hall, 1997. Ch. 2.
C-UAS Detection Range
Estimates maximum passive RF detection range for a UAS signal. Assumes free-space, line-of-sight propagation.
UAS Signal Parameters
Frequency MHz
868 MHz (EU ISM), 915 MHz (LTE/O3), 1.2 GHz (FPV video), 2.4 GHz (control/video), 5.8 GHz (FPV video)
UAS Transmit Power dBm
Typical consumer drones: 20-30 dBm (100 mW-1 W). DJI O3: ~26 dBm.
UAS Antenna Gain dBi
Most consumer drones: 0-3 dBi omnidirectional.
Detection System Parameters
Receiving Antenna
Rx Antenna Gain dBi
Selected antenna may not cover this frequency. Verify coverage before use.
Receiver Sensitivity dBm
SDR: -110 to -130 dBm. Dedicated RF sensor: -120 to -140 dBm.
System / Cable Losses dB
Include cable, connector, and filter losses. Typical: 1-3 dB.
Fade Margin dB
Real-world margin for multipath, terrain, atmospheric effects. Typical: 10 dB (benign), 15-20 dB (operational).
Detection Link Budget Results
Max Detection Range
-
kilometers
Path Loss (max range)
-
dB
EIRP (UAS)
-
dBm
System Gain
-
dB link budget
Rx Power at 500 m
-
dBm
Detection Range Indicator -
100 m 500 m 2 km 10 km 20 km+
Free-space, line-of-sight assumption. Results include the specified fade margin. Antenna gain data from typical measured performance - see product datasheets .
Equations
EIRP:    \( \text{EIRP} = P_{tx} + G_{tx} \quad \text{(dBm)} \)
Free-space path loss:    \( \text{FSPL} = 20\log_{10}(f_\text{MHz}) + 20\log_{10}(d_\text{km}) + 32.44 \quad \text{(dB)} \)
Link budget:    \( \text{LB} = \text{EIRP} + G_{rx} - L_\text{sys} - L_\text{fade} - S_\text{min} \)
Max detection range:    \( d_\text{max} = 10^{\,(\text{LB} - 32.44 - 20\log_{10}f_\text{MHz})\,/\,20} \quad \text{km} \)
References
[1] J. D. Parsons, The Mobile Radio Propagation Channel , 2nd ed., Wiley, 2000. Ch. 2.
[2] ITU-R P.525-4, “Calculation of free-space attenuation,” 2019.
[3] NATO STANAG 4671, UAV system airworthiness requirements, 2009.
VSWR / Return Loss
Enter any one impedance mismatch metric and all others are calculated instantly, including mismatch loss.
Enter Any One Value
VSWR :1
Return Loss dB
Reflection Coefficient |Γ|
Reflected Power %
Reference
VSWR 1.0:1 RL = ∞ Perfect match
VSWR 1.5:1 RL = 14.0 dB Excellent
VSWR 2.0:1 RL = 9.5 dB Good
VSWR 2.5:1 RL = 7.4 dB Acceptable
VSWR 3.0:1 RL = 6.0 dB Marginal
Results
VSWR
-
:1
Return Loss
-
dB
|Γ|
-
refl. coeff.
Mismatch Loss
-
dB lost
AntenX antennas are designed for minimum VSWR - see product datasheets for typical VSWR curves.
Equations
Reflection coefficient:    \( |\Gamma| = \dfrac{\text{VSWR}-1}{\text{VSWR}+1} \)
VSWR from \(|\Gamma|\):    \( \text{VSWR} = \dfrac{1+|\Gamma|}{1-|\Gamma|} \)
Return Loss:    \( \text{RL} = -20\log_{10}|\Gamma| \quad \text{dB} \)
Reflected Power:    \( P_\text{refl} = |\Gamma|^2 \times 100\% \)
Mismatch Loss:    \( \text{ML} = -10\log_{10}(1-|\Gamma|^2) \quad \text{dB} \)
References
[1] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 2.
[2] IEEE Std 100-1992, IEEE Standard Dictionary of Electrical and Electronics Terms .
Near / Far Field Boundary
Calculates reactive near field, Fresnel, and Fraunhofer (far field) boundaries for a given aperture.
Antenna Parameters
Frequency MHz
Largest Aperture Dimension (D) mm
Largest physical dimension of a single antenna element. For arrays, use the full array aperture.
Region Definitions
Reactive Near Field
\( \style{color:white}{r < 0.62\sqrt{D^3/\lambda}} \)

Fresnel (Radiating Near Field)
\( \style{color:white}{0.62\sqrt{D^3/\lambda} < r < \dfrac{2D^2}{\lambda}} \)

Fraunhofer (Far Field)
\( \style{color:white}{r > \dfrac{2D^2}{\lambda}} \)   (phase error < 22.5°)
Field Region Boundaries
Wavelength
-
mm
D / λ
-
aperture in wavelengths
Reactive Near Field Ends
-
mm
Far Field Begins
-
mm
Far Field Begins
-
meters
Far Field Begins
-
feet
AntenX antenna datasheets include gain measurements made in the far field of each antenna.
References
[1] C. A. Balanis, Antenna Theory: Analysis and Design , 4th ed., Wiley, 2016. Ch. 2.
[2] IEEE Std 149-1979, IEEE Standard Test Procedures for Antennas .
[3] R. C. Johnson, Antenna Engineering Handbook , 3rd ed., McGraw-Hill, 1993.
Wavelength
Converts between frequency and wavelength for any transmission medium. Enter either to calculate the other.
Enter Frequency or Wavelength
Frequency MHz
Wavelength mm
Velocity Factor
Common Frequencies
433 MHz: λ = 692 mm
915 MHz: λ = 328 mm
2.4 GHz: λ = 125 mm
5.8 GHz: λ = 51.7 mm
10 GHz: λ = 30 mm
24 GHz: λ = 12.5 mm
77 GHz: λ = 3.9 mm
Results
Wavelength
-
mm
Half Wavelength
-
mm
Quarter Wavelength
-
mm
Wavenumber (k)
-
rad/m
Period
-
ps
Angular Freq.
-
Grad/s
Wavelength in medium = (c x VF) / f. Quarter-wave used for monopole and stub design; half-wave for dipoles.
Equations
Wavelength in medium:    \( \lambda = c \cdot \text{VF} / f \)
Wavenumber:    \( k = 2\pi/\lambda \quad \text{(rad/m)} \)
Period:    \( T = 1/f \quad \text{(s)} \)
Angular frequency:    \( \omega = 2\pi f \quad \text{(rad/s)} \)
Velocity factor:    \( \text{VF} = 1/\sqrt{\varepsilon_r} \quad \text{(non-magnetic media)} \)
References
[1] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 1.
[2] NIST CODATA 2018: \(c\) = 299,792,458 m/s (exact).
Rectangular Waveguide Calculator
Calculates cutoff frequencies and operating parameters for standard rectangular waveguide sizes or custom dimensions.
Waveguide Selection
Standard Size
Width (a) mm
Height (b) mm
Operating Frequency GHz
Mode Reference
TE 10 (dominant mode):
\( \style{color:white}{f_{c10} = \dfrac{c}{2a}} \)
TE 20 : \( \style{color:white}{f_{c20} = \dfrac{c}{a}} \)
TE 01 : \( \style{color:white}{f_{c01} = \dfrac{c}{2b}} \)

Single-mode: \( \style{color:white}{f_{c10} < f < f_{c20}} \)
Recommended: \( \style{color:white}{1.25\,f_{c10} \sim 1.9\,f_{c10}} \)
Results
TE10 Cutoff
-
GHz
TE20 Cutoff
-
GHz
TE01 Cutoff
-
GHz
Guide Wavelength
-
mm
Phase Velocity
-
x c
Mode Status
-
at operating freq.
AntenX stocks waveguide components in WR-137 and WR-90, including standard gain horns, exponential horns, adapters, and probes.
Equations
TE\(_{mn}\) cutoff:    \( f_{c,mn} = \dfrac{c}{2}\sqrt{\left(\dfrac{m}{a}\right)^2+\left(\dfrac{n}{b}\right)^2} \)
TE\(_{10}\) cutoff:    \( f_{c10} = c/(2a) \)
Guide wavelength:    \( \lambda_g = \lambda_0\Big/\sqrt{1-(f_c/f)^2} \)
Phase velocity:    \( v_p = c\Big/\sqrt{1-(f_c/f)^2} \)
TE\(_{10}\) wave impedance:    \( Z_{TE_{10}} = \eta_0\Big/\sqrt{1-(f_{c10}/f)^2}, \quad \eta_0=377\,\Omega \)
References
[1] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 3.
[2] C. G. Montgomery, R. H. Dicke, E. M. Purcell, Principles of Microwave Circuits , MIT Rad. Lab., 1948.
[3] EIA/JEDEC Standard, WR-series rectangular waveguide designations.
RF Power Unit Converter
Convert between all common RF power units instantly. Enter any value and all others update automatically.
Enter Any Power Value
dBm
dBW
Watts W
Milliwatts mW
Microwatts uW
Reference
+40 dBm = 10 W
+30 dBm = 1 W
+20 dBm = 100 mW
+10 dBm = 10 mW
  0 dBm = 1 mW
-10 dBm = 100 uW
-30 dBm = 1 uW
-60 dBm = 1 nW

+3 dB = double power
+10 dB = 10x power
Equivalent Power Levels
dBm
-
dBm
dBW
-
dBW
Watts
-
W
Milliwatts
-
mW
AntenX antennas are rated for up to 10W (40 dBm) continuous input power. See individual datasheets for power handling specs.
Equations
dBm → Watts:    \( P_W = 10^{P_\text{dBm}/10}/1000 \)
Watts → dBm:    \( P_\text{dBm} = 10\log_{10}(P_W \times 1000) \)
dBm → dBW:    \( P_\text{dBW} = P_\text{dBm} - 30 \)
+3 dB rule:    \( +3\,\text{dB} \Rightarrow 2\times\text{power}; \quad +10\,\text{dB} \Rightarrow 10\times\text{power} \)
References
[1] IEEE Std 100-1992, IEEE Standard Dictionary of Electrical and Electronics Terms .
[2] ITU-R V.574-4, “Use of the decibel and the neper in telecommunications,” 2015.
Antenna Gain & Effective Aperture
Converts between gain and effective aperture. Estimates gain from physical aperture or half-power beamwidths.
Gain to Effective Aperture
Antenna Gain dBi
Frequency GHz
Estimate Gain from Beamwidth
E-Plane HPBW deg
H-Plane HPBW deg
G (dBi) ~= 10*log10(27000 / (HPBW_E * HPBW_H)). Valid for pencil-beam antennas.
Results
Effective Aperture
-
cm 2
Effective Aperture
-
mm 2
Gain (linear)
-
x isotropic
Gain (beamwidth est.)
-
dBi
Directivity
-
linear
Solid Angle
-
steradians
Ae = G*λ 2 /(4*π). Beamwidth estimate has +/-3 dB accuracy. See AntenX datasheets for measured gain and beamwidth.
Equations
Effective aperture:    \( A_e = G\lambda^2/(4\pi) \)
Gain from beamwidths:    \( G \approx 27000\,/\,(\theta_E \cdot \theta_H) \quad (\theta\text{ in degrees}) \)
Solid angle:    \( \Omega_A = \theta_{E,\text{rad}}\,\theta_{H,\text{rad}} \quad \text{(sr)} \)
Directivity:    \( D = 4\pi/\Omega_A \)
References
[1] C. A. Balanis, Antenna Theory: Analysis and Design , 4th ed., Wiley, 2016. Ch. 2.
[2] W. L. Stutzman, G. A. Thiele, Antenna Theory and Design , 3rd ed., Wiley, 2012. Ch. 2.
[3] IEEE Std 145-2013, IEEE Standard for Definitions of Terms for Antennas .
Transmission Line Impedance
Calculates characteristic impedance of coaxial and microstrip lines from physical dimensions.
Coaxial Cable
Outer Conductor ID mm
Inner diameter of outer conductor (shield). RG-58: 2.95 mm, RG-8: 7.24 mm
Inner Conductor OD mm
Outer diameter of center conductor. RG-58: 0.9 mm, RG-8: 2.17 mm
Dielectric ε r
ε r D d Outer conductor (shield)
D = inner diam. of outer conductor   d = outer diam. of center conductor
Microstrip
Trace Width (W) mm
50-Ω trace on FR4 (1.6 mm): ~1.9 mm wide
Substrate Height (H) mm
PCB substrate thickness. Standard FR4: 1.6 mm, 0.8 mm, 0.4 mm
Substrate ε r
Ground Plane ε r Trace W H
W = trace width   H = substrate height
Results
Coax Impedance
-
Ω
Coax Velocity Factor
-
x c
Microstrip Z 0
-
Ω
Coax: Z 0 = (138/√ε r ) × log 10 (D/d). Microstrip uses the Hammerstad closed-form approximation. Use the AntenX waveguide adapters for reliable 50-Ω coax-to-waveguide transitions.
Equations
Coaxial impedance:    \( Z_0 = \dfrac{138}{\sqrt{\varepsilon_r}}\log_{10}\!\dfrac{D}{d} \quad (\Omega) \)
Coax velocity factor:    \( \text{VF} = 1/\sqrt{\varepsilon_r} \)
Microstrip \(\varepsilon_{r,\text{eff}}\):    \( \varepsilon_{r,\text{eff}} = \dfrac{\varepsilon_r+1}{2} + \dfrac{\varepsilon_r-1}{2}\left(1+\dfrac{12h}{W}\right)^{-1/2} \)
Microstrip \(Z_0\) (\(W/h\leq 1\)):    \( Z_0 = \dfrac{60}{\sqrt{\varepsilon_{r,\text{eff}}}}\ln\!\left(\dfrac{8h}{W}+\dfrac{W}{4h}\right) \)
Microstrip \(Z_0\) (\(W/h > 1\)):    \( Z_0 = \dfrac{120\pi}{\sqrt{\varepsilon_{r,\text{eff}}}\!\left[W/h + 1.393 + 0.667\ln(W/h+1.444)\right]} \)
References
[1] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 3.
[2] E. O. Hammerstad, “Equations for microstrip circuit design,” 5th Eur. Microw. Conf. , 1975.
[3] I. J. Bahl, R. Trivedi, “A designer’s guide to microstrip line,” Microwaves , May 1977.
Coaxial Cable Loss Calculator
Calculates coaxial cable attenuation at a given frequency and length.
Cable Parameters
Cable Type
Attenuation at 1 GHz dB/m
Auto-populated from cable type. Edit for custom cable.
Cable Length m
Frequency GHz
Cable Reference (dB/m @ 1 GHz)
RG-58: 0.072 dB/m
RG-8/213: 0.046 dB/m
LMR-100: 0.033 dB/m
LMR-200: 0.022 dB/m
LMR-400: 0.013 dB/m
LMR-600: 0.008 dB/m

Attenuation scales with sqrt(f).
At 4 GHz: multiply by sqrt(4) = 2x
Results
Total Cable Loss
-
dB
Loss per Meter
-
dB/m at frequency
Power at Output
-
% of input
Signal Voltage Ratio
-
Vout/Vin
Cable loss scales approximately with sqrt(frequency). Connector losses (typically 0.1-0.3 dB per SMA connector) should be added separately.
Equations
Attenuation at frequency \(f\):    \( \alpha(f) = \alpha_{1\,\text{GHz}}\sqrt{f_\text{GHz}} \quad \text{(dB/m)} \)
Total loss:    \( L = \alpha(f)\times\ell \quad \text{(dB)} \)
Output power fraction:    \( P_\text{out}/P_\text{in} = 10^{-L/10} \)
Voltage ratio:    \( V_\text{out}/V_\text{in} = 10^{-L/20} \)
Skin-effect scaling:    \( \alpha_\text{skin} \propto \sqrt{f}; \quad \alpha_\text{diel} \propto f \)
References
[1] Times Microwave Systems, LMR Cable Technical Data , timesmicrowave.com.
[2] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 2.
[3] MIL-C-17, Military Specification: Cables, Radio Frequency, Flexible and Semirigid .
Fresnel Zone Calculator
Calculates Fresnel zone radii for an RF link. At least 60% of the first zone should be clear for reliable performance.
Link Parameters
Frequency MHz
Total Link Distance (d) km
Distance to Obstruction (d1) km
Distance from transmitter to obstruction. Max radius occurs at d/2 (midpoint).
Fresnel Zone Reference
\( \style{color:white}{r_n = \sqrt{\dfrac{n\,\lambda\,d_1\,d_2}{d}}} \)

Zone 1: 60% must be clear
Zone 2: destructive interference
Zone 3: constructive

Min. antenna height above terrain ≈ r 1 at midpoint of link
Results
Wavelength
-
mm
1st Fresnel Zone Radius
-
meters
2nd Fresnel Zone Radius
-
meters
3rd Fresnel Zone Radius
-
meters
For a reliable RF link, the first Fresnel zone should be at least 60% clear of obstructions. Full clearance gives an additional 6 dB fade margin.
Equations
\(n\)-th Fresnel zone radius:    \( r_n = \sqrt{\dfrac{n\,\lambda\,d_1\,d_2}{d}} \)
where:    \( d = \text{total link distance},\quad d_1 = \text{Tx to obstruction},\quad d_2 = d-d_1 \)
Max radius (midpoint):    \( r_{1,\text{max}} = \tfrac{1}{2}\sqrt{\lambda\,d} \quad (d_1 = d/2) \)
60% clearance rule:    \( \text{obstruction} \leq r_1 \Rightarrow < 0.4\,\text{dB additional loss} \)
References
[1] T. S. Rappaport, Wireless Communications , 2nd ed., Prentice Hall, 2002. Ch. 3.
[2] ITU-R P.526-15, “Propagation by diffraction,” 2019.
[3] A. J. Fresnel, Mémoire sur la diffraction de la lumière , Ann. Chim. Phys., 1818.
Doppler Shift Calculator
Calculates Doppler frequency shift due to relative motion. Covers one-way (passive detection) and two-way (radar) modes.
Parameters
Carrier Frequency GHz
Target Velocity m/s
Velocity Unit
Radar Mode
Radar return: signal travels to target and back, doubling the Doppler shift.
Drone Velocity Reference
DJI Mini 4 Pro: 16 m/s (58 km/h)
DJI Mavic 3: 21 m/s (75 km/h)
DJI FPV: 39 m/s (140 km/h)
Racing FPV drone: 50+ m/s (180+ km/h)
Fixed-wing UAV: 30-70 m/s

Positive shift = approaching
Negative shift = receding
Results
Velocity (m/s)
-
m/s
Velocity (km/h)
-
km/h
Doppler Shift
-
Hz
Received Frequency
-
GHz (f₀ + fₐ approaching)
fd = (2*v*f0)/c for radar (two-way), fd = (v*f0)/c for one-way detection. Doppler processing can distinguish moving drones from stationary clutter in C-UAS detection systems.
Equations
One-way Doppler shift:    \( f_d = v\,f_0 / c \)
Two-way (radar) shift:    \( f_d = 2v\,f_0 / c \)
Received frequency:    \( f_{rx} = f_0 \pm f_d \quad (+\text{approaching},\;-\text{receding}) \)
Valid for:    \( v \ll c \quad \text{(non-relativistic)} \)
References
[1] M. A. Richards, J. A. Scheer, W. A. Holm, Principles of Modern Radar , SciTech, 2010. Ch. 3.
[2] N. Levanon, E. Mozeson, Radar Signals , Wiley-IEEE, 2004. Ch. 1.
[3] C. Doppler, “Über das farbige Licht der Doppelsterne,” Abh. Königl. Böhm. Ges. Wiss. , 1842.
Attenuator Pad Calculator
Calculates resistor values for π-pad and T-pad attenuators for a matched, calibrated signal reduction.
Parameters
Attenuation dB
System Impedance Ω
Topologies
π-PAD In R 1 R2 R 1 Out R1 shunt - R2 series - R1 shunt T-PAD In R1 R 2 R1 Out R1 series - R2 shunt - R1 series
Both are passive, reciprocal, and bidirectional
Resistor Values
π -pad R1 (shunt)
-
Ω
π -pad R2 (series)
-
Ω
T-pad R1 (series)
-
Ω
T-pad R2 (shunt)
-
Ω
All resistor values are for a symmetric, matched attenuator (equal source and load impedance). Use 1% tolerance or better resistors for accurate attenuation.
Equations
Voltage ratio:    \( K = 10^{A_\text{dB}/20} \)
\(\pi\)-pad shunt \(R_1\):    \( R_1 = Z_0\,\dfrac{K+1}{K-1} \)
\(\pi\)-pad series \(R_2\):    \( R_2 = Z_0\,\dfrac{K^2-1}{2K} \)
T-pad series \(R_1\):    \( R_1 = Z_0\,\dfrac{K-1}{K+1} \)
T-pad shunt \(R_2\):    \( R_2 = \dfrac{2Z_0 K}{K^2-1} \)
Both topologies:    \( Z_\text{in} = Z_\text{out} = Z_0 \quad \text{(matched, passive, bidirectional)} \)
References
[1] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 2.
[2] H. W. Bode, Network Analysis and Feedback Amplifier Design , Van Nostrand, 1945.
[3] Mini-Circuits, Attenuator Design Notes , minicircuits.com.
Cascaded Noise Figure (Friis Formula)
Calculates total system noise figure for a receiver chain using the Friis formula.
Receiver Chain (Antenna to Receiver)
Stage 1 - Cable/Filter Loss dB
Enter as negative dB for a lossy stage (e.g. -2 for 2 dB cable loss). Positive dB for amplifier gain.
Stage 1 Noise Figure dB
For a passive loss of L dB, NF = L dB (noise figure equals loss).
Stage 2 - LNA Gain dB
Stage 2 LNA Noise Figure dB
Stage 3 - Receiver NF dB
Friis Formula
\( \style{color:white}{F_{\!total} = F_1 + \dfrac{F_2-1}{G_1} + \dfrac{F_3-1}{G_1 G_2}} \)

F = linear noise factor   G = linear gain

Key insight:
First stage dominates system NF.
Low-NF LNA before lossy cable
dramatically improves sensitivity.

For a passive stage, NF (dB) = insertion loss (dB).
Results
Stage 1 Contribution
-
dB NF contribution
Stage 2 Contribution
-
dB NF contribution
Stage 3 Contribution
-
dB NF contribution
Total System NF
-
dB
Total Gain
-
dB
Noise Temperature
-
K
Friis formula shows that the first stage dominates system NF. Placing a low-noise LNA before a lossy cable dramatically improves system performance. Browse AntenX antennas.
Equations
Friis formula:    \( F_\text{total} = F_1 + \dfrac{F_2-1}{G_1} + \dfrac{F_3-1}{G_1 G_2} + \cdots \)
Noise factor:    \( F = 10^{NF_\text{dB}/10} \quad \text{(linear)} \)
Passive loss \(L\) dB:    \( NF_\text{dB} = L_\text{dB},\quad F = 10^{L_\text{dB}/10} \)
Noise temperature:    \( T_e = T_0(F-1), \quad T_0 = 290\,\text{K} \)
Total gain:    \( G_\text{total} = G_1 + G_2 + \cdots \quad \text{(dB, additive)} \)
References
[1] H. T. Friis, “Noise figures of radio receivers,” Proc. IRE , vol. 32, pp. 419–422, 1944.
[2] D. M. Pozar, Microwave Engineering , 4th ed., Wiley, 2011. Ch. 10.
[3] IRE Standards on Methods of Measuring Noise, Proc. IRE , vol. 48, 1960.
Receiver Sensitivity & Noise Figure
Calculates receiver sensitivity from noise figure, bandwidth, and required SNR.
System Parameters
Noise Figure dB
Total system NF including LNA, cables, receiver. SDR: 6-10 dB.
Bandwidth MHz
Required SNR dB
Detection only: 3-6 dB. Signal decoding: 10-20 dB.
Temperature K
Standard: 290 K (room temperature, ITU-R standard).
Reference
Sensitivity = -174 + NF + 10log(BW) + SNR

Typical noise figures:
Low-noise amplifier: 0.5-2 dB
RTL-SDR dongle: 6-10 dB
HackRF: 8-12 dB
USRP: 5-8 dB
Dedicated RX: 3-5 dB
Results
Thermal Noise/Hz
-
dBm/Hz
Noise in Bandwidth
-
dBm
Receiver Sensitivity
-
dBm
Add antenna gain to convert sensitivity into effective isotropic sensitivity (EIS). See AntenX antennas for gain figures.
Equations
Thermal noise floor:    \( N_0 = kT \quad \text{(W/Hz)} = -174\,\text{dBm/Hz at }290\,\text{K} \)
Noise in bandwidth \(B\):    \( N = -174 + NF + 10\log_{10}B_\text{Hz} \quad \text{(dBm)} \)
Receiver sensitivity:    \( S_\text{min} = -174 + NF + 10\log_{10}B + \text{SNR}_\text{min} \quad \text{(dBm)} \)
Boltzmann constant:    \( k = 1.381\times10^{-23}\,\text{J/K} \)
References
[1] H. T. Friis, “Noise figures of radio receivers,” Proc. IRE , vol. 32, pp. 419–422, 1944.
[2] J. B. Johnson, “Thermal agitation of electricity in conductors,” Phys. Rev. , vol. 32, 1928.
[3] ITU-R P.372-14, “Radio noise,” 2019.
Dipole & Monopole Antenna
Calculates half-wave dipole and quarter-wave monopole dimensions with velocity factor correction.
Parameters
Frequency MHz
Velocity Factor
Practical antennas are ~3–10% shorter than the theoretical length due to end effects.
Wire / Element Diameter mm
Thicker elements have slightly shorter resonant length. Typical wire: 0.5–2 mm.
Reference
Half-wave dipole:
• Gain: 2.15 dBi
• Feedpoint Z: ~73 Ω (resistive)
• Radiation pattern: figure-8, broadside

Quarter-wave monopole:
• Gain: 5.19 dBi (over ground plane)
• Feedpoint Z: ~36 Ω
• Requires ground plane or counterpoise
Antenna Dimensions
Half-Wave Dipole (total)
-
mm
Each Dipole Arm
-
mm
Quarter-Wave Monopole
-
mm
5/8 λ Monopole
-
mm
Free-Space λ
-
mm
Dipole Gain
2.15
dBi
Dimensions account for velocity factor. Feedpoint impedance of a half-wave dipole is ~73 Ω; use a balun or matching network for 50 Ω systems. Monopole over a ground plane has ~36 Ω feedpoint impedance.
Equations
Free-space wavelength:    \( \lambda = c/f \)
Half-wave dipole length:    \( L = \tfrac{1}{2}\lambda \cdot \text{VF} \)
Quarter-wave monopole:    \( L = \tfrac{1}{4}\lambda \cdot \text{VF} \)
5/8λ monopole:    \( L = \tfrac{5}{8}\lambda \cdot \text{VF} \)
Dipole gain:    \( G = 2.15\,\text{dBi} \)
Monopole gain (over GP):    \( G = 5.19\,\text{dBi} \)
References
[1] C. A. Balanis, Antenna Theory: Analysis and Design , 4th ed., Wiley, 2016. Ch. 4–5.
[2] W. L. Stutzman, G. A. Thiele, Antenna Theory and Design , 3rd ed., Wiley, 2012. Ch. 3.
[3] IEEE Std 145-2013, IEEE Standard for Definitions of Terms for Antennas .
Horn Antenna
Estimates gain, beamwidth, and aperture size for pyramidal horn antennas. Convert between gain and dimensions.
Parameters
Mode
Frequency GHz
Target Gain dBi
Aperture Efficiency
Reference
Pyramidal horn:
• Optimum gain when flare angles satisfy Balanis Ch. 13 conditions
• η ≈ 0.511 for optimum-gain design
• H-plane wider than E-plane

Typical gains:
• SGH-WR90 (X-band): ~15 dBi
• SGH-WR62 (Ku-band): ~15 dBi
• DRHA (1–18 GHz): 8–16 dBi
Horn Parameters
Gain
-
dBi
Aperture Width (a)
-
mm
Aperture Height (b)
-
mm
Effective Aperture
-
cm²
E-Plane HPBW (est.)
-
degrees
H-Plane HPBW (est.)
-
degrees
Optimum-gain horn theory (Balanis Ch. 13). Beamwidth estimated from aperture size; actual HPBW depends on flare angle and waveguide mode. See AntenX horn antennas for measured data.
Equations
Gain from aperture:    \( G = \eta\,\dfrac{4\pi A}{\lambda^2} \)
Aperture area:    \( A = a \times b \)
Aperture from gain:    \( A = \dfrac{G\lambda^2}{4\pi\eta} \)
E-plane HPBW:    \( \theta_E \approx 0.886\,\lambda/b \quad\text{(rad)} \)
H-plane HPBW:    \( \theta_H \approx 1.189\,\lambda/a \quad\text{(rad)} \)
Optimum-gain efficiency:    \( \eta = 0.511 \)
References
[1] C. A. Balanis, Antenna Theory: Analysis and Design , 4th ed., Wiley, 2016. Ch. 13.
[2] W. L. Stutzman, G. A. Thiele, Antenna Theory and Design , 3rd ed., Wiley, 2012. Ch. 7.
[3] A. W. Love, Electromagnetic Horn Antennas , IEEE Press, 1976.
Radar Range Equation
Calculates maximum detection range for a monostatic radar using the radar range equation.
Radar Parameters
Frequency GHz
Transmit Power dBm
37 dBm = 5 W. Typical small radar: 30–40 dBm.
Antenna Gain dBi
Same antenna used for Tx and Rx (monostatic). Horn antennas: 15–25 dBi.
Radar Cross Section dBsm
Small drone: −10 to −20 dBsm. Car: ~10 dBsm. Person: ~0 dBsm.
Minimum Detectable Signal dBm
Receiver sensitivity. Typical: −90 to −110 dBm.
System Losses dB
RCS Reference
Small drone (DJI Mini): −20 to −15 dBsm
Medium drone (DJI Mavic): −15 to −10 dBsm
Large drone / UAV: −10 to 0 dBsm
Person (standing): ∼0 dBsm
Car: ∼+10 dBsm
Small aircraft: +10 to +20 dBsm

RCS depends on aspect angle, frequency, and material. Values are rough estimates.
Radar Performance
Max Detection Range
-
km
Max Detection Range
-
meters
EIRP
-
dBm
Wavelength
-
mm
Radar Constant K
-
dB
Monostatic radar (same antenna Tx/Rx), free-space propagation assumed. Add atmospheric attenuation (~0.01 dB/km at 10 GHz) for long-range estimates. See the C-UAS Detection Range calculator for passive RF detection (non-radar).
Equations
Radar range equation:    \( R_{\max} = \left(\dfrac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 S_{\min} L}\right)^{1/4} \)
In dB form:    \( 40\log_{10}R = P_t + 2G + 20\log_{10}\lambda + \sigma_{\text{dBsm}} - 30\log_{10}(4\pi) - S_{\min} - L \)
RCS:    \( \sigma\,(\text{m}^2) = 10^{\sigma_{\text{dBsm}}/10} \)
References
[1] M. A. Richards, J. A. Scheer, W. A. Holm, Principles of Modern Radar , SciTech, 2010. Ch. 2.
[2] N. J. Willis, Bistatic Radar , Artech House, 1991.
[3] M. I. Skolnik, Introduction to Radar Systems , 3rd ed., McGraw-Hill, 2001. Ch. 2.
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