RF Calculator — Equations and Implementation Notes
These are the equations used by each calculator tile. They document the current browser implementation. Where a value is solved backwards, the same equation is rearranged or solved numerically as noted.
Constants and conventions
c = 299792458 m/s
k = 1.380649 × 10−23 J/K
μ0 = 4π × 10−7 H/m
ε0 = 8.8541878128 × 10−12 F/m
η0 = 376.730313668 Ω
T0 = 290 K
k = 1.380649 × 10−23 J/K
μ0 = 4π × 10−7 H/m
ε0 = 8.8541878128 × 10−12 F/m
η0 = 376.730313668 Ω
T0 = 290 K
Unless stated otherwise, logarithms in dB equations are base 10, angles used internally by trigonometric functions are radians, and impedances are in ohms.
Page 1 — RF circuits, matching, power and cascades
1. Pi-pad Attenuator (symmetric)
K = 10A/20
Rseries = Z0(K² − 1)/(2K)
Rshunt = Z0(K + 1)/(K − 1)
Rseries = Z0(K² − 1)/(2K)
Rshunt = Z0(K + 1)/(K − 1)
Analysis uses ABCD matrices:
Mshunt(Y) = [1 0; Y 1], Mseries(Z) = [1 Z; 0 1]
Mπ = Mshunt(1/Rshunt) Mseries(Rseries) Mshunt(1/Rshunt)
DS = A + B/Z0 + CZ0 + D
S11 = (A + B/Z0 − CZ0 − D)/DS
S21 = 2/DS
Loss = −20 log10|S21|
S11 dB shown = 20 log10|S11|
Zin = (AZ0 + B)/(CZ0 + D)
Mπ = Mshunt(1/Rshunt) Mseries(Rseries) Mshunt(1/Rshunt)
DS = A + B/Z0 + CZ0 + D
S11 = (A + B/Z0 − CZ0 − D)/DS
S21 = 2/DS
Loss = −20 log10|S21|
S11 dB shown = 20 log10|S11|
Zin = (AZ0 + B)/(CZ0 + D)
2. T-pad Attenuator (symmetric)
K = 10A/20
Rseries = Z0(K − 1)/(K + 1) (each series arm)
Rshunt = 2Z0K/(K² − 1)
Rseries = Z0(K − 1)/(K + 1) (each series arm)
Rshunt = 2Z0K/(K² − 1)
The same ABCD-to-S equations as the Pi-pad are used, with:
MT = Mseries(Rseries) Mshunt(1/Rshunt) Mseries(Rseries)
3. L-pad Match (2 resistors)
For Z2 > Z1 (shunt at input):
Δ = Z2 − Z1
Rseries = √(Z2Δ)
Rshunt = Z1√(Z2/Δ)
Rseries = √(Z2Δ)
Rshunt = Z1√(Z2/Δ)
For Z1 > Z2 (shunt at output):
Δ = Z1 − Z2
Rseries = √(Z1Δ)
Rshunt = Z2√(Z1/Δ)
Rseries = √(Z1Δ)
Rshunt = Z2√(Z1/Δ)
Ra∥b = RaRb/(Ra + Rb)
Γ = (Z − Zref)/(Z + Zref)
S11 dB shown = 20 log10|Γ|
Pavs = Vs²/(4Z1)
PL = V2²/Z2
Loss = −10 log10(PL/Pavs)
Γ = (Z − Zref)/(Z + Zref)
S11 dB shown = 20 log10|Γ|
Pavs = Vs²/(4Z1)
PL = V2²/Z2
Loss = −10 log10(PL/Pavs)
4. Potential Divider (Vout, I, R1, R2)
Rtotal = R1 + R2
I = Vin/Rtotal
Vout = VinR2/(R1 + R2)
r = Vout/Vin
R2 = R1r/(1 − r)
R1 = R2(1 − r)/r
I = Vin/Rtotal
Vout = VinR2/(R1 + R2)
r = Vout/Vin
R2 = R1r/(1 − r)
R1 = R2(1 − r)/r
5. RF Power Converter
P(W) = 10(dBm−30)/10
dBm = 10 log10P(W) + 30
dBW = dBm − 30
P(mW) = 1000P(W)
Vrms = √(PZ)
Vpk = √2 Vrms
Vpp = 2√2 Vrms
Irms = √(P/Z)
dBm = 10 log10P(W) + 30
dBW = dBm − 30
P(mW) = 1000P(W)
Vrms = √(PZ)
Vpk = √2 Vrms
Vpp = 2√2 Vrms
Irms = √(P/Z)
6. VSWR / Return Loss / Γ
|Γ| = (VSWR − 1)/(VSWR + 1)
VSWR = (1 + |Γ|)/(1 − |Γ|)
Return Loss = −20 log10|Γ|
Mismatch Loss = −10 log10(1 − |Γ|²)
|Γ| = √(1 − 10−ML/10)
VSWR = (1 + |Γ|)/(1 − |Γ|)
Return Loss = −20 log10|Γ|
Mismatch Loss = −10 log10(1 − |Γ|²)
|Γ| = √(1 − 10−ML/10)
7. Complex Z ↔ S11 / Γ
Z = R + jX
Γ = (Z − Z0)/(Z + Z0)
Z = Z0(1 + Γ)/(1 − Γ)
Γ = |Γ|ejθ
Return Loss = −20 log10|Γ|
VSWR = (1 + |Γ|)/(1 − |Γ|)
Γ = (Z − Z0)/(Z + Z0)
Z = Z0(1 + Γ)/(1 − Γ)
Γ = |Γ|ejθ
Return Loss = −20 log10|Γ|
VSWR = (1 + |Γ|)/(1 − |Γ|)
8. Reactive L-Match Designer
Rlow = min(Rs,RL), Rhigh = max(Rs,RL)
Q = √(Rhigh/Rlow − 1)
Xs = QRlow
Xp = Rhigh/Q
ω = 2πf
Ls = Xs/ω, Cp = 1/(ωXp)
Cs = 1/(ωXs), Lp = Xp/ω
Q = √(Rhigh/Rlow − 1)
Xs = QRlow
Xp = Rhigh/Q
ω = 2πf
Ls = Xs/ω, Cp = 1/(ωXp)
Cs = 1/(ωXs), Lp = Xp/ω
9. Reactive Pi-Match Designer
Rv = Rs/(Q1² + 1)
Q2 = √(RL/Rv − 1)
Xp1 = Rs/Q1
Xp2 = RL/Q2
Xs = (Q1 + Q2)Rv
Cin = 1/(ωXp1)
Lseries = Xs/ω
Cout = 1/(ωXp2)
Q2 = √(RL/Rv − 1)
Xp1 = Rs/Q1
Xp2 = RL/Q2
Xs = (Q1 + Q2)Rv
Cin = 1/(ωXp1)
Lseries = Xs/ω
Cout = 1/(ωXp2)
10. Reactive T-Match Designer
Rv = Rs(Q1² + 1)
Q2 = √(Rv/RL − 1)
Xs1 = Q1Rs
Xs2 = Q2RL
Xp1 = Rv/Q1, Xp2 = Rv/Q2
Xp = [1/Xp1 + 1/Xp2]−1
L1 = Xs1/ω, Csh = 1/(ωXp), L2 = Xs2/ω
Q2 = √(Rv/RL − 1)
Xs1 = Q1Rs
Xs2 = Q2RL
Xp1 = Rv/Q1, Xp2 = Rv/Q2
Xp = [1/Xp1 + 1/Xp2]−1
L1 = Xs1/ω, Csh = 1/(ωXp), L2 = Xs2/ω
11. RLC Impedance / Reactance
ω = 2πf
XL = ωL
XC = 1/(ωC)
Zseries = R + j(XL − XC)
|Z| = √(R² + (XL − XC)²)
∠Z = atan2(XL − XC,R)
Yparallel = 1/R + j(1/XC − 1/XL)
Zparallel = 1/Yparallel
XL = ωL
XC = 1/(ωC)
Zseries = R + j(XL − XC)
|Z| = √(R² + (XL − XC)²)
∠Z = atan2(XL − XC,R)
Yparallel = 1/R + j(1/XC − 1/XL)
Zparallel = 1/Yparallel
12. LC Resonance / Q / Bandwidth
f0 = 1/(2π√(LC))
C = 1/[(2πf0)²L]
Q = 2πf0L/R
BW = f0/Q
Zchar = √(L/C)
C = 1/[(2πf0)²L]
Q = 2πf0L/R
BW = f0/Q
Zchar = √(L/C)
13. Cascaded Gain / Loss / RF Power
P1(dBm) = Pin + G1
P2 = P1 + G2 …
Gtotal(dB) = ΣGi
Pout(dBm) = Pin(dBm) + Gtotal(dB)
P2 = P1 + G2 …
Gtotal(dB) = ΣGi
Pout(dBm) = Pin(dBm) + Gtotal(dB)
14. Cascaded Noise Figure — Friis
Gi = 10gi/10
Fi = 10NFi/10
FT = F1 + (F2−1)/G1 + (F3−1)/(G1G2) + (F4−1)/(G1G2G3)
NFT = 10 log10FT
Te = (FT − 1) × 290 K
Fi = 10NFi/10
FT = F1 + (F2−1)/G1 + (F3−1)/(G1G2) + (F4−1)/(G1G2G3)
NFT = 10 log10FT
Te = (FT − 1) × 290 K
15. Cascaded IP3 / P1dB Budget
IIP3i,dBm = OIP3i,dBm − Gi,dB
Ii(mW) = 10IIP3i,dBm/10
Gi,lin = 10Gi,dB/10
1/IIP3T = 1/I1 + G1/I2 + G1G2/I3
IIP3T,dBm = 10 log10(IIP3T,mW)
OIP3T = IIP3T + Gtotal
P1dBin ≈ min(OP11−G1, OP12−G1−G2, OP13−Gtotal)
P1dBout ≈ P1dBin + Gtotal
Ii(mW) = 10IIP3i,dBm/10
Gi,lin = 10Gi,dB/10
1/IIP3T = 1/I1 + G1/I2 + G1G2/I3
IIP3T,dBm = 10 log10(IIP3T,mW)
OIP3T = IIP3T + Gtotal
P1dBin ≈ min(OP11−G1, OP12−G1−G2, OP13−Gtotal)
P1dBout ≈ P1dBin + Gtotal
Page 2 — PCB, transmission lines and RF interconnect
1. Ground-backed CPW Impedance
a = w, b = w + 2g
k = a/b, k′ = √(1−k²)
k1 = sinh(πa/4h)/sinh(πb/4h), k′1 = √(1−k1²)
K(k) = π/[2 AGM(1,√(1−k²))]
εeff = 1 + (εr−1)/2 · [K(k)/K(k′)] · [K(k′1)/K(k1)]
Z0 = (30π/√εeff) · K(k′)/K(k)
k = a/b, k′ = √(1−k²)
k1 = sinh(πa/4h)/sinh(πb/4h), k′1 = √(1−k1²)
K(k) = π/[2 AGM(1,√(1−k²))]
εeff = 1 + (εr−1)/2 · [K(k)/K(k′)] · [K(k′1)/K(k1)]
Z0 = (30π/√εeff) · K(k′)/K(k)
When Z0 is edited, width w is obtained by monotonic bisection of the same Z0(w) equation.
2. Microstrip Impedance
u = w/h
εeff = (εr+1)/2 + (εr−1)/2 · 1/√(1+12/u), u ≥ 1
εeff = (εr+1)/2 + (εr−1)/2 · [1/√(1+12/u) + 0.04(1−u)²], u < 1
εeff = (εr+1)/2 + (εr−1)/2 · 1/√(1+12/u), u ≥ 1
εeff = (εr+1)/2 + (εr−1)/2 · [1/√(1+12/u) + 0.04(1−u)²], u < 1
Finite copper-thickness correction:
Dt = (t/h)² + [(1/π)/(w/t + 1.1)]²
Δw = (t/π) ln(4e/√Dt)
weff = w + Δw(1 + 1/εr)/2
r = 4h/weff
A = [(14 + 8/εr)/11]r
B = √[A² + π²(1 + 1/εr)/2]
C = 60/√[2(εr+1)]
Z0 = C ln[1 + r(A+B)]
Δw = (t/π) ln(4e/√Dt)
weff = w + Δw(1 + 1/εr)/2
r = 4h/weff
A = [(14 + 8/εr)/11]r
B = √[A² + π²(1 + 1/εr)/2]
C = 60/√[2(εr+1)]
Z0 = C ln[1 + r(A+B)]
Target Z0 → width is solved by bisection.
3. Microstrip Optimal Mitre
r = w/h
D = w√2
m = 0.52 + 0.65e−1.35r
X = Dm
M = 2X
Mitre (%) = 100m
D = w√2
m = 0.52 + 0.65e−1.35r
X = Dm
M = 2X
Mitre (%) = 100m
4. Stripline Impedance (asymmetric)
b = h1 + h2 + t
W = w/b, T = t/b
Weff = W − (0.35−W)²/(1+12T), W<0.35
Weff = W, W≥0.35
W = w/b, T = t/b
Weff = W − (0.35−W)²/(1+12T), W<0.35
Weff = W, W≥0.35
Cf = (2/π)ln[1/(1−T)+1] − (T/π)ln[1/(1−T)²−1] (0<T<1)
Z0,air = 30π(1−T)/(Weff+Cf)
Asym = 0.5 − 0.5(2h1+t)/(h1+h2+t)
ΔZ0,air = 0.0325πZ0,air²|Asym|2.2[(t+w)/(h1+h2+t)]2.9
Z0 = (Z0,air − ΔZ0,air)/√εr
Z0,air = 30π(1−T)/(Weff+Cf)
Asym = 0.5 − 0.5(2h1+t)/(h1+h2+t)
ΔZ0,air = 0.0325πZ0,air²|Asym|2.2[(t+w)/(h1+h2+t)]2.9
Z0 = (Z0,air − ΔZ0,air)/√εr
For the zero/thin-copper branch used by the implementation, Cf is set to zero. Target Z0 → width is solved by bisection.
5. Coaxial Cable Characteristic Impedance
Ln = ln(D/d)
Z0 = 60Ln/√εr
C′ = 2πε0εr/Ln
L′ = μ0Ln/(2π)
VF = 1/√εr
Z0 = 60Ln/√εr
C′ = 2πε0εr/Ln
L′ = μ0Ln/(2π)
VF = 1/√εr
6. Transmission-Line Electrical Length
λ = c·VF/f
θ(deg) = 360l/λ
td = l/(c·VF)
lλ/4 = λ/4
θ(deg) = 360l/λ
td = l/(c·VF)
lλ/4 = λ/4
7. Transmission-Line Input Impedance
β = 2πf/(c·VF)
ZL = RL + jXL
Zin = Z0[ZL + jZ0tan(βl)]/[Z0 + jZLtan(βl)]
|Zin| = √(Re²+Im²)
∠Zin = atan2(Im,Re)
ZL = RL + jXL
Zin = Z0[ZL + jZ0tan(βl)]/[Z0 + jZLtan(βl)]
|Zin| = √(Re²+Im²)
∠Zin = atan2(Im,Re)
8. Quarter-Wave Transformer Designer
Zt = √(RsRL)
λ = c·VF/f
l = λ/4
td = l/(c·VF)
λ = c·VF/f
l = λ/4
td = l/(c·VF)
9. Open / Short Shunt Stub Tuner
zL = (RL + jXL)/Z0
θ = βd
z(d) = [zL + jtanθ]/[1 + jzLtanθ]
y(d) = 1/z(d)
θ = βd
z(d) = [zL + jtanθ]/[1 + jzLtanθ]
y(d) = 1/z(d)
The calculator numerically finds the first two positions where Re{y(d)} = 1. Then:
bstub = −Im{y(d)}
θopen = atan(bstub) mod π
θshort = atan(−1/bstub) mod π
λ = c·VF/f
d = θλ/(2π)
lopen = θopenλ/(2π)
lshort = θshortλ/(2π)
θopen = atan(bstub) mod π
θshort = atan(−1/bstub) mod π
λ = c·VF/f
d = θλ/(2π)
lopen = θopenλ/(2π)
lshort = θshortλ/(2π)
10. Differential Microstrip Impedance
ZSE = Zmicrostrip(w,h,t,εr)
Zdiff = 2ZSE[1 − 0.48e−0.96s/h]
Zdiff = 2ZSE[1 − 0.48e−0.96s/h]
εeff is the effective permittivity returned by the single-ended microstrip equation above.
11. Edge-Coupled Differential Stripline
h = (b − t)/2
ZSE = Zstripline(w,h,h,t,εr)
Zdiff = 2ZSE[1 − 0.347e−2.9s/b]
ZSE = Zstripline(w,h,h,t,εr)
Zdiff = 2ZSE[1 − 0.347e−2.9s/b]
12. Coplanar Waveguide — No Back Ground
k = w/(w+2g), k′ = √(1−k²)
εeff = (εr+1)/2
Z0 = (30π/√εeff) · K(k′)/K(k)
K(k) = π/[2 AGM(1,√(1−k²))]
εeff = (εr+1)/2
Z0 = (30π/√εeff) · K(k′)/K(k)
K(k) = π/[2 AGM(1,√(1−k²))]
13. Rectangular Waveguide TE10
fc10 = c/(2a)
fc20 = c/a
fc01 = c/(2b)
fnext = min(fc20,fc01)
q = √[1 − (fc10/f)²]
λ0 = c/f
λg = λ0/q
ZTE = η0/q
vg/c = q
fc20 = c/a
fc01 = c/(2b)
fnext = min(fc20,fc01)
q = √[1 − (fc10/f)²]
λ0 = c/f
λg = λ0/q
ZTE = η0/q
vg/c = q
14. Skin Depth & Surface Resistance
μ = μ0μr
ω = 2πf
δ = √[2/(ωμσ)]
Rs = √(πfμ/σ)
R′ ≈ Rs/w
ω = 2πf
δ = √[2/(ωμσ)]
Rs = √(πfμ/σ)
R′ ≈ Rs/w
15. RF Via Inductance & Reactance
hin = hmm/25.4
L(nH) = 5.08hin[ln(4h/d) + 1]
XL = 2πfL
Leq = L/N
Xeq = XL/N
L(nH) = 5.08hin[ln(4h/d) + 1]
XL = 2πfL
Leq = L/N
Xeq = XL/N
The N-via result treats identical vias as ideal parallel inductors.
Page 3 — Everyday RF engineering quick calculators
1. Frequency ↔ Wavelength
λ = c·VF/f
f = c·VF/λ
εr,eq = 1/VF²
f = c·VF/λ
εr,eq = 1/VF²
2. Frequency ↔ Period ↔ Angular Frequency
T = 1/f
f = 1/T
ω = 2πf
cycles/µs = f/106
f = 1/T
ω = 2πf
cycles/µs = f/106
3. dB Ratio Converter
dB = 10 log10(P2/P1)
P2/P1 = 10dB/10
dB = 20 log10(V2/V1)
V2/V1 = 10dB/20
P2/P1 = 10dB/10
dB = 20 log10(V2/V1)
V2/V1 = 10dB/20
4. Free-Space Path Loss (FSPL)
λ = c/f
FSPL(dB) = 20 log10(4πd/λ)
td = d/c
FSPL(dB) = 20 log10(4πd/λ)
td = d/c
5. Friis RF Link Calculator
FSPL = 20 log10(4πdf/c)
Pr(dBm) = Pt + Gt + Gr − FSPL − L
Link Margin = Pr − Sensitivity
EIRP = Pt + Gt
Pr(dBm) = Pt + Gt + Gr − FSPL − L
Link Margin = Pr − Sensitivity
EIRP = Pt + Gt
6. EIRP / ERP / Conducted Power
EIRP(dBm) = Pt(dBm) − Lfeeder(dB) + Gantenna(dBi)
ERP(dBm) = EIRP(dBm) − 2.15
EIRP(W) = 10(EIRPdBm−30)/10
Power density at 1 m = EIRP(W)/(4π)
ERP(dBm) = EIRP(dBm) − 2.15
EIRP(W) = 10(EIRPdBm−30)/10
Power density at 1 m = EIRP(W)/(4π)
7. Antenna Gain ↔ Effective Aperture
λ = c/f
G = 10GdBi/10
Ae = Gλ²/(4π)
G = 4πAe/λ²
Dequiv = √[4Ae/(πη)]
G = 10GdBi/10
Ae = Gλ²/(4π)
G = 4πAe/λ²
Dequiv = √[4Ae/(πη)]
8. Thermal Noise Floor — kTB
N = kTB
N(dBm) = 10 log10N(W) + 30
N0(dBm/Hz) = 10 log10(kT) + 30
Vrms,50Ω = √(50N)
N(dBm) = 10 log10N(W) + 30
N0(dBm/Hz) = 10 log10(kT) + 30
Vrms,50Ω = √(50N)
9. Noise Figure ↔ Noise Temperature
F = 10NF/10
Te = T0(F − 1)
F = 1 + Te/T0
NF = 10 log10F
Te = T0(F − 1)
F = 1 + Te/T0
NF = 10 log10F
10. Receiver Sensitivity / MDS
Nin(dBm) = −173.975187 + 10 log10B(Hz) + NF
Sensitivity(dBm) = Nin + SNRrequired + Lother
P(W) = 10(Sensitivity−30)/10
Vrms = √(PZ0)
Sensitivity(dBm) = Nin + SNRrequired + Lother
P(W) = 10(Sensitivity−30)/10
Vrms = √(PZ0)
11. Doppler Shift Calculator
Δf = fvr/c
vr = Δfc/f
vr = Δfc/f
This is the first-order radial-velocity approximation used by the tile.
12. Mixer / LO / IF / Image Planner
LOlow = RF − IF
Imagelow = RF − 2IF
LOhigh = RF + IF
Imagehigh = RF + 2IF
Spur1 = 2RF − LOlow
Spur2 = 2LOhigh − RF
Imagelow = RF − 2IF
LOhigh = RF + IF
Imagehigh = RF + 2IF
Spur1 = 2RF − LOlow
Spur2 = 2LOhigh − RF
13. Frequency Accuracy — ppm ↔ Hz
Δf(Hz) = f · ppm · 10−6
ppm = (Δf/f)106
fmin = f − |Δf|
fmax = f + |Δf|
ppm = (Δf/f)106
fmin = f − |Δf|
fmax = f + |Δf|
14. Phase ↔ Time Delay ↔ Path Length
Δt = φ/(360f)
φ(deg) = 360fΔt
Δl = c·VF·Δt
Δt = Δl/(c·VF)
φ(deg) = 360fΔt
Δl = c·VF·Δt
Δt = Δl/(c·VF)
15. ADC Clock-Jitter SNR
SNRjitter(dB) = −20 log10(2πftj,rms)
tj,rms = 10−SNR/20/(2πf)
ENOB = (SNR − 1.76)/6.02
tj,rms = 10−SNR/20/(2πf)
ENOB = (SNR − 1.76)/6.02