Example 14 mil spacing
Ground spacing 14 mil, width 6.25 mil, and copper 0.70 mil.
Worked example
Formulas use these lengths in mil: B 14 mil, W 6.25 mil, t 0.7 mil.
Width correction uses the printed line for t/B much less than 1. This t/B is 0.05.
Cohn impedance 47.8796 ohm
Formula
Z0 = 30 * pi / sqrt(er) * K(k-prime) / K(k). The ratio uses the arithmetic-geometric mean.
Width ratio 0.4699
Formula
W / (B - t)
IPC-2141 impedance 50.1254 ohm
IPC-2141 does not apply because W/(B-t) is not below 0.35. Cohn is the impedance model when the elliptic ratio is defined.
Formula
Z0 about 60 / sqrt(er) * ln(1.9 * B / (0.8 * W + t)), valid when W/(B-t) < 0.35. It is not substituted for Cohn.
Delay and loss
Phase velocity 162583954.873637 m/s
Formula
vp = 2.9979e8 / sqrt(er)
Delay per mm 6.150542 ps/mm
Formula
tpd = 3.3356 * sqrt(er) ps/mm
Delay per inch 156.215961 ps/inch
Formula
tpd = 84.72 * sqrt(er) ps/inch
Dielectric loss 3.358265 dB/m
Formula
alpha_d = 27.3 * sqrt(er) * tand * f / 2.9979e8 dB/m
Dielectric loss per inch 0.085557 dB/inch
Formula
alpha_d = 2.32 * f_GHz * sqrt(er) * tand dB/inch
Skin depth 6.608549e-7 m
Formula
delta = 1 / sqrt(pi * f * mu0 * sigma), mu0 = 4 * pi * 1e-7
Printed copper skin shortcut 0.000661 um
Formula
66.1 / sqrt(f_Hz) um, printed copper shortcut. It is not substituted for the general skin depth.
66.1/sqrt(f) micrometers is the copper shortcut. It is not substituted for skin depth when conductivity is typed.
Surface resistance 0.02609 ohm
Formula
Rs = sqrt(pi * f * mu0 / sigma)
Conductor loss 0.506166 Np/m
Formula
alpha_c = Rs / (2 * pi * Z0 * B) * [1 + 2W/(B-t) + (1/pi) * ln((B+t)/(B-t))] / (1 - t/B) Np/m. Z0 is the Cohn value.
Conductor loss in dB 4.396502 dB/m
Formula
dB/m = Np/m * 20 / ln(10)
Summed loss 7.754767 dB/m
Formula
alpha_total dB/m = alpha_d dB/m + alpha_c dB/m
Width correction 1.869114e-5 m
Formula
dW = (t/pi) * (1 - ln(2t/B)), for t/B much less than 1. 1 mil = 0.0000254 m.
Effective width 0.000177 m
Formula
W-prime = W + dW
Modulus k 0.755724
Formula
k = sech(pi * W-prime / (2 * B))
Complement k-prime 0.65489
Formula
k-prime = tanh(pi * W-prime / (2 * B))
The worked example uses 14 mil, 6.25 mil, 0.70 mil, permittivity 3.40, loss tangent 0.002, and 10 GHz. Conductivity 5.8e7 S/m is the copper value in the source equation.