# Impedance of transmission line

A customer requested a 50 ohm loss-less planar line, with the restriction  that the width of the strip cannot exceed 3 mm. what should the distance, ‘d’ be? However due to error during manufacturing, if the final characteristic impedance value turns out to be 52 ohm, the how much would the ‘d’ differ

Impedance of loss-less line is $Z0 = \sqrt{(L/C)}$

Inductance of line parallel to conducting wall is

$L = [\mu*l/(2*pi)]*ln(d/a)$

$l$ is length, $d$ is distance to wall, $a$ is wire radius

Capacitance of line parallel to conducting wall is

$C = (2*\pi*\epsilon*l)/[ln(d/a)]$

$Z^2 = L/C = [\mu/(4*pi*\epsilon)]*ln^2(d/a)$

$Z =50$ ohm, $a = 3$ mm, $\mu =4*\pi*10^{-7}$ H/m, $1/(4*\pi*\epsilon) =9*10^9$

$ln^2(d/a) = 50^2/(4*10^{-7}*9*10^9) =0.221$

$d/a =1.6$

$d = 1.6*3 =4.8 mm$

for $Z = 524$ ohms

$ln^2(d/a) = 0.239$

$d/a =1.63$ so that $d =4.892$ mm

difference in d is $4.892-4.8 =0.092$ mm