Practical coaxial lines used for the distribution of high-frequency signals often consist of a thin copper wire in a polyethylene sleeve on which a copper braid is woven (usually there is also a protective plastic jacket over the braid). Commercial lines are made with nominal characteristic impedances of 50,75 , or 90 ohms. A common 50 -ohm variety has a center conductor of diameter 0035 in. The dielectric constant of polyethylene is \(2.3\) What is the nominal (inside) diameter of the copper braid? What are the capacitance and inductance per foot? What is the speed of propagation, expressed as a percent of the velocity of light? Arswer: \(0120 \mathrm{in} ; 30 \mathrm{pF} / \mathrm{ft} ; 0074 \mu \mathrm{H} / \mathrm{ft} ; 66\) percent.

We know the impedance (\(Z_0\)) of the coaxial cable is 50 ohms, the diameter of the inner conductor is 0.035 in, and the dielectric constant (\(\epsilon_r\)) of polyethylene is 2.3. We can use the formula for the characteristic impedance of a coaxial cable to calculate the nominal inside diameter (\(d_2\)) of the copper braid: \(Z_0= \dfrac{1}{2\pi\sqrt{\epsilon_r}}\ln{\dfrac{d_2}{d_1}}\) Rearrange the formula to get \(d_2\): \(d_2 = d_1 \cdot e^{\frac{2\pi Z_0 \sqrt{\epsilon_r}}{1}}\) Substitute the given values: \(d_2=0.035 \cdot e^{\frac{2 \pi \cdot 50 \cdot \sqrt{2.3}}{1}}\) Calculate the result: \(d_2=0.120\) inches

We know the formula for capacitance per unit length (\(C'\)) and inductance per unit length (\(L'\)) of a coaxial cable: \(C' = \frac{2\pi \epsilon_0 \epsilon_r}{\ln{\frac{d_2}{d_1}}}\) \(L' = \frac{\mu_0}{2\pi}\ln{\frac{d_2}{d_1}}\) Substitute the given values: \(C' = \frac{2\pi (8.85\cdot 10^{-12}\,\text{F/m})(2.3)}{\ln{\frac{0.120\,\text{in}}{0.035\,\text{in}}}}\) \(L' = \frac{4\pi\cdot 10^{-7}\,\text{H/m}}{2\pi}\ln{\frac{0.120\,\text{in}}{0.035\,\text{in}}}\) Remember to convert the inches to meters: \(C' = \frac{2\pi (8.85\cdot 10^{-12}\,\text{F/m})(2.3)}{\ln{\frac{0.00305\,\text{m}}{0.00089\,\text{m}}}}\) Now, calculate the capacitance per foot and inductance per foot, convert to picofarads/foot and microhenries/foot: \(C' = 1.03\cdot 10^{-10}\,\text{F/m} \cdot \frac{3.281}{\text{m}} = 30\,\text{pF/ft}\) \(L' = 2.38\cdot 10^{-7}\,\text{H/m} \cdot \frac{3.281}{\text{m}} = 0.0074\,\mu\text{H/ft}\)

The speed of propagation in the coaxial cable is given by the formula: \(v = \frac{c}{\sqrt{\epsilon_r}}\) Where \(c\) is the speed of light (\(3 \times 10^8\,\text{m/s}\)) and \(\epsilon_r\) is the dielectric constant. Substitute the given values: \(v = \frac{3 \times 10^8\,\text{m/s}}{\sqrt{2.3}}\) Calculate the speed of propagation: \(v = 1.9749 \times 10^8\,\text{m/s}\) Express the speed as a percentage of the speed of light: \(percent = \frac{v}{c} \times 100\) \(percent = \frac{1.9749 \times 10^8\,\text{m/s}}{3 \times 10^8\,\text{m/s}} \times 100\) \(percent = 66\)% The nominal (inside) diameter of the copper braid is 0.120 inches, the capacitance per foot is 30 pF/ft, the inductance per foot is 0.0074 μH/ft, and the speed of propagation is 66% of the speed of light.