# Coaxial waveguide-Cont. (Homework 2-323)

Fist part of this question is here

4. Consider a cylindrical coaxial waveguide of inner radius r1 and outer radius r2. The space between conductors is vacuum.

b. Find the time-average power transmitted down the waveguide.

c. What is the ratio of magnitude of the transverse components of E and H? (In this context called the “wave impedance”.)

d. What is the ratio of the voltage across the conductor and the current down one of the conductors? (In this context called the “characteristic impedance”.)

b)

Power is

$P=VI=πV_0^2 \sqrt{(ϵ/μ)}*\cos^2⁡(ωt)$

And since over a period

$cos^2 (ωt) ) ̅=1/2 ( 1/2π*∫_0^2π\cos^2⁡x dx=1/2$

one has the average power

$P ̅=π/2*V_0^2*\sqrt{(ϵ/μ)}$

c)

Magnetic field is

$H ⃗=B/μ=ϕ ̂*V_0/μc*(ln⁡(r))/r*\cos⁡(kz-ωt)$

And since in the coaxial cable B and H are transversal (TEM modes) we have

$E ⃗/H ⃗ =(V_0*(ln⁡(r))/r)/(V_0/μc*(ln⁡(r))/r)=μc=\sqrt{(μ/ϵ)}$ (wave impedance)

d) The characteristic impedance is (from blue equations)

$Z_0=V/I=1/π*\sqrt{(μ/ϵ)}$

Observation: given the cable has radius R and r the characteristic impedance is in fact (wiki)

$Z_0=1/2π*ln⁡(R/r) \sqrt{(μ/ϵ)}$