(Solved): A lossless transmission line has characteristic impedance Z0=50. It is 80m long and connect ...

A lossless transmission line has characteristic impedance $$Z_0=50?$$. It is $$80\mathrm{m}$$ long and connected at one end to a source of $$v_g=120\cos (2?ft)$$ and at the other a load of $$80?$$. 1. Suppose the operating frequency $$f$$ is $$60\mathrm{Hz}$$ with a propagation velocity $$u_p=2c/3$$, where $$c$$ is the speed of light. Find the wavelength of the signals on the traveling wave. Compare it with the length of the line. Find the voltage across the load. 2. Suppose the operating frequency is instead $$500\mathrm{kHz}$$ with the same propagation velocity $$u_p=2c/3$$. (a) Find the voltage $${\tilde{V}}_L$$ and the current $${\tilde{I}}_L$$ across the load. (b) Find the input voltage $${\tilde{V}}_i$$ and the input current $${\tilde{I}}_i$$. (c) Compute the time-average power at the input and at the load. Compare the two powers. Note: For a pair of voltage $$\tilde{V}$$ and current $$\tilde{I}$$ the time-average power can be computed with $$P_{av}=\frac{1}{2}\mathrm{Re}\left[\tilde{V}\times {\tilde{I}}^{?}\right]$$ where $${\tilde{I}}^{?}$$ is the complex conjugate of the current phasor $$\tilde{I}$$.