A long lossless transmission line has a unity power factor (UPF) load at the receiving end and an ac voltage source at the sending end. The parameters of the transmission line are as follows:
Characteristic impedance $${Z_c} = 400\Omega ,\,\,$$, propagation constant $$\,\beta = 1.2 \times {10^{ - 3}}\,\,rad/km,\,\,$$ and length $$\,l = 100\,km.\,\,$$ The equation relating sending and receiving end questions is $${V_s} = {V_r}\,\cosh \,\,\left( {\beta l} \right) + j\,Z{}_c\,\,\sinh \left( {\beta l} \right){{\rm I}_R}$$ Complete the maximum power that can be transferred to the UPF load at the receiving end if $$\left| {{V_s}} \right| = 230\,\,kV.\,\,$$

Consider a long, two-wire line composed of solid round conductors. The radius of both conductors is $$0.25$$ cm and the distance between their centers is $$1$$m. If this distance is doubled, then the inductance per unit length.

A single input single output system with $$y$$ as output and $$u$$ as input, is described by $$${{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + 10y = 5{{d\,u} \over {dt}} - 3\,u$$$
For the above system find an input $$u(t),$$ with zero initial condition, that produces the same output as with no input and with the initial conditions.
$${{d\,y\left( {{0^ - }} \right)} \over {dt}} = - 4,\,\,\,y\left( {{0^ - }} \right) = 1$$

What is the $$rms$$ value of the voltage waveform shown in Fig?