An island $($receiving end$)$ is connected to a power generation station $($sending end$)$ by three$-$phase transmission cables. To reduce power transmission losses, a step$-$up transformer $($n $=4)$ is installed on the sending end, and a step$-$down transformer $($n $=0\mathrm{.}25)$ is installed on the receiving end. Each cable has an impedance of $(0\mathrm{.}1+$ j$0\mathrm{.}05).$ Power is delivered to the island at $220$ V $($phase r$.$m$.$s$.$ voltage$)$ by a $50$ Hz$,$ balanced three$-$phase supply. The island presents a balanced starconnected load, consuming a total power of $240$ kW at $0\mathrm{.}92$ lagging power factor. Assume that both transformers are ideal. a$)$ Calculate the line current in the island, and the current in the transmission cables. $(3)$ b$)$ What would be the increase in active power losses in the transmission cables if transformers were not used? $(3)$ c$)$ For the step$-$down transformer, the maximum flux density in the core is $2\mathrm{.}2$ T and the core cross$-$sectional area is $0\mathrm{.}06$ m$2.$ What is the required number of turns in the primary winding and the secondary winding? $(6)$ d$)$ The step$-$down transformer is damaged and a radial air$-$gap is found in the core. After testing, $1000$ additional ampere$-$turns are required to reach the same flux density $(2\mathrm{.}2$ T$)$ in the core. How wide is the air$-$gap? Assume that the flux density remains constant throughout the core and the air$-$gap, $$r of the core is $540,$ and $0$ is $4$$10-7$ Hm$-1.(4)$ e$)$ Describe the losses that would be present and possible methods for loss reduction if the transformers were not ideal.