Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 2 / e

Table of contents
- Solution e

Solution

Substitution into

ϵ = A_{0} U^{2} / (N ω)

gives

ϵ = \frac{A _{0} B ^{2} D _{0}}{[( A + A _{0} ) ( D + D _{0} ) - B ^{2} ] [( A + A _{0} ) D - B ^{2} ]} .

When

B^{2} ≪ A D

and

D ≪ D_{0}

,

ϵ ≃ \frac{A _{0} B ^{2}}{D ( A + A _{0} ) ^{2}} .

After scaling body size so that

A_{0} \mapsto λ A_{0}

, differentiation with respect to

λ A_{0}

shows that the optimum impedance match is

λ A_{0} = A

. Therefore

λ_{m a x} = \frac{A}{A _{0}}, ϵ_{m a x} = \frac{B ^{2}}{4 A D} .

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