Past exam of the physics 2 paper of the University of Oxford / 2013 resit / 10 / ii

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- Solution ii

Solution

The source figure has a series capacitor followed by

R

in parallel with

L

. Its complex impedance is

Z = - \frac{i}{ω C} + \frac{iω L R}{R + iω L} = \frac{R ω ^{2} L ^{2}}{R ^{2} + ω ^{2} L ^{2}} + i [\frac{ω L R ^{2}}{R ^{2} + ω ^{2} L ^{2}} - \frac{1}{ω C}] .

The imaginary part vanishes at a finite positive frequency only if

R^{2} C > L

, in which case

ω_{*}^{2} = \frac{R ^{2}}{L ( R ^{2} C - L )} = \frac{1}{L C - L ^{2} / R ^{2}} .

At that frequency, substituting

R^{2} + ω_{*}^{2} L^{2} = ω_{*}^{2} L R^{2} C

into the real part gives

Z (ω_{*}) = \frac{L}{RC} .

If

R^{2} C \leq L

, no finite positive-frequency cancellation exists; the impedance merely tends toward its high-frequency limit.

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