Simple High-Pass Filter

Electricity and Magnetism Level 3

The circuit below is a simple high-pass filter which takes a sinusoidal input V i n V_{in} and produces a sinusoidal output V o u t V_{out} . The angular frequency (in rad/s) of the input sinusoid is ω \omega .

For what value of ω \omega is ∣ V o u t ∣ ∣ V i n ∣ = 1 2 \frac{|V_{out}|}{|V_{in}|} = \frac{1}{\sqrt{2}} ?


The answer is 666.667.

Steven Chase by Steven Chase , 37, USA

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1 solution

We note that V o u t V i n = j ω L R + j ω L = 1 1 − j R ω L \dfrac {V_{out}}{V_{in}} = \dfrac {j\omega L}{R + j\omega L} = \dfrac 1{1- j \frac R{\omega L}} . For ∣ V o u t ∣ ∣ V i n ∣ = 1 2 \dfrac {|V_{out}|}{|V_{in}|} = \dfrac 1{\sqrt 2} , we have:

∣ 1 1 − j R ω L ∣ = 1 2 ∣ 1 − j R ω L ∣ = 2 1 + R 2 ω 2 L 2 = 2 ⟹ ω = R L = 2 3 × 1 0 − 3 ≈ 666.667 rad/s \begin{aligned} \left| \frac 1{1- j \frac R{\omega L}} \right| & = \frac 1{\sqrt 2} \\ \left|1- j \frac R{\omega L}\right| & = \sqrt 2 \\ 1 + \frac {R^2}{\omega^2 L^2} & = 2 \\ \implies \omega & = \frac RL = \frac 2{3 \times 10^{-3}} \approx \boxed{666.667} \text{ rad/s} \end{aligned}

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