Section 8

Filters and frequency response

viz

H(j\omega)=\frac{1}{1+j\omega RC}

The RC low-pass |H|=1/√(1+(ωRC)²) rolls off after cutoff on the Bode plot.

Linear time-invariant filter; light load unless stated.

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RC cutoff frequency

viz

f_c=\frac{1}{2\pi RC}

fc=1/(2π RC) is the cutoff where |H| is 1/√2 (−3 dB) on the Bode plot.

Linear time-invariant filter; light load unless stated.

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RC high-pass filter

viz

H(j\omega)=\frac{j\omega RC}{1+j\omega RC}

The RC high-pass rises +20 dB/dec below fc and approaches 0 dB above cutoff.

Linear time-invariant filter; light load unless stated.

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RL low-pass filter

viz

H(j\omega)=\frac{R}{R+j\omega L}

The RL low-pass has the same first-order Bode shape as the RC low-pass.

Linear time-invariant filter; light load unless stated.

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Gain in decibels

G_{\mathrm{dB}}=20\log_{10}|H|

Computes Gain in decibels from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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First-order slope

m=-20\ \mathrm{dB/dec}

The −20 dB/dec slope applies in the roll-off band of a first-order low-pass; a high-pass rises +20 dB/dec below fc.

The −20 dB/dec slope applies in the roll-off band of a first-order low-pass; a high-pass rises +20 dB/dec below fc.

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RLC band-pass magnitude

|H(j\omega)|=\frac{R}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}

Computes RLC band-pass magnitude from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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Filter bandwidth

\mathrm{BW}=\frac{f_0}{Q}

Computes Filter bandwidth from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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Filter quality factor

Q=\frac{f_0}{\mathrm{BW}}

Computes Filter quality factor from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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RLC center frequency

f_0=\frac{1}{2\pi\sqrt{LC}}

Computes RLC center frequency from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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Section 8

Filters and frequency response

RC low-pass filter

viz

H(j\omega)=\frac{1}{1+j\omega RC}

The RC low-pass |H|=1/√(1+(ωRC)²) rolls off after cutoff on the Bode plot.

Linear time-invariant filter; light load unless stated.

View detail →

RC cutoff frequency

viz

f_c=\frac{1}{2\pi RC}

fc=1/(2π RC) is the cutoff where |H| is 1/√2 (−3 dB) on the Bode plot.

Linear time-invariant filter; light load unless stated.

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RC high-pass filter

viz

H(j\omega)=\frac{j\omega RC}{1+j\omega RC}

The RC high-pass rises +20 dB/dec below fc and approaches 0 dB above cutoff.

Linear time-invariant filter; light load unless stated.

View detail →

RL low-pass filter

viz

H(j\omega)=\frac{R}{R+j\omega L}

The RL low-pass has the same first-order Bode shape as the RC low-pass.

Linear time-invariant filter; light load unless stated.

View detail →

Gain in decibels

G_{\mathrm{dB}}=20\log_{10}|H|

Computes Gain in decibels from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

View detail →

First-order slope

m=-20\ \mathrm{dB/dec}

The −20 dB/dec slope applies in the roll-off band of a first-order low-pass; a high-pass rises +20 dB/dec below fc.

The −20 dB/dec slope applies in the roll-off band of a first-order low-pass; a high-pass rises +20 dB/dec below fc.

View detail →

RLC band-pass magnitude

|H(j\omega)|=\frac{R}{\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}}

Computes RLC band-pass magnitude from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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Filter bandwidth

\mathrm{BW}=\frac{f_0}{Q}

Computes Filter bandwidth from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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Filter quality factor

Q=\frac{f_0}{\mathrm{BW}}

Computes Filter quality factor from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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RLC center frequency

f_0=\frac{1}{2\pi\sqrt{LC}}

Computes RLC center frequency from the circuit model and the stated conditions.

Linear time-invariant filter; light load unless stated.

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