Calculate RL Low-pass Filter

Calculator and formulas for calculating the parameters of an RL low-pass filter

RL Low-pass Filter Calculator

RL Low-pass Filter

This function calculates the properties of a low-pass filter made from a resistor and an inductor. It calculates the output voltage, attenuation, and phase shift at the given frequency.

Results
Reactance XL:
Output voltage:
Attenuation dB:
Phase shift φ:

Circuit Diagram & Parameters

RL Low-pass Filter
Parameters
\(\displaystyle L\) = Inductance [H]
\(\displaystyle R\) = Resistance [Ω]
\(\displaystyle U_1\) = Input voltage [V]
\(\displaystyle U_2\) = Output voltage [V]
\(\displaystyle X_L\) = Inductive reactance [Ω]
\(\displaystyle f_c\) = Cutoff frequency [Hz]
\(\displaystyle φ\) = Phase angle [°]

Example Calculations

Practical Calculation Examples

Example 1: Audio Low-pass Filter

Given: R = 8Ω, L = 1mH, f = 3kHz, U₁ = 10V

\[X_L = 2\pi \cdot 3000 \cdot 0.001 = 18.8Ω\]
\[U_2 = 10 \cdot \frac{8}{\sqrt{8^2 + 18.8^2}} = 3.92V\]
\[V_u = 20 \cdot \lg\left(\frac{3.92}{10}\right) = -8.1dB\]
High frequencies are attenuated
Example 2: Cutoff Frequency Calculation

Given: R = 100Ω, L = 10mH

\[f_c = \frac{R}{2\pi L} = \frac{100}{2\pi \cdot 0.01} = 1592Hz\]
At fc: \[X_L = R = 100Ω\]
\[U_2 = U_1 \cdot \frac{1}{\sqrt{2}} = 0.707 \cdot U_1\]
-3dB attenuation at cutoff frequency
Example 3: Phase Shift

Given: R = 50Ω, L = 5mH, f = 1kHz

\[X_L = 2\pi \cdot 1000 \cdot 0.005 = 31.4Ω\]
\[φ = \arctan\left(\frac{X_L}{R}\right) = \arctan\left(\frac{31.4}{50}\right) = 32.1°\]
Current lags voltage by 32.1°
Typical inductive behavior
Low-pass Filter Characteristics
Frequency Response:
f << fc: Pass-through (0dB)
f = fc: -3dB attenuation
f >> fc: Strong attenuation
Roll-off: -20dB/decade
Phase Response:
f = 0: φ = 0°
f = fc: φ = -45°
f → ∞: φ → -90°
Inductive: Current lags voltage

Formulas for RL Low-pass Filter

Calculate Voltage Ratio

The output voltage U2 of an RL low-pass filter is calculated using the following formula.

Output Voltage
\[\displaystyle U_2=U_1 \cdot\frac{R} {\sqrt{R^2 + (2 \cdot \pi \cdot f \cdot L)^2}}\]

or simpler, when XL is known:

\[\displaystyle U_2=U_1 \cdot\frac{R}{\sqrt{R^2 + X_L^2}}\]
Reactance
\[\displaystyle X_L=2 \pi \cdot f \cdot L\]

The inductive reactance increases proportionally with frequency.

Attenuation in Decibels

The attenuation is 3dB at the cutoff frequency. It can be calculated for different frequencies using the formulas below.

Attenuation (simple)
\[\displaystyle V_u=20 \cdot \lg \left(\frac{U_2}{U_1} \right)\]

When input and output voltages are known.

Attenuation (complex)
\[\displaystyle V_u=20\cdot\lg\left(\frac{R} {\sqrt{R^2 + (\omega \cdot L)^2}}\right)\]

Direct calculation from R, L and ω.

Phase Shift

In an RL low-pass filter, the output voltage lags the input voltage by 0° to -90° depending on frequency. At the cutoff frequency, the phase shift is -45°.

Phase angle (simple)
\[\displaystyle φ=\arccos \left(\frac{U_2}{U_1} \right)\]
Phase angle (complex)
\[\displaystyle φ= \arctan \left(\frac{X_L}{R}\right)\]
Low-pass Phase Response

Cutoff Frequency

At cutoff frequency fc, the value of the amplitude-frequency response equals 0.707. This corresponds to –3dB.

Cutoff Frequency Formulas
\(\displaystyle \omega_c= \frac{R}{L} \Rightarrow f_c=\frac{R}{2\cdot\pi\cdot L}\)
\(\displaystyle R=2\cdot\pi\cdot f_c\cdot L\)
\(\displaystyle L=\frac{R}{2\cdot\pi\cdot f_c}\)

Practical Applications

Audio Technology
  • Subwoofer filters
  • Crossover networks
  • Noise filters
  • Anti-aliasing
Signal Processing
  • Smoothing filters
  • Integrator
  • Interference suppression
  • EMC filters
Power Electronics
  • Motor chokes
  • Mains filters
  • Rectifier circuits
  • Switch-mode power supplies


Other induction calculators

Inductance  •  Reactance of a coil  •  Cutoff frequency R/L  •  Differentiator R/L  •  Highpass filter R/L  •  Lowpass filter R/L  •  Series circuit R/L  •  Parallel circuit R/L  •  Transformer  •