RL low pass filter
Calculator and formulas for calculating the parameters of an RL low pass filter
This function calculates the properties of a low-pass filter consisting of a resistor and a coil. The output voltage, attenuation and phase shift are calculated for the given frequency.
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\(\displaystyle L\) = Inductance [H]
\(\displaystyle R\) = Resistance [Ω]
\(\displaystyle U_1\) = Input voltage [V]
\(\displaystyle U_2\) = Output voltage [V]
\(\displaystyle X_L\) = Reactance [Ω]
\(\displaystyle f_g\) = Cutoff frequency [Hz]
\(\displaystyle φ\) = Phase angle [°]
Formulas for the RL low pass
Calculate the voltage ratio
The output voltage U2 of an RL low pass is calculated using the following formula.
\(\displaystyle U_2=U_1 ·\frac{R} {\sqrt{R^2 + (2 · π · f · L)^2}}\)
or simply if XL is known
\(\displaystyle U_2=U_1 ·\frac{R}{\sqrt{R^2 + X_L^2}}\)
\(\displaystyle X_L=2 π · f ·L\)
Attenuation in decibels
At the resonance frequency, the damping is 3 dB. It can be calculated for the various frequencies using the formulas below. If the input and output voltage are known, the attenuation can easily be calculated using the following formula.
\(\displaystyle V_u=20 · lg \left(\frac{U_2}{U_1} \right) \)
If the voltages are not known, the following formula is used.
\(\displaystyle V_u=20·lg\left(\frac{R} {\sqrt{R^2 + (2 · π · f · L)^2}}\right)\)
or simply shown
\(\displaystyle V_u=20·lg\left(\frac{R} {\sqrt{R^2 + (ω · L)^2}}\right)\)
Phase shift
In an RL low pass, the output voltage lags behind the input voltage by 0 ° to -90 °, depending on the frequency. At the resonance frequency, the phase shift is -45 °. At low frequencies it tends towards 0. At high frequencies the phase rotates towards -90 °. The phase shift can be calculated using the following formula.
\(\displaystyle φ=acos \left(\frac{U_2}{U_1} \right)\)
\(\displaystyle φ= atan \left(ω · \frac{L}{R}\right)\)
Cutoff frequency
At the cutoff frequency fg or ωg the value of the amplitude-frequency response (ie the magnitude of the transfer function) is 0.707, which corresponds to -3dB.
\(\displaystyle 0.707= \frac{1}{\sqrt{2}}\)
Cutoff frequency formulas
\(\displaystyle ω_g= \frac{R}{L} ⇒ f_g=\frac{R}{2·π·L}\)
\(\displaystyle R= 2·π·f_g·L\)
\(\displaystyle L=\frac{R}{2·π·f_g}\)
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