Calculate RC Low Pass Filter

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

Calculate RC Low Pass

RC Low Pass Filter

This function calculates the properties of a low pass filter from resistor and capacitor values. For the given frequency, the output voltage, attenuation, and phase shift are calculated.

Results
Reactance XC:
Output Voltage U₂:
Attenuation [dB]:
Phase Angle φ:
Cutoff Frequency f₀:
Impedance Z:
Voltage UR:
Current I:
Time Constant τ:

RC Low Pass Circuit

RC Low Pass Circuit
Symbol Explanations
C = Capacitance [F]
R = Resistance [Ω]
U₁ = Input Voltage [V]
U₂ = Output Voltage [V]
XC = Capacitive Reactance [Ω]
φ = Phase Angle [°]
Z = Input Impedance [Ω]
I = Current [A]
UR = Voltage across Resistor [V]
Low Pass Characteristics
  • Passes low frequencies
  • Attenuates high frequencies
  • -3dB at cutoff frequency
  • -20dB/decade roll-off
  • Phase shift 0° to -90°
Cutoff Frequency
\[f_g = \frac{1}{2\pi RC}\]

At the cutoff frequency, the attenuation is -3dB.

RC Low Pass - Theory and Formulas

RC Low Pass Basics

An RC low pass is a first-order filter that passes low frequencies and attenuates high frequencies. The output is taken across the capacitor. At high frequencies, the capacitor has a low resistance, at low frequencies, a high resistance.

Important Formulas

Voltage Ratio
\[U_2 = U_1 \cdot \frac{1}{\sqrt{1 + (2\pi fRC)^2}}\]

or simpler with XC:

\[U_2 = U_1 \cdot \frac{X_C}{\sqrt{R^2 + X_C^2}}\]
Reactance
\[X_C = \frac{1}{2\pi fC}\]

The capacitive reactance decreases with increasing frequency.

Attenuation and Phase

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

or directly:

\[V_u = 20 \cdot \lg\left(\frac{1}{\sqrt{1 + (\omega RC)^2}}\right)\]
Phase Shift
\[\phi = \arccos\left(\frac{U_2}{U_1}\right)\]

or:

\[\phi = \arctan(\omega RC)\]

Cutoff Frequency and Characteristic Values

Cutoff Frequency
\[f_g = \frac{1}{2\pi RC}\]

At fg: Attenuation = -3dB, Phase = -45°

Impedance
\[Z = \sqrt{X_C^2 + R^2}\]

Total impedance of the circuit

Time Constant
\[\tau = RC\]

Characteristic time of the circuit

Frequency Response

Frequency Response Characteristics
  • Low frequencies (f ≪ fg): No attenuation, phase → 0°
  • Cutoff frequency (f = fg): -3dB attenuation, phase = -45°
  • High frequencies (f ≫ fg): Strong attenuation, phase → -90°
  • Slope: -20dB/decade above fg
  • Transfer function: H(jω) = 1/(1 + jωRC)

Practical Applications

Audio Filters:
• Subwoofer filters
• Anti-aliasing
• Noise suppression
• Treble attenuation
Smoothing Filters:
• Power supplies
• Signal smoothing
• Interference suppression
• PWM filters
Timing Elements:
• RC delay
• Integration
• Averaging
• Smoothing circuits

Design Guidelines

Important Design Aspects
  • Cutoff frequency selection: Should be well above the highest frequency to be transmitted
  • Capacitance selection: Larger C → lower fg, but larger components
  • Resistance selection: Compromise between input impedance and signal level
  • Loading effects: Following stage should have high input impedance
  • Tolerances: Component variations affect the cutoff frequency

Mathematical Relationships

Basic Formulas
\[\tau = R \times C\] \[f_g = \frac{1}{2\pi \tau}\]

Relationship between time constant and cutoff frequency

Conversions
\[R = \frac{1}{2\pi f_g C}\] \[C = \frac{1}{2\pi f_g R}\]

Calculation of components for a given cutoff frequency

Low Pass vs. High Pass

Differences
Low Pass (RC):
  • Output at the capacitor
  • Attenuates high frequencies
  • Phase: 0° to -90°
  • Smoothing characteristic
High Pass (CR):
  • Output at the resistor
  • Attenuates low frequencies
  • Phase: +90° to 0°
  • Differentiating characteristic

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Capacitor functions

Series connection with capacitors  •  Series connection with 2 capacitors  •  Reactance Xc of a capacitor  •  Time constant of an R/C circuit  •  Capacitor charging voltage  •  Capacitor discharge voltage  •  R/C for the charging voltage  •  Series circuit R/C  •  Parallel circuit R/C  •  Low pass-filter R/C  •  High pass-filter R/C  •  Integrator R/C  •  Differentiator R/C  •  Cutoff-frequency R,C  •  R and C for a given impedance  •