Calculate RL Cutoff Frequency

Calculator and formulas for calculating the cutoff frequency of a coil and a resistor

RL Cutoff Frequency Calculator

RL Cutoff Frequency

With this function, the cutoff frequency of a coil and a resistor, or the inductance or resistance can be calculated. Two of the values must be known to calculate the third.

Results
Inductance:
Cutoff frequency:
Resistance:

Cutoff Frequency & -3dB Point

What is the Cutoff Frequency?

The cutoff frequency is the point where the output voltage has dropped to 70.7% of the input voltage (-3dB attenuation). In RL circuits, it determines the transition behavior between resistive and inductive regions.

Basic Formula
\[f_c = \frac{R}{2\pi L}\]

At this frequency XL = R

Parameter Meaning
fcCutoff frequency (-3dB point)
ROhmic resistance
LInductance
XLInductive reactance
ωcCutoff angular frequency = 2πfc

Example Calculations

Practical Calculation Examples

Example 1: Audio Crossover

Given: L = 2.2 mH, R = 8 Ω (Speaker)

\[f_c = \frac{R}{2\pi L} = \frac{8}{2\pi \cdot 2.2 \times 10^{-3}}\]
\[f_c = \frac{8}{0.0138} = 579 \text{ Hz}\]
Crossover frequency for midrange
Example 2: Motor Choke

Given: L = 50 mH, R = 25 Ω (Winding resistance)

\[f_c = \frac{25}{2\pi \cdot 50 \times 10^{-3}}\]
\[f_c = \frac{25}{0.314} = 79.6 \text{ Hz}\]
At 50 Hz: XL = 15.7 Ω < R
Resistive behavior at mains frequency
Example 3: RF Choke

Given: L = 10 µH, R = 0.5 Ω (ESR)

\[f_c = \frac{0.5}{2\pi \cdot 10 \times 10^{-6}}\]
\[f_c = \frac{0.5}{62.8 \times 10^{-6}} = 7.96 \text{ kHz}\]
Above 8 kHz acts inductively
RF decoupling from 8 kHz
Understanding Frequency Behavior
Below cutoff frequency (f < fc):
XL < R: Resistive behavior dominates
Phase: φ < 45°
Current: Little frequency dependence
Application: Ohmic load
Above cutoff frequency (f > fc):
XL > R: Inductive behavior dominates
Phase: φ > 45°
Current: Strongly frequency dependent
Application: Inductive choke

RL Cutoff Frequency - Theory and Formulas

What is the RL Cutoff Frequency?

The cutoff frequency of an RL combination is the point where the inductive reactance XL equals the ohmic resistance R. This corresponds to a phase shift of 45° and an attenuation of -3dB. It divides the frequency spectrum into a resistive and an inductive region.

Calculation Formulas

Basic Calculation Formulas
Cutoff Frequency
\[f_c = \frac{R}{2\pi L}\]

Frequency at XL = R

Calculate Resistance
\[R = 2\pi f_c L\]

Resistance for desired cutoff frequency

Calculate Inductance
\[L = \frac{R}{2\pi f_c}\]

Inductance for desired cutoff frequency

Angular Frequency
\[ω_c = \frac{R}{L} = 2\pi f_c\]

Cutoff angular frequency in rad/s

Frequency-Dependent Behavior
Low Frequencies (f << fc)
\[X_L = 2\pi f L << R\] \[Z ≈ R\] \[φ ≈ 0°\]

Resistive behavior dominates

High Frequencies (f >> fc)
\[X_L = 2\pi f L >> R\] \[Z ≈ X_L\] \[φ ≈ 90°\]

Inductive behavior dominates

At Cutoff Frequency (f = fc)
\[X_L = R\]

Equal impedances

\[φ = 45°\]

Phase shift

\[|Z| = R\sqrt{2}\]

Total impedance

Filter Properties
RL Low-pass (Output at R)
\[H(f) = \frac{R}{\sqrt{R^2 + (2\pi f L)^2}}\]
  • Low frequencies pass
  • High frequencies are attenuated
  • -20dB/decade above fc
RL High-pass (Output at L)
\[H(f) = \frac{2\pi f L}{\sqrt{R^2 + (2\pi f L)^2}}\]
  • High frequencies pass
  • Low frequencies are attenuated
  • +20dB/decade below fc
-3dB Point

At the cutoff frequency, the output voltage is reduced by the factor 1/√2 ≈ 0.707. This corresponds to a power reduction by half (-3dB).

\[20 \log_{10}(0.707) = -3.01 \text{ dB}\]
Practical Applications
Audio Technology
  • Speaker crossovers: Separation of different frequency ranges
  • EMI suppression: RF suppression in audio signals
  • Equalizers: Frequency response correction
  • Microphone filters: Wind noise suppression
Power Electronics
  • Motor chokes: Current limiting and smoothing
  • Mains filters: EMC compliance
  • Switching regulators: Output inductors
  • PFC chokes: Power factor correction
RF Technology
  • Decoupling: RF chokes for DC supplies
  • Impedance matching: Antenna tuning
  • Filters: Harmonic suppression
  • Baluns: Balanced/unbalanced conversion
Measurement & Control
  • Anti-aliasing: Filters before A/D converters
  • Sensor filters: Interference signal suppression
  • Control loops: Stability optimization
  • Oscilloscopes: Bandwidth limitation

Design Guidelines and Optimization

Important Design Aspects
  • Quality factor: Q = XL/R determines filter sharpness
  • Temperature stability: Both L and R are temperature dependent
  • Saturation behavior: Coil core must not saturate at high currents
  • Parasitic effects: Self-capacitance of coil at high frequencies
  • Losses: ESR of coil reduces filter effectiveness
  • Current rating: Both coil and resistor must be adequately dimensioned

Cutoff Frequency in Different Ranges

Audio (20 Hz - 20 kHz):
L: 1 mH - 100 mH
R: 4 - 16 Ω
fc: 100 Hz - 2 kHz
Mains (50/60 Hz):
L: 10 mH - 1 H
R: 1 - 100 Ω
fc: 1 - 100 Hz
Switching PSU (kHz):
L: 10 µH - 10 mH
R: 0.1 - 10 Ω
fc: 100 Hz - 10 kHz
RF (MHz):
L: 1 nH - 100 µH
R: 0.1 - 50 Ω
fc: 1 kHz - 100 MHz

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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  •