Low Pass Filter — RC & RL Circuit Explained with Formulas
What is a Low Pass Filter?
A low pass filter is a circuit that allows signals with frequencies lower than a certain cutoff frequency (fc) to pass through while attenuating signals with frequencies higher than fc. The range of frequencies below fc is called the passband, and the range above is called the stopband.
Low pass filters are fundamental building blocks in electronics — used in audio systems, power supplies, communication receivers, and signal conditioning circuits. They remove unwanted high-frequency noise while preserving the desired low-frequency signal.
Types of Low Pass Filter
There are two basic passive low pass filter configurations:
- Low-Pass RC Filter — uses a resistor and capacitor
- Low-Pass RL Filter — uses a resistor and inductor
Both achieve the same filtering function but use different reactive components. The choice depends on the application, frequency range, and component availability.
Low-Pass RC Filter
In a low-pass RC filter, the output voltage is taken across the capacitor. The capacitive reactance XC = 1/(2πfC) decreases as frequency increases. At low frequencies, XC is high so most of the input voltage appears across the capacitor (output). At high frequencies, XC drops to near zero, shorting the output.
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| Fig-1: Low Pass RC Filter Circuit |
Key Formulas for RC Low Pass Filter
At the cutoff frequency:
- Output voltage = 70.7% of input voltage (Vo = 0.707 × Vi)
- Output power is reduced by half (−3 dB point)
- Phase angle between Vo and Vi is −45°
- Resistance equals capacitive reactance: R = XC
Frequency Response of RC Filter
The frequency response curve shows how the output voltage magnitude varies with signal frequency. The output remains nearly constant in the passband (f < fc) and rolls off at −20 dB/decade in the stopband (f > fc).
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| Fig-2: Frequency Response of Low Pass RC Filter |
Low-Pass RL Filter
In a low-pass RL filter, the output voltage is taken across the resistor. The inductive reactance XL = 2πfL increases with frequency. At low frequencies, XL is small so the inductor offers little opposition and most voltage appears across R (output). At high frequencies, XL becomes large and drops most of the input voltage, reducing the output.
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| Fig-3: Low Pass RL Filter Circuit |
Key Formulas for RL Low Pass Filter
At the cutoff frequency:
- Output voltage = 70.7% of input (Vo = 0.707 × Vi)
- Output is at −3 dB
- Phase angle between Vo and Vi is +45°
- Resistance equals inductive reactance: R = XL
Frequency Response of RL Filter
The RL low pass filter has the same magnitude response shape as the RC filter — flat in the passband and rolling off at −20 dB/decade. The key difference is the phase response: the RC filter produces a lagging phase shift while the RL filter produces a leading phase shift.
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| Fig-4: Frequency Response of Low Pass RL Filter |
RC vs RL Low Pass Filter — Comparison Table
Applications of Low Pass Filters
- Audio Systems: Subwoofer crossover networks use LPF to send only bass frequencies (below 200 Hz) to the woofer
- Power Supplies: Smoothing capacitors in rectifier circuits act as low pass filters to remove AC ripple
- Communication Receivers: Anti-aliasing filters before ADC converters prevent frequency folding
- Sensor Signal Conditioning: Removing high-frequency noise from temperature, pressure, and strain gauge readings
- DAC Output Smoothing: Reconstructing smooth analog signals from digital staircase outputs
- EMI/EMC Compliance: RL filters on power lines suppress conducted high-frequency interference
Design Example — RC Low Pass Filter
Problem: Design an RC low pass filter with a cutoff frequency of 1 kHz.
Choose C = 0.1 μF (standard value)
R = 1/(2π × fc × C)
R = 1/(2π × 1000 × 0.1×10⁻⁶)
R = 1591 Ω ≈ 1.6 kΩ (nearest standard value)
With R = 1.6 kΩ and C = 0.1 μF, the filter passes all frequencies below 1 kHz and attenuates frequencies above it at 20 dB per decade.
Frequently Asked Questions
1. What is the difference between a low pass filter and a high pass filter?
A low pass filter allows frequencies below the cutoff frequency to pass and blocks higher frequencies. A high pass filter does the opposite — it blocks low frequencies and allows high frequencies to pass. In an RC high pass filter, the output is taken across the resistor instead of the capacitor.
2. Why is the cutoff frequency called the −3 dB point?
At the cutoff frequency, the output voltage drops to 0.707 times the input voltage. In decibels, 20×log₁₀(0.707) = −3.01 dB. This corresponds to the half-power point where output power is exactly half the input power.
3. Can I cascade two low pass filters to get a steeper roll-off?
Yes. Cascading two identical first-order RC filters gives a second-order filter with −40 dB/decade roll-off. However, the effective cutoff frequency shifts lower, and you need a buffer (op-amp) between stages to prevent loading effects that alter the response.
4. Which is better for low-frequency filtering — RC or RL?
RC filters are preferred for low-frequency applications because inductors at low frequencies require large inductance values, making them physically bulky and expensive. Capacitors are compact, inexpensive, and available in a wide range of values, making RC filters the practical choice for most signal-level applications.
5. What happens to the phase shift at frequencies much higher than fc?
For an RC low pass filter, the phase shift approaches −90° as frequency increases far beyond fc. At f = fc the phase is −45°, and at f = 10×fc it is approximately −84°. The output increasingly lags the input as frequency rises.