The fundamental band pass filter equation for the center frequency (fc) of a second-order passive RLC filter is fc = 1 / (2π√(LC)). This formula dictates the exact resonant peak where the filter passes signals with minimal attenuation, while the associated bandwidth equation (BW = R / 2πL) defines the width of that passband. Below is the complete derivation framework, symbol definitions, and practical design math you need to select real-world components.

The Core Band Pass Filter Equations & Symbol Table

A standard passive series RLC band pass filter relies on the resonance between the inductor and capacitor. At resonance, their reactances cancel out, leaving only the resistance to limit current. The two governing equations for center frequency and bandwidth are:

Center Frequency:
fc = 1 / (2π√(L × C))

Bandwidth (-3dB points):
BW = R / (2π × L)

Quality Factor:
Q = fc / BW = (1 / R) × √(L / C)

Symbol Parameter Standard SI Unit Practical Component Range
fc Center (Resonant) Frequency Hertz (Hz) 20 Hz (Audio) to 2.4 GHz (RF)
BW Bandwidth (fH - fL) Hertz (Hz) 10 Hz (Narrow IF) to 20 kHz (Wideband)
Q Quality Factor (Dimensionless) None 0.5 (Broad) to 100+ (Crystal/Narrow)
R Total Series Resistance Ohms (Ω) 1 Ω (Low loss) to 10 kΩ (Loaded)
L Inductance Henries (H) 10 nH (RF) to 100 mH (Audio/Subwoofer)
C Capacitance Farads (F) 1 pF (RF) to 100 μF (Audio)
π Archimedes' Constant ~3.14159 N/A

Assumptions, Realistic Magnitudes, and Unit Traps

When the Formula Applies (Assumptions)

These equations assume ideal, linear components operating in a sinusoidal steady state. In reality, inductors have parasitic parallel capacitance and DC winding resistance (DCR), while capacitors have Equivalent Series Resistance (ESR) and Equivalent Series Inductance (ESL). For audio frequencies (under 20 kHz), the ideal equations hold up well. For RF frequencies (above 10 MHz), parasitics dominate, and you must use S-parameter models or vendor-specific RF simulation tools rather than raw algebraic formulas.

Realistic Answer Magnitudes

  • Audio Crossovers: fc typically falls between 80 Hz (subwoofer integration) and 3,500 Hz (tweeter handoff). BW is usually wide, with Q factors between 0.7 (Butterworth) and 1.0 (Chebyshev).
  • AM Radio IF Filters: fc is strictly 455 kHz or 10.7 MHz (FM). BW is narrow, around 10 kHz, requiring high-Q inductors (Q > 50).
  • SDR / RF Front Ends: fc ranges from 433 MHz to 2.4 GHz. BW might be 200 kHz for narrowband telemetry or 20 MHz for WiFi.

Unit Mistakes That Break the Math

The most common reason a calculated band pass filter yields garbage results on the bench is failing to convert sub-multiples to base SI units before calculating.
The Trap: Plugging 10 mH and 100 nF directly into the calculator as "10" and "100".
The Fix: You must convert milli (m) to 10-3, micro (μ) to 10-6, nano (n) to 10-9, and pico (p) to 10-12. A 100 nF capacitor is always 0.0000001 F (1 × 10-7 F) in the equation.

Rearranged Forms for Component Selection

When designing a filter on the bench, you rarely solve for fc from scratch; you usually have a target frequency and need to find the right inductor or capacitor. Here are the rearranged forms solving for each variable:

  • Solve for Inductance (L): L = 1 / (4π² × fc² × C)
  • Solve for Capacitance (C): C = 1 / (4π² × fc² × L)
  • Solve for Resistance (R) to set Bandwidth: R = 2π × L × BW
  • Solve for Inductance (L) based on Bandwidth: L = R / (2π × BW)

Worked Examples with Unit Tracking

Problem 1: Analyzing an Existing Audio Crossover Filter

Scenario: You are repairing a vintage passive speaker crossover. The band pass section for the midrange driver uses a 10 mH radial inductor (e.g., Bourns 78F series), a 1 μF film capacitor (e.g., WIMA MKS2), and the driver's voice coil presents an 8 Ω resistive load. Find the center frequency and the -3dB bandwidth.

Step 1: Convert to base SI units.

  • L = 10 mH = 0.01 H
  • C = 1 μF = 1 × 10-6 F
  • R = 8 Ω

Step 2: Calculate Center Frequency (fc).

  • fc = 1 / (2π√(L × C))
  • fc = 1 / (2 × 3.14159 × √(0.01 × 1 × 10-6))
  • fc = 1 / (6.28318 × √(1 × 10-8))
  • fc = 1 / (6.28318 × 0.0001)
  • fc = 1 / 0.000628318
  • fc = 1591.55 Hz (A standard midrange crossover point)

Step 3: Calculate Bandwidth (BW).

  • BW = R / (2π × L)
  • BW = 8 / (2 × 3.14159 × 0.01)
  • BW = 8 / 0.0628318
  • BW = 127.32 Hz

Problem 2: Designing a 455 kHz AM Radio IF Filter

Scenario: You are building a custom superheterodyne receiver and need a 455 kHz Intermediate Frequency (IF) band pass filter with a 10 kHz bandwidth to pass audio sidebands while rejecting adjacent channels. Your circuit presents a 50 Ω system impedance. Calculate the required L and C values.

Step 1: Define knowns in base SI units.

  • Target fc = 455 kHz = 455,000 Hz
  • Target BW = 10 kHz = 10,000 Hz
  • R = 50 Ω

Step 2: Calculate Inductance (L) using the BW equation.

  • L = R / (2π × BW)
  • L = 50 / (2 × 3.14159 × 10,000)
  • L = 50 / 62,831.8
  • L = 0.0007957 H
  • L ≈ 796 μH (Select a standard 820 μH inductor and trim with a variable capacitor)

Step 3: Calculate Capacitance (C) using the rearranged fc equation.

  • C = 1 / (4π² × fc² × L)
  • C = 1 / (39.4784 × (455,000)² × 0.0007957)
  • C = 1 / (39.4784 × 2.07025 × 1011 × 0.0007957)
  • C = 1 / (39.4784 × 164,730,000)
  • C = 1 / 6,503,276,000
  • C = 1.537 × 10-10 F
  • C ≈ 154 pF (Use a 100 pF fixed ceramic + 60 pF trimmer capacitor for exact tuning)

Frequently Asked Questions

How does the band pass filter equation change for active RC op-amp circuits?

In active filters (like the Multiple Feedback or Sallen-Key topologies), inductors are replaced by op-amp integrator networks using resistors and capacitors. The center frequency equation shifts to fc = 1 / (2π√(R1R2C1C2)). While the fundamental physics of resonance remain the same, the Q-factor and bandwidth are controlled by the ratio of the resistors rather than a physical inductor's wire resistance. Active filters are preferred for audio (under 100 kHz) because physical inductors at those frequencies are bulky, expensive, and prone to picking up electromagnetic interference (EMI).

Why is my calculated center frequency different from the measured oscilloscope reading?

If your scope shows a shifted fc, you are likely encountering component tolerance and parasitics. Standard ceramic capacitors (especially X7R and Y5V dielectrics) can lose up to 50% of their capacitance when DC bias is applied or when operating near their voltage rating. Furthermore, inductors have a Self-Resonant Frequency (SRF); if your target fc is close to the inductor's SRF, the parasitic parallel capacitance of the windings will pull the resonant peak lower than the ideal equation predicts. Always measure your physical L and C values with an LCR meter at the target operating frequency before soldering.

Can I use the band pass filter equation for digital FIR/IIR DSP filters?

No. The equations provided here apply strictly to continuous-time analog circuits. Digital filters (implemented on DSPs, FPGAs, or microcontrollers like the Teensy 4.1 or STM32) operate in discrete time. To design a digital band pass filter, you must first define your analog prototype using the equations above, and then apply a mathematical transformation—typically the Bilinear Transform with pre-warping—to convert the analog s-domain transfer function into a discrete z-domain difference equation. The sampling rate (fs) becomes a hard constraint, and your center frequency cannot exceed the Nyquist limit (fs/2).