RLC Resonance Filter Guide | f0, Q & Bandwidth

Resonance is a fundamental phenomenon in AC circuits where energy oscillates between electric and magnetic fields. For a quick series RLC check, 100 mH with 10 uF resonates at f0 = 159.2 Hz; with R = 10 ohms, Q = 10, bandwidth is about 15.9 Hz, and a 120 V source can create about 1200 V across L or C. Use the Impedance Calculator for RLC operating points and the RC Circuit Calculator when the real task is timing or cutoff frequency.

Introduction to Resonance

Definition of Resonance

Electrical Resonance: A condition in AC circuits where the inductive reactance equals the capacitive reactance, resulting in a purely resistive impedance.

Mathematical Condition: X_L = X_C ωL = 1/(ωC)

Resonant Frequency: f₀ = 1/(2π√LC) ω₀ = 1/√LC

Physical Interpretation

Energy Exchange:

• Energy oscillates between electric field (capacitor) and magnetic field (inductor)
• At resonance, reactive components cancel each other
• Maximum energy transfer occurs
• Circuit exhibits purely resistive behavior

Historical Context:

• Discovered by Heinrich Hertz in 1886
• Foundation for radio and wireless communication
• Enabled development of tuned circuits
• Critical for modern electronics and communications

Series Resonance

Series RLC Circuit Analysis

Circuit Configuration: R, L, and C connected in series with AC source

Total Impedance: Z = R + j(ωL - 1/ωC) = R + j(X_L - X_C)

At Resonance (ω = ω₀): Z = R + j(ω₀L - 1/ω₀C) = R + j0 = R

Characteristics of Series Resonance

Impedance Minimum:

• Z_min = R (purely resistive)
• Minimum impedance occurs at resonant frequency
• Impedance increases on either side of resonance

Current Maximum:

• I_max = V/R (at resonance)
• Current is maximum and in phase with voltage
• Power factor = 1 (unity power factor)

Voltage Magnification:

• Voltage across L: V_L = jω₀LI = jQ·V
• Voltage across C: V_C = -j(1/ω₀C)I = -jQ·V
• |V_L| = |V_C| = Q·|V| (can exceed source voltage!)

Example 1: Series Resonance Circuit: R = 10Ω, L = 100mH, C = 10μF, V = 120V

Resonant Frequency: f₀ = 1/(2π√(0.1 × 10×10⁻⁶)) = 159.2 Hz

Quality Factor: Q = ω₀L/R = (2π × 159.2 × 0.1)/10 = 10

At Resonance:

• Current: I = 120V/10Ω = 12A
• Voltage across L: V_L = Q × V = 10 × 120V = 1200V
• Voltage across C: V_C = 1200V (opposite polarity)

Quality Factor (Q) in Series Circuits

Definition: Q = ω₀L/R = (1/R)√(L/C) = X_L/R = X_C/R

Physical Meaning:

• Ratio of reactive power to real power
• Measure of energy storage vs. energy dissipation
• Indicates sharpness of resonance
• Higher Q = sharper resonance peak

Bandwidth Relationship: BW = f₀/Q = R/(2πL)

Half-Power Frequencies: f₁ = f₀ - BW/2 (lower half-power frequency) f₂ = f₀ + BW/2 (upper half-power frequency)

Parallel Resonance

Parallel RLC Circuit Analysis

Circuit Configuration: R, L, and C connected in parallel with AC source

Total Admittance: Y = 1/R + j(ωC - 1/ωL) = G + j(B_C - B_L)

At Resonance: Y = G + j0 = 1/R (minimum admittance)

Characteristics of Parallel Resonance

Impedance Maximum:

• Z_max = R (at resonance)
• Maximum impedance occurs at resonant frequency
• Often called "anti-resonance"

Current Minimum:

• I_min = V/R (total current from source)
• Branch currents can be much larger
• Circulating current between L and C

Current Magnification:

• Current through L: I_L = V/(jω₀L) = -jV/(ω₀L)
• Current through C: I_C = jω₀CV
• |I_L| = |I_C| = Q·|I_total|

Example 2: Parallel Resonance Same components as Example 1 in parallel:

At Resonance:

• Total current: I = 120V/10Ω = 12A
• Current through L: I_L = 120V/(2π × 159.2 × 0.1) = 1.2A
• Current through C: I_C = 2π × 159.2 × 10×10⁻⁶ × 120V = 1.2A
• Circulating current = 1.2A (10 times smaller than series case)

Quality Factor in Parallel Circuits

Definition: Q = R/ω₀L = ωCR = R√(C/L)

Note: For parallel circuits, higher resistance gives higher Q (opposite of series)

Frequency Response Analysis

Magnitude Response

Series RLC Magnitude: |H(jω)| = |Z(jω)|/R = √[1 + Q²((ω/ω₀) - (ω₀/ω))²]

Parallel RLC Magnitude: |H(jω)| = R|Y(jω)| = 1/√[1 + Q²((ω/ω₀) - (ω₀/ω))²]

Phase Response

Series RLC Phase: θ(ω) = arctan[Q((ω/ω₀) - (ω₀/ω))]

Characteristics:

• Phase = 0° at resonance
• Phase = +90° at high frequencies (inductive)
• Phase = -90° at low frequencies (capacitive)

Bandwidth and Selectivity

3-dB Bandwidth: BW = f₂ - f₁ = f₀/Q

Selectivity:

• Higher Q = narrower bandwidth = more selective
• Lower Q = wider bandwidth = less selective
• Trade-off between selectivity and bandwidth

Half-Power Points: At f₁ and f₂, power is half of maximum (3-dB down)

Filter Design Principles

Filter Classifications

By Frequency Response:

• Low-pass filters (LPF)
• High-pass filters (HPF)
• Band-pass filters (BPF)
• Band-stop filters (BSF)

By Implementation:

• Passive filters (R, L, C only)
• Active filters (include amplifiers)
• Digital filters (software implementation)

Low-Pass Filters

First-Order RC Low-Pass: H(jω) = 1/(1 + jωRC)

Cutoff Frequency: f_c = 1/(2πRC)

Characteristics:

• Passes frequencies below f_c
• Attenuates frequencies above f_c
• -20 dB/decade rolloff
• -3 dB at cutoff frequency

Example 3: RC Low-Pass Design Design for f_c = 1 kHz with R = 1.6 kΩ

C = 1/(2πRf_c) = 1/(2π × 1600 × 1000) = 99.5 nF

Use standard value: C = 100 nF

Second-Order LC Low-Pass: H(jω) = 1/(1 - ω²LC + jωRC)

Advantages:

• Steeper rolloff (-40 dB/decade)
• Better stopband attenuation
• Can achieve specific response shapes

High-Pass Filters

First-Order RC High-Pass: H(jω) = jωRC/(1 + jωRC)

Cutoff Frequency: f_c = 1/(2πRC)

Characteristics:

• Passes frequencies above f_c
• Attenuates frequencies below f_c
• +20 dB/decade rolloff below f_c
• -3 dB at cutoff frequency

Example 4: RC High-Pass Design Design for f_c = 100 Hz with C = 1 μF

R = 1/(2πCf_c) = 1/(2π × 1×10⁻⁶ × 100) = 1.59 kΩ

Use standard value: R = 1.6 kΩ

Band-Pass Filters

Series RLC Band-Pass: H(jω) = jωRC/(R + jωL + 1/(jωC))

Center Frequency: f₀ = 1/(2π√LC)

Bandwidth: BW = R/(2πL) = f₀/Q

Example 5: RLC Band-Pass Design Design for f₀ = 10 kHz, BW = 1 kHz

Q = f₀/BW = 10 kHz/1 kHz = 10

Choose L = 1 mH: C = 1/(4π²f₀²L) = 1/(4π² × (10×10³)² × 1×10⁻³) = 253 nF R = 2πf₀L/Q = 2π × 10×10³ × 1×10⁻³/10 = 6.28 Ω

Band-Stop (Notch) Filters

Parallel LC in Series with Load: Provides high impedance at resonant frequency

Series LC in Parallel with Load: Provides low impedance path at resonant frequency

Applications:

• Power line interference rejection (50/60 Hz)
• Radio frequency interference (RFI) suppression
• Audio hum elimination
• Spurious signal rejection

Advanced Filter Concepts

Filter Approximations

Butterworth Response:

• Maximally flat passband
• Monotonic response
• -20n dB/decade rolloff (n = order)

Chebyshev Response:

• Ripple in passband
• Steeper rolloff than Butterworth
• Better stopband performance

Elliptic (Cauer) Response:

• Ripple in both passband and stopband
• Steepest possible rolloff
• Most complex implementation

Active Filters

Advantages:

• No inductors required
• Gain possible
• Easier to tune
• Better performance at low frequencies

Sallen-Key Topology:

• Popular active filter configuration
• Uses op-amp as buffer
• Easy to design and implement

Multiple Feedback Topology:

• Higher Q achievable
• Good stability
• Inverting configuration

Summary

Resonance and filters are fundamental concepts in electrical engineering:

1. Series Resonance: Minimum impedance, maximum current, voltage magnification
2. Parallel Resonance: Maximum impedance, minimum current, current magnification
3. Quality Factor: Determines bandwidth and selectivity
4. Filter Types: Low-pass, high-pass, band-pass, and band-stop configurations
5. Design Methods: Passive, active, and digital implementations
6. Applications: Communications, power electronics, audio, and measurement systems

Understanding these concepts enables effective design of frequency-selective circuits for diverse applications.

Next Steps

Continue your filter and resonance education with these related topics:

• Advanced Filter Design: Butterworth, Chebyshev, and elliptic responses
• Active Filter Circuits: Op-amp based filter implementations
• Digital Signal Processing: Software-based filtering techniques
• RF Circuit Design: High-frequency filter applications

Mastering resonance and filter concepts is essential for modern electronic system design and signal processing applications.