Filters and Resonance

Summary

This chapter provides practical coverage of filter circuits used in audio and signal processing applications. Students will learn to design first and second-order filters using RC, RL, and RLC configurations, understand filter specifications like order and roll-off rate, and apply these concepts to audio tone control circuits. The chapter covers both passive filters using only RLC components and introduces active filters using operational amplifiers. By the end, students will be able to design filters to meet specific frequency response requirements.

Concepts Covered

This chapter covers the following 22 concepts from the learning graph:

First-Order Filter
Second-Order Filter
RC Low-Pass Filter
RC High-Pass Filter
RL Low-Pass Filter
RL High-Pass Filter
RLC Band-Pass Filter
Passive Filter
Active Filter
Filter Design
Audio Tone Control
Bass Filter
Treble Filter
Amplifier Gain
Decibels in Audio
dBV
dBu
Headroom
Dynamic Range
Audio Signal
Microphone
Speaker

Prerequisites

This chapter builds on concepts from:

title: Filters and Resonance description: Design practical filter circuits for audio and signal processing applications generated_by: chapter-content-generator v0.03 date: 2026-01-30

Introduction: Sculpting Sound with Circuits

Every time you adjust the bass and treble on your stereo, you're using filters. Every time you talk on a phone that somehow makes your voice clear despite background noise, filters are working behind the scenes. Every time a radio picks out one station from thousands of signals crowding the airwaves—you guessed it, filters.

Filters are the sculptors of the frequency spectrum. They let you decide which frequencies to keep and which to discard, which to emphasize and which to suppress. In the previous chapter, you learned the theory of frequency response. Now it's time to put that theory to work designing real filter circuits.

This chapter bridges the gap between mathematical transfer functions and practical component selection. You'll learn to design first-order RC and RL filters, second-order RLC filters, and get your first taste of active filters that can amplify while filtering. Along the way, we'll explore audio applications where filters transform raw signals into the sounds we want to hear.

First-Order Filters: The Building Blocks

First-order filters contain one reactive element (capacitor or inductor) and produce a -20 dB/decade roll-off. They're simple, predictable, and form the foundation for more complex filter designs.

Transfer function form:

[H(j\omega) = \frac{K}{1 + j\omega/\omega_c} \text{ (low-pass)}] [H(j\omega) = \frac{K \cdot j\omega/\omega_c}{1 + j\omega/\omega_c} \text{ (high-pass)}]

Characteristics:

Property	First-Order Value
Roll-off rate	20 dB/decade
Phase shift range	0° to 90°
Order	1
Reactive elements	1 (C or L)

RC Low-Pass Filter

The RC low-pass filter is the most common first-order filter. Output is taken across the capacitor.

Circuit: Series resistor, capacitor to ground

Transfer function: [H(j\omega) = \frac{1}{1 + j\omega RC}]

Cutoff frequency: [f_c = \frac{1}{2\pi RC}]

Design equations: Given \(f_c\), choose R and C such that \(RC = \frac{1}{2\pi f_c}\)

Example: For \(f_c = 1\) kHz: [RC = \frac{1}{2\pi \times 1000} = 159.2 \text{ μs}]

Choose C = 100 nF: \(R = \frac{159.2 \text{ μs}}{100 \text{ nF}} = 1.592 \text{ kΩ}\) Use standard value R = 1.5 kΩ → actual \(f_c = 1.06\) kHz

RC High-Pass Filter

The RC high-pass filter has the same components as the low-pass, but output is taken across the resistor.

Circuit: Series capacitor, resistor to ground (output across R)

Transfer function: [H(j\omega) = \frac{j\omega RC}{1 + j\omega RC}]

Cutoff frequency: [f_c = \frac{1}{2\pi RC}]

At low frequencies, the capacitor blocks; at high frequencies, it passes.

RL Low-Pass Filter

The RL low-pass filter uses an inductor instead of a capacitor.

Circuit: Resistor to ground, series inductor (output across R)

Cutoff frequency: [f_c = \frac{R}{2\pi L}]

Why RL is less common: - Inductors are larger, heavier, and more expensive than capacitors - Inductors have parasitic resistance - Inductors can pick up magnetic interference

RL filters are mainly used in power applications where inductors handle high currents better than capacitors.

RL High-Pass Filter

Circuit: Inductor to ground, series resistor (output across L)

Cutoff frequency: [f_c = \frac{R}{2\pi L}]

Diagram: First-Order Filter Comparison

First-Order Filter Comparison

Type: microsim Bloom Level: Analyze (L4) Bloom Verb: compare Learning Objective: Students will compare RC and RL implementations of low-pass and high-pass filters by observing their frequency responses with the same cutoff frequency. Visual elements: - Four circuits: RC LP, RC HP, RL LP, RL HP - Bode magnitude plot showing all four responses overlaid - Cutoff frequency marker shared by all - Component values displayed for each circuit Interactive controls: - Slider: Target cutoff frequency (100 Hz to 10 kHz) - Toggle: Show RC only / RL only / All - Toggle: Normalize component values - Display: Calculated R, C, L for each circuit - Input: Fix R or C/L value Observations highlighted: - "Same cutoff, same response shape" - "RC more practical at audio frequencies" - "RL used in power applications" Default parameters: - f_c = 1 kHz - R = 1 kΩ for all - C = 159 nF, L = 159 mH (calculated) Canvas layout: - Circuit schematics: 400 × 200 pixels (top) - Bode plot: 600 × 250 pixels (bottom) - Control area: 100 pixels below Implementation: p5.js

Second-Order Filters: Adding Resonance

Second-order filters contain two reactive elements and produce -40 dB/decade roll-off. They can exhibit resonance, producing a peak in the response.

Key parameters:

Cutoff frequency (\(f_c\) or \(\omega_0\)): Where roll-off begins
Quality factor (Q): Sharpness of response
Damping ratio (\(\zeta = 1/2Q\)): Controls peak behavior

Standard form transfer function: [H(s) = \frac{\omega_0^2}{s^2 + (2\omega_0/Q)s + \omega_0^2}]

Q Value	Damping	Response Shape
< 0.5	Overdamped	No peak, gradual rolloff
0.707	Butterworth	Maximally flat passband
1	Slightly underdamped	Small peak (≈3 dB)
> 1	Underdamped	Pronounced peak
10	Highly underdamped	Sharp resonance

RLC Band-Pass Filter

The RLC band-pass filter passes a band of frequencies centered at the resonant frequency.

Series RLC band-pass (output across R): [f_0 = \frac{1}{2\pi\sqrt{LC}}] [Q = \frac{1}{R}\sqrt{\frac{L}{C}} = \frac{f_0}{BW}] [BW = \frac{R}{2\pi L}]

Design process:

Choose center frequency \(f_0\)
Choose desired Q (affects bandwidth)
Calculate LC product: \(LC = \frac{1}{(2\pi f_0)^2}\)
Choose L or C, calculate the other
Calculate R: \(R = \frac{2\pi f_0 L}{Q}\)

Diagram: Second-Order Filter Designer

Second-Order Filter Designer

Type: microsim Bloom Level: Apply (L3) Bloom Verb: design Learning Objective: Students will design second-order RLC filters by specifying center frequency and Q, then observing the resulting component values and frequency response. Visual elements: - Circuit schematic (series RLC) - Bode magnitude and phase plots - Peak height indicator for underdamped responses - Bandwidth markers on plot - Component values display Interactive controls: - Slider: Center frequency f₀ (100 Hz to 100 kHz, log) - Slider: Q factor (0.5 to 20) - Radio: Filter type (LP, HP, BP) - Toggle: Fix L or Fix C for design - Input: Fixed component value - Button: Calculate components Design calculations displayed: - LC = 1/(2πf₀)² - R = 2πf₀L/Q (or R = Q/(2πf₀C)) - BW = f₀/Q - Peak height (dB) if Q > 0.707 Default parameters: - f₀ = 1 kHz - Q = 5 (narrow band-pass) - L = 10 mH (fixed) - Calculated: C = 2.53 μF, R = 12.6 Ω Canvas layout: - Circuit/specs: 250 × 400 pixels - Bode plots: 350 × 400 pixels - Control area: 100 pixels below Implementation: p5.js

Passive vs. Active Filters

Passive Filters

Passive filters use only resistors, capacitors, and inductors—no power supply or amplification.

Advantages:

Simple, no power supply needed
High reliability
Can handle high power
No active device noise

Disadvantages:

Cannot amplify (gain ≤ 1)
Loading effects change response
Large inductors for low frequencies
Limited Q in some topologies

Active Filters

Active filters use amplifying devices (usually op-amps) along with RC networks.

Advantages:

Can provide gain (amplify while filtering)
No inductors needed (cheaper, smaller)
Easy to cascade without loading
High Q achievable
Adjustable gain and cutoff

Disadvantages:

Requires power supply
Bandwidth limited by op-amp
More components
Potential for noise and distortion

Property	Passive	Active
Gain	≤ 1 (loss)	Any value
Inductors	Often needed	Not needed
Power supply	None	Required
Cascading	Loading issues	No loading
High Q	Difficult	Easy
High frequency	Excellent	Limited by op-amp

When to Choose Active Filters

Use active filters for audio frequencies when you need gain, precise response shapes, or want to avoid large inductors. Use passive filters for RF, high-power applications, or when simplicity and reliability are paramount.

Filter Design Process

Designing a filter involves translating specifications into component values:

Step 1: Define specifications

Filter type (LP, HP, BP, BR)
Cutoff frequency/frequencies
Passband ripple (if any)
Stopband attenuation
Order required

Step 2: Choose topology

First-order RC/RL for simple roll-off
Second-order RLC for steeper roll-off or resonance
Active for gain or inductor-free design

Step 3: Calculate component values

Using design equations for the chosen topology.

Step 4: Select standard values

Components come in standard values (E12, E24 series). Choose nearest values.

Step 5: Verify response

Simulate or measure to confirm specifications are met.

Audio Tone Control: Bass and Treble

Audio tone control circuits adjust the frequency balance of sound. The classic Baxandall tone control (named after Peter Baxandall) provides independent bass and treble adjustment.

Bass Filter

A bass filter emphasizes or attenuates low frequencies (typically below 200-500 Hz).

Bass boost: Low-pass shelving filter with gain at low frequencies

Bass cut: Attenuate low frequencies while passing midrange and treble

Typical bass control range: ±12 dB at 100 Hz

Treble Filter

A treble filter emphasizes or attenuates high frequencies (typically above 2-5 kHz).

Treble boost: High-pass shelving filter with gain at high frequencies

Treble cut: Attenuate high frequencies while passing bass and midrange

Typical treble control range: ±12 dB at 10 kHz

Shelving Filters

Unlike the simple filters that continue rolling off forever, shelving filters level off to a constant gain in the stop band. They're like stepping up or down to a shelf, not sliding off a cliff.

Low-frequency shelf: Adjusts gain below a corner frequency, flat response above

High-frequency shelf: Adjusts gain above a corner frequency, flat response below

Audio Signal Levels and Decibels

Audio engineers use specific decibel references:

dBV (decibels relative to 1 volt)

\[dBV = 20\log_{10}\left(\frac{V_{rms}}{1V}\right)\]

Reference: 1 V RMS = 0 dBV

Common levels:

dBV	Voltage	Application
+4 dBV	1.58 V	Professional line level
0 dBV	1.0 V	Consumer line level (some)
-10 dBV	316 mV	Consumer line level (typical)
-60 dBV	1 mV	Microphone level

dBu (decibels relative to 0.775 volts)

\[dBu = 20\log_{10}\left(\frac{V_{rms}}{0.775V}\right)\]

Reference: 0.775 V RMS = 0 dBu (originally referenced to 600Ω termination = 1 mW)

Why 0.775V? It's the voltage that produces 1 mW into 600Ω.

dBu	dBV	Voltage
+4 dBu	+1.8 dBV	1.23 V
0 dBu	-2.2 dBV	0.775 V
-10 dBu	-12.2 dBV	245 mV

Headroom and Dynamic Range

Headroom is the safety margin between normal operating level and maximum level (clipping).

\[\text{Headroom (dB)} = \text{Maximum level} - \text{Nominal level}\]

Typical professional equipment has 20+ dB of headroom.

Dynamic range is the ratio between the loudest and quietest signals a system can handle:

\[\text{Dynamic Range (dB)} = \text{Maximum level} - \text{Noise floor}\]

Medium	Typical Dynamic Range
CD audio	96 dB
Vinyl record	60-70 dB
FM radio	50-60 dB
Telephone	30-40 dB

Audio Transducers: Microphones and Speakers

Microphones

A microphone converts sound (acoustic energy) to electrical signals.

Types:

Dynamic: Moving coil in magnetic field, rugged, no power needed
Condenser: Capacitor with moving plate, sensitive, needs phantom power
Ribbon: Thin metal ribbon in magnetic field, delicate, warm sound

Typical output levels: -60 to -40 dBV (1-10 mV)

Amplifier gain needed: 40-60 dB to reach line level

Speakers

A speaker (loudspeaker) converts electrical signals to sound.

Basic operation:

Current through voice coil creates magnetic field
Interacts with permanent magnet
Cone moves, pushing air
Air pressure waves = sound

Impedance: Typically 4Ω, 8Ω, or 16Ω (mostly resistive at low frequencies, complex at higher frequencies)

Sensitivity: Sound pressure level (SPL) produced for 1W input at 1 meter distance

Crossover networks: Filter circuits that direct different frequencies to appropriate drivers (woofer, tweeter)

Diagram: Audio Signal Chain

Audio Signal Chain

Type: microsim Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain the signal levels and gain stages in a typical audio system from microphone to speaker. Visual elements: - Signal flow diagram: Mic → Preamp → Tone Control → Power Amp → Speaker - Level meters at each stage showing dBV or dBu - Gain stages with amplification/attenuation values - Frequency response modification at tone control - Headroom indication at each stage Interactive controls: - Slider: Input level (mic sensitivity) - Slider: Preamp gain (20-60 dB) - Slider: Bass adjustment (±12 dB) - Slider: Treble adjustment (±12 dB) - Slider: Volume (power amp gain) - Display: Level at each stage Annotations: - "Mic level: ~-50 dBV" - "Line level: ~0 dBV" - "Power amp output: Watts to speaker" - Clipping indicator if any stage overloads Default parameters: - Typical vocal recording chain - Moderate levels with good headroom Canvas layout: - Signal chain diagram: 600 × 350 pixels - Control area: 100 pixels below Implementation: p5.js

Amplifier Gain in Filter Circuits

When using active filters, amplifier gain becomes a design parameter alongside cutoff frequency.

First-order active low-pass (inverting): [H(j\omega) = -\frac{R_f/R_i}{1 + j\omega R_f C}]

DC gain: \(-R_f/R_i\)
Cutoff: \(f_c = \frac{1}{2\pi R_f C}\)

Sallen-Key second-order (unity gain):

A popular topology for second-order active filters using a single op-amp with gain of 1.

\[f_c = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}\]

For equal-value components (\(R_1 = R_2 = R\), \(C_1 = C_2 = C\)): [f_c = \frac{1}{2\pi RC}]

Practical Design Example

Design a tone control circuit for a guitar amplifier:

Specifications:

Bass control: ±15 dB at 100 Hz
Treble control: ±15 dB at 3 kHz
Unity gain at 1 kHz (midpoint)

Approach:

Use shelving filter topology
Calculate component values for corner frequencies
Add potentiometers for variable boost/cut
Buffer input and output with op-amp stages

Component guidelines:

Bass control: RC time constant for ~100 Hz
Treble control: RC time constant for ~3 kHz
Use log-taper potentiometers for perceptually linear adjustment

Self-Check Questions

1. Design an RC low-pass filter with cutoff frequency 5 kHz. If C = 10 nF, what value of R is needed?

Using the cutoff frequency formula: [f_c = \frac{1}{2\pi RC}]

Solving for R: [R = \frac{1}{2\pi f_c C} = \frac{1}{2\pi \times 5000 \times 10 \times 10^{-9}}] [R = \frac{1}{3.14 \times 10^{-4}} = 3183 \text{ Ω}]

Use standard value R = 3.3 kΩ (E24 series)

Actual cutoff with R = 3.3 kΩ: [f_c = \frac{1}{2\pi \times 3300 \times 10 \times 10^{-9}} = 4.82 \text{ kHz}]

Close enough for most applications!

2. A band-pass filter has center frequency 10 kHz and Q = 20. What is the bandwidth?

Bandwidth formula: [BW = \frac{f_0}{Q} = \frac{10000}{20} = 500 \text{ Hz}]

The filter passes frequencies from approximately: - Lower cutoff: \(f_0 - BW/2 = 10000 - 250 = 9750\) Hz - Upper cutoff: \(f_0 + BW/2 = 10000 + 250 = 10250\) Hz

This 500 Hz window centered at 10 kHz is quite narrow, making it useful for selecting a specific frequency.

3. Why are active filters preferred over passive LC filters at audio frequencies?

Several practical reasons favor active filters for audio:

No inductors: At audio frequencies (20 Hz - 20 kHz), inductors would need to be very large (henries!) and expensive. Capacitors and op-amps are much cheaper and smaller.
Gain is possible: Active filters can amplify the signal while filtering, eliminating the need for separate amplifier stages.
No loading issues: Op-amp buffers provide high input impedance and low output impedance, so cascading multiple filter stages doesn't change their individual responses.
High Q easily achieved: Getting high Q with passive RLC requires either very large L or very small R, both problematic. Active filters achieve high Q with just RC components.
Adjustability: Active filter parameters can be made adjustable with potentiometers without complex switching.

4. A microphone outputs -50 dBV. What voltage is this, and how much gain (in dB) is needed to reach +4 dBV professional line level?

Converting -50 dBV to voltage: [V = 10^{-50/20} \times 1\text{V} = 10^{-2.5} \text{V} = 3.16 \text{ mV}]

Gain needed: [\text{Gain (dB)} = +4 - (-50) = 54 \text{ dB}]

This is a voltage gain of: [A_V = 10^{54/20} = 10^{2.7} = 501]

The microphone preamp needs about 500× voltage gain to bring the signal to professional line level.

Summary

This chapter equipped you with practical filter design skills:

First-order filters are building blocks - RC and RL configurations provide -20 dB/decade roll-off
Second-order filters add resonance capability - RLC circuits can have peaked responses with high Q
Passive filters are simple but limited - No gain, loading effects, large inductors at low frequencies
Active filters overcome passive limitations - Gain, no inductors, easy cascading, but need power
Design starts with specifications - Cutoff frequency, Q, gain, filter type
Standard component values constrain designs - Adjust calculations to nearest E12 or E24 values
Audio tone controls use shelving filters - Bass and treble adjust low and high frequencies
Decibel references matter in audio - dBV and dBu have specific voltage references
Headroom prevents clipping - Design for adequate margin above normal levels
The signal chain has multiple gain stages - Microphone to speaker involves several amplification and filtering steps

With these tools, you can design filters for any frequency response specification. Whether you're building an audio equalizer, removing noise from a sensor signal, or selecting a radio channel, the principles are the same: choose the right topology, calculate component values, and verify the response. Filter design is where theory meets practice—and now you're equipped to make it happen.