lecture_EE1L1_lec1_introduction_2025-11-10

📌 Overview

IP-1 Project: “The Booming Bass”
Analyzing, designing, simulating, building, and testing an audio system:

• Symmetric power supply
• Audio power amplifier
• Loudspeaker measurements
• Passive 3-way loudspeaker filter
• Linkwitz Transform for bass enhancement

Project Flow (Mermaid diagram):

flowchart TD
    A[Symmetric Power Supply] --> B[Audio Power Amplifier]
    B --> C[Loudspeaker Measurements]
    C --> D[Passive 3-Way Filter Design]
    D --> E[Linkwitz Transform: Booming Bass]
    E --> F[Build & Test Full System]

Core Focus: Convert electrical signals to acoustic waves via loudspeakers, achieving flat frequency response (20 Hz–20 kHz) for balanced sound.

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🎯Learning Objectives

• Understand sound as pressure waves: propagation, frequency, wavelength.
• Grasp loudspeaker operation: electro-mechanical transduction, Lorentz force, mass-spring mechanics.
• Analyze acoustic theory: sound pressure, volume flow, displacement vs. frequency.
• Explain multi-driver systems and impedance models.
• Describe measurement setups for impedance and parameters (f_c, Q factors).
• Apply concepts to project: design for flat SPL response.

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💡Key Concepts & Definitions

Sound Fundamentals

• Sound: Pressure variation propagating at $v\approx 345$ m/s (depends on temperature, humidity, pressure).
• Pitch: Frequency $f$ in Hz (cycles/second).
• Wavelength: $\lambda =\frac{v}{f}$ (distance per cycle).
  • Example: At $f=1000$ Hz, $\lambda =\frac{345}{1000}=0.345$ m.

Wave Propagation Analogy (Water waves):
Excitation creates ripples; sound similarly propagates pressure waves with amplitude $A$, losing energy over distance.

Sine Wave Characteristics:

$$p(t)=\hat{p}\sin (2\pi ft)$$

Volume flow $U=Av$, piston speed $v=2\pi f\hat{x}$, acceleration $a=-(2\pi f{)}^2\hat{x}$.

Loudspeaker Basics

• Driver: Electro-mechanical transducer (electrical → mechanical/acoustic energy).
• Electrical Side: Voice coil in magnetic field $B$: Lorentz force $\overrightarrow{F_l}=I\vec{l}\times \vec{B}$. Linear motor!
• Mechanical Side: Cone movement creates pressure; forward → +front/-back pressure (needs enclosure to avoid acoustic short-circuit).
  • Mass-spring system (spider/surround): Resonance frequency $f_c$.

Acoustic Theory
Sound pressure at distance $d$:

$$p_{rms}=\frac{\rho Uf}{2\pi d}=\rho a_{rms}(\rho \approx 1.19{\text{kg/m}}^3)$$

• $U_{rms}=A\cdot 2\pi f{\hat{x}}_{rms}$, so $p_{rms}\approx \rho \cdot 2\pi f^2{\hat{x}}_{rms}$.
• Key Insight: For constant $p_{rms}$, ${\hat{x}}_{rms}\propto \frac{1}{f^2}$ (lower $f$ needs 4x displacement at half frequency).
• Sound Intensity: $I\approx \frac{p_{rms}^2}{\rho c}$ (c = speed of sound).
• SPL: $SPL=20{\log }_{10}\left(\frac{p_{rms}}{p_{ref}}\right)$ dB, $p_{ref}=2\times 10^{-5}$ N/m².

Behavior: Low frequencies require larger drivers (more air displacement); high $f$ → directive, resonances (cone breakup).

• Multi-Driver Systems: 2-way (bass+treble), 3-way (bass+mid+treble); $f_{min}\ll 10/100\times f_{max}$.
  • Aim: Flat SPL response (e.g., 2.83V/1m → 1W/8Ω).

Mermaid: Multi-Driver System

graph LR
    Amp[Amplifier] --> Filter[Passive Crossover]
    Filter --> Woofer[Bass Driver<br/>20-300 Hz]
    Filter --> Midrange[Mid Driver<br/>300-5k Hz]
    Filter --> Tweeter[Treble Driver<br/>5k-20k Hz]
    Woofer --> Sound[Booming Bass<br/>Flat SPL]
    Midrange --> Sound
    Tweeter --> Sound

Impedance Model

• Full: $Z=R_e+j\omega L_e$ in series with parallel ($R_a$, $R_v$, back-EMF branch $R_p||L_p||C_p$).
  • $R_e$: DC resistance; $L_e$: Inductance; $R_a$: Radiation; $R_v$: Mechanical loss; Branch: Suspension effects.
• Simplified: Most power to $R_e$ (efficiency ~1%, $R_a,R_v\ll R_e$).

Frequency Response Examples:

• Woofer: Peak at resonance, roll-off high $f$.
• Tweeter: High-pass, impedance peaks at resonance.

Measurements

• Setup: PC soundcard (Line Out/In), white noise (equal power all $f<24$ kHz), reference resistor $R_{ref}$.
• Impedance: $Z_{unknown}(f)=R_{ref}\cdot \frac{V_{ref}(f)}{V_{in}(f)-V_{ref}(f)}$.
• Parameters: From $|Z(f)|,\varphi (f)$: $f_c$ (resonance), $Q_m$ (mech), $Q_e$ (elec), $Q_{tc}$ (total), $R_e$.
  • Software: Input $R_{dc}$ (Ohmmeter), speaker type, $R_{ref}=100\Omega$, $f_{min/max}$, generate figures/Excel data.
• Mounted vs. Free: Box alters parameters slightly.

When Allowed/Rare Cases:

• Assumptions: Linear operation (small signals); far-field ($d\gg \lambda$); ideal piston (no breakup).
• Rare: Nonlinear distortion (large $x$), near-field interference, temperature effects on $v$. Multi-driver: Phase alignment critical to avoid cancellation.

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✍️ Notes

Project Teasers: Build from power supply to full 3-way system with bass boost. Measure real drivers; simulate filters. End goal: “Booming Bass” via Linkwitz Transform (boosts low $f$ response).

Gaps in Lecture: No explicit exercises; focus on theory for measurements/project design. Respect: Enclosure prevents short-circuit; efficiency low → heat in $R_e$.

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🔗 Resources

• Presentation:
• Dr. G.J.M. Janssen: g.j.m.janssen@tudelft.nl, HB 17.060.

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❓ Post lecture

Example Question 1: Wavelength Calculation
Q: Compute $\lambda$ for a 440 Hz tone (A4 note) at 20°C ($v=343$ m/s). When is $\lambda =v/f$ valid?

Stepwise Solution:

1. Recall: $\lambda =\frac{v}{f}$. Valid for plane waves in uniform medium (far-field, non-dispersive). Not for near-field or nonlinear propagation.
2. Plug in: $\lambda =\frac{343}{440}\approx 0.78$ m.
3. Rare: At high amplitude, harmonics distort sine; humidity/temperature shifts $v$ by ~0.6 m/s/°C.

Example Question 2: Displacement for Constant SPL
Q: For a woofer ($A=0.02$ m²), maintain $p_{rms}=0.1$ Pa at $d=1$ m, $f=100$ Hz vs. 50 Hz. Find ${\hat{x}}_{rms}$ ratio. Assumptions?

Stepwise Solution:

1. From $p_{rms}\approx \rho \cdot (2\pi f{)}^2{\hat{x}}_{rms}$ (piston approx., $\rho =1.19$). Valid: Low $f$ ($\lambda \gg$ driver size), linear. Rare: Cone breakup at high $f$.
2. ${\hat{x}}_{rms}\approx \frac{p_{rms}}{\rho (2\pi f{)}^2}$.
3. At 100 Hz: ${\hat{x}}_{rms,100}\approx \frac{0.1}{1.19\cdot (2\pi \cdot 100{)}^2}\approx 2.7\times 10^{-6}$ m.
4. At 50 Hz: ${\hat{x}}_{rms,50}\approx \frac{0.1}{1.19\cdot (2\pi \cdot 50{)}^2}\approx 1.1\times 10^{-5}$ m (4x larger).
5. Ratio: $4:1$ (as $1/f^2$). For project: Explains need for larger bass drivers/enclosures.

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📖 Homework

• Review impedance measurement: Simulate $Z(f)$ for simple $RLC$ model using Python/MATLAB.
• Calculate SPL for given $x_{rms},f$; plot vs. $f$ (show $1/f^2$ roll-off).
• Sketch 3-way crossover basics (high/low-pass filters).