ELEC-E5650 Electroacoustics

ELEC-E5650ELECElectroacoustics-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Lecture 2: SteadyRaimundo -GonzalezState Analysis / Dynamic Analogies Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering Raimundo2 2GonzalezFebruary 2018

Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

March 7, 2019

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez 1 Aalto, Signal Processing & Acoustics ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 LectureRaimundo Gonzalez 2: Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018 I. Steady State Analysis

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez 2 Aalto, Signal Processing & Acoustics Steady state sinusoidal response

When designing Electroacoustics systems we are usually more interestedELEC in the-E5650 steady state behavior of the system. This will LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 lead into Raimundomostly Gonzalez working in the frequency domain. Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 3 Phasors to represent sinusoidal signals

ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez In audio, we are interested in signals which are Department of Signal Processing and Acoustics Aalto University School sinusoidal, meaning the its frequency components of Electrical Engineering always have a magnitude and phase. 22 February 2018

x(t) = !"#$%"#& = !"#$%'& where is a phasor ! = ! "#& which solely contains the amplitude and phase.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 4 Complex Number

From physics and signal processing we know that it is easier to work with complex numbers for solutions to systemsELEC with- E5650sinusoidal behavior. LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 5 A time domain attempt would require solving…. ELECRC circuit-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering 22 February 2018 with our solution being…

But we are not interested in the transients response, the homogenous solution, therefore we can try an easier approach.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 6 Our solution

RC circuit – steady state solution Solving ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018 Steady state solution

Our governing equation magnitude phase

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 7 Phasor Domain – Voltage-Current Relation

The imaginary basis j equals 90 degree shift counterclockwise in the complex plane which will affect the phase of the output signal. ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 ResistorsRaimundo Gonzalez Inductor Capacitor Department of Signal Processing and Acoustics Aalto University School ! V = Iof RElectrical Engineering V = I jωL V = I 22 February 2018 jωC

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 8 RLC Circuit 2nd order form !"#(%) ( !#(%) #(%) 1 + + = - /01234 !% ) !% )* ) . ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Electrical Impedance Raimundo Gonzalez -. 1 Department of Signal Processing and Acoustics Aalto University School = ( + + /0) = 78 of Electrical Engineering # /0* 22 February 2018

Natural Frequency Damping coefficient 1 ( * Governing equation 00 = )* : = 1 !#(%) 2 ) - % = (# t + 6 # % !% + ) . * !% And if… Forced Frequency where " 234 : < 1 0! = 00 1 − : -. % = -. 1

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 9 ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 LectureRaimundo Gonzalez 2: Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018 II. Dynamic Analogies

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez 10 Aalto, Signal Processing & Acoustics Transducers ELEC-E5650 Rectilinear Lecture 1: Overview, Electroacoustics introduction & Circuit Elements pt1 ElectroacousticsElectronics Mechanics Acoustics Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering Differential Differential 22 February 201Algebra8 Analytic tools: Equations Equations

To work with all three we need a common denominator. Hopefully one that makes things easier!

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 11 Method: Dynamic Analogies

• Use the simplicityELECof-E5650electronic circuit analysis to model problems in LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 mechanicalRaimundoand Gonzalezacoustics domains as Department of Signal Processing and Acoustics Aalto University School well as moreof ElectricalcomplexEngineering domain-coupled

systems. 22 February 2018

• We can do this because the equations of motion for each domain are equivalent.

From “Dynamical Analogies”, Harry F. Olson (1943)

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 12 Benefits ELEC-E5650 • Model domain-coupled systems under single paradigm LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez • Math: AlgebraicDepartment of Signal Processingtools and Acousticsavailable Aalto University Schoolfor electronics can be applied to systems of other domainsof Electrical Engineering. 22 February 2018 • Analysis: the physical behavior of each element in the system is conserved. • Boundaries of electric elements are clearly defined. • We can apply well-known representation methods (Circuit Diagrams). • Easier visualization.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 13 Physical and Mathematical Meaning of Electronic Quantities and Circuit Elements

ElementsELEC-E5650 • LectureElectroacousticsResistance 1: Overview, Electroacoustics-type introduction & Circuit Elements pt1 • RaimundoCapacitance Gonzalez-type Department of Signal Processing and Acoustics Aalto University School • ofInductance Electrical Engineering-type

• 2Transformation2 February 2018 -type • Gyration-type

Quantities • Potential across the circuit element ("!) • Flow through the circuit element ($#) • Magnitude of the circuit element (&̃)

Figure adapted from [2]

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 14 Resistance

Property of an element which dissipates energy.

ELEC-E5650 Electrical Mechanical Acoustical Resistance (RE ) Resistance (RM ) Resistance (RA ) LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Frictional Viscosity DepartmentElement of Signal Processing… and AcousticsResistor Aalto University School of Electrical Engineering Surface/Element property of air 22 February 2018 $ ' * Defined by… ! = ! = ! = " % & ( ) +

Representation…

Figures from [1] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 15 Inductance, Mass, Inertance

Property of an element which opposes change in flow. ELEC-E5650 Inductance (L ) Mass (MM ) Inertance (MA ) LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Uncompressed DepartmentElement of Signal Processing… and AcousticsInductor Aalto University School Mass of Electrical Engineering volume of air

22 February 2018 $% $* $- Defined by… ! = # ' = ( + = ( $& ) $& , $&

Representation… *

Figures from [1] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 16 Electrical Capacitance, Compliance, Acoustic Capacitance

Property of an element which opposes change in potential.

ELEC-E5650 Electrical Mechanical Acoustic Compliance (C ) Capacitance (CA ) LectureElectroacoustics 1: Overview, Electroacoustics Capacitanceintroduction & Circuit Elements (CE pt1 ) M Raimundo Gonzalez Compressive DepartmentElement of Signal Processing… and AcousticsCapacitor Aalto University School Spring of Electrical Engineering property of air 22 February 2018 1 1 1 Defined by… ! = & ' () * = & , () - = & / () $% $+ $.

Representation…

Figures from [1] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 17 Mechanical and Acoustic Sources (ideal)

MechanicELEC- E5650Sources Acoustic Sources LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School Velocity Sourceof Electrical Engineering Volume Velocity Source

Wheel forcing22 February constant2018 Velocity of piston converts sinusoidal velocity to volume velocity

Force Source Pressure Source Electrodynamic force Surface pushing air generator driven by piston

Figures from [2] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 18 Impedance Vs Admittance

ELECImpedance-E5650 (Z) Admittance (Y) The complexLectureElectroacoustics 1: Overview, ratioElectroacoustics of introductionpotential & Circuit Elementsover pt1 The inverse of impedance which flow whichRaimundo Gonzalezdescribes how a system describes how well a system allows Department of Signal Processing and Acoustics Aalto University School resists of( impedesElectrical Engineering ) flow through it once (admits) flow through it once there is a

there 2is2 February a potential2018 energy across it. potential energy across it

%&'()'*+, 1 $ = ! = -,&. $ Admittance can be more adequate to describe the behavior of mechanical systems where a force generates velocity.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 19 Duality Principal Every element and concept has an admittance type pair impedance ↔ admittance ELEC-E5650 resistance ↔ conductance LectureElectroacoustics 1: Overview, Electroacoustics introduction &capacitance Circuit Elements pt1 ↔ inductance Raimundo Gonzalez voltage source ↔ current source Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering voltage ↔ current 22 February 2018 series connection ↔ parallel connection

Figures from [1] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 20 Example

ELECInductor-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez • What is the property of its Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering impedance? 22 February 2018 • What element is its mechanical analogy? • What element is its admittance mechanical analogy?

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 21 WhyELEC would-E5650 it be useful to model a system in the mechanicalLectureElectroacoustics 1: Overview, Electroacoustics or introduction acoustic & Circuit Elements pt1 domain using electrical Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering admittances?

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 22 Dynamic Analogies ELEC-E5650 ImpedanceLectureElectroacoustics 1: Overview, Electroacoustics Analogy introduction & Circuit Elements pt1 Admittance Analogy Raimundo Gonzalez • Pro: It expressesDepartment of Signal mechanical Processing and Acoustics impedances Aalto University School • Pro: Maintains the topology of the as electricalof Electrical impedances.Engineering mechanical system. 22 February 2018 • Con: Does not represent the topology of • Con: It expresses mechanical the mechanical system impedances as electrical admittances. • Also known as the Mobility Analogy.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 23 To keep in mind when modelling using analogies…

• We mustELECdefine -theE5650 number of degrees of freedom. • AnalogiesLectureElectroacoustics must 1: Overview, Electroacousticsmaintain introduction the & Circuit principle Elements pt1 of conservation of energy. Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School • The total ofenergy Electrical Engineering of a driven system is:

22 February 2018 Kinetic Energy + Potential Energy + Losses = Driving Source

Impedance Analogy • Kinetic Energy -> Inductance, Mass & Inertance • Potential Energy -> Capacitance, Mechanical Compliance & Ac. Capacitance • Thermal Losses -> Resistance

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 24 In class exercise

ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 25 Same system can be modelled in other ways…

22 February 2018 =

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 26 Impedance Analogy

We can use solutions from electrical circuits for systems of other domains with similar behavior ELEC-E5650 LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

Electrical Impedance (ZE ) Mechanical Impedance (ZM ) Acoustical Impedance (ZA ) ! 1 H 1 > 1 " = % + + ()+ = ,- = %A + + ()AA = ,A = %@ + + ()A@ = ,@ # ()* I ()*A ? ()*@

v" = voltage F = force p = air pressure i = current u = velocity U = volume velocity

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 27 Admittance Analogy for the previous system

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 28 The Admittance Analogy allows to model more complex systems!

22 February 2018

Figures from [2] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 29 Mechanical Analogies Acoustical Analogies

22 February 2018

Figures from [1] ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 30 Transformer

• Device formed by two or more inductors sharing the ELECsame lossless-E5650 core. • A primary coilLectureElectroacoustics (input) 1: Overview, Electroacousticsinduces introduction a higher & Circuit orElements lower pt1 voltage on theRaimundo secondary Gonzalez coil (output) while Department of Signal Processing and Acoustics Aalto University School while powerof at Electrical bothEngineering sides is maintained. • The ratio of 2transformation2 February 2018 depends on the number of turns in the primary and secondary coils. • Heavily used in audio to match the impedance of amplifiers with speakers. • When modelling we use it as an impedance converter.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 31 Mechanical & Acoustical Transformers

ELECLever-E5650 Horn LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 32 MechanicalELEC-E5650 Analogy Acoustic Analogy LectureElectroacoustics 1: Overview, Electroacoustics introduction & Circuit Elements pt1 Raimundo Gonzalez Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 33 Gyrator

Hypothetical linear element made of active components ELEC(op-amps-E5650 / transistors). They are used in modelling for they convert between Lecture 1: Overview, Electroacoustics introduction & Circuit Elements pt1 impedance andElectroacoustics admittance through its Raimundo Gonzalez parameter G.Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering

22 February 2018

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 34 Bibliography ELEC-E5650 [1] - KleinerLectureElectroacoustics 1:, Overview,Mendel. Electroacoustics introduction Electroacoustics. & Circuit Elements pt1 CRC Press, 2013. Raimundo Gonzalez [2] – Beranek,Department of Signal Leo Processing andL. Acoustics Acoustics Aalto University School Sound Fields and Transducer. of Electrical Engineering Academic22 FebruaryPress,2018 2012.

ELEC-E5650 Electroacoustics, Lecture 1 Raimundo Gonzalez Aalto, Signal Processing & Acoustics 35