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Lumped Models and Systems Overview
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Lumped Models and Systems Overview
05 Electroacoustics
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1
Question
What is a lumped model?
Answer
A lumped model is a: - Simplification of the description of the behavior of a spatial physical system. - It discretizes elements to approximate the system - Is useful for electrical systems, mechanical bodily systems, acoustics, electromagnetism, etc. - It transfers systems from partial differential equations into ordinary differential equations with a finite number of elements.
2
Question
What condition applies in lumped models?
Answer
In lumped models, the wavelength (m) must be larger than the dimension of the system (m).
3
Question
What is the formula to calculate frequency range for the human ear?
Answer
f = c / λ, where c is the speed of sound in m/s and λ is the wavelength.
4
Question
What is the speed of sound?
Answer
The speed of sound is 343 m/s.
5
Question
How can the maximum wavelength for the lumped model of the human ear be calculated?
Answer
max λ = c / f_min = 300 m/s / 100 Hz = 3 m.
6
Question
How can the minimum wavelength for the lumped model of the human ear be calculated?
Answer
min λ = c / f_max = 300 m/s / 10,000 Hz = 3 cm.
7
Question
What are the fundamental variables of electrical systems?
Answer
The fundamental variables are Voltage (U or V in volts), Current (I in amperes), and Resistance (R in ohms).
8
Question
What does the Law of Ohm state?
Answer
The Law of Ohm defines the relation between voltage (U), current (I), and resistance (R) as U = I * R.
9
Question
How is impedance defined in electrical systems?
Answer
Impedance (Z) can be defined as Z = U / I.
10
Question
What are the two forms of current in electrical systems?
Answer
The two forms of current are direct current (DC) and alternating current (AC).
11
Question
How is resistance (R) defined graphically?
Answer
Current goes through an element at voltage goes over it. WHen their values are pluted (in AC current) the difference between the 2 lines at their peak amplitudes makes up the resistance
12
Question
What is a phasor?
Answer
A phasor is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (φ) are time-invariant.
13
Question
What is the relationship between current and voltage in an AC circuit?
Answer
In an AC circuit, the current (I) and voltage (U) may be out of phase, and this relationship can be described with impedance (Z).
14
Question
How can the complex number of Impedence (Z) be described by the summation of two vectors?
Answer
The complex impedance (Z) can be described as the summation of resistance (R) and reactance (X) represented as vectors: Z = R + jX, where R is the real part (resistive component) and jX is the imaginary part (reactive component), with "j" being the imaginary unit.
15
Question
What is the general form of electrical impedances?
Answer
There are three types of impedances: - Resistor (R) - Capacitor (C) - Inductor (L).
16
Question
Give the expressions of resistor, capacitor, and inductor as phasors
Answer
1. Z_R = u(t)/i(t) = R 2. Z_C = u(t)/i(t) = 1/jwC (phase difference of -90° = current leads voltage by 90°) 3. Z_L = u(t)/i(t) = jwL (phase difference of +90° = voltage leads current by +90°)
17
Question
What does conductance (G) represent?
Answer
Conductance (G) is the reciprocal of resistance (R), expressed as G = 1/R.
18
Question
What is Hook's law?
Answer
Hook's law states that the force (F) exerted by a spring is proportional to its displacement (x), expressed as F = k * x.
19
Question
What are the fundamental variables of mechanical systems?
Answer
The fundamental variables are Force (F in newtons), Velocity (v in m/s), Displacement (x in meters), and Acceleration (a in m/s²).
20
Question
What are the three elements of a mechanical system?
Answer
Damper (resistor), mass (comparable to capacitor), spring (comparable to inductor)
21
Question
What is the equation relating force and acceleration in mechanical systems?
Answer
F = m * (dv/dt), where m is mass. (F = m*a)
22
Question
What are the fundamental variables of acoustical systems?
Answer
The fundamental variables are Sound Pressure (P in pascals) and Volume Velocity (q in m/s).
23
Question
What are the three main elements of acoustic systems?
Answer
Damper (Resistance) - Converts acoustic energy into heat Acoustic Compliance - Compressed air (no temp. change) Acoustic Mass (Inertance) - Accelerated but not compressed air
24
Question
What is a Helmholtz resonator?
Answer
A Helmholtz resonator is a device that is used to determine the resonance frequency of a system and consists of a bottle-like shape with specific dimensions. The "A" refers to the area of teh bottle opening
25
Question
What is the ideal pressure conditions in an ideal (healthy) ear?
Answer
The auditory canal is modelled as a small tube. In ideal conditions there is no pressure difference between the auditory canal (external ear canal [EAC] and the middle ear (i.e. they are equal)
26
Question
How would flexibility be returned if you have an increased pressure in one of the sides? (either external auditory canal or the inner ear)
Answer
If there is a pressure increase on one side, there must be a pressure increase on the other too. And vice versa This is regulated by the Eustachian tube in the middle ear Unequal pressures can lead to a number of pathologies like tympanic membrane retraction
27
Question
How does the resonance frequency relate to the acoustic resistance?
Answer
The resonance frequency corresponds to the frequency at which the impedance of the system is minimized.
28
Question
What is the significance of the Bode diagram in acoustics?
Answer
The Bode diagram depicts the magnitude and phase of a complex impedance as a function of frequency, providing insights into the frequency response of a system.
29
Question
What do lumped models let us do?
Answer
They allow us to translate different engineering modalities using an element where we define an ACROSS and a THROUGH variable