ECE 403  ExamII – Version 1.00 April 18, 2017  Spring 2017
Univ. of Illinois  Tue, April 18, 2017; 7-10PM  Prof. Allen

Topics: Acoustics, Transforms, Two-port networks

Instructions:

• If you need more space for calculation, use the back of any page.

• NO Cell phones.

• NO Calculators

• Note that each problem has points assigned. More points means “harder.” You may open the exam at 7:00 PM; You must close it by 9:00 PM.

NAME: NETID: I will not share information during this exam, and I will discretely report others if I observe cheating: (sign here)

# value score
1 15
2 7
3 16
4 17
5 15
6 15

Terminology, constants & transforms

Definitions of common acoustic variables,^{1} the mathematical symbols and the units, as used in the text (see p. 180-181).

Variable  Symbol  [Units]
Total pressure  P P_{o} p t  [N/m^{2} P a]
temperature  T T_{o} τ t  [K^{}] Kelvin
density  ρ ρ_{o} δ t  [kg/m^{3}]
adiabatic law  p δ^{γ} Const.
Boyle’s Law  P_{o} ρ_{o}T_{o} Const.

Constants (air)

Atm Pressure  P_{o} 10^{5}  [Pa]
Abs temperature  T_{o} 273  [K^{}]
density  ρ_{o} 1.18  [kg/m^{3}]
sound speed  c 345 \overline{γP_{o} ρ_{o}} [m/s]
specific impedance  ρ_{o}c 407  [Rayls]
viscosity  μ 1.86 10^{5}  [Ns/m^{2}] (Poiseuille)
thermal conductivity κ 25.4 10^{3}  [N/sK]
specific heat cap @V_{o} c_{v}  [J/kg]
specific heat cap @P_{o} c_{p}  [J/kg]
Boltzman’s const. k 1.38 10^{23} J/K
ratio of specific heats γ c_{p} c_{v} 1.4  (unitless)
Transform  Time Frequency  Frequency Time
Laplace Transform (LT)  F s _{0}^{}f t e^{st}dt  f t \frac{1}{2πj} _{C_{L}} F s e^{st}ds

C_{L} is the Laplace contour

Fourier Transform (FT)  F ω _{}^{}f t e^{jωt}dt  f t \frac{1}{2π} _{}^{}F ω e^{jωt}dω
Fourier Series (FS)  F k \frac{1}{2π} _{π}^{π} f t e^{jnt}dt f t  _{k}^{}F k e^{jnt}
Z-Transform (ZT)  F z  _{n 0}^{}f n z^{n}  f n \frac{1}{2πj} _{C_{z}} F z z^{n 1}dz

C_{z} is the Z-transform contour

Discrete-Time FT (DTFT) F ω  _{}^{}f n e^{jωn}  f n \frac{1}{2π} _{π}^{π} F ω e^{jωn}dω
Discrete FT (DFT)  F k _{n 0}^{N 1} x n e^{\frac{2πj}{N} kn}  f n \frac{1}{N} _{k 0}^{N 1} F k e^{\frac{2πj}{N} kn}
Discrete convolution: f n g n _{k}^{}f k g n k
Continuous convolution: f t g t _{}^{}f τ g t τ dτ

1c_{p}, c_{v}: https://en.wikipedia.org/wiki/Heat_capacity#Specific_heat_capacity

B_{o}l  ^{f t}^{u t}^{i t}^{ϕ t} Z_{e} s  Z_{m} s

Figure 1: The Hunt model is composed of an electrical impedance Z_{e} s , a Gyrator having parameter T lB_{o}, and a mechanical impedance Z_{m} s . The electrical side is driven by a voltage ϕ t and a current i t , while the mechanical side produces a force f t and a particle velocity u t .

1 (15pts) Transducer Thevenin Parameters

1. (3pts) Find the ABCD (Transmission) T s matrix for the Hunt transducer model in Figure 1

\begin{matrix}Φ ω \\ I ω \end{matrix} T s \begin{matrix}F ω \\ U ω \end{matrix} .

2. (2pts) Find the impedance matrix Z s .

\begin{matrix}Φ ω \\ F ω \end{matrix} Z s \begin{matrix}I ω \\ U ω \end{matrix} .

3. (2pts) Compute Δ_{Z} and Δ_{T}. What is the significance of each of these determinants?

4. (4pts) Define, and then find the Thevenin equivalent force F_{thev} s [N], given an input voltage Φ_{o}. Note: If you are not sure of your answers to the previous parts, you can solve this problem in terms of an arbitrary ABCD matrix.

5. (4pts) Define, and then find the Thevenin equivalent mechanical impedance in mechanical Ohms, Z_{thev} s [Ω].

2 (7pts) Nyquist Theorem on Thermal noise

A stub of transmission line having characteristic impedance z_{0} is terminated in each end with this Thevenin model, with R z_{0}. Then at t 0, the resistance is short or open circuited. This setup is shown in the figure.

R  R V_{m} x 0, t 0  V_{m} L, t 0 z_{0}  z_{0}  V_{m} t 0

Figure 2: On the left an ideal lossless transmission line, driven by two resistors. Each resistor has a noise source in series with it, that generates a voltage equal to the thermal noise in the resistor. At time t 0 the resistors are switched out, leaving the voltage in the transmission line frozen in place.

The transmission line stores the voltage as a function of x at t 0 once the switch is opened removing the resistors (and also the Thevenin source). At that point voltage V_{m} L, t becomes frozen.

1. (1pt) Discuss what happens after t 0.

2. (2pt) What is the fundamental period of the noise voltage, v_{m} L, t 0 ?

3. (2pts) Every periodic signal has a Fourier series. If the period is T [s], what can you say about the Fourier series frequencies?

4. (2pts) The noise spectrum will differ depending on the time at which the switch is opened. Why?

3 (16pts) Hilbert transform

Analyze the real, causal impulse response

h t e^{t τ_{0}} u t ,

with τ_{0} 10 [ms], in terms of its Hilbert transform (integral) relations. Note: Use the notation h t H s and H ω H s _{s jω}.

1. (2pts) Find H s , the Laplace transform of h t .

2. (1pt) Where are the poles of H s e^{t τ_{0}} u t ?

3. (4pts) Find and sketch the real and imaginary parts of H ω H s _{s jω}.

4. (1pt) Write out the even h_{e} t h t h t and odd h_{o} t h t h t functions. Describe why these equations are symmetric and antisymmetric functions.

5. (4pts) Find the Fourier transforms of h_{e} t H_{e} ω and h_{o} t H_{o} ω . How do they relate to the real and imaginary parts of H ω ?

6. (4pts) Find the Hilbert (integral) relations between H_{r} ℜH ω (real part) and H_{i} ℑH ω (imag part) of H ω . Hint: These integrals come from a frequency-domain convolution.

4 (17pts) Wave equation

4.1 History of the wave equation

1. (1pt) What year did d’Alembert derive his solution to the wave equation?

2. (1pt) What is the form of D’Alembert’s solution?

3. (1pt) Who was the first person to calculate the speed of sound, and what was the result?

4.2 The Webster horn equation:

In the time domain, in 2x2 matrix form, the Webster horn equation is given by

\frac{∂}{∂x} \begin{matrix}p x, t \\ ν x, t \end{matrix}  \begin{matrix}0 & \frac{ρ_{o}}{A x} \\ \frac{A x}{γP_{o}} & 0 \end{matrix} \frac{∂}{∂t} \begin{matrix}p x, t \\ ν x, t \end{matrix} .  (1)
where p x, t  x, ω is the pressure and ν x, t  x, ω is the volume velocity.

1. (2pts) Transform the Horn equation to the frequency domain.

2. (4pts) Assuming a conical horn, having area A x A_{o} x x_{o} ^{2} with A_{o} 4π, rewrite Equation 1 as a second order equation solely in terms of the pressure x, ω (remove ), and thereby find the frequency domain solutions P_{}x, s for the conical horn equation.

3. (4pts) Assuming an exponential area function

A x A_{0}e^{2mx}

(m is a positive constant, called the horn flair parameter), derive the exponential horn equation for the pressure.

\frac{∂^{2}p x, t}{∂x^{2}} 2m\frac{∂p x, t}{∂x} \frac{1}{c^{2}} \frac{∂^{2}p x, t}{∂^{2}t}  (2)

4. (4pts) In general the solution to a wave equation is of the form

p x, t P^{}κ e^{κ s x}e^{st} P^{}κ e^{κ s x}e^{st}

where s σ ȷω is the Laplace frequency and κ s is the complex “wave number.”

(a) What is κ s for the conical horn?

(b) What is the significance of κ s ?

(c) Why is it a function of s?

(d) What is the role of P^{}κ, s ?

5 (15pts) Reflectance

1. (6pts) A tube transmission line with characteristic impedance z_{0} and length L is terminated in a load impedance Z_{L} s .

(a) The reflectance at any location x along the tube transmission line is

Γ x, s \frac{Z x, s z_{0}}{Z x, s z_{0}} .

Starting with this formula, show that the impedance

Z x, s \frac{x, s}{x, s} .

Hint: What is z_{0} in terms of the forward and reverse traveling waves ^{}and ^{}?

(b) Find the formula for the reflectance at the load (simplify your answer, if applicable), Γ x, s _{x L} Γ L, s Γ_{L} s for i. Z_{L} s r [Nt-s/m^{5}]

ii. Z_{L} s 1 sC [Nt-s/m^{5}]

iii. Z_{L} s r sM [Nt-s/m^{5}] 2. (3pts) For the transmission line described in the previous problem, let L 1, Z_{L} s 1

and z_{0} 2.

(a) Find the frequency domain reflectance Γ 0, s at x 0.

(b) Find the time-domain reflectance γ 0, t Γ 0, s at x 0.

3. (3pts) Two transmission lines are in cascade, the first one having an area of 1 [cm^{2}] and a second having an area of 2 [cm^{2}], with lengths L_{1} and L_{2} respectively, terminated with a resistor r ρc A, where A 2 [cm^{2}]. Find R x 0, s .

4. (3pts) What is the inverse Laplace transform of

(a) H s 1 s 1 ? Find h t .

(b) H s s s 1 ?

6 (15pts) Model of the middle ear

As shown in the figure, the free field sound pressure, defined as P_{0} ω acts as a source in series with the radiation resistance R_{rad}. The total radiation impedance Z_{rad} s is a combination of the resistance and a reactive component L_{rad}, which represents the local stored field. The two impedances are in parallel

Z_{rad} s sL_{rad}R_{rad} sL_{rad} R_{rad} 1 Y_{rad} s .

where s σ jω is the Laplace complex frequency variable. The radiation admittance for a sphere is

Y_{rad} 1 Z_{rad} \frac{A_{rad}}{sr_{c}ρ_{o}} \frac{A_{rad}}{ρ_{o}c} \frac{1}{sL_{rad}} \frac{1}{R_{rad}} ,

where r_{c} is the radius of the sphere and A_{rad} is the effective area of the radiation.

z_{0}^{}^{U_{ec} ω}^{P_{tragus} ω}^{P_{0} ω}^{R_{rad}}  L  C_{al} R_{c} P_{c} ω^{L_{rad}}

Figure 3: Model of the ear canal, terminated by the radiation impedance Z_{rad} s at the tragus (x 0), and by the eardrum and cochlea at x L.

On the left we terminate the ear in a radiation impedance

Z_{rad} s \frac{sL_{rad}R_{rad}}{sL_{rad} R_{rad}} .

At the cochlear end we terminate the line with an impedance

Z_{c} R_{c} 1 sC_{al} ,

where R_{c} is the cochlear impedance and C_{al} is the stiffness of the annular ligament, which is the ligament that holds the stapes in the oval window. The cochlear resistance (R_{c}) is assumed to be twice the characteristic impedance of the ear canal.

To Do:

1. (4pts) Find the formula for the reflection coefficient at x L. Sketch the magnitude R L, s as a function of frequency. Hint: You should label any constants, or set them equal to 1 for plotting.

2. (4pts) Find the formula for the reflection coefficient at x 0 (looking out towards the radiation impedance). Sketch its magnitude as a function of frequency.

3. (3pts) Consider the radiation impedance looking out the ear canal, Y_{rad}.

(a) What is the frequency for which its real and imaginary parts are equal (in terms of L_{rad} and R_{rad})?

(b) Describe the dependence of Y_{rad} on L_{rad} and R_{rad} below and above this frequency.

4. (4pts) Find the formula for the input impedance Z 0, s of the middle ear at the entrance of the ear canal, when the cochlea is “blocked” (Z_{c} or Γ L, s 1)?

Version 1.00 April 18, 2017 ˜/403/Assignments/ExamII – Version 1.00 April 18, 2017