Appunti Electromagnetic Micro Nano Devices

List of Questions

A. Waves

1. Solution for the one-dimensional wave equation in z and t variables
2. Solution for the one-dimensional wave equation in sinusoidal steady state
3. Transmission line: telegraph equations
4. Transmission line: sinusoidal steady state

B. Lattices & Crystals

1. Point Symmetry operations for a plane lattice
2. Point groups in three dimensions
3. Three-dimensional crystals and lattices: triclinic, monoclinic, orthorombic
4. Three-dimensional crystals and lattices: tetragonal, hexagonal, rhombohedral, trigonal, cubic

C. XRD

1. X-Rays and Bragg's Law

D. Piezoelectricity

1. Theory
• Piezoelectricity (and ferroelectricity) and direct piezoelectric effect
• Converse piezoelectric effect and linear relations for a piezoelectric material (T, E)
• Linear relations for a piezoelectric material: (S, E), (T, D), (S, D)
• Piezoelectric ceramics - ferroelectric material
• Electromechanical coupling factors
• Mechanical waves in solids: solution for an elastic material
2. Applications
• Resonance with a piezoelectric ceramic
• Kinetic energy harvester: model and piezoelectric converters
• Piezoelectric actuators
• Inchworm motor, piezo legs motor and stick-slip motor
• Langevin vibrator and applications
• Travelling-wave motors

E. Waves in medium

1. Uniform plane waves in lossless medium: from Maxwell’s equations to one dimensional wave equation
2. Uniform plane waves in a lossless medium: solution in sinusoidal steady state. Lossy media: complex permittivity
3. Uniform plane waves in lossy media and materials
4. Retarded potentials
5. Electric dipole
6. Magnetic dipole and wave impedence

F. Shielding

1. Shielding effectiveness: definition and shielding at low frequency
2. Shielding effectiveness: analysis with uniform plane waves
3. Absorption loss and reflection loss with uniform plane waves and with near field sources

1) Solution of the One Dimensional Wave Equation in z and t Variables

Order 2nd Homogeneous Linear

∂²ψ(z,t) / ∂z² = 1 / v² ∂²ψ(z,t) / ∂t²

Change Variables:

• k = t / 2 - z / v
• w = t / 2 + z / v

z = 1/2 (k-w) , t = 1/2 (k+w)

1st order:

∂ψ / ∂t = ∂ψ / ∂k ∂k / ∂t + ∂ψ / ∂w ∂w / ∂t = ψ (∂k / ∂t) + ψ (∂w / ∂t)

∂ψ / ∂t = ∂ψ / ∂k + ∂ψ / ∂w

2nd order:

∂²ψ / ∂t² = ∂ / ∂t (∂ψ / ∂k + ∂ψ / ∂w)

= ∂²ψ / ∂k² + 2 ∂²ψ / ∂k∂w + ∂²ψ / ∂w²

= -1/v² (∂²ψ / ∂z² - 2 ^ψ ∂² / ∂k∂w)

∂²ψ(k,w) / ∂k∂w = 0 = ∂/∂k (∂/∂w ψ(k,w))

ψ(k,w) = α₁(w) + α₂(k)

ψ(z,t) = α₁(t/2+z/v) + α₂(t/2-z/v)

q.e.d. Quantized in a Binomial

Point Symmetry Operations for a Plane Lattice

Rotation for a plane lattice:

a_1x = a₁cosφ   a_1y = a₁sinφ   a_x = a_y   a_yy = 0

If rotation of φ is lattice point (l) then also -φ is lattice point (l).

If lattice is invariant with rotation φ, it is also invariant with rotation -φ.

|2a_xcosφ| = M*a_y,   M > 0

In order to have a symmetry of rotation 5 values are possible:

0 {2π} π   3π   π/3   2   1/3

Expressed as:

2π/m,   m = 1,2,3,4,6

PIEZOELECTRICITY (AND FERROELECTRICITY)

"ELECTRICITY BY PRESSURE"

• 32 CRYSTAL CLASSES -> 11 CENTROSYMMETRIC
• 21 NOT
• 20 OF THESE NON POLAR
• 10 POLAR
• ONLY WITH MECHANICAL LOAD
• SPONTANEOUS
• FERROELECTRIC CRYSTALS

SIMPLE MODEL

MECHANICAL FORCE

MOMENT OF THE BARYCENTER OF THE POSITIVE AND NEGATIVE CHARGES

=> ELECTRIC DIPOLE MOMENT

[IN LARGE SCALE]

UNIFORM POLARIZATION

LINEAR MODEL IS CONSIDERED

P_p = d_ijT

PZT CONST [C/N]

PIEZ POLARIZATION

d = [_3x6

d₁₁ d₁₂ d₁₃ d₁₄ d₁₅ d₁₆

d₂₁ d₂₂ d₂₃ d₂₄ d₂₅ d₂₆

d₃₁ d₃₂ d₃₃ d₃₄ d₃₅ d₃₆]

Mechanical Waves in Solid: Solution for an Elastic Material

d_x, d_y, d_z

f₁₁ = T₁₁ + ∂T₁₁/∂x d_x/2 + (T₁₁ + ∂T₁₂/∂y)d_y/2

f₁₃ + ∂T₁₃/∂z d_z/2

(∂T₁₁/∂x + ∂T₁₂/∂y + ∂T₁₃/∂z)dxdydz

N.B. Same for other two directions

Longitudinal Vibrations

if uk/1 infinitesimal displacement

S₁ = (1 / ∂x) u_x(x,t)

d²/dx²(u_x(x,t))

u(x,t) = A₁ sin [w(t-x/v_n)] + α] + A₂ sin [w(t+x/v_n)] + β]

S₁ = ∂ / ∂x (u_x(x,t)) = A₁ w_n cos [w(t-x/v_n) + α] + A₂ w_n cos [w(t+x/v_n) + β]

Boundary Conditions

LANGEVIN VIBRATOR AND APPLICATIONS

W_j = T/L = λ/2

ρ_L = N/2L + 5cm/2 in steps→ ρ_m @ 40 kHz → L = 5.25 cm

PAIR OF DISKS PIEZOELECTRIC

N(t) SINUSOIDAL AT RESONANT FREQUENCYRESPONSE POLARIZED AND PARALLEL CONNECTS

SUITABLE FOR ULTRASONIC TRANSMISSION IN SEA

• Navantia - 5 km/s
• L = 5.25 cm @ 40 kHz
• 0.5 cm bronze piezoceramic PHDZ plates

Retarded Potentials

B(p, t) = ∇ × A(p, t)

E(p, t) = -∇^*A(p, t) - ∂/∂t A(p, t)

-∇²E(p, t) - ∂/∂t (∇·A(p, t)) = -ρ_free(p, t)/ε

Set ∇^*A(p, t) = 0

∇²V(p, t) = -ρ_free(p, t)/ε

∇²A(p, t) = -μ₀J(p, t)

V(p, t) = ∫ (ρ_free(q, t_ret)/ε) d_V

A(p, t) = ∫ (μ₀J(q, t_ret)/(4π)) d_V

30

Absorption loss and deflection loss with uniform plane waves and with near field sources

_Adr = 20 log ^d⁄_δ ≈ 1⁄8 _log ^e ≈ 8.7 ^d⁄_δ δ = 1⁄^√_πfμ0

σ >> μ in magnitude (more relevant) conductor better than magnets assuming this function

for Q_ab assume that _E⁰ = in ^v_μ negligible in medium 2

Combination of boundary 1 and 2

→ [_σ(z=0) - _σ(z=d)] → [^E⁄_H] _z=0 · [^E⁄_H] _z=d = (M₀ + M) (M₀ + m)⁄Z_μ

→

Pol_s = -20 log [M_S + M]²⁄[^M⁄_m]

Good conductor

σ >> WE → η = ⁄

→

Pol_s ≈ 20 log [M_S + M]²⁄[^M⁄_m] - 10 log