ELECTROMAGNETICS
AND APPLICATIONS
Electromagnetics and Applications
• Maxwell’s equations: statics, quasistatics, and wave phenomena
• Applications: wireless, media, circuits, forces and generators,
computer speed, microwaves, antennas, photonics, acoustics, etc.
Mathematical Methods
• Partial differential and difference equations, phasors, vector calculus
Problem Solving Techniques
• Perturbation, boundary-value, and energy methods; duality
Academic Review
• Mechanics, quantum phenomena, devices, circuits, signals, linear
systems
Capstone Subject—Professional Preparation
Follow-on Subjects:
Electromagnetic waves: 6.632, Quasistatics: 6.641
L24-1
ACOUSTIC ANTENNAS
∇p = − ρo ∂u [Nm−3 ] [kg m-2s−2 ]
∂t
2
2
Wave equation:
(∇ + k )p = 0*
(k = ω/cs)
∂/∂θ = ∂/∂φ = 0
(radial source)
Yields:
d2p/dr2 + (2/r)dp/dr + k2p = 0
Equivalent to:
d2(rp)/dr2 + k2(rp) = 0
General solution: rp ∝ e±jkr
Radiation outward: p(r) = (A/r)e-jkr
∇p
A ⎛
1 ⎞ − jkr
Velocity field u:
u(r) = −
= rˆ
1
+
⎜
⎟e
Monopole Radiators:
jωρo
ηsr ⎝
jkr ⎠
Far-Field: kr >> 1 ⇒ r >> λ/2π):
p(r) = (A/r)e-jkr
u(r) = (A/rηs)e-jkr = p(r)/ηs
Near-Field Radiation: Since ∇ p = − jω u therefore:
p(r) = (A/r)e-jkr
u(r) = (-jA/r2kηs)e-jkr = -jp(r)/ρoωr
Velocity
microphone
“Velocity mikes” close to the lips boost lows; need ×ω compensation
1
∂2 ⎞
* ⎛ ∇2 = 1 ∂ (r 2 ∂ ) + 1 ∂ (sin θ ∂ ) +
⎜
2 ∂r
2
2
2
2 ⎟
r
∂
∂θ
∂θ
∂φ
r
r
sinθ
r
sin
θ
⎝
⎠
L24-2
ACOUSTIC ANTENNAS (2)
Antenna Gain G(θ,φ), Effective Area A(θ,φ) [m2]:
Pr (θ, φ)
G(θ, φ) =
(Pt / 4πr 2 )
Preceived = I(θ,φ) A(θ,φ) [W]
Antenna (Loudspeaker, Microphone) Configurations:
Monopole
Lense
Parabolic dish
Dipole
-
+ ⇒
Horn
G(θ,φ)
Baffled
monopole
Array (end-fire
or broadside)
θnull
λ/2
L24-3
ACOUSTIC RESONATORS
Amnp Resonances of a Box:
λy
m( ) = a ⇒ λ y = 2a
2
m
λ
λ
n( x ) = b, p( z ) = d
2
2
( ∇2 + k 2 ) p = 0
e.g., p = po e
− jk x x − jk y y − jk z z
x
b
y
λz
u
z
d
2
ω
k =
= k x 2 + k y 2 + k z 2 = ( 2π )2 + ( 2π )2 + ( 2π )2
λx
λy
λz
c s2
2
Resonant Frequencies of the Amnp Mode in a Box:
⎛
⎞
ω
1
1
2
2⎜ 1
⎟ , where λ x = 2b
+
+
fmnp = ( ) = c s
2π
n
⎜ λ 2x λ 2y λ 2z ⎟
⎝
⎠
p
fmnp = c s ( m )2 + ( n )2 + ( )2 [Hz]
2a
2b
2d
f000
= 0 Hz (constant pressure)
f001
= 340/2d ⇒ 170 Hz for a one-meter closed pipe
2
L24-4
RESONATOR MODAL DENSITY
Modal Density in Rectangular Resonators:
p
Recall: fmnp = c s ( m )2 + ( n )2 + ( )2 [Hz]
2a
2b
2d
pcs/2d
Each cube has volume = cs3/8V
Δf
where V = abd (volume of resonator)
Number of modes in Δf ≅
(Volume of shell)/(vol. of cell) ≅
4πf2 Δf/[8(cs3/8V)] ≅
4πf2 Δf V/cs3 modes in Δf
p=2
fmnp
m=2
3
mcs/2a
Example:
ncs/2b
Bathroom 3×3×3 meters
⇒ lowest f100 =cs/2a ≅ 340/6 ≅ 57 Hz
Modal density at 500 Hz ≅ 4π × 5002 × 1 × 33/3403 ≅ 2 modes/Hz
How can we select just one mode when we sing a single note?
L24-5
EXCITATION OF RESONATORS
TEM Resonators (with loss):
I(t) = Io cos ωot
V(δ,t) = Vo cos(ωot + φ) sin(2πδ/d)
I(t)
φ = 0 exactly at resonance
Pin(t) ≅ IoVo cos2(ωot) sin(2πδ/d) [W]
= 0 at voltage nulls
Vo Voltage null
Zo,c
V(z,t)
d
δ
Cannot excite TEMm modes by driving current into voltage nulls! (Or
by voltage sources in series at current nulls). Pin(t) = 0 in both cases.
Acoustic Resonators:
I[ Wm−2 ] = pu • nˆ
Put u here
Cannot excite acoustic modes with:
velocity sources at pressure nulls (pk = 0), or
pressure sources at velocity nulls (vk = 0)
Bathroom Opera:
Mouth ≈ velocity source
Place mouth near a pressure maximum of desired mode
L24-6
HUMAN ACOUSTIC RESONATORS
Human Vocal Tract:
f1 = cs/λ1 = cs/4d
= 340/(4 × 0.16)
= 531 Hz
Higher Resonances:
p(t,z)
u(t,z)
0
z
d
d
u
f2 = 3f1 = 1594 Hz
f3 = 5f1 = 2655 Hz
Energy Densities at
Location “ ”
At f1:
wp ≅ wu
At f2:
wu >> wp
At f3:
wp << wu
vocal chords
f2
p(t,z)
u(t,z)
0
p(t,z)
u(t,z)
0
d
z
f3
z
L24-7
RESONANCE SHIFTS IN HUMAN VOICES
Human Vocal Tract:
Average force exerted by waves:
Outward at maximum p (max wp)
Inward at maximum u (max wu)
(Bernoulli force)
d
Resonator Total Energy wT = nhfo:
Pressing inward at pmax increases wT and fo
(Phonon number n = constant for slow changes)
Recall: pressure {N m-2] ∝ energy density [J m-3]
Resonance Perturbations:
Δf = Δ(w p − w u )
f
wT
fo ∝ c s ∝
γPo
ρo
vocal chords
wp >> wu at f3
f3
wu >> wp at f2
f2
f1
0
Tongue position wp ≅ wu at f1
determines vowel
t
vowels
L24-8
MIT OpenCourseWare
http://ocw.mit.edu
6.013 Electromagnetics and Applications
Spring 2009
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