# Electro-Mechano-Acoustical_20Circutis_3_ by xiangpeng

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```									         Electro - Mechano - Acoustical
1) Mechanical Circuits
- Physical and Mechanical Meaning of Circuit Elements
- Mechanical Circuits
2) Acoustical Circuits
- Acoustical Elements
3) Transducer
- Transformation Type Element
- Gyration Type Element
4) Circuit Theorems and Energy and Power
- Conversion from Mobility-Type Analogies to Impedance-Type Analogies
- Thevenin’s Theorem
- Energy and Power Relations
- Transducer Impedances
- Combinations of Electrostatic and Electromagnetic Transducer
1) Mechanical Circuits
 Physical and Mechanical Meaning of Circuit Elements

- Generators of Two Type

- Circuit Elements of Four Type

Acoustics Chap 3 (Leo L. Beranek)   2
- a  e(Voltage, f(Force), u(Velocity), p(Pressure), U(Volume Velocity)
- b  i(Current), u(Velocity), f(Force), U(Volume Velocity), p(Pressure)
- c  Mass, Compliance, Inductance, Resistance……
( 여기서 c 는 a,b가 선택된 의미에 따라 결정된다. )

Acoustics Chap 3 (Leo L. Beranek)                    3
 Mechanical Circuits

- Velocity is an
Analog to Voltage in
that it can be
Measured without
Interrupting the
Flow Quantity. (Not
Force and Current)

Acoustics Chap 3 (Leo L. Beranek)   4
Force : Voltage
Direct(Impedance) Analogy
Velocity : Current

Velocity : Voltage
Inverse(Mobility) Analogy
Force : Current

a) Mechanical Impedance ZM, and Mechanical Mobility zM
- Mechanical Impedance, ZM si Complex Ratio of Force to Velocity
 N  sec 
          SI   mech.ohm.
 m 
- Mechanical Mobility, zM si Complex Ratio of Velocity to Force
 m 
          SI   mech.mohm.
 N  sec 

Acoustics Chap 3 (Leo L. Beranek)                        5
b) Mass MM
- Newton’ Second Law
du(t )
f (t )  M M
dt

- 전제조건 : 주기적인 성질을 갖는 시스템을 고찰 u  u0 e jt                   
Steady State (  2 )

f  jM M u           (Impedance Type)

f
u                    (Mobility Type)
jM M

Acoustics Chap 3 (Leo L. Beranek)           6
c) Mechanical Compliance CM

1
f (t )  kx 
CM  u (t )dt
Where    u (t )  u1  u2

이전과 동일한 전제조건 하에서

u
f                     (Impedance Type)
j C M

u  jCM f            (Mobility Type)

Acoustics Chap 3 (Leo L. Beranek)   7
d) Mechanical Resistance RM, and Mechanical Responsiveness rM

f  RM u

1    m 
rM                   SI       mech.mohm.
RM    N  sec 

f  RM u      (Impedance Type)

u  rM f      (Mobility Type)

Acoustics Chap 3 (Leo L. Beranek)         8
e) Mechanical Generators

Constant Velocity                             Constant Force
Very Strong Motor Attached to a           Electromagnetic Transducer
Shuttle Mechanism                         (e.g. Moving-Coil Loudspeaker)

Acoustics Chap 3 (Leo L. Beranek)                     9
e) Levers (Transformer)
f1l1  f 2l2

u1l1  u2l2     (Assuming Small Displacement)
2
u1  l1                                             l1 
zM 1     zM 2                                           :1
f1  l2                                            l 
(Mobility Type)
                                                2
 l2 
2
f1  l2 
ZM1     ZM 2               (Impedance Type)             :1
l 
u1  l1 
                                                 1

Acoustics Chap 3 (Leo L. Beranek)            10
f) Mechanical Impedance and Mobility

Mechanical Impedance in Series           Mechanical Impedance in Parallel
( u=Const )                               ( f=Const )

+                                               +
+                      u            u1        u2
Z1      f1
-
f        u                                      f               Z1        Z2
+
Z2      f2
-
-                                               -

f1  uZ1 , f 2  uZ 2                               f  u1Z1  u2 Z 2
f  f1  f 2                                     u  u1  u2
f1  f 2                                 1    u u1  u2 1  1
ZTotal
f
            Z1  Z 2                                   
u     u                              ZTotal      f    f    Z1 Z 2
Acoustics Chap 3 (Leo L. Beranek)                            11
Mechanical Mobility in Series               Mechanical Mobility in Parallel
( f=Const )                                  ( u=Const )

+                                               +
+                       f                f1        f2
z1      u1
-
u         f                                     u                    z1        z2
+
z2     u2
-
-                                               -
u1  fz1 , u2  fz2                                 u  f1 z1  f 2 z2
u  u1  u2                                         f  f1  f 2
u u1  u2                                 1     f  f1  f 2 1 1
zTotal             z1  z2                                       
f    f                                zTotal    u     u     z1 z2

Acoustics Chap 3 (Leo L. Beranek)                                 12
2) Acoustical Circuits
 Acoustical Elements
- Sound Pressure p(N/m2=Pa) is Measured without Modifying the Device.
It is Desirable to make Pressure Analogous to Voltage.
- Acoustical Impedance, ZA is the Complex Ratio of Pressure to Volume
Velocity, v(Nsec/m5) or SI Acoustic ohm.
- Acoustic Mobility zA, is Complex Ratio of Velocity to Pressure.

p ( N / m2 )                                   v
ZA   3                                      zA 
v (m / sec)                                    p

* The Flow Quantity : v   u ds     ( U= 입자의 속도 )

Acoustics Chap 3 (Leo L. Beranek)               13
a) Basic Acoustical Elements Can be Realized by Considering the
Following Device

x0                               x l
Piston
Cross Section Area = S

u  u0j t

- Simple Plane Wave Propagating to and fro in X-Direction.
- Constant Velocity Source.
- 피스톤 부분의 공기 입자는 피스톤과 함께 움직인다.

d2p
* Helmholtz Eq is                    k2 p  0
dx 2
Acoustics Chap 3 (Leo L. Beranek)             14
* Euler’s Eq is
 1 p        u        
u ( x)                   0   p 
j  0 x      t        
dp                    p 1 D dp 
  j  0u             
  
dx                    x      dx 

Let        p( x)  A sin kx  B cos kx
dp
 k ( A cos kx  B sin kx)
dx
j
 u ( x)      ( A cos kx  B sin kx)
0c
From Boundary Condition
jA
u ( x  0)  u 0           or     A   j (  0 cu0 )
0c
Acoustics Chap 3 (Leo L. Beranek)             15
i) Close End Problem
jB
u (l )  0  u0 cos kl      sin kl
0c
 B   j 0 cu0 cot kl , u ( x)  u0 (coskx  cot kl  sin kx)
p( x)   j 0 cu0 (sin kx  cos kx  cot kl)
“Input” Acoustic Impedance Z A @ x  0
p        p x 0  j 0 cu0 cot kl  j 0 c cot kl
ZA                                  
v x 0 u0 S            u0 S              S
  c
  j  0  cot kl
 S 
For Small kl(i.e. kl  1 , l   / 8) and cot kl  1  kl  A
kl       3

  0 c   1 kl   j 0 c     j 0l
2
Z A   j                        
 S   kl 3       Sl          3S
A         B
Acoustics Chap 3 (Leo L. Beranek)             16
0c 2                      1        1
        if l                      
jSI         16            SI  jC A
j 
  c2 

 0 
Volume
CA                                  Acoustic Compliance
0c

Again, we Write

  c                1          l
Z A   j  0  cot kl                 j 0           (kl  1)
 S                 V         3S
j 
  c2 

 0 
1        MA                             0l
        j            where M A 
jC A      3                              S

Acoustics Chap 3 (Leo L. Beranek)               17
ii) Open End Problem
Assume that Negligible Sound Radiates from that End (i.e. p=0 @ x=)
p( x  l )   j 0 cu0 sin kl  B cos kl  0
B  j 0 cu0 tan kl
p( x)  j 0 cu0  sin kl  cos kx  tan kl
“Input” Acoustic Impedance Z A @ x  0
p       p x 0     0c 
ZA                   j      tan kl
v x  0 u0 S        S 
For k <<1
 l                               tan kl  (kl) 
(kl)3
 kl
Z A  j   0   jM A              Where
3
 S 
1        MA
 ZA that Open End and Close End is          jM A  Z A         j
jC A      3

Acoustics Chap 3 (Leo L. Beranek)                      18
b) Acoustic Mass, MA (kg/m4)

- Associated with an Open Tube of Fluid, the Entrained Fluid will be
Accelerated with Negligible Compression.
- Effective Length “” is Longer than  Because
Particle of Fluid in the Neighborhood of either
End are also Accelerated.
- “p” is the Pressure Difference from End to End.
- “U” is Volume Velocity Owing through the Tube.

 0l
p  jM AU  j            U
S

Acoustics Chap 3 (Leo L. Beranek)                19
c) Acoustic Compliance, CA (m5/N)

- Associated with a Closed Rigid-Wall Container
of Fluid all of whose Dimensions are Small
Compared to “” , the Enclosed Fluid is Compressed
with Negligible Acceleration.
- “p” is the Pressure Only to Compress the Volume “V”
- “U” is the Instantaneous Volume Velocity Flowing
into “V”

U                          V
p               Where C A 
jC A                      0c 2

Acoustics Chap 3 (Leo L. Beranek)   20
d) Acoustic Compliance, RA (Nsec/m5)
- Losses Due to the Viscous Flow of a Fluid through some Materials.

- Acoustic Responsiveness
1
rA 
RA
p  RAU           (Impedance Type)
U  rA p          (Mobility Type)

e) Acoustic Generator
Constant Pressure                     Constant Volume Velocity

Acoustics Chap 3 (Leo L. Beranek)                 21
2) Transducer
 Transformation Type Element

Lorent Force
f  Bli
e  Blv

a
b
b   C:1       d                         c
+                        bc  d
+
a                  b                        a    2 g
-                  -                          c
b      d
Acoustics Chap 3 (Leo L. Beranek)                     22
Acoustics Chap 3 (Leo L. Beranek)   23
Acoustics Chap 3 (Leo L. Beranek)   24
Acoustics Chap 3 (Leo L. Beranek)   25
Acoustics Chap 3 (Leo L. Beranek)   26
Acoustics Chap 3 (Leo L. Beranek)   27
Acoustics Chap 3 (Leo L. Beranek)   28
Acoustics Chap 3 (Leo L. Beranek)   29

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