Linear Electrical and Acoustic Systems
– Some Basic Concepts
Learning Objectives
Two port systems
transfer matrices
impedance matrices
reciprocity
1-D compressional waves in a solid
equation of motion/constitutive equation
acoustic transfer matrix of a layer
Single Input-Single Output systems
LTI systems
impulse response, transfer functions
convolution/ deconvolution
Wiener filter
Two Port Systems
Consider our simple system again:
R
Vi
C
If we take off the voltage
source we are left with:
R
C
This is an example of a
two port system:
V1
I1
I2
V2
I1
V1
Transfer matrix:
I2
[T ]
V2
⎧V1 ⎫ ⎡T11 T12 ⎤ ⎧V2 ⎫
⎨ ⎬=⎢
⎨ ⎬
⎥
⎩ I1 ⎭ ⎣T21 T22 ⎦ ⎩ I 2 ⎭
Alternately, we can express this two port system
in terms of an impedance matrix:
I1
V1
I'2 = - I2
[Z ]
V2
⎧V1 ⎫ ⎡ Z11
⎨ ⎬=⎢
⎩V2 ⎭ ⎣ Z 21
Z12 ⎤ ⎧ I1 ⎫
⎨ ⎬
⎥
Z 22 ⎦ ⎩ I 2′ ⎭
I1(1)
A
State (1):
(1)
I2
[T ]
(1)
V1
(1)
V2
(2)
I1
I2 (2)
C
State (2):
[T ]
(2)
V1
B
(2)
D
V2
Reciprocity
(1) ( 2 )
( 2 ) (1)
(1)
V1 I1 − V1 I1 = V2 I 2
( 2)
( 2)
− V2 I 2
(1)
For reciprocal systems, the impedance matrix
is symmetric, i.e.
and the determinant of the
transfer matrix is equal to one
Z 21 = Z12
⎧V1 ⎫ ⎡ Z11
⎨ ⎬=⎢
⎩V2 ⎭ ⎣ Z 21
I1
V1
det [T ] = T11T22 − T12T21 = 1
⎧V1 ⎫ ⎡T11 T12 ⎤ ⎧V2 ⎫
⎨ ⎬=⎢
⎥ ⎨I ⎬
I
T
T
⎩ 1 ⎭ ⎣ 21 22 ⎦ ⎩ 2 ⎭
Z12 ⎤ ⎧ I1 ⎫
⎨ ⎬
⎥
Z 22 ⎦ ⎩ I 2′ ⎭
I1
I'2
[Z ]
V2
V1
I2
[T ]
V2
Relationship between transfer matrix components and
impedance matrix components (reciprocal system)
Z11
, T12 =
T11 =
Z12
2
Z
Z
Z
−
( 11 22 12 )
Z12
1
Z 22
, T22 =
T21 =
Z12
Z12
Note:
T12 ≠ T21
1-D plane compressional wave in an elastic solid
dy
∂σ
σ x + x dx
∂x
dz
σx
dx
σ x …stress
u x … displacement
∂σ x ⎞
∂ 2u x
⎛
dx ⎟ dydz − σ x dydz = ρ dxdydz 2
⎜σ x +
∂x
∂t
⎝
⎠
∂σ x
∂ 2u x
=ρ 2
∂x
∂t
compressional (P)
wave speed
Constitutive equation
E (1 − ν ) ∂u x
σx =
(1 + ν )(1 − 2ν ) ∂x
∂u x
= ρc
∂x
2
P
E (1 −ν )
cP =
(1 + ν )(1 − 2ν ) ρ
∂ 2u x 1 ∂ 2u x
− 2
=0
2
2
∂x
cP ∂t
displacement
u x ( x ) = A exp [ik P x − iω t ] + B exp [ −ik P x − iω t ]
velocity v x = −iω A exp ik x − iω t − iω B exp −ik x − iω t
[ P
]
[ P
]
x( )
vx = −iω u x
σ x ( x ) = iωρ cP A exp [ik P x − iω t ] − iωρ cP B exp [ −ik P x − iω t ]
stress
A
waves in a
solid layer
B
compressive force
x=0
x= l
l
Fx = −σ x S
⎧⎪ Fx ( 0 ) ⎫⎪ ⎡ cos ( k P l )
⎨
⎬=⎢
a
v
0
i
sin
k
l
/
Z
−
(
)
(
)
⎪⎩ x
⎪⎭ ⎣
0
P
−iZ 0a sin ( k P l ) ⎤ ⎧⎪ Fx ( l ) ⎫⎪
⎬
⎥⎨
cos ( k P l ) ⎦ ⎪⎩ vx ( l ) ⎪⎭
acoustic
impedance of
the layer
Z 0a = ρ cP S
Equivalent transfer matrices
I1
V1
I2
[T1]
[T2]
I1
V1
[TN]
I2
[Tg]
V2
⎡⎣Tg ⎤⎦ = [T1 ][T2 ] [TN ]
det ⎡⎣Tg ⎤⎦ = det [T1 ] det [T2 ] det [TN ] = 1
V2
When we specify termination conditions at both
ports of a two port system, we end up with a system
where single inputs and outputs are related:
(terminated with
voltage source)
Vi
Vi
I = 0 (open circiut
termination)
R
C
V0
V0
R
Vi
C
V0
Vi ( t ) − V0 ( t ) = i ( t ) R
dV0 ( t )
i (t ) = C
dt
so
dV0 ( t ) V0 ( t ) Vi ( t )
+
=
dt
RC
RC
Note: V0(t) is defined here in terms of Vi(t) only
implicitly as the solution of this differential equation, i.e.
V0 ( t ) = L ⎡⎣Vi ( t ) ⎤⎦
L … linear operator
An important class of these single input-output systems
is a linear time-shift invariant (LTI) system
i(t)
o(t)
L
o1 ( t ) = L ⎡⎣i1 ( t ) ⎤⎦
if
o2 ( t ) = L ⎡⎣i2 ( t ) ⎤⎦
then
if
o ( t ) = L ⎡⎣i ( t ) ⎤⎦
then
o ( t ) = L ⎡⎣ a1i1 ( t ) + a2i2 ( t ) ⎤⎦
o ( t − t0 ) = L ⎡⎣i ( t − t0 ) ⎤⎦
= a1 L ⎡⎣i1 ( t ) ⎤⎦ + a2 L ⎡⎣i2 ( t ) ⎤⎦
time-shift invariance
linearity
Impulse response of LTI systems and the convolution
integral
delta δ(t)
function
g(t)
L
i(t)
impulse response
function
o(t)
L
o (t ) =
+∞
∫ i (τ )g ( t − τ ) dτ
−∞
+∞
=
∫ g (τ ) i ( t − τ ) dτ
−∞
convolution of g and i
i(t)
≅ i (τ ) Δτ δ ( t − τ )
i(τ)
Δτ
t
τ
Δo ( t ) ≅ i (τ ) Δτ g ( t − τ )
linearity and time
shift invariance
o ( t ) ≅ ∑ i (τ ) Δτ g ( t − τ )
+∞
=
∫ i (τ )g ( t − τ ) dτ
−∞
linearity
(additivity)
If
I (ω ) =
+∞
∫ i ( t ) exp ( iω t ) dt
−∞
O (ω ) =
+∞
∫ o ( t ) exp ( iω t ) dt
−∞
G (ω ) =
+∞
∫ g ( t ) exp ( iω t ) dt
−∞
and
o (t ) =
+∞
∫ i (τ )g ( t − τ ) dτ
−∞
then
O (ω ) = G (ω ) I (ω )
o (t ) =
+∞
∫ i (τ )g ( t − τ ) dτ
−∞
+∞
=
∫ g (τ ) i ( t − τ ) dτ
−∞
O (ω ) = G (ω ) I (ω )
convolution in the frequency domain is just
(complex-valued) multiplication
I(ω)
G1(ω)
G2(ω)
GN(ω)
O(ω)
O (ω ) = G1 (ω ) G2 (ω ) ⋅⋅⋅ GN (ω ) I (ω )
The frequency components of the impulse response function
of an LTI system are also called the transfer function, t(ω), for the
system since this function "transfers" the inputs to the outputs:
I(ω)
t(ω)
O (ω )
t (ω ) =
I (ω )
O(ω)
input voltage
Vi(ω)
flaw signal
VR(ω) output voltage
Pulser
Receiver
cabling
cabling
Transducer
(transmitter)
Ft(ω)
output force
FB(ω)
flaw
Transducer
(receiving)
force on receiver
VR (ω ) FB (ω ) Ft (ω )
VR (ω ) =
Vi (ω )
FB (ω ) Ft (ω ) Vi (ω )
= t R (ω ) t A (ω ) tG (ω ) Vi (ω )
Deconvolution
deconvolution in the frequency domain is just
(complex-valued) division … but it must be
done with care
O (ω )
G (ω ) =
I (ω )
Generally, a Wiener filter is used in ultrasonics applications
to desensitize the division process to noise
G (ω ) =
O (ω ) I * (ω )
{
I (ω ) + ε max I (ω )
2
2
small "noise" constant
2
}
( )*= complex
conjugate
I (ω ) = I (ω ) I * (ω )
2
It is easier to see the Wiener filter if we rewrite the
deconvolution in the form
I (ω )
O (ω )
G (ω ) =
I (ω ) I (ω ) 2 + ε 2 max I (ω ) 2
2
{
}
O (ω )
W (ω )
=
I (ω )
where
W (ω ) =
I (ω )
2
{
I (ω ) + ε max I (ω )
Wiener filter
2
2
2
}
MATLAB example showing effects of choice of ε
5
4.5
4
3.5
3
I
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
7
8
9
10
1
0.9
0.8
ε =. 1
0.7
ε =.01
0.6
W
W
>> f = linspace(0, 10, 200);
>> I= f.*(f<5) +(10-f).*(f>5);
>> plot(f, I)
>> e =.01;
>> W = I.^2./(I.^2 + e^2*max(I.^2));
>> plot(f, W)
>> hold on
>> e= .1;
>> W = I.^2./(I.^2 + e^2*max(I.^2));
>> plot(f, W,'--')
>> xlabel(' frequency, f')
>> ylabel(' W')
>> hold off
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
frequency, f
f