joshua ottosen presentation - computer action...
TRANSCRIPT
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Periodic Structuresand
Filter Design by the Image
Parameter Method
ECE531: Microwave Circuit Design I
Pozar Chapter 8, Sections 8.1 & 8.2
Josh Ottosen
2/24/2011
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Microwave Filters(Chapter Eight)
• “A microwave filter is a two-port network used to
control the frequency response at a certain point
in a microwave system by providing
transmission at frequencies within the passband
of the filter and attenuation in the stopband of
the filter. Typical frequency responses include
low-pass, high-pass, bandpass, and band-reject
characteristics. Applications can be found in
virtually any type of microwave communication,
radar, or test and measurement system.”
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Periodic Structures(Section 8.1)
“An infinite transmission line or waveguide
periodically loaded with reactive elements
is referred to as a periodic structure...
Periodic structures can take various forms,
depending on the transmission line media
being used. Often the loading elements are
formed as discontinuities in the line, but in
any case they can be modeled as lumped
reactances across a transmission line...
Periodic structures… have passband and
stopband characteristics similar to those of
filters; they find applications in traveling-
wave tubes, masers, phase shifters, and
antennas.”
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Unit cell“Each unit cell of this line consists of a
length d of transmission line with a
shunt susceptance across the
midpoint of the line; the susceptance b
is normalized to the characteristic
impedance, Zo. If we consider the
infinite line as being composed of a
cascade of identical two-port networks,
we can relate the voltages and
currents on either side of the nth unit
cell using the ABCD matrix:”
=
+
+
1n
1n
n
n
I
V
I
V
DC
BA
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Table 4.1 (or inside
cover of Pozar)
If a network is reciprocal,
AD – BC = 1
Reciprocal Networks: “A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port
1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. Exchanging voltage and
current results in an equivalent definition of reciprocity. In general, a network will be reciprocal if it consists entirely of
linear passive components (that is, resistors, capacitors and inductors). In general, it will not be reciprocal if it contains
active components such as generators.”
p.311 , Mahmood Nahvi, Joseph Edminister, Schaum's outline of theory and problems of electric circuits, McGraw-Hill
Professional, 2002
*
*
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221
221
IVI
IVV
DC
BA
+=
+=
=
+
+
1n
1n
n
n
I
V
I
V
DC
BA
(note direction of I2)
⇒(8.1)
:
/
examplerefresher
tionmultiplica
matrix
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=
2cos
2sin
2sin
2cos
1
01
2cos
2sin
2sin
2cos
θθ
θθ
θθ
θθ
j
j
jbj
j
DC
BA
22
θβ =⋅=⋅ l
dk
22
θβ =⋅=⋅ l
dk
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=
2cos
2sin
2sin
2cos
1
01
2cos
2sin
2sin
2cos
θθ
θθ
θθ
θθ
j
j
jbj
j
DC
BA
−++
−+−=
)sin2
(cos)2
cos2
(sin
)2
cos2
(sin)sin2
(cos
θθθθ
θθθθ
bbbj
bbj
b
DC
BA
c
(8.2) 9
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( ) ( )( ) ( ) z
z
eIzI
eVzV
γ
γ
−
−
=
=
0
0For a wave propagating in the +z direction,
Since the structure is infinitely long, the voltage and current at the nth
terminals can differ from the voltage and current at the n+1 terminals only
by the propagating factor, de γ−
d
nn
d
nn
eII
eVV
γ
γ
−+
−+
=
=
1
1⇒
=
=
+
+
+
+
d
n
d
n
n
n
n
n
eI
eV
I
V
DC
BA
I
Vγ
γ
1
1
1
1
(8.3)
(8.4)
01
1 =
−
−
+
+d
n
d
n
d
d
eI
eV
eDC
BeAγ
γ
γ
γ
From (8.1),
⇒
For a nontrivial solution, the determinant of the above matrix must vanish:
0)(2 =−+−+ BCeDAeAD dd γγ
Since AD – BC =1,
0)(1 2 =+−+ dd eDAe γγ
(8.5)
(8.6)
=
+
+
+
+
+
+
+
d
n
d
n
n
n
n
n
eI
eV
DI
BI
CV
AVγ
γ
1
1
1
1
1
1⇒
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0)(1 2 =+−+ dd eDAe γγ
)( DAee dd +=+− γγ
θθγγγ
sin2
cos2
)(
2cosh
bDAeed
dd
−=+
=+
=−
βαγ j+=&
θθβαβαγ sin2
cossinsinhcoscoshcoshb
ddjddd −=+=⇒
From (8.2),
(8.7)
(8.8)
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2cosh
dd eed
γγ
γ+
=−
Hyperbolic Function Refresher
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θθβαβαγ sin2
cossinsinhcoscoshcoshb
ddjddd −=+=
Since the right-hand side of (8.8) is purely real, or0=α 0=β
Case#1: Propagating, Non-Attenuating => PASSBAND
θθβ sin2
coscosb
d −=
Case#2: Attenuating, Non-Propagating => STOPBAND
0
0
≠
=
βα
1sin2
coscosh ≥−= θθαb
d
πβα
,0
0
=
≠
⇒ Depending on frequency and normalized susceptance, the periodically
loaded line will exhibit either passbands or stopbands and therefore act as
a filter.
Remember that the equations are for V & I waves defined at terminals of
unit cells and don’t necessarily describe conditions at other points along
the line. These are similar to Bloch waves.
(8.9b)
(8.9a)
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Bloch gives his name to the characteristic impedance of these waves:
1
10
+
+⋅=n
nB
I
VZZ
( ) 011 =+− ++ nn
d BIVeA γFrom (8.5),
dBeA
BZZ γ−
−⋅=⇒ 0
From (8.6),
( ) 42
2
2
0
−+−−
−=⇒ ±
DADAA
BZZ B
m
( ) ( )2
42 −+±+
=DADA
e dγ
0)(1 2 =+−+ dd eDAe γγSo we can solve for
deγ
⇒
So we can solve for the two solutions of the Bloch impedance:
12
0
−
±=⇒ ±
A
BZZ BSince the unit cell is symmetrical, A=D
(8.10)
(8.11)
(8.12)
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If (passband), then, for symmetrical networks: 0,0 ≠= βα
1sin2
coscos ≤=−= Ab
d θθβ
+−= θθ cos22
sinbb
jB
θθβαβαγ sin2
cossinsinhcoscoshcoshb
ddjddd −=+=
θθγ sin2
cos2
)(cosh
bA
DAd −==
+=
From (8.2) we see that B is always purely imaginary.
12
0
−
±=±
A
BZZ B(8.12) shows that ZB will be real.
If (stopband), then, for symmetrical networks: 0,0 =≠ βα
1sin2
coscoshcosh ≥=−== Ab
dd θθαγ
12
0
−
±=±
A
BZZ B(8.12) shows that ZB will be imaginary.
This situation is similar to that for the wave impedance of a
waveguide, which is real for propagating modes and imaginary for
cutoff, or evanescent, modes.
⇒
⇒
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We earlier assumed that the structure was infinitely long, but to implement
this filter we will need to terminate the line. If the load impedance doesn’t
match our Bloch impedance, there will be reflections, which will invalidate
our earlier work.
ndjndj
n
ndjndj
n
eIeII
eVeVV
ββ
ββ
−−+
−−+
+=
+=
00
00
(8.4)
BL
BL
ZZ
ZZ
+−
=Γ
d
nn
d
nn
eII
eVV
γ
γ
−+
−+
=
=
1
1
⇒
To avoid reflections, ZL must match ZB, which is real for a lossless structure
operating in a passband. If necessary, a quarter-wave transformer can be
used between the periodically loaded line and the load.
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ββω k
cvp ==
ββω
d
dkc
d
dvg ==
(Brillouin diagram)
diagramsk −β
ckk −= 2β
ckk < βFor , there is no solution for
(Waveguide)
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θθβ sin2
coscosb
d −= (8.9)
diagramk −β (Periodically Loaded Line Example)
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Figure 8.7 shows an arbitrary, reciprocal two-port
network with image impedances defined as follows:
Zi1 = input impedance at 1 when 2 is terminated with Zi2
Zi2 = input impedance at 2 when 1 is terminated with Zi1
DCZ
BAZ
DICV
BIAV
I
VZ
i
iin +
+=
++
==2
2
22
22
1
11
221
221
IVI
IVV
DC
BA
+=
+=
112
112
IVI
IVV
AC
BD
+−=
−=
Since AD – BC = 1
ACZ
BDZ
AICV
BIDV
I
VZ
i
iin +
+=
+−−
=−
=1
1
11
11
2
22
11 iin ZZ =
22 iin ZZ = CD
ABZi =1
AC
BDZi =2We want ⇒ and
Image Parameter Method of Filter Design(Section 8.2)
21 ii ZZ =⇒If symmetric, A=D
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1
1
112 VIVV
−=−=
iZ
BDBD
( )BCADA
D
AB
CDBD
Z
BD
i
−=−=
−=
11
2
V
V
( )BCADD
AACZA
I
VC i −=+−=+−= 1
1
1
1
2
I
I
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Two important types of two-port
networks are the T and π
circuits, which can be made in
symmetric form. Table 8.1 list
the image impedances and
propagation factors, along with
other useful parameters, for
these two networks.
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122
12
2
22
2
−+−=ccc
eωω
ωω
ωωγ
41
1,
2
21
LC
C
LZ
CjZLjZ iT
ωω
ω −=⇒==
kC
LR
LCc === 0,
2ω
2
2
0 1c
iT RZωω
−=⇒
0RZ iT =0=ω ⇒when
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kC
LR
LCc === 0,2
1ω
kC
LR
LCc === 0,
2ω
There are only two parameters to choose (L and C), which are deteremined by the cutoff
frequency and the image impedance at zero frequency.
These results are only valid when the filter section is terminated in its image impedance,
which is a function of frequency and is not likely to mach a given source or load impedance.
Its attenuation isn’t very good in the stopband.
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To improve our design from the constant-k filter, we
are going to try the m-derived filter.
Replace Z1 with Z’1 and Z2 with Z’2 where Z’1=mZ1
Choose Z’2 to keep ZiT the same:
4'
4
'''
4
2
121
2
121
2
121
mZZmZ
ZZZ
ZZZZ iT +=+=+=
( )1
2
21122
4
1
44' Z
m
m
m
ZmZ
m
Z
m
ZZ
−+=−+=
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CjZLjZ
ωω
1, 21 ==
For a low-pass filter,
( )Lj
m
m
CmjZLmjZ ω
ωω
4
11','
2
21
−+==
So the m-derived impedances will be:
+++=
2
1
2
1
2
1
'4
'1
'
'
'2
'1
Z
Z
Z
Z
Z
Zeγ
( ) ( )2
2
2
22
1
11
2
411'
'
−−
−
=
−+
=
c
c
m
m
mmLj
Cmj
Lmj
Z
Z
ωω
ωω
ωω
ωLC
c
2=ω
( )2
2
2
2
1
11
1
'4
'1
−−
−
=+⇒
c
c
mZ
Z
ωω
ωω
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+++=
2
1
2
1
2
1
'4
'1
'
'
'2
'1
Z
Z
Z
Z
Z
Zeγ
( )2
2
2
2
1
11
1
'4
'1
−−
−
=+
c
c
mZ
Z
ωω
ωω
If we restrict 0 < m < 1, then these results show that is real and >1 for >
Thus the stopband begins at , as for the constant-k section.
However, when , where ,
ωcωω =
∞=ωω21 m
c
−=∞
ωω
γe γe
γe
cω
becomes infinite.
The m-derived section has a very
sharp cutoff but then the attenuation
decreases as
To have infinite attenuation as
we can cascade it with a constant-k
section.
∞→ωω∞→ωω
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The m-derived T-section was designed so
that its image impedance was identical to
that of the constant-k section (independent
of m), so we still have the problem of a
nonconstant image impedance. But a π-
section’s image impedance does depend on
m. By adjusting m as needed, we can use
this to optimize our match.
( )0
2
2
2
1
11
R
m
Z
c
c
i
−
−−
=
ωω
ωω
π
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To benefit from the π-section’s ability to
keep a relatively constant image impedance
but still match up with constant-k or sharp
cutoff T-section, we will bisect a π-section.
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Backup Slides
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=
2cos
2sin
2sin
2cos
1
01
2cos
2sin
2sin
2cos
θθ
θθ
θθ
θθ
j
j
jbj
j
DC
BA
+
=
2cos
2sin
2sin
2cos
2cos
2cos
2sin
2sin
02
sin
02
cos 2
θθ
θθ
θθ
θθ
θ
θ
j
j
jb
jbj
jDC
BA
+
−=
2cos
2sin
2sin
2cos
2cos
2cos
2sin
2sin
2sin
2cos
θθ
θθ
θθθ
θθθ
j
j
bj
jb
DC
BA
+
+
+
−−=
2cos
2sin2
cos
2sin2
cos2
sin
2sin
2sin2
cos2
cos2
sin2
cos
2sin
2sin2
cos2
sin2
cos2
cos
2
22
22
22
θθθ
θθθ
θθθθθθ
θθθθθθ
j
jj
jjbjbj
jbjb
DC
BA
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−++
−+−=
)sin2
(cos)2
cos2
(sin
)2
cos2
(sin)sin2
(cos
θθθθ
θθθθ
bbbj
bbj
b
DC
BA
+
+
+
−−=
2cos
2sin2
cos
2sin2
cos2
sin
2sin
2sin2
cos2
cos2
sin2
cos
2sin
2sin2
cos2
sin2
cos2
cos
2
22
22
22
θθθ
θθθ
θθθθθθ
θθθθθθ
j
jj
jjbjbj
jbjb
DC
BA
θθθθθθθθθθθθ
sin2
cossin22
sin212
sin2
cos222
sin2
cos2
sin2
sin2
cos2
cos 222222 bbbjbA −=
−
−=
−
−=
+
−=⇒
+−=
−−=
+
−=⇒ θθθθθθθθθθ
cos22
sin2
cos1
2sin2
cos22
sin2
cos2
sin2
sin2
cos 2 bbjbjjjbjB
++=
++=+
+=⇒ θθθθθθθθθθ
cos22
sin2
cos1
2sin2
cos22
sin2
cos2
cos2
sin2
cos 2 bbjbjjbjC
θθθθθθθθθθθθθ
sin2
cos2
sin2
cos222
sin212
sin2
cos2
sin2
cos2
cos2
sin2
sin2
cos 22222 bbbjjbjD −=
−
−=
−
−=+
+=⇒