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Stepped Impedance Low-Pass Filterekim/e194rfs01/lec19aek.pdf · 2001-04-24 · Stepped Impedance...
Transcript of Stepped Impedance Low-Pass Filterekim/e194rfs01/lec19aek.pdf · 2001-04-24 · Stepped Impedance...
EEE 194 RF 1
Stepped Impedance Low-Pass Filter• Relatively easy (believe that?) low-pass
implementation• Uses alternating very high and very low
characteristic impedance lines• Commonly called Hi-Z, Low-Z Filters• Electrical performance inferior to other
implementations so often used for filtering unwanted out-of-band signals
EEE 194 RF 2
Approximate Equivalent Circuits for Short Transmission line Sections
• Using Table 4-1, approximate equivalent circuits for a short length of transmission line with Hi-Z or Low-Z are found
EEE 194 RF 3
Approximate Equivalent Circuits for Short Transmission line Sections
• The equivalent circuits are:jX / 2 jX / 2
jB
XL=Zo βl
BC=Yo βl
T-Equialent circuit for transmission line sectionβ l << π / 2
Equialent circuit for small β l and large Zo
Equialent circuit for small β l and small Zo
EEE 194 RF 4
Approximate Equivalent Circuits for Short Transmission line Sections
• Series inductors of a low-pass prototype replaced with Hi-Z line sections (Zo= Zh)
• Shunt capacitors replaced with Low-Z line sections (Zo= Zl)
• Ratio Zh/Zl should be as high as possible
( )
( )
inductor
capacitor
g
h
l
g
LRl
Z
CZl
R
β
β
=
=
EEE 194 RF 5
Stepped Impedance Low-Pass Filter• Select the highest and lowest practical line
impedance; e.g. the highest and lowest line impedances could be 150 and 10 Ω, respectively
• For example, given the low-pass filter prototype, solve for the lengths of the microstriplines:
glowLn n Cn n
g high
RZl g ; l g
R Zβ β= =
EEE 194 RF 6
6th Order Low-Pass Filter Prototype
Stepped Impedance Implementation
Microstripline Layout of Filter
L1 L2
C2C1 C3
L3
Zo Zlow Zhigh ZoZlow ZlowZhigh Zhigh
l1 l2 l3 l4 l5 l6
Stepped Impedance Low-Pass Filter -Implementation
EEE 194 RF 7
Bandstop Filter• Require either maximum or minimal
impedance at center frequency fo
• Let line lengths l = λo /4• Let Ω = 1 cut-off frequency of the low-
pass prototype transformed into upper and lower cut-off frequencies of bandstopfilter via bandwidth factor :
( )1
2 2 2U LL
o o
sbwbf cot cot ; sbw
ω ωπ ω πω ω
− = = − =
EEE 194 RF 8
Bandstop Filter: Implementation1. Find the low-pass filter prototype2. The L’s and C’s replaced by open and short
circuit stubs, respectively as in Low-Pass filter design with
ZLn = (bf ) gn and YCn = (bf ) gn
3. Unit lengths of λo /4 are inserted and Kuroda’s Identities are used to convert all series stubs into shunt stubs
4. Denormalize the unit elements
EEE 194 RF 9
Coupled Filters: Bandpass• Even and Odd mode excitations resulting in
1 1Oe Oo
pe e po od
Z ; Zv C v C
= =
EEE 194 RF 10
Coupled Filters: Even & Odd Impedances
EEE 194 RF 11
Bandpass Filter Section
( ) ( ) ( ) ( )2 2 212in Oe Oo Oe OoZ Z Z Z Z cos l
sin lβ
β= − − +
EEE 194 RF 12
Bandpass Filter Section• According to Figure 5-47, the characteristic
bandpass filter performance attained when l = λ /4 or β l = π /2 .
EEE 194 RF 13
Bandpass Filter Section• The upper and lower frequencies are
( ) 11 21 2
Oe Oo,,
Oe Oo
Z Zl cos
Z Zβ θ − −
= = ± +
5th Order coupled line Bandpass Filter
EEE 194 RF 14
Bandpass Filter: Implementation1. Find the low-pass filter prototype2. Identify normalized bandwidth, uper, and lower
frequencies
• Allowing:
U L
O
BWω ω
ω−
=
0 1 1 11 11
1 1 12 22, i,i N ,N
O O O O N Ni i
BW BW BWJ ; J ; J
Z g g Z Z g gg gπ π π
+ +++
= = =
EEE 194 RF 15
Bandpass Filter: Implementation• This allows determination of the odd and
even characteristic line impedances:
• Indices i, i+1 refer to the overlapping elements and ZO is impedance at ends of the filter structure
( )
( )
21 11
21 11
1
and
1
Oo O O i,i O i,ii,i
Oe O O i,i O i,ii,i
Z Z Z J Z J
Z Z Z J Z J
+ ++
+ ++
= − +
= + +
EEE 194 RF 16
Bandpass Filter: Implementation• Determine line dimensions and S and W of
the coupled lines from the graph on Figure 5-45 p256.
• Length of each coupled line segment at the center frequency is λ /4.
• Normalized frequency is
c c
U L c
ω ωωω ω ω ω
Ω = − −