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Prof. T. L. Wu
Microwave Filter Design
Chp4. Transmission Lines and Components
Prof. Tzong-Lin Wu
Department of Electrical Engineering
National Taiwan University
Prof. T. L. Wu
Microstrip Lines
Microstrip Structure
� Inhomogeneous structure:
Due to the fields within two guided-wave media, the microstrip does not support
a pure TEM wave.
� When the longitudinal components of the fields for the dominant mode of a microstrip
line is much smaller than the transverse components, the quasi-TEM approximation is
applicable to facilitate design.
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Prof. T. L. Wu
Microstrip Lines
- Transmission Line Parameters
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
� For thin conductors (i.e., t → 0), closed-form expression (error ≤ 1 % ):
� W/h ≤ 1:
εre
� W/h ≥ 1:
� For thin conductors (i.e., t → 0), more accurate expressions:
� Effective dielectric constant (error ≤ 0.2 % ): � Characteristic impedance (error ≤ 0.03 % ):
Prof. T. L. Wu
Microstrip Lines
- Transmission Line Parameters
� Guided wavelength
� Propagation constant
� Phase velocity
� Electrical length
0g
re
λλ
ε=
300
( )g
re
mmf GHz
λε
=or
2
g
πβ
λ=
p
re
cωυ
β ε= =
θ β= ℓ
βoZ
,
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Prof. T. L. Wu
Microstrip Lines
- Transmission Line Parameters
� Losses� Conductor loss
� Dielectric loss
� Radiation loss
� Dispersion� εre(f)
� Zo(f)
� Surface Waves and higher-order modes� Coupling between the quasi-TEM mode and surface wave mode become
significant when the frequency is above fs
� Cutoff frequency fc of first higher-order modes in a microstrip
� The operating frequency of a microstrip line < Min (fs, fc)
1tan
2 1
−
=−r
s
r
cf
h
επ ε
( )2 0.8=
+c
r
cf
W hε
Prof. T. L. Wu
Microstrip Lines
- Tx-Line
� Synthesis of transmission line – electrical or physical parameters
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Prof. T. L. Wu
Coupled Lines
Coupled line Structure
� The coupled line structure supports two quasi-TEM modes: odd mode and even mode.
Electrical wallOdd mode
Magnetic Wall
Electric fieldMagnetic field
Even mode
Prof. T. L. Wu
Coupled Lines
– Odd- and Even- Mode
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
� Odd- and Even- Mode:
The characteristic impedances (Zco and Zce) and effective dielectric
constants (εore and εe
re) are obtained from the capacitances (Co and Ce):
� Odd-Mode: � Even-Mode:
Odd mode Even mode
� Caoand Ca
eare even- and odd-mode capacitances for the coupled microstrip
line configuration with air as dielectric.
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Prof. T. L. Wu
Coupled Lines
– Odd- and Even- Mode
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
� Odd- and Even- Mode Capacitances:
� Odd-Mode: � Even-Mode:
Odd mode
Even mode
� Cp denotes the parallel plate capacitance between the strip and the ground plane:
� Cf is the fringe capacitance as if for an uncoupled single microtrip line:
� Cf’ accounts for the modification of fringe capacitance Cf :
� Cgd may be found from the corresponding coupled stripline geometry:
� Cga can be modified from the capacitance of the corresponding coplanar strips:
,
, ,
Prof. T. L. Wu
Discontinuities And Components
– Discontinuities
� Microstrip discontinuities commonly encountered in the layout of practical filters
include steps, open-ends, bends, gaps, and junctions.
� The effects of discontinuities can be accurately modeled by full-wave EM simulator
or closed-form expressions and taken into account in the filter designs.
� Steps in width: � Open ends:
� Gaps: � Bends:
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Prof. T. L. Wu
Discontinuities
– Steps in width
Note : Lwi for i = 1, 2 are the inductances per unit length of the appropriate
micriostrips, having widths W1 and W2, respectively.
Zci and εrei denote the characteristic impedance and effective dielectric
constant corresponding to width Wi, and h is the substrate thickness in
micrometers.
where
Prof. T. L. Wu
Discontinuities
– Open ends
� The fields do not stop abruptly but extend slightly further due to the effect of the
fringing field.
� Closed-form expression:
where
� The accuracy is better than 0.2 % for the range of 0.01 ≤ W/h ≤ 100 and εr ≤ 128
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Prof. T. L. Wu
Discontinuities
– Gaps
where
� The accuracy is within 7 % for 0.5 ≤ W/h ≤ 2 and 2.5 ≤ εr ≤ 15
Prof. T. L. Wu
Discontinuities
– Bends
� The accuracy on the capacitance is quoted as within 5% over the ranges of 2.5 ≤
εr ≤ 15 and 0.1 ≤ W/h ≤ 5.
� The accuracy on the inductance is about 3 % for 0.5 ≤ W/h ≤ 2.
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Prof. T. L. Wu
Components
– lumped inductors and capacitors
� Lumped inductors and capacitors
The elements whose physical dimensions are much smaller than the free
space wavelength λ0 of the highest operating frequency (smaller than 0.1 λ0).
� Design of inductors
� High-impedance line � Meander line
� Circular spiral � Square spiral
� Circuit representation
� Initial design formula for straight-line inductor
Prof. T. L. Wu
Components
– lumped inductors and capacitors
� Design of capacitors
� Interdigital capacitor
Assuming the finger width W equals to the space and empirical formula for capacitance
is shown as follow
� Metal-insulator-metal (MIM) capacitor
Estimation of capacitance and resistance is approximated by parallel-plate
� Circuit representation
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Prof. T. L. Wu
Components
– Quasilumped elements (1)
� Quasilumped elements
Physical lengths are smaller than a quarter of guided wavelength λg.
� High-impedance short line element
gl
λλλλ<<<<8
cos sin
1sin cos
c
c
jZA B
jC DZ
β β
β β
=
ℓ ℓ
ℓ ℓ
( )11 12
21 22
cos 1
sin sin
1 cos1
sin sin
c c
c c
AD BCDjZ jZY Y B B
Y Y A
jZ jZB B
ββ β
ββ β
− − − = = −−
ℓ
ℓ ℓ
ℓ
ℓ ℓ
2 2 22 2
11 12
cos sin cos sin 2sin tancos 1 2 2 2 2 2 2
sin 22sin cos 2sin cos2 2 2 2
c cc c
BY Y j j
jZ ZjZ jZ
β β β β β ββ
β β β ββ
− − + − − + = = = = =
ℓ ℓ ℓ ℓ ℓ ℓ
ℓ
ℓ ℓ ℓ ℓℓ
� Derivation
12
1 1
sinc
YjZ jxβ
− = =ℓ
inductive element:
capacitive element:
Y11+Y12
-Y12
Y22+Y12
Prof. T. L. Wu
Components
– Quasilumped elements (2)
� Quasilumped elements
� Low-impedance short line element
gl
λλλλ<<<<8
cos sin
1sin cos
c
c
jZA B
jC DZ
β β
β β
=
ℓ ℓ
ℓ ℓ
( )11 12
21 22
cos
sin sin
cos1
sin sin
c c
c c
Z ZAD BCAZ Z j jC C
Z Z Z ZD
j jC C
ββ β
ββ β
− = =
ℓ
ℓ ℓ
ℓ
ℓ ℓ
12
1
sin
cZZj jBβ
= =ℓ
� Derivation
11 12
cos 1tan
sin 2 2
cc
Z xZ Z jZ j
j
β ββ−
− = = =ℓ ℓ
ℓinductive element:
capacitive element:
tan2 2
c
xZ
β=
ℓ
sin
c
BZ
β=
ℓ
Zc, β
ι
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Prof. T. L. Wu
Components
– Quasilumped elements (3)
� Quasilumped elements
� Open- and short-circuited stubs
(assuming the length L is smaller than a quarter of guided wavelength λg)
8
gl
λ<
8
gl
λ<
C L
Prof. T. L. Wu
Components
– Resonators
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Prof. T. L. Wu
Loss Considerations for Microstrip Resonators
� Unloaded quality factor Qu is served as a justification for whether or not the required
insertion loss of a bandpass filter can be met.
� The total unloaded quality factor of a resonator can be found by adding conductor,
dielectric, and radiation loss together.
� Quality factors Qc and Qd for a microstrip line
EM simulator
or