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.
Prof. T. L. Wu
Microstrip Lines
- Transmission Line Parameters
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
εre
For thin conductors (i.e., t → 0), closed-form expression (error ≤ 1 % ):
W/h ≤ 1:
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
λ
λg = 0
ε re
Propagation constant
β=
λg =
300
mm
f (GHz ) ε re
2π
λg
Phase velocity
υp =
or
ω
c
=
β
ε re
Zo , β
Electrical length
θ = βℓ
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
c tan −1 ε r
fs =
2π h ε r − 1
Cutoff frequency fc of first higher-order modes in a microstrip
fc =
c
ε r ( 2W + 0.8h )
The operating frequency of a microstrip line < Min (fs, fc)
Prof. T. L. Wu
Microstrip Lines
- Tx-Line
Synthesis of transmission line – electrical or physical parameters
Prof. T. L. Wu
Coupled Lines
Coupled line Structure
The coupled line structure supports two quasi-TEM modes: odd mode and even mode.
Odd mode
Electrical wall
Even mode
Magnetic Wall
Electric field
Magnetic field
Prof. T. L. Wu
Coupled Lines
– Odd- and Even- Mode
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
Odd mode
Even mode
Odd- and Even- Mode:
The characteristic impedances (Zco and Zce) and effective dielectric
constants (εore and εere) are obtained from the capacitances (Co and Ce):
Odd-Mode:
Even-Mode:
Cao and Cae are even- and odd-mode capacitances for the coupled microstrip
line configuration with air as dielectric.
Prof. T. L. Wu
Coupled Lines
– Odd- and Even- Mode
Odd mode
Effective Dielectric Constant (εre) and Characteristic Impedance(ZC)
Odd- and Even- Mode Capacitances:
Even mode
Even-Mode:
Odd-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:
Prof. T. L. Wu
Discontinuities
– Steps in width
where
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.
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
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.
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
Circuit representation
Circular spiral
Square spiral
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
Prof. T. L. Wu
Components
– Quasilumped elements (1)
Quasilumped elements
Physical lengths are smaller than a quarter of guided wavelength λg.
l<
High-impedance short line element
Derivation
8
-Y12
Y11+Y12
D
Y11 Y12   B
Y Y  = 
 21 22   −1
 B
 cos β ℓ
A B 
=
C D   j 1 sin β ℓ


 Z c
inductive element:
λg
jZ c sin β ℓ 

cos β ℓ 

1
1
−Y12 =
=
jZ c sin β ℓ jx
cos β ℓ − 1
capacitive element: Y11 + Y12 =
=
jZ c sin β ℓ
cos2
βℓ
2
− sin
2
Y22+Y12
− ( AD − BC )   cos β ℓ
  jZ c sin β ℓ
B
=
−1
A
 
  jZ c sin β ℓ
B
βℓ  2 βℓ
βℓ 
−  cos
+ sin

2
2
2
2

jZ c 2 sin
βℓ
2
cos
βℓ
−2 sin
=
jZ c 2 sin
2
2
βℓ
2
βℓ
−1

jZ c sin β ℓ 

cos β ℓ 
jZ c sin β ℓ 
tan
2
βℓ
2
Components
– Quasilumped elements (2)
Quasilumped elements
l<
Low-impedance short line element
Derivation
λg
8
ι
Zc, β
 cos β ℓ
A B 
C D  =  j 1 sin β ℓ


 Z c
jZ c sin β ℓ 

cos β ℓ 

capacitive element: Z12 =
 Z11

 Z 21
Zc
1
=
j sin β ℓ jB
inductive element: Z11 − Z12 =
B=
A
Z12   C
=
Z 22   1
 C
 Z c cos β ℓ
  j sin β ℓ
=
  Zc
  j sin β ℓ
( AD − BC ) 
C
D
C
Zc

j sin β ℓ 

Z c cos β ℓ 
j sin β ℓ 
sin β ℓ
Zc
Z c cos β ℓ − 1
βℓ
x
= jZ c tan
= j
j sin β ℓ
2
2
B
2 = j
= j
β ℓ Prof. T.Z cL. Wu 2
cos
x
βℓ
= Z c tan
2
2
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)
C
l<
λg
8
L
l<
λg
8
Prof. T. L. Wu
Components
– Resonators
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.
EM simulator
Quality factors Qc and Qd for a microstrip line
or
Prof. T. L. Wu