Microwave Filter Design
Chp. 5 Bandpass Filter
Prof. Tzong-Lin Wu
Department of Electrical Engineering
National Taiwan University
Prof. T. L. Wu
End-Coupled, Half-Wavelength Resonator Filters
Each open-end microstrip resonator is approximately a half guided wavelength long at the
midband frequency f0 of the bandpass filter.
Prof. T. L. Wu
End-Coupled, Half-Wavelength Resonator Filters
- Equivalent model
What kind of resonator ? Why?
Prof. T. L. Wu
End-Coupled, Half-Wavelength Resonator Filters
- Design theory
1. Find J-inverter values (half-wavelength resonator)
Zin
ZL
Prof. T. L. Wu
End-Coupled, Half-Wavelength Resonator Filters
- Design theory
Zin
ZL
 ω 
Z L + jZ0 tan  π

1
 ω0 
Zin = Z0
=
ω
 ω 
jY0 tan(π )
Z0 + jZ L tan  π

ω
ω
0
0  ω near ω

0
=
1
jB (ω )
Z L =∞
Susceptance slope parameters
bi =
ω0 dB (ω )
ω
1
π
= 0 Y0π
= Y0
ω0
2 d ω ω =ω
2
2
J 01 =
0
J i ,i +1 =
Y0b1 FBW
Y FBW
π
= 0
Y0
Ω c g 0 g1
Ω c g 0 g1
2
FBW
Ωc
bi bi +1
gi gi +1
= Y0
i ,i +1
= Y0
Ωc =1
π FBW
2
Y b FBW
Y0 FBW
π
J n ,n +1 = 0 n
=
Y0
Ω c g n g n +1
Ω c g n g n +1
2
π FBW
2 g 0 g1
1
gi gi +1
= Y0
Ωc =1
π FBW
2 g n g n +1
2. Find the susceptance of series gap capacitance
Prof. T. L. Wu
End-Coupled, Half-Wavelength Resonator Filters
- Design theory
The physical lengths of resonators
The second term on the righthand side of the 2nd equation indicates the absorption of the
negative electrical lengths of the J-inverters associated with the jth half-wavelength resonator.
Prof. T. L. Wu
Example
a microstrip end-coupled bandpass filter is designed to have a fractional bandwidth FBW =
0.028 or 2.8% at the midband frequency f0 = 6 GHz. A three-pole (n = 3) Chebyshev lowpass
prototype with 0.1 dB passband ripple is chosen.
1.
2.
Prof. T. L. Wu
Example
3. Microstrip implementation
How to determine the physical dimension of the coupling gap?
•
Use the closed-form expressions for microstrip gap given in Chapter 4.
However, the dimensions of the coupling gaps for the filter seem to be outside
the parameter range available for these closed-form expressions. This will be
the case very often when we design this type of microstrip filter.
•
Utilize the EM simulation
Arrows indicate the reference planes for deembedding
to obtain the two-port parameters of the microstrip gap.
Prof. T. L. Wu
Example
by interpolation
Prof. T. L. Wu
Example
Prof. T. L. Wu
Parallel-Coupled, Half-Wavelength Resonator
Filters
They are positioned so that adjacent resonators are parallel to each other along half of their
length.
This parallel arrangement gives relatively large coupling for a given spacing between resonators,
and thus, this filter structure is particularly convenient for constructing filters having a wider
bandwidth as compared to the structure for the endcoupled microstrip filters described in the
last section.
Prof. T. L. Wu
Parallel coupled line filters
−j
( Zoe + Zoo ) cot θ
2
−j
Z12 = Z 21 = Z 34 = Z 43 =
( Zoe − Z oo ) cot θ
2
−j
Z13 = Z 21 = Z 24 = Z 42 =
( Zoe + Zoo ) csc θ
2
−j
Z14 = Z 41 = Z 23 = Z 32 =
( Zoe − Z oo ) csc θ
2
Z11 = Z 22 = Z 33 = Z 44 =
Apply boundary
condition (open-end)
 Z11
Z
 41
V1   Z11
V  =  Z
 4   41
− j
Z + Z oo ) cot θ
Z14   2 ( oe
=

Z 44   − j
( Zoe − Zoo ) csc θ
 2
 Z11

 A B   Z 41
C D  = 
1


Z
 41
1
3
Zoe, Zoo
2
4
Z14   I1 
Z 44   I 4 
−j
( Z oe − Z oo ) csc θ 
2

−j
( Zoe + Zoo ) cot θ 
2

Z   Z oe + Z oo
cos θ

Z 41   Z oe − Z oo
=
Z 44  
j
sin θ
2
Z 41   Z oe − Z oo
θ
( Zoe + Zoo )
θ
1
Zoe, Zoo
4
cot 2 β l − ( Z oe − Z oo ) csc 2 θ 

2 j ( Z oe − Z oo ) csc θ


Z oe + Z oo
cos θ

Z oe − Z oo

2
2
Prof. T. L. Wu
Parallel coupled line filters
θ
θ
1
J
Zo
1
≈
Zoe, Zoo
θ
Zo
4
4
Left
Right
 Z +Z
oo
cos θ
 oe
 A B   Z oe − Z oo
=
C D  
j


sin θ
2
Z
−
 oe Z oo
 cos θ
A B 
=
C D   j 1 sin θ


 Z o
j
( Zoe − Zoo )
csc 2 θ − ( Z oe + Z oo ) cot 2 θ 

2 ( Z oe − Z oo ) csc θ


Z oe + Z oo
cos θ

Z oe − Z oo

jZ o sin θ  
 0
cos θ  
  − jJ
2
1   cos θ
− j 
1
J
  j sin θ
0   Z o
2
jZ o sin θ 

cos θ 



1 
cos2 θ  
j  JZ o2 sin 2 θ −
  JZ o +

 cos θ sin θ
JZ
J  

o 

=
  1

 
1 
 j  2 sin 2 θ − J cos2 θ   JZ o +
 cos θ sin θ 
JZ o 
 
  JZ o

θ ≈ π/2 =>
1
 Z oe + Z oo
 Z − Z = JZ o + JZ
oe
oo
o

 Z oe − Z oo = JZ 2
o

2
=>
2 3
2 2
 Z oe + Z oo = 2 J Z o + 2 Z o = 2 Z o (1 + J Z o )

2
 Z oe − Z oo = 2 JZ o
=>
 Z oe = Z o (1 + JZ o + J 2 Z o2 )


2 2
 Z oo = Z o (1 − JZ o + J Z o )
Prof. T. L. Wu
Parallel-Coupled, Half-Wavelength Resonator Filters
-Theory
Prof. T. L. Wu
Parallel-Coupled, Half-Wavelength Resonator Filters
-Example
Step 1:
Let us consider a design of five-pole (n = 5) microstrip bandpass filter that has a fractional
bandwidth FBW = 0.15 at a midband frequency f0 = 10 GHz. Suppose a Chebyshev prototype
with a 0.1-dB ripple is to be used in the design.
Prof. T. L. Wu
Parallel-Coupled, Half-Wavelength Resonator Filters
-Example
Step 2:
Find the dimensions of coupled microstrip lines that exhibit the desired even- and odd-mode
impedances.
Assume that the microstrip filter is constructed on a substrate with a relative dielectric
constant of 10.2 and thickness of 0.635 mm.
Prof. T. L. Wu
Parallel-Coupled, Half-Wavelength Resonator Filters
-Example
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
They may conceptually be obtained by folding the resonators of parallel-coupled, half-wavelength
resonator filters.
This type of “U” shape resonator is the so-called hairpin resonator.
The same design equations for the parallel-coupled, half-wavelength resonator filters may be used.
However, to fold the resonators, it is necessary to take into account the reduction of the coupled-line
lengths, which reduces the coupling between resonators.
If the two arms of each hairpin resonator are closely spaced, they function as a pair of coupled line
themselves, which can have an effect on the coupling as well.
To design this type of filter more accurately, a design approach employing full-wave EM simulation will
be described.
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
- Example
A microstrip hairpin bandpass filter is designed to have a fractional bandwidth of 20% or FBW
= 0.2 at a midband frequency f0 = 2 GHz. A five-pole (n = 5) Chebyshev lowpass prototype with
a passband ripple of 0.1 dB is chosen.
We use a commercial substrate (RT/D 6006) with a relative dielectric constant of 6.15 and a
thickness of 1.27 mm for microstrip realization.
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
- Example
Using a parameter-extraction technique described in Chapter 8, we then carry out full-wave
EM simulations to extract the external Q and coupling coefficient M against the physical
dimensions.
The hairpin resonators used have a line width of 1 mm and a separation of 2 mm between the
two arms.
Dimension of the resonator as indicated by L is about λg0/4 long and in this case, L =
20.4 mm.
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
- Example
The filter is quite compact, with a substrate size of 31.2mm by 30 mm.
The input and output resonators are slightly shortened to compensate for the effect of the
tapping line and the adjacent coupled resonator.
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
- Example
A design equation is proposed for estimating the tapping point t as
Derivation on external Qe
equivalent
L=
λg
4
Yin = jωC +
t
1
=
jω L
1 

j  ωC −
= jB
ω L 

External quality factor for parallel L/C resonator:
b=
ω0 ∂B
ω 
1  ω
= 0  C + 2  = 0 ( C + C ) = ω0C
ω0 L  2
2 ∂ω ω =ω
2 
ω0C b
0
Q=
G
=
G
Prof. T. L. Wu
Hairpin-Line Bandpass Filters
- Example
Yin
External quality factor for hairpin line:
L=
λg
4
t
{
}
Yin = jYr tan  β ( L − t ) + jYr tan  β ( L + t ) = jYr tan  β ( L − t )  + tan  β ( L + t )  = jB (ω )
b=
ω0 ∂B
ω ∂B ∂β
ω Y
= 0
= 0 r {( L − t ) sec 2  β 0 ( L − t ) + ( L + t ) sec2  β 0 ( L + t )}
2 ∂ω ω =ω
2 ∂β ∂ω β = β
2 up
0
=
1
ωL
π
( L − t ) csc2 ( β 0t ) + ( L + t ) csc 2 ( β 0t ) = Yr 0
=
2
2 up
u p sin ( β 0t ) 2
2
Qe =
b
=
G
Example:
β0 L =
π
2
0
ω0 Yr
π
condition:
Yr
 πt 
sin 2 

π Z0
 2L  =
1
 πt 
2 Z r sin 2 

Z0
 2L 
t=
2L
π
sin −1
Yr
 2π
sin 2 
λ
 g

t


=
π
Yr
2
 πt 
sin 2 

 2L 
π Z0 Z r
2
Qe
the hairpin line is 1.0 mm wide, which results in Zr = 68.3 ohm
on the substrate used. Recall that L = 20.4 mm, Z0 = 50 ohm,
and the required Qe = 5.734. Substituting them into the eq.
yields a t = 6.03 mm, which is close to the t of 7.625 mm found
from the EM simulation above.
Prof. T. L. Wu
Interdigital Filter
- Concept
The filter configuration, as shown, consists of an array of n TEM-mode or quasi-TEM-mode
transmission line resonators, each of which has an electrical length of 90° at the midband frequency
and is short-circuited at one end and open-circuited at the other end with alternative orientation.
Parallel coupled-line BPF
Interdigital line filter with tapped line
Folding λ/2
resonator
Y1 = Yn denotes the single microstrip characteristic
Prof. T. L. Wu
impedance of the input/output resonator.
Interdigital Filter
- Concept
θ
Yoe, Yoo
θ
Yoo − Yoe
2
Yoe
Yoe
θ
θ
BPF with short-circuited stubs
Prof. T. L. Wu
Interdigital Filter
- Design Equations
θ
J
Yu
−Yu
−Yu
θ
 1
 jY
 u
 tan θ
0  cos θ

1 
  jYu sin θ
θ
j sin θ   1
Yu   jYu

cos θ   tan θ

0  cos θ
=
1  jYu
 
 sin θ
j sin θ 
 1
Yu  
 jYu

0   tan θ


0  0
=
1  jYu
 
 sin θ
j sin θ 

Yu   0
=
 
0   jJ

j
J

0
J=
Yu
sin θ
Prof. T. L. Wu
Interdigital Filter
- Design Equations
Transform the lowpass prototype filter to the highpass prototype filter
Highpass
Transformation
BPF Response
J 0,1 =
Y0
Lp1 g 0 g1
J i ,i +1 =
1
Lpi Lp(i +1) gi gi +1
J n ,n +1 =
i =1 to n =1
Yn +1
Lpn g n g n +1
Richard’s Transformation to decide the inductor value at edge frequency
θ
of passband
1
Y=
pLpi
Lpi
Y1
t
t = j tan θ
Y=
p = jω
ω1 ω2
Y1
ω1 =ω0 − ω0
Y=
1
Y1
=
jω1L pi j tan θ
1
Y
= 1
L pi tan θ
FBW
ω
ω  FBW 
=> θ = 1 l = 0 l  1 −

2
2 
up
up 
=
π  FBW 
1 −

2  Prof.
2 T.L. Wu
Interdigital Filter
- Design Equations
J-inverter value for the BPF
J i ,i +1 =
1
L pi L p(i +1) gi gi +1
=
i =1 to n =1
Y1
tan θ gi gi +1
=
i =1 to n =1
Y
gi gi +1
i =1 to n =1
Y1 denotes the characteristic
impedance of the short-circuited stubs.
From Richard’s transformation and previous derivations on interdigital filter
Input part: Y1 = Ya1 + Y12
Internal part: Y1 = Yai + Yi −1,i + Yi ,i +1
Output part: Y1 = Yan + Yn −1,n
Ya1 = Y1 -Y12
Yai = Y1 -Yi −1,i -Yi ,i +1
Yan = Y1 -Yn −1,n
Yai denotes the characteristic
impedance you have to find
from the interdigital filter .
Prof. T. L. Wu
Interdigital Filter
- Design Equations
Self- and mutual capacitances per unit length are used to find the corresponding physical dimensions
Ya1 = Y1 -Y12
Yai = Y1 -Yi −1,i -Yi ,i +1
Yan = Y1 -Yn −1,n
Connecting line Yi,i+1
C1
C
Y -Y
× 1 = u pC1 = Y1 -Y12 => C1 = 1 12
L1
C1
up
Y1 -Yi −1,i -Yi ,i +1
Yai = Y1 -Yi −1,i -Yi ,i +1 => Ci =
up
Y1 -Yn −1,n
Yan = Y1 -Yn −1,n => Cn =
up
Yi ,i +1
Ci ,i +1 =
for i=1 to n-1
up
Ya1 =
Approximate design approach to find a pair of parallel-coupled lines
Prof. T. L. Wu
Interdigital Filter
- Design theory
Prof. T. L. Wu
Interdigital Filter
- Design theory
Full-wave electromagnetic (EM) simulation technique can be employed to extract such even- and odd-mode
impedances for the filter design,
Some commercial EM simulators such as em can automatically extract εre and Zc.
For the even-mode excitation, the average
even-mode impedance is then found by
the average odd-mode impedance is
determined by
Prof. T. L. Wu
Interdigital Filter
- Design Example
the design is worked out using an n = 5 Chebyshev lowpass prototype with a passband ripple 0.1 dB. The bandpass
filter is designed for a fractional bandwidth FBW = 0.5 centered at the midband frequency f0 = 2.0 GHz.
The prototype parameters are
using the design equations above given the table 5.7
A commercial dielectric substrate (RT/D 6006) with a relative dielectric constant of 6.15 and a thickness of
1.27 mm is chosen for the filter design, and table 5.8 is obtained.
Prof. T. L. Wu
Design Example with asymmetric coupled lines
Next, we need to decide the lengths of microstrip interdigital resonators.
unequal effective dielectric constants for the both modes
there is a capacitance Ct that needs to be loaded to the input and output resonators
capacitive loading may be achieved by an open-circuit stub, namely, an extension in length of the
resonators.
Prof. T. L. Wu
Design Example with asymmetric coupled lines
Prof. T. L. Wu
Combline Filters
The combline bandpass filter is comprised of an array of coupled resonators.
The resonators consist of line elements 1 to n, which are short-circuited at one end, with a
lumped capacitance CLi loaded between the other end of each resonator line element and
ground.
Not resonator, but input and output of the filter
through the coupled line elements.
The input and output of the filter are
through coupled-line elements 0 and n + 1, which are not resonators.
The larger the loading capacitances CLi, the shorter
the resonator lines, which results in a more compact
filter structure with a wider stopband between the
first passband (desired) and the second passband
(unwanted).
The minimum resonator line length could be limited
by the decrease of the unloaded quality factor of
resonator and a requirement for heavy capacitive
loading.
resonators
Prof. T. L. Wu
Combline Filters
- Design Concept
Prof. T. L. Wu
Combline Filters
- Design Concept
θ
Y −Y
= υ C ab
2
a
oo
Yoea = υ Ca
θ
a
oe
Yoeb = υ Cb
θ
Prof. T. L. Wu
Combline Filters
- Design Concept
YinL1
1
YinL1 = YA +
Yins1
θ
Y =
θ
Yin1
Y +Y
= YA
2
a
oo

Yoea 
a
 Y + Yoe + j tan θ  + jY tan θ


Y + Yoea + jYA tan θ

1 
Y (1 + j tan θ ) + Yoea  1 +
j tan θ  Y 2
YYoea

=Y
=
+
YA (1 + j tan θ )
YA YA j tan θ
Yooa − Yoea
2
Constraint:
2
θ
Yoeb = υ Cb

Yoea 
 YA + j tan θ  + jY tan θ
Y + jY tan θ

Yins1 = Y
=Y 
=Y
Y + jYinL1 tan θ

Ya 
Y + j  YA + oe  tan θ
j tan θ 

L
in1
Yooa − Yoea
= υ C ab
2
Yoea = υ Ca
Yoea
j tan θ
a
oe
Yin1 =

Y2
YYoea
Yb
Y2
1  YYoea
+
+ oe =
+
+ Yoeb 

YA YA j tan θ j tan θ YA j tan θ  YA

Yin 2 = GT / /Yins 2 =
Yins 2
Yin2
Yin1 = Yin 2
YA
Ys
+
N 2 j tan θ
N=
YA
2YA
=
Y Yooa − Yoea
Ys =

YYoea
YY a
Y 
+ Yoeb = oe + Yoea − Yoea + Yoeb = Yoea  1 +  − Yoea + Yoeb
YA
YA
 YA 
Y +Y
= (Y A − Y )  A
 YA

 YA2 − Y 2 
 N 2 −1
a
b
a
b
a
b
−
Y
+
Y
=
−
Y
+
Y
=
Y
 oe oe 
 oe oe
A
 − Yoe + Yoe
2
Y
N
Prof.
T.
L.
Wu





A
Combline Filters
- Design Concept
a
b
Yoea = υ Ca
Yoeb = υ Cb
Yooa = υ ( Ca + 2Cab )
2YA
N=
υ ( Ca + 2Cab − Ca )
=
YA
υ Cab
Design formula for coupled-line:
 N 2 −1
a
b
Ya1 = YA 
 − Yoe + Yoe + Y12
2
N


  υ C 2 
= YA 1 −  01   + υ ( −C0 + C1 + C12 )
  YA  
Yaj
j =2 to n −1
= Y j + Y j −1, j + Y j , j +1 = υ ( C j + C j −1, j +Prof.
C j , j +T.1 )L. Wu
Combline Filters
- Formulation
A design procedure starts with choosing the resonator susceptance slope parameters bi
(Parallel LC)
Bi (ω ) = ωCLi − Yai cot θ
Q B (ω0 ) = ω0CLi − Yai cot θ 0 ≡ 0
∴ CLi = Yai
cot θ 0
ω0
Prof. T. L. Wu
Combline Filters
- Formulation
Prof. T. L. Wu
Design Procedure (1/3)
Find the gi (i = 1 to n) coefficients for the chosen prototype filter.
Choose the admittance of each resonator, Yai.
Choose the midband electrical length of each resonator, θ0.
Obtain the susceptance slope parameter of each resonator using
Obtain the required lumped capacitance using
Prof. T. L. Wu
Design Procedure (2/3)
Obtain the J-inverter values using the formula
Obtain the required self and mutual capacitance for n + 2 lines using
Prof. T. L. Wu
Design Procedure (3/3)
Find the dimensions of the coupled lines, namely the width and
spacing that gives the required self and mutual capacitance. (similar to
the interdigital BPF)
Prof. T. L. Wu
Combline filters
- Another design approach
Instead of working on the self- and mutual capacitances, an alternative design approach is to
determine the dimensions in terms of another set of design parameters consisting of external
quality factors and coupling coefficients.
Prof. T. L. Wu
Combline filters
- Another design approach
As mentioned above, the combline resonators cannot be λg0/4 long if they are realized with TEM
transmission lines. However, this is not necessarily the case for a microstrip combline filter because the
microstrip is not a pure TEM transmission line.
For demonstration, we also let the microstrip resonators be λg0/4 long and require no capacitive loading for
this filter design.
The microstrip filter is designed on a commercial substrate (RT/D 6010) with a relative dielectric
constant of 10.8 and a thickness of 1.27 mm.
With the EM simulation we are able to work directly on the dimensions of the filters. Therefore, we
first fix a line width W = 0.8 mm for the all line elements except for the terminating lines.
The terminating lines are 1.1mm wide, which matches to 50 ohm terminating impedance.
Prof. T. L. Wu
Combline filters
It is interesting to note that there is an attenuation pole near the high
edge of the passband, resulting in a higher selectivity on that side.
This attenuation pole is likely due to cross couplings between the
nonadjacent resonators.
Prof. T. L. Wu
Pseudocombline Filters
Pseudocombline bandpass filter that is comprised of an array of coupled resonators. The resonators consist
of line elements 1 to n, which are open-circuited at one end, with a lumped capacitance CLi loaded between
the other end of each resonator line element and ground.
Prof. T. L. Wu
Pseudocombline Filters
Similar to the combline filter, if the capacitors were not present, the resonator lines would be a full λg0/2
long at resonance.
The filter structure would have no passband at all when it is realized in stripline, due to a total cancellation
of electric and magnetic couplings. For microstrip realization, this would not be the case
The type of filter in Figure 5.22 can be designed with a set of bandpass design parameters consisting of
external quality factors and coupling coefficients.
Prof. T. L. Wu
Pseudocombline Filters
- Example
Design on commercial substrate (RT/D 6010) with a relative dielectric
constant of 10.8 and a thickness of 1.27 mm. Fix a line width W = 0.8 mm.
The tapped lines are 1.1mm wide, which matches to 50 ohm terminating
impedance.
Using the parameter extraction technique described in Chapter 8, the design curves for external quality
factor and coupling coefficient against spacing s can be obtained
Prof. T. L. Wu
Pseudocombline Filters
- Example
Similar to the combline filter discussed previously, there is an attenuation pole near the high edge of the
passband, resulting in a higher selective on that side.
This attenuation pole is likely to be due to cross couplings between the nonadjacent resonators.
However, the second passband of the pseudocombline filter is centered at about 4 GHz, which is only
twice the midband frequency. This is expected because the λg0/2 resonators are used without any lumped
capacitor loading.
Prof. T. L. Wu
Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
Since quarter-wave short-circuited transmission line stubs look like parallel resonant
circuits, they can be used as the shunt parallel LC resonators for bandpass filters.
Quarter wavelength connecting lines between the stubs will act as admittance
inverters, effectively converting alternate shunt stubs to series resonators.
For a narrow passband bandwidth (small ∆), the response of such a filter using N
stubs is essentially the same as that of a lumped element bandpass filter of order N.
The circuit topology of this filter is convenient in that only shunt stubs are used, but a
disadvantage in practice is that the required characteristic impedances of the stub lines
are often unrealistically low. A similar design employing open-circuited stubs can be
used for bandstop filters
How to design Z0n ?
Prof. T. L. Wu
Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
Note that a given LC resonator has two degrees of freedom: L and C, or equivalently, ω0,
and the slope of the admittance at resonance.
For a stub resonator the corresponding degrees of freedom are the resonant length and
characteristic impedance of the transmission line Z0n .
Prof. T. L. Wu
Bandpass Filters Using Quarter-Wave Coupled
Quarter-Wave Resonators
Prof. T. L. Wu
These two results are exactly equivalent for all frequencies if the following conditions are satisfied:
Prof. T. L. Wu
Prof. T. L. Wu
Prof. T. L. Wu
Prof. T. L. Wu
Prof. T. L. Wu
Stub Bandpass Filters
h is a dimensionless constant which may be assigned to
another value so as to give a convenient admittance level in
the interior of the filter.
Prof. T. L. Wu
Derivation concept (1/6)
Lowpass filter with only one kind of reactive element
Cai can be arbitrary chosen and the
inverters J01 and Jn, n+1 are eliminated!
C
h
C = a = g1
2 2
"
1
h

C1' =  1 −  g1
2

C1" =
gn gn+1 − ( h / 2 ) g1 Rn+ 1
Rn+ 1
g
C n = gn n+ 1 = Cn' + C n"
Rn+ 1
Ca = hg1
C n' =
C1 = g1 = C1' + C1"
J 12 =
Cai is scaled by terminal
impedance YA!
C1Ca
g1 g 2
= YA
hg1
g2
Ca
2
J k ,k +1 2 to n−1 =
Ca
gk gk + 1
= YA
J n − 1, n =
hg1
gk g k + 1
C aC n
g n− 1 g n
= YA
hg1 gn+ 1
gn−1
Prof. T. L. Wu
Derivation concept (2/6)
For interior sections
C1" =
Using image impedance
Definition of image impedance
Ca h
= g1
2 2
C1" =
Ca
2
Ca = hg1
Yimag1
Yimag2
Yimag1 =
1
C1D1
=
Z imag1
A1B1
Yimag 2 =
1
C2 D2
=
Z imag 2
A2 B2
Prof. T. L. Wu
Derivation concept (3/6)
Derivations on the image parameters
Ca
2
Ca
2
Jk, k+1
 A1
C
 1
1
B1  
= ωCa
D1   j
 2
0  0

1 
  jJ k ,k +1
j
 ωCa 
C1D1
Yimag1 =
= J k ,k +1 1 − 

A1B1
 2 J k ,k +1 
θ
 A2
C
 2
Yk ,k +1
s
k ,k +1
Y
s
k ,k +1
θ
Y
θ=
π ω
2 ω0
θ
1
B2  
=
D2   − jYks,k +1 cot θ
−ωCa

0 
2 J k ,k +1
=
2
1 
ωCa )
(
  jJ k ,k +1 − j
J k ,k +1

1  1
J k ,k +1   ωC
a
 j
0   2

0  cos θ
1 
 jYk ,k +1 sin θ
1 
J k ,k +1 

−ωCa 

2 J k ,k +1 
j
2
j

sin θ  
1
Yk ,k +1
  − jYks,k +1 cot θ
cos θ  

Yks,k +1
cos
cos θ
θ
+

Yk ,k +1

=
2
2
Yks,k +1 ) cos2 θ
(
 −2 jY s cos θ + jY
k ,k +1
k ,k +1 sin θ − j

Yk ,k +1 sin θ
sin θ

0
1


Yk ,k +1


s
Yk ,k +1
cos θ +
cos θ 

Yk ,k +1

j
Yk2,k +1 − (Yk ,k +1 + Yks,k +1 ) cos2 θ
C2 D2
=
=
sin θ
A2 B2
sin θ
2
Yimag 2
Prof. T. L. Wu
Derivation concept (4/6)
Design equations for interior sections
Lowpass
Bandpass
(
(ω
)
=ω )=Y
'
1. Yimag1 ω = 0 = Yimag 2 (ω = ω0 )
2
2. Yimag1
'
'
1
imag 2
(ω = ω1 )
1
(
)
'
1. Yimag1 ω = 0 = Yimag 2 (ω = ω0 )
Yk2,k +1 − (Yk ,k +1 + Yks,k +1 ) cos2 θ
 ωCa 
1− 
 =
sin θ
 2 J k ,k +1 
2
J k ,k +1
2. Yimag1 (ω = ω ) = Yimag 2 (ω = ω1 )
'
1
J k ,k +1 = Yk ,k +1
Yk2,k +1 − (Yk ,k +1 + Yks,k +1 ) cos2 θ1
 ω 'C 
J k ,k +1 1 −  1 a  =
sin θ1
 2 J k ,k +1 
2
'




2
2
( J k ,k +1 sin θ1 ) 1 −  2ωJ1Ca   = Yk2,k +1 − (Yk ,k +1 + Yks,k +1 ) cos2 θ1
  k ,k +1  


2
'
2
2
'
2
( J k ,k +1 + Yks,k +1 ) cos2 θ1 = J k2,k +1 (1 − sin 2 θ1 ) +  ω1Ca2sin θ1 


2
Prof. T. L. Wu
Derivation concept (5/6)
(
)
(J
2. Yimag1 ω ' = ω1' = Yimag 2 (ω = ω1 )
C1" =
2
 ω1'Ca sin θ1 
2
2
2
s
+
θ
=
−
θ
+
Y
cos
J
1
sin
)
(
)
1
1
k ,k +1
k ,k +1
k ,k +1


2


'
2
( J k ,k +1 + Yks,k +1 ) = J k2,k +1 +  ω1Ca 2tan θ1 


Ca h
= g1
2 2
2
2
Assume ω1’=1
2
2
 ω 'C tan θ1 
 YAhg1 tan θ1 
2
Yks,k +1 = J k2,k +1 +  1 a
 − J k ,k +1
 − J k ,k +1 = J k ,k +1 + 
2
2




Ca = hg1
2
J
  hg tan θ1 2
= YA  k ,k +1  +  1
 − J k ,k +1 = YA N k ,k +1 − J k ,k +1
2

 YA  
Note: θ1 =
π ω1 π (ω0 − ω0 FBW / 2 ) π  FBW 
=
= 1 −

2 ω0 2
2
2 
ω0
Required stub admittance
Yk = Yks−1,k + Yks,k +1 = (YA N k -1,k − J k -1,k ) + (YA N k ,k +1 − J k ,k +1 )


J
J
=YA  N k -1,k +N k ,k +1 - k -1,k - k ,k +1 
YA
YA 

for k = 2 to n − 1
Prof. T. L. Wu
Derivation concept (6/6)
Design equations for end sections
Yin2
Yin1
Take input end for example
Im {Yin1} Im {Yin 2 }
=
Re {Yin1} Re {Yin 2 }
ω1'C1'
1 / R0
=
Y1' cot θ1
YA
h
Y1' = Y A g0ω1'C1' tan θ1 = YA g0 g1 (1 − ) tan θ
2
Required stub admittance for Y1

h
h
J 
Y1 = Y1' + Y12s = YA g 0 g1 (1 − ) tan θ + Y A N12 − J 12 = Y A g 0 g1 (1 − ) tan θ + YA  N12 − 12 
2
2
YA 

Since h can be assigned to be arbitrary number, the best way to simplify the calculation process is h = 2.
Prof. T. L. Wu
Stub Bandpass Filters
- Example
Design a stub bandpass filter using five-order chebyshev prototype with a passband ripple
LAr = 0.1 dB at f0 = 2 GHz with a fractional bandwidth of 0.5. A 50 ohm terminal impedance
is chosen.
Using previous equations to find the required Yi and Yi, i+1
fabricated on a substrate with a relative dielectric constant of 10.2 and a
thickness of 0.635 mm.
Prof. T. L. Wu
Stub Bandpass Filters
- Example
f0
3f0
2f0
Prof. T. L. Wu
Stub Bandpass Filters without shorting via
Additional passbands in the vincity of f = 0
and f = 2f0
Better passband response with transmission
zeros
If Yia = Yib
at f0/2 and 3f0/2
If Yia ≠ Yib
transmission zeros are controlled
Design equation:
where fzi is the assigned transmission zero at the lower edge of passband
Prof. T. L. Wu
Design Equation
Yin2
Yin1
YL
YL = jYib tan θ
Yi
Yin1 =
j tan θ
Yin 2 = Yia
Equivalence of the two input admittance using assumption of Yib = αiYia
Yin1 = Yin 2
Yi
jY tan θ + jYia tan θ
= Yia ib
j tan θ
Yia − Yib tan 2 θ
j (α i + 1) tan θ
Yi
= Yia
j tan θ
1 − α i tan 2 θ
Yi
jα Y tan θ + jYia tan θ
= Yia i ia
j tan θ
Yia − αiYia tan 2 θ
Yia =
Transmission zero in the λg/2 case
Yin 2 = ∞
Yia − Yib tan 2 θ = 0
YL + jYia tan θ
jY tan θ + jYia tan θ
= Yia ib
Yia + jYL tan θ
Yia − Yib tan 2 θ
Yia − αiYia tan 2 θ = 0
Yi (αi tan 2 θ − 1)
(1 + αi ) tan 2 θ
 2π λ0 
 π fz 
= cot 2
 λ 4 Prof. T. L. 2Wuf 
0 

 g

αi = cot 2 θ = cot 2 
Stub Bandpass Filters without shorting via
- Example
Specifications for the λg/2 stubs are the same as the ones using λg/4 stubs . Choose the fzi = 1
GHz.
Using previous equations to find the required Yia, Yib and Yi, i+1
fabricated on a substrate with a relative dielectric constant of 10.2 and a
thickness of 0.635 mm.
Prof. T. L. Wu
Stub Bandpass Filters without shorting via
- Example
f0
0
2f0
Open-end effect
f0/2
3f0/2
Prof. T. L. Wu
HW III
Please design a BPF based on Chebyshev prototype with following specifications using endcoupled and parallel-coupled half-wavelength resonators:
1.
Passband ripple < 0.1 dB
2.
Center frequency : 5 GHz
3.
FBW : 0.1
4.
Insertion loss > 20 dB at 4 GHz and 6 GHz.
The properties of the substrate is εr = 4.4 and loss tangent of 0. The substrate thickness is 1.6 mm.
a.
b.
c.
d.
e.
Please decide the order of the BPF and calculate the J-inverter values.
Find out the required series capacitances for end-coupled resonator case and the even- and
odd-mode characteristic impedances of the coupled-line for parallel-coupled resonator case.
Plot the return loss and insertion loss for the designed BPF using HFSS and ADS.
Considering the discontinuities in the two cases, and using HFSS and ADS again to compare
the results with those in (c). Please also discuss the reasons for the discrepancy between them.
Discuss the sizes and frequency responses using the two methods.
Prof. T. L. Wu
HW IV
1. According to the specification on HW III, design a interdigital filter. Let Y1 = 1/49.74 mho.
a. Calculate the per unit length self- and mutual-capacitances.
b. Obtain the required even- and odd-mode impedance.
c. Using EM solver to find corresponding dimensions based on problem (b).
d. Considering the discontinuities and using EM solver to plot the return loss and insertion
loss.
e. Discuss the sizes and frequency responses for the three methods. (interdigital filter, endcoupled, and parallel-coupled resonator)
2. Please derive the Ys and turns ratio N for the equivalence of the short-circuted parallel
coupled-line and a transformer with short-circuited stub in combline filter.
θ
Y −Y
= υ Cab
2
a
oo
Yoea = υ Ca
θ
a
oe
Yoeb = υ Cb
θ
Prof. T. L. Wu