Prof. Tzong-Lin Wu Department of Electrical Engineering...

Prof. T. L. Wu

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


- Equivalent model

What kind of resonator ? Why?

Prof. T. L. Wu


- Design theory

1. Find J-inverter values (half-wavelength resonator)

ZLZin

Prof. T. L. Wu


- Design theory

2. Find the susceptance of series gap capacitance

( )

0

0

0

0

000 =0

Z Z tan1 1

Z Z

tan( )Z Z tanL

L

in

L

Znear

j

jBjYj

ω ω

ωπω

ω ωω ππ ωω ∞

+

= = =

+

ZLZin

Susceptance slope parameters

( )

0

0 00 0

0

1b Y Y

2 2 2i

dB

dω ω

ωω ω ππ

ω ω=

= = = 0 1 001 0 0

0 1 0 1 0 11

J Y Y2 2

cc c

Y b FBW Y FBW FBW

g g g g g g

π π

Ω =

= = =Ω Ω

1, 1 0

1 1, 1

1J Y

2

i ii i

c i i i ii i

FBW b b FBW

g g g g

π++

+ ++

= =Ω

0 0, 1 0 0

1 1 11

J Y Y2 2

c

nn n

c n n c n n n n

Y b FBW Y FBW FBW

g g g g g g

π π+

+ + +Ω =

= = =Ω Ω

Prof. T. L. Wu


- 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 simulationArrows 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

θ

Zoe, Zoo

1

2

3

4

θ

Zoe, Zoo

1

4

( )

( )

( )

( )

11 22 33 44

12 21 34 43

13 21 24 42

14 41 23 32

cot2

cot2

csc2

csc2

−= = = = +

−= = = = −

−= = = = +

−= = = = −

oe oo

oe oo

oe oo

oe oo

jZ Z Z Z Z Z

jZ Z Z Z Z Z

jZ Z Z Z Z Z

jZ Z Z Z Z Z

θ

θ

θ

θ

( ) ( )

( ) ( )

11 14

41 44

cot csc2 2

csc cot2 2

− − + − = − − − +

oe oo oe oo

oe oo oe oo

j jZ Z Z Z

Z Z

Z Z j jZ Z Z Z

θ θ

θ θ

1 11 14 1

4 41 44 4

V Z Z I

V Z Z I

=

( ) ( )( )

2 22 2

11

41 41

44

41 41

cot csccos

2 csc

12 sin cos

+ − −+ − − = = +

− −

loe oo oe oooe oo

oe oo oe oo

oe oo

oe oo oe oo

Z Z Z Z ZZ Z Z

A B Z Z Z Z j Z Z

C D Z j Z Z

Z Z Z Z Z Z

β θθ

θ

θ θ

Apply boundary

condition (open-end)

Prof. T. L. Wu

Parallel coupled line filters

θ

Zoe, Zoo

1

4

θ

Zo1 4J

θ

Zo≈

( ) ( )( )

2 22 2csc cotcos

2 csc

2 sin cos

− − ++

− − = +

− −

oe oo oe oooe oo

oe oo oe oo

oe oo

oe oo oe oo

Z Z Z ZZ Zj

A B Z Z Z Z

C D j Z Z

Z Z Z Z

θ θθ

θ

θ θ

22 2

2 2

2

cos sin cos sin10

1 1sin cos sin cos

0

1 coscos sin sin

1 1sin cos cos sin

− = −

+ −

= − +

o o

o o

o o

o

o

o o

jZ jZA B j

Jj jC D

jJZ Z

JZ j JZJZ J

j J JZJZ JZ

θ θ θ θ

θ θ θ θ

θθ θ θ

θ θ θ θ

Left

Right

θ ≈ π/2 =>2

1

2

+ = + −

− =

oe ooo

oe oo o

oe ooo

Z ZJZ

Z Z JZ

Z ZJZ

=>( )2 3 2 2

2

2 2 2 1

2

+ = + = +

− =

oe oo o o o o

oe oo o

Z Z J Z Z Z J Z

Z Z JZ=>

( )( )

2 2

2 2

1

1

= + +

= − +

oe o o o

oo o o o

Z Z JZ J Z

Z Z JZ J Z

Prof. T. L. Wu

Parallel-Coupled, Half-Wavelength Resonator Filters

-Theory

Prof. T. L. Wu


-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


-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


-Example

Prof. T. L. Wu

Hairpin-Line Bandpass FiltersThey 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


- 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


- 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


- Example

A design equation is proposed for estimating the tapping point t as

Derivation on external Qe

t

4= g

Lλλλλ

equivalent

External quality factor for parallel L/C resonator:

( )0

0 0 002

0

1

2 2 2

Bb C C C C

Lω ω

ω ω ωω

ω ω=

∂= = + = + = ∂

1 1 = + = − =

inY j C j C jBj L L

ω ωω ωω ωω ωω ωω ωω ωω ω

0C bQ

G G

ω= =

Prof. T. L. Wu


- Example

Example:

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.

External quality factor for hairpin line: t

4= g

LλλλλYin

( ) ( ) ( ) ( ) ( )tan tan tan tanin r r r

Y jY L t jY L t jY L t L t jBβ β β β ω= − + + = − + + =

2

0

2

0

2sin

2

12 sin

2

r

e

r

Y

t

b ZLQ

tGZ

Z L

ππ

ππ

= = =

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

0 0

2 20 0 00 0

2 20 00 0 2

20 2

sec sec2 2 2

1csc csc

2 sin 2 22 sinsin2

r

p

r r rr

p p

g

B B Yb L t L t L t L t

u

Y L Y YL t t L t t Y

tu u tt

L

ω ω β β

ω ω β ωβ β

ω β ω

ω ω π πβ β

πβ πλ

= =

∂ ∂ ∂= = = − − + + + ∂ ∂ ∂

= − + + = = =

1 02sin

2

r

e

L Z Zt

Q

ππ

−=

02

=Lπ

β

condition:

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.

Y1 = Yn denotes the single microstrip characteristic

impedance of the input/output resonator.

Folding λ/2

resonator

Parallel coupled-line BPF Interdigital line filter with tapped line

Prof. T. L. Wu

Interdigital Filter

- Concept

θ

Yoe, Yoo

θ

θθ

2

oo oeY Y−

oeYoeY

BPF with short-circuited stubs

Prof. T. L. Wu

Interdigital Filter

- Design Equations

θ

θθ

uY−uY−

uY J

sin sinsin cos 01 0 1 0 1 0

cos 0

1 1 10sin cos 0 0tan tan tan

sin sin

u uuu u u

u uu

j jj j

Y YY JjY jY jY

jY jYjJjY

θ θθ θθ

θ θθ θ θθ θ

= = =

=sin

uYJ

θ

Prof. T. L. Wu

Interdigital Filter

- Design Equations Transform the lowpass prototype filter to the highpass prototype filter

00,1

1 0 1p

YJ

L g g=

( ), 1

111 1

1i i

pi i ip ii to n

JL L g g

+

++= =

= 1, 1

1

nn n

pn n n

YJ

L g g

++

+

=

Highpass

Transformation

Richard’s Transformation to decide the inductor value at edge frequency

of passband

1

1

1= =

L tanpi

YY

j jω θ11

=L tanpi

Y

θ

BPF Response

ω1ω2

1 01 0 0= = 1

2 2

12 2

p p

FBW FBW

u u

FBW

ω ωω ω ω θ

π

− => = −

= −

l l

Lpi

1

pi

YpL

=

p jω=

1YY

t=

1Y

tant j θ=

θ

Prof. T. L. Wu

Interdigital Filter

- Design Equations

J-inverter value for the BPF

( )

1, 1

1 1 11 1 1 1 11 1

1

tani i

pi i i i i i ip i i to n i to ni to n

Y YJ

L L g g g g g gθ+

+ + ++ = = = == =

= = = Y1 denotes the characteristic

impedance of the short-circuited stubs.

1 1 12aY Y Y= +

From Richard’s transformation and previous derivations on interdigital filter

1 1,an n nY Y Y −= +

1 1, , 1− += + +ai i i i iY Y Y Y

Input part:

Internal part:

Output part:

1 1 12-aY Y Y=

1 1, , 1- -ai i i i iY Y Y Y− +=

1 1,-an n nY Y Y −=

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

1 1 12-aY Y Y=

1 1, , 1- -ai i i i iY Y Y Y− +=

1 1,-an n nY Y Y −=

1 1 1 121 1 1 12 1

1 1

--a p

p

C C Y YY u C Y Y C

L C u= × = = => =

1 1, , 1

1 1, , 1

- -- - i i i i

ai i i i i i

p

Y Y YY Y Y Y C

u

− +− += => =

1 1,

1 1,

-- n n

an n n n

p

Y YY Y Y C

u

−−= => =

Connecting line Yi,i+1, 1

, 1

i i

i i

p

YC

u

++ = for i=1 to n-1

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 averageeven-mode impedance is then found by

the average odd-mode impedance isdetermined by

Prof. T. L. Wu

Interdigital Filter

- Design Examplethe 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

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.

using the design equations above given the table 5.7

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.

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.

Not resonator, but input and output of the filterthrough the coupled line elements.

resonators

Prof. T. L. Wu

Combline Filters

- Design Concept

Prof. T. L. Wu

Combline Filters

- Design Concept

a

oe aY Cυυυυ= b

oe bY Cυυυυ=

2

−=

a a

oo oeab

Y YCυυυυ

θθθθ

θθθθ θθθθ

Prof. T. L. Wu

Combline Filters

- Design Concept

1

2 2 2 2/ /

tan= = +s A s

in T in

Y YY G Y

N j θYin2

2

s

inY

1tan

= +a

L oein A

YY Y

j θ

( )

( )

11

1

2

tan tantan tantan

tan tantan

tan

11 tan 1

tan

1 tan tan

+ + + + + + = = =

+ + + + +

+ + +

= = ++

a aaoe oe

A oeLs in

in L aa

in oe AoeA

a

oe a

oe

A A A

Y YY jY Y Y jY

j jY jYY Y Y Y

Y jY Y Y jYYY j Y

j

Y j Yj Y YY

YY j Y Y j

θ θθ θθ

θ θθ

θ

θθ

θ θ

2 2

1

1

tan tan tan

= + + = + +

a b aboe oe oe

in oe

A A A A

Y YY Y Y YYY Y

Y Y j j Y j Yθ θ θ

1 2=in inY Y2

= =−

A A

a a

oo oe

Y YN

Y Y Y

( )2 2 2

2

1

1

= + = + − + = + − +

+ − −= − − + = − + = − +

a ab a a b a a boe oe

s oe oe oe oe oe oe oe

A A A

a b a b a bA AA oe oe oe oe A oe oe

A A

YY YY YY Y Y Y Y Y Y Y

Y Y Y

Y Y Y Y NY Y Y Y Y Y Y Y Y

Y Y N

2

+=

a a

oo oeA

Y YYConstraint:

Yin1

1

s

inY1

L

inY

2

−=

a a

oo oeY YY

a

oe aY Cυυυυ= b

oe bY Cυυυυ=

2

−=

a a

oo oeab

Y YCυυυυ

θθθθ

θθθθ θθθθ

Prof. T. L. Wu

Combline Filters

- Design Concept

Design formula for coupled-line:

( )

2

1 122

2

010 1 12

1

1

−= − + +

= − + − + +

a b

a A oe oe

A

A

NY Y Y Y Y

N

CY C C C

Y

υυ

( )1, , 1 1, , 12 1 − + − += −= + + = + +aj j j j j j j j j j jj to n

Y Y Y Y C C Cυ

=a

oe aY Cυ

=b

oe bY Cυ

( )2= +a

oo a abY C Cυ

( )2

2=

+ −

=

A

a ab a

A

ab

YN

C C C

Y

C

υ

υ

a

b

Prof. T. L. Wu

Combline Filters

- Formulation

( ) coti Li aiB C Yω ω θ= −

A design procedure starts with choosing the resonator susceptance slope parameters bi

(Parallel LC)

0 0 0

0

0

( ) cot 0

cot

Li ai

Li ai

B C Y

C Y

ω ω θ

θω

= − ≡

∴ =

Q

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


Obtain the J-inverter values using the formula

Obtain the required self and mutual capacitance for n + 2 lines using

Prof. T. L. Wu


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

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.

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.

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 FiltersPseudocombline 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 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.

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

Prof. T. L. Wu


- 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


- ExampleSimilar 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

Since quarter-wave short-circuited transmission line stubs look like parallel resonantcircuits, 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

Bandpass Filters Using Quarter-Wave Coupled

Quarter-Wave Resonators

How to design Z0n ?

Prof. T. L. Wu

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 andcharacteristic impedance of the transmission line Z0n .



Prof. T. L. Wu



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)

a 1C hg='

1 1

hC 1 g

2

= −

' "

1 1 1 1C g C C= = +

( )' /n n 1 1 n 1

n

n 1

g g h 2 g RC

R

+ +

+

−=

' "n 1n n n n

n 1

gC g C C

R

+

+

= = +

1 a12

1 2

1A

2

C CJ

g g

hgY

g

=

=

,

ak k 1 2 to n 1

k k 1

1A

k k 1

CJ

g g

hgY

g g

+ −+

+

=

=

,a n

n 1 n

n 1 n

1 n 1A

n 1

C CJ

g g

hg gY

g

−−

+

−

=

=

" a1

CC

2=

" a1 1

C hC g

2 2= =

Cai can be arbitrary chosen and the

inverters J01 and Jn, n+1 are eliminated!

Lowpass filter with only one kind of reactive element

Cai is scaled by terminal

impedance YA!

Prof. T. L. Wu


Yimag1

Yimag2

Using image impedance For interior sections

Definition of image impedance

1 11

1 1 1

1= =imag

imag

C DY

Z A B

2 22

2 2 2

1= =imag

imag

C DY

Z A B

a 1C hg=

" a1

CC

2=

" a1 1

C hC g

2 2= =

Prof. T. L. Wu


Derivations on the image parameters

2

1 11 , 1

1 1 , 1

12

++

= = −

aimag k k

k k

C D CY J

A B J

ω

( )22 2

, 1 , 1 , 12 2

2

2 2

cos

sin

+ + +− += =

s

k k k k k k

imag

Y Y YC DY

A B

θ

θ

, 1

s

k kY +

θθθθ

θθθθ θθθθ, 1

s

k kY +

, 1k kY +

Jk, k+12

aC

2

aC

( ), 1 , 1

1 1, 1 2

1 1

, 1 , 1

, 1 , 1

111 0 1 0 20

1 102 2

2

+ +

+

+ ++ +

−

= = − −

a

k k k k

k ka a

a ak k k k

k k k k

Cj

J JjA BJC C

C D j j C CjJ jJ j

J J

ω

ω ωω ω

( )

2 2, 1

, 1 , 12 2

, 1

, 1

, 1 , 1

22 2

, 1 , 1

, 1 , 1

, 1

cos sin1 0 1 0

cot 1 cot 1sin cos

cos cos sin

cos cos2 sin cos

sin sin

+

+ ++

+

+ +

+ ++ +

+

= − −

+

=

− + − +

k ks s

k k k k

k k

s

k k

k k k k

s

k ks k k

k k k k

k k

jA B

YjY jYC D

jY

Y j

Y Y

Y YjY jY j

Y

θ θ

θ θθ θ

θ θ θ

θ θθ θ

θ θ , 1

cos+

s

k kYθ

02=π ω

θω

Prof. T. L. Wu

2

1


Design equations for interior sections

( ) ( )'

1 2 00= = =imag imagY Yω ω ω

( ) ( )' '

1 1 2 1= = =imag imagY Yω ω ω ω

1.

2.

Lowpass Bandpass

( )22 2 2

, 1 , 1 , 1

, 1

, 1

cos1

2 sin

+ + +

++

− + − =

s

k k k k k ka

k k

k k

Y Y YCJ

J

θωθ

, 1 , 1+ +=k k k kJ Y

( ) ( )'

1 2 00= = =imag imagY Yω ω ω1.

( ) ( )' '

1 1 2 1= = =imag imagY Yω ω ω ω2.( )22 2 2

', 1 , 1 , 1 1

1, 1

, 1 1

cos1

2 sin

+ + +

++

− + − =

s

k k k k k ka

k k

k k

Y Y YCJ

J

θωθ

( ) ( )2

'22 2 21

, 1 1 , 1 , 1 , 1 1

, 1

sin 1 cos2

+ + + ++

− = − +

sak k k k k k k k

k k

CJ Y Y Y

J

ωθ θ

( ) ( )2

'2

2 2 2 1 1, 1 , 1 1 , 1 1

sincos 1 sin

2+ + +

+ = − +

s ak k k k k k

CJ Y J

ω θθ θ

Prof. T. L. Wu


( ) ( )' '

1 1 2 1= = =imag imagY Yω ω ω ω2. ( ) ( )2

'2

2 2 2 1 1, 1 , 1 1 , 1 1

sincos 1 sin

2+ + +

+ = − +

s ak k k k k k

CJ Y J

ω θθ θ

( )2

'2

2 1 1, 1 , 1 , 1

tan

2+ + +

+ = +

s ak k k k k k

CJ Y J

ω θ

2 2'2 21 1 1 1

, 1 , 1 , 1 , 1 , 1

2 2

, 1 1 1, 1 , 1 , 1

tan tan

2 2

tan

2

+ + + + +

++ + +

= + − = + −

= + − = −

s a Ak k k k k k k k k k

k k

A k k A k k k k

A

C Y hgY J J J J

J hgY J Y N J

Y

ω θ θ

θ

Required stub admittance

( ) ( )1, , 1 -1, -1, , 1 , 1

-1, , 1

-1, , 1= + - -

− + + +

++

= + = − + −

s s

k k k k k A k k k k A k k k k

k k k k

A k k k k

A A

Y Y Y Y N J Y N J

J JY N N

Y Y 2 1= −for k to n

Assume ω1’=1

( )0 011

0 0

/ 21

2 2 2 2

− = = = −

FBW FBWω ωπ ω π πθ

ω ωNote:

a 1C hg=

" a1 1

C hC g

2 2= =

Prof. T. L. Wu


Yin1

Yin2

Design equations for end sections

1 2

1 2

Im Im

Re Re=in in

in in

Y Y

Y Y

' ' '

1 1 1 1

0

cot

1 /=

A

C Y

R Y

ω θ ' ' '

1 0 1 1 1 0 1tan (1 ) tan2

= = −A A

hY Y g C Y g gω θ θ

Take input end for example

Required stub admittance for Y1

' 121 1 12 0 1 12 12 0 1 12(1 ) tan (1 ) tan

2 2

= + = − + − = − + −

s

A A A A

A

h h JY Y Y Y g g Y N J Y g g Y N

Yθ θ

Since h can be assigned to be arbitrary number, the best way to simplify the calculation process is h = 2.

Prof. T. L. Wu


- 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


- 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

Yin1Yin2

1tan

= iin

YY

j θ

tan=L ibY jY θ

YL

2 2

tan tan tan

tan tan

+ += =

+ −L ia ib ia

in ia ia

ia L ia ib

Y jY jY jYY Y Y

Y jY Y Y

θ θ θθ θ

1 2=in inY Y

Equivalence of the two input admittance using assumption of Yib = αiYia

2

tan tan

tan tan

+=

−i ib ia

ia

ia ib

Y jY jYY

j Y Y

θ θθ θ

2

tan tan

tan tan

+=

−i i ia ia

ia

ia i ia

Y j Y jYY

j Y Y

α θ θθ α θ

( )2

1 tan

tan 1 tan

+=

−ii

ia

i

jYY

j

α θθ α θ

( )( )

2

2

tan 1

1 tan

−=

+i i

ia

i

YY

α θ

α θ

Transmission zero in the λg/2 case

2 = ∞inY 2tan 0− =ia ibY Y θ 2tan 0− =ia i iaY Yα θ 2 2 20

0

2cot cot cot

4 2

= = =

zi

g

f

f

π λ πα θ

λ

Prof. T. L. Wu


- 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


- Example

f0 2f0

f0/2

Open-end effect

0

3f0/2

Prof. T. L. Wu

HW IIIPlease design a BPF based on Chebyshev prototype with following specifications using end-

coupled 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. Please decide the order of the BPF and calculate the J-inverter values.

b. 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.

c. Plot the return loss and insertion loss for the designed BPF using HFSS and ADS.

d. 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.

e. Discuss the sizes and frequency responses using the two methods.

Prof. T. L. Wu

HW IV1. 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, end-

coupled, 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.

a

oe aY Cυυυυ= b

oe bY Cυυυυ=

2

−=

a a

oo oeab

Y YCυυυυ

θθθθ

θθθθ θθθθ

Prof. Tzong-Lin Wu Department of Electrical Engineering...

Documents

Transcript of Prof. Tzong-Lin Wu Department of Electrical Engineering...