EKT 441 MICROWAVE COMMUNICATIONS Engineering/833/crs-14244... · EKT 441 MICROWAVE COMMUNICATIONS MICROWAVE FILTERS . 2 INTRODUCTION What is a Microwave filter ? Circuit that controls

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EKT 441

MICROWAVE COMMUNICATIONS

MICROWAVE FILTERS

2

INTRODUCTION What is a Microwave filter ? Circuit that controls the frequency response at a certain

point in a microwave system

provides perfect transmission of signal for frequencies in

a certain passband region

infinite attenuation for frequencies in the stopband region

Attenuation/dB

0

1

3

10

20

30

40 2

Attenuation/dB

0

c

3

10

20

30

40

3

FILTER DESIGN METHODS

Filter Design Methods

Two types of commonly used design methods:

- Image Parameter Method

- Insertion Loss Method

•Image parameter method yields a usable filter

Filters designed using the image parameter method consist of a

cascade of simpler two port filter sections to provide the desired cutoff frequencies and attenuation characteristics but do not allow the

specification of a particular frequency response over the complete operating range. Thus, although the procedure is relatively simple, the

design of filters by the image parameter method often must be iterated

many times to achieve the desired results.

Filter Design by The Insertion Loss Method

4

The perfect filter would have zero insertion loss in the pass-

band, infinite attenuation in the stop-band, and a linear

phase response in the pass-band.


5

6


7


8

Practical filter response:

Maximally flat:

• also called the binomial or Butterworth response,

• is optimum in the sense that it provides the flattest possible

passband response for a given filter complexity.

N

c

LR kP

21

– frequency of filter

c – cutoff frequency of filter N – order of filter

Equal ripple also known as Chebyshev.

- sharper cutoff

221 NLR TkP


9


• The insertion loss method (ILM) allows a systematic way to design

and synthesize a filter with various frequency response.

• Design usually begins by designing a normalized low-pass

prototype (LPP). The LPP is a low-pass filter with source and load

resistance of 1 and cutoff frequency of 1 Radian/s.

• Impedance transformation and frequency scaling are then applied

to denormalize the LPP and synthesize different type of filters with

different cutoff frequencies.

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THE INSERTION LOSS METHOD

Low pass filter prototype, N = 2

Low Pass Filter Prototype

11

= 1.4142

12


Ladder circuit for low pass filter prototypes and their element definitions. (a)

begin with shunt element. (b) begin with series element.

Low Pass Filter Prototype – Ladder Circuit

For an arbitrary number of elements N.

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g0 = generator resistance, generator conductance.

gk = inductance for series inductors, capacitance for shunt

capacitors. (k=1 to N)

gN+1 = load resistance if gN is a shunt capacitor, load

conductance if gN is a series inductor.

14


Element values for maximally flat LPF prototypes

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Attenuation versus normalized frequency for maximally flat filter prototypes.

Low Pass Filter Prototype – Maximally Flat

To determine the order (size) of the filter. Is usually determined by the

specification of the insertion loss at some frequency in the stopband of the

filter.

16


221 NLR TkP

For an equal ripple low pass filter with a cutoff frequency ωc =

The power loss ratio is:

Low Pass Filter Prototype – Equal Ripple

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Element values for equal ripple LPF prototypes (0.5 dB ripple level)

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Element values for equal ripple LPF prototypes (3.0 dB ripple level).

19


Attenuation versus normalized frequency for equal-ripple filter prototypes.

(0.5 dB ripple level)

20


Attenuation versus normalized frequency for equal-ripple filter prototypes

(3.0 dB ripple level)

Scaling

and TRANSFORMATIONS

21

Scaling

22

23

= 1.4142

Example 1: LPF

24

LL

s

RRR

RR

R

CC

LRL

0

'

0

'

0

'

0

'

Low Pass Filter Prototype – Impedance Scaling

25

'

'

kk

c

k

kk

c

k

CjCjjB

LjLjjX

The new element values of the prototype filter:

c

Frequency scaling for the low pass filter:

26

The new element values are given by:

kk

kk

CC

LL

'

'

c

c

27

finally

The new element values are given by:

c

kk

c

kk

R

CC

LRL

0

'

0'

28

= 1.4142

Example 1: LPF

TRANSFORMATIONS

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Low-pass to high-pass transformation

31

FILTER TRANSFORMATIONS

c

Low pass to high pass transformation

The frequency substitution:

kc

k

kc

k

C

RL

LRC

0'

0

' 1

The new component values are given by:

33

BANDPASS & BANDSTOP

TRANSFORMATIONS

0

0

0

012

0 1

0

12

210

Where,

The center frequency is:

Low pass to Bandpass transformation

34

BANDPASS & BANDSTOP

TRANSFORMATIONS

k

k

kk

LC

LL

0

'

0

'

The series inductor, Lk, is transformed to a series LC circuit with

element values:

The shunt capacitor, Ck, is transformed to

a shunt LC circuit with element values:

0

'

0

'

kk

k

k

CC

CL

[8.22a]

[8.22b]

0

12

210

35

BANDPASS & BANDSTOP

TRANSFORMATIONS

1

0

0

0

12

210

Where,

The center frequency is:

Low pass to Bandstop transformation

36

BANDPASS & BANDSTOP

TRANSFORMATIONS

k

k

kk

LC

LL

0

'

0

'

1

The series inductor, Lk, is transformed to a parallel LC circuit with

element values:

The shunt capacitor, Ck, is transformed to a series LC circuit with

element values:

0

'

0

' 1

kk

k

k

CC

CL

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BANDPASS & BANDSTOP

TRANSFORMATIONS

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EXAMPLE 5.1 Example 2:

Design a maximally flat low pass filter with a cutoff

freq of 2 GHz, impedance of 50 Ω, and at least 15 dB

insertion loss at 3 GHz. Compute and compare with

an equal-ripple (3.0 dB ripple) having the same order.

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EXAMPLE 5.1 (Cont)

Solution:

First find the order of the maximally flat filter to satisfy the

insertion loss specification at 3 GHz.

We can find the normalized freq by using: 5.012

31

c

40 618.0

618.1

0.2

618.1

618.0

5

4

3

2

1

g

g

g

g

g

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EXAMPLE 5.1 (Cont)

The ladder diagram of the LPF prototype to be used is as follow:

LL

s

RRR

RR

R

CC

LRL

0

'

0

'

0

'

0

'

C3 C5

L2 L4

C1

cR

gC

0

11

c

gRL

20

2

cR

gC

0

33

c

gRL

40

4

cR

gC

0

55

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EXAMPLE 5.1 (Cont)

984.0

102250

618.09

0

11

cR

gC

438.6

1022

618.1509

202

c

gRL

183.3

102250

00.29

0

3

3

cR

gC

438.6

1022

618.1509

404

c

gRL

984.0

102250

618.09

0

55

cR

gC

pF

nH

pF

nH

pF

LPF prototype for maximally flat filter

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EXAMPLE 5.1 (Cont)

541.5

102250

4817.39

0

11

cR

gC

031.3

1022

7618.0509

202

c

gRL

223.7

102250

5381.49

0

33

cR

gC

031.3

1022

7618.0509

404

c

gRL

541.5

102250

4817.39

0

55

cR

gC

pF

nH

pF

nH

pF

4817.3

7618.0

5381.4

7618.0

4817.3

5

4

3

2

1

g

g

g

g

g

LPF prototype for equal ripple filter:

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EXAMPLE 5.2 Example:

Design a band pass filter having a 0.5 dB

equal-ripple response, with N = 3. The center

frequency is 1 GHz, the bandwidth is 10%,

and the impedance is 50 Ω.

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EXAMPLE 5.2 (Cont)

Solution: The low pass filter (LPF) prototype ladder diagram is

shown as follow:

LRg

Lg

Cg

Lg

000.1

5963.1

0967.1

5963.1

4

33

2

1

2

1

46

EXAMPLE 5.2 (Cont)

Transforming the LPF prototype to the BPF prototype

= 0.1 N = 3 = 1 GHz

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EXAMPLE 5.2 (Cont)

pF

LZC 199.0

5963.1102250

1.0'

9100

1

nH

ZLL 0.127

1.01012

505963.1'

9

0

1

1

0

pF

Z

CC 91.34

50)1.0(1012

0967.1'

900

22

nH

C

ZL 726.0

0967.11012

501.0'

920

02

49

EXAMPLE 5.2 (Cont)

pF

LZC 199.0

5963.1102250

1.0'

9300

3

nH

ZLL 0.127

1.01012

505963.1'

9

0

3 03

Thank you

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EKT 441 MICROWAVE COMMUNICATIONS Engineering/833/crs-14244... · EKT 441 MICROWAVE COMMUNICATIONS MICROWAVE FILTERS . 2 INTRODUCTION What is a Microwave filter ? Circuit that controls