chapter 4: microwave filters - unimap portalportal.unimap.edu.my/portal/page/portal30/lecturer...

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CHAPTER 4:

MICROWAVE FILTERS

EKT 345

MICROWAVE ENGINEERING

2

INTRODUCTION

What is a Microwave filter ?

linear 2-port network

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 stopbandregion

a linear phase response in the passband (to reduce

signal distortion).f2

3

INTRODUCTION

The goal of filter design is to approximate the ideal requirements within acceptable tolerance with

circuits or systems consisting of real components.

f1

f3

f2

Commonly used block Diagram of a Filter

4

INTRODUCTION

Why Use Filters?

RF signals consist of:

1. Desired signals – at desired frequencies

2. Unwanted Signals (Noise) – at unwanted frequencies

That is why filters have two very important bands/regions:

1. Pass Band – frequency range of filter where it passes all signals

2. Stop Band – frequency range of filter where it rejects all signals

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INTRODUCTION

Categorization of Filters Low-pass filter (LPF), High-pass filter (HPF), Bandpass filter

(BPF), Bandstop filter (BSF), arbitrary type etc. In each category, the filter can be further divided into active

and passive types. In active filter, there can be amplification of the of the signal

power in the passband region, passive filter do not provide power amplification in the passband.

Filter used in electronics can be constructed from resistors, inductors, capacitors, transmission line sections and resonatingstructures (e.g. piezoelectric crystal, Surface Acoustic Wave (SAW) devices, and also mechanical resonators etc.).

Active filter may contain transistor, FET and Op-amp.

Filter

LPF BPFHPF

Active Passive Active Passive

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INTRODUCTION

Types of Filters

1. Low-pass Filter

Passes low freq

Rejects high freq

f1

f2

f1

2. High-pass Filter

Passes high freq

Rejects low freq

f1

f2

f2

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INTRODUCTION

3. Band-pass Filter

Passes a small range

of freq

Rejects all other freq

4.Band-stop Filter

Rejects a small range of

freq

Passes all other freq

f1

f3

f2f1

f3

f1

f2 f2

f3

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INTRODUCTION

Filter Parameters

Pass bandwidth; BW(3dB) = fu(3dB) – fl(3dB)

Stop band attenuation and frequencies,

Ripple difference between max and min of

amplitude response in passband

Input and output impedances

Return loss

Insertion loss

Group Delay, quality factor

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INTRODUCTION

Low-pass filter (passive).

A Filter

H(ω)V1(ω) V2(ω)

ZL

A(ω)/dB

ω0

ωc

3

10

20

30

40

50

( )( )

−=

ω

ω

1

21020A

V

VLognAttenuatio (1.1b)

( )( )( )ω

ωω

1

2

V

VH = (1.1a)

ωc

|H(ω)|

ω

1Transfer

function

Arg(H(ω))

ω

10

INTRODUCTION

For impedance matched system, using s21 to observe the filter response is more convenient, as this can be easily measured using Vector Network Analyzer (VNA).

Zc

01

221

01

111

22 ==

==

aaa

bs

a

bs

Transmission line

is optional

ωc

20log|s21(ω)|

ω

0dB

Arg(s21(ω))

ω

FilterZcZc

ZcVs

a1 b2

Complex value

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INTRODUCTION

A(ω)/dB

ω0

ωc

3

10

20

30

40

50

A Filter

H(ω)V1(ω) V2(ω)

ZL

Passband

Stopband

Transition band

Cut-off frequency (3dB)

Low pass filter response (cont)

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INTRODUCTION

High Pass filter

A(ω)/dB

ω0

ωc

3

10

20

30

40

50

ωc

|H(ω)|

ω

1

Transfer

function

Stopband

Passband

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INTRODUCTION

Band-pass filter (passive). Band-stop filter.

ω

A(ω)/dB

40

ω1

3

30

20

10

0 ω2ωo

ω1

|H(ω)|

ω

1 Transfer

function

ω2ωo

ω

A(ω)/dB

40

ω1

3

30

20

10

0ω2ωo

ω1

|H(ω)|

ω

1

Transfer

function

ω2ωo

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INTRODUCTION

6 8 10 12 14

Frequency (G Hz)

Filter R esponse

-50

-40

-30

-20

-10

0

12.124 GHz-3.0038 dB7.9024 GHz

-3.0057 dB

Input R eturn Loss

Insertion Loss

Figure 4.1: A 10 GHz Parallel Coupled Filter Response

Pass BW (3dB)

Stop band frequencies and attenuation

Q factor

Insertion Loss

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FILTER DESIGN METHODS

Filter Design Methods

Two types of commonly used design methods:1. Image Parameter Method2. Insertion Loss Method

• Image parameter method yields a usable filter• However, no clear-cut way to improve the design i.e to control the

filter response

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Filter Design Methods

•The insertion loss method (ILM) allows a systematic way to design and synthesize a filter with various frequency response.

•ILM method also allows filter performance to be improved in a straightforward manner, at the expense of a ‘higher order’ filter.

•A rational polynomial function is used to approximate the ideal |H(ω)|, A(ω) or |s21(ω)|.

•Phase information is totally ignored.Ignoring phase simplified the actual synthesis method. An LC network is then derived that will produce this approximated response.

•Here we will use A(ω) following [2]. The attenuation A(ω) can be cast into power attenuation ratio, called the Power Loss Ratio, PLR, which

is related to A(ω).

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( ) ( ) 211

1

211

Load todeliveredPower network source from availablePower

ωω Γ−

Γ−

===

=

AP

AP

LoadPincP

LRP

PLR large, high attenuation

PLR close to 1, low attenuation

For example, a low-pass

filter response is shown

below:

PLR large, high attenuation

PLR close to 1, low attenuation

For example, a low-pass

filter response is shown

below:

ZLVs

Lossless

2-port network

Γ1(ω)

Zs

PAPin

PL

PLR(f)

Low-Pass filter PLRf

1

0

Low

attenuation

High

attenuation

fc

(2.1a)

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PLR and s21

In terms of incident and reflected waves, assuming ZL=Zs = ZC.

ZcVs

Lossless

2-port network

Zc

PAPin

PL

a1

b1

b2

221

1

2

2

12

221

212

1

sLR

b

a

b

a

LPAP

LR

P

P

=

===

(2.1b)

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FILTER RESPONSES

Filter Responses

Several types filter responses:- Maximally flat (Butterworth)- Equal Ripple (Chebyshev)- Elliptic Function- Linear Phase

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THE INSERTION LOSS METHOD

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.- no ripple is permitted in its attenuation profile

N

c

LR kP

+=

ω

ω21[8.10]

ω – frequency of filter

ωc – cutoff frequency of filter

N – order of filter

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Equal ripple- also known as Chebyshev.- sharper cutoff- the passband response will have ripples of amplitude 1 +k2

+=

c

NLR TkPω

ω221 [8.11]

ω – frequency of filter

ωc – cutoff frequency of filter

N – order of filter

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Figure 5.3: Maximally flat and equal-ripple low pass filter response.

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Elliptic function:

- have equal ripple responses in the passband and stopband.

- maximum attenuation in the passband.- minimum attenuation in the stopband.

Linear phase:- linear phase characteristic in the passband

- to avoid signal distortion- maximally flat function for the group delay.

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Figure 5.4: Elliptic function low-pass filter response

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Filter Specification

Low-pass Prototype Design

Scaling & Conversion

Filter Implementation

Optimization & Tuning

Normally done using simulators

Figure 5.5: The process of the filter design by the insertion

loss method.

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Figure 5.6: Low pass filter prototype, N = 2

Low Pass Filter Prototype

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Figure 5.7: 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

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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.

As a matter of practical design procedure, it will be

necessary to determine the size, or order of the filter. This is

usually dictated by a specification on the insertion loss at

some frequency in the stopband of the filter.

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Figure 4.8: Attenuation versus normalized frequency for maximally flat

filter prototypes.

Low Pass Filter Prototype – Maximally Flat

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Figure 4.9: Element values for maximally flat LPF prototypes

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( )ω221 NLR TkP +=

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

1, The power loss ratio is:

( )

=1

0ωNT

[5.12]

Where 1 + k2 is the ripple level in the passband. Since the Chebyshev polynomials have the property that

[5.12] shows that the filter will have a unity power loss ratio at

ω = 0 for N odd, but the power loss ratio of 1 + k2 at ω = 0 for N even.

Low Pass Filter Prototype – Equal Ripple

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Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter

prototypes. (0.5 dB ripple level)

33


Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple

level)

34


Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter

prototypes (3.0 dB ripple level)

35


Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple

level).

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FILTER TRANSFORMATIONS

LL

s

RRR

RR

R

CC

LRL

0

'

0

'

0

'

0

'

=

=

=

= [8.13a]

[8.13b]

[8.13c]

[8.13d]

Low Pass Filter Prototype – Impedance Scaling

37


'

'

kk

c

k

kk

c

k

CjCjjB

LjLjjX

ωω

ω

ωω

ω

==

==

The new element values of the prototype filter:

cω

ωω ←

Frequency scaling for the low pass filter:

[8.14]

[8.15a]

[8.15b]

38


The new element values are given by:

c

kkk

c

kkk

R

CCC

LRLL

ωω

ωω

0

'

0'

==

== [8.16a]

[8.16b]

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ω

ωω c−←

Low pass to high pass transformation

The frequency substitution:

kc

k

kc

k

C

RL

LRC

ω

ω

0'

0

' 1

=

=

The new component values are given by:

[8.17]

[8.18a]

[8.18b]

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BANDPASS & BANDSTOP TRANSFORMATIONS

−

∆=

−

−←

ω

ω

ω

ω

ω

ω

ω

ω

ωω

ωω 0

0

0

012

0 1

0

12

ω

ωω −=∆

210 ωωω =

Where,

The center frequency is:

[8.19]

[8.20]

[8.21]

Low pass to Bandpass transformation

41


k

k

kk

LC

LL

0

'

0

'

ω

ω

∆=

∆=

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

'

ω

ω

∆=

∆=

kk

k

k

CC

CL

[8.22a]

[8.22b]

[8.23a]

[8.23b]

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1

0

0

−

−∆−←

ω

ω

ω

ωω

0

12

ω

ωω −=∆

210 ωωω =

Where,

The center frequency is:

[8.24]

Low pass to Bandstop transformation

43


k

k

kk

LC

LL

∆=

∆=

0

'

0

'

1

ω

ω

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

ω

ω

kk

k

k

CC

CL

∆=

∆=

[8.25a]

[8.25b]

[8.26a]

[8.26b]

44


45

EXAMPLE 5.1

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.

46


Filter Specification

Low-pass Prototype Design

Scaling & Conversion

Filter Implementation

Optimization & Tuning

Normally done using simulators

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SUMMARY OF STEPS IN FILTER

DESIGN

A. Filter Specification

1. Max Flat/Equal Ripple,

2. If equal ripple, how much pass band ripple allowed? If max

flat filter is to be designed, cont to next step

3. Low Pass/High Pass/Band Pass/Band Stop

4. Desired freq of operation

5. Pass band & stop band range

6. Max allowed attenuation (for Equal Ripple)

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SUMMARY OF STEPS IN FILTER DESIGN (cont)

B. Low Pass Prototype Design

1. Min Insertion Loss level, No of Filter

Order/Elements by using IL values

2. Determine whether shunt cap model or series

inductance model to use

3. Draw the low-pass prototype ladder diagram

4. Determine elements’ values from Prototype Table

49


C. Scaling and Conversion

1. Determine whether if any modification to the

prototype table is required (for high pass, band

pass and band stop)

2. Scale elements to obtain the real element values

50


D. Filter Implementation

1. Put in the elements and values calculated from

the previous step

2. Implement the lumped element filter onto a

simulator to get the attenuation vs frequency

response

51

EXAMPLE 5.2

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.

52

EXAMPLE 5.3

Design a five-section high pass lumped

element filter with 3 dB equal-ripple

response, a cutoff frequency of 1 GHz, and

an impedance of 50 Ω. What is the resulting

attenuation at 0.6 GHz?

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Master formula of FilterTransformation

chapter 4: microwave filters - unimap portalportal.unimap.edu.my/portal/page/portal30/lecturer...

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