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EKT 356
MICROWAVE COMMUNICATIONS
CHAPTER 4:
MICROWAVE FILTERS
1
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 stopband
region
a linear phase response in the passband (to reduce
signal distortion).
2
INTRODUCTION
The goal of filter design is to approximate the ideal
requirements within acceptable tolerance with
circuits or systems consisting of real components.
f1
f2
f3
Commonly used block Diagram of a Filter
3
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
4
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 resonating
structures (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
Active
Passive
HPF
BPF
Active
Passive
5
INTRODUCTION
Types of Filters
1.
Low-pass Filter
f1
2.
f1
f1
f2
Passes low freq
Rejects high freq
High-pass Filter
f2
f2
Passes high freq
Rejects low freq
6
INTRODUCTION
3.
f1
Band-pass Filter
4.Band-stop
f1
f2
Filter
f1
f2
f2
f3
f3
Passes a small range
of freq
Rejects all other freq
Rejects
f3
a small range of
freq
Passes all other freq
7
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
8
INTRODUCTION
|H(ω)|
Low-pass filter (passive).
Transfer
function
1
V1(ω)
A Filter
H(ω)
V2(ω)
V2 (ω )
(
)
Hω =
(1.1a)
V1 (ω )
ZL
Arg(H(ω))
ωc
ω
A(ω)/dB
50
ω
40
30
20
10
3
0
 V2 (ω ) 

Attenuation A = −20 Log10 

 V1 (ω ) 
(1.1b)
ω
ωc
9
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).
a1
Vs
b2
Zc
Zc
20log|s21(ω)|
Zc
Filter
Zc
Arg(s21(ω))
Transmission line
is optional
0dB
ωc
ω
ω
b
b
s11 = 1
s21 = 2
a1 a =0
a1 a =0
2
2
Complex value
10
INTRODUCTION
Low pass filter response (cont)
A(ω)/dB
Passband
Transition band
50
40
30
20
10
3
0
Stopband
ω
ωc
Cut-off frequency (3dB)
V1(ω)
A Filter
H(ω)
V2(ω)
ZL
11
INTRODUCTION
High Pass filter
|H(ω)|
A(ω)/dB
Transfer
function
Passband
50
40
1
30
ω
ωc
20
10
3
0
ω
ωc
Stopband
12
INTRODUCTION
Band-pass filter (passive).
Band-stop filter.
A(ω)/dB
A(ω)/dB
40
40
30
30
20
20
10
3
0
10
3
0
ω1 ωo ω2
ω
|H(ω)|
1
ω1
|H(ω)|
ωo
ω2
ω
Transfer
function
1
Transfer
function
ω
ω1 ωo ω2
ω
ω1
ωo
ω2
13
INTRODUCTION
Insertion Loss
Pass BW (3dB)
Filter R esp on se
0
Q factor
-1 0
12 .1 24 G H z
-3 .0 0 3 8 d B
7 .9 0 24 G H z
-3 .0 0 5 7 dB
-2 0
-3 0
Inp ut R eturn L oss
-4 0
Ins ertio n Lo ss
-5 0
6
8
10
F re q u e n cy (G H z)
12
14
Figure 4.1: A 10 GHz Parallel Coupled Filter Response
Stop band frequencies and attenuation
14
FILTER DESIGN METHODS
Filter Design Methods
Two types of commonly used design methods:
1. Image Parameter Method
2. 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
15
FILTER DESIGN METHODS
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)|.
16
Filter Design Methods
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).
FILTER DESIGN METHODS
Zs
Lossless
2-port network
Vs
PA
Pin
ZL
PL
Γ1(ω)
PLR = Power available from source network
Power delivered to Load
P
PA
1
= inc =
=
PLoad
2
2
PA 1− Γ1 (ω )  1− Γ1 (ω )


(2.1a)
PPLR large,
high attenuation
LR large, high attenuation
to 1, low attenuation
PPLR close
LR close to 1, low attenuation
For
Forexample,
example,aalow-pass
low-pass
filter
filterresponse
responseisisshown
shown
below:
below:
PLR(f)
High
attenuation
1
Low-Pass filter PLR
0
Low
attenuation
fc
f
18
PLR and s21
In terms of incident and reflected waves, assuming ZL=Zs = ZC.
b1
a1
b2
Zc
Lossless
2-port network
Vs
PA
Pin
Zc
PL
1a 2
2
1
PA
a1
2
PLR =
=
=
PL
b2
1b 2
2
2
PLR = 1
(2.1b)
2
s21
19
FILTER RESPONSES
Filter Responses
Several types filter responses:
- Maximally flat (Butterworth)
- Equal Ripple (Chebyshev)
- Elliptic Function
- Linear Phase
20
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
ω 
PLR = 1 + k  
 ωc 
N
2
[4.1]
ω – frequency of filter
ωc – cutoff frequency of filter
N – order of filter
21
THE INSERTION LOSS METHOD
Equal ripple
- also known as Chebyshev.
- sharper cutoff
- the passband response will have ripples of amplitude 1 +k2
ω
PLR = 1 + k T  
 ωc 
2
2
N
4.2]
ω – frequency of filter
ωc – cutoff frequency of filter
N – order of filter
22
THE INSERTION LOSS METHOD
Figure 4.2: Maximally flat and equal-ripple low pass filter response.
23
THE INSERTION LOSS METHOD
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.
24
THE INSERTION LOSS METHOD
Figure 4.3: Elliptic function low-pass filter response
25
THE INSERTION LOSS METHOD
Filter
Specification
Low-pass
Prototype
Design
Normally done using
simulators
Optimization
& Tuning
Scaling &
Conversion
Filter
Implementation
Figure 4.4: The process of the filter design by the insertion
loss method.
26
THE INSERTION LOSS METHOD
Low Pass Filter Prototype
Figure 4.5: Low pass filter prototype, N = 2
27
THE INSERTION LOSS METHOD
Low Pass Filter Prototype – Ladder Circuit
Figure 4.6: Ladder circuit for low pass filter prototypes and their
element definitions. (a) begin with shunt element. (b) begin with
series element.
28
THE INSERTION LOSS METHOD
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.
29
THE INSERTION LOSS METHOD
Low Pass Filter Prototype – Maximally Flat
Figure 4.7: Attenuation versus normalized frequency for maximally flat
filter prototypes.
30
THE INSERTION LOSS METHOD
Figure 4.8: Element values for maximally flat LPF prototypes
31
THE INSERTION LOSS METHOD
Low Pass Filter Prototype – Equal Ripple
For an equal ripple low pass filter with a cutoff frequency ωc =
1, The power loss ratio is:
PLR = 1 + k T (ω )
2
2
N
[4.3]
Where 1 + k2 is the ripple level in the passband. Since the
Chebyshev polynomials have the property that
0
TN (ω ) = 
1
[4.3] 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 : two cases to consider depending on N
32
THE INSERTION LOSS METHOD
Figure 4.9: Attenuation versus normalized frequency for equal-ripple filter
33
prototypes. (0.5 dB ripple level)
THE INSERTION LOSS METHOD
Figure 4.10: Element values for equal ripple LPF prototypes (0.5 dB ripple
level)
34
THE INSERTION LOSS METHOD
Figure 4.11: Attenuation versus normalized frequency for equal-ripple filter
prototypes (3.0 dB ripple level)
35
THE INSERTION LOSS METHOD
Figure 4.12: Element values for equal ripple LPF prototypes (3.0 dB ripple
level).
36
EXAMPLE 4.1
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.
37
FILTER TRANSFORMATIONS
Impedance scaling: In the prototype design, the
source and load resistance are unity (except for equal
ripple filters with even N, which have non unity load
resistance).
38
FILTER TRANSFORMATIONS
Low Pass Filter Prototype – Impedance Scaling
'
L = R0L
C
C =
R0
'
[4.4b]
'
s
[4.4c]
R = R0
R
[4.4a]
'
L
= R0RL
[4.4d]
39
FILTER TRANSFORMATIONS
Low Pass Filter Prototype – Frequency Scaling
Frequency scaling: To change the cut-off
frequency of a LP prototype from unity to ωc
requires to scale the frequency dependence of
the filter by the factor 1/ ωc which is
accomplished by replacing ω by ω/ωc
Frequency scaling for the low pass filter:
ω ←
ω
ωc
40
FILTER TRANSFORMATIONS
The new Power Loss Ratio, P’LR
P’LR (ω) = PLR (ω/ωc)
[4.5]
Cut off occurs when ω/ωc = 1 or ω = ωc
The new element values of the prototype filter:
jX
jB
k
k
ω
L k = j ω L 'k
ωc
ω
= j
C k = j ω C k'
ωc
= j
[4.6]
[4.7]
41
FILTER TRANSFORMATIONS
The new element values are given by:
'
k
L =
Lk
ω
[4.8a]
ωc
Ck
C =
=
ω R0ωc
'
k
Ck
=
R0 Lk
[4.8b]
42
FILTER TRANSFORMATIONS
Low pass to high pass transformation
The frequency substitution:
ωc
ω←−
ω
[4.9]
The new component values are given by:
'
k
C =
1
R 0ω c L k
R0
L =
ω cC k
'
k
[4.10a]
[4.10b]
43
BANDPASS & BANDSTOP TRANSFORMATIONS
Low pass to Bandpass transformation
 ω ω 0  1  ω ω0 
 −  =  − 
ω←
ω2 − ω1  ω0 ω  ∆  ω0 ω 
ω0
Where,
ω2 − ω1
∆=
ω0
[4.11]
Centre freq.
[4.12]
Edges of
passband
The center frequency is:
ω0 = ω1ω2
[4.13]
44
BANDPASS & BANDSTOP TRANSFORMATIONS
The series inductor, Lk, is transformed to a series LC circuit with
element values:
L
L 'k =
C
k
[4.14a]
∆ω0
∆
=
ω 0Lk
'
k
[4.14b]
The shunt capacitor, Ck, is transformed to a shunt LC circuit with
element values:
∆
L 'k =
C
'
k
ω 0C
Ck
=
∆ω0
[4.15a]
k
[4.15b]
45
BANDPASS & BANDSTOP TRANSFORMATIONS
Low pass to Bandstop transformation
 ω ω0 
ω ← −∆ − 
 ω0 ω 
Where,
−1
[4.16]
ω2 − ω1
∆=
ω0
The center frequency is:
ω0 = ω1ω2
46
BANDPASS & BANDSTOP TRANSFORMATIONS
The series inductor, Lk, is transformed to a parallel LC circuit with
element values:
∆L
L'k =
k
ω0
1
C =
ω0 ∆Lk
'
k
[4.17a]
[4.17b]
The shunt capacitor, Ck, is transformed to a series LC circuit with
element values:
1
L'k =
'
k
C =
ω 0 ∆C k
∆C k
ω0
[4.18a]
[4.18b]
47
BANDPASS & BANDSTOP TRANSFORMATIONS
48
THE INSERTION LOSS METHOD
Filter
Specification
Low-pass
Prototype
Design
Normally done using
simulators
Optimization
& Tuning
Scaling &
Conversion
Filter
Implementation
49
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)
50
SUMMARY OF STEPS IN FILTER
DESIGN (cont)
B.
Low Pass Prototype Design
1. Min Insertion Loss level, Number of Filter
Order/Elements by using IL values
2. Determine whether shunt capacitance model or
series inductance model to use
3. Draw the low-pass prototype ladder diagram
4. Determine elements’ values from Prototype Table
51
SUMMARY OF STEPS IN FILTER
DESIGN (cont)
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
52
SUMMARY OF STEPS IN FILTER
DESIGN (cont)
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
53
EXAMPLE 4.2
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.
54
EXAMPLE 4.3
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?
55
Filter Realization Using Distributed Circuit
Elements (1)
Lumped-element filter realization using surface
mounted inductors and capacitors generally works
well at lower frequency (at UHF, say < 3 GHz).
At higher frequencies, the practical inductors and
capacitors loses their intrinsic characteristics.
A limited range of component values : available from
manufacturer – difficult design at microwave freq.
Therefore for microwave frequencies (> 3 GHz),
passive filter is usually realized using distributed
circuit elements such as transmission line sections.
56
Cont’d…
Distributed elements : open cct TL stubs or short cct TL stubs.
At microwave freq, the distance between filter components is not
negligible.
Richard’s transformation:
Can be used to convert lumped elements to TL sections
Kuroda’s identities:
Can be used to physically separate the filter elements by using TL
sections.
The four kuroda’s identities operations:
Physically separate transmission line stubs
Transform series stubs into shunt stubs or vice versa
Change impractical characteristic impedances into more
realizable values.
57
Filter Realization Using Distributed Circuit
Elements (2)
Recall in the study of Terminated Transmission Line Circuit that a
length of terminated Tline can be used to approximate an inductor and
capacitor.
This concept forms the basis of transforming the LC passive filter into
distributed circuit elements.
l
Zc , β
≅
L
≅
≅
C
≅
l
Zc , β
Zo
Zo
Zo
≅
Zo
58
Filter Realization Using Distributed Circuit
Elements (3)
This approach is only approximate. There will be deviation between the
actual LC filter response and those implemented with terminated Tline.
Also the frequency response of distributed circuit filter is periodic.
Other issues are shown below.
How do we implement series Tline
connection ? (only practical for
Zo
certain Tline configuration)
Connection physical
length cannot be
ignored at
microwave region,
comparable to λ
Zo
Thus
Thussome
sometheorems
theoremsare
areused
usedto
to
facilitate
the
transformation
of
LC
facilitate the transformation of LC
circuit
circuitinto
intostripline
striplinemicrowave
microwavecircuits.
circuits.
Chief
Chiefamong
amongthese
theseare
arethe
theKuroda’s
Kuroda’s
Identities
Identities(See
(SeeAppendix)
Appendix)
59
More on Approximating L and C with
Terminated Tline: Richard’s Transformation
l
≅
Zin
Z in = jZ c tan (βl ) = jωL = jLω
tan (βl ) = ω
L
Zc , β
Zc = L
(3.1.1a)
l
≅
Zin
Zc , β
C
Yin = jYc tan (βl ) = jωC = jCω
tan (βl ) = ω
(3.1.1b)
Yc = 1 = C
For LPP design, a further requirment is
that:
tan (βl ) = ωc = 1
⇒ tan  2π
 λc
Zc
Wavelength at
cut-off frequency
λc (3.1.1c)

l  =1⇒ l =
8

60
More on Approximating L and C with
Terminated Tline: Richard’s Transformation
61
Kuroda’s Identities
As taken from [2].
l
1
Z2
β
Z1
Z2
2
n =1+
Z1
l
Z2/n2
β
Z2
n2Z1
β
Z2
Z1
1
2
n Z2
β
Z2
/n2
1: n2
β
Z1
n2
l
l
1
Z2
Z1
n
l
l
Z1
β
l
l
Z1
Note: The inductor represents
shorted Tline while the capacitor
represents open-circuit Tline.
β
n2Z1
n2: 1
β
1
2
n Z2
62
Example – LPF Design Using Stripline
Design a 3rd order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc =
1.5GHz.
Step 1 & 2: LPP
g1
1.000H
Zo=1
g3
1.000H
g4
1
g2
2.000F
1 = 0.500
2.000
Length = λc/8
for all Tlines
at ω = 1 rad/s
63
Example – LPF Design Using Stripline
Design a 3rd order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc =
1.5GHz.
Step 3: Convert to Tlines using Richard’s Transformation
Length = λc/8
for all Tlines
at ω = 1 rad/s
1 = 0.500
2.000
64
Example Cont…
Step 4: Add extra Tline on the series
connection
Extra T-lines
Length = λc/8
for all Tlines
at ω = 1 rad/s
65
Example Cont…
Step 5: apply Kuroda’s
1st Identity.
Step 6: apply Kuroda’s
2nd Identity.
Similar operation is
performed here
66
Example Cont…
After applying Kuroda’s Identity.
Length = λc/8
for all Tlines
at ω = 1 rad/s
Since
Sinceall
allTlines
Tlineshave
havesimilar
similarphysical
physical
length,
length,this
thisapproach
approachto
tostripline
striplinefilter
filter
implementation
implementationisisalso
alsoknown
knownas
as
Commensurate
CommensurateLine
LineApproach.
Approach.
67
Example Cont…
Step 5: Impedance and frequency denormalization.
Microstrip line using double-sided FR4 PCB (εr = 4.6, H=1.57mm)
Zc/Ω
50
25
100
λ/8 @ 1.5GHz /mm
13.45
12.77
14.23
W /mm
2.85
8.00
0.61
Length = λc/8
for all Tlines at
f = fc = 1.5GHz
68
Example Cont…
Step 6: The layout (top view)
69
Example 2
Design a low pass filter for fabrication using microstrip
lines. The specifications are: cutoff freq of 4 GHz, third
order, impedance of 50 ohms and a 3dB equal ripple
characteristics
g1 = 3.3487 = L1
g2 = 0.7117 = C2
g3 = 3.3487 = L3
g4 = 1.0000 = RL
70
Example 2: Richard’s Transformation
71
Example 2: Cont’d…
UE =1 (Z2 =1)
Z1 =3.3487
Z1 =3.3487
UE =1 (Z2 =1)
72
Cont’d…
UE
UE
73
Example 5.8 (cont) Kuroda’s Identity
74
Example 5.8 (cont)
75
Kuroda Identities
76