EKT 441
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
f2 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
f1
f2
Passes low freq
Rejects high freq
2.
High-pass Filter
f1
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
Passes a small range of
freq
Rejects all other freq
f3
f3
Rejects 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()|
Low-pass filter (passive).
Transfer
function
1
V1()
A Filter
H()
V2()
V2  
H   
(1.1a)
V1  
ZL
Arg(H())
c

A()/dB
50

40
30
20
10
3
0
 V2   

Attenuation A  20 Log10 

 V1   
c
(1.1b)

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()|
Zc
Filter
Zc
Arg(s21())
Transmission line
is optional
0dB
c


b
b
s11  1
s21  2
a1 a 0
a1 a 0
2
2
Complex value
10
INTRODUCTION
Low pass filter response (cont)
A()/dB
Passband
Transition band
50
40
30
20
10
3
0
Stopband

c
Cut-off frequency (3dB)
V1()
A Filter
H()
V2()
ZL
11
INTRODUCTION
High Pass filter
|H()|
A()/dB
Transfer
function
Passband
50
40
1

c
30
20
10
3
0
c

Stopband
12
INTRODUCTION
Band-pass filter (passive).
Band-stop filter.
A()/dB
A()/dB
40
40
30
30
20
20
10
3
0
10
3
0
1 o 2

|H()|
1
1
|H()|
Transfer
function
o
2

Transfer
function
1

1 o 2

1
o
2
13
INTRODUCTION
Insertion Loss
Pass BW (3dB)
Filter Response
0
Q factor
-10
12.124 GHz
-3.0038 dB
7.9024 GHz
-3.0057 dB
-20
-30
Input Return Loss
-40
Insertion Loss
-50
6
8
10
Frequency (GHz)
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:
- Image Parameter Method
- 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()|,
A() or |s21()|.
•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() following [2]. The attenuation A() can be cast
into power attenuation ratio, called the Power Loss Ratio, PLR, which
16
is related to A()2.
FILTER DESIGN METHODS
Zs
Lossless
2-port network
Vs
PA
Pin
ZL
PL
1
PLR  Power available from source network
Power delivered to Load
P
PA
1
 inc 

PLoad
2
2
PA 1 1    1 1  


(2.1a)
PLR large, high attenuation
PLR close to 1, low attenuation
For example, a low-pass
filter response is shown
below:
PLR(f)
High
attenuation
1
Low-Pass filter PLR
0
Low
attenuation
fc
f
17
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
s 21
18
FILTER RESPONSES
Filter Responses
Several types filter responses:
- Maximally flat (Butterworth)
- Equal Ripple (Chebyshev)
- Elliptic Function
- Linear Phase
19
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
 
PLR  1  k  
 c 
N
2
[8.10]
 – frequency of filter
c – cutoff frequency of filter
N – order of filter
20
THE INSERTION LOSS METHOD
Equal ripple
- also known as Chebyshev.
- sharper cutoff
- the passband response will have ripples of amplitude 1 +k2

PLR  1  k T  
 c 
2
2
N
[8.11]
 – frequency of filter
c – cutoff frequency of filter
N – order of filter
21
THE INSERTION LOSS METHOD
Figure 5.3: Maximally flat and equal-ripple low pass filter response.
22
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.
23
THE INSERTION LOSS METHOD
Figure 5.4: Elliptic function low-pass filter response
24
THE INSERTION LOSS METHOD
Filter
Specification
Low-pass
Prototype
Design
Normally done using
simulators
Optimization
& Tuning
Scaling &
Conversion
Filter
Implementation
Figure 5.5: The process of the filter design by the insertion
loss method.
25
THE INSERTION LOSS METHOD
Low Pass Filter Prototype
Figure 5.6: Low pass filter prototype, N = 2
26
THE INSERTION LOSS METHOD
Low Pass Filter Prototype – Ladder Circuit
Figure 5.7: Ladder circuit for low pass filter prototypes and their
element definitions. (a) begin with shunt element. (b) begin with
series element.
27
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.
28
THE INSERTION LOSS METHOD
Low Pass Filter Prototype – Maximally Flat
Figure 4.8: Attenuation versus normalized frequency for maximally flat
filter prototypes.
29
THE INSERTION LOSS METHOD
Figure 4.9: Element values for maximally flat LPF prototypes
30
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  
2
2
N
[5.12]
Where 1 + k2 is the ripple level in the passband. Since the
Chebyshev polynomials have the property that
0
TN    
1
[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.
31
THE INSERTION LOSS METHOD
Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter
prototypes. (0.5 dB ripple level)
32
THE INSERTION LOSS METHOD
Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple
level)
33
THE INSERTION LOSS METHOD
Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter
prototypes (3.0 dB ripple level)
34
THE INSERTION LOSS METHOD
Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple
level).
35
FILTER TRANSFORMATIONS
Low Pass Filter Prototype – Impedance Scaling
L  R0 L
'
C
C 
R0
'
[8.13a]
[8.13b]
R  R0
[8.13c]
R  R0 RL
[8.13d]
'
s
'
L
36
FILTER TRANSFORMATIONS
Frequency scaling for the low pass filter:


c
[8.14]
The new element values of the prototype filter:

jX k  j
Lk  jL'k
c

jBk  j Ck  jCk'
c
[8.15a]
[8.15b]
37
FILTER TRANSFORMATIONS
The new element values are given by:
L 
'
k
Lk

c
Ck
C 

 R0c
'
k
Ck

R0 Lk
[8.16a]
[8.16b]
38
FILTER TRANSFORMATIONS
Low pass to high pass transformation
c


The frequency substitution:
[8.17]
The new component values are given by:
C 
'
k
1
R0c Lk
R0
L 
c Ck
'
k
[8.18a]
[8.18b]
39
BANDPASS & BANDSTOP
TRANSFORMATIONS
Low pass to Bandpass transformation
  0  1   0 
      

2  1  0     0  
[8.19]
2  1

0
[8.20]
0
Where,
The center frequency is:
0  12
[8.21]
40
BANDPASS & BANDSTOP
TRANSFORMATIONS
The series inductor, Lk, is transformed to a series LC circuit with
element values:
L
L'k 
k
 0

C 
0 Lk
'
k
[8.22a]
[8.22b]
The shunt capacitor, Ck, is transformed to a shunt LC circuit with
element values:

L'k 
0C k
Ck
C 
 0
'
k
[8.23a]
[8.23b]
41
BANDPASS & BANDSTOP
TRANSFORMATIONS
Low pass to Bandstop transformation
  0 
    
 0  
Where,
1
[8.24]
2  1

0
The center frequency is:
0  12
42
BANDPASS & BANDSTOP
TRANSFORMATIONS
The series inductor, Lk, is transformed to a parallel LC circuit with
element values:
L
L'k 
k
0
1
C 
0 Lk
'
k
[8.25a]
[8.25b]
The shunt capacitor, Ck, is transformed to a series LC circuit with
element values:
1
L'k 
C 
'
k
 0 C k
C k
0
[8.26a]
[8.26b]
43
BANDPASS & BANDSTOP
TRANSFORMATIONS
44
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.
45
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:
g 1  0.618

3
 1   1  0.5
c
2
g 2  1.618
g 3  2 .0
g 4  1.618
g 5  0.618
46
EXAMPLE 5.1 (Cont)
The ladder diagram of the LPF prototype to be used is as follow:
L2
C1
L'  R0 L
C
C 
R0
'
R  R0
'
s
R  R0 RL
'
L
L4
C3
C5
g1
C1 
R0c
C3 
g3
R0c
C5 
g5
R0c
L2 
R0 g 2
L4 
R0 g 4
c
c
47
EXAMPLE 5.1 (Cont)
LPF prototype for maximally flat filter
g1
0.618
C1 

 0.984 pF
9
R0c 50 2  2 10 
R0 g 2
50 1.618
L2 

 6.438 nH
9
2  2 10 
c
g3
2.00
C3 

 3.183pF
9
R0c 50 2  2  10
R0 g 4
50 1.618
L4 

 6.438 nH
9
2  2 10 
c
g5
0.618
C5 

 0.984 pF
9
R0c 50 2  2 10 


48
EXAMPLE 5.1 (Cont)
LPF prototype for equal ripple filter:
g1  3.4817
g 2  0.7618
g 3  4.5381
g 4  0.7618
g 5  3.4817
g1
3.4817
C1 

 5.541 pF
9
R0c 50 2  2 10 
R0 g 2
50  0.7618
L2 

 3.031 nH
9
2  2 10 
c
g3
4.5381
C3 

 7.223 pF
9
R0c 50 2  2 10 
R0 g 4
50  0.7618
L4 

 3.031 nH
9
2  2 10 
c
g5
3.4817
C5 

 5.541 pF
9
R0c 502  2 10 
49
THE INSERTION LOSS METHOD
Filter
Specification
Low-pass
Prototype
Design
Normally done using
simulators
Optimization
& Tuning
Scaling &
Conversion
Filter
Implementation
50
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)
51
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
52
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
53
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
54
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.
55
EXAMPLE 5.2 (Cont)
Solution: The low pass filter (LPF) prototype ladder diagram is
shown as follow:
 = 0.1
 = 1 GHz
N=3
RS
L1
L3
C2
RL
56
EXAMPLE 5.2 (Cont)
From the equal ripple filter table (with 0.5 dB ripple), the filter
elements are as follow;
g1  1.5963  L1
g 2  1.0967  C 2
g 3  1.5963  L 3
g 4  1.000  RL
57
EXAMPLE 5.2 (Cont)
Transforming the LPF prototype to the BPF prototype
RS
L1
C1
L2
L3
C2
C3
RL
58
EXAMPLE 5.2 (Cont)
L1Z 0
1.5963  50
L1 

 127.0nH
9
0  2 110  0.1

0.1
C1 

 0.199 pF
9
Z 00 L1 50 2  2 10 1.5963


Z 0
0.1 50
L2 

 0.726nH
9
0C 2 2 110 1.0967 
C2
1.0967
C2 

 34.91 pF
9
0 Z 0 2 110 (0.1)50
59
EXAMPLE 5.2 (Cont)
L3Z 0
1.5963  50
L3 

 127.0nH
9
0  2 110  0.1

0.1
C3 

 0.199 pF
9
Z 00 L3 502  2 10 1.5963
60
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?
61
EXAMPLE 5.3 (Cont)
Solution: The high pass filter (HPF) prototype ladder diagram is
shown as follow:
N=5
 = 1 GHz
At c = 0.6 GHz,

1
1 
 1  0.667 ; referring back to Fig 4.12
c
0.6
The attenuation for N = 5, is about 41 dB
RS
C2
L1
L3
C3
L5
RL
62
EXAMPLE 5.3 (Cont)
From the equal ripple filter table (with 3.0 dB ripple), the filter
elements are as follow;
g1  3.4817  L1
g 2  0.7618  C 2
g 3  4.5381  L3
g 4  0.7618  C 4
g 5  3.4817  L5
g 6  1.000  RL
63
EXAMPLE 5.3 (Cont)
Impedance and frequency scaling:
Z0
50
L'1 

 2.28nH
9
c L1 2 110  3.4817 
1
1
C '2 

 4.18 pF
9
Z 0c C 2 502 110  0.7618
Z0
50
L'3 

 1.754nH
9
c L3 2 110  4.5381
64
EXAMPLE 5.3 (Cont)
1
1
C '4 

 4.18 pF
9
Z 0c C 4 502 110  0.7618
Z0
50
L'5 

 1.754nH
9
c L5 2 110  4.5381
65
EXAMPLE 5.4

Design a 4th order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc =
1.5GHz.
Step 1&2: LPP
L1=0.7654H
g0= 1
L2=1.8478H
C1=1.8478F C2=0.7654F
Step 3: Frequency scaling
and impedance denormalization L =4.061nH
1
g0=1/50
 c  2 1.5GHz  9.4248  10 9 rad/s
Zo  50
RL= 1
R  Z o Rn
L
L  Zo n
c
L2=9.803nH
C1=3.921pF C2=1.624pF
C
Cn
Z o c
RL= 50
66
EXAMPLE 5.5

Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. Rs
= 50Ohm, fc = 1.5GHz.
Step 1&2: LPP
L1=1.6703H
g0= 1
L2=2.3661H
C1=1.1926F C2=0.8419F
Step 3: Frequency scaling
and impedance denormalization L =8.861nH
1
g0=1/50
 c  2 1.5GHz  9.4248  10 9 rad/s
Zo  50
RL=
1.9841
R  Z o Rn
L
L  Zo n
c
L2=12.55nH
C1=2.531pF C2=1.787pF
C
Cn
Z o c
RL=
99.2
67
EXAMPLE 5.6




Design a bandpass filter with Butterworth (maximally flat)
response.
N = 3.
Center frequency fo = 1.5GHz.
3dB Bandwidth = 200MHz or f1=1.4GHz, f2=1.6GHz.
68
EXAMPLE 5.6 (cont)

From table, design the Low-Pass prototype (LPP) for 3rd order
Butterworth response, c=1.
Step 1&2: LPP
2<0o
g2
2.000H
Zo=1
g1
1.000F
g3
1.000F
g4
1
 c  2f c  1
 f c  21  0.1592 Hz
69
EXAMPLE 5.6 (cont)

1  2 1.4GHz
 2  2 1.6GHz
LPP to bandpass transformation.
Impedance denormalization.

Step 3: Frequency scaling
and impedance denormalization
f1 f 2  1.497 GHz
 1
  2
o
LZ o
o 

 o LZ o
50
Vs
fo 
C
 o Z o
79.58nH
 0.133
Z o
 oC
0.1414pF
RL
15.916pF
0.7072nH
15.916pF
50
0.7072nH
70