1. EKT 441 MICROWAVE COMMUNICATIONS
CHAPTER 4:
MICROWAVE FILTERS

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 stopband region
a linear phase response in the passband (to reduce signal distortion). f2

3. Commonly used block Diagram of a Filter
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:
Desired signals – at desired frequencies
Unwanted Signals (Noise) – at unwanted frequencies
That is why filters have two very important bands/regions:
Pass Band – frequency range of filter where it passes all signals
Stop Band – frequency range of filter where it rejects all signals

5. 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
BPF
HPF
Active
Passive

6. INTRODUCTION Types of Filters Low-pass Filter Passes low freq
Rejects high freq
High-pass Filter
Passes high freq
Rejects low freq f1 f1 f1 f2 f2 f2

7. INTRODUCTION Band-pass Filter Band-stop Filter
Passes a small range of freq
Rejects all other freq
Band-stop Filter
Rejects a small range of freq
Passes all other freq f1 f1 f1 f2 f2 f2 f3 f3 f3

8. 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

9. INTRODUCTION (1.1a) (1.1b) Low-pass filter (passive). c  1 A Filter
|H()|  1
Transfer function
Arg(H())
Low-pass filter (passive).
A Filter
H()
V1()
V2() ZL
A()/dB  c 3 10 20 30 40 50
(1.1b)

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).
Filter Zc Vs a1 b2
Transmission line
is optional Zc c
20log|s21()| 
0dB
Arg(s21())
Complex value

11. INTRODUCTION Low pass filter response (cont) Transition band Passband
A()/dB  c 3 10 20 30 40 50
Passband
Stopband
Transition band
Cut-off frequency (3dB)
A Filter
H()
V1()
V2() ZL

12. INTRODUCTION High Pass filter Passband Stopband Transfer function
A()/dB  c 3 10 20 30 40 50 c
|H()|  1
Transfer function
Stopband
Passband

13. INTRODUCTION Band-pass filter (passive). Band-stop filter.  A()/dB40 1 3 30 20 10 2 o 
A()/dB 40 1 3 30 20 10 2 o 1
|H()|  1
Transfer function 2 o 1
|H()|  1
Transfer function 2 o

14. INTRODUCTION Figure 4.1: A 10 GHz Parallel Coupled Filter Response
Insertion Loss
Pass BW (3dB)
Q factor
Figure 4.1: A 10 GHz Parallel Coupled Filter Response
Stop band frequencies and attenuation

15. 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

16. 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 is related to A()2.

17. FILTER DESIGN METHODS (2.1a) ZL Vs Lossless 2-port network 1 Zs PA
Pin PL
PLR large, high attenuation
PLR close to 1, low attenuation
For example, a low-pass
filter response is shown
below:
PLR(f)
(2.1a) f 1
Low
attenuation
High fc
Low-Pass filter PLR

18. PLR and s21
In terms of incident and reflected waves, assuming ZL=Zs = ZC. Zc Vs
Lossless
2-port network PA
Pin PL a1 b1 b2
(2.1b)

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
[8.10]
 – frequency of filter
c – 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
[8.11]
 – frequency of filter
c – cutoff frequency of filter
N – order of filter

22. THE INSERTION LOSS METHOD
Figure 5.3: 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 5.4: Elliptic function low-pass filter response

25. THE INSERTION LOSS METHOD
Low-pass Prototype Design
Filter Specification
Scaling & Conversion
Normally done using simulators
Optimization & Tuning
Filter Implementation
Figure 5.5: The process of the filter design by the insertion loss method.

26. THE INSERTION LOSS METHOD
Low Pass Filter Prototype
Figure 5.6: Low pass filter prototype, N = 2

27. 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.

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 4.8: Attenuation versus normalized frequency for maximally flat filter prototypes.

30. THE INSERTION LOSS METHOD
Figure 4.9: 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:
[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.

32. THE INSERTION LOSS METHOD
Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter prototypes. (0.5 dB ripple level)

33. THE INSERTION LOSS METHOD
Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple level)

34. THE INSERTION LOSS METHOD
Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter prototypes (3.0 dB ripple level)

35. THE INSERTION LOSS METHOD
Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple level).

36. FILTER TRANSFORMATIONS
Low Pass Filter Prototype – Impedance Scaling
[8.13a]
[8.13b]
[8.13c]
[8.13d]

37. FILTER TRANSFORMATIONS
Frequency scaling for the low pass filter:
[8.14]
The new element values of the prototype filter:
[8.15a]
[8.15b]

38. FILTER TRANSFORMATIONS
The new element values are given by:
[8.16a]
[8.16b]

39. FILTER TRANSFORMATIONS
Low pass to high pass transformation
The frequency substitution:
[8.17]
The new component values are given by:
[8.18a]
[8.18b]

40. BANDPASS & BANDSTOP TRANSFORMATIONS
Low pass to Bandpass transformation
[8.19]
Where,
[8.20]
The center frequency is:
[8.21]

41. BANDPASS & BANDSTOP TRANSFORMATIONS
The series inductor, Lk, is transformed to a series LC circuit with element values:
[8.22a]
[8.22b]
The shunt capacitor, Ck, is transformed to a shunt LC circuit with element values:
[8.23a]
[8.23b]

42. BANDPASS & BANDSTOP TRANSFORMATIONS
Low pass to Bandstop transformation
[8.24]
Where,
The center frequency is:

43. BANDPASS & BANDSTOP TRANSFORMATIONS
The series inductor, Lk, is transformed to a parallel LC circuit with element values:
[8.25a]
[8.25b]
The shunt capacitor, Ck, is transformed to a series LC circuit with element values:
[8.26a]
[8.26b]

44. BANDPASS & BANDSTOP TRANSFORMATIONS

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. 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:

47. EXAMPLE 5.1 (Cont)
The ladder diagram of the LPF prototype to be used is as follow:

48. EXAMPLE 5.1 (Cont) pF nH pF nH pF
LPF prototype for maximally flat filter pF nH pF nH pF

49. EXAMPLE 5.1 (Cont) pF nH pF nH pF
LPF prototype for equal ripple filter: pF nH pF nH pF

50. THE INSERTION LOSS METHOD
Low-pass Prototype Design
Filter Specification
Scaling & Conversion
Normally done using simulators
Optimization & Tuning
Filter Implementation

51. SUMMARY OF STEPS IN FILTER DESIGN
Filter Specification
Max Flat/Equal Ripple,
If equal ripple, how much pass band ripple allowed? If max flat filter is to be designed, cont to next step
Low Pass/High Pass/Band Pass/Band Stop
Desired freq of operation
Pass band & stop band range
Max allowed attenuation (for Equal Ripple)

52. SUMMARY OF STEPS IN FILTER DESIGN (cont)
Low Pass Prototype Design
Min Insertion Loss level, No of Filter Order/Elements by using IL values
Determine whether shunt cap model or series inductance model to use
Draw the low-pass prototype ladder diagram
Determine elements’ values from Prototype Table

53. SUMMARY OF STEPS IN FILTER DESIGN (cont)
Scaling and Conversion
Determine whether if any modification to the prototype table is required (for high pass, band pass and band stop)
Scale elements to obtain the real element values

54. SUMMARY OF STEPS IN FILTER DESIGN (cont)
Filter Implementation
Put in the elements and values calculated from the previous step
Implement the lumped element filter onto a simulator to get the attenuation vs frequency response

55. 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.

56. EXAMPLE 5.2 (Cont) RS L1 L3 C2 RL
Solution: The low pass filter (LPF) prototype ladder diagram is shown as follow:
= 0.1 N = 3  = 1 GHz RS L1 L3 C2 RL

57. EXAMPLE 5.2 (Cont)
From the equal ripple filter table (with 0.5 dB ripple), the filter elements are as follow;

58. EXAMPLE 5.2 (Cont) RS RL C1 C2 C3 L1 L2 L3
Transforming the LPF prototype to the BPF prototype RS RL C1 C2 C3 L1 L2 L3

59. EXAMPLE 5.2 (Cont)

60. EXAMPLE 5.2 (Cont)

61. 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?

62. EXAMPLE 5.3 (Cont) RS RL L1 L3 L5 C2 C3
Solution: The high pass filter (HPF) prototype ladder diagram is shown as follow:
N = 5  = 1 GHz
At c = 0.6 GHz, ; referring back to Fig 4.12
The attenuation for N = 5, is about 41 dB RS RL L1 L3 L5 C2 C3

63. EXAMPLE 5.3 (Cont)
From the equal ripple filter table (with 3.0 dB ripple), the filter elements are as follow;

64. EXAMPLE 5.3 (Cont)
Impedance and frequency scaling:

65. EXAMPLE 5.3 (Cont)

66. EXAMPLE 5.4
Design a 4th order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc = 1.5GHz.
L1=0.7654H
L2=1.8478H
C1=1.8478F
C2=0.7654F
RL= 1
g0= 1
Step 1&2: LPP
Step 3: Frequency scaling
and impedance denormalization
L1=4.061nH
L2=9.803nH
C1=3.921pF
C2=1.624pF
RL= 50
g0=1/50

67. EXAMPLE 5.5
Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. Rs = 50Ohm, fc = 1.5GHz.
L1=1.6703H
L2=2.3661H
C1=1.1926F
C2=0.8419F
RL=
g0= 1
Step 1&2: LPP
Step 3: Frequency scaling
and impedance denormalization
L1=8.861nH
L2=12.55nH
C1=2.531pF
C2=1.787pF
RL= 99.2
g0=1/50

68. 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.

69. EXAMPLE 5.6 (cont)
From table, design the Low-Pass prototype (LPP) for 3rd order Butterworth response, c=1.
Zo=1 g1
1.000F g3 g2
2.000H g4 1
2<0o
Step 1&2: LPP

70. EXAMPLE 5.6 (cont) LPP to bandpass transformation.
Impedance denormalization.
Step 3: Frequency scaling
and impedance denormalization 50 Vs
15.916pF
0.1414pF
79.58nH
0.7072nH RL