1. Design of RF and Microwave Filters
2009학년도 여파기 설계
Design of RF and Microwave Filters

2. Contents 1. Introduction ; types of Filters
2. Characterization of Filters
3. Approximate Design Methods
4. Lowpass Prototype Network
5. Impedance Scaling and
Frequency Mapping
6. Immittance Inverters

3. 1. Introduction 1.1 Types of Filters
A. Lowpass Filters    B. Highpass Filters
C. Bandpass Filters   D. Bandstop Filters

4. 2. Filter Characterization(1)
Two-port Network ;
Fig. 1 Two-port Network

5. 2. Filter Characterization(2)
Characteristics of ideal bandpass filters ;
Fig. 2 Characteristics of
ideal bandpass filter
→ not realizable
→ approximation required

6. 2. Filter Characterization(3)
Practical specifications ;
1) Passband
; lower cutoff frequency - upper cutoff frequency
2) Insertion loss :
; must be as small as possible
3) Return Loss :
; degree of impedance matching
4) Ripple
; variation of insertion loss within the passband

7. 2. Filter Characterization(4)
5) Group delay
; time to required to pass the filter
6) Skirt frequency characteristics
; depends on the system specifications
7) Power handling capability

8. 3. Approximate Design Methods
1) based on Amplitude characteristics
A. Image parameter method
B. Insertion loss method
                a) J-K inverters
                b) Unit element - Kuroda identity
 2) based on Linear Phase characteristics

9. 3.1 Filter design(the insertion loss method)
Definition of Power Loss Ratio (PLR)
; impedance matching as well as frequency selectivity
Fig. 3  General filter network
← network synthesis procedures are required

10. 3.1 Filter Design(2) Approximation methods :
1) Maximally Flat (Butterworth) response
2) Chebyshev response
3) Elliptic Function response

11. 3.2 Approximation Methods
A. Maximally flat response
Where, ; passband tolerance
; order of filter
Usually
→ degree of freedom=1 (order N)
Fig. 4 Comparison Between Maximally
Flat and Chebyshev response

12. 3.2 Approximation Methods(2)
B. Chebyshev response
: equal ripple response in the passband
: Chebyshev Polynomial of order

13. 3.2 Approximation Methods(3)
; ripple (0.01 dB, 0.1 dB, etc.)
; order of filter
→ degree of freedom=2 (ripple and order)
Fig. 5  Chebyshev and Elliptic Function response

14. 3.2 Approximation Methods(4)
C. Elliptic Function response
 equal ripple passband in both passband and stopband
: stopband minimum attenuation
: transmission zero at stopband
        degree of freedom=3 (order N, ripple, transmission zero at stopband )

15. 4. Lowpass Prototype Filter
; normalized to 1
Fig. 5  Lowpass prototype

16. 4. Lowpass Prototype Filter(2)
   Maximally Flat response ;                 
                                      
           
Equal Ripple response ;

17. 4. Lowpass Prototype Filter(3)
Element No
Butterworth
0.1 dB ripple
Chebyshev
0.5 dB ripple 1
0.6180
1.1468
1.7058 2
1.6180
1.3712
1.2296 3
2.0000
1.9750
2.5408 4 5
Table1. Element values for Butterworth and chebyshev filters

18. 5. Impedance and freq. mapping
5.1 Impedance Scaling
Impedance level × 50 
; same reflection coefficient maintained
       series branch(impedance) elements ;
      shunt branch(admittance) elements ; 

19. 5. Impedance and freq. mapping(2)
5.2 Frequency Expansion
cutoff frequency 1 → lowpass cutoff frequency
mapping function ;
series and shunt branch elements ;

20. 5. Impedance and freq. mapping(3)
Fig. 6  Various mapping relations derived from lowpass prototype network

21. 5.3 Lowpass to Highpass transformation
5. Impedance and freq. mapping(4)
5.3 Lowpass to Highpass transformation
(lowpass cutoff freq. 1 → highpass cutoff freq. )
        mapping function ;
          series branch(impedance) elements ;
       shunt branch(admittance) elements ; 
Fig. 7  Highpass filter derived from lowpass prototype

22. 5.4 Lowpass to bandpass transformation
5. Impedance and freq. mapping(5)
5.4 Lowpass to bandpass transformation
(low cutoff freq. , high cutoff freq. )
mapping function ;

23. 5. Impedance and freq. mapping(6)
series branch element : impedance
shunt branch element : admittance
Fig. 8 Bandpass filter derived from the lowpass prototype

24. 5. Impedance and freq. mapping(7)
Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω.
step 1 : from the element values of lowpass prtotype
(0.5dB ripple Chebyshev)
step 2 : apply impedance scaling
                

25. 5. Impedance and freq. mapping(8)
step 3 : apply bandpass transformation

26. 5.5 Lowpass to bandstop transformation
5. Impedance and freq. mapping(9)
5.5 Lowpass to bandstop transformation
(low cutoff freq. , high cutoff freq. )
mapping function ;
inverse of bandpass mapping function

27. 5. Impedance and freq. mapping(10)
series branch element : admittance
shunt branch element : impedance
Fig. 9 Bandstop network derived from the lowpass prototype

28. 5. Impedance and freq. mapping(11)
5.6 Immitance Inverters
Fig. 10 Immitance inverter
K ; impedance inverter  →   
J ; admittance inverter  → 
ex. simplest form of inverter : λ/4 transformer
    series LC  →  J-inverter + shunt LC
    shunt LC  →  K-inverter + series LC

29. 5.7 Bandpass filters using J-, K-inverters
5. Impedance and freq. mapping(12)
5.7 Bandpass filters using J-, K-inverters
Fig. 11 Equivalent Network for lowpass prototype and bandpass network
Reflection coefficient ;
lowpass :
bandpass :
If (mapping relation)

30. 5. Impedance and freq. mapping(13)
Fig. 12  Lowpass network and bandpass network

31. 5. Impedance and freq. mapping(14)
From the partial fraction expansion including bandpass mapping relation
              
           : fractional bandwidth,  : center freq.
In the same manner, J-inverter values are derived as

32. 5. Impedance and freq. mapping(15)
Typical immittance inverters ;
Fig. 13  Impedance(K-) inverters

33. 5. Impedance and freq. mapping(16)
Fig. 14  Admittance(J-) inverters

34. 6. LC filters, Distributed filters
A. C-coupled bandpass filters
Fig. 14  Bandpass filter network using ideal J-inverters
Fig. 15 Bandpass filter network containing practical inverters

35. 6. LC filters, Distributed filters(2)
Fig. 16  Inverter of first and last stages
By equating the real and imaginary part of and

36. 6. LC filters, Distributed filters(3)
Fig.17 C-coupled
  Bandpass filter
B. L-coupled bandpass filter
Fig.18 L-coupled
  Bandpass filter

37. 6. LC filters, Distributed filters(4)
Design a LC bandpass filter. The f0 is 2.8 GHz, bandwidth is 500 MHz, and the input and output impedance 50Ω.
step 1 : from the element values of lowpass
prototype
step 2 : apply impedance scaling     
step 3 : apply bandpass transformation using
J-inverters
Step 4 : simulation

38. 6. LC filters, Distributed filters(3)
Simulated results:

39. Step 5 : Realization Insertion loss < 3.1 dB
Retrun loss > 15.5 dB
3.3 GHz : 15 dB

40. Step 6. improvement
C-coupling
LC filter +
L-coupling
LC filter =

41. Measured results
27 dB

42. 6. LC filters, Distributed filters(3)
  At microwave frequencies :
     Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators
     Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides

43. 6. LC filters, Distributed filters(3)
A. Combline filters
: cellular base stations as well as portable phone
Fig. 19 (a) Top view of
Combline Filter
Fig. 19 (b) Side view of
Combline Filter

44. Instead of lumped element inductors distributed
inductors (L < λ/4) are used.
Overall equivalent circuit :
Fig. 20 Coupled line
Fig. 21 Equivalent circuit of Fig.20
Fig. 22 Equivalent circuit
of Fig. 19