1. Microwave
Filter
Microwave Engineering
CHO, Yong Heui

2. Microwave Engineering
Circuit Resonator
2 EM Wave Lab

3. Microwave Engineering
1. LC resonator
Applications
 

Filter
Oscillator
 Frequency meter
 Tuned amplifier
3 EM Wave Lab

4. Microwave Engineering
1. LC resonator
LC resonator: ideal resonator
  Input impedance
j
Z in = jωL −
ωC
 Input power
1 * 1 2 1 2 j 
Pin = VI = Z in I = I  jωL − 
2 2 2  ωC 
 Resonant frequency: Wm = We
1
ω=
LC
4 EM Wave Lab

5. Microwave Engineering
1. LC resonator
Series resonator
  R, L, C
 Input impedance
j
Z in = R + jωL −
ωC
 Input power
1 * 1 2 1 2 j 
Pin = VI = Z in I = I  R + jωL − 
2 2 2  ωC 
 Resonant frequency
1
ω=
LC
5 EM Wave Lab

6. Microwave Engineering
1. LC resonator
Quality factor
  Definition
Average energe stored
Q =ω
Energy loss/second
 3 dB bandwidth
f0
Q=
BW
 Q in terms of R, L, C
2Wm ω 0 L 1
Q = ω0 = =
Pl R ω 0 RC
6 EM Wave Lab

7. Microwave Engineering
1. LC resonator
Perturbation
  Input impedance
 ω 2 − ω0 
2
Z in = R + jωL
 ω 2  ≈ R + j 2 L∆ω

 
7 EM Wave Lab

8. Microwave Engineering
1. LC resonator
Parallel resonator
  R, L, C
 Input admittance
1 j
Yin = − + j ωC
R ωL
 Input power
1 * 1 * 2 1 2 1 j 
Pin = VI = Yin V = V  + − j ωC 
2 2 2  R ωL 
 Resonant frequency
1
ω=
LC
8 EM Wave Lab

9. Microwave Engineering
1. LC resonator
Quality factor
  Q in terms of R, L, C
2Wm R
Q = ω0 = = ω 0 RC
Pl ω0 L
9 EM Wave Lab

10. Microwave Engineering
1. LC resonator
Perturbation
  Input admittance
1  ω 2 − ω0  1
2
Yin = + jωC 
 ω 2  ≈ R + j 2C∆ω

R  
10 EM Wave Lab

11. Microwave Engineering
1. LC resonator
Loaded Q
  Unloaded Q: resonant circuit itself
 Loaded Q: External load resistor
1 1 1
= +
QL Qe Q
11 EM Wave Lab

12. Microwave Engineering
2. Tx line resonator
Short-circuited half-wave line
  Transmission line
 Input impedance: lossy medium
Z in = Z 0 tanh (α + jβ ) l
tanh(αl ) + j tan( βl )
= Z0
1 + j tan( βl ) tanh(αl )
12 EM Wave Lab

13. Microwave Engineering
2. Tx line resonator
Approximation
  Low-loss transmission line
tanh(αl ) ≈ αl
 Phase: ω = ω 0 + ∆ω , l = λ / 2
∆ωπ ∆ωπ
tan( βl ) = tan(π + )≈
ω0 ω0
13 EM Wave Lab

14. Microwave Engineering
2. Tx line resonator
Equivalence
  Input impedance
αl + j (∆ωπ / ω 0 )  ∆ωπ 
Z in ≈ Z 0 ≈ Z 0 αl + j
 

1 + j (∆ωπ / ω 0 )αl  ω0 
= R + 2 jL∆ω
 Quality factor
ω0 L π β
Q= = =
R 2αl 2α
14 EM Wave Lab

15. Microwave Engineering
2. Tx line resonator
Open-circuited half-wave line
  Transmission line
 Input impedance: lossy medium
Z in = Z 0 coth (α + jβ ) l
1 + j tan( βl ) tanh(αl )
= Z0
tanh(αl ) + j tan( βl )
15 EM Wave Lab

16. Microwave Engineering
2. Tx line resonator
Approximation
  Low-loss transmission line
tanh(αl ) ≈ αl
 Phase: ω = ω 0 + ∆ω , l = λ / 2
∆ωπ ∆ωπ
tan( βl ) = tan(π + )≈
ω0 ω0
16 EM Wave Lab

17. Microwave Engineering
2. Tx line resonator
Equivalence
  Input impedance
1 + j (∆ωπ / ω 0 )αl Z0
Z in ≈ Z 0 ≈
αl + j (∆ωπ / ω 0 ) αl + j (∆ωπ / ω 0 )
1
=
1 / R + 2 jC∆ω
 Quality factor
π β
Q = ω 0 RC = =
2αl 2α
17 EM Wave Lab

18. Microwave Engineering
3. Waveguide cavity
Rectangular waveguide
  Metallic wall
 Propagation constant
2 2
 mπ   nπ 
β mn = k −2
 − 
 a   b 
 Resonant condition
β mn d = lπ
18 EM Wave Lab

19. Microwave Engineering
3. Waveguide cavity
Resonant wavenumber
  Resonant wavenumber
2 2 2
 mπ   n π   l π 
k mnl =   +  + 
 a   b  d 
 TE101 mode and TM110 mode
 Q of cavity
2We
Q = ω0
Pl
19 EM Wave Lab

20. Microwave Engineering
3. Waveguide cavity
Circular waveguide
  Metallic wall
 Propagation constant
 Resonant condition
β mn d = lπ
 TE111 mode and TM110 mode
20 EM Wave Lab

21. Microwave Engineering
4. Dielectric cavity
Dielectric material
 

High Q
Fringing field
 High permittivity: magnetic wall
 Mechanical tuning
 TE01δ mode
 Notation
δ = 2 L / λg < 1
21 EM Wave Lab

22. Microwave Engineering
5. Mirror
Fabry-Perot resonator
 

Two mirrors
High Q
 Laser
 Millimeter and optical applications
22 EM Wave Lab

23. Microwave Engineering
Microwave Filter
23 EM Wave Lab

24. Microwave Engineering
1. Filter
Characteristics
 2 port network: S parameters
Pass band and stop band
Return loss and insertion loss
Ripple and selectivity (skirt)
Pole and zero
Group delay
24 EM Wave Lab

25. Microwave Engineering
1. Filter
Characteristics
  Phase response
 Signal distortion
25 EM Wave Lab

26. Microwave Engineering
1. Filter
Classification
 

LPF (Low Pass Filter)
HPF (High Pass Filter)
 BPF (Band Pass Filter)
 BSF (Band Stop Filter): notch filter
26 EM Wave Lab

27. Microwave Engineering
1. Filter
Filter response
 

Maximally flat (Butterworth) filter
Chebyshev filter
 Elliptic function filter
 Bessel function filter
27 EM Wave Lab

28. Microwave Engineering
2. Filter design
Design process
 

Filter specifications
Design of low pass filter
 Scaling and conversion
 Design of transmission line
 Implementation
28 EM Wave Lab

29. Microwave Engineering
2. Filter design
Insertion loss method
  Precise design method
 Power loss ratio: transducer gain
Power available from source 1
PLR = = 2
Power delivered to load 1 − Γ(ω )
 Reflection coefficient
2 M (ω 2 )
Γ(ω ) =
M (ω 2 ) + N (ω 2 )
 Results: M (ω 2 )
PLR = 1 +
N (ω 2 )
29 EM Wave Lab

30. Microwave Engineering
2. Filter design
Filter responses: LPF
  Maximally flat response
2N
ω 
PLR = 1 + k  
2
ω 
 c
 Equal ripple response
ω 
PLR = 1 + k T  
2
ω 
2
N
 c
 Chebyshev polynomial
TN (cosθ ) = cos(nθ )
30 EM Wave Lab

31. Microwave Engineering
2. Filter design
Example
  Design 2-poles low pass filter in terms of the
insertion loss method where ω c = 1, Z S = 1
PLR = 1 + ω 4
Z L (1 − jωZ L C )
Z in = jωL +
1 + (ωZ L C ) 2
31 EM Wave Lab

32. Microwave Engineering
2. Filter design
Impedance scaling
Z L (1 − jωZ L C )
 L′ = Z 0 L Z in = jωL +
1 + (ωZ L C ) 2
C
C′ = Example
Z0 ω0 L
Q=
′
Zs = Z0 R
′
Z L = Z0Z L 1
ω0 =
LC
Series RLC resonator
32 EM Wave Lab

33. Microwave Engineering
2. Filter design
Frequency scaling for LPF
  Basic equation
ω 
PLR (ω ) = PLR  
′ ω 
 c
ω L
jX = j L = j ωL ′  L′ =
ωc ωc
ω C
jB = j C = jωC ′ C′ =
ωc ωc
33 EM Wave Lab

34. Microwave Engineering
2. Filter design
Frequency scaling for HPF
  Basic equation
 ωc 
′
PLR (ω ) = PLR  − 
 ω 
ωc 1 1
jX = − j L=  L′ =
ω j ωC ′ ω cC
ωc 1 1
jB = − j C = C′ =
ω j ωL ′ ωc L
34 EM Wave Lab

35. Microwave Engineering
2. Filter design
Frequency scaling for BPF
  Basic equation: ω 0 = ω1ω 2
  ω ω0  
′  Q
PLR (ω ) = PLR   −  , Q = ω 0
  ω0 ω  
 ω 2 − ω1
 ω ω0  1
 jX = jQ
 ω − ω  L = jωL1′ − jωC ′

 0  1
 ω ω0  1
jB = jQ
ω − C = jωC2 − ′
 0 ω 
 jωL2′
35 EM Wave Lab

36. Microwave Engineering
2. Filter design
Frequency scaling for BSF
  Basic equation: ω 0 = ω1ω 2
 1 ω ω  
−1
′
PLR (ω ) = PLR   0 −  , Q = ω 0
 Q  ω ω0  
   ω 2 − ω1

−1 −1
j  ω0 ω   1 
 jX =  −  L =  ωL′ − ωC1′ 
j 
Q  ω ω0 
   1 
−1 −1
j  ω0 ω   1 
jB =  −  C= j ′
 ωC ′ − ωL2 
Q  ω ω0 
   2 
36 EM Wave Lab

37. Microwave Engineering
2. Filter design
Example
  Design 5-poles low pass filter with a cutoff
frequency of 2 [GHz], impedance = 50 [Ohms],
insertion loss = 15 dB at 3 [GHz]
g1 = 0.618
g 2 = 1.618
g3 = 2
g 4 = 1.618
g 5 = 0.618
Maximally flat response
37 EM Wave Lab

38. Microwave Engineering
2. Filter design
Richard’s transformation
  Transformation
 ωl 
Ω = tan( βl ) = tan 
v 
 p
 Input impedance: stub
jX = jΩL = jL tan( βl )
jB = jΩC = jC tan( βl )
38 EM Wave Lab

39. Microwave Engineering
2. Filter design
LC to stubs
 jX = jΩL = jL tan( βl )
 jB = jΩC = jC tan( βl )
39 EM Wave Lab

40. Microwave Engineering
2. Filter design
Stub characteristics
  Resonance: wavelength/8 related to the cutoff
frequency
Ω = 1 = tan( βl )
 Attenuation pole: wavelength/4
 Period: wavelength/2
40 EM Wave Lab

41. Microwave Engineering
2. Filter design
Kuroda’s identity
  Stub transformation: shunt and series stub
 Series to shunt stub transform: microstrip line
Z2
N = 1+
Z1
 Implementation
41 EM Wave Lab

42. Microwave Engineering
2. Filter design
Kuroda’s identity

Stub transformation
42 EM Wave Lab

43. Microwave Engineering
2. Filter design
Equivalent transmission line
  Series to shunt stub transform: microstrip line
 Implementation: realization
43 EM Wave Lab

44. Microwave Engineering
3. Implementation
Materials
 

Microstrip line
Dielectric resonator
 Waveguide
 Semiconductor
 MEMS (Micro ElectroMechanical System)
 LTCC (Low Temperature Cofired Ceramic)
 SAW (Surface Acoustic Wave)
 FBAR (Film Bulk Acoustic Resonator)
 Superconductor
44 EM Wave Lab