WMB-2-color-Hunter
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WMB-2
PRACTICAL ASPECTS OF MICROWAVE FILTER DESIGN AND REALIZATION Ian Hunter,Dan Swanson,Andy Guyette
University of Leeds Leeds, United Kingdom, LS2 9JT
Outline
•Low pass prototype •Impedance inverters •Circuit transformations •Effect of losses •Input coupling measurement •Inter-resonator coupling •Cross-couplings •Combline filters •Conclusions
1
Lowpass Prototypes
Ladder Networks (Minimum Phase)
Impedance Inverters
[T ] =
0 j/K jK 0
2
Impedance Inverters
Narrowband Approximation
Negative capacitances absorbed in resonators
Circuit Transformations
Impedance Scaling •Convert from 1 Inductors: impedance level to Zo
Z = Lp
Capacitors:
Z 0 Lp = ( Z 0 L) p
Z=
1 Cp
Z0 = Cp
1 C p Z0
3
Circuit Transformations
Bandpass Transform
ω →α ω ω0 − ω0 ω
C
1
α=
ω2 − ω1
ω0
ω0 = (ω1ω2 )
C' =
1
2
αC ω0
L' =
L
L' =
α Cω0
αL ω0
C' =
1
α Lω0
Bandpass Transform
4
Effect of Losses
•Increased passband insertion loss •Decreased selectivity
Effect of Losses
Bandpass Filter Lowpass prototype
Bandpass equivalent
Equivalent circuit of lossy bandpass at midband Midband loss
L= 4.343 f 0 ∆fQu
N r =1
gr
5
Effect of Losses
Bandstop Filter
•Zeroes shift onto real axis •Stopband insertion loss is reduced
Input Coupling Measurement using Reflected Group Delay
1st bandpass resonator based on lowpass prototype values:
C=
α C1 ω0
L=
α=
ω0 ∆ω
1 α C1ω0
C
L
C1 is 1st capacitor in LPP
6
Admittance of C1 considered in isolation:
Y ( jω ) = jωC1
1 − jωC1 1 − ω 2C12 − j 2ωC1 = S11 ( jω ) = 1 + jωC1 1 + ω 2C12
Phase of S11: with
ψ (ω ) = tan −1
2ωC1 ω 2C12 − 1
In bandpass resonator:
ω' = α (
and
ω ω0 − ) ω0 ω
−1
Reflected delay maximum at resonant frequency( ω ' = 0 and ω = ω0 )
ψ (ω ) = tan
'
2ω 'C1 ω '2C12 − 1
Tg max =
4α C1
ω0
=
4C1 ∆ω
Reflected group delay:
−dψ ' (ω ) 2C1 1 ω0 Tg (ω ) = = α + dω 1 + ω '2C12 ω0 ω 2
7
Inter-Resonator Coupling Input impedance of two coupled resonators: jωC2 Z ( jω ) = 2 K12 − ω 2C1C2 Poles occur when:
ωa ,b = ±
K12
Then:
( C1C2 )
1
2
α α
Applying bandpass transformation: ω ω0 ω →α − ω0 ω
ω a ω0 − K12 − = 1 ω0 ω a ( C1C2 ) 2 ωb ω0 K12 − = 1 ω0 ωb ( C1C2 ) 2
Therefore:
ωb − ωa =
or:
K12ω0
α ( C1C2 )
1
2
∆C =
K12 ∆ω
( C1C2 )
1
2
(coupling bandwidth)
8
Inter-Resonator Coupling Measurement of coupling bandwidth
Measurement of Unloaded Q Unloaded Q of resonator:
Qu =
ω0 C
G
ω0C
Input impedance of resonator:
Z in ( jω ) =
ω02C + j ωC − Qu ω
K2 Adjust input coupling K for perfect match at resonance:
K2 =
ω0 C
Qu
9
ω ω0 Z in ( jω ) = 1 + jQu − ω0 ω
3-dB frequencies a and are measured. Imaginary part of Z=2 at these frequencies:
b
Hence:
Solving quadratic equations:
2ω0 ωb − ωa = Qu
Or:
ω ω Qu a − 0 = −2 ω0 ωa ω ω Qu b − 0 = 2 ω0 ωb
Qu =
2ω0 ∆ω
Measurement of Unloaded Q
10
Positive coupling --> 1 zero on imaginary axis above passband Negative coupling --> 1 zero on imaginary axis below passband
Positive coupling: real axis zeroes (group delay equalization) Negative coupling: symmetrically located pair of imaginary-axis zeroes
Top view
1 capacitive (negative) coupling Top view Side view
Coaxial-Resonator Combline (photograph courtesy of Filtronic plc)
H
Dielectric puck
Alumina tube Dielectric Resonator Filters
Input Coupling (Magnetic)
Dielectric resonator
Puck
Wall Support Dielectric Resonator Filters
Opposite Sign Coupling
Coaxial resonator。
