on the tuning of multiband microwave filters

ON THE TUNING OF MULTIBAND MICROWAVE FILTERS Petrie Meyer Stellenbosch University, Stellenbosch, South Africa

Contents • Introduction • Reactance Mapping for Multiband Filters • Coupled Resonator Implementations • Tuning of Prototypes using CST and Circuit Analysis • Conclusion

Introduction • There is a growing need to increase the complexity of receivers whilst reducing

size • Multi-band operation is an emerging requirement • Current Approaches

• interconnection of single-band filters • resonators which allow higher order resonant modes • multi-band approximation functions

• New Approach using Reactance Function Transformation • is mathematically rigorous • requires no optimisation, • is unlimited in the number of bands which can be created, • allows pass-band frequencies of each band to be chosen completely

arbitrarily, • results in a set of pass-bands of which each is an exact bandwidth-scaled

and frequency-translated copy of the original low-pass function • leads to circuit topologies which are simple to realise in practice, • is very well suited to coupled-resonator implementations.

• For waveguide filters in particular, the constraints imposed by the physical size and geometries of the structures make it quite challenging to apply most of the current techniques

Reactance Mapping for Multiband Filters

𝐶𝑚 Ω𝑚 = 𝐶𝑠 Ω𝑠 Ω𝑚

• |Ω_s |≤1 should map to pass-bands. • |Ω_s |>1 should not map to pass-bands. • |Ω_s |=0 should map to ±Ω_0n. • |Ω_s |=±∞ should preferably map to frequencies between every pass-band

and to ±∞, as such a mapping will ensure a point of infinite isolation between each adjacent set of pass-bands for a low-pass function with attenuation poles at ±∞.

Reactance Mapping for Multiband Filters

Ω𝑠𝑠(Ω𝑚𝑚 ) = Ωm

2𝑁𝑁 + 𝛼𝛼2𝑁𝑁−2Ωm2𝑁𝑁−2 + ⋯+ 𝛼𝛼2Ωm

2 + 𝛼𝛼0

𝛽𝛽2𝑁𝑁−1Ωm2𝑁𝑁−1 + ⋯+ 𝛽𝛽1Ω𝑚𝑚

Reactance function of a lossless passive one-port network • monotonically increasing, • with poles and zeros interspersed on the frequency axis

Reactance Mapping for Multiband Filters • There is no theoretical limit on the number of pass-bands which can be obtained. • Each pass-band has the exact same reflection and transmission responses as the original

low-pass transfer function in terms of amplitude and phase. • The centre frequency and bandwidth for each band can be chosen completely arbitrarily. • By simply choosing the correct poles and zeros, or equivalently the pass-band edges for

each band, a valid mapping function can be obtained using a single matrix equation. No optimisation is required.

• Given a realisable low-pass filter function, a realisable multi-band filter function is guaranteed due to the nature of the reactance function.

• The multi-band circuit can simply be realised by replacing each reactive element of any low-pass network with a network obtained by synthesizing the mapping function as a scaled impedance or admittance one-port. The process places no restrictions on the low-pass function, therefore any low-pass network can be used.

• No complicated cross-couplings are incurred by the low-pass to multi-band mapping, as each reactive element is realised in isolation.

• The final multi-band function needs never be realised directly, only the low-pass function and an impedance or admittance in the form of the mapping function.

• The mapping function lends itself very well to implementation as a coupled-resonator multi-band filter, as shown in the next section.

• The proposed function simply reduces to the classical low-pass to band-pass transform for the N=1 case.

Coupled Resonator Implementations

𝑠𝑠′ = s2𝑁+ 𝛼2𝑁−2𝑠2𝑁−2+⋯+ 𝛼2s2+ 𝛼0

𝛽2𝑁−1s2𝑁−1+⋯+ 𝛽1s

= 𝑠𝑠𝑘𝐴∞ +1

𝑠𝑠𝑘𝐴𝐴+

1

𝑠𝑠𝑘𝐵∞ + 1𝑠𝑠𝑘𝐵𝐴

+ 1𝑠𝑠𝑘𝐶∞ + 1

𝑠𝑠𝑘𝐶𝐴

+ ⋯

A capacitor with Y(s’)=s’C or an inductor with Z(s’)=s’L is therefore expanded as a series of LC admittances or impedances coupled by impedance inverters

Coupled Resonator Implementations

Realization of Prototypes

Problems • Degrees of freedom is

multiplied by N • Normal coupling

calculation cannot be used due to N frequencies

For a three-band, third order filter we have to optimize 19 variables to get to the ideal

Realization of Prototypes

Approach • Design cavities using

equal width waveguide • Use subdivision of the

circuit

Realization of Prototypes using Subdivision

Realization of Prototypes using Subdivision

Problems • Due to the wide bandwidth,

coupling apertures are very large and sections cannot be isolated by simply grounding the tuning screws

• Adding new sections change the behaviour at all frequencies

• Optimizing for amplitude response is not unique

A Combined Realization and Tuning Approach • Ground plates • Group Delay • Parameter extraction

Tuning of Prototypes using CST and Circuit Analysis

CST Analysis Circuit optimization to fit EM analysis

Update EM model


Tuning of first branch


Problem for error function


Even worse


A lot better if the order of the main chain and the branches are kept the same


However, if one resonator is detuned by 2%, very strong spikes appear This is due to the arms being identical, with small imbalances causing large effects Can only be mitigated by choosing other values randomly until the spikes disappear

Results

8.5 9 9.5 10 10.5Frequency (GHz)

-40

-30

-20

-10

0

|S11

, S21

|dB 10 GHz

-20.03 dB

9.8 GHz-20 dB

9.6 GHz-20.03 dB9.4 GHz

-20.22 dB9.2 GHz-20.33 dB

9 GHz-19.97 dB

|S11|dB - Ideal

|S21|dB - ideal

|S11|dB - CST

|S21|dB - CST

Conclusion

A rigorous mapping technique for creating multiband filters was developed Realizing these filters is difficult, as • Resonators are at different frequencies • Coupling coefficients cannot be calculated in the normal way We propose a systematic design loop using sections of the fllter Care must be taken to • Use certain specific sections together • Ensure unwanted spikes in the group delay do not occur

Thank You

on the tuning of multiband microwave filters

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