ELCT564 Spring 2012

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Chapter 8: Microwave Filters

Filters

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• Two-port circuits that exhibit selectivity to frequency: allow some frequencies to go through while block the remaining

• In receivers, the system filters the incoming signal right after reception

• Filters which direct the received frequencies to different channels are called multiplexers

• In many communication systems, the various frequency channels are very close, thus requiring filters with very narrow bandwidth & high out-of band rejection

• In some systems, the receive/transmit functions employ different frequencies to achieve high isolation between the R/T channels.

• In detector, mixer and multiplier applications, the filters are used to block unwanted high frequency products

• Two techniques for filter design: the image parameter method and the insertion loss method. The first is the simplest but the second is the most accurate

Periodic Structures

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Passband Stopband

Bloch Impedance

Terminated Periodic Structures

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Symmetrical network

Analysis of a Periodic Structure

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Consider a periodic capacitively loaded line, as shown below. If Zo=50 Ω, d=1.0 cm, and Co=2.666 pF, compute the propagation constant, phase velocity, and Bloch impedance at f=3.0 GHz. Assume k=k0.

Image Parameter Method

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Constant-k Filter

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m-derived section

Composite Filter

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Summary of Composite Filter Design

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Example of Composite Filter Design

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Design a low-pass composite filter with a cutoff frequency of 2MHz and impedance of 75 Ω, place the infinite attenuation pole at 2.05 MHz, and plot the frequency response from 0 to 4 MHz.

Insertion Loss Method

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Filter response is characterized by the power loss ratio defined as:

Where Γ(ω) is the reflection coefficient at the input port of the filter, assuming the the output port is matched.

Low-pass & Band-pass filter Insertion Loss:

Filter Responses

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Maximally Flat, Equal Ripple, and Linear Phase

Maximally Flat: Provides the flattest possible pass band response for a given complexity.

Cutoff frequency is the freqeuncy point which determines the end of the pass band. Usually, where half available power makes it through.

Cut-off frequency is called the 3dB point

Equal Ripple or Chebyshev Filter: Power loss is expressed as Nth order Chebyshev polynomial TN(ω) TN(x)= cos (Ncos-1x), |X| ≤1

TN(x)= cosh (Ncosh-1x), |X|≥ 1

Much better out-of-band rejection than maximallyflat response of the same order. Chebyshev filtersare preferred a lot of times.

Filter Responses

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Linear Phase Filters

• Need linear phase response to reduce signal distortion (very important in multiplexing)

• Sharp cut-off incompatible with linear phase– design specifically for phase linearity

• Inferior amplitude performance• If φ(ω) is the phase response then filter group delay

Filter Design Method

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• Development of a prototype (low-pass filter with fc=1Hz and is made of generic lumped elements)

• Specify prototype by choice of the order of the filter N and the type of its response• Same prototype used for any low-pass, band pass or band stop filter of a given

order.• Use appropriate filter transformations to enter specific characteristics• Through these transformations prototype changes – low-pass, band-pass or band-

stop• Filter implementation in a desired from (microstrip or CPW)

use implementation transformations.

Maximally Flat Low-Pass Filter

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g0=1,ωc=1, N=1 to 10

Equal-Ripple Low-Pass Filter

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g0=1,ωc=1, N=1 to 10

Maximally-Flat Time Delay Low-Pass Filter

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g0=1,ωc=1, N=1 to 10

Filter Transformations

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• Impedance Scaling

• Frequency Scaling for Low-Pass Filters

• Low-Pass to High-Pass Transformation

Filter Implementation

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• Richards’ Transformation

• Kuroda’s Identities

• Physically separate transmission line stubs• Transform series stubs into shunt stubs, or

vice versa• Change impractical characteristic

impedances into more realizable ones

Design Steps

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• Lumped element low pass prototype (from tables, typically)

• Convert series inductors to series stubs, shunt capacitors to shunt stubs

• Add λ/8 lines of Zo = 1 at input and output

• Apply Kuroda identity for series inductors to obtain equivalent with shunt open stubs with λ/8 lines between them

• Transform design to 50Ω and fc to obtain physical dimensions (all elements are λ/8).

Low-pass Filters Using Stubs

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• Distributed elements—sharper cut-off• Response repeats due to the periodic nature of stubs

Design a low-pass filter for fabrication using microstrip lines. The specifications include a cutoff frequency of 4GHz, and impedance of 50 Ω, and a third-order 3dB equal-ripple passband response.

Bandpass and Bandstop Filters

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A useful form of bandpass and bandstop filter consists of λ/4 stubs connected by λ/4transmission lines.

Bandpass filter

Stepped Impedance Low-pass Filters

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• Use alternating sections of very high and very low characteristics impedances• Easy to design and takes-up less space than low-pass filters with stubs• Due to approximations, electrical performance not as good – applications where

sharp cut-off is not required

Stepped Impedance Low-pass Filter Example

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Design a stepped-impedance low-pass filter having a maximally flat response and a cutoff frequency of 2.5 GHz. It is necessary to have more than 20 dB insertion loss at 4 GHz. The filter impedance is 50 Ω; the highest practical line impedance is 120 Ω, and the lowest is 20 Ω. Consider the effect of losses when this filter is implemented with a microstrip substrate having d = 0.158 cm, εr =4.2, tanδ=0.02, and copper conductors of 0.5 mil thickness.

Coupled Line Theory

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Coupled Line Bandpass Filters

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• This filter is made of N resonators and includes N+1coupled line sections

• dn ≈ λg/4 = (λge + λgo)/8

• Find Zoe, Zoo from prototype values and fractional bandwidth

• From Zoe, Zoo Calculate conductor and slot width

• N-order coupled resonator filter N+1 coupled line sections

•Use 2 modes to represent line operation

Coupled Line Bandpass Filters

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1. Compute Zoe, Zoo of 1st coupled line section from

2. Compute eve/odd impedances of nth coupled line section

3. Compute even/odd impedances of (N+1) coupled line section

4. Use ADS to find coupled line geometry in terms of w, s, & βe, βo or εeff,e , εeff,o

5. Compute

Coupled Line Bandpass Filters Example I

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Design a 0.5dB equal ripple coupledline BPF with fo=10GHz, 10%BW & 10-dB attenuation at 13 GHz. Assume Zo=50Ω.

From atten. Graph N=4 ok But use N=5 to have Zo=50 Ω

go=ge=1, g1=g5=1.7058, g2=g4=1.229, g3=2.5408

Coupled Line Bandpass Filters Example II

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Design a coupled line bandpass filter with N=3 and 0.5dB equal ripple response. The center frequency is 2GHz, 10%BW & Zo=50Ω. What is the attenuation at 1.8 GHz

Capacitively Coupled Resonator Filter

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• Convenient for microstrip or stripline fabrication• Nth order filter uses N resonant sections of transmission line with N+1 capacitive

gaps between then.• Gaps can be approximated as series capacitors• Resonators are ~ λg/2 long at the center frequency

Capacitively Coupled Resonator Filter

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Design a bandpass filter using capacitive coupled series resonators, with a 0.5 dBequal-ripple passband characteristic. The center frequency is 2.0 GHz, the bandwidthis 10%, and the impedance is 50 Ω. At least 20 dB of attenuation is required at 2.2GHz

Bandpass Filters using Capacitively Shunt Resonators

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Bandpass Filters using Capacitively Shunt Resonators

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Design a third-order bandpass filter with a 0.5 dB equal-ripple response usingcapacitively coupled short-circuited shunt stub resonators. The center frequencyIs 2.5 GHz, and the bandwidth is 10%. The impedance is 50 Ω. What is the resultingattenuation at 3.0 GHz?