ELCT564
Spring 2012
Chapter 8: Microwave Filters
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Filters
• 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
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Periodic Structures
Passband
Stopband
Bloch Impedance
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Terminated Periodic Structures
Symmetrical network
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Analysis of a Periodic Structure
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.
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Image Parameter Method
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Constant-k Filter
m-derived section
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Composite Filter
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Summary of Composite Filter Design
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Example of Composite Filter Design
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.
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Insertion Loss Method
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
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Insertion Loss:
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Filter Responses
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)
T (x)= cos (Ncos-1x), |X| ≤1
N
TN(x)= cosh (Ncosh-1x), |X|≥ 1
Much better out-of-band rejection than maximally
flat response of the same order. Chebyshev filters
are preferred a lot of times.
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Filter Responses
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
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Filter Design Method
• 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 bandstop
• Filter implementation in a desired from (microstrip or CPW)
use implementation transformations.
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Maximally Flat Low-Pass Filter
g0=1,ωc=1, N=1 to 10
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Equal-Ripple Low-Pass Filter
g0=1,ωc=1, N=1 to 10
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Maximally-Flat Time Delay Low-Pass Filter
g0=1,ωc=1, N=1 to 10
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Filter Transformations
• Impedance Scaling
• Frequency Scaling for Low-Pass Filters
• Low-Pass to High-Pass Transformation
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Filter Implementation
• 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
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Design Steps
• 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).
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Low-pass Filters Using 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.
•
•
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Distributed elements—sharper cut-off
Response repeats due to the periodic nature of stubs
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Bandpass and Bandstop Filters
A useful form of bandpass and bandstop filter consists of λ/4 stubs connected by λ/4
transmission lines.
Bandpass filter
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Stepped Impedance Low-pass Filters
• 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
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Stepped Impedance Low-pass Filter Example
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.
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Coupled Line Theory
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Coupled Line Bandpass Filters
• 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
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Coupled Line Bandpass Filters
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
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Coupled Line Bandpass Filters Example I
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
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Coupled Line Bandpass Filters Example II
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
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Capacitively Coupled Resonator Filter
• 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
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Capacitively Coupled Resonator Filter
Design a bandpass filter using capacitive coupled series resonators, with a 0.5 dB
equal-ripple passband characteristic. The center frequency is 2.0 GHz, the bandwidth
is 10%, and the impedance is 50 Ω. At least 20 dB of attenuation is required at 2.2GHz
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Bandpass Filters using Capacitively Shunt Resonators
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Bandpass Filters using Capacitively Shunt Resonators
Design a third-order bandpass filter with a 0.5 dB equal-ripple response using
capacitively coupled short-circuited shunt stub resonators. The center frequency
Is 2.5 GHz, and the bandwidth is 10%. The impedance is 50 Ω. What is the resulting
attenuation at 3.0 GHz?
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