Filters

Select the frequencies you want to receive.

Avoid transmitting out of band energy

1

2

Major Ideal Filter Types

LPF HPF

BPF BSF

3

Major Filter Implementations

• Active • Inductor & Capacitor (Lumped Element) • Stripline and microstrip (Distributed Element) • Ceramic LTCC • Cavity • Piezoelectric

– Quartz crystal – Ceramic – SAW – BAW – FBAR

• YIG

4

Active Filters • Generally incorporate op-amps

• Limited to relatively low frequencies due to gain-bandwidth limitations

• Examples: – Sallen-Key

– State Variable

– Biquad

• Refer to the Analog Devices Linear Circuit Design Handbook Chapt 8 for excellent discussion.

https://www.analog.com/en/education/education-library/linear-circuit-design-handbook.html 5

Conversions

• It is common for filters to be initially thought of as low pass filters.

• The low pass filter model can then be converted through various techniques to other filter forms.

6

LC Filters

• Well suited to mid-range frequencies

• Excellent design flexibility

• Limitations: – Component parasitics and dimensions

• Component dimensions < λ/10

• Inductors operating well below their SRF

– Temperature coefficients, particularly capacitors

– Losses at high frequencies

– Space requirements may be larger than other options

• Concepts may apply to other filter implementations

7

A general LC filter topology

ZP1 ZP2 ZPn

ZS2 ZS1 ZS(n-1) ZSn

Optional Optional

8

Inductor Z and Y 𝑍 = 𝑗𝜔𝐿

𝑌 =1

𝑗𝜔𝐿=−𝑗

𝜔𝐿

L1 chosen such that Z = 100Ω @ 10 MHz

9

Capacitor Z and Y 𝑍 =1

𝑗𝜔𝐶=−𝑗

𝜔𝐶

𝑌 = 𝑗𝜔𝐶

C1 chosen such that Z = 100Ω @ 10 MHz

10

Series LC Z and Y

𝑍 = 𝑗𝜔𝐿 − 𝑗𝜔𝐶

𝑍 = 0 when 𝜔0𝐿 =1

𝜔0𝐶

𝜔02 =1

𝐿𝐶 𝜔0 =

1

𝐿𝐶

𝑌 =𝑗𝜔𝐶

1 − 𝜔2𝐿𝐶 If 𝜔 = 𝜔0 then 𝑌 → ∞ 11

Parallel LC Z and Y

𝑌 = 𝑗𝜔𝐶 − 𝑗𝜔𝐿

𝑍 =𝑗𝜔𝐿

1 − 𝜔2𝐿𝐶

𝑌 = 0 when 𝜔0𝐿 =1

𝜔0𝐶

𝜔02 =1

𝐿𝐶 𝜔0 =

1

𝐿𝐶

If 𝜔 = 𝜔0 then 𝑍 → ∞

12

Fundamental Definition of Q

𝑄 = 2𝜋𝑓0Energy stored in the resonator at 𝑓0Average power lost in the resonator

𝑄 = 2𝜋Energy stored in a resonator

Energy dissipated per cycle

13

Definitions of Q First order circuit

𝑄 =𝑋𝐿𝑅=𝜔𝐿

𝑅 𝑄 =

𝐵𝐶𝐺=𝜔𝐶

𝐺

Series L-R Parallel C-G

14

Definitions of Q for Second Order Circuit

(a) Parallel (b) Series

𝑄 =1

𝑅

𝐿

𝐶 𝑄 = 𝑅

𝐶

𝐿

Z =𝑅

1 + 𝑗𝑄𝜔𝜔0−𝜔0𝜔

Y =1𝑅

1 + 𝑗𝑄𝜔𝜔0−𝜔0𝜔

15

Magnitude and Phase vs. Q & ζ

Q =1

2휁

𝑄 =𝜔02

d𝜙

d𝜔

16

Definitions of Q

Q =𝑓0𝐵𝑊

17

Typical LC Filter Topologies

18

General Filter Model

Network

→ ←

Γ1 =𝑍1 − 𝑅𝑆𝑍1 + 𝑅𝑆

h(t)

H(s)

Power delivered to load: 𝑃𝐿 =𝑉22

𝑅𝐿

Power delivered by source into a matched load: 𝑃𝑆 =𝑉𝑆2

4𝑅𝑆=𝑉12

𝑅𝑆

19

A little more analysis

The Transfer Function 𝑇 =𝑃𝐿𝑃𝑆= 2𝑅𝑆𝑅𝐿

𝑉2𝑉𝑆=𝑅𝑆𝑅𝐿

𝑉2𝑉1

Γ1 =𝑍1 − 𝑅𝑆𝑍1 + 𝑅𝑆

𝑃𝐿 =𝑉22

𝑅𝐿

From the previous page:

Note that: 𝑇 =𝑉2𝑉1= 𝑆21 and if Γ1 = 𝑆11

𝑃𝑆 =𝑉𝑆2

4𝑅𝑆=𝑉12

𝑅𝑆

𝑅𝐿 = 𝑅𝑆 then

If the filter is lossless, comprised of purely ideal reactive elements, then all power supplied at it input is either passed to the load or reflected back to the source. That means that:

Γ12 + 𝑇2 = 1 20

An example LPF

https://incompliancemag.com/article/s-parameters-and-emi-filters-response/ 21

With different termination impedances

Response of the filter with non 50 ohms terminal impedances.

https://incompliancemag.com/article/s-parameters-and-emi-filters-response/ 22

Primary Classical Filter Types

• Butterworth – Maximally Flat Amplitude

• Bessel – Maximally Flat Group Delay

• Chebyshev filter – Provides steeper slope with ripple in the response

– Type I fast rolloff, ripple in the passband

– Type II slower rolloff, ripple in the stopband

• Elliptic – Fastest rolloff, ripple in both bands

23

A Comparison of Types

https://en.wikipedia.org/wiki/Elliptic_filter 24

Bessel Filter Based on work done by German mathematician Friedrich Bessel (1784 – 1846) W.E. Thomas applied the Besssel functions to filter design in 1949 Maximally flat group delay but slow rolloff. Commonly used in audio crossover circuits

https://en.wikipedia.org/wiki/Bessel_filter https://www.rane.com/note147.html 25

Bessel Filter

𝐻 𝑠 =Θ𝑛 0

Θ𝑛𝑠𝜔0

The low-pass filter transfer function is:

where:

Θ𝑛 0 = lim𝑥→0Θ𝑛 𝑥 Θ𝑛 𝑠 = 𝑎𝑘𝑠

𝑘

𝑛

𝑘=0

𝑎𝑘 =2𝑛 − 𝑘 !

2𝑛−𝑘𝑘! 𝑛 − 𝑘 ! 𝑘 = 0, 1, … , 𝑛

https://en.wikipedia.org/wiki/Bessel_filter 26

Θ𝑛 𝑠 is a reverse Bessel polynomial

where:

𝑛 = number of poles

Butterworth Filter Described in 1930 by British engineer and physicist Stephen Butterworth.

𝐺 𝜔 =1

1 +𝜔𝜔𝑐

2𝑛

https://en.wikipedia.org/wiki/Butterworth_filter 27

A – E => 1-5 poles

Characteristic gain & delay plot for the Butterworth filter

https://en.wikipedia.org/wiki/Butterworth_filter#/media/File:Butterworth3_GainDelay.png 28

Comparison of the Butterworth and Bessel filters

https://en.wikipedia.org/wiki/Bessel_filter#/media/File:Bessel4_GainDelay.png

https://en.wikipedia.org/wiki/Bessel_filter#/media/File:Bessel4_GainDelay.png

29

Butterworth Bessel

Chebyshev Type I Filter

https://en.wikipedia.org/wiki/Chebyshev_filter#/media/File:Chebyshev_Type_I_Filter_Response_(4th_Order).svg

30

ε is the ripple factor

Response of the Type I Chebyshev Filter

𝐺𝑛 𝜔 = 𝐻𝑛 𝑗𝜔 =1

1 + 휀2𝑇𝑛2 𝜔𝜔0

https://en.wikipedia.org/wiki/Chebyshev_filter

휀 = 10𝛿10 − 1

𝑇𝑛 is a Chebyshev polynomial of the 𝑛𝑡ℎ order

31

𝛿 is the passband ripple in dB

Chebyshev Type II Filter

https://en.wikipedia.org/wiki/Chebyshev_filter#/media/File:Chebyshev_Type_I_Filter_Response_(4th_Order).svg

Named after Pafnuty Chebyshev, derived from the Chebyshev polynomials. Type II filters are usually called “inverse Chebyshev filters.”

32

Response of the Type II Chebyshev Filter

https://en.wikipedia.org/wiki/Chebyshev_filter

휀 = 10𝛿10

𝑇𝑛 is a Chebyshev polynomial of the 𝑛𝑡ℎ order

𝐺𝑛 𝜔 = 𝐻𝑛 𝑗𝜔 =1

1 +1

휀2𝑇𝑛2 𝜔0𝜔

33

Elliptic (Cauer) Filter

https://en.wikipedia.org/wiki/Elliptic_filter#/media/File:Elliptic_Filter_Response_(4th_Order).svg

34

Cauer Topologies

https://www.radio-electronics.com/info/rf-technology-design/rf-filters/elliptic-cauer-rf-filter-basics.php

35

Example of an exotic LC Filter

36 http://www.nickc.com/uploaded/NIC_catalog.pdf

Transition between types

The elliptic filter is characterized by the ripple in both pass-band and stop-band as well as the fastest transition between pass-band and ultimate roll-off of any RF filter type. The levels of ripple in the pass-band and stop-band are independently adjustable during the design. As the ripple in the stop-band approaches zero, the filter becomes a Chebyshev type I filter, and as the ripple in the pass-band approaches zero, it becomes a Chebyshev type II filter. If the ripple in both stop-band and pass-band become zero, then the filter transforms into a Butterworth filter.

https://www.radio-electronics.com/info/rf-technology-design/rf-filters/elliptic-cauer-rf-filter-basics.php 37

Distributed Element Filters

• Stripline or Microstrip

• Typically used for frequencies > 1 GHz (λ < 30 cm)

• PCB geometry limitations and losses can be problematic in the multi-GHz range (> 10 GHz)

More info available at https://en.wikipedia.org/wiki/Distributed_element_filter

38

Stripline Filters

https://www.mwrf.com/rf-classics/design-strip-line-band-pass-filters

λ/2 end-coupled filter

λ/2 side-coupled filter

39

Microstrip Examples

Figure 1. A circuit featuring many of the filter structures described in this article. The operating frequency of the filters is around 11 gigahertz (GHz). This circuit is described in the box below.

The PCB inside a 20GHz Agilent N9344C spectrum analyser showing various microstrip distributed element filter technology elements

https://en.wikipedia.org/wiki/Distributed_element_filter 40

Equivalent Circuits

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_filter_lumped_equivalents.svg 41

K and J are impedance and admittance transformers respectively.

42

Low Pass Filter Example

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Microstrip_Low_Pass_Bowtie_Stub_Filter.jpg 43

LPF and equivalent lumped circuit

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_filter_low-pass_form_1.svg 44

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_filter_low-pass_form_2.svg 45

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_filter_low-pass_stubs.svg 46

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_capacitive_gap_filter.svg

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_parallel-coupled_lines_filter.svg

Capacitive gap BPF Limited BW (Q>5) for practical implementation

47

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_hairpin_filter_2.svg 48

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Microstrip_Hairpin_Filter_And_Low_Pass_Stub_Filter.jpg

49

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Stripline_stub_filter.svg

50

Coupled Line filters from a

spectrum analyzer PCB

https://en.wikipedia.org/wiki/Distributed_element_filter#/media/File:Interdigital_Filter.jpg 51

Example of Combline Filter

52

LTCC Ceramic Filters

• Ceramic substrates provide high ε’ and low ε”

• This allows for smaller geometries and lower losses

• ε’ can range from 4.5 to around 100.

• Tradeoffs include temperature coefficients and dielectric losses (ε”).

• As frequencies increase, lower ε’ becomes more appropriate.

• Designs are proprietary

http://www.magneticsgroup.com/pdf/p18-25%20Dielectr.pdf 53

Mini-Circuits BFCN-1445+ LTCC Band Pass Filter, 1420-1470 MHz

https://www.minicircuits.com/pdfs/BFCN-1445+.pdf https://www.minicircuits.com/WebStore/dashboard.html?model=BFCN-1445%2B

Quantity Unit Price

20 $3.95

50 $3.75

100 $3.55

200 $3.35

500 $3.25

54

Mini-Circuits

BFCN-8650+

LTCC Band Pass Filter,

8550-8750 MHz

https://www.minicircuits.com/pdfs/BFCN-8650+.pdf

Quantity Unit Price

20 $3.95

50 $3.75

100 $3.55

200 $3.35

500 $3.25

https://www.minicircuits.com/WebStore/dashboard.html?model=BFCN-8650%2B 55

Cavity Filter

56

http://rfcas.eps.uam.es/web/sites/default/files/trabajos_academicos/AnaMoranLopez_Filter_Design_In_CoaxialCavities.PDF 57

http://rfcas.eps.uam.es/web/sites/default/files/trabajos_academicos/AnaMoranLopez_Filter_Design_In_CoaxialCavities.PDF

Combline Filter Design

58

ZVBP-11G3-S+ Connectorized Band Pass Filter, 11200-11400 MHz Connector Type: SMA

Quantity Unit Price

1 - 4 $340.00

5 or more $340.00

https://www.minicircuits.com/pdfs/ZVBP-11G3+.pdf

https://www.minicircuits.com/WebStore/dashboard.html?model=ZVBP-11G3-S%2B

59

Quartz Crystal Filter

https://www.radio-electronics.com/info/rf-technology-design/receiver-selectivity/monolithic-crystal-filters.php

Very high Q, 10,000 to 100,000 Frequencies up to ~ 75 MHz

60

Murata examples of crystal filters

61 https://www.murata.com/en-us/products/filter/crystalfil/xdca

Examples of Murata Crystal Filter Performance

62

Ceramic Resonator Filters

https://www.radio-electronics.com/info/rf-technology-design/receiver-selectivity/ceramic-if-rf-bandpass-filters.php 63

64

65

CSBP-D1228+ Ceramic Resonator Band Pass Filter, 1203-1253 MHz

Quantity Unit Price

1 - 4 $29.95

5 - 9 $29.95

10 - 24 $28.95

25 - 49 $26.95

50 - 99 $25.95

100 or more

$23.95

https://www.minicircuits.com/pdfs/CSBP-D1228+.pdf https://www.minicircuits.com/WebStore/dashboard.html?model=CSBP-D1228%2B 66

Typical Ceramic Resonator filter Performance

67

N = Number of Sections

http://www.nickc.com/uploaded/NIC_catalog.pdf

BAW-SAW Filter Options

https://cdn.embedded.com/ContentEETimes/Images/EDN1Wireless/13MayJune/C1000Figure%202%20Alternative_Apr%2024%202013.jpg 68

SAW Resonator

• Utilizes Surface Acoustic Waves

• “Acoustic” in this context refers to energy transferred through mechanical vibrations.

• The acoustic waves travel on the surface of the piezoelectric material

• Typical materials: – Lithium Niobate – LiNbO3

– Lithium Tantalate – LiTaO3

– Aluminum Nitrate – AlN

69

SAW Resonator Operation

• Wavelength of traveling wave is

• Constructive interference happens when the

finger spacing 𝑑𝐹 =𝜆

2 or 𝑓 =

𝑣

2𝑑𝐹

Fd

vf

2

𝜆 =𝑣

𝑓

70

SAW Properties

• Generally bandpass filters

• Typical frequency range is 0.1 to 2.5 GHz

• Typical insertion loss 1 dB to 3 dB

71

TDK B3714

https://en.rf360jv.com/inf/40/ds/ae/B3714.pdf 72

Murata SF2124E

73

Bulk Acoustic Wave Resonator

• BAW-SMR Solidly Mounted Resonator

• FBAR – Film Bulk Acoustic Resonator

• Both achieve very high Q values, e.g. 2500 @ 2 GHz

• Acoustic waves travel vertically through substrate

74

BAW

https://m.eet.com/media/1182779/c1000fig1new.jpg 75

BAW

https://m.eet.com/media/1182779/c1000fig1new.jpg 76

https://www.qorvo.com/products/p/QPQ1237

https://www.qorvo.com/products/d/da006106

77

Qorvo BAW Frequency Response

78

79

FBAR

https://pdfs.semanticscholar.org/358b/55d82445129e3d47a132292cee213cd09c58.pdf 80

FBAR Improvements

The quality factor Qs at the series resonant frequency in the free-standing air-backed FBAR is calculated to be 1,322. It is about six times larger than that in the FBAR with 0.9m thick Si3N4 as a support diaphragm. The quality factor Qp at the parallel resonant frequency in the case (e) is calculated to be 513, three times that in the case (a). Figure of Merit (FOM) of an FBAR is defined as the product of Q and kt

2. Insertion loss in an FBAR-based filter is inversely proportional to the FOM of the FBARs. In the free-standing air-backed FBAR, we have increased kt

2 twice and the quality factor six times, and filters made out of this kind of resonator are expected to have larger bandwidth and much lower insertion loss compared to that made by an FBAR built on a support diaphragm

http://mems.usc.edu/fbar.htm 81

https://pdfs.semanticscholar.org/358b/55d82445129e3d47a132292cee213cd09c58.pdf 82

http://www.rfwireless-world.com/Terminology/Advantages-and-Disadvantages-of-FBAR-Filter.html

Following are the benefits or advantages of FBAR Filter: ➨It offers lower insertion loss. As a result it consumes less current. This leads to long battery life and fewer dropped calls. ➨It offers steeper filtering. Hence better coexistence with adjacent bands can be achieved. ➨It offers better out-of-band rejection. Hence multi-band capability can be achieved. ➨It offers ultra small size devices. Hence it can be fit easily with other semiconductor chips. ➨FBAR filters operate reliably in high power and worst temperature conditions.

Following are the drawbacks or disadvantages of FBAR Filter: ➨Thermal path for heat generated in the device is crucial in the design of FBAR and BAW-SMR filters. In BAW-SMR type, heat has conduction path into the substrate from which heat can be spread. However in FBAR type, there is air gap on each side of the resonator. Hence in FBAR designs, thermal conduction path is weaker.

83

The ACFM-7024 features a single antenna connection, which eliminates the need for antenna switching. All ports are matched to 50 ohms. The ACFM-7024 is designed with Avago Technologies’ film bulk acoustic resonator (FBAR) technology. The ACFM-7024 also utilizes Avago Technologies’ innovative microcap bonded-wafer, chip scale packaging technology. This process allows the filters to be assembled in a module with a footprint of only 3.6 mm × 2 mm with a maximum height of 0.80 mm.

84

YIG sphere resonator

https://en.wikipedia.org/wiki/YIG_sphere

https://www.microlambdawireless.com/resources/ytfdefinitions2.pdf

85

Features of YIG resonators

• Resonant frequency linearly related to magnetic field

• Wide tuning range – Generally 1 – 3+ octaves

• Can resonate in the range of 1 – 50 GHz

• High Q, typically 100 to 200

• Commonly used for either a filter or oscillator

• Temperature sensitive, commonly use heater

• Filters may have multiple (e.g. 2 – 8) stages

https://www.microlambdawireless.com/components/band-pass-filters/

https://www.microlambdawireless.com/components/wide-tuning-range-oscillators/ 86