Radio frequency (RF) bulk acoustic wave (BAW) filters of assessed quality - Part 2: Guidelines for the use

This part of IEC 62575 gives practical guidance on the use of RF BAW filters which are used in telecommunications, measuring equipment, radar systems and consumer products. General information, standard values and test conditions will be provided in a future IEC standard1. This part of IEC 62575 includes various kinds of filter configurations, of which the operating frequency range is from approximately 500 MHz to 10 GHz and the relative bandwidth is about 1 % to 5 % of the centre frequency.

Akustische Volumenwellenfilter für Hochfrequenzanwendungen (HF-BAW-Filter) - Teil 2: Leitfaden für die Anwendung

Filtres radiofréquences (RF) à ondes acoustiques de volume (OAV) sous assurance de la qualité - Partie 2: Lignes directrices d'emploi

La CEI 62575-2:2012 donne des conseils pratiques sur l'utilisation des filtres RF à OAV utilisés dans le domaine des télécommunications, des équipements de mesure, des systèmes radar et des produits de grande consommation. Les informations générales, les valeurs normalisées et les conditions d'essai seront fournies dans une future norme CEI. La présente partie de la CEI 62575 inclut différents types de configurations de filtres dont la gamme de fréquences de fonctionnement est comprise entre environ 500 MHz et 10 GHz et la largeur de bande relative est d'environ 1 % à 5 % de la fréquence centrale.

Radiofrekvenčni filtri (RF) za zelo visoke zvočne frekvence (BAW) določene kakovosti - 2. del: Smernice za uporabo

Ta del standarda IEC 62575 določa praktične napotke za uporabo radiofrekvenčnih filtrov (RF) za zelo visoke zvočne frekvence (BAW), ki se uporabljajo na področju telekomunikacij, merilne opreme, radarskih sistemov in potrošniških proizvodov. Splošne informacije, standardne vrednosti in pogoji testiranja bodo vključeni v prihodnji standard IEC standard1. Ta del standarda IEC 62575 zajema različne vrste konfiguracij filtrov s frekvenčnim območjem delovanja od približno 500 MHz do 10 GHz in relativno pasovno širino približno od 1 % do 5 % srednje frekvence.

General Information

Status
Published
Publication Date
29-Oct-2012
Technical Committee
Current Stage
6060 - National Implementation/Publication (Adopted Project)
Start Date
15-Oct-2012
Due Date
20-Dec-2012
Completion Date
30-Oct-2012

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2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.Akustische Volumenwellenfilter für Hochfrequenzanwendungen (HF-BAW-Filter) - Teil 2: Leitfaden für die AnwendungFiltres radiofréquences (RF) à ondes acoustiques de volume (OAV) sous assurance de la qualité - Partie 2: Lignes directrices d'emploiRadio frequency (RF) bulk acoustic wave (BAW) filters of assessed quality - Part 2: Guidelines for the use31.140Piezoelectric and dielectric devicesICS:Ta slovenski standard je istoveten z:EN 62575-2:2012SIST EN 62575-2:2012en01-december-2012SIST EN 62575-2:2012SLOVENSKI
STANDARD



SIST EN 62575-2:2012



EUROPEAN STANDARD EN 62575-2 NORME EUROPÉENNE
EUROPÄISCHE NORM September 2012
CENELEC European Committee for Electrotechnical Standardization Comité Européen de Normalisation Electrotechnique Europäisches Komitee für Elektrotechnische Normung
Management Centre: Avenue Marnix 17, B - 1000 Brussels
© 2012 CENELEC -
All rights of exploitation in any form and by any means reserved worldwide for CENELEC members.
Ref. No. EN 62575-2:2012 E
ICS 31.140
English version
Radio frequency (RF) bulk acoustic wave (BAW) filters
of assessed quality -
Part 2: Guidelines for the use (IEC 62575-2:2012)
Filtres radiofréquences (RF) à ondes acoustiques de volume (OAV)
sous assurance de la qualité -
Partie 2: Lignes directrices d'emploi (CEI 62575-2:2012)
Volumenwellenfilter für Hochfrequenzanwendungen
(HFBAW-Filter) -
Teil 2: Leitfaden für die Anwendung (IEC 62575-2:2012)
This European Standard was approved by CENELEC on 2012-08-29. CENELEC members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration.
Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CENELEC member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation under the responsibility of a CENELEC member into its own language and notified to the CEN-CENELEC Management Centre has the same status as the official versions.
CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Croatia, Cyprus, the Czech Republic, Denmark, Estonia, Finland, Former Yugoslav Republic of Macedonia, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the United Kingdom.
SIST EN 62575-2:2012



EN 62575-2:2012 - 2 - Foreword The text of document 49/994/FDIS, future edition 1 of IEC 62575-2, prepared by IEC/TC 49 "Piezoelectric, dielectric and electrostatic devices and associated materials for frequency control, selection and detection" was submitted to the IEC-CENELEC parallel vote and approved by CENELEC as EN 62575-2:2012.
The following dates are fixed: • latest date by which the document has to be implemented at national level by publication of an identical national standard or by endorsement (dop) 2013-05-29 • latest date by which the national standards conflicting with the document have to be withdrawn (dow) 2015-08-29
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. CENELEC [and/or CEN] shall not be held responsible for identifying any or all such patent rights.
Endorsement notice The text of the International Standard IEC 62575-2:2012 was approved by CENELEC as a European Standard without any modification. In the official version, for Bibliography, the following notes have to be added for the standards indicated:
IEC 60862-1:2003 NOTE Harmonised as EN 60862-1:2003 (not modified). IEC 62047-7:2011 NOTE Harmonised as EN 62047-7:2011 (not modified).
SIST EN 62575-2:2012



IEC 62575-2 Edition 1.0 2012-07 INTERNATIONAL STANDARD NORME INTERNATIONALE Radio frequency (RF) bulk acoustic wave (BAW) filters of assessed quality –
Part 2: Guidelines for the use
Filtres radiofréquences (RF) à ondes acoustiques de volume (OAV) sous assurance de la qualité –
Partie 2: Lignes directrices d’emploi
INTERNATIONAL ELECTROTECHNICAL COMMISSION COMMISSION ELECTROTECHNIQUE INTERNATIONALE T ICS 31.140 PRICE CODE CODE PRIX ISBN 978-2-83220-248-7
® Registered trademark of the International Electrotechnical Commission
Marque déposée de la Commission Electrotechnique Internationale ®
Warning! Make sure that you obtained this publication from an authorized distributor.
Attention! Veuillez vous assurer que vous avez obtenu cette publication via un distributeur agréé. SIST EN 62575-2:2012 colourinside



– 2 – 62575-2 © IEC:2012 CONTENTS FOREWORD . 3 INTRODUCTION . 5 1 Scope . 6 2 Normative references . 6 3 Technical considerations . 6 4 Fundamentals of RF BAW filters . 7 4.1 General . 7 4.2 Fundamentals of RF BAW resonators . 8 4.3 RF resonator structures . 13 4.4 Ladder filters . 15 4.4.1 Basic structure . 15 4.4.2 Principle of operation . 16 4.4.3 Characteristics of ladder filters . 17 5 Application guide . 18 5.1 Application to electronics circuits. 18 5.2 Availability and limitations . 18 5.3 Input levels . 18 6 Practical remarks. 18 6.1 General . 18 6.2 Feed-through signals . 19 6.3 Load and source impedance conditions . 19 7 Miscellaneous . 19 7.1 Soldering conditions . 19 7.2 Static electricity . 19 8 Ordering procedure . 19 Bibliography . 22
Figure 1 – Frequency response of a RF BAW filter . 7 Figure 2 – Applicable range of frequency and relative
bandwidth of the RF BAW filter and the other filters . 8 Figure 3 – Basic BAW resonator structure. 9 Figure 4 – BVD model . 9 Figure 5 – Typical impedance characteristics . 10 Figure 6 – Typical impedance characteristics of RF BAW devices . 12 Figure 7 – Modified BVD model. 13 Figure 8 – FBAR structures . 14 Figure 9 – SMR structure . 15 Figure 10 – Structure of ladder filter . 15 Figure 11 – Equivalent circuit of basic section of ladder filter . 16 Figure 12 – Basic concept of ladder filter . 16 Figure 13 – Typical characteristics of a 1,9 GHz range ladder filter . 17
SIST EN 62575-2:2012



62575-2 © IEC:2012 – 3 – INTERNATIONAL ELECTROTECHNICAL COMMISSION ____________
RADIO FREQUENCY (RF) BULK ACOUSTIC
WAVE (BAW) FILTERS OF ASSESSED QUALITY –
Part 2: Guidelines for the use
FOREWORD 1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising all national electrotechnical committees (IEC National Committees). The object of IEC is to promote international co-operation on all questions concerning standardization in the electrical and electronic fields. To this end and in addition to other activities, IEC publishes International Standards, Technical Specifications, Technical Reports, Publicly Available Specifications (PAS) and Guides (hereafter referred to as “IEC Publication(s)”). Their preparation is entrusted to technical committees; any IEC National Committee interested in the subject dealt with may participate in this preparatory work. International, governmental and non-governmental organizations liaising with the IEC also participate in this preparation. IEC collaborates closely with the International Organization for Standardization (ISO) in accordance with conditions determined by agreement between the two organizations. 2) The formal decisions or agreements of IEC on technical matters express, as nearly as possible, an international consensus of opinion on the relevant subjects since each technical committee has representation from all interested IEC National Committees.
3) IEC Publications have the form of recommendations for international use and are accepted by IEC National Committees in that sense. While all reasonable efforts are made to ensure that the technical content of IEC Publications is accurate, IEC cannot be held responsible for the way in which they are used or for any misinterpretation by any end user. 4) In order to promote international uniformity, IEC National Committees undertake to apply IEC Publications transparently to the maximum extent possible in their national and regional publications. Any divergence between any IEC Publication and the corresponding national or regional publication shall be clearly indicated in the latter. 5) IEC itself does not provide any attestation of conformity. Independent certification bodies provide conformity assessment services and, in some areas, access to IEC marks of conformity. IEC is not responsible for any services carried out by independent certification bodies. 6) All users should ensure that they have the latest edition of this publication. 7) No liability shall attach to IEC or its directors, employees, servants or agents including individual experts and members of its technical committees and IEC National Committees for any personal injury, property damage or other damage of any nature whatsoever, whether direct or indirect, or for costs (including legal fees) and expenses arising out of the publication, use of, or reliance upon, this IEC Publication or any other IEC Publications.
8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is indispensable for the correct application of this publication. 9) Attention is drawn to the possibility that some of the elements of this IEC Publication may be the subject of patent rights. IEC shall not be held responsible for identifying any or all such patent rights. International Standard IEC 62575-2 has been prepared by IEC technical committee 49: Piezoelectric, dielectric and electrostatic devices and associated materials for frequency control, selection and detection. The text of this standard is based on the following documents: FDIS Report on voting 49/994/FDIS 49/999/RVD
Full information on the voting for the approval of this standard can be found in the report on voting indicated in the above table. This publication has been drafted in accordance with the ISO/IEC Directives, Part 2. SIST EN 62575-2:2012



– 4 – 62575-2 © IEC:2012 A list of all the parts in the IEC 62575 series, published under the general title Radio frequency (RF) Bulk acoustic wave (BAW) filters of assessed quality, can be found on the IEC website.
The committee has decided that the contents of this publication will remain unchanged until the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data related to the specific publication. At this date, the publication will be
• reconfirmed, • withdrawn, • replaced by a revised edition, or • amended.
IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates that it contains colours which are considered to be useful for the correct understanding of its contents. Users should therefore print this document using a colour printer.
SIST EN 62575-2:2012



62575-2 © IEC:2012 – 5 – INTRODUCTION RF BAW filters are now widely used in mobile communications. While the RF BAW filters have various specifications, many of them can be classified within a few fundamental categories. Standard specifications, given in IEC 62575, and national specifications or detail specifi-cations issued by manufacturers, define the available combinations of nominal frequency, pass bandwidth, ripple, shape factor, terminating impedance, etc. These specifications are compiled to include a wide range of RF BAW filters with standardized performances. It cannot be over-emphasized that the user should, wherever possible, select his RF BAW filters from these specifications, when available, even if it may lead to making small modifications to his circuit to enable standard filters to be used. This applies particularly to the selection of the nominal frequency. This standard has been compiled in response to a generally expressed desire on the part of both users and manufacturers for guidance on the use of RF BAW filters, so that the filters may be used to their best advantage. To this end, general and fundamental characteristics have been explained in this part of IEC 62575. It is not the aim of this standard to explain theory, nor to attempt to cover all the eventualities which may arise in practical circumstances. This standard draws attention to some of the more fundamental questions, which should be considered by the user before he places an order for an RF BAW filter for a new application. Such a procedure will be the user's insurance against unsatisfactory performance. SIST EN 62575-2:2012



– 6 – 62575-2 © IEC:2012 RADIO FREQUENCY (RF) BULK ACOUSTIC
WAVE (BAW) FILTERS OF ASSESSED QUALITY –
Part 2: Guidelines for the use
1 Scope This part of IEC 62575 gives practical guidance on the use of RF BAW filters which are used in telecommunications, measuring equipment, radar systems and consumer products.
General information, standard values and test conditions will be provided in a future IEC standard1. This part of IEC 62575 includes various kinds of filter configurations, of which the operating frequency range is from approximately 500 MHz to 10 GHz and the relative bandwidth is about 1 % to 5 % of the centre frequency. 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. None. 3 Technical considerations It is of prime interest to a user that the filter characteristics should satisfy a particular specification. The selection of tuning networks and RF BAW filters to meet that specification should be a matter of agreement between user and manufacturer. Filter characteristics are usually expressed in terms of insertion attenuation as a function of frequency, as shown in Figure 1. A standard method for measuring insertion attenuation is described in IEC 60862-1:2003, 5.5.2. Insertion attenuation characteristics are further specified by nominal frequency, minimum insertion attenuation or maximum insertion attenuation, pass-band ripple and shape factor. The specification is to be satisfied between the lowest and highest temperatures of the specified operating temperature range and before and after environmental tests. ___________ 1 This standard (under consideration) is expected to bear the reference number IEC 62575-1. SIST EN 62575-2:2012



62575-2 © IEC:2012 – 7 –
Figure 1 – Frequency response of a RF BAW filter 4 Fundamentals of RF BAW filters 4.1 General The features of RF BAW filters are their small size, light weight, adjustment-free, high stability and high reliability. RF BAW filters add new features and applications to the field of surface acoustic wave (SAW) filters and dielectric resonator filters. Nowadays, RF BAW filters with low insertion attenuation are widely used in various applications in the gigahertz range. RF BAW filters are becoming rapidly popular as miniature and low insertion attenuation filters for mobile communication application. RF BAW resonator filters can realize low insertion attenuation easily and of a smaller size than that of the RF SAW filters with the same bandwidth. Their feasible bandwidth is, however, limited by employing piezoelectric materials, design methods and so on. It is desirable for users to understand these factors for RF BAW resonator filters. This standard explains the principles and characteristics of RF BAW resonator filters.
RF BAW filters usually employ a filter configuration called the ladder filter, which is composed of multiple RF BAW resonators. They are classified into two types: film bulk acoustic resonators and solidly mounted resonators. In Figure 2, the applicable frequency range and relative bandwidth of the RF BAW filters are shown in comparison with those of ceramic, crystal, dielectric, helical, SAW and stripline filters. Frequency
(GHz)
Pass-band ripple
Specified stop-band relative attenuation
Minimum insertion
attenuation
Nominal insertion
attenuation
Reference frequency Centre Cut-off Cut-off Attenuation 0 Specified
pass-band
Attenuation
(dB) IEC
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– 8 – 62575-2 © IEC:2012
Figure 2 – Applicable range of frequency and relative
bandwidth of the RF BAW filter and the other filters 4.2 Fundamentals of RF BAW resonators a) Acoustic resonance When a mechanical impact is applied to a solid surface, acoustic waves are generated, and a portion of their energy is transmitted by propagation of the acoustic waves in the bulk. This type of wave is called the bulk acoustic wave (BAW). Remaining energy may be transferred by acoustic waves propagating along the surface. This type of wave is called the surface acoustic wave (SAW). There are two types of BAWs: the longitudinal or dilatational BAW, with the displacement toward the propagation direction, and the transverse or shear BAW, with the displacement normal to the propagation direction. Acoustic wave velocities in solids are a few hundreds of meters per second to twenty thousands of meters per second. Usually the longitudinal BAW is few times faster than the shear BAW for a given material and orientation. In the case of acoustic wave propagation in a parallel plate, it is known that the plate causes a mechanical resonance (thickness resonance) when the plate thickness h is half-integer times the wavelength λ of acoustic waves propagating in the plate normal to the plate surface, i.e. 2λnh=, where n is an integer and called the order of modes. We obtain mechanical resonance frequencies fr as
)2(rhnVVf==λ
(1) where V is the acoustic wave velocity. Equation (1) indicates that in addition to a lowest-order resonance (n=1) called the fundamental resonance, a series of higher-order (n≠1) ones might be excited. Since fr for n≠1 will be integer times fr for n=1 in this case, higher-order resonances are often called harmonics or harmonic resonances. When the longitudinal BAW is responsible for the thickness resonance, it is called the thickness extensional (TE) resonance but when the shear wave is responsible, it is called the thickness shear (TS) resonance. Stripline filters Relative bandwidth
(%) Dielectric filters Crystal filters Ceramic filters Helical filters SAW filters Frequency
(Hz) 1 M
10 M
100 M
1 G
10 G
100 G 10–3 10–2 10–1 100 101 102 RF BAW filters IEC
1445/12 SIST EN 62575-2:2012



62575-2 © IEC:2012 – 9 – There are also acoustic waves propagating along the plate top surface. When wave energy is well confined near the top surface and influence of the back surface is negligible, the waves are called the surface acoustic waves (SAWs). On the other hand, when wave energy penetrates into the plate and influence of the back surface is not negligible, the waves are called plate waves or Lamb waves. b) Piezoelectric excitation and detection In the case where a piezoelectric plate is sandwiched between two parallel electrodes (see Figure 3), when an electrical voltage E is applied between two electrodes, mechanical force is generated through the piezoelectricity, and acoustic motion will be induced. On the other hand, electrical charges will be induced to the electrodes by electric fields associated with propagating acoustic waves.
Figure 3 – Basic BAW resonator structure An electromechanical equivalent circuit shown in Figure 4 may be deduced from these relations. In Figure 4, C0 is the clumped capacitance originating from the electrostatic coupling between two electrodes, and C1, L1 and R1 are the motional capacitance, inductance and resistance, respectively, originating from mechanical reaction, i.e. elasticity, inertia and damping, respectively. This circuit is called the Butterworth-Van Dyke (BVD) model.
Figure 4 – BVD model Figure 4 implies that mechanical resonances described above can be excited and detected electrically through the electrodes. Namely, this device serves as an electrical resonator. This type of resonator is called the BAW resonator. Proper choice of the piezoelectric material offers small acoustic attenuation, which results in long duration of the mechanical vibration. This mechanical property influences the electrical one as large quality (Q) factor of the electrical resonance circuit. Figure 5 shows typical resonance characteristics calculated by the BVD model. It is seen that a series resonance occurs at a frequency fr where the electrical impedance Z between two electrodes becomes pure resistive and very small. From the BVD model, fr is given by
11r21CLfπ≈
(2) On the other hand, at a frequency fa slightly above fr, a parallel resonance occurs where Z becomes pure resistive and very large. From the BVD model, fa is given by
h Piezoelectric material Upper electrode Lower electrode
L1 C1 R1 C0 IEC
1446/12 IEC
1447/12 SIST EN 62575-2:2012



– 10 – 62575-2 © IEC:2012
110111a)(21−−−+≈CCLfπ
(3) These frequencies are called the resonance and the anti-resonance frequencies, respectively2
Figure 5 – Typical impedance characteristics The capacitance ratio r is often used as a measure of the resonator performance, and is defined by
()[]12ra1−−=ffr
(4) From the BVD model, and r is given by
10CCr=
(5) In the filter design discussed later, r limits achievable fractional frequency bandwidth for filter applications. At frequencies much lower than fr, the resonator is equivalent to a capacitor with the capacitance of )1(1010−+=+rCCC, which is given by hS/ε, where ε is the dielectric constant and S is the electrode area. Thus C0 is adjustable only by S because h is mostly determined by the frequency setting. It is clear from Equation 5 that r indicates weakness of the piezoelectricity. In fact, full wave analysis gives a relation between fr/fa and the electromechanical coupling factor kt2 for the thickness-longitudinal vibration of the piezoelectric material as
)2/tan(/)2/(arar2tffnffnkππ=
(6) When fr ≅ fa, Equations (3) and (5) become ___________ 2
Frequencies fm and fn giving minimum |Z| and maximum |Z| are the frequencies of maximum and minimum admittance or those of minimum and maximum impedance. When Q is large, fm and fn are almost equal to fr and fa, respectively. 104 0,98 0,99 1,00 1,01 1,02 1,03 1,04 Frequency
(GHz)
BVD mBVD fr fa Impedance
| Z| (Ω) 103 102 101 100 10–1 IEC
1448/12 SIST EN 62575-2:2012



62575-2 © IEC:2012 – 11 –
≅≅−even :0odd :/42/1/)(222tarannnkfffπγ
(7) This indicates three important facts: 1) achievable r is limited by kt2 of employed piezoelectric material; 2) even-order overtones cannot be excited electrically; and
3) γ increases rapidly with an increase in n. It should be noted that Equations (6) and (7) are only valid when a uniform piezoelectric layer is sandwiched between two infinitesimally thin electrodes with infinite conductance. Since influence of electrodes is not negligible as will be discussed later, piezoelectric strength of the resonator structure is often characterized by the effective electromechanical coupling factor defined by
)2/tan(/)2/(arar2efftffffkππ=
(8) From the BVD model, the Q factor at fr is given by
11rr2RLfQπ=
(9) and is often referred to as the resonance Q or Qr. We can also evaluate the Q factor at the anti-resonance frequency, and the value is called the anti-resonance Q or Qa. In the filter design, Qr and Qa determine steepness of the pass-band edges for filter applications. For resonator characterization, the figure of merit, M is defined as
rQMr=
(10) In the filter design, M determines achievable minimum insertion attenuation. It is interesting to note that the BVD model indicates that
minrmaxaZCfZCfM00212ππ≅≅
(11) where Zmax and Zmin are electrical impedances of the resonator at fn (≈fa) and fm (≈fr), respectively. Thus Zmax/Zmin called the impedance ratio is also used for the resonator characterization. NOTE 1 This approximated form is valid only when Qr and r are large. c) Secondary effects Basic operation of BAW resonators is simulated fairly well by the use of the BVD model described above. In real devices, however, various secondary effects occur, and their influences shall be well-controlled for device design and production. Significant secondary effects are: 1) Lateral wave propagation At frequencies close to the resonance, Lamb waves are excited and propagate along the surface. If their wave energy is dissipated, it will cause Q reduction of the main resonance. If the resonator structure is designed to confine the wave energy, the resonance Q might be SIST EN 62575-2:2012



– 12 – 62575-2 © IEC:2012 preserved, while it may cause unwanted resonances often called spurious resonances. Since lateral structural size is significantly larger than the BAW wavelength in general, frequency separation between the resonances is narrow. From this property, these spurious resonances are called inharmonics. The top surface of the resonator is sometimes shaped in an irregular polygon to smear out spurious resonance peaks. 2) Parasitic impedances
Ohmic resistances, parasitic capacitances, and inductances of the electrodes and pads are not negligible in radio frequencies (RF). Figure 6 shows polar plot (Smith chart) of the return coefficient Γ of two RF BAW resonators. Γ is given by (Z-R0)/(Z+R0), where Z is the device impedance and R0 is the characteristic impedance of the measurement system. The trace rotates clockwise with the frequency, and leftmost and rightmost points of the trace correspond to the resonance and anti-resonance frequencies, respectively. Series of inharmonics are seen in Figure 6 a). It should be noted that the overtones are only seen above the resonance frequency, and this property called the cut-off indicates they are due to lateral wave propagation. Application of an appropriate technology enables to suppress inharmonics almost completely as shown in Figure 6 b). In addition, it is seen that the trace approaches to the outermost circle, namely |Γ| close to unity. This indicates the lateral wave propagation can be one of the most significant loss mechanisms.
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