6-9 GHz Low-Noise Amplifier Design and Implementation

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(1)LiU-ITN-TEK-A--10/047--SE. 6-9 GHz Low-Noise Amplifier Design och Implementering Mohammad Billal Hossain 2010-06-14. Department of Science and Technology Linköping University SE-601 74 Norrköping, Sweden. Institutionen för teknik och naturvetenskap Linköpings Universitet 601 74 Norrköping.

(2) LiU-ITN-TEK-A--10/047--SE. 6-9 GHz Low-Noise Amplifier Design och Implementering Examensarbete utfört i Electronics vid Tekniska Högskolan vid Linköpings universitet. Mohammad Billal Hossain Handledare Adriana Serban Examinator Adriana Serban Norrköping 2010-06-14.

(3) Upphovsrätt Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare – under en längre tid från publiceringsdatum under förutsättning att inga extraordinära omständigheter uppstår. Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut enstaka kopior för enskilt bruk och att använda det oförändrat för ickekommersiell forskning och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande. För att garantera äktheten, säkerheten och tillgängligheten finns det lösningar av teknisk och administrativ art. Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart. För ytterligare information om Linköping University Electronic Press se förlagets hemsida http://www.ep.liu.se/ Copyright The publishers will keep this document online on the Internet - or its possible replacement - for a considerable time from the date of publication barring exceptional circumstances. The online availability of the document implies a permanent permission for anyone to read, to download, to print out single copies for your own use and to use it unchanged for any non-commercial research and educational purpose. Subsequent transfers of copyright cannot revoke this permission. All other uses of the document are conditional on the consent of the copyright owner. The publisher has taken technical and administrative measures to assure authenticity, security and accessibility. According to intellectual property law the author has the right to be mentioned when his/her work is accessed as described above and to be protected against infringement. For additional information about the Linköping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: http://www.ep.liu.se/. © Mohammad Billal Hossain.

(4) 6-9 GHz Low-Noise Amplifier Design and Implementation Mohammad Billal Hossain June 14, 2010.

(5) Preface This report is the result of Master of Science Thesis carried out at the Department of Science and Technology (ITN) in Linköping University. I would like to take a chance to thank the people who have helped and encouraged me during this Master Thesis work. First of all, I want to express my gratitude to Adriana Serban, for giving me the opportunity to perform my Master Thesis at ITN department and for being my supervisor. This thesis work could not have been accomplished without her guidance and full assistance. I would also like to thank professors in ITN department Shaofang Gong, Magnus Karlsson and Allan Huynh for their suggestion and comments.. i.

(6) Abstract Low-noise amplifier design (LNA) is a critical step when designing a receiver frontend. For the broadband technologies and particularly ultra-wideband (UWB) system, designing the LNA becomes more challenging. This master thesis mainly focuses on the LNA design for the European UWB recommendation, i.e. LNA covering the 6 - 9 GHz spectrum. Moreover, better understandings of the design process in correlation with the implementing of the LNA on a printed circuit board (PCB) were expected. The LNA was manufactured, assembled and measured with network analyzer. This report presents a complete functional design of an UWB LNA.. ii.

(7) List of Abbreviations ADS BPFs BW CAD EDA EMI EIRP FCC FM-UWB GaAs HBTs HEMTs JFETs LNA MC-UWB MW NF OFDM PSD RF RFI RFID RSC SMDs S-parameters UMTS UWB VSWR WPAN. Advanced Design System Bandpass Filters Bandwidth Computer Aided Design Electronic Design Automation Electromagnetic Interference Equivalent Isotropically Radiated Power Federal Communication Commission Frequency Modulation UWB Gallium Arsenide Hetero-junction Bipolar Transistors High Electron Mobility Transistors Junction Field Effect Transistors Low-Noise Amplifier Multi Carrier UWB Microwave Noise Figure Orthogonal Frequency Division Multiplexing Power Spectral Densities Radio Frequency Radio Frequency Interference Radio Frequency Identification Radio Spectrum Committee Surface Mount Devices Scattering Parameters Universal Mobile Telecommunication System Ultra-wideband Voltage Standing Wave Ratio Wireless Personal Area Network. iii.

(8) List of Figures Figure 2- 1 Diagram of band allocation [7] Figure 2- 2 FCC emission limit for outdoor UWB communications [3] Figure 2- 3 FCC emission limit for indoor UWB communications [3] Figure 2- 4 Spectrum of the Main Interfering Communication Standards for UWB Communication System [11] Figure 2- 5 Skin depth area of a wire [13]. Figure 2- 6 Electric equivalent circuit representation of the resistor [15]. Figure 2- 7 Absolute impedance value of a 500 ohm thin-film resistor as a function of frequency [15]. Figure 2- 8 Electric equivalent circuit for a high frequency capacitor [15]. Figure 2- 9 Absolute value of the capacitor impedance as a function of frequency [15]. Figure 2- 10 Distributed capacitance and series resistance in the inductor coil [4]. Figure 2- 11 Equivalent circuit model of the HF inductor [15]. Figure 2- 12 Frequency response of the impedance of an RFC [15]. Figure 2- 13 Biasing effect of n-channel JFET [21] Figure 2- 14 IV characteristic of FET [23]. Figure 2- 15 GaAs MESFET [5] Figure 2- 16 HEMT [16] Figure 2- 17 High frequency FET model [22] Figure 2- 18 Segment of transmission line expressed with distributed parameters R, L, C and G, where all parameters are given in terms of unit length [4]. Figure 2- 19 Terminated transmission line at location z=0. Figure 2- 20 (a) Microstrip line; (b) end view of microstrip line [9]. Figure 2- 21 Line voltages reference to the load end [21] Figure 2- 22 Smith Chart. Figure 2- 23 Parametric representation of the normalized resistance r [15]. Figure 2- 24 Parametric representation of the normalized reactance x [15]. Figure 2- 25 Smith chart by combining r and x circles for Γ ≤ 1 [15]. Figure 2- 26 Reflection coefficient: A = (0.8-j1.6), angle BOC=-55.5 degree [21]. Figure 2- 27 T network connected to the base-emitter input impedance of a bipolar transistor. Assuming Z 0 = 50 ohm and f c = 2 GHz [15] Figure 2- 28 Computation of the normalized input impedance of the T network Figure 2- 29 Two-port network [15]. Figure 2- 30 Two port scattering network with source and load [21] Figure 3- 1 RF receiver using a heterodyne architecture [5]. Figure 3- 2 Thermal noise [7] Figure 3- 3 Shot noise [7] Figure 3- 4 Representation of noise by input noise generators [7]. Figure 3- 5 Simplified single stage amplifier [3] Figure 3- 6 Eight possible two components networks [3]. Figure 3- 7 Impedance effects of series and shunt connections of L and C [3]. Figure 4- 1 Minimum noise figure, associated gain vs. frequency characteristics [3].. iv.

(9) Figure 4- 2 ADS Simulation setup for the I-V characteristic using electrical model of NE3512S02. Figure 4- 3 Simulated I-V Characteristic of NE3512S02. Figure 4- 4 I-V Characteristic of NE3512S02 according to data sheet [3] Figure 4- 5 Simulation setup for the Electrical and S-Parameter model of NE3512S02. Figure 4- 6 S-Parameters are estimated using Electrical (Thick line) and S-Parameter model (Thin line) at the Q-point (ID = 20 mA and VDS = 2 V). Figure 4- 7 Fixed-bias Configuration [7] Figure 4- 8 Self-bias Configuration [7] Figure 4- 9 Active-bias Configuration [8]. Figure 4- 10 Layout component of footprint for the transistor (NE3512S02) and three types of packages such as 0402, 0603 and 0805. Figure 4- 11 Different layout components of via hole model and ADS via model. Figure 4- 12 ADS set-up for via simulation. Figure 4- 13 Reflection coefficients for different via models. Figure 4- 14 Impedance vs Frequency. Solid line represents for ATC100A101 (100pF) and dot line for ideal 100 pF capacitor [11] .Figure 4- 14 Impedance vs Frequency. Solid line represents for ATC100A101 (100pF) and dot line for ideal 100 pF capacitor [11] . Figure 4- 15 Insertion loss (S21) of ATC100A101 (100 pF) capacitor [11]. Figure 4- 16 Kemet COG ceramic capacitor model schematic [12]. Figure 4- 17 Kemet X7R ceramic capacitor model schematic [12]. Figure 4- 18 Murata Monolithic ceramic SMT Capacitor model [12]. Figure 4- 19 CAPP2 (Chip capacitor) model for ATC [12]. Figure 4- 20 Forward transmission vs frequency characteristics for 1 pF capacitor dofferent companies such as Kemet, ATC, Philips and Murata. Figure 4- 21 Forward Transmission vs Frequency characteristic of Kemet capacitor model with different values. Figure 4- 22 Forward Transmission vs Frequency for Bypassing Kemet capacitor model. Thick line for C1 and thin line for C2 to C4 Figure 4- 23 Three types of RF choke [13]. Figure 4- 24 ADS set-up for RF choke using quarter wave stub. Figure 4- 25 RF choke using radial stub. Figure 4- 26 RF choke using butterfly stub. Figure 4- 27 Forward transmission vs frequency characteristics of different types of RF chokes (Thick line for Butterfly, Thin line for quarter wave line and Das line for radial). Figure 4- 28 Forward transmission vs frequency with different terminated values in port 3 (Das line for 5 ohm, star line for 10 ohm thin line for 20 ohm and thick line for 50 ohm). Figure 4- 29 Complete Schematic of RF choke with bias arrangement. Figure 4- 30 Forward transmission vs frequency characteristic for the schematic of Figure 4- 29. Figure 4- 31 Schematic of the NE3512S02 S-parameter model before stabilization. Figure 4- 32 Stability factor vs frequency before stabilization. Figure 4- 33 Power gain and noise figure before stabilization. v.

(10) Figure 4- 34 Annotation of DC simulation of stabilized FET (Electrical model) fixed bias (IDS=20 mA and VDS=2V, VGS=-0.17V) circuit without matching network. Figure 4- 35 Schematic of stabilized FET(S-parameter model) without matching network. Figure 4- 36 Stability factor vs frequency characteristic after stabilization with Sparameter model. Figure 4- 37 Power gain and noise figure after stabilization with S-parameter model. Figure 4- 38 Layout component. Figure 4- 39 Matching for noise figure at 8.5 GHz. Figure 4- 40 Smith chart of input matching network by lumped elements. Figure 4- 41 Input matching network where L1=1.54 nH, L2=1.87 nH, L3=1.02 nH, L4=1.11 nH,C1=1.2 pF, C2=1.5 pF, C3=1.55 and C4=0.42 pF pF. Figure 4- 42 Matching condition at 8.5 GHz after putting input matching network. Figure 4- 43 Comparison between Noise figure (star line) and minimum noise (solid line). Figure 4- 44 Input matching network with microstrip, L1=2.1 mm, L2=2 mm, L3=2 mm, L4=3.5 mm, L5=3.57 mm, L6=1.25 mm, L7=2.38 mm and W=0.524 mm. Figure 4- 45 Power gain and noise figure (star line) vs frequency with input matching network. Figure 4- 46 Matching condition at 8.5 GHz after putting input microstrip matching network. Figure 4- 47 Output matching network with microstrip line, L1=2 mm, L2=7 mm, L3=2 mm, L4=2.5mm, L5=3mm, L6=2 mm, L7=4mm and W=0.524mm.Figure 4- 1 Minimum noise figure, associated gain vs. frequency characteristics [3]. Figure 4- 48 Power gain and noise figure (star line) vs frequency with input and output network. Figure 4- 49 LNA before matching network. Figure 4- 50 Simulation result of LNA at 8.5 GHz matching point without matching network. Figure 4- 51 Smith chart for IMN design at 8.5 GHz. Figure 4- 52 IMN at 8.5 GHz before optimize. Figure 4- 53 IMN at 9 GHz after optimize. Figure 4- 54 Simulation result of LNA with IMN after optimization. Figure 4- 55 Smith chart for OMN design at 9 GHz. Figure 4- 56 OMN at 9 GHz before optimize. Figure 4- 57 Simulation result of LNA with optimized IMN and unutilized OMN. Figure 4- 58 OMN at 9 GHz after optimization. Figure 4- 59 LNA with optimize IMN and OMN at 9 GHz. Figure 4- 60 Simulation result of LNA with IMN and OMN. Figure 4- 61 complete layout look like LNA with IMN and OMN. Figure 4- 62 Forward transmission (solid line) and transducer gain (dot line). Figure 4- 63 Layout of RF choke with bias circuit; C1=10pF, C2=100 pF, C3=220 pF, C4=100nF and R1=43 Ω. Figure 5- 1 Schematic for VIA model simulation. Figure 5- 2 Input reflection coefficient of different via models of Figure 5- 1. Figure 5- 3 Schematic for SMT capacitor model simulation.. vi.

(11) Figure 5- 4 Forward transmission vs frequency characteristics for 10 pF capacitor, Kemet-solid line star, ATC-circle line, Philips-star line and Murata-triangle line. Figure 5- 5 Layout of RF choke with bias circuit; C1=10pF, C2=100 pF, C3=220 pF, C4=100nF and R1=43 Ω. Figure 5- 6 Forward transmission vs frequency simulation result of Figure 5- 5. Figure 5- 7 Forward transmission vs frequency simulation result of Figure 5- 5. Figure 5- 8 Input reflection coefficient vs frequency simulation result of Figure 5- 5. Figure 5- 9 Simulation setup of the LNA module-1with input and out put matching networks. Figure 5- 10 Power gain vs frequency of LNA module-1. Figure 5- 11 NF vs frequency (LNA module-1); star line- actual noise and solid lineminimum noise. Figure 5- 12 Simulation setup of the LNA module-2 with input and output matching network at 8.5 GHz Figure 5- 13 Power gain vs frequency of LNA module-2 Figure 5- 14 NF vs frequency (LNA-modele-2); star line- actual noise and solid lineminimum noise Figure 5- 15 Simulation setup of the LNA with input and out put matching networks. Figure 5- 16 Power gain vs frequency. Figure 5- 17 Noise figure vs frequency; star line- actual noise and solid lineminimum noise. Figure 5- 18 Power gain vs frequency; solid line at 8.5 GHz matching and dot line at 9 GHz matching. Figure 5- 19 Noise figure vs frequency; dot line-for 9 GHz and solid line-8.5 GHz Figure 5- 20 Complete layout of LNA module-1; C1, C2, C3, C7=10 pF; C4, C8=100 pF; C5, C9=220 pF; C6, C10=100 nF. Figure 5- 21Complete layout of LNA module-2; C1, C2,C3, C7=1.5 pF; C4, C8=3 pF; C5, C9=10 pF; C6, C10=100 pF. R1=3.9 ohm, R2=10 ohm, R3=150 ohm and Q1= NE3512S02 Figure 5- 22 Schematic for data display of measured LNA modele-1. Figure 5- 23 Power gain of the LNA after implementation. Figure 5- 24 Schematic for data display of measured LNA modele-2 Figure 5- 25 LNA module-2 Figure 5- 26 Comparison between LNA module -1 and -2. Solid line for LNA module -1 and star line for module -2. Figure 5- 27 LNA post manufactured simulation Figure 5- 28 Power gain vs frequency Figure 5- 29 Noise figure. vii.

(12) Contents Preface …………………………………………………………………………...……i Abstract …………………………………………………………………………..…. ii List of Abbreviations ……………………………………………………………......iii List of Figures …………………………………………………………………...…. iv Contents ……………………………………………………………………...…….viii. 1. Introduction …………………………………………………………………….1 1.1. Background …………………………………………………………….…….1 1.2. Purpose …………………………………………………………………….....1 1.3. Task ……………………………………………………………………..……2 1.4. Outline ………………………………………………………………….…….2 References 2. Ultra-wideband and General RF Theory ……………………………….…3 2.1.. 2.2.. 2.3.. 2.4.. 2.5. 2.6.. 2.7.. 2.8.. Ultra-wideband ……………………………………………………………….3 2.1.1. History of UWB ………………………………………………...……3 2.1.2. UWB Theory …………………………………………………………4 2.1.3. Regulations …………………………………………………………...6 2.1.4. Applications …………………………………………………………..6 RF Passive Components ……………………………………………………...7 2.2.1. Wire …………………………………………………………………..7 2.2.2. Resistors …………………………………………………………..….8 2.2.3. Capacitor ……………………………………………………………..9 2.2.4. Inductor ……………………………………………………………..10 Active Devices ……………………………………………………………...12 2.3.1. JFETs ………………………………………………………………..12 2.3.2. GaAs MESFETs …………………………………………………….13 2.3.3. HEMTs ……………………………………………………………...13 2.3.4. FET Transistor Modeling …………………………………………...14 2.3.4.1. High Frequency Model of FET…………………………..….15 Transmission Line …………………………………………………………..15 2.4.1. Lossless Transmission Line ……………………………………...….17 2.4.2. Voltage Reflection Coefficient …………………………………...…17 2.4.3. Standing Wave Ratio …………………………………………….….18 Microstrip Transmission Line ……………………………………………....18 Transmission Line as Electrical Elements ……………………………….…19 2.6.1. Transmission Line as a Reactance ………………………………….21 2.6.2. Transmission Line as a Transformer …………………………….….21 Smith Chart ………………………………………………………………....22 2.7.1. Smith Chart Theory ……………………………………………..…..23 2.7.2. Smith Chart Applications ………………………………………..….25 Networks model …………………………………………………………….26. viii.

(13) 2.8.1. 2.8.2. References. Two-port Networks …………………………………………………27 S-parameters ………………………………………………………...28. 3. Low-Noise Amplifier ……………………………………..…………….…..31 3.1. 3.2. 3.3.. Receiver Overview ………………………………………………….…...….31 Stability ……………………………………………………………………..32 Noise Analysis ……………………………………………………………....32 3.3.1. Internal Noise Sources ………………………………………………32 3.3.2. Noise Figure ………………………………………………………...34 3.4. Power Gain ………………………………………………………………….35 3.5. Matching Network …………………………………………………………..36 3.5.1. Microstrip Matching Network ………………………………………38 References. 4. Design of LNA …………………………………………................................39 4.1. 4.2. 4.3. 4.4. 4.5.. LNA Specification ……………………………………………………….….39 Transistor Selection …………………………………………………………40 Transistor Characteristics ………………………………………………..….40 Transistor Models Comparison ……………………………………………..41 Transistor Biasing Network Design …………………………………….…..43 4.5.1. Fixed-bias Configuration …………………………………………....43 4.5.2. Self-bias Configuration ……………………………………………..43 4.5.3. Active-bias Network …………………………………………….…..44 4.6. Microstrip Footprint ………………………………………………………...44 4.7. VIA Hole Model ………………………………………………………….....45 4.8. Broadband Chip Capacitors Selection ………………………………………46 4.9. ADS Capacitor Model ………………………………………...………...…..48 4.10. DC Blocking and Decoupling ………………………………………………50 4.10.1. DC Blocking Capacitor Selection …………………………………..50 4.10.2. Dcoupling Capacitor Selection ……………………………..……….51 4.11. Microstrip RF Choke ………………………………………………………..52 4.11.1. RF Choke Design and Simulation …………………………………..52 4.11.2. RF Choke with Bias Arrangement ………………………………….54 4.12. LNA Design …………………………………………………………….…..55 4.12.1. Matching Network Design at 8.5 GHz …………………………..….59 4.12.1.1. Matching Network by Lumped Elements ……………….….60 4.12.1.2. Microstrip Matching network ………………………….……61 4.12.2. Matching Network Design at 9 GHz …………………………..……62 4.12.2.1. IMN Design …………………………………………………64 4.12.2.2. OMN Design ………………………………………………..66 4.12.3. Layout of RF Choke …………………………………………….…..70 References. 5. LNA Implementation: Simulation Results and Measurements ……...72 ix.

(14) 5.1.. Simulation Results …………………………………………………..………72 5.1.1. VIA Model Simulation ………………………………………...……72 5.1.2. DC Blocking Capacitor Simulation ……………………...………….74 5.1.3. RF Choke …………………………………………………..………..75 5.1.4. LNA with Matching Network at 8.5 GHz …………………………..77 5.1.5. LNA with Matching Network at 9 GHz …………………….………82 5.1.6. Comparison of Two LNA ……………………………………….…..83 5.1.7. LNA Layout ……………………………………………….………..85 5.2. Measurement ………………………………………………………………..87 5.3. LNA Post-manufactured Simulation ……………………………………….89. 6. Conclusion and Further Work ……………………………….………….....92 Appendix …………………………………………………………………...……..93. x.

(15) 1 Introduction This chapter is intended to give an overall idea of this Master Thesis work. The report starts with describing the background and is continues by explaining the purpose, task and outline.. 1.1 Background Most of the today’s radio systems operate within 1 to 40 GHz which is a part of the microwave spectrum defined by Radio Society of Great Britain [1]. Ultra-wideband radio (UWB) is a wireless communication technology based on either orthogonal frequency division multiplexing (OFDM) or spread spectrum technologies [2]. The UWB spectrum in the range 3.1 GHz to 10.6 GHz is defined by the Federal Communication commission (FCC) in USA [3]. Europe, Japan and recently China have put restriction on 3.1 to 4.8 GHz frequency band that causes problems regarding from the coexistence of the UWB system with other narrowband wireless system [4]. Consequently, here is a great interest in the higher part of the UWB European Spectrum 6-9 GHz (6-8.5 GHz long term range and 8.5-9 GHz short term range) applications [5]. Howerever, at high frequency and over a wide frequency band, it is a challenging task to design radio receiver circuits [4]. A simple receiver front-end consists of a Low-Noise Amplifier (LNA), filter and mixer. All active and passive components contribute to process the signal but also they can degrade the original signal. At each circuit, the signal must be handled carefully over a wide frequency band to meet the design specifications [6]. For example, as the operation frequency increases above 1 GHz, the lumped element models as those used in SPICE and SPICE-like simulators are no longer valid. In order to provide accurate models, new design techniques based on electromagnetic simulations should be considered. To support the new challenges, electronic design automation (EDA) and computer aided design (CAD) vendors have continuously improved their tools so that the entire design from system simulation to every circuit design can be performed under a single simulation environment [6]. Despite design difficulties, the UWB technology is considered to be one of the promising technologies to design indoor data communication for high data rates [4].. 1.2 Purpose The purpose of this Master thesis is to design and implement a 6-9 GHz Low-Noise Amplifier (LNA) and to understand the design process of the LNA module from schematic to the LNA module layout and prototype. Another purpose is to acquire skills using the Advanced Design System (ADS) tool from Agilent Technologies, a complex and important software for any engineer developing circuits and systems operating at high frequencies.. 1.

(16) 1.3 Task The main task was the design of a broadband 6-9 GHz LNA with the help of ADS. At first, special theoretical knowledge about designing RF circuits was needed. The necessary information sources were various books, scientific papers, internet sources and useful discussions with the supervisor. The advanced ADS skills were step-by-step learned by working and trying to solve different problems. This thesis work also involves the passive component selection, layout design, simulation techniques, and understanding of high frequency effects when implementing the LNA in a PCB process.. 1.4 Outline The report is organized with six chapters including the introduction. And the report is ended with appendix. Chapter two starts with the theoretical concepts of the UWB and radio frequency (RF) passive components. This chapter also gives same RF theory that was used in this work. Chapter three covers the theoretical design consideration for the LNA. Chapter four gives the overall design process of the LNA. This starts with design specification then covers the component selection to matching network design. Chapter five contains the final LNA simulation and measurement results. In the same time other necessary simulation results for component selection are given. Chapter six is for the conclusion and further work.. 1.5 References [1] Wikipedia, http://en.wikipedia.org/wiki/Ku_band. [2] Eric Ottosson, Design and Implementation of a Ultra wide-band low noise amplifier 3.1-4.8 GHz, Thesis LITH-ITN-ED-EX-06/017-SE. [3] Federal Communication commission (FCC), Revision of part 15 of the Commission’s Rules Regarding Ultra Wideband Transmission Systems, First Report and Order ET Docket 98-153, Feb. 2002. [4] Adriana Serban, Ultra-Wideband Low-Noise Amplifier and Six-Port Transceiver for High Speed Data Transmission. LiU-Tryck Linkoping, Sweden, 2010. [5] Radio Spectrum Committee, RSCOMO7-23 Final CEPT Report on UWB Mandate, March 2007. http://circa.europa.eu/Public/irc/infso/Home/main?index. [6] Adriana Serban Craciunescu, Low-Noise Amplifier Design for UltraWideband Systems, Thesis LiU-TEK-LIC-2006:62.. 2.

(17) 2 Ultra-wideband and General RF Theory The main purpose of this chapter is to describe UWB and general RF theory which are necessary for this diploma work. It is assumed that the reader has basic knowledge of electrical engineering. The motivated reader, who would like to get deeper understanding, can see the books and articles listed in the references.. 2.1 Ultra-wideband Ultra-wideband (UWB) communication technology promises a huge opportunity to impact the future communication world. Large available bandwidth, the wide scope of the data rate/range trade-off, and low-cost operation which will lead to massive usages, all present a unique opportunity for UWB systems to impact the way people and intelligent machines communicate and interact with their environment. In particularly, UWB will give huge advantages for short-range communications. In the past 20 years, UWB has been used for different areas e.g. radar, sensing, military communications [1]. Even though the development and advancement of UWB system is not faster as other wireless system, UWB will be the next best technology for all types of wireless systems [2].. 2.1.1 History of UWB UWB history is generally perceived to start after 1960 with the development of Linear Time Invariant System description via impulse stimuli. On the contrary, UWB transmissions history is much older and goes back to the end of XIX century. The history of wireless communications can be considered to start at the end of XIX century with the work carried by Guglielmo Marconi. From the end of XIX century until nowadays, three eras can be devised in the history of development of UWB systems development: pioneering era (18861906), subnanosecond era (1939-1994) contemporary standardization and commercialization era (1998-2007) [3]. Despite its renewed interest during the past decade, UWB has a history as long as radio. When invented by Guglielmo Marconi more than a century ago, radio communications utilized enormous bandwidth as information was conveyed using spark-gap transmitters [4]. Until 1960s communications were dominated by continuous wave radio transmissions [3]. The next milestone of UWB technology came in the late 1960s, when the high sensitivity to scatterers and low power consumption motivated the introduction of UWB radar systems [4]. During the 1980s, UWB technology was referred alternately to as impulse, carrier-free or baseband. The term ‘‘UltraWideBand’’ was first coined by the U.S. Department of Defense in 1989. After the great technical developments, related to subnanosecond pulses in the years from the sixties until the end of the century, another rush started with the world-wide activities for technology standardization. Nowadays, Multi Carrier UWB (MC-UWB), Orthogonal Frequency Division Multiplexing (OFDM) UWB and Frequency Modulation UWB (FM-UWB) are the strongest candidates for future UWB communication systems [3].. 3.

(18) 2.1.2 Theory When UWB technology was proposed for civilian applications, there were no definitions for the signal. According to the FCC definition, the signal is characterized as UWB if the signal bandwidth (BW) is 500 MHz or more or a fractional bandwidth Bf of more than 20% [5]. The fractional bandwidth is defined as Bf =. BW f − fL =2 H fc fH + fL. (2.1). Where fL is the lower and fL is the higher -10 dB emission point, respectively. As for an example, Universal mobile Telecommunication system (UMTS) operates around 2 GHz with a bandwidth of 5 MHz. This system is often called wideband, however according to Equation (2.1), the fractional bandwidth of UMTS is 0.0025, which is much smaller than 0.2 (i.e., 80 times smaller)! Channel capacity of a communication system is defined by the Shannon’s capacity theorem. The channel capacity (C bit/s) of a system relates to the following equation. C = BW ⋅ log 2 (1 + SNR ) Where signal to noise ratio, SNR =. (2.2) S N. Where S is the signal power and N is the noise power respectively. It can be seen that channel capacity increases linearly with the bandwidth (BW) and logarithmically with SNR. So channel bandwidth is the main route to get the high data rate. UWB wireless personal area network (WPAN) physical (PHY) layer standard divides the whole available ultra wideband spectrum between 3.1- 10.6 GHz into 14 sub-bands belonged to 6 band groups as show in Fig. 1 [1]. Band group 1 is mandatory, remaining groups are optional.. Figure 2- 1 Diagram of band allocation [7]. FCC Mask To avoid interference with existing communication systems, various regions of the spectrum should have different allowed power spectral densities (PSD). FCC has assigned the effective isotropic radiated power (EIRP) allowed for each frequency band [6]. EIRP is the equivalent isotropically radiated power. 4.

(19) which is the power radiated by an omnidirectional antenna with gain 1. The level of –41.3 dBm/MHz in the frequency range of 3.1–10.6 GHz is set to limit the interference to existing communication systems, and to protect the existing radio services [8]. Figure 2- 2 and Figure 2- 3 shows the FCC emission limit for outdoor and indoor UWB communications respectively. These figures also called FCC mask. Table 1 FCC emission limits for indoor and outdoor UWB. Frequency Ranges 960 MHz-1.61 GHz 1.61 GHz-1.99 GHz 1.99 GHz-3.1 GHz 3.1 GHz-10.6 GHz Above 10.6 GHz. Indoor EIRP (dBm/MHz) -75.3 -53.3 -51.3 -41.3 -51.3. Outdoor EIRP (dBm/MHz) -75.3 -63.3 -61.3 -41.3 -51.3. Figure 2- 2 FCC emission limit for outdoor UWB communications [3]. Figure 2- 3 FCC emission limit for indoor UWB communications [3]. 5.

(20) 2.1.3 Regulations One of the important issues in UWB communication is the frequency of operation. There are many systems operating under allocated bands in the UWB signal band. So existing narrowband allocated services to be protected from possible interference generated in UWB systems. Figure 2- 4 shows the possible interferes for the UWB system. It can be seen that UWB has huge bandwidth and less PSD, whereas other narrow-band e.g. GSM, GPS, WiMAX have high PSD and less data rate.. Figure 2- 4 Spectrum of the Main Interfering Communication Standards for UWB Communication System [11]. In USA, the FCC committee is responsible for all kind of regulations and legal requirements of UWB system. FCC has given permission to design and operation of low power UWB system within 3.1 to 10.6 GHz frequency spectrum [5]. In Europe there are number of key organizations are recognize by the European Commission (EC). Currently ETSI, ECC and EC have all recommended and approved the use of UWB devices within 6-8.5 GHz subject to mitigate the technical problems arise by the FCC in USA [9]. And the extended range 8.5 to 9 GHz band which is same as US but this band is considered for UWB impact analysis on surveillance radars. Furthermore, EC has allowed to operate in EU 4.2 – 4.8 GHz band with -41.3 dBm/MHz and the maximum peak EIRP of 0 dBm measured in 50 MHz [6].. 2.1.4 Applications UWB technology was first used in the Second World War by US Army for their communication system. Since the signal at any particular frequency is incomplete, the enemies could not able to intercept the entire message [6]. Thus far the UWB technology has been mainly applied to military (especially radar) appliances [9]. UWB has a number of features which make it attractive for consumer communications applications. In particular, UWB systems. • have potentially low complexity and low cost; • have a noise-like signal spectrum; • are resistant to severe multipath and jamming; 6.

(21) •. have very good time-domain resolution allowing for location and tracking applications. Even with the significant power restrictions, UWB holds enormous potential for wireless ad-hoc and peer-to-peer networks [1]. Some of the commercial applications of UWB are given below [3]:. • • • • • •. Adhoc Networking e.g. WPANs Wireless sensor networks e.g. smart highway Radio Frequency Identification or RFID e.g. tag, barcode Consumer Electronics e.g. wireless DVD player Asset Location e.g. inventory items Medical applications e.g. medical imaging. 2.2 RF Passive Components The RF passive and active components do not behave according to simple mathematical models; they have size, shape and are manufactured using non ideal materials [12]. Capacitors at certain frequencies may not be capacitors at all, but may look inductive, while inductors may look like capacitors, and resistors may tend to be a little of both. In this chapter we will discuss about RF passive components. But, first we will look the simplest components of any system and examine its problem at radio frequency [13].. 2.2.1 Wires Wires used in an RF circuit can take many forms. Wire-wound resistors, inductors, and axial- and radial-leaded capacitors all use a wire of some size and length either in their leads, or in the actual body of the component, or both. Wires are also used in many interconnect applications in the lower RF spectrum. The behavior of a wire in the RF spectrum depends to a large extent on the wire’s diameter and length [13]. Wires at low frequencies, utilizes its entire cross-sectional area as a transport medium for charge carriers. As the frequency is increased, an increased magnetic field at the center of the conductor presents an impedance to the charge carriers, thus decreasing the current density at the center of the conductor and increasing the current density around its perimeter. This increased current density near the edge of the conductor is known as skin effect. It occurs in all conductors including resistor leads, capacitor leads, and inductor leads. The depth into the conductor at which the charge-carrier current density falls to l/e, or 37% of its value along the surface, is known as the skin depth and is a function of the frequency and the permeability and conductivity of the medium. The net result of skin effect is an effective decrease in the cross-sectional area of the conductor and, therefore, a net increase in the ac resistance of the wire as shown in Figure 25 [13]. For copper, the skin depth is approximately 0.65 µm at 10 GHz [14]. In the medium surrounding any current-carrying conductor, there exists a magnetic field. If the current in the conductor is an alternating current, this magnetic field is alternately expanding and contracting and, thus, producing a voltage on the wire which opposes any change in current flow. This opposition to change is called self-inductance and we call anything that possesses this quality 7.

(22) an inductor. Straight-wire inductance might seem trivial, but the higher the frequency is the more important this effect becomes. The inductance of a straight wire depends on both its length and its diameter [13].. Figure 2- 5 Skin depth area of a wire [13].. 2.2.2 Resistors Resistors are used everywhere in circuits, as transistor bias networks, pads etc. Behaviors of high frequency resistors are different from the world of direct current (dc). There are different types of resistors such as carbon composite, wirewound, metal film and thin-film chip resistors [15]. Of these types, mainly the thin-film chip resistors are found application in RF and MW circuits as surface mount devices (SMDs). The electric equivalent circuit of a high frequency resistor’s R is more complicated and parasitic components have to be considered. Figure 2- 6 represents the equivalent circuit of a RF resistor. The model includes two inductances L, modeling the inductor’s leads, the stray capacitance Ca and inter-lead capacitance Cb [15].. Figure 2- 6 Electric equivalent circuit representation of the resistor [15].. Figure 2- 7 represents the example of 500 ohm thin-film resistor as a function of frequency. This example underscores the care that is required when dealing with RF resistors. Not all resistors behave as shown in the figure, often multiple resonance point occurs when the frequency reaches GHz range [15].. Figure 2- 7 Absolute impedance value of a 500 ohm thin-film resistor as a function of frequency [15].. 8.

(23) 2.2.3 Capacitor A capacitor typically consists of two conducting surfaces or plates separated by dielectric insulation material that permits the storage of energy in the electric field between the plates. The dielectric is usually ceramic, air, paper, mica, plastic, film, glass, or oil. Dielectric prevents current flow when applied voltage is constant, but a time-varying voltage produces a current proportional to the rate of voltage change [16]. The current in a capacitor is given by I =C. dv dt. (2.3). where C is the capacitance measured in farads (F). One farad is the capacitance that will store one coulomb of electrical charge (6.28×1018 electrons) at an electrical potential of one volt. Or, in math form: C farads =. Qcoulombs Vvolts. (2.4). The capacitance of the parallel plate structure is given by C =ε. A A = ε oε r d d. (2.5). Where. ε = absolute permittivity of the dielectric = εoεr A = area of parallel plates d = spacing of plates εo= permittivity of free space εr = relative permittivity or dielectric constant of dielectric medium There are widespread applications of chip capacitors in the RF circuits for the tuning and matching networks as well as for biasing active components such as transistors. At high frequency dielectric becomes lossy. The impedance of a capacitor must be written as a parallel combination of conductance Ge and susceptance ωC: Z=. 1 Ge + jωC. (2.6). Figure 2- 8 represents the equivalent circuit for a high frequency capacitor with parasitic lead inductance L, series resistor Rs describing losses in the lead conductors and dielectric loss resistance [15] Re =. 1 Ge. (2.7). Figure 2- 8 Electric equivalent circuit for a high frequency capacitor [15].. 9.

(24) In the Figure 2- 9, the capacitor reveals a similar resonance behavior due to the presence of dielectric losses and finite lead wires [15].. Figure 2- 9 Absolute value of the capacitor impedance as a function of frequency [15].. 2.2.4 Inductor An inductor is nothing more than a wire wound or coiled in such a manner as to increase the magnetic flux linkage between the turns of the coil. This increased flux linkage increases the wire’s self-inductance. Inductors are used extensively in RF design in resonant circuits, filters, radio frequency interference (RFI)/electromagnetic interference (EMI) suppression, phase shift and delay networks, and as RF chokes used to prevent, or at least reduce, the flow of RF energy along a certain path [13]. Figure 2- 10 shows a RF coil [15]. It is known from the previous discussion that the windings represent an inductance in addition to the frequency dependent wire resistance Rd and parasitic capacitance Cd [15].. Figure 2- 10 Distributed capacitance and series resistance in the inductor coil [4].. Inductance L is a property of electrical circuits that opposes changes in the flow of current. An inductor stores energy in a magnetic field. The unit of inductance is the Henry (H). A Henry is the inductance that creates an electromotive force (EMF) of one volt when the current in the inductor is changing at a rate of one ampere per second or in math form: V =L. ∆I ∆t. (2.8). Where V = created EMF in volts (V) L = inductance in henrys (H) I = current in amperes (A) t = time in seconds (s). 10.

(25) ∆ indicates a small change in. Several factors affect the inductance of a coil. Perhaps the most obvious are the length, the diameter and the number of turns in the coil. Also affecting the inductance is the nature of the core material and its cross-sectional area [17]. Well known formula for the inductance of an air core solenoid: L=. πr 2 µo N 2. (2.9). l. Where N = number of turns L = length of the coil r = radius of the coil core µ o = permeability in vacuum= 4π×107 H/m. The equivalent circuit model of the RF inductor is shown in Figure 2- 11 [15]. The parasitic shunt capacitance Cs and series resistance Rs represent composite effect of distribution capacitance Cd and resistance Rd respectively.. Figure 2- 11 Equivalent circuit model of the HF inductor [15].. Figure 2- 12 Frequency response of the impedance of an RFC [15].. Figure 2- 12 shows the frequency response of the RFC impedance which deviates from the expected behavior of an ideal inductance at high frequencies. Frequency dependency can form complicated resonance conditions with additional elements in an RF system [15]. The ratio of an inductor’s reactance to its series resistance is often used as a measure of the quality of the inductor. Q=. X Rs. (2.10). The larger the ratio, the better is the inductor. This quality factor is referred to as the Q of the inductor. If the inductor were wound with a perfect 11.

(26) conductor, its Q would be infinite and we would have a lossless inductor. Of course, there is no perfect conductor and, thus, an inductor always has some finite Q. At low frequencies, the Q of an inductor is very good because the only resistance in the windings is the dc resistance of the wire-which is very small. But as the frequency increases, skin effect and winding capacitance begin to degrade the quality of the inductor [13].. 2.3 Active Devices: Generally Field Effect Transistors (FETs) are used in the RF and MW systems due to high gain and low noise figure. There are different types of FETs family e.g. junction field effect transistors (JFETs), high electron mobility transistors (HEMTs), metal semiconductor barrier junction transistor (MESFET). MW transistor amplifiers are always rugged, low-cost, reliable and can be integrated in both hybrid and monolithic integrated circuits with mixer, oscillator and related components [18]. Basic structures of FETs for high frequency applications are discussed in these subsections.. 2.3.1 JFETs It is the most common FET. JFET has high input impedance (on the order of 107 to 1012 Ω) compare to BJT. Unlike BJT, a JFET has a negative temperature coefficient so that thermally runway is not a problem. Due to robustness, the JFET is used as a power transistor [19]. Its operation depends on control of majority carrier in a channel by applying voltage. This voltage, control the currents by means of an electric field. Thus JFET is a voltage controlled current source. Figure 2- 13 shows the biasing effect of n-channel JFET, where electrons flow from the source (S), past the gate (G), to the drain (D). If a negative voltage is applied at the gate terminal, its negative electric field will try to pinch the electrons flow and confine it to a smaller cross-section of the n-channel. This affects the resistance of the n-channel and limits the current flow. Hence by varying the gate-source voltage it is possible to control the current flow [21].. Figure 2- 13 Biasing effect of n-channel JFET [21]. 12.

(27) Cut-off. Figure 2- 14 IV characteristic of FET [23].. This figure shows the three regions such as triode, saturation and cut-off. When the gate voltage is zero the maximum carrier flows through the channel from source to drain. Cut off is the off state of the FET. It needs minimum drain to source voltage (VDS) to turn on the FET. Once the FET is biased, the drain current (IDS) increases linearly with VDS up to saturation level at a given value of gate-source voltage (VGS). The power amplifier is design in the triode region whereas the LNA is designed at the saturation region.. 2.3.2 GaAs MESFET Metal semiconductor barrier junction transistor (MESFET) is similar to FET except that junction is a metal semiconductor barrier much as is the case Schottkey diodes [20]. GaAs MESFET is used in high performance circuits of communications, computer, and military systems. Specific functions for MESFET include MW power amp, oscillator, switches, and mixer [19]. Due to higher electron mobility GaAs is used instead of Si. That, coupled with the use of a Schottky-barrier gate with a length of only about 1 µm, allows its use as a microwave amplifier with very good operating characteristics [16].. Figure 2- 15 GaAs MESFET [5]. 2.3.3 HEMTs HEMTs are important recent developments in microwave and millimeterwave transistors. These devices make use of heterojunctions for their operation.. 13.

(28) The heterojunctions are formed between semiconductors of different compositions and bandgaps, for example, GaAs/AlGaAs and InGaAs/InP. These relatively new types of devices offer significant improvements for low-noise amplifiers and microwave power amplifiers [16].. Figure 2- 16 HEMT [16]. Figure 2- 16 shows the cross-section of an HEMT structure using GaAs and AlGaAs. The conventional HEMT is similar to a GaAs MESFET. As seen in figure HEMT has two ohmic contacts (source and drain) and a Schottky gate. The difference between the two types of devices and the key to the HEMT’s improved performance is in the underlying semiconductor material. The HEMT has superior electron transport properties and much higher sheet charge density than the MESFET because of a two-dimension electron gas layer that is formed in a thin layer between the AlGaAs and the undoped GaAs layers. HEMPTs have demonstrated unprecedented noise performance at cryogenic temperatures (within a few degrees of absolute zero) and good microwave and millimeter-wave noise and power performance at room temperature at frequencies up to 60 GHz. Typical noise figures at 12 GHz for commercially available low-noise HEMTs are about 1.0 dB In addition to lower noise figure, HEMTs also have several characteristics that make them more attractive for low-noise applications. They are easier to provide impedance matching, and they have a larger gain-bandwidth product [16].. 2.3.4 FET Transistor Modeling FET Transistor is nonlinear device and for circuits analysis different types of models are used e.g. small signal, large signal. Moreover these small and large signal models are divided into low and high frequency applications. Small-signal modeling is a common analysis technique is used to approximate the behavior of nonlineaer devices with linear equations. This linearization is formed about the DC bias point of the device (that is, the voltage/current levels present when no signal is applied). Nothing changes because the assumption is that the signal is so small that the operating point (gain, capacitance etc) doesn't change. Large-signal modeling is a common analysis method used in electrical engineering to describe nonlinear devices in terms of the underlying nonlinear equations. This model takes into account the fact that the large signal actually affects the operating point and takes into account that elements are non-linear and circuits can be limited by power supply values.. 14.

(29) 2.3.4.1 High Frequency model of FET It is necessary to take some considerations for the high frequency model of FET. FET structure acts as a parallel capacitor when viewed from the gate and source terminals. Frequency dependent components are: Cgs – gate to source capacitance, Cgd -gate to drain capacitance and Cds -drain to source capacitance. Drain to source capacitance is small and less effect of high frequency. Capacitance can be modeled as voltage dependent in the following ways [22]. C gs =. C gso  VGS 1 + ψo .   . m. and C gd =. C gdo  VGD 1 + ψo .   . m. where. Cgso and Cgdo are the zero bias gate-source and gate-drain junction capacitance; VGS and VDS are the quiescent gate-source and drain-source voltage; m is the gate p-n grading coefficient (SPICE default is 0.5) and ψo is the gate junction (barrier) potential typically 0.6 V [22].. Figure 2- 17 High frequency FET model [22]. The maximum operating frequency ωT, is the frequency at which the FET no longer amplifies the input signal i.e. the dependent current source gmvgs is equal to the input current [22].. ωT =. gm (Cgs + Cds ). (2.11). 2.4 Transmission Line In the conventional low frequency circuit analysis the Kirchhoff’s laws can be applied where voltages and currents are uniform all over the conductor. At the higher frequencies the Kirchhoff’s laws can not be applied directly due to spatial behavior of the voltage and current. A new approach is needed to explain the transmission line which is called distributed circuit theory. The transmission lines considered here are system of two or more parallel conductors [3]. The line is subdivided into infinitesimal length ∆z, over which voltage and current can be assumed to remain constant depicted in Figure 2- 18 [4].. 15.

(30) Figure 2- 18 Segment of transmission line expressed with distributed parameters R, L, C and G, where all parameters are given in terms of unit length [4].. Applying Kirchhoff’s voltage laws in the Figure 2- 18 ( R + jωL) I ( z ) ∆z + V ( z + ∆z ) = V ( z ). (2.12). which is re-expressed as a differential equation −. dV ( z ) = ( R + jωL) I ( z ) dz. (2.13). Applying Kirchhoff’s current laws to the node a in Figure 2- 18 yields. −. dI ( z ) = −(G + jωC )V ( z ) dz. (2.14). From equation (2.13) and (2.14) we can get the standard 2nd order differential equation. d 2V ( z ) − k 2V ( z ) = 0 2 dz. (2.15). d 2 I ( z) − k 2 I ( z) = 0 2 dz. (2.16). Where k is known as a complex propagation constant. k = k r + jk i = ( R + jωL)(G + jωC ). (2.17). Solutions of equation (2.15) and (2.16) are two exponential functions for the voltage and current. V ( z ) = V + e − kz + V − e + kz. (2.18). I ( z ) = I + e − kz + I − e + kz. (2.19). Equation (2.18) and (2.19) are the general solutions for the transmission lines aligned along the z-axis. From these two equations, the characteristic line impedance Z0 can be defined as. Z0 =. ( R + jωL) = k. ( R + jωL) (G + jωC ). (2.20). Characteristic impedance can be written as. Z0 =. V+ V− = − I+ I−. (2.21). 16.

(31) Z0 is not impedance in the conventional circuit sense. Its definition is based on the positive and negative travelling voltage and current waves.. 2.4.1 Lossless Transmission Line The characteristic line impedance defined in equation 2.20 is a complex quantity and therefore there will be losses always in the realistic lines [4]. There are two regions where Z0 tends to be resistive and constant [9]. The first region occurs at very low frequencies when R >> jωL and G >> jωC . These results in. Z0 =. R G. (2.22). The second region occurs at very high frequencies when jωC >> G . These result in. Z0 =. L C. jωL >> R and. (2.23). which is a constant factor and the transmission line is said to be lossless because there are no dissipative elements in the line. Equations (2.22) and (2.23) are very important because under these conditions the line impedance tends to remain frequency independent and a state known as ‘distortion-less transmission’ [21].. 2.4.2 Voltage Reflection Coefficient High frequency electric circuits can be viewed as a collection of finite transmission line sections connected to various discrete active and passive devices. Therefore consider the terminated line of length l shown in Figure 2- 19 [4]. We know the voltage along the line is given by (2.18). The second term in (2.18) has the meaning of a reflection from the terminating load impedance for values z < 0. Voltage reflection coefficient Γ0 indicates the amount of reflected wave with respect to incident wave at the load z = 0 . Γ0 =. V− V+. (2.24). Figure 2- 19 Terminated transmission line at location z=0.. Reflection coefficient can be written as. 17.

(32) Γ0 =. Z L − Z0 Z L + Z0. (2.25). which involves known circuit quantities and independent of voltage wave. For the case where load impedance matches the line impedance i.e. Z 0 = Z L , no reflection occurs and Γ0 = 0 .. 2.4.3 Standing Wave Ratio Standing wave ratio (SWR) is defined as the ratio of the maximum voltage (current) over the minimum voltage (current). This is the best way to find the mismatch of a transmission line.. SWR =. Vmax I = max Vmin I min. (2.26). Another form of SWR is. SWR =. 1 + Γ0 1 − Γ0. (2.27). which has a range of l ≤ SWR < ∞ For the matched termination SWR → 1 and for the worst case of either open or short circuit results in SWR → ∞ .. 2.5 Microstrip Transmission Line Microstrip line is one of the most popular types of planner transmission lines. It is cheap to manufacture, easily integrated with passive and active devices [8]. Geometry of microstrip line is shown in the Figure 2- 20 [4] where W is the width of the line, d is the thickness of the dielectric and εr is the relative permittivity of dielectric. The phase velocity and propagation constant of a microstrip line can be expressed as Phase velocity v p =. c. (2.28). ε eff. Propagation constant β = k0 ε eff. (2.29). where εeff is the effective dielectric constant of the microstrip line which satisfies the relation, 1 < ε eff < ε r and is independent of on the substrate thickness d and conductor width W.. Figure 2- 20 (a) Microstrip line; (b) end view of microstrip line [9].. 18.

(33) By neglecting the thickness of the conductor, t compare to the substrate height, d, the characteristic impedance can be represent with the line dimension (W and d). For narrow strip line, W / d < 1 , the line impedance,. Zf. Z 0=. ln(8. 2π ε eff. d W + ) W 4d. (2.30). Where Z f = ( µ0 / ε 0 ) = 376.8 Ω, is the wave impedance in free the space and the effective dielectric constant is given by. ε eff =. ε r + 1 ε r − 1  2. +. 2. d  1 + 12  W . −1 / 2. 2 d    + 0.041 −    W  . (2.31). For the wide line W/d > 1, line impedance is. Zf. Z0 =.  d 2 d  ε eff 1.393 + + ln + 1.444   W 3 W  . (2.32). With. ε eff =. εr + 1 εr −1 2. +. d  1 + 12  2  W. −1 / 2. (2.33). With the knowledge of the effective dielectric constant we can compute the expression for the wavelength of. λ=. vp λ c = = 0 f f ε eff ε eff. (2.34). where c is the speed of light and f is the operating frequency.. 2.6 Transmission Line as Electrical Components It is possible to design transmission line that will behave like electrical components e.g. capacitor, inductor, resistor, transformer. These components are made by careful choice of transmission line characteristic impedance (Z0), line length (l) and termination (ZL). The properties of these components can be calculated by using well known expressions for calculating the input impedance of a transmission line. From the transmission line equation the voltage reflection coefficient can be written as [1] Γv =. Z L − Z0 Z L + Z0. (2.35). 19.

(34) Figure 2- 21 Line voltages reference to the load end [21]. It is more convenient to take voltage and current references from the terminating or load end of the line. This is shown in Figure 2.6. From the definition of line attenuation and for a distance l from the load, we have incident and reflected voltages [21]. vi = ViL e + γl and vr = VrL e + γl And using the definition for voltage reflection coefficient Γv V Γvl = rl = ΓL e − 2γl Vil where l = line length Γv= voltage reflection coefficient at load Γvl= voltage reflection coefficient at load distance l from load γ = propagation constant = α+jβ nepers/m. (2.36) (2.37) (2.38). At any point on a transmission line of distance l from the load. vl = vi + vr = vi + vi Γv e −2γl il = ii + ir = ii + ii Γi e. (2.39). −2 γl. (2.40). Dividing Equation (2.39) by (2.40) and defining Zl at point l and Zo. 1 + Γv e −2γl  Zl = Zo  − 2γl  1 − Γv e . (2.41). If the total length of the line is l, the impedance at point l becomes the input impedance (Zin) of the line. After doing some calculation, Zin can be written as  Z sinh γl + Z L cosh γl  Z in = Z o  o   Z o cosh γl + Z L sinh γl . (2.42). We know propagation constant, γ = α + jβ . When the line is considered as low loss i.e. α << β, then propagation constant becomes γ = jβ . Since we know β = 2π / λ where λ is the electrical length at the frequency of operations, so Equation (2.42) becomes. 20.

(35) 2πl 2πl    jZ o sin λ + Z L cos λ  Z in = Z o  2πl 2πl   jZ L sin  + Z o cos λ λ  . (2.43). This equation will investigate the property of transmission line.. 2.6.1 Transmission Line as a Reactances A transmission line can be made to behave like a reactance by making the terminating load a short circuit ( Z L = 0 ). In this case, Equation (2.43) becomes. 2πl 2πl      jZ o sin λ + 0   j sin λ  2πl Z in = Z o  = Zo  = jZ o tan 2πl  2πl  λ  0 + Z o cos   cos  λ  λ    2πl When l < λ / 4 , Z in = jZ o tan λ is an inductive. 2πl When λ / 4 < l > λ / 2 , Z in = − jZ o tan λ is a capacitive.. (2.44). (2.45a). (2.45b). Similar reactive effects can be produced by open-circuited load.. Z in = − jZ o cot. 2πl. (2.46). λ. At the radio frequency any unterminated transmission has a stray capacitance with an open circuit. This stray capacitance can be ignored for our frequencies of operation, its reactance is extremely high [21].. 2.6.2 Transmission Line as a Transformer: When l = λ / 2 , Equation (2.43) becomes.  jZ sin π + Z L cos π  Z in = Z o  o  = ZL  jZ L sin π + Z o cos π  So transmission line acts as a 1:1 transformer. A resistor dissipating a lot of heat adjacent to a transistor can cause the latter to malfunction. With a 1:1 transformer, the resistor can be physically moved away from the transistor without upsetting electrical operating conditions. when l = λ / 4 , Equation (2.43) becomes Z in =. Z o2 ZL. (2.47). Input impedance becomes higher when length of transmission line becomes quarter wave. This concept can be used in the bias circuit of active device.. 21.

(36) 2.7 Smith Chart Smith chart is a very useful tool for RF circuit design e.g. amplifier, oscillator. Gain circles, noise circles, matching network design, impedance and admittance determination, and finding reflection coefficients and voltage standing wave ratio can be represented using the Smith chart [21]. The Smith chart was developed by P.H. Smith in the late 1930s. Figure 2- 22 shows a simplified Smith chart.. Wavelengths to generator. Inductive domain. Short circuit. Capacitive domain. Wavelengths to load. Open circuit. Figure 2- 22 Smith Chart.. The Smith chart is a phasor diagram of the reflection coefficient, Γ, on which constant-r and constant-x circles are drawn, where r and x are the normalized values of the series resistive and reactive parts of the load impedance. The horizontal and vertical axes of the chart are the real and imaginary axes of the reflection coefficient. Any circle centered on the Smith chart centre is a constant|Γ| circle and a constant VSWR circle too. The Smith chart, described so far as a family of impedance coordinates, can easily be used to convert any impedance (Z) to an admittance (Y), and viceversa. In mathematical terms, an admittance is simply the inverse of an impedance, or Y=. 1 Z. (2.48). where, the admittance (Y) contains both a real and an imaginary part, similar to the impedance (Z). Thus Y = G ± jB. (2.49). G = Conductance in Siemens (S) B = Susceptance in Siemens (S). 22.

(37) 2.7.1 Smith Chart Theory This section deals with the derivation of the resistance (R) and reactance (X) circles of the Smith chart. The normalized load impedance is z=. Z L R + jX = = r + jx Z0 Z0. (2.50). And the reflection coefficient is Γ = Γr + jΓi. (2.51). From the Equation (2.25) we can write z=. 1+ Γ 1− Γ. (2.52). Substituting z and Г in Equation (2.50) r + jx =. 1 + Γr + jΓi 1 − Γr − jΓi. (2.53). can be separated into r=. 1 − Γr2 − Γi2 (1 − Γr ) 2 + Γi2. and x =. (2.54). 2Γi (1 − Γr ) 2 + Γi. (2.55). The Equation (2.54) and (2.55) are the transformations rules of finding z if the reflection coefficient is specified in term of Гr and Гi. We can derive the parametric equations of circles from (2.54) and (2.55) as 2. r    1  2  Γr −  + Γi =   r +1   r +1. (Γr − 1)2 +  Γi − 1  . 2. 1 =  x  x. 2. (2.56). 2. (2.57). Both (2.56) and (2.57) are parametric equations of circles in Г-plane. Figure 2- 23 represents the parametric circle equations (2.56) for various normalized resistances. For example, if the normalized resistance r is zero, the circle is centered at the origin. And in the limit for r → ∞ , the circles radius approaches r /( r + 1) → 0 . This mapping is for fixed values of r only and does not involve x. thus for a fixed value of r, an infinite range of reactance values x as indicated by straight line in z-plane [15].. 23.

(38) Figure 2- 23 Parametric representation of the normalized resistance r [15].. Figure 2- 24 represents the parametric circle Equations (2.57) for various normalized reactance. Here the centers of the circles reside along a line perpendicular to the Γr = 1 point. For x = ∞ , the circle of radius becomes zero. It is observed that negative x-values refer to capacitive impedances residing in the lower half of the Г-plane [15].. Figure 2- 24 Parametric representation of the normalized reactance x [15].. Individually equation (2.56) and (2.57) does not construct unique mapping from normalized impedance into the reflection coefficient plane. Figure 2- 25 represents the smith chart by combining r and x circles for Γ ≤ 1 . This Smith chart is a one-to-one mapping between normalized impedance and the reflection coefficient plane. It is also noticed that resistance circles r have a range 0 ≤ r < ∞ and the reactance circles x can be either negative (capacitive) or positive (inductive) values in the range, − ∞ < x + ∞ . For the computation of the input impedance of a terminated transmission line, the motion is always away from the load impedance or towards the generator. This rotation is indicated by an arrow on the periphery of the chart [15].. 24.

(39) Figure 2- 25 Smith chart by combining r and x circles for. Γ ≤ 1 [15].. 2.7.2 Smith Chart Applications The basics of Smith chart and theory have been discussed before. Some of the applications of Smith chart are given in this section.. a) Reflection Coefficients Evaluation Smith chart can be used to find the reflection coefficient at any point, in phasor form. Figure 2- 26 shows a point A (08-j1.6). The line OA is extended to B and the resulting angle BOC is about -55.50. The modulus of the reflection can be found from equation (2.27) which states that the voltage standing wave ratio, 1 + Γv VSWR = 1 − Γv where Γv is the voltage reflection coefficient. VSWR is obtained by completing the circle enclosing the point A. It is then read off the intersection between the circle and the real axis and in this case the value is 5. So Γv = (5 − 1) / (5 + 1) = 0.667 and hence the reflection coefficient is 0.667∠ − 55.50 [21].. Figure 2- 26 Reflection coefficient: A = (0.8-j1.6), angle BOC=-55.5 degree [21].. 25.

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