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ABSTRACT HEO, SEUNG KYUN. The Performance of Passive Lumped Element. (Under the direction of Dr. Michael B. Steer). Lumped elements are the most widely used passive components in Radio Frequency (RF) circuits. The characteristics vary over frequency, however, limiting their application. This limitation stems from the physical dimensions of lumped elements, which are, in general, 0603 (i.e. 3 mm by 1.5 mm), 0402 (i.e. 2 mm by 1 mm), and 0201 (i.e. 1 mm by 0.5 mm) in size in order to avoid the phase shift between the input and the output. By examining the quality factor and self-resonant frequency, the performance of lumped elements can be quantified. The quality factor contains the energy storing and dissipation terms, which vary according to frequency. This quality factor can be obtained using the S-parameters of lumped elements. It is difficult to obtain accurate measurements of a non-coaxial packaged devices such as a lumped inductor, because it cannot be measured using a wide band measurement system. A Print Circuit Board (PCB) fixture is thus considered to connect a device under test to 50 ohm connectors, and the effects of the fixture are then removed by performing a standards calibration. The combination of port extension and calibration using a manufacturer’s calibration kit is employed to collect one-port S-parameters. The best result is achieved using an SMA fixture that has a short line and good impedance matching. This thesis presents an extraction method to derive the quality factor of lumped elements, and the quality factor is obtained using one-port S-parameters. The Performance of Passive Lumped Element by SeungKyun Heo A thesis submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering Raleigh, North Carolina 2010 APPROVED BY: _______________________________ Professor Michael B. Steer Committee Chair ________________________________ Professor David Schurig ______________________________ Professor Paul D. Franzon DEDICATION This dissertation is dedicated to my wife, Mi-Young Kim, and to my parents in Korea. ii BIOGRAPHY Seung-Kyun Heo was born on 4th August, 1977 in Korea. He received the Bachelor of Science degree in Electrical and Computer Engineering from Chung-Ang University located in Seoul, Korea, in 2003. He began his master degree in the Electrical and Computer Engineering Department at North Carolina State University in Raleigh, NC, in 2008. From 2003 to 2007, he worked as an RF engineer in Samsung Electronics Co., Suwon, Korea, and from June 2009 to January 2010 he worked as an intern engineer in RFMD, Greensboro, NC. His current interest is focused in the Field of RF circuit design. iii TABLE OF CONTENTS LIST OF TABLES ................................................................................................................v LIST OF FIGURES..............................................................................................................vi CHAPTER 1 Introduction ...................................................................................................1 1.1 Motivation ...............................................................................................................1 1.2 Contribution.............................................................................................................2 1.3 Thesis Organization .................................................................................................3 CHAPTER 2 Basic Characteristic of Lumped Elements ......................................................4 2.1 Introduction .............................................................................................................4 2.2 Basic Parameters ......................................................................................................4 2.3 Quality Factor ..........................................................................................................8 2.3.1 Definition of Quality Factor ..........................................................................8 2.3.2 Effect of Quality Factor in an RF Circuit .....................................................10 2.3.3 Extraction of the Quality Factor...................................................................11 2.4 Self Resonant Frequency........................................................................................13 2.5 Summary ...............................................................................................................13 CHAPTER 1 Measurement Method .................................................................................. 17 3.1 Introduction ...........................................................................................................17 3.2 Calibration .............................................................................................................18 3.2.1 Errors in VNA Measurement .......................................................................18 3.2.2 Errors Characteristic by Fixture Measurements ...........................................19 3.2.3 SOLT Calibration ........................................................................................20 3.2.4 TRL Calibration ..........................................................................................22 3.2.5 Port Extension .............................................................................................23 3.2.6 Time Gating ................................................................................................23 3.3 Quality Factor from Two-Port S-parameters ...........................................................24 3.5 Summary ...............................................................................................................28 CHAPTER 4 Measurement Results ..................................................................................... 31 4.1 Introduction ...........................................................................................................31 4.2 One-Port S-Parameter Results ................................................................................32 4.3 Two-Port S-Parameter Results from Manufacturer .................................................34 4.4 Summary ...............................................................................................................36 CHAPTER 5 Conclusions ................................................................................................... 38 5.1 Conclusions ...........................................................................................................38 5.2 Future Work ..........................................................................................................39 APPENDIX......................................................................................................................... 40 Appendix A: Plot of Measurement Result ....................................................................41 iv LIST OF TABLES Table 2.1 Basic parameters of resistor ...................................................................................5 Table 2.2 Critical parameter by material [10][17] ..................................................................6 Table 2.3 Basic parameters of capacitors ...............................................................................7 Table 2.4 Basic parameters of inductors ................................................................................8 Table 3.1 Summary of quality factor calculation.................................................................. 28 Table 4.1 TOKO inductor one-port quality factor and SRF measurement result ................... 33 Table 4.2 Johanson capacitor one-port SRF measurement result .......................................... 33 Table 4.3 Panasonic resistor one-port measurement result ................................................... 34 Table 4.4 Johanson capacitor two-port SRF measurement result .......................................... 35 Table 4.5 TOKO inductor two-port Q and SRF measurement result .................................... 35 Table 4.6 Panasonic resistor two-port measurement result ................................................... 36 v LIST OF FIGURES Figure 1.1 RF block of LG300G ............................................................................................1 Figure 2.1 Relationship between I and V in a capacitor .........................................................5 Figure 2.2 Series RLC circuit and parallel RLC circuit ..........................................................8 Figure 2.3 Common source amplifier .................................................................................. 10 Figure 2.4 Resonant circuit method ..................................................................................... 11 Figure 2.5 Reflection coefficient ......................................................................................... 12 Figure 3.1 DUT with fixture ................................................................................................ 19 Figure 3.2 Through standard S11,S22.................................................................................. 20 Figure 3.3 User calibration kit ............................................................................................. 21 Figure 3.4 12-terms error model [2] ..................................................................................... 22 Figure 3.5 Port extension method ........................................................................................ 23 Figure 3.6 Time domain reflection response of a DUT with fixture ..................................... 24 Figure 3.7 ABCD parameter of series impedance ................................................................ 25 Figure 3.8 One-port VNA error model [5] [6] ...................................................................... 26 Figure 3.9 SMA connector fixture ....................................................................................... 28 Figure 4.1 Quality factor of Johanson capacitor [2] ............................................................. 31 Figure 4.2 SRF of Johanson capacitor [2] ............................................................................ 32 Figure A.1 Inductance and phase of impedance ................................................................... 42 Figure A.2 Quality factor of 2.2 nH inductor ....................................................................... 42 Figure A.3 Inductance and phase of impedance ................................................................... 43 Figure A.4 Quality factor of 3.3 nH inductor ....................................................................... 44 Figure A.5 Inductance and phase of impedance ................................................................... 44 Figure A.6 Quality factor of 6.8 nH inductor ....................................................................... 44 Figure A.7 Inductance and phase of impedance ................................................................... 45 Figure A.8 Quality factor of 10 nH inductor ........................................................................ 45 Figure A.9 Inductance and phase of impedance ................................................................... 46 Figure A.10 Quality factor of 33 nH inductor ...................................................................... 46 Figure A.11 Inductance and phase of impedance ................................................................. 47 Figure A.12 Quality factor of 56 nH inductor ...................................................................... 47 Figure A.13 Inductance and phase of impedance ................................................................. 48 Figure A.14 Quality factor of 100 nH inductor .................................................................... 48 Figure A.15 Inductance and phase of impedance ................................................................. 49 Figure A.16 Quality factor of 270 nH inductor .................................................................... 49 Figure A.17 Capacitance and phase of impedance ............................................................... 50 Figure A.18 Quality factor of 2.2 pF capacitor..................................................................... 50 Figure A.19 Capacitance and phase of impedance ............................................................... 51 Figure A.20 Quality factor of 3.3 pF capacitor..................................................................... 51 Figure A.21 Capacitance and phase of impedance ............................................................... 52 Figure A.22 Quality factor of 10 pF capacitor...................................................................... 52 vi Figure A.23 Capacitance and phase of impedance ............................................................... 53 Figure A.24 Quality factor of 15 pF capacitor...................................................................... 53 Figure A.25 Capacitance and phase of impedance ............................................................... 54 Figure A.26 Quality factor of 33 pF capacitor...................................................................... 54 Figure A.27 Capacitance and phase of impedance ............................................................... 55 Figure A.28 Quality factor of 47 pF capacitor...................................................................... 55 Figure A.29 Capacitance and phase of impedance ............................................................... 56 Figure A.30 Quality factor of 82 pF capacitor...................................................................... 56 Figure A.31 Capacitance and phase of impedance ............................................................... 57 Figure A.32 Quality factor of 10 Ω resistor.......................................................................... 57 Figure A.33 Resistance, phase of impedance, and reactance ................................................ 58 Figure A.34 Quality factor of 39 Ω resistor.......................................................................... 58 Figure A.35 Resistance, phase of impedance, and reactance ................................................ 59 Figure A.36 Quality factor of 68 Ω resistor.......................................................................... 59 Figure A.37 Resistance, phase of impedance, and reactance ................................................ 60 Figure A.38 Quality factor of 200 Ω resistor........................................................................ 60 Figure A.39 Resistance, phase of impedance, and reactance ................................................ 61 Figure A.40 Quality factor of 300 Ω resistor........................................................................ 61 Figure A.41 Resistance, phase of impedance, and reactance ................................................ 62 Figure A.42 Quality factor of 560 Ω resistor........................................................................ 62 Figure A.43 Resistance, phase of impedance, and reactance ................................................ 63 Figure A.44 Quality factor of 1 kΩ resistor.......................................................................... 63 Figure A.45 Resistance, phase of impedance, and reactance ................................................ 64 Figure A.46 Quality factor of 2 kΩ resistor.......................................................................... 64 Figure A.47 Resistance, phase of impedance, and reactance ................................................ 65 Figure A.48 Quality factor of 3 kΩ resistor.......................................................................... 65 vii CHAPTER 1 Introduction 1.1 Motivation Lumped elements, such as inductors, capacitors, and resistors, are the most common and important components in radio frequency (RF) and microwave circuits. To design RF circuits correctly and efficiently, knowledge about the lumped elements is essential. Over the last few decades suppliers have been able to meet this demand by overcoming RF circuit performance limitations. Lumped elements, however, cannot always be applied in high frequency applications because of variations in their characteristics. Figure 1.1 shows the RF section of a mobile handset. Most of the parts, such as a switching module, power amplifier module, low noise amplifier, mixer, and filters are merged into one package with the exception of the lumped elements. Lumped elements none the less are still widely used in biasing and matching networks because of their advantages including small size and low loss. Figure 1.1 RF block of LG300G Lumped elements are not the core part of an RF circuit. However, they constitute a basic block of a big module. A robust system can be built by using high performance elements that exhibit low variation over temperature, time, frequency, and in the biasing environment. Characteristics that vary in frequency are often significant factors in RF 1 circuits. Identifying the quality factor and Self-Resonant Frequency (SRF) is useful in defining and understanding the characteristics of lumped elements over frequency. The quality factor contains the energy storing and dissipation terms, which are vary by frequency, and this characteristic affects the performance of the circuit. If a high quality factor inductor is used as the RF choke in a bias circuit, the gain of the amplifier can be improved. However, a high quality factor does not always mean high performance. A high performance quality factor varies according to the purpose of the circuit in which lumped elements are used. By examining the characteristics of lumped elements over frequency, the lumped elements can be better understood. 1.2 Contribution Numerous articles about enhancing the performance of lumped elements have been published. However, it is difficult to find recent articles that focus on characterizing the lumped element itself. It is useful, therefore, to investigate the parameters of lumped elements, especially their frequency-dependent characteristics. The starting point in RF circuit design is understanding and knowing the limitation of lumped elements. This thesis focuses on the characterization of lumped element parameters and establishing RF measurement method. Finally, measurements are taken to obtain the quality factor and SRF of a number of lumped element components. This thesis lays out the RF measurement approach. Many data sheets of lumped element do not contain the quality factor over frequency. By measuring major commercial lumped elements under the same testing conditions, an understanding of the frequency-dependant characteristics of lumped elements 2 can be acquired. This thesis presents the performance of lumped elements over frequency extended from S-parameter. 1.3 Thesis Organization Chapter 2 reviews the basic parameters of lumped elements and the effects of parasitics, and investigates the quality factor and SRF using S-parameters. Chapter 3 presents a way to remove the error that is caused in using the vector network analyzer (VNA), which uses a fixture, and method to obtain the quality factor using the S-parameters of lumped elements. Chapter 4 summarizes the measurement results. 3 CHAPTER 2 Basic Characteristic of Lumped Elements 2.1 Introduction The performance of lumped elements has been improved through invention and applying numerous methods. Many manufacturers have tried to reduce the effects of parasitics on the ideal inductance, capacitance, and resistance. Lumped elements have a frequency limitation because of their dimensions and parasitics. Generally, the dimensions of a lumped element should be smaller than λ/20 to guarantee phase equality at the input and output ports. In spite of this problem, lumped elements are used widely because of their small size, wide bandwidth, and low cost. Lumped elements have a large number of applications such as impedance matching, DC blocking, RF chocking, bypass circuit, and degeneration in RF circuit. The characteristic parameters of lumped elements must be known in order to use lumped elements correctly and efficiently. In this chapter the basic parameters of lumped elements are reviewed including quality factor (Q) and self resonant frequency (SRF). 2.2 Basic Parameters Resistors are used in lossy impedance matching networks, attenuators, damping resistors, and termination circuits. The most common resistor is a tick film resistor. When a resistor is used as an RF component, the thermal noise from the resistor should be considered. Because of this thermal noise, a resistor is rarely used in an RF matching network. A resistance value can be calculated using sheet resistance with physical dimensions. Table 2.1 shows the basic parameters of a resistor. 4 Table 2.1 Basic parameters of resistor. Parameter Description Maximum power that guarantees the basic characteristic and Power rating reliability. Maximum overload voltage Maximum voltage that can applied for 5 seconds. Resistance tolerance Variation of the resistance value. Temperature coefficient of resistance (TCR) Variation of the degree of resistance value over temperature. Operating temperature range Temperature range that guarantees the performance. Rated current Continuous maximum current allowed. A capacitor stores energy in an electric field. A typical unit range for a lumped capacitor used in RF and microwave circuits is from picofarads to microfarads. The impedance of an ideal capacitor can be expressed as 1 jωC . The impedance of the capacitor contains ‘-j’ terms, which refers to the current leading the voltage by 90 degrees. This basic concept is useful when a lumped element is applied to an RF circuit to cancel out the effects of reactance. By using this concept, the phase relationship between the current and the voltage in the RF circuit can be predicted easily. j 1 0.8 I 0.6 -1 1 Amplitude V 0.4 0.2 0 -0.2 V I -0.4 -0.6 -0.8 -1 -j 0 2 4 6 8 Time Figure 2.1 Relationship between I and V in a capacitor. 5 10 12 14 Parameters such as the dielectric constant, break-down voltage, temperature coefficient, insulation resistance, and dissipation factor determine the basic performance of a capacitor [14]. Table 2.2 shows the characteristics of a capacitor for several dielectric materials, which serve as the dominant performance determining factor of a capacitor. For example, power loss in a capacitor comes from leakage of current and dissipation of power in the dielectric. Furthermore, the temperature variation characteristic is also affected by the dielectric constant. The NP0 material has the lowest dielectric constant has an advantage in terms of temperature variation and power loss. However, high capacitance values cannot be obtained by using NP0 material. To obtain a high capacitance value, a high dielectric constant material is needed, which results in large temperature variations and low efficiency. The appropriate tolerance to guarantee the performance for capacitance in an RF matching circuit is ±5%. Table 2.3 shows the basic parameters of a capacitor. Table 2.2 Critical parameter by material [10] [17]. Material Dielectric Constant % Capacitance Change Dissipation Factor NP0 15-100 < 0.4% (-55 to 125C) 0.1% X7R 2000-4000 +/-15% (-55 to 125C) 3.5% +/-15%(-55 to 85C) 5% Up to 82% (-30 to 85C) 9% X5R Y5V >16000 6 Table 2.3 Basic parameters of capacitors. Parameter Description Dielectric constant Constant factor of material used for dielectrics. Category temperature range Operating temperature. Rated voltage Maximum voltage can be applied. Rated current Maximum current can be applied. Capacitance tolerance Variation of capacitance value. Withstand voltage Voltage that a capacitor can endure for a short time (above rated voltage). Dissipation factor The ratio of loss resistance to capacitance. Insulation resistance DC resistor (related to leakage current). An inductor is used as an RF choke in the bias circuit, and part of the matching network. The current flowing through a line generates magnetic field, so that a single line has inductance. For example, a typical bond wire has around 1 nH of inductance. The voltage across an inductor leads the current by 90 degrees. Table 2.4 shows the basic characteristics of an inductor. Table 2.4 Basic parameters of inductors. Parameter Description Maximum current rating Maximum current can be applied. Maximum power rating Maximum power can be applied. Operating temperature range Temperature range that guarantees the performance. Temperature coefficient of inductance Inductance variation over temperature. Time constant Voltage charge time in an inductor. 7 2.3 Quality Factor 2.3.1 Definition of Quality Factor Most literature sources cite the definition of the quality factor as “The definition of quality factor (Q) is 2π times the ratio of energy stored in a system to the energy dissipated per cycle” [2] [3] [4]. QQ = 2π Max Energy Stored Total Energy Stored =ω Average Power Loss Energy Loss per Cycle (Equation 2-1) This definition of a dimensionless quality factor can be applied to all energy stored systems. AC L AC V i C G L C R Figure 2.2 Series RLC circuit and parallel RLC circuit. The definition of quality factor can be applied to a simple series RLC: 1 2 Li fL ωL XL Q = 2π 2 = 2π = = 1 2 R R R Ri T 2 (Equation 2-2) Similar to Equation 2-2, a parallel RLC circuit quality factor is 1 2 CV fC ωC 1 R Q = 2π 2 = 2π = = = 1 G G ωGL XC 2 2 GV T 8 (Equation 2-3) The loss of lumped elements can be determined by observing the quality factor. In reality, the loss of lumped elements varies by frequency, so it is not possible to calculate the loss using a multimeter. The loss which varies with frequency is due to radiation and skin effect. In general a high quality factor value means less loss and more energy stored. In a reactive circuit, the ratio of the reactance in Ωs to the resistance in Ωs is defined as the quality factor. The quality factor of an inductor can be represented using Equation 2-1 [5]. This general representation of a quality factor is sufficient for practical application. Y11 Q = ω � m | − |W � e |) (|W Max Energy Stored Im|Y11 | ≅ 2ω = − Power Loss Re|Y11 | Average Power Loss (Equation 2-4) This definition can be applied when the operating frequency is much lower than the selfresonant frequency. In a filter, the quality factor is usually taken as the ratio of the center frequency to the bandwidth [1] and represents the sharpness of the response. Q= f0 ∆f (Equation 2-5) Where f0 is resonant frequency and ∆f is 3dB bandwidth. The quality factor of a lumped element is referred to as the unloaded quality factor. The unloaded quality factor is the quality factor of a component or device that does not have an external source and load. This concept considers only the ratio of the reactive part and resistive part of the network itself. The loaded quality factor is affected by the choice of load values and bandwidth. The unloaded quality factor of a lumped element is measured in this thesis. In a resonant circuit, the quality factor is affected by load conditions rather than the quality factor of the component itself. 9 2.3.2 Effect of Quality Factor in an RF Circuit The quality factor is a frequency-defendant parameter and thereby affects RF circuits. An inductor is used widely as an RF choke in amplifier bias circuits. Figure 2.3 shows a simple common source Low Noise Amplifier (LNA) structure. Sufficient gain and input impedance cannot be obtained for a LNA using a single common source amplifier; however, it is sufficient for observing the effect of an inductor RF choke. The gain of an LNA is 1 1 1 |Av | = gm � 2 + + � Q RD jωLd RL -1 (Equation 2-6) Where gm is the transconductance of M1. The gain of the LNA is proportional to the quality factor of Ld . To maximize the gain of the LNA, a high quality factor RF choke inductor should be used. Ld Rd Ld Vout Vin M1 D G Vout + RL Vgs gmVgs - S Figure 2.3 Common source amplifier. 10 Ld RL Rd 2.3.3 Extraction of the Quality Factor The quality factor can be extracted by applying its definition to the circuit or component. One of the oldest and most common methods used is a Q-meter measurement. A Q-meter can measure the quality factor using a resonant circuit. This method is implemented by applying the specified frequency current source to a DUT and a variable capacitor. Then the maximum voltage across a variable capacitor, when its value is adjusted, is measured. The reactance of the DUT is derived from the maximum voltage. Figure 2.5 shows this resonant circuit method. DUT L(XD) RD V C Milivoltmeter Figure 2.5 Resonant circuit method. Determining the quality factor from the critical point method using the equivalent model and impedance locus is also possible [15]. The quality factor is the ratio of the reactive energy stored to the dissipated energy. In the case of an inductor, the reactive energy stored is the difference between the magnetic energy and the electric energy. The quality factor is then derived using this concept. 11 Lumped Inductor I1 I2 + + V1 V2 - - ZL ΓL Γin Figure 2.6 Reflection coefficient. Im|Z| This general definition of quality factor,𝑠𝑠Q= Re|Z| , is sufficient for lumped element measurements since the operating frequency is below the self resonant frequency. From the general definition of quality factor, the quality factor can be obtained in terms of impedance. The impedance of a lumped element can be obtained by measuring the input impedance of the lumped element while placing a perfect short at the output. In this case, 𝑆𝑆11 is the reflection coefficient and can be converted to the input impedance. By using one port measurement with a short at the output, the quality factor can be generated. In terms of twoport S-parameter, the input impedance of a network can be derived: Γin = S11 + Zin = 50 S12 S21 ΓL S12 S21 = S11 − 1 − S22 ΓL 1 + S22 1 − Γin = R + jX 1 + Γin Z = Zin = R + jX Q= (Equation 2-7) (Equation 2-8) (Equation 2-9) |X | R (Equation 2-10) Where Load is a short (ΓL = −1). If the load is a short, the input impedance is considered to be the impedance of the network. Equations 2-7 to 2-10 show the derivation of the quality factor using S-parameter. 12 One of the advantages of using S-parameter is that it is easy to obtain data up to high frequencies. For example, the Anritsu ME7828A VNA can measure up to 100 GHz. Also, they can be converted to other parameters such as Z-parameter, Y-parameter, and ABCDparameter easily. 2.4 Self Resonant Frequency A lumped element has parasitics because of the not insignificant physical dimensions. If a lumped element has large dimensions, it has more parasitic characteristics and a lower SRF. The effects of the parasitic capacitance and inductance increase as the operating frequency increasing. The parasitic capacitance and inductance effects are dominant at frequencies above the SRF. A lumped element must be used below the SRF to guarantee nominal performance. The SRF can be determined by measuring the S-parameter. The point at which the sign of the imaginary part of the impedance is changed is the SRF of the lumped element. By measuring the S21 , the SRF also can be found. 2.5 Summary The basic parameters of lumped elements and the key parameters of an RF circuit were reviewed in this chapter. Because of their benefits, lumped elements are used widely. However, parasitics, which stem from the physical dimensions of lumped elements, limit their operating frequency. These parasitics introduce loss and affect the characteristics of the lumped elements over frequency. To characterize their effects the quality factor and self resonant frequency are used. The definition of quality factor represents the ratio of power 13 storage to power loss in a network, and the SRF indicates the frequency limitations of the lumped element. The performance of the RF circuit, such as gain and frequency selectivity, varies according to the quality factor of lumped elements. It was shown that the quality factor can be generated using measured S-parameters. 14 REFERENCES [1] I. J. Bahl, Lumped elements for RF and microwave circuits, Artech House, 2003. [2] M. Sucher, Handbook of microwave and measurements, Polytechnic Press of the Polytechnic Institute of Brooklyn, 1963. [3] M. B. Steer, Microwave and RF Design: A Systems Approach, SciTech Publishing, 2010. [4] D. M. Pozar, Microwave Engineering, 3rd ed. John Wiley & Sons, Inc., 2005. [5] K. O, “Estimation methods for quality factors of inductors fabricated in silicon integrated circuit process technologies,” IEEE J. Solid-State Circuits, vol. 33, pp. 1249–1252, Aug. 1998. [6] H. Jiang, Y. Wang, J. A. Yeh, and N. C. Tien, “On-chip spiral inductors suspended over deep copper-lined cavities,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 2415–2423, Dec. 2000. [7] Y. K. Koutsoyannopoulos and Y. Papananos, “Systematic analysis and modeling of integrated inductors and transformers in RF IC design,” IEEE Trans. Circuits System II, vol. 47, pp. 699–713, Aug. 2000. [8] T. S. Horng, K. C. Peng, J. K. Jau, and Y. S. Tsai, “S-parameter formulation of quality factor for a spiral inductor in generalized two-port configuration”, IEEE Trans. Microwave Theory Tech., vol. 51, pp. 2197–2202, Nov. 2003. [9] C. P. Yue and S. S. Wong, “On-chip spiral inductors with patterned ground shields for Sibased RF IC’s”, IEEE J. Solid-State Circuits, vol. 33, pp. 743–752, May. 1998. 15 [10] Johanson Dielectrics, “Basics of Ceramic Chip Capacitors”, http://www.johansondielectrics.com/technical-notes/product-training/basics-of-ceramic-chipcapacitors.html. [11] K. Naishadham and T. Durak, “Measurement-based closed-form modeling of surfacemounted RF components”, IEEE Trans. Microwave Theory Tech., vol. 50, pp. 2276–2286, Oct. 2002. [12] L. F. Tiemeijer, D. Leenaerts, N. Pavlovic, and R. J. Havens, “Record Q spiral inductors in standard CMOS”, in Int. Electron Devices Meeting Tech. Dig., 2000, pp. 40.7.1–40.7.3, Dec. 2001. [13] R. K. Ulrich, Integrated Passive Component Technology, NJ: Wiley, 2003. [14] J. G. Webster, Electrical Measurement, Signal Processing, and Displays, Boca Raton: CRC Press, 2004. [15] P. Wang, L. H. Chua, D. Mirshekar-Syahkal, “Accurate characterization of low-Q microwave resonator using critical-points method”, IEEE Trans. Microwave Theory Tech., vol. 53, pp. 349–353, Jan. 2005. [16] E. Sun, S. Chao , “ Unloaded Q measurement-the critical points method”, IEEE Trans. Microwave Theory Tech., vol. 43, pp. 1983–1986, Aug. 1995. [17] ATC capacitor, “Capacitor Dielectric Characteristics”, http://www.atceramics.com/pdf/technotes/capacitor_dielectric.pdf. 16 CHAPTER 1 Measurement Method 3.1 Introduction The most important part of taking RF circuit measurements is setting up the correct test environment that can be calibrated to remove errors. This is a time-consuming aspect of the RF measurement process. A Vector Network Analyzer (VNA) is the most suitable instrument to use for RF measurements when a passive element is to be measured over frequency. To measure the S-parameter of a lumped element, the interface between the lumped element and the VNA must be provided. A suitable fixture must be found to provide a connection to a non-coaxial device such as a lumped element. An ideal fixture is a material having no loss, no electrical length, no reflection, and a wide frequency response. A fixture should guarantee the bandwidth and offer low loss to provide accurate test results. In this thesis, Printed Circuit Boards (PCB) and SMA connector fixtures are used. To accurately measure the S-parameters, the effects of the fixture should be removed by performing calibration fixture standards. The added fixture causes measurement problems such as insertion loss, phase shift, and mismatch. For example, the transmission lines in a PCB fixture and in a connector generate insertion loss and phase shift. The mismatch problem occurs when the connections between the networks have different characteristic impedance. Fixture error problems can be removed by performing a full calibration. Using a Through-Reflection-Load (TRL) calibration, better source match and load match corrections can be achieved than via a Short-Open-Load-through (SOLT) calibration because of the capacitance variation found in an open standard. However, the SOLT calibration is applied 17 here instead of the TRL calibration because of the absence of TRL calibration standards; the line standard cannot be implemented in the test fixture. 3.2 Calibration 3.2.1 Errors in VNA Measurement Three major errors occur in VNA measurements [1], [5], [6]. Drift error occurs when a test environment is changed. One such environmental variation factor is a change in temperature. To reduce this effect, most of the laboratory environment contains equipment that can maintain a stable temperature and humidity. Calibrations should be executed regularly to minimize the effect of test environment variations. The second major source of errors is random error. Errors from noise cannot be predicted and removed. A VNA is an instrument that itself is a noise source, so VNA measurements are affected by such noise. These errors can be reduced not by calibration but by increasing the number of times measurements are taken. Third, systematic errors, such as mismatch, insertion loss, and phase shift, are the dominant types of error in VNA measurement. The root causes of these systematic errors are signal leakage and unwanted reflection. Such systematic errors could be removed by using their repeatable characteristics. Systematic errors can be determined during the calibration procedure. In this thesis, of the three types of common error, only systematic errors are considered. 18 Figure 3.1 DUT with fixture. 3.2.2 Errors Characteristic by Fixture Measurements The characteristic impedance of an instrument that is connecting to a lumped element causes mismatch. If the reflection performance is measured, the reflection should be only from the Device Under Test (DUT). However, a reflection from connector or trace line could also occur, which causes measurement error. This error can be removed using a time gating method. In this thesis, however, the time gating method cannot be applied, because a VNA does not have a time gating function option. Figure 3.2 shows the input and output reflections of the through standards used for two-port S-parameter measurements. A mismatch ripple is evident at around 3.5 GHz. By getting rid of the gap that causes this mismatch reflection between the SMA connector and PCB board fixture, the mismatch ripple is mitigated slightly. However, the reduction in the ripple is not enough to obtain an accurate measurement. This error can be removed using a time gating method that can be applied in a window of the time domain. The transmission lines in the fixture and connector generate insertion loss and phase shift. Insertion loss and phase shift would be small if a short transmission line is used in a fixture. To avoid these errors a fixture should maintain zero electrical length. However, it is 19 not possible to maintain a zero electrical length in real measurement. The trace line should be short to minimize the phase shift if an accurate calibration is not possible. S11 0 1000 2000 3000 4000 5000 -50 -100 Thru S22 0 0 6000 dB dB 0 Freq(MHz) 2000 3000 4000 5000 6000 -50 -100 Modified Thru 1000 Freq(MHz) Thru Modified Thru Figure 3.2 Through standard S11 , S22. 3.2.3 SOLT Calibration The SOLT calibration method uses a 12-term error model, which is calculated by measuring calibrations standards. To obtain the 12 unknown parameters, 12 measurements are needed. In order to perform SOLT calibration for an extended reference plane, work for this thesis required specialized fixture calibration standards unique to this study. Figure 3.3 shows the through board of the RF3025 from RF Micro Devices that is employed as the user standard for the SOLT calibrations performed in this thesis. However, these user standards could not predict the correct error terms that cause measurement error. The through should be zero length. However, because the zero length could not be implemented, the through standard has added length, and the short standard does not provide an ideal short. The load must guarantee the broadband response. At the DUT reference point, two 100 Ω resistors were used to reduce inductance. However, the load standard contains error at high frequency because of the frequency variations of the resistor. These errors led to inaccurate calibration. 20 Two-port calibration is also possible by using only Open-Short-Load (OSL) standards calibration [3]. Each standard has its own characteristics. The short standard is a unity reflection with a 180° phase shift. The short standard contains inductance and should be specified. A short line interface is needed to reduce the inductance. The open standard is a unity reflection with no phase shift. However, fringing capacitance generates the phase shift. This fringing capacitance should be determined in order to insert the calibration standard data. The characteristics of the fringing capacitance of the open standard can be generated by using a suitably calibrated instrument. The characteristics of all the standards should be known in order to execute the correct calibration. However, fringing capacitance varies by frequency, so an SOLT calibration is useful only below 3 GHz. Open Short Load DUT Through Figure 3.3 User calibration kit. 21 3.2.4 TRL Calibration The TRL calibration method uses a 12-terms error model. By measuring through, reflection, and line standard the error terms can be determined. The through standard is a 0 direct connection between Port 1 and Port 2 (S= � 1 1 �). the reflection standard is a short or 0 open response (S11 =S22 =Γshort or Γopen ). The line is a short line inserted between Port 1 and port 2 ( S= � 0 e-βl e-βl �). 0 Figure 3.4 12-terms error model, after [2]. If the fixture is symmetric, the through-line (TL) calibration can be applied instead of the TRL calibration. Short or open standard measurements can be obtained from through measurements [4]. The advantage of TL calibration is that the random effect, which causes glitches in measurements, of the arbitrary reflection standard can be removed. The reflection 22 of the short standard is derived from the S-parameter measurement of the through standard. A detailed derivation is presented in [4]. Γshort = S11 Through − S21Through (Equation 3-1) 3.2.5 Port Extension The port extension method is useful when calibration standards are difficult to implement. This method cannot correct mismatch and losses. So, it is useful when a short well-matched fixture is available. First, the coaxial reference plane should be a known plane by performing a full 2-port calibration using manufacturer calibration standards. Phase shift and added delay are corrected by doing the port extension of a short or open standard. DUT 2-Port port extension 2-Port calibration using manufacture Cal kit Figure 3.5 Port extension method. 3.2.6 Time Gating The frequency domain can be converted to the time domain by using the inverseFourier transform. If the through standard is measure using the VNA’s time domain setting, 23 more than one reflection can be observed. One reflection occurs from the DUT and the other occurs from mismatch. These two reflections are illustrated in Figure 3.6. The first one is from an SMA-transmission line connection, and the second one is from a DUT. So, only the second response is the expected measurement. By capturing only the desired reflection, the reflection from the mismatch error can be removed. Figure 3.6 Time domain reflection response of a DUT with fixture. 3.3 Quality Factor from Two-Port S-parameters Network parameters, such as the S-parameter, Z-parameter, Y-parameter, and ABCDparameters represent the characteristics of passive elements. If the two-port S-parameter of a lumped element is measured, the quality factor of the lumped element can be generated. The first step of the quality factor calculation is finding the impedance. From [2] Yij = Ii (Vk = 0 for k ≠ j) Vj (Equation 3-2) “Yij can be determined by driving Port j with voltage Vj , short-circuiting all other ports (so Vk =0 for k≠j), and measuring the short-circuit current at Port i” [2]. From this definition Y11 is the admittance with an ideal short load at the output port; 𝑌𝑌11 is the 24 admittance of the input port while the output port is shorted to ground. As mentioned in Chapter 2, S-parameters can be converted to Z-parameters, Y-parameters, and ABCDparameters. By converting the S-parameters to Y-parameters, the quality factor also can be generated. Y11 = (1 − S11 )(1 + S22 ) + S12 S21 (1 + S11 )(1 + S22 ) − S12 S21 𝑄𝑄Q Q = − Y11 B = Z0 Y11 Q= (Equation 3-3) Im|Y11 | Re|Y11 | (Equation 3-4) (1 + S11 )(1 + S22 ) − S12 S21 2S21 Im|B| Re|B| (Equation 3-5) (Equation 3-6) The impedance of the network can be derived from the ABCD parameter. Series impedance z is the B of the ABCD parameter according to an ABCD parameter definition [7], see Figure 3.7. z 1 2 𝐴𝐴 = 1 𝐵𝐵 = 𝑧𝑧 � � 𝐶𝐶 = 0 𝐷𝐷 = 1 Figure 3.7 ABCD parameters of a series impedance. 25 3.4 Quality Factor from One-Port S-Parameter A one-port measurement is required if the characteristic of the DUT is determined using only reflection. The error in one–port measurements can be represented using a threeterm error, see Figure 3.8. These error terms can be determined by examining the SOL standards. 𝐸𝐸𝑅𝑅𝑅𝑅 𝑆𝑆11𝑀𝑀 𝐸𝐸𝐷𝐷𝐷𝐷 𝐸𝐸𝑆𝑆𝑆𝑆 1 𝑆𝑆11 Reference Figure 3.8 One-port VNA error model [5] [6]. By using a signal flow graph applied to the network in Figure 3.8: S11M = EDF + S11 = S11 ERF 1-S11 ESF S11M − EDF ESF (S11M − EDF ) + ERF (Equation 3-7) (Equation 3-8) The three unknown term, EDF , ESF , and ERF , can be derived by measuring three calibration standards, and measured value, S11M , convert to the actual value, S11 , using Equation 3-8. 26 In the case of an inductor, the stored reactive energy is the difference between magnetic energy and electric energy. The quality factor is derived using this concept when the load is an ideal short. The following calculation is derived from [6]: Peak Magnetic Energy-Peak Electric Energy Energy Loss per One Cycle Q = 2π = 2π = �Wm,avg − We,avg � �2×Wm,avg − 2×We,avg � =2ω Pl,avg ×T Pl,avg Im{Zin } Re{Zin } where Zin=Z0 1 + S11 1 − S11 )� = 1+S Re(50× 1 − S11 ) 11 �Im(50× (1 + Γin ) (1 − Γin ) (Equation 3-9) So, the quality factor can be derived from a one-port S-parameter using an ideal short. To obtain the one-port S-parameter of a lumped element, the manufacturer calibration standards and port extension method are implemented in this thesis. The port extension is not a perfect calibration method, so the direct SMA connector fixture is applied to minimize measurement errors. As shown in Figure 3.9, the SMA connector fixture has a very short length and direct connection to the DUT. 27 Figure 3.9 SMA connector fixture. 3.5 Summary The test environment is the most important factor in obtaining accurate RF measurements. Unknown terms and variations in the instruments should be removed. The appropriate instrument to employ is the VNA to measure the characteristics of lumped elements over frequency. Possible errors in VNA measurements include drift error, random error, and systematic error. Several calibration methods, such as SOLT calibration, TRL calibration, port extension, and time gating, are considered for making the VNA and fixtures known terms. From the S-parameter, the quality factor can be determined. Table 3.1 presents a summary of the quality factor calculation methods. Table 3.1 Summary of quality factor calculation. One-Port S-parameter Using term Q factor Two-Port S-parameter (with virtual short load) Zin Zin Im|Zin | Re|Zin | Im|Zin | Re|Zin | 28 Y11 − Im|Y11 | Re|Y11 | B Im|B| Re|B| The best approach to obtaining a one-port S-parameter with the limited source available is the combination of the port extension and manufacturer calibration kit. To minimize measurement errors, a direct SMA connector fixture is applied. It has a very short length and direct connection to the DUT. 29 REFERENCES [1] Agilent, “In-fixture Measurements Using Vector Network Analyzers,” Application Note AN 1287-9, 2006. [2] Agilent, “Agilent Network Analysis Applying the 8510 TRL Calibration for NonCoaxial Measurement,” Product Note 8510-8AZ, 2006. [3] Z. Y. Chen, Y. L. Wang, Y. Liu, and N. H. Zhu, “Two-port Calibration of test fixtures with OSL method,” in International Conference on Microwave and Millimeter Wave Technology Proceedings, pp. 138–141, 2002. [4] M. B. Steer, S. B. Goldberg, G. Rinne, P. D. Franzon, I. Turlik, and J. S. Kasten, “Introducing Through-Line Deembedding Procedure,” in 1992 IEEE MTT-S International Microwave Symposium Digest, vol.3 pp. 1455–1458, 1992. [5] L. F. Chen, Micorwave Electronics: Measurement and Materials Characterization, Chichester: John Wiley & Sons, Ltd, 2004. [6] J. M. Golio, The RF and Microwave Handbook, 2nd ed. CRC Press, 2008. [7] M. B. Steer, Microwave and RF Design: A Systems Approach, SciTech Publishing, 2010. 30 CHAPTER 4 Measurement Results 4.1 Introduction An Anritsu MS4623 VNA was used in this study to collect S-parameters up to 5 GHz. To obtain two-port S-parameters, a user fixture, which provides the interface between the lumped element and the VNA, was employed. Correct measurement result, however, could not be achieved because of the inaccuracy of the user fixture calibration standards. The proposed alternative method used was to obtain a one-port measurement, which combines port extension and calibration using manufacturer’s calibration kit. To improve the measurement accuracy, a SMA fixture, which has a short line and good impedance matching, was used. A lumped element was soldered directly to the SMA connector. The quality factor and SRF could then be obtained using one-port S-parameters and two-port S-parameters from the manufacturer. Figure 4.1 Quality factor of Johanson capacitor [2]. 31 Figure 4.2 SRF of Johanson capacitor [2]. The part numbers for the lumped elements are TOKO LL1608FSL for the inductor, Johanson 251R15S for the capacitor, and Panasonic ERJ-3GEYJ100 for the resistor. The size of each lumped element is 1.6 × 0.8 mm. 4.2 One-Port S-Parameter Results From one-port parameters, which were collected using a short load, the impedance values of lumped elements are obtained, and the quality factor and SRF of the lumped elements are obtained from their impedance value. Table 4.1 shows a summary of quality factors obtained from one-port S-parameter measurements. Detailed measurement plots are presented in Appendix A. Each capacitance, inductance, resistance, SRF, and quality factor of the lumped elements is plotted and presented in Appendix A. 32 Table 4.1 TOKO Inductor one-port quality factor and SRF measurement results. 0603 Toko Inductor SRF(MHz) Data sheet Data sheet 9100±15% above 5000 5800±15% above 5000 4700±15% Measurement 2.2nH 13 300 MHz 26.5 6.469 32.765 29.732 48.698 53.038 42.082 13.8 27.8 36 46.4 52.9 68 7.271 32.494 28.032 43.966 40.799 41.539 15.8 29.4 39 49.3 55.8 67.5 4970 10.634 30.026 37.295 42.685 44.776 47.182 Data sheet 4000±15% 17.3 29.3 38.6 48.1 53.8 56.7 Measurement 4060 13.436 35.263 40.47 43.162 44.069 41.005 Data sheet 2000±15% 19.6 33.2 42.2 48 47.9 15.9 Measurement 2016 18.47 30.42 38.074 33.397 29.951 13.437 Data sheet 1500±15% 21.5 31.7 38.6 38.6 33.5 Measurement 1455 19.532 25.67 32.146 26.901 20.204 Data sheet 900±15% 20.5 35.5 39.1 20.4 Measurement 910 19.6 20.549 20.739 4.454 Data sheet 470±15% 21.9 21.1 Measurement 430 19.476 10.728 Measurement Data sheet 3.3nH 6.8nH 10nH 33nH 56nH 100nH 270nH Quality Factor 500 800 MHz MHz 35.4 44.2 Measurement 100 MHz 1000 MHz 51.9 1800 MHz 63.9 Table 4.2 Johanson Capacitor one-port SRF measurement results. SRF(MHz) 0603 Johanson Capacitor 2.2pF 3.3pF 10pF 15pF 33pF 47pF 82pF 4500 3500 2000 1600 1100 900 700 The measurement data presented in Table 4.2 are the measurement results obtained from the Johanson 251R15S capacitors. These results do not include the quality factor results because of instrument resolution problems. The instrumentation uncertainty of the MS4623 is ±0.15dB [1]. In the case of a high quality factor, that is, above 1000, the results can be 33 inaccurate due to this instrumentation uncertainty. For example, the quality factor of the Johanson 3.3pF capacitor is about 3000 at 300 MHz, as shown in Figure 4.1. The Equivalent Series Resistance (ESR) value is very low; a typical value of ESR is from 0.05 to 0.15. So, the instrumentation uncertainty affects the measurement results, and the measurement results of capacitor include the instrument error. The ripples caused by instrument error are shown in the capacitor result plots presented in Appendix A. Table 4.3 Panasonic resistor one-port measurement results Quality Factor (at 1 GHz) 0603 Panasonic Resistor 10 Ω 39 Ω 0.7 0.15 68 Ω 200 Ω 300 Ω 0.075 0.025 0.06 560 Ω 0.15 1000 Ω 2000 Ω 3000 Ω 0.3 0.6 1 Table 4.3 shows the quality factor value of the resistors at 1 GHz. Resistors have very low quality factor values compare those of capacitors and inductors. Inductance is dominant from 10 Ω to 68 Ω, and capacitance is dominant from 200 Ω to 3000 Ω as parasitics. Capacitance parasitics values are 0.1 to 0.3 pF, and inductance parasitics values are around 1 nH. Most of the resistor manufacturers do not provide the quality factor values of their resistors, so the results cannot be compared with the datasheet. 4.3 Two-Port S-Parameter Results from Manufacturer To obtain the impedance value of lumped elements, two-port S-parameters are converted to one-port S-parameters using a virtual short to ground plane. Errors that stem from the calibration standards, however, affect the results, especially at high frequencies. So, 34 instead of the measurement data, S-parameters provided by the manufacturer are used to compare the one-port and two-port measurement data. Table 4.4, Table 4.5, and Table 4.6 provide a summary of the inductor results obtained from the calculations. Table 4.4 Johanson capacitor two-port SRF measurement results. SRF(MHz) 0603 Johanson Capacitor 2.2pF 3.3pF 10pF 15pF 33pF 47pF 82pF 4000 3600 2000 1600 1200 1100 800 Table 4.5 TOKO inductor two-port Q and SRF measurement results. Quality Factor 500 800 MHz MHz 35.4 44.2 0603 Toko Inductor SRF(MHz) Data sheet 13 9.493 15.992 22.59 Data sheet 9100±15% above 6000 5800±15% 300 MHz 26.5 13.8 27.8 Calculation 5290 11.198 Data sheet 4700±15% Calculation 2.2nH 3.3nH 6.8nH 10nH 33nH 56nH 100nH 270nH 1000 MHz 51.9 1800 MHz 63.9 29.418 31.493 39.069 36 46.4 52.9 68 17.999 24.392 32.699 36.625 43.362 15.8 29.4 39 49.3 55.8 67.5 3850 11.975 21.619 30.031 38.985 42.513 40.708 Data sheet 4000±15% 17.3 29.3 38.6 48.1 53.8 56.7 Calculation 2980 13.821 23.737 32.08 40.254 42.662 29.637 Data sheet 2000±15% 19.6 33.2 42.2 48 47.9 15.9 Calculation 1570 16.903 30.154 36.574 34.386 25.237 Data sheet 1500±15% 21.5 31.7 38.6 38.6 33.5 Calculation 1160 18.023 29.632 32.251 19.843 8.01 Data sheet 900±15% 20.5 35.5 39.1 20.4 Calculation 820 20.875 31.083 26.546 1.825 Data sheet 470±15% 21.9 21.1 Calculation 400 20.363 12.4 Calculation 100 MHz 35 Table 4.6 Panasonic resistor two-port measurement results. 0603 Panasonic Resistor 10 Ω 39 Ω 0.7 0.15 Quality Factor (at 1 GHz) 68 Ω 560 Ω 1000 Ω 0.075 0.1 0.2 3000 Ω 0.6 4.4 Summary This chapter presents the quality factors and SRFs of lumped elements using Sparameters. Several methods were applied to obtain the correct S-parameters of the lumped elements. The two-port measurements, which were taken first, are not appropriate for deriving the quality factor because of calibration standards errors. To overcome the error of a fixture, a port extension and calibration using a manufacturer’s calibration kit were applied for a one-port measurement. In the case of inductors, the measurement results match the values in the data sheet very well. However, the instrumentation uncertainty affects the results of capacitors that have low ESR values; the result, ripples are shown in the quality factor plot. 36 REFERENCES [1] Anritsu, “MS4622 A/B/C/D, MS4623, MS4624 A/B/C/D Vector Network Measurement Systems,” 2006. Available at http://www.us.anritsu.com/Downloads/Datasheet-Configuration%20Guide/MS462xASeriesMS462xCSeries-ME7840A-ME7842B-ME7840/4-MS462xB/DSeriesVariousWirelessBaseStationTransmitFrequencies_downloadmaster.aspx?fileName=1141000288.pdf&fileID=1270&fileType=12 [2] Johanson, “Capacitor Catalog,” Available at http://www.johansontechnology.com/images/stories/catalog/JTI_MLCC_HighQ_201004.pdf. [3] Anritsu, “Inductor catalog,” Available at http://www.toko.co.jp/products/pdf/inductors/ll1608-fsl.pdf. 37 CHAPTER 5 Conclusions 5.1 Conclusions To observe the characteristics of lumped elements in terms of frequency change, the quality factor, which represents the ratio of power storage to power loss in a network, is the appropriate parameter. In this thesis, the quality factors of the lumped elements are derived from one-port S-parameters. One-port S-parameters, which are useful in RF circuit analysis, can be obtained over frequency using a VNA. A test fixture should be employed to provide the interface between the DUT and the VNA, and the effects of the test fixture should be removed using standard calibration. To minimize the errors caused by a port extension method, the SMA connector is used which has a very short length and direct connection to the DUT, is applied. Good measurement results for the inductors are achieved compared to the resultant data in the data sheet. However, the results for the capacitors, which have low ESR values show ripples in the quality factor plot. This error comes from instrumentation uncertainty. The most challenging part of this research was collecting accurate measurement data, because the experiments were executed within a limited test environment. If TRL standards could be implemented, then more accurate results could be achieved at high frequencies. The quality factor and SRF of commercial lumped RF microwave elements are presented in this thesis 38 5.2 Future Work Two-port VNA measurement could not provide accurate data for this research due to imperfect calibration standards. If correct user fixture calibration standards are employed, correct two-port S-parameters of lumped elements can be obtained, and a comparison of the calibration methods can then be exhibited. To apply the correct user fixture calibration standards, new PCB fixtures or commercial fixture are needed. Experimental results show the discrepancies among the data sheet data, especially in the case of the capacitor. Another method should be considered. For instance, a resonant circuit method, which is based on the resonant system with a variable impedance load, can be employed as the solution to measure the high quality factors. 39 APPENDIX 40 Appendix A: Plot of Measurement Result Inductance, capacitance, resistance, phase of impedance, and Q factor was plotted using S-parameters of lumped elements. Each lumped element has two figures. Reactance and phase of impedance were plotted in the upper figure and Q factor was plotted in the lower figure. 41 1. Inductor 10 100 9 90 8 80 7 70 6 60 5 50 4 40 3 30 2 Inductance Degree 1 0 Degree Inductance (nH) 0603 Toko 2.2 nH Inductor (LL1608FSL) 20 10 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.1 Inductance and phase of impedance. 0603 Toko 2.2 nH Inductor (LL1608FSL) 100 90 1-Port S-Parameter 80 2-Port S-Parameter from manufacturer Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.2 Quality factor of 2.2 nH inductor. 42 10000 10 100 9 90 8 80 7 70 6 60 5 50 4 40 3 30 2 Inductance Degree 1 Degree Inductance (nH) 0603 Toko 3.3 nH Inductor (LL1608FSL) 20 10 0 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.3 Inductance and phase of impedance. 0603 Toko 3.3 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.4 Quality factor of 3.3 nH inductor. 43 10000 20 100 18 90 16 80 14 70 12 60 10 50 8 40 6 30 4 20 Inductance Degree 2 0 0 1000 Degree Inductance (nH) 0603 Toko 6.8 nH Inductor (LL1608FSL) 10 0 2000 3000 4000 5000 6000 Freq (MHz) Figure A.5 Inductance and phase of impedance. 0603 Toko 6.8 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.6 Quality factor of 6.8 nH inductor. 44 10000 0603 Toko 10 nH Inductor (LL1608FSL) 100 30 80 60 40 20 20 Degree Inductance (nH) 25 0 15 -20 10 -40 -60 Inductance Degree 5 -80 -100 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.7 Inductance and phase of impedance. 0603 Toko 10 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.8 Quality factor of 10 nH inductor. 45 10000 100 100 90 80 80 60 70 40 60 20 50 0 40 -20 30 Inductance Degree 20 10 Degree Inductance (nH) 0603 Toko 33 nH Inductor (LL1608FSL) -40 -60 -80 -100 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.9 Inductance and phase of impedance 0603 Toko 33 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.10 Quality factor of 33 nH inductor. 46 10000 0603 Toko 56 nH Inductor (LL1608FSL) 150 100 130 80 60 40 90 20 70 0 50 -20 Inductance Degree 30 10 Degree Inductance (nH) 110 -40 -60 -80 -10 0 1000 2000 3000 4000 5000 6000-100 Freq (MHz) Figure A.11 Inductance and phase of impedance. 0603 Toko 56 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.12 Quality factor of 56 nH inductor. 47 10000 0603 Toko 100 nH Inductor (LL1608FSL) 100 300 Inductance Degree 80 60 40 200 20 Degree Inductance (nH) 250 0 150 -20 100 -40 -60 50 -80 -100 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.13 Inductance and phase of impedance. 0603 Toko 100 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.14 Quality factor of 100 nH inductor. 48 10000 0603 Toko 270 nH Inductor (LL1608FSL) 100 500 Inductance Degree 450 60 350 40 300 20 250 0 200 -20 150 -40 100 -60 50 -80 0 -100 0 1000 2000 3000 4000 5000 Degree Inductance (nH) 400 80 6000 Freq (MHz) Figure A.15 Inductance and phase of impedance. 0603 Toko 270 nH Inductor (LL1608FSL) 100 1-Port S-Parameter 2-Port S-Parameter from manufacturer 90 80 Q Factor 70 60 50 40 30 20 10 0 10 100 1000 Freq (MHz) Figure A.16 Quality factor of 270 nH inductor. 49 10000 2. Capacitor 0603 Johanson 2.2 pF Capacitor (251R15S) 9 8 Capacitance (pF) 100 Capacitance Degree 80 60 7 40 6 20 5 0 4 -20 3 -40 2 -60 1 -80 0 -100 0 1000 2000 3000 4000 5000 Degree 10 6000 Freq (MHz) Figure A.17 Capacitance and phase of impedance. 0603 Johanson 2.2 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.18 Quality factor of 2.2 pF capacitor. 50 5000 6000 20 100 18 80 16 60 14 40 12 20 10 0 8 -20 6 -40 4 -60 Capacitance Degree 2 Degree Capacitance (pF) 0603 Johanson 3.3 pF Capacitor (251R15S) -80 -100 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.19 Capacitance and phase of impedance. 0603 Johanson 3.3 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.20 Quality factor of 3.3 pF capacitor. 51 5000 6000 50 100 45 80 40 60 35 40 30 20 25 0 20 -20 15 -40 10 -60 Capacitance Degree 5 0 0 1000 2000 3000 4000 5000 Degree Capacitance (pF) 0603 Johanson 10 pF Capacitor (251R15S) -80 -100 6000 Freq (MHz) Figure A.21 Capacitance and phase of impedance. 0603 Johanson 10 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.22 Quality factor of 10 pF capacitor. 52 5000 6000 0603 Johanson 15 pF Capacitor (251R15S) 70 100 80 60 50 40 40 20 0 30 -20 20 -40 -60 Capacitance Degree 10 0 0 1000 2000 3000 4000 5000 Degree Capacitance (pF) 60 -80 -100 6000 Freq (MHz) Figure A.23 Capacitance and phase of impedance. 0603 Johanson 15 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.24 Quality factor of 15 pF capacitor. 53 5000 6000 100 100 90 80 80 60 70 40 60 20 50 0 40 -20 30 -40 20 -60 Capacitance Degree 10 0 0 1000 2000 3000 4000 5000 Degree Capacitance (pF) 0603 Johanson 33 pF Capacitor (251R15S) -80 -100 6000 Freq (MHz) Figure A.25 Capacitance and phase of impedance. 0603 Johanson 33 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.26 Quality factor of 33 pF capacitor. 54 5000 6000 0603 Johanson 47 pF Capacitor (251R15S) 100 120 80 60 40 80 20 0 60 -20 40 Degree Capacitance (pF) 100 -40 -60 Capacitance Degree 20 0 0 1000 2000 3000 4000 5000 -80 -100 6000 Freq (MHz) Figure A.27 Capacitance and phase of impedance 0603 Johanson 47 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.28 Quality factor of 47 pF capacitor. 55 5000 6000 200 100 180 80 160 60 140 40 120 20 100 0 80 -20 60 -40 40 -60 Capacitance Degree 20 0 0 1000 2000 3000 4000 5000 Degree Capacitance (pF) 0603 Johanson 82 pF Capacitor (251R15S) -80 -100 6000 Freq (MHz) Figure A.29 Capacitance and phase of impedance. 0603 Johanson 82 pF Capacitor (251R15S) 10000 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 1000 100 10 1 0 1000 2000 3000 4000 Freq (MHz) Figure A.30 Quality factor of 82 pF capacitor. 56 5000 6000 3. Resistor 0603 Panasonic 10 ohm Resistor (ERJ-3GEYJ100) 80 11.5 11 40 10.5 Resistance 10 20 Degree 0 9.5 Inductance (nH) Degree 60 1.2 3.5 3 2.5 2 1.5 1 0.5 0 1 0.8 Inductance Capacitance 0.6 0.4 0.2 Capacitance (pF) Resistance (ohm) 12 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.31 Resistance, phase of impedance, and reactance. 0603 Panasonic 10 ohm Resistor (ERJ-3GEYJ100) Q Factor 4 1-Port S-Parameter 3 2 1 0 10 100 1000 Freq (MHz) Figure A.32 Quality factor of 10 Ω resistor. 57 10000 0603 Panasonic 39 ohm Resistor (ERJ-3GEYJ100) 40 20 40 10 Resistance 39 0 Degree -10 38 3 Inductance (nH) Degree 30 41 1.2 Inductance Capacitance 2.5 2 1 0.8 1.5 0.6 1 0.4 0.5 0.2 0 Capacitance (pF) Resistance (ohm) 42 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.33 Resistance, phase of impedance, and reactance. 0603 Panasonic 39 ohm Resistor (ERJ-3GEYJ100) 1 1-Port S-Parameter 2-Port S-Parameter from… Q Factor 0.8 0.6 0.4 0.2 0 10 100 1000 Freq (MHz) Figure A.34 Quality factor of 39 Ω resistor. 58 10000 0603 Panasonic 68 ohm Resistor (ERJ-3GEYJ100) 20 71 69 10 68 Resistance 67 5 Degree 66 0 3 Inductance (nH) Degree 15 70 1.2 Inductance Capacitance 2.5 2 1 0.8 1.5 0.6 1 0.4 0.5 0.2 0 Capacitance (pF) Resistance (ohm) 72 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.35 Resistance, phase of impedance, and reactance. 0603 Panasonic 68 ohm Resistor (ERJ-3GEYJ100) 0.5 1-Port S-Parameter 2-Port S-Parameter from manufacturer Q Factor 0.4 0.3 0.2 0.1 0 10 100 1000 Freq (MHz) Figure A.36 Quality factor of 68 Ω resistor. 59 10000 0603 Panasonic 200 ohm Resistor (ERJ-3GEYJ100) 5 Resistance Degree 195 0 190 -5 185 -10 180 Degree 200 -15 175 170 -20 1.2 60 Inductance Capacitance 1 0.8 50 40 0.6 30 0.4 20 0.2 10 0 Capacitance (pF) Inductance (nH) Resistance (ohm) 205 0 0 1000 2000 3000 4000 5000 6000 Freq (MHz) Figure A.37 Resistance, phase of impedance, and reactance. 0603 Panasonic 200 ohm Resistor (ERJ-3GEYJ100) 0.3 1-Port S-Parameter Q Factor 0.25 0.2 0.15 0.1 0.05 0 10 100 1000 Freq (MHz) Figure A.38 Quality factor of 200 Ω resistor. 60 10000 5 300 0 250 -5 200 -10 150 -15 100 -20 Resistance Degree 50 0 -25 -30 60 5.0 Inductance (nH) Degree 350 Inductance Capacitance 4.0 50 40 3.0 30 2.0 20 1.0 10 0.0 0 0 1000 2000 3000 4000 5000 Capacitance (pF) Resistance (ohm) 0603 Panasonic 300 ohm Resistor (ERJ-3GEYJ100) 6000 Freq (MHz) Figure A.39 Resistance, phase of impedance, and reactance. 0603 Panasonic 300 ohm Resistor (ERJ-3GEYJ100) 0.6 1-Port S-Parameter Q Factor 0.5 0.4 0.3 0.2 0.1 0 10 100 1000 Freq (MHz) Figure A.40 Quality factor of 300 Ω resistor. 61 10000 0603 Panasonic 560 ohm Resistor (ERJ-3GEYJ100) 500 0 -10 400 -20 300 -30 200 100 -40 0 -50 70 60 50 40 30 20 10 0 Inductance Capacitance 4.0 3.0 2.0 1.0 0.0 0 1000 2000 3000 4000 5000 Degree 600 5.0 Inductance (nH) 10 Resistance Degree Capacitance (pF) Resistance (ohm) 700 6000 Freq (MHz) Figure A.41 Resistance, phase of impedance, and reactance. 0603 Panasonic 560 ohm Resistor (ERJ-3GEYJ100) 1.2 1-Port S-Parameter 2-Port S-Parameter from… Q Factor 1 0.8 0.6 0.4 0.2 0 10 100 1000 Freq (MHz) Figure A.42 Quality factor of 560 Ω resistor. 62 10000 Resistance Degree 1000 800 600 400 200 0 Inductance (nH) 5.00 Inductance Capacitance 4.00 3.00 2.00 1.00 0.00 0 1000 2000 3000 4000 5000 10 0 -10 -20 -30 -40 -50 -60 -70 Degree Resistance (ohm) 1200 35 30 25 20 15 10 5 0 Capacitance (pF) 0603 Panasonic 1000 ohm Resistor (ERJ-3GEYJ100) 6000 Freq (MHz) Figure A.43 Resistance, phase of impedance, and reactance. 0603 Panasonic 1000 ohm Resistor (ERJ-3GEYJ100) Q Factor 2 1-Port S-Parameter 2-Port S-Parameter from… 1.5 1 0.5 0 10 100 1000 Freq (MHz) Figure A.44 Quality factor of 1 kΩ resistor. 63 10000 0603 Panasonic 2000 ohm Resistor (ERJ-3GEYJ100) 20 Resistance Degree 0 1500 -20 1000 -40 500 -60 0 -80 5.00 8 7 6 5 4 3 2 1 0 Inductance Capacitance 4.00 3.00 2.00 1.00 0.00 0 1000 2000 3000 4000 5000 Degree 2000 Capacitance (pF) Inductance (nH) Resistance (ohm) 2500 6000 Freq (MHz) Figure A.45 Resistance, phase of impedance, and reactance. 0603 Panasonic 2000 ohm Resistor (ERJ-3GEYJ100) 3 1-Port S-Parameter Q Factor 2.5 2 1.5 1 0.5 0 10 100 1000 Freq (MHz) Figure A.46 Quality factor of 2 kΩ resistor. 64 10000 0603 Panasonic 3000 ohm Resistor (ERJ-3GEYJ100) Resistance Degree 3000 2500 0 Degree -20 2000 -40 1500 -60 1000 -80 500 0 -100 5.00 Inductance (nH) 20 14 12 10 8 6 4 2 0 Inductance Capacitance 4.00 3.00 2.00 1.00 0.00 0 1000 2000 3000 4000 5000 Capacitance (pF) Resistance (ohm) 3500 6000 Freq (MHz) Figure A.47 Resistance, phase of impedance, and reactance. 0603 Panasonic 3000 ohm Resistor (ERJ-3GEYJ100) 5 1-Port S-Parameter 2-Port S-Parameter from… Q Factor 4 3 2 1 0 10 100 1000 Freq (MHz) Figure A.48 Quality factor of 3 kΩ resistor. 65 10000