Wireless Power Transmission Utilizing A Phased Array Of
`Tesla Coils
`
`by
`Joseph C. Stark III
`
`Submitted to the Department of Electrical Engineering and Computer Science
`in partial fulfillment of the requirements for the degree of
`
`Masters of Engineering in Electrical Engineering
`
`at the
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`MASSACHUSETTS INSTITUTE OF TECHNOLOGY
`
`May2004 L3V-'ne 20c, i7
`@ Joseph C. Stark III, MMIV. All rights reserved.
`
`The author hereby grants to MIT permission to reproduce and distribute publicly
`paper and electronic copies of this thesis document in whole or in part.
`
`Author....
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`................ ............
`Department of Electrical Engineering and Computer Science
`May 13, 2004
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`A
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`Certified by.....
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`...............
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`...
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`..........
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`..........
`Chathan M. Cooke
`Lecturer and Principal Research Engineer
`Thesis Supervisor
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`Accepted by ............
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`Arthur C. Smith
`Chairman, Department Committee on Graduate Students
`
`MASSACHUSETTS INSTITUTE
`OF TECHNOLOGY
`
`JUL 2 0 2004
`
`LIBRARIES
`
`BARKER
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 001
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`

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`2
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 002
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 002
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`

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`Wireless Power Transmission Utilizing A Phased Array Of Tesla Coils
`
`by
`
`Joseph C. Stark III
`
`Submitted to the Department of Electrical Engineering and Computer Science
`on May 13, 2004, in partial fulfillment of the
`requirements for the degree of
`Masters of Engineering in Electrical Engineering
`
`Abstract
`
`This thesis discusses the theory and design of coupled resonant systems and how they
`can be linked in a phased array for the wireless transmission of electrical power. A detailed
`derivation of their operational theory is presented with a strong emphasis on the current and
`voltage waveforms produced. Formulas are presented relating the features of the waveforms
`to specific parameters of the system. They provide a theoretical basis for the design of
`the Tesla coil systems. Unloaded and loaded operating efficiency is considered from both
`a power and energy perspective with emphasis on maximizing the two quantities. With
`these design formulas, a working set of two distinct coupled resonant systems were locked
`in frequency and controllable in phase to produce a phased array capable of wireless power
`transmission. The operational details and practical design considerations are presented and
`explained. The measured output waveforms were found to closely agree with the predicted
`models.
`
`Thesis Supervisor: Chathan M. Cooke
`Title: Lecturer and Principal Research Engineer
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`3
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 003
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 004
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 004
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`

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`Acknowledgments
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`First and foremost, I would like to thank my parents for all their support and encourage-
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`ment. Without their help, I would not be who I am today. To Tetazoo, much thanks for
`
`providing the necessary distractions. And finally, to my advisor, Dr. Chathan Cooke who
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`took me in as a wet-behind-the-ears college grad and showed me the real world of electrical
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`engineering.
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`5
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 005
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 006
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`Momentum Dynamics Corporation
`Exhibit 1008
`Page 006
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`Contents
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`1 Introduction
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`1.1 Overview
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`2 RLC Circuit Analysis
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`2.1 History of the Coupled Resonant System .
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`2.6 Summary of Results
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`3 Coupled Resonate RLC Networks
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`3.1 Derivation of Dynamics in the Frequency Domain .
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`3.2.4 Perturbation Analysis
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`5 Tesla Coil Design Constraints and Tradeoffs
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`5.1 Explanation of Variables, Parameters, and Assumptions
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`6 Switching Needs for Frequency and Phase Control of Coupled Resonant
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`Systems
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`7 Building the Coupled Resonators and Control Circuitry
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`7.2.1 Anti-Aliasing and Low Pass Filtering Circuitry .
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`7.3.1 Mechanical Switches .
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`166
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`7.3.3 Electronic Switches . .
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`7.4 Resonant System Construction and Switching .
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`7.4.1 Coil Construction
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`7.4.2 Response Waveforms from Mechanically Switched Systems
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`7.4.3 The Effects of k .
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`7.5 Tuning and Frequency Feedback Control Circuitry
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`7.5.1
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`Feedback System for Frequency Matching
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`8 Comparisons and Analysis of Coupled Resonant Systems
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`179
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`190
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`205
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`8.1 Comparisons of Waveforms
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`8.1.1 Method of Determining Parameters
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`8.2 Analysis of Quality Factors
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`205
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`9
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 009
`
`

`

`9 Project Conclusion
`
`9.1 Wireless Power Transmission
`
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`9.2 Future W ork
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`9.3 Project Conclusion
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`221
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`225
`
`A Appendix A: Derivation of Time and Frequency Domain Relations
`
`A.1 Necessary Requirements of the Laplace Transforms ..................
`
`A.2 Why k is Between 0 And 1
`
`.......
`
`...........................
`
`A.3 Derivation of Perturbation Analysis
`
`A.4 Efficiency Integration .
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`B Appendix B: Compilations of Scripts
`
`227
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`227
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`228
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`229
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`230
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`231
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`10
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 010
`
`

`

`Chapter 1
`
`Introduction
`
`1.1 Overview
`
`Power is important to modern systems. Prom the smallest MEMS sensors and bionic im-
`
`plants to satellites and oil platforms, it is important to be able to deliver power by means
`
`other than wires or transmission lines. In the case of biological implants, there must be a
`
`battery or energy storage element present that can receive and hold energy. This element
`
`takes up valuable space inside a person. In the case of satellites and oil platforms, either so-
`
`lar panels, fuel cells, or combustion engines are currently used to supply power. Solar panels
`
`take up a great deal of weight and bulk in terms of energy density and must be constantly
`
`repositioned to maximize exposure to the sun. Fuel cells and combustion engines require
`
`fuel and maintenance to be delivered on-site. The use of wireless power transmission, on
`
`a scale much larger than used by magnetic induction devices, would allow for systems to
`
`operate remotely without the need for relatively large energy storage devices or routine
`
`maintenance.
`
`This thesis will explore the theory, design, and construction of a method to transmit
`
`wireless electrical power through space. To this end, the Tesla coil configuration is used
`
`as the basis to generate high voltage, high frequency electrical power. Multiple Tesla coils,
`synchronized in frequency, are considered to increase delivered power and provide direction-
`
`ality. The generated power can be radiated to a receiver through an antenna array whose
`
`design will vary depending on the needs of the application. For some applications, a focused
`
`radiation pattern would be optimal, while for others, such as nanosensors spaced about a
`
`wide area, an omnidirectional radiation pattern would be appropriate. The receiver is itself
`
`11
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 011
`
`

`

`a resonant system tuned to the high frequency of the radiated power for maximum power
`
`transfer, much as a radio receiver must be tuned to a given frequency for higher reception
`
`gain.
`
`One of the core requirements of this project is to completely characterize the output
`
`waveforms of a Tesla coil. With a complete characterization, designers can tailor the output
`
`to meet specifications for power supply and other requirements. Another goal is to char-
`
`acterize the energy and power efficiency of the Tesla coil such that the tradeoffs between
`
`efficiency and component size, coupling coefficient, and other parameters is known. Finally,
`
`and most importantly, this project considers a new topology of multiple, separate Tesla
`
`coils operated in synchronization so as to create an array with directional attributes.
`
`Besides theoretical design considerations, this thesis aims to construct a working phased
`
`array. To this end, a method of driving and controlling the two coupled resonant systems
`
`is explored and designed. Considerations such as switching frequencies, types of switches,
`the uses of MOSFETs versus IGBTs, and feedback methods for controlling the system are
`
`examined. Ancillary issues such as parasitic losses and inductor coil construction are also
`
`considered. Essentially, this thesis aims to demonstrate and explain key elements of the
`
`design and construction process.
`
`This thesis is broken down into eight subsequent chapters. The first starts with the
`
`lumped parameters, second order circuits that comprise half of the Tesla coil structure and
`
`their theory of operation. The operation of capacitors and inductors is explored as well as
`
`their properties and physical limits. The Tesla coil itself is represented by two coupled series
`
`circuits. Two different modeling approaches are presented allowing a thorough analysis of
`
`performance criteria as well as frequency and time domain solutions.
`
`The second chapter discusses how two second order systems can be coupled to yield a
`
`single fourth order circuit with a mutual inductance coupling the inductors of the two cir-
`
`cuits. These two second order circuits are known as the primary and secondary of the Tesla
`
`coil. As the traditional Tesla coil is comprised of two coupled second order systems sharing
`
`a resonant frequency, the Tesla coil is also referred to as a coupled resonant system through-
`
`out this thesis. The issues of matched resonant frequencies and the interaction of quality
`
`factors are presented. The frequency and time domain current and voltage waveforms are
`
`heavily discussed and characterized.
`
`The third chapter discusses the operating efficiency of the coupled resonant system.
`
`12
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 012
`
`

`

`Concepts such as energy and power efficiency are distinguished and explored. Furthermore,
`
`the concept of matched output resistors and their effect on the operational efficiency is
`
`shown. Various phenomenon for energy and power losses are characterized.
`
`The fourth chapter discusses the some practical design issues when constructing resonant
`
`systems. The chapter emphasizes the characterization of inductors and capacitors and
`
`the losses associated with them due to their electrical operation and geometry. Design
`
`procedures for optimizing the voltage on the secondary coil are shown as well as procedures
`
`for designing other optimizations. The chapter is concluded with a design scenario showing
`
`the practical limitations of physical components.
`
`The fifth chapter discusses methods of driving the coupled resonant system. The pulsed
`
`system for driving the coils is explored and the theory behind its operation is shown. Meth-
`
`ods of synchronizing the resonant frequency between the primary and secondary coils of
`
`each resonant system are shown. The theory listed here leads directly to the circuit built
`
`in the following chapter.
`
`The sixth chapter shows the building of a phased array, pulse driven, coupled resonant
`
`system. Two separate, but nearly identical, systems are built and coupled via frequency and
`
`phase control. Various methods are used to control the operation of the coils. Experimental
`
`measurements and waveforms are shown.
`
`The seventh chapter gives a comparison between the waveforms measured from the
`
`constructed system and the theoretical waveforms predicted. The sensitivity of the system
`
`to the coupling coefficient is shown. The accuracy of the theoretical model is shown when
`
`their network parameters are correctly chosen.
`
`The eighth chapter is the culmination of the thesis tying together the results of the
`
`theoretical design with the constructed system. The benefits of the pulsed, coupled resonant
`
`system are weighed against the traditional methods of transmitting electromagnetic energy.
`
`Recommendations for future system designs and more generalized resonators are given.
`
`Several concerns involving large scale operation are addressed.
`
`1.2 Motivation
`
`When Nikola Tesla began his Colorado Springs experiments, the concept of the Tesla coil
`
`was unveiled to the scientific world. Yet, over fifty years later, these remarkable devices are
`
`13
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 013
`
`

`

`no more a part of modern equipment than Tesla's infamous death rays. The issue is not
`
`a lack of utility, since the Tesla coil is a remarkable device able to generate high voltage,
`
`high frequency waveforms with little control circuitry.[34] Instead, one might reason that
`
`its lack of use is due to the ill-understood nature of its operation. There are presently
`
`very few references that approach the design of the Tesla coil and its method of operation
`
`from an engineering standpoint. While there are many builders of Tesla coils that give step
`
`by step instructions on how to build a working coil, there is little design theory present in
`
`their explanations. Furthermore, these builders are interested in producing electric arcs and
`
`visible effects suitable for displays and general amusement, not in producing power supplies
`
`and power transmission units which may have significant practical importance.
`
`The primary goal of this thesis is to introduce the theory behind the operation of Tesla
`
`coils as well as all coupled resonant systems. Combined with methods of driving the resonant
`
`systems and practical issues associated with their construction, it is hoped that future work
`
`on this fascinating topic will be encouraged.
`
`14
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 014
`
`

`

`Chapter 2
`
`RLC Circuit Analysis
`
`A coupled resonant circuit is a collection of circuits, each oscillating or resonating at one or
`
`more frequencies, mutually coupled by electromagnetic influences. These influences can be
`
`either electromagnetic, as observed in a transmitting antennae, inductive (magnetostatic),
`
`as seen between the two coils of a transformer, or capacitative (electrostatic). This thesis
`
`deals primarily with two inductively coupled second order circuits with one natural resonant
`
`frequency.
`
`With this restriction, the operational theory of coupled resonant circuits lies in the
`
`analysis of second order RLC circuits and their mutual coupling to one another. A straight-
`
`forward method of analysis is to decompose the resonant system via an electrodynamic
`
`lumped-parameter model into three generic electrical elements: resistances, inductances,
`
`and capacitances. One elementary yet fundamental model of a coupled resonant system is
`
`shown in Figure 2-1 below:
`
`M
`+ M+
`
`CP
`
`L
`
`V
`
`Vs
`
`L
`
`Cs
`
`Voltage
`Source
`
`R P
`
`R S
`
`Primary Circuit
`(Series)
`
`Secondary Circuit
`(Series)
`
`Figure 2-1: A lumped parameter model of a coupled resonant system showing resistances,
`inductances, and capacitances. Both circuits show a series RLC topology.
`
`15
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 015
`
`

`

`This model shows two series circuits, comprised of a resistor, inductor, and capacitor
`
`(RLC), magnetostatically coupled by the mutual flux connecting their two inductors. The
`
`designation of series versus parallel topology comes from the location of the loss mechanism
`
`in the circuit diagram relative to the energy storage elements. In the coupled circuit of
`
`Figure 2-1, the loss elements, the resistors, are in series with the energy storage elements,
`
`the inductors and capacitors.
`
`By convention, the leftmost circuit will be called the primary, the rightmost circuit
`
`the secondary. These designations follow from conventions established in power electronics
`
`terminology. These designations make this modelled system identical to that of a Tesla coil.
`
`A Tesla coil almost always is comprised of two second order systems coupled through the
`
`flux of the inductors. In the case of the Tesla coil, the secondary circuit is often a physically
`
`large inductive coil with a parasitic resistance and capacitance modelled as shown above.
`
`Regardless of the physical realization of the components, if the models are identical, the
`
`the mathematics used to describe their behavior is also identical. To understand both the
`
`steady state and transient operation of coupled resonant circuits, it is first necessary to
`
`understand the constituent nature of their parts.
`
`2.1 History of the Coupled Resonant System
`
`Magnetic fields and inductive coupling have been studied extensively since the discovery of
`
`the transformer by Hans Oersted and Michael Faraday. In 1886, William Stanley, working
`
`for Westinghouse, developed the first commercial AC transformer. It was not until the
`
`invention of radio by Nikola Tesla in 1891 and its commercial utilization in the early 1900's
`
`that the theory of coupled, second order systems assumed the form it has today[10], as will
`
`be explored in this and the following chapters.
`
`While Tesla invented and patented the schematic for a coupled resonant radio trans-
`
`mitter system in 1891, he did not publish the mathematical details of its workings. [35] The
`
`early years of the twentieth century saw great excitement as Guglielmo Marconi transmitted
`
`radio signals across the Atlantic in 1901. A number of radio transmitters were established
`
`for communication during World War I, but a complete mathematical understanding of the
`
`coupled circuits used to make the transmitter and receiver were not published until the
`
`beginning of the 1930's. [2] [28] Frederick Terman, in 1935, published his first of a series of
`
`16
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 016
`
`

`

`handbooks that described the mathematics behind the operation of coupled second order
`
`systems utilizing frequency domain analysis.[32] [33]
`
`By the end of the 1930's, commercial radio and communication systems appeared rapidly
`
`across the United States and Europe, making use of the now understood coupled systems.
`
`With the mathematics of the circuits understood, engineers utilized this powerful theory
`
`when designing radar transmitters and modelling power transmission systems.[11]
`
`2.2 Lumped Parameter Analysis
`
`2.2.1 The Capacitor
`
`The capacitors in this project take on two forms. One is a physical device with two terminals
`
`and a designated capacitance. The other is a distributed capacitance that exists between a
`
`two surfaces of different electrical potentials. A prominent example of distributed capaci-
`
`tance is found between the windings of a large inductor coil and ground. The ground could
`
`be an electrical ground or even the floor, the walls, or anything that happens to be around
`
`the coil including a person.
`
`Capacitance is a definition of charge storage per unit voltage. All capacitors, by defini-
`
`tion, obey the following constitutive relation:
`
`Q=C-V
`
`(2.1)
`
`For a fixed value capacitor whose charge storage follows a linear relationship relative to
`
`the applied voltage, the following time domain and frequency domain relations holds:
`
`I(t) =
`
`dQ(t)
`0 dv(t)
`=C - dt)(2.2)
`dt
`dt
`I(s) = V(s) - Cs
`
`(2.3)
`
`where 's' is the Laplace complex frequency operator expressible as s = a + jw. By Ohm's
`law, he impedance of a capacitor is given by:
`
`Ze = --
`Cs
`
`(2.4)
`
`The capacitor is considered a passive, energy-storage element that fixes the relationship
`
`17
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 017
`
`

`

`between the voltage and current waveforms passing through it. At steady state, the voltage
`
`waveform lags the current waveform by 90 electrical degrees.
`
`The energy stored by a capacitor is given by:
`
`1
`Ecp= 1CV2
`
`(2.5)
`
`When the voltage is increasing across a capacitor, it is absorbing current over time and
`
`charging, thus increasing its stored energy. When the voltage across the capacitor terminals
`
`drops, the capacitor will discharge (i.e. release current) and attempt to raise the voltage
`
`across its terminals to maintain electrical equilibrium.
`
`All linear circuits can be described with a differential equation whose order is equal to
`
`the number of inductors and capacitors in the circuit. In order to fully solve the differential
`
`equation and find a dynamic model of a circuit, it is necessary to identify the boundary
`
`condition(s) associated with the capacitor. One boundary condition is that for finite cur-
`
`rents, the voltage across a capacitor cannot change instantaneously. The current passing
`
`through it, however, as well as the time derivative of the voltage, dvt), can instantaneously
`
`change.
`
`The physical limitations of a capacitor are found in both its physical size and the mate-
`
`rials used to construct it. The smaller a capacitor is, the more likely it is for high voltage to
`
`cause an electrical arc across its terminals. This arc can happen through the air surrounding
`
`the capacitor or directly through the dielectric inside a capacitor. When such an arc occurs
`
`inside a capacitor, it is said to have reached breakdown voltage. The breakdown potential
`
`of still, dry air at STP is approximately 28kV/cm.
`
`Another form of capacitor failure is caused by driving excessive current through it.
`
`Excessive current through the dielectric causes heating, changing the electrical properties of
`
`the capacitor. In extreme cases, this heating will melt or vaporize the dielectric. The failure
`
`mode will vary from capacitor to capacitor depending on physical dimensions, construction
`
`techniques, and, most importantly, the type of dielectric used. A subtle but important effect
`
`is the stress of electric fields on the dielectric. [18][27] Effects known as dielectric absorption
`
`and polarization are the result of some nonlinearities in the charging and discharging cycle
`
`of a capacitor. The amount of nonlinearity in the charge and discharge cycle is a direct
`
`result of the dielectric used and varies considerably with construction, temperature, and
`
`18
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 018
`
`

`

`Stray
`Inductance
`
`Capacitor
`Plates
`
`R
`
`ESR
`of dielectric
`
`Stray
`Inductance
`
`Figure 2-2: Model of a capacitor, with dielectric parasitics and stray inductance from
`terminal leads.
`
`other environmental factors. Another important caveat with capacitors is their equivalent
`
`series resistance, known as ESR which is also directly related to the bulk material properties
`
`of the dielectric. Figure 2-2 illustrates how this effect is modelled.
`
`2.2.2 The Inductor
`
`The inductor is the workhorse of the Tesla coil, providing both a means of energy storage,
`
`voltage transformation, and a means of coupling two resonant circuits. Inductance is a
`
`magnetic effect that is self induced over a given geometry of current flow. For instance, a
`
`single current carrying wire in the vacuo of space has a self-inductance. Thus, every length
`
`of wire in the final design of the coupled resonant system contributes to the total inductance
`
`of the system. However, to keep the mathematics tractable, current carrying wires are bent
`
`into highly symmetric cylindrical coils or other geometric objects so that the inductance of
`
`the structure dominates over the parasitics in wire lengths joining elements of the circuit.
`
`The time and frequency domain relations quantifying inductance are given below:
`
`v(t) = L -di(t)
`dt
`V(s) = 1(s) - Ls
`
`(2.6)
`
`(2.7)
`
`The expression for inductive impedance is given by dividing the voltage by the current:
`
`ZL = L - s
`
`(2.8)
`
`Notice the symmetry between the relations for an inductor and capacitor. This symmetry
`
`19
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 019
`
`

`

`follows from the symmetry of Maxwell's equations and the creative assignment of elec-
`
`tromagnetic definitions. Also, the inductor causes the phase of the current and voltage
`
`waveforms passing through it to shift in a similar fashion to a capacitor. The inductor in
`
`AC steady state fixes the waveforms such that the voltage leads the current by 90 electrical
`
`degrees.
`
`The energy stored by an inductor is given by:
`
`Einducto = -LI2
`2
`
`(2.9)
`
`As the current passing through an inductor increases, the magnetic field induced by the
`
`inductor will increase in magnitude and store energy. Since the energy in an inductor cannot
`
`change instantaneously, the current flowing from its terminals must be continuous, i.e. it
`
`cannot change magnitude instantaneously. This inability to instantly change is seen often
`
`in switched mode power supplies. When inductors are in series with switching transistors,
`as soon as the transistors attempt to switch the current on or off, thus changing the current
`
`flow "instantaneously", the inductors will induce a voltage to maintain the continuity in
`
`current. This voltage transient is responsible for destroying many transistor switches.
`
`An inductor adds another order to the differential equation describing the dynamics of
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`the circuit. Again, a boundary condition is needed to fully solve the circuit's dynamics. A
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`boundary condition associated with an inductor is that the current passing through it at
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`any point in time (i.e. i(t = 0+)) cannot change instantly. As is evident from equation
`
`2.6, the voltage across an inductor as well as the current's rate of change, di/dt can change
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`instantly, but the current itself cannot.
`
`The inductor has performance limitations due to both the fundamental device physics of
`
`its operation as well as the physical materials used in its construction. As seen from equation
`
`2.8, at low frequencies, the inductor behaves as a short circuit, i.e. as a straight wire. Note
`
`that the wire itself will still have its characteristic resistance per unit length, so the voltage
`
`across the inductor will not be exactly OV, even at DC. As the frequency increases, the
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`inductor behaves more like an open circuit. Most inductors are built by wrapping wires
`
`around a structure, sometimes magnetic, in order to concentrate the magnetic flux and
`
`the energy inside the physical dimensions of the inductor. These wires, however, at higher
`
`frequencies will begin to behave as capacitors with charge being stored between adjacent
`
`20
`
`Momentum Dynamics Corporation
`Exhibit 1008
`Page 020
`
`

`

`Inductor
`Coil
`
`Parasitic
`Resistance
`
`Parasitic
`Capacitance
`
`Figure 2-3: Model of an Inductor Coil, with Parasitics
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`windings. This situation is compounded