LIBRARY Michigan State University This is to certify that the thesis entitled ADAPTIVE RESONANCE TUNING presented by Jeffrey Andrew Jeltema has been accepted towards fulfillment of the requirements for the MS. degree in Mechanical Engineering (940M yéjor Professor’s Signature 0 e / .Zi / 03“ Date MSU is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN Box to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 c:/CIFiC/DateDue.p6&p. 15 ADAPTIVE RESONANCE TUNING By Jeffrey Andrew Jeltema A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 2005 ‘ 'fl.s ABSTRACT ADAPTIVE RESONANCE TUNING By Jeffrey Andrew J eltema Resonant systems arise in many areas of science and engineering. Some examples include ultrasonic motors, piezoelectric transducers, induction heating loads, resonant inverter loads, vibrational gyroscopes, cavity resonators and cyclotrons. For optimal performance, these systems must be excited at their resonant frequencies. However, even if resonant and excitation frequencies are initially matched, over time they can drift due to disturbances such as environmental change, load variation, manufacturing variability, fatigue damage, microphonics and electromagnetic detuning, resulting in a loss of performance. This necessitates employment of a resonance tuning control system that maintains lock between the excitation and resonant frequencies. In this thesis, three resonance tuning methods for lightly damped second order passive loads are investigated. Each method uses the error between the excitation and resonant frequencies obtained by a phase detector to adaptively match these frequencies. In the first method, the excitation to the load is provided by a voltage controlled oscillator, the frequency of which is adaptively tuned. The second method adaptively tunes the resonant frequency of the resonator by changing its structure or element values. Finally, the third method uses proportional feedback around the resonator and adaptively adjusts the feedback gain to tune the closed loop resonant frequency to the excitation frequency. Using the frozen time approach, nonlinear time-varying models that accurately predict the performance of these control systems are developed, and subsequently linearized, to obtain linear time-invariant models that facilitate both analysis and design. Furthermore, stability issues are considered and design guidelines are provided. The results are illustrated through examples. To my family, Thomas, Linda and Laura iii ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Cevat Gokgek, whose expertise and guidance have added considerably to my graduate experience. I truly appreciate the time and effort he has given to me as a research advisor and mentor. I would also like to thank all of my friends and family for their constant support and words of encouragement. iv TABLE OF CONTENTS LIST OF FIGURES VII 1 Introduction 1 1.1Motivation.......................... ....... 1 1.2 Problem Formulation ........................... 2 1.3 Resonance Tuning Systems ........................ 3 1.3.1 RTSl ................................ 3 1.3.2 RTS2 ................................ 5 1.3.3 RTS3 ................................ 6 1.4 Thesis Overview .............................. 7 1.4.1 Organization ........................... 7 1.4.2 Original Contributions ...................... 7 2 Modeling 9 2.1 Introduction . .. .............................. 9 2.2 Resonator Model ............................. 10 2.2.1 Resonant Frequency ....................... 10 2.2.2 Effects of Detuning ........................ 14 2.3 Phase Detectors .............................. 17 2.3.1 Multiplication Phase Detector .................. 17 2.3.2 Exor Phase Detector ....................... 19 2.4 Resonance Tuning Systems ........................ 22 2.4.1 RTSl ................................ 22 2.4.2 RTS2 ................................ 24 2.4.3 RTS3 ................................ 25 3 Analysis 28 3.1 Introduction ................................ 28 3.2 Analysis .................................. 28 3.2.1 RTSl ................................ 29 3.2.2 RTS2 ................................ 32 3.2.3 RT83 ................................ 35 3.3 Stability .................................. 37 3.3.1 Linear Models ........................... 37 3.3.2 Nonlinear Models ......................... 4 Design 4.1 Design ................................... 4.2 Simulation ................................. 4.2.1 RTSl ................................ 4.2.2 RTS2 ................................ 4.2.3 RTS3 ................................ 5 Extension - Adaptively‘ Enhanced PLL 5. 1 Introduction ................................ 5.2 Modeling .................................. 5.3 Analysis and Design ........................... 5.4 Simulation ................................. 6 Conclusions 6.1 Summary ................................. 6.2 Future Work ................................ BIBLIOGRAPHY vi 38 48 48 50 50 52 55 59 59 59 62 64 70 70 71 73 LIST OF FIGURES 1.1 Plasma ignition system. ......................... 5 2.1 Model of the RLC circuit. ........................ 11 2.2 Bode plot of the Hc(jw) .......................... 13 2.3 Bode plot of the H 3(jw) .......................... 13 2.4 Bode plot of the H L(jw) ....... l ................... 14 2.5 Bode plot of RLC example. ....................... 15 2.6 Output of uncontrolled resonator excited at wn(t) ............ 16 2.7 Output of uncontrolled resonator with 1 percent drop in capacitance. 16 2.8 Model of the multiplication phase detector ................ 18 2.9 Multiplication phase comparator characteristic. ............ 19 2.10 Model of the exor phase detector. .................... 20 2.11 Exor phase comparator characteristic ................... 21 2.12 Model of RTSl. .............................. 22 2.13 Model of the voltage controlled oscillator ................. 23 2.14 Model of the RTS2 ............................. 24 2.15 Model of RTS3. .............................. 26 3.1 Linearized model of RTSl ......................... 31 3.2 Linearized resonance tuning system .................... 35 3.3 Plot of arctangent and saturation functions. .............. 38 3.4 Feedback connection of linear system and nonlinear element. ..... 43 3.5 Sector bounded nonlinearity of RTS2 and RTS3 with exor PD ..... 44 3.6 Sector bounded nonlinearity of RTS3 with multiplication PD. . . . . 46 3.7 RTSl nonlinearity crossing the sector bound near the origin. ..... 47 4.1 Simulation results of RTSl with can = 1005.04 .............. 51 4.2 Simulation results of RTSl with w" = 1054.1. ............. 52 4.3 Simulation results with F(s) = 2.827/(3 + 1). ............. 54 4.4 Simulation results with F(s) = 1.571 / s .................. 54 4.5 Simulation results with F(s) = 62.832/[s(s + 20)] ............ 55 4.6 Simulation results of RTS3 for vibrational gyroscope example. . . . . 56 4.7 Simulation results of RTS3 with F(s) = figfoi ............ 58 5.1 Model of the enhanced PLL system .................... 60 vii 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Model of the frequency estimator system ................. 61 Simplified model of the resonance tuning system. ........... 63 Frequency-phase error results with small initial frequency difference. . 65 Input and output signals with small initial frequency difference. . . . 66 Frequency - phase error results with wo = 1100 rad /s .......... 67 Input and output signals with we = 1100 rad/s. ............ 67 Frequency - phase error result with tag = 4000 rad/s. ......... 68 Input and output signals with we = 4000 rad/s. ............ 69 viii CHAPTER 1 Introduction Resonant systems arise in many areas of science and engineering. In some cases, these systems must be excited at their resonant frequencies to achieve optimal perfor- mance (e.g. resonators) while in some others, excitation of their resonant frequencies may be catastrophic and must be avoided (e. g. bridges). Control systems that achieve the former objective will be referred to as resonance tuning systems in this thesis. 1 .1 Motivation Ultrasonic motors, piezoelectric transducers, induction heating loads, some res- onant inverter loads, microelectromechanical gyroscopes, cavity resonators and cy- clotrons can be modeled as lightly damped second order passive loads [1]—[16]. In order to maximize the average power delivered to such a load or maximize the res— onant amplification, it is necessary to drive the load at its resonant frequency [1]— [16]. However, disturbances such as temperature or humidity change, load variation, manufacturing variability, fatigue damage, microphonics and electromagnetic detun- ing [1]—[16] can cause the resonant frequency or excitation frequency of the load to drift over time and significantly impair the achieved performance. This mandates employment of a resonance tuning system that maintains lock between the excitation frequency and the resonant frequency of the load. The resonance tuning control problem has received significant attention recently in the contexts of the specific applications listed above and many others. Several useful analysis and design methods for such systems have been developed and many such systems have been practically implemented [1]—[16]. Despite these encouraging results, development of simple but accurate analysis methods as well as robust and optimal design methods for resonance tuning systems are still needed. The aim of this thesis is to provide effective tools for modeling, analysis and design of resonance tuning systems. Specifically, resonant systems that can be modelled as lightly damped second order loads will be studied. Since the phase of a second order load takes a constant value (—1r/2, 0 or +1r / 2 rad) at its resonant frequency, the main idea in designing the resonance tuning system is to use a phase detector to extract the phase information and use it for feedback purpose. This thesis will investigate three separate resonance tuning methods. 1 .2 Problem Formulation Based on the above motivation, the following three problems are formulated. Modeling Problem: Given a resonator, develop an overall model that includes the resonator and an appropriate resonance tuning system. Analysis Problem: Given an overall model of a resonator and a resonance tuning system, determine the performance of the feedback system (stability, robustness, dis- turbance rejection, reference tracking, rise time, settling time, maximum overshoot, etc.); also determine performance limitations and tradeoffs. Design Problem: Given an overall model of a resonance tuning system, design the parameters so that the system achieves certain desired performance objectives. Although the resonators mentioned above can be similarly modeled, they differ from each other as to what part of the system actuation is possible. In this the- sis, resonance tuning systems will be classified into three groups based on actuation possibilities. Group 1: This group consists of those systems that allow actuation on the excita- tion frequency. That is, these systems contain means to adaptively tune the excitation frequency to the resonant frequency of the resonator. It should be noted that with this case, actuation on the resonant frequency may or may not be possible. Group 2: This group of systems consists of those that allow actuation on the resonant frequency. This requires the ability to adjust a parameter or element value of the resonator on which the resonant frequency is dependant. It should be noted that with this case, actuation on the excitation frequency may or may not be possible. Group 3: Finally, the last group consists of systems where actuation on neither the excitation frequency, nor the resonant frequency is practical. For each group of resonators, this thesis will present an appropriate resonance tuning system and provide solutions to the three problems formulated above. The resonance tuning systems will be demonstrated with simulations. 1.3 Resonance Tuning Systems This section introduces the three resonance tuning systems considered in this thesis. Some representative examples are also provided. 1.3.1 RTSl The first resonance tuning system assumes the resonator fits into actuation group 1 above. That is, actuation on the excitation frequency is possible. In this resonance tuning system, the source of excitation is a voltage controlled oscillator (VCO). A phase detector is used to produce an error signal that is proportional to the error between excitation and resonant frequencies. This error signal is then used to adap- 3 tively tune the output frequency of the VCO in a direction to reduce the error between excitation and resonant frequencies. The combination of a phase detector and a VCO is known as a phase locked loop (PLL). The PLL is a common tool in many commu- nication systems, however, use of the PLL for the purpose of resonance tuning is a fairly recent idea. Some physical systems where RTSl may be implemented include some fluorescent lighting loads, induction heating loads, piezoelectric transducers or ultrasonic motors. The exact frequency at which these systems resonate is not important, only that they achieve resonance within a reasonable range. Thus RTSl is well suited for these applications. One physical system that currently implements RTSl is a plasma ignition system. Although this ignition system is still being developed, early results have shown that when RTSl is implemented, the system can maintain resonance despite changes in its resonant frequency. The basic principle of this plasma ignition system is to use resonance boosting to achieve the required high voltage for the breakdown and sustain the plasma by continuing RF excitation. This ignition system generates a strong RF plasma torch for a prescribed duration and hence it is expected to achieve faster, more uniform and more complete combustion. Prior to discharge, a spark plug is basically a very small cylindrical capacitor. Connecting an inductor in series with the spark plug forms a series resonant circuit. A model of the plasma ignition system is shown in Figure 1.1. In this figure, the lightly damped second order load is the series RLC circuit and the rest of the circuit provides the excitation. Driving this circuit with a suitable RF generator exactly at its resonant frequency yields a very high RF voltage across the spark plug and results in an RF plasma torch. However, due to environmental changes, the resonant frequency may drift with time, thus impairing the the performance of the system significantly. Hence a resonance tuning system is II BI "l 4| l Figure 1.1. Plasma ignition system. required. As the plasma ignition system fits into actuation group 1 above, RTSl is implemented. 1.3.2 RTSZ The second resonance tuning system considered assumes the resonator fits into actuation group 2. That is, actuation is possible on a parameter of the resonator that changes the resonant frequency. Again, a phase detector is used to determine the error between excitation and resonant frequencies. This error is then used to adaptively tune a parameter of the resonator so that the resonant frequency varies in a direction that reduces the error between the resonant and excitation frequencies. RTS2 is well suited for use in wireless communication devices. In these devices, for instance cellular phones, the frequency of an incoming signal cannot be adjusted. Hence, to achieve optimal performance, the center frequency of the bandpass filter within the phone must be adjusted. This can be accomplished by implementing RTS2. Another physical system well suited for RTSZ is a cavity resonator. The resonant frequencies of these resonators are required to be precisely tuned to their excitation frequencies as they typically have high quality factors. However, their resonant fre- quencies can easily drift during operation due to factors such as temperature change, humidity change or microphonics. The result is a large decrease in the performance of these resonators. Thus, a resonance tuning system is required. RTS2 is well suited to cavity resonators due to the fact that the resonant frequency of a cavity resonator de— pends on the cavity dimensions. Thus, the resonant frequency of these systems can be tuned by altering their physical dimensions. These adjustments can be accomplished, for instance, with piezoelectric actuators [8]—[12]. 1.3.3 RTS3 The third resonance tuning system considered assumes the resonator fits into actuation group 3 above. That is, actuation is possible on neither the excitation frequency nor the resonant frequency. Instead, a feedback loop is introduced around the resonator to allow for the tuning of the closed loop resonant frequency. Similar to RTSl and RTS2, a phase detector is used to determine the error between excitation and closed loop resonant frequencies. This error is then used to adaptively tune the feedback gain, and thus tune the closed loop resonant frequency, in a direction that reduces the error between excitation and closed loop resonant frequencies. An example of a physical system where this method is well suited is a vibrational gyroscope. Vibrational gyroscopes measure the angular velocity of a rotating system by sensing the Coriolis force on the system. They are basically a mass spring damper system driven to vibrate in one direction. When the gyroscope rotates, the Coriolis force causes the mass to move in a direction perpendicular to the drive axis and rotation axis [6]. In order to obtain a large response, most gyroscopes must be driven at their resonant frequency. However, due to manufacturing variability, temperature and aging the resonance condition can easily be lost. Thus, a resonance tuning 6 system is required. Typical MEMS gyroscopes utilize a PLL to tune the excitation frequency to the resonant frequency of the gyroscope [6]. However, this method does not allow for the operating frequency to be chosen by the designer, which means this frequency cannot be used in the signal processing design. Instead RTSB, which creates a feedback loop around the resonator and adaptively tunes the feedback gain to bring the closed 100p resonant frequency to match the excitation frequency can be utilized. 1 .4 Thesis Overview 1.4.1 Organization This thesis is organized as follows: Chapter 2 provides the models of the three resonance tuning systems considered in this work as well as the models of a generic resonator and two types of phase detectors. Chapter 3 provides analysis of the three resonance tuning systems including model simplification and stability results. Chap- ter 4 discusses the issues surrounding the design of these resonance tuning systems and provides simulation results of each resonance tuning system. Chapter 5 presents a byproduct of this study, adaptively enhanced phase locked loops. Finally, in Chapter 6, conclusions are given as well as some future research directions. 1.4.2 Original Contributions The main contribution of this thesis is the development of systematic analysis and design methods for the adaptive resonance tuning systems described above. Specif- ically, assuming that the parameters of a lightly damped second order passive load (including its resonant frequency) and its excitation frequency vary slowly with time, nonlinear time-varying models that accurately predict the performance of these res- onance tuning systems are developed. These developed models are subsequently lin- earized to obtain linear time-invariant models that predict the performance of these systems. Furthermore, stability analysis of the nonlinear and linear models is car- ried out. Finally, based on the developed linear time-invariant models, guidelines for designing the resonance tuning systems are provided. The research in this thesis has generated three published works [19]-[21]. Also, another paper providing a summary of the research on adaptive resonance tuning systems presented in this thesis is currently in preparation. CHAPTER 2 Modeling 2.1 Introduction In this chapter, models are presented for the three resonance tuning systems in- troduced in Chapter 1. Also presented is a review of the phase detectors used in this thesis and the model of a generic resonator. As stated earlier, the resonance tuning systems considered in this thesis apply to resonators that are lightly damped second order passive loads. Since any lightly damped second order load can be modeled as a series RLC circuit, this thesis will repeatedly use series RLC circuits for the purpose of illustration. It should be noted that in most of the examples provided in this work, the nominal element values of this circuit will not accurately represent any physical system. However, this fact does not devalue the examples since time can be scaled without changing the qualitative response of the resonance tuning systems. The response of these system is, however, dependant on the quality factor of the resonator. Therefore, examples covering a variety of quality factors will be considered throughout this work. The remainder of this chapter is organized as follows. In Section 2.2, the model of a resonator is considered. In Section 2.3, two types of phase detectors are discussed. Finally, in Section 2.4, the three methods of resonance tuning are investigated. 2.2 Resonator Model The dynamics of a lightly damped second order system are governed by the dif- ferential equation W) + 2C(t)wn(t)?)(t) + «fifth/(t) = 1990030110): (2-1) where u(t) is the input, y(t) is the output, wn(t) is the resonant frequency, C (t) is the damping ratio and kg is the input gain. The input to the system u(t) is assumed to be in the form u(t) = A cos [wot + 0(t)] , (2.2) where A is the amplitude, we is the nominal angular frequency and 6(t) is the instan- taneous phase of the input. It then follows that the instantaneous frequency of the input is w,(t) = we + 9(t). (2.3) Thus, optimal performance requires w3(t) to be equal to the resonant frequency of the resonator. 2.2. 1 Resonant Frequency The way in which the resonant frequency of a lightly damped second order sys- tem is defined varies throughout the different disciplines of science and engineering. Certain applications consider a system in resonance when maximum power transfer is achieved, while in others, maximum signal amplification defines resonance. In this thesis, we will define the resonant frequency w, as the frequency at which maximum power transfer occurs. For lightly damped second order systems, it follows that w, = w... (2.4) 10 We will define the frequency at which maximum signal amplification occurs as the peak frequency wp. For lightly damped second order systems, the value of top depends on where the system output is taken. To help illustrate this, consider a series RLC circuit as the resonator. This circuit is as shown in Figure 2.1, where R, L and G are a resistor, inductor and capacitor, respectively, and u(t) and y(t) are the input and output, respectively. In Figure 2.1, Figure 2.1. Model of the RLC circuit. the output y(t) is taken as the voltage across the capacitor. The resulting transfer function 110(3) is (.02 110(3) 2 52 + 2(wns + wfi’ (25) where 1 n — 2.6 w "_LC ( ) and R C C — -2— I. (2.7) From the definition above, it follows that the peak frequency is the frequency at which IHc(jw)| is a maximum. By taking the derivative of |H0(jw)| with respect to jw and 11 setting it equal to zero, it follows that the the peak frequency top is (up = nun/1 — 26. (2.8) However, the output can also be taken as the voltage across the resistor or inductor. Taking y(t) as the voltage across the resistor results in the transfer function 2Cwns H = . 2.9 MS) 32 + 2Cwns + (cg, ( ) Again defining the peak frequency as before, it follows that top = w... (2.10) Furthermore, taking 3; as the voltage across the inductor results in the transfer func- tion H 32 = . 2.11 L(s) 32 + 2Cwns + w}: ( ) In this case, the peak frequency is w — ——“’—"—- (2.12) ,,— ,/1—2(2' To illustrate these transfer functions, consider the RLC circuit with values of R, L and C as R=5 f2, L=10 mH and C=100 pF. Thus, wn=1000 rad/s and C = 0.25. The bode plots of Hc(jw), H R(jw) and HL(jw) are shown in Figure 2.2, Figure 2.3 and Figure 2.4, respectively. As seen in these plots, it is evident that the phase at which w, or to, occur, is dependant on where the system output is taken. This value determines what type of phase detector may be used in the resonance tuning system. More discussion of phase detectors is provided in Section 2.3. 12 0. 500 600 l l t—c————__-—-————t 1 700 800 I l l 900 1000 1 100 1200 1300 1400 1500 1.1 _0.9 p-———_--—— l I 900 1000 1100 1200 1300 1400 1500 500 600 700 800 900 1000 1100 1200 1300 1400 1500 Figure 2.3. Bode plot of the H R(jw). 13 M N 0" q -i q d .t .1 l 052M IHL l l I I l I I I l l 1 l l I l I 500 600 700 800 900 1000 1100 1200 1300 1400 1500 6‘. lHLUl-U) deg 8 p—-- l l l l 1100 1200 1300 1400 1500 go 0') s V 8 § 900 1000 w, rad/s Figure 2.4. Bode plot of the HL(jw). Despite the different definitions provided above, for all resonance tuning applica- tions, this thesis assumes optimal performance always occurs when (2.4) is satisfied. This is possible due the nature of the systems we will be considering. In these sys- tems, the damping ratio is very small and therefore cur, wp and wn are all practically equal. For the remainder of this work, we will refer to any of these three frequencies simply as the resonant frequency. 2.2.2 Effects of Detuning Depending on the quality factor of a resonator operating at resonance, a very slight change in either the resonant or excitation frequency can cause a very large decrease in the system output. To illustrate this fact, consider a series RLC circuit and assume the nominal values of R, L and C are R0 = 0.25 0, L0 = 0.1 H and Co = 10 HF, respectively, so that 14 the nominal values of can, C and Q are too = 1000 rad/s, Co = 0.00125 and Q0 = 400, respectively. A Bode plot of this resonator is shown in Figure 2.5. Due to the 400 r f If 1 r I 1 r _300 " A g... o l l l l I I l 500 600 700 300 900 1000 1100 1200 1300 1400 1500 o I fl I I j‘ ‘F 6 fir T 8? .45. "U 3 '90- '5 v ”I: .435 - J V L -1” .— 1 1 I l l l 500 600 700 800 1200 1300 1400 1500 900 1000 1100 w, rad/s Figure 2.5. Bode plot of RLC example. sharpness of the peak in this figure, even slight detuning from the resonant frequency is expected to cause a significant decrease in the resonator output. This can be seen in Figures 2.6 and 2.7 which are plots of output voltage across the capacitor versus time. In Figure 2.6 the resonator is excited at its resonant frequency which results in a steady state output of 400 V across the capacitor. However, in Figure 2.7, the value of C is decreased by 1 percent, so that the resonant frequency can (t) becomes 1005.04 rad/s, while the excitation frequency remains 1000 rad / 3. These figures show that a change of only 0.5 percent in the resonant frequency causes the steady state output voltage to decrease by 75 percent from 400 V to 100 V. 15 ‘ I'm“l'lllllll'lllllvll""ll alll"”""‘l‘ll ”ll l l lllfllllllllllllll‘llhllln llll5 Iii ULl llulll illlll. lllilllulllL H"lllmlllu‘ W mill ‘ l L Lil llllil llllllul O 1 2 3 4 6 7 B 9 10 Figure 2.7. Output of uncontrolled resonator with 1 percent drop in capacitance. 16 With such a decrease in output, the resonator is obviously far from optimal per- formance. Therefore, a method is clearly needed to control the resonant or excitation frequency. 2.3 Phase Detectors The previous discussion necessitates employment of a resonance tuning system that maintains w, (t) and wn(t) as close to each other as possible despite changes in either frequency. In order to achieve this task, the controller must have means to sense the error between these frequencies. As mentioned earlier, a phase detector can be used to sense the error. A phase detector compares the phases of two signals applied to its inputs and generates an output signal whose average value is related to the phase difference between these input signals [24, 25]. This output is also a measure of frequency difference, as signals with differing frequencies cannot be locked in phase unless one is a harmonic of the other. There exist several types of phase detectors with different characteristics. The phase detectors considered in this thesis are a multiplication type analog phase de- tector and an exor type digital phase detector. Below, these two types of phase detectors will be reviewed briefly. 2.3.1 Multiplication Phase Detector A multiplication type phase detector uses an analog multiplier phase comparator followed by a lowpass filter to extract the phase information at its inputs as shown in Figure 2.8. In this figure, the signals u1(t) and u2(t) are the inputs, u(t) is the output, F (s) is the transfer function of the lowpass filter and x is the multiplication operation. 17 ’Ul(t) F (S) "—" ”(t) v200 Figure 2.8. Model of the multiplication phase detector. Assuming that v1(t) and u2(t) are in the forms ul (t) = A1(t) cos [wot + 91(t)] (2.13) and ’Ug(t) = A205) COS [wot + 6205)] , (2.14) where A1(t) > 0 and A2(t) > 0 are the instantaneous amplitudes, 01(t) and 62(t) are the instantaneous phases and too is the nominal angular frequency, it follows that = Ara/art) 3(t) “—3—— cos [62(t) — 6105)] + M 2 COS [2wot + 92(t) + 91(t)]. (2.15) Assuming further that A1(t), A2(t), 61(t) and 02(t) vary slowly with time compared to the time variation of wot (i.e., |A1(t)| << coo, |A2(t)| << wo, |91(t)| << we and |92(t)| << too) and that the lowpass filter completely removes the high frequency term around 2%, the phase detector output u(t) can be written as ”(t) : f0) * (pm [62“) — 610)] 1 (2'16) where from lgltll = ———-A1(t)2A2(t) cos [6(t)] , (2-17) 18 f (t) is the impulse response of the lowpass filter and * is the convolution operation. Figure 2.9 shows gom [6(t)] as a function of 6(t) = 62(t) — 01(t), where the amplitude [A1(t)A2(t)] / 2 is normalized to A. rm(0) ll +A : 9 —21r —1r 0 —1r +27r —A Figure 2.9. Multiplication phase comparator characteristic. The nonlinear equation (2.17) describes the operation of the multiplication type phase detector very accurately provided that the assumptions made earlier are satis- fied. Note that the amplitudes of both input signals affect the output of the phase detector. This must be accounted for when designing. the gains in the system. As will be shown next, the amplitudes of the input signals do not affect the output of the exor type phase detector. 2.3.2 Exor Phase Detector An exor type phase detector uses a digital exor phase comparator followed by a lowpass filter to extract the phase information at its inputs as shown in Figure 2.10. In this figure, the signals ul (t) and u2(t) are the inputs, u(t) is the output, F(s) is the transfer function of the lowpass filter, 69 is the exor logic gate with logic 0 and 1 19 levels being —V and +V, respectively, and HLs are hard limiters with limiting levels :EV that convert the signals u1(t) and u2(t) at the input of phase detector into the digital signals u1(t) and u2(t). {u(t) z(t) U2(t) --—>‘ HL “2“) Figure 2.10. Model of the exor phase detector. Assuming that the inputs u1(t) and v2(t) are as before, the signals u1(t) and u2(t) can be expressed as u1(t) = ngn (cos [wot + 91(0)) (2.18) and um) = ngn (cos [wot + 02(2)]). (2.19) Thus, it follows that z(t) = ma) a; ma). (2.20) Similar to the previous case, assuming that both 01 (t) and 02(t) vary slowly with time compared to the time variation of wot (i.e., |91(t)| << we and |62(t)| << we), the signal z(t) can be expressed as 2(t) = 9% [920) - 910)] + 7‘0), (2-21) where cps [6(t)] as a function of 9(t) = 62(t) —- 01(t) is as shown in Figure 2.11 and the 20 term r(t) includes the high frequency components that occur around 2am, 4w0, and so on. Assuming further that the lowpass filter F (3) completely removes the high 906(0) ll +V ; l9 _27r —7r 0 +7l' +277 —V Figure 2.11. Exor phase comparator characteristic. frequency term r(t), the output of the phase detector u(t) is given by ’00?) = f (t) * Soc [920) - 910)]. (2-22) where f (t) is the impulse response of the lowpass filter and at is the convolution operation. Like the previous case, this nonlinear equation governs the operation of the phase detector very accurately provided that the assumptions stated above are satisfied. Moreover, note that the output of phase detector u(t) is also a slowly time-varying signal. 21 2.4 Resonance Tuning Systems The following subsections provide models of the three resonance tuning systems introduced in Chapter 1. 2.4.1 RTSl As discussed earlier, RTSl is a PLL based resonance tuning system. A block diagram of RTSl is shown in Figure 2.12. In this figure, SYS is a resonator, PD is a phase detector, VCO is a voltage controlled oscillator, u(t) is the VCO output, y(t) is the resonator output and u(t) is the phase detector output. VCO 7 SYS > y(t) PD u(t) Figure 2.12. Model of RTSl. A model of the VCO is shown in Figure 2.13, where we is the center frequency and kw is the angular frequency gain. Defining 910) = av“). (2-23) 22 the signal u(t) may be written as u(t) = V cos [wot + 61(t)] , (2.24) where V is the amplitude. If a digital VCO is used, this signal becomes the Walsh function u(t) = ngn (cos [wot + 01(t)]). (2.25) It should be noted that the narrow bandpass nature of a resonator will essentially reject all but the first harmonic of the input signal. Thus, if a sinusoidal function or Walsh function are input to a resonance tuning system, the response of the resonator will remain generically the same. u(t)——- k... —j9— s-l Vcos(°) L—-u(t) wo Figure 2.13. Model of the voltage controlled oscillator. Assuming the resonator model described above, the output y(t) satisfies the dif- ferential equation f(t) + 2C(t)wn(t)3)(t) + w3.(t)y(t) = W3(t)u(t), (2-26) where wn(t) is the natural frequency of the resonator and C (t) is the damping ratio. Assuming the instantaneous frequency of the input is w,(t), the goal of RTSl is to keep w5(t) as close to can (t) as possible despite disturbances due to environmental 23 changes, by adaptively tuning w,(t). 2.4.2 RTSZ As mentioned in Chapterl, RTS2 adaptively controls the resonant frequency of a resonator by changing the structure of the resonator. To help illustrate RTS2, a series RLC circuit is again considered as the resonator. A block diagram of RTSZ is shown in Figure 2.14, where R(t), L(t) and D(t) form the tunable resonator and PD is a phase detector. ’01“) PD '00) ’U2(t)r R(t) L(t) ——vav—/mn— + r(t) i) y(t) 7‘ pm Figure 2.14. Model of the RTS2. The resonance tuning is achieved by adaptively controlling the resonant frequency through the variable capacitor D(t) using the error between the excitation frequency and resonant frequency. As stated before, the resonant frequency of a series RLC circuit is w..(t) = ——1— (2.27) ,/L(t)C(t)' However, in the resonator being considered, it is assumed that the capacitance D(t) 24 is controlled by the output of the phase detector u(t) about its uncontrolled value C (t) according to the expression C(t) D(t) 2 1+ kwu(t)’ (2.28) where kw is an appropriate constant. With this tunable capacitor, the new resonant frequency of the resonator becomes wc(t) = wn(t) \/1 + kwu(t). (2.29) Similarly, the new damping ratio of the resonator becomes _ ((0 {(t) _ m. (2.30) Thus, assuming R(t), L(t) and D(t) vary slowly with time, the voltage y(t) across the capacitor satisfies the differential equation f(t) + 2C (0014090) + Math/(t) = w:(t)$(t)- (231) Hence, the goal of the resonance tuning system is to keep wc(t) as close to w,(t) as possible through u(t), despite disturbances due to environmental changes. 2.4.3 RTS3 As previously stated, RTS3 does not directly actuate on resonant or excitation frequency. Instead, this system places the resonator in a feedback loop, and the feedback gain is adaptively adjusted to bring the resonant frequency of the closed-loop system to match the excitation frequency. A model of RTS3 is shown in Figure 2.15. 25 y(t) V —-—--1b——— P?‘ L... {XI -—-——-—--b- A ' l...— I' I I I I I I I I I I I I I I l I I I L I I PD v(t) Figure 2.15. Model of RTS3. In this figure, SYS is the lightly damped second order system, PD is the phase detector, kf is the feedback gain and z(t) is the external input to the system. It follows from this figure that W) = $05) - kr’v(t)y(t), (232) where u(t) is the error signal generated by the phase detector. Thus, the differential equation governing the closed-loop system inside the dashed box can be written as f(t) + 2C(t)wn(t)3)(t) + l1 + krkgv(t)l w3.(t)y(t) = kgw3.(t)$(t)- (2-33) The resonant frequency of this system is wc(t) = wn(t)\/1 + kwu(t), (2.34) 26 where kw = k 1kg. Hence, the goal of the resonance tuning system is to keep wc(t) as close to w,(t) as possible through u(t), despite changes in the excitation frequency or resonant frequency of the resonator. 27 CHAPTER 3 Analysis 3.1 Introduction In this section, the models developed in Chapter 2 are considered and analysis methods are developed. For this purpose, the slow time-varying approach [26, 27, 28] is adopted. Since it is assumed that the resonator parameters are slowly time-varying compared to its excitation, this approach is justified. For each resonance tuning system, a nonlinear time-varying model that accurately predicts the performance of the resonance tuning system is developed. This devel- oped model is subsequently linearized to obtain a linear time-invariant model that facilitates both analysis and design of the resonance tuning system. Stability results of the developed models are also provided. The remainder of this chapter is organized as follows. In Section 3.2, the developed models are simplified. In Section 3.3, stability issues are considered. 3.2 Analysis In this section, models of the three resonance tuning systems are analyzed and simplified. For each resonance tuning system, either a digital exor phase detector or an analog multiplication phase detector is used. However, the methods presented in this work can be easily extended if a phase detector other than the one illustrated here is used. 28 3.2.1 RTSl Consider the resonance tuning system shown in Figure 2.12 and assume that PD is an exor type phase detector. Assume further that the VCO output u(t) is in the form of (2.25), where we is the angular frequency and 01(t) is the instantaneous phase. The instantaneous frequency of the input is then ws(t) = we + 91(t). (3.1) Under these assumptions, the output y(t) satisfies the differential equation (2.26). Since this equation is time varying, it is impossible to find the voltage y(t) analytically. However, with the frozen time assumption, the parameters wn(t), C (t), and 61(t) can be treated as constant but unknown, and the transfer function from u(t) to y(t) can be expressed as 0120) H = " . .2 (S) s2 + 2C(t)w,,(t)s + w3,(t) (3 ) The output y(t) can then be approximated at steady state as y(t) = A2(t) cos [wot + 92(t)] + e(t), (3.3) where A2(t) = 4V|H [jws (t)] I / 7r, 02(t) = 61(t)+AH[jw,(t)] and e(t) includes the higher order harmonics that occur at 3w,(t), 5w,(t) and so on. Assuming the quality factor of the resonator is sufficiently high (greater than 10), these higher order harmonics can be neglected, and y(t) can be approximated as y(t) = A2(t) COS [wot + 620)] . (3.4) 29 Thus, with u(t) and y(t) the inputs to the exor phase detector, it follows from (2.22) that the output u(t) is ’00) = f(t) * welHliws(t)l, (3.5) where f (t) is the impulse response of the lowpass filter and * is the convolution opera- tion. Since —7r < [H [10250)] S 0, it follows from the phase comparator characteristic that ’00) = f(t) * k9 (-4Hljws(t)l - 7r/2). (3-5) where [679 = 2V/ 7r is the gain of the phase detector. Combining (2.23) with (3.6), and using zHij,(t)] = — arctan [Zigilt‘g’ftlj’éil] , (3.7) it follows that ' _ 023.0) - £030) 01(t) — wf(t) * k9 arctan [2C(t)wn(t)w,(t)] . (3.8) Combining this equation with (3.1) results in _ w3.(t) - u(f(t) w,(t) — wo — kwf(t) * k9 arctan [2C(t)wn(t)w,(t)] . (3.9) Although simulation shows that (3.9) characterizes the RTSl very accurately, it is nonlinear and time-varying, and thus, it has limited value. Hence, this equation is linearized below to obtain a more tractable linear time-invariant model. Let 6wn(t) = 02,,(t) —wo and 6w,(t) = w,(t) -wo be the deviations of wn(t) and w,(t) from the nominal natural frequency 020, respectively. Then, (3.9) can be linearized about the nominal frequency 010 as 6ws(t) = 1.ka f(t) at 3515175 [6wn(t) — 50.0)]. (3.10) Moreover, letting C0 be the nominal value of C (t), the system described by (3.25) can 30 be further approximated by the linear time-invariant system 1 C0600 (5018(t) = kwk9f(t) * [6wn(t) -— 6ws(t)]. (3.11) Taking the Laplace transform of both sides of this equation, it follows that _ kwkg Cowo A9,,(s) F(s) [Afln(s) —- A93(s)] , (3.12) where A0,,(s) and Ail,(s) are the Laplace transforms of 6wn(t) and 6ws(t), respec- tively. Thus, the transfer function from 6wn(t) to 602, (t) is AQ,(s) _ kF(s) A9,,(s) _ 1 + kF(s)’ (3'13) where k = (mica/((0010). The block diagram of (3.13) is shown in Figure 3.1. Having obtained this simplified model, the performance of the RSTl can be analyzed using standard linear control system analysis methods. It should be noted that this linear approximation is quite accurate provided that [dwn(t) — 6w,(t)| < (0000 _ 1. (3.14) This inequality arises from the approximation arctan[a:] z :2: around the origin. Clearly this approximation is only accurate around a small region, as the arctan- gent function is similar to the saturation function. 60),, (t)—§f——- kF(s) =6ws(t) Figure 3.1. Linearized model of RTSl. 31 3.2.2 RTSZ Consider the resonance tuning system shown in Figure 2.14 and assume that PD is an exor type phase detector. Further assume that the excitation source 22(t) is in the form $(t) = A1(t) COS [Wot 'I' 61(t)] , (3.15) where A1(t) > 0 is the amplitude, 010 is the angular frequency and 01(t) is the instan- taneous phase of the input a:(t). The amplitude A1(t) and the phase 01(t) are also assumed to be slowly time-varying parameters. The instantaneous frequency ws(t) of the input a:(t) is defined as in (3.1). Under these assumptions, the voltage y(t) across the capacitor satisfies the differ- ential equation (2.31). Since this equation is time varying, it is impossible to find the voltage y(t) analytically. However, again using the frozen time approach, wc(t), wn(t), C (t), A1(t) and 01 (t) can be treated as constant but unknown. Hence, the transfer function from r(t) to y(t) is H(s) — ”f(t) (3 16) " s2 + 2C(t)w,,(t)s + w3(t)' ‘ Thus, the voltage y(t) can be expressed approximately as y(t) = A2(t) cos [wot + 62(t)] , (3.17) where A2(t) = lHLiw,(t)]|A1(t) and 62(t) = 61(t) + AHij,(t)]. It then follows that 9.0) — 0.0) = 411100401. (3.18) 32 Using this in (2.22), the output of the phase detector becomes v0) = f (t) * we (111 liws(t)l) - (3-19) Since —7r < AHLy'wsUII S 0, it follows from the phase comparator characteristic that u(t) = -kgf(t) * (AHij3(t)] + 7r/2) , (3.20) where kg = 2V/7r is the gain of the phase detector. Hence, using 211mm] = — arctan [ffifi‘g’ftl‘jztgl] (3.21) in (3.20) yields u(t) = kof(t) * arctan [zfigzjn—(SSJfiZJ . (3.22) Moreover, it follows from (2.29) that 00.2.0) - 01.2.0) 1 u(t) = E 023, (t) (3.23) Thus, combining (3.23) and (3.22) results in 0130) - 013.0) __ 01.20) - 0130) aura) ‘ mm * ma“ l2c——>6wc(t) Figure 3.2. Simplified model of the resonance tuning system. noted that this approximation is quite accurate provided that |6w3(t) — 6wc(t)| Col-00 < 1. (3.30) 3.2.3 RTS3 Consider the resonance tuning system shown in Figure 2.15 and assume that the phase detector is a multiplication type. Again, assume that the input r(t) is in the form of (3.15). The output y(t) satisfies the differential equation (2.33). As this equation is time varying, it is impossible to find y(t) analytically, but again, with the frozen time approach the transfer function from :1:(t) to y(t) can be written as _ kgw,2,(t) H“) " 32 + 2C(t)w,,(t)s + 030) (3'31) The output y(t) can be approximated using its steady-state part as y(t) = A2“) COS [wot + 020)] , (3.32) where A2(t) = [H Uw,(t)]|A1(t) and 02(t) = 01(t) + AHUws(t)]. It then follows that 92(t) - 010) = ZHUwsUH- (3-33) 35 Using this in (2.17) yields 710) = f (t) * 90m (AHij,(t)]). (334) Thus, the output of the phase detector becomes «20) = f(t) * 511—,(t—llH0wrtm cos (41100401). (3.35) Since (Human (AHLiw.(t)l) = Re{H0w.(t)1}. (3.36) the phase detector output can be rewritten as Ai(t)kgw3.(t) [w3(t) - w3(t)l ’U = at: . 3.37 (t) f(t) 2 ([w§(t) — 013m]? + [2C(t)wn(t)w.(t)12) ( ) Next, solving (2.34) for u(t) and substituting into (3.37), it follows that w§(t) -— w,2,(t) kwkgAi(t)w3.(t) [w3(t) - w§(t)] (3.33) wart) = f‘ l * 2 (twat) — «230012 + [2<(t)w.(t)w. 1. (3.54) Hence, rewriting (3.50) as _ K(A + BK)'1B — 1 “0(le _ K(A + Bro-13 Kr, (3.55) we see that 3: satisfies this equation only when K a: = 0. Hence, once again equation (3.49) implies that a: = 0, and this completes the proof. CI Note that since standard control system design requires Re(A) _<_ O, the first condition of theorem (3.45) is generally satisfied. Also note that the system (3.45) may have either one isolated equilibrium (the origin), or three isolated equilibria (one being the origin). Moreover, since the Jacobian linearization of (3.45) around the origin is 61: = (A + BK)6$, (3.56) it follows from the Lyapunov’s indirect method [30] that if A + BK is Hurwitz, then the origin is asymptotically stable. Combining this result with that of Theorem 3.1, we obtain the following 41 Corollary 3.1 Under the assumptions of Theorem 3.1, the origin :1: = 0 is the unique asymptotically stable equilibrium of the closed-loop system (3.45). Proof: The proof of this corollary follows from the Lyapunov’s indirect method and Theorem 3.1. D Next, assuming A is Hurwitz, we show that the solution of (3.45) remains bounded for all time. Theorem 3.2 Consider the system (3.45), and assume that A is Hurwitz. Then, 33(t) is bounded for all time, i.e., there exists an m > 0 such that ||r(t)|| S m for all t 2 0. Proof: The solution of (3.45) starting from r(0) satisfies 6 r(t) = exp(At):1:(0) + / exp[A(t — r)]B0 (36) From equation 3.61, G(s) = -—1—C(sI — 20-13, (3.63) 2C0 which can be rewritten as G(s) = imam). (3.64) 2C0 Hence, 1 + fikFUw) Re [1+ akF(jw)] > 0, (3.65) where k = 2—20kwke. The remainder of the proof then clearly follows from the proof of the circle criterion in [30], which uses the fact that that the storage function V(:z:) = ézTPat (3.66) for the linear dynamical system satisfies the Kalman-Yakubovich-Popov equations. CI Using a multiplication phase detector, and following the same process as before, the nonlinear model of RST3 (3.38) can be written as 6w? t + 26wC t w ( l 2 ( ) ° = («wing/mm) * “’0 6w§(t) + 26wc(t)we (6w§(t) + 2564060)? + (261%)” (3.67) 45 With the change of variables 2(t) = 6w3(t) + 26wc(t)we, (3.68) this equation can be written e(t) 220) + (2Cow3l z(t) = f(t) * K (3.69) Again using the circle criterion with the requirement that a > 0, it follows that (3.69) will only remain sector bounded in a finite domain. This can be seen from Figure 3.6. Hence, the same method as above can be applied to show stability with a finite domain. Figure 3.6. Sector bounded nonlinearity of RTS3 with multiplication PD. Finally, consider the nonlinear model of RTSI given by (3.11). Let the output ws(t) equal its nominal value we plus some perturbation term 6ws. Also, let wn(t) 46 equal the nominal value we and C (t) = (0 Thus (3.24) becomes 6w, = kwkef(t) =1: arctan — 6% (6% _ 2%) ]. (3.70) 2Cow0 (6w, + we) It should be noted that the nonlinearity of this equation is not bounded by a sector with an upper bound equal to the slope of the arctangent function, as is the case with RTSZ and RTS3. Instead, as 6w, moves away from the origin in the negative direction, this nonlinearity will always start outside the sector. However, circle criterion may still be applied to this case by simply increasing the value of 6 to include this portion of the nonlinearity in the sector. For illustrative purposes, an exaggerated view of this nonlinearity is shown in Figure 3.7. u A fl I, au/ \Il[u] I // / / / / / / / / / I ’ r u 7 Figure 3.7. RTSl nonlinearity crossing the sector bound near the origin. 47 CHAPTER 4 Design In this chapter, the design problem formulated above is considered and some guide- lines for designing the resonance tuning system are provided. Using these guidelines, the actual resonance tuning systems, their nonlinear time-varying models and their simplified linear time-invariant models are simulated using the parameters of the pre- viously discussed applications. These simulation results illustrate the performance of the resonance tuning systems and verify the accuracy of the pr0posed simplified models. All simulations are carried out in MATLAB/Simulink. 4.1 Design Based on the developed simplified linear time-invariant models, the resonance tuning systems can be designed using standard control system design methods. For RTSl, the goal of the design is to find the “controller” F(s) so that the “output” 6w,(t) tracks the “reference” 6w,,(t) satisfactorily. For RTS2 and RTS3, the goal of the design is to find the “controller” F(s) so that the “output” 6wc(t) tracks the “reference” 6w, (t) satisfactorily. In designing the resonance tuning system, the following guidelines should be taken into consideration. As a direct consequence of the internal model principle [29], to achieve perfect asymptotic reference tracking the filter F(s) must contain the model of the reference (6wn(t) for RTSl or 6w5(t) for RTS2 and RTS3). Similarly, the filter F(s) must contain the model of the disturbance 6wn(t) to achieve perfect 48 asymptotic disturbance rejection (RTS2 and RTS3). In particular, for the output to asymptotically track any step change in the reference and asymptotically reject any step change in the disturbance, the filter F(s) must contain at least one integrator. Moreover, the bandwidth of the lowpass filter F(s) should be sufficiently small to filter out the higher order harmonics at the output of the phase comparator. In addition, the bandwidth of kF(s) / [1 + kF(s)] should be sufficiently large for good reference tracking whereas the gain of 1 / [1 + kF(s)] should be sufficiently small for good disturbance rejection. In most practical applications, a first order filter of the form F(s) = 9, (4.1) S where B is a design parameter, usually gives adequate results. With this filter, the settling time for 6w, (t) in RTSl or 6wc(t) in RTS2 and RTS3 calculated from the linearized model is r = 4 / (k6). If the desired performance is not achievable by a first order filter, a second order filter of the form _ a F(s) — 3(3 + a), (4.2) where or and 6 are design parameters, may be used. For this case, the settling time for 6w,(t) or 6wc(t) calculated from the linearized model is r = 8/01 provided that 012 3 41:6. These well established linear system design methods are generally suitable. How- ever, in certain applications, for instance when (3.14), (3.30) and (3.41) are not satis- fied, the linear system will not accurately represent the resonance tuning system. In this instance, the linear system is still valuable in that it acts as a starting point for the design process, however, some guidelines pertaining to the nonlinear system are 49 helpful. Currently, more research is required to provide definite methods for determin- ing parameters such as rise time and settling time for the nonlinear models. However, using the theorems provided in Section 3.3, the resonance tuning systems can be eas- ily designed so that stability is guaranteed even in the absence of an accurate linear model. 4.2 Simulation 4.2.1 RTSl In this section, the original RTSl, its nonlinear time-varying model and its sim- plified linear time-invariant model are simulated. These simulation results illustrate the performance of RTSl and verify the accuracy of the proposed simplified model. Assume that the nominal values of the parameters R(t), L(t) and C(t) of this system are Re = 0.25 {2, Le = 0.1 H and Ge = 10 HF, respectively, so that the nominal values of wn(t), ((t) and Q(t) are we = 1000 rad/s, Co = 0.00125 and Qe = 400. Assume further that V = 2.5 V so that kg = 1.592 V/rad. Let kw = 50 V‘1 and assume the center frequency of the VCO is we = 1000 rad/s. Hence, the system is tuned to its resonant frequency. However, as before assume the value of C (t) decreases by 1 percent, so that the resonant frequency of the system becomes 1005.04 rad/s. For the second order filter (4.3) the response of the actual RTSI shown in Figure 2.12, the nonlinear model (3.9) and the linear model (3.11) to this input step change is shown in Figure 4.1. In this figure the solid line is the response of the actual resonance tuning system, the dashed line is the response of the nonlinear model, and the dashdot line is the response of the linear model. 50 It is evident from this figure that RTSl is working satisfactorily and that the two simplified models are reasonably accurate approximations of the actual RTSl system. However, to demonstrate the range of this resonance tuning system, a second simulation is considered. Assume the same parameters as before, but now with a 10 percent decrease in C (t), so that the resonant frequency of the system becomes 1054.1 rad/s. The response is shown in Figure. 4.2. From this figure, it is clear that RTSl successfully tunes w,(t) to wn (t). Hewever, it is also clear that the linear time-invariant model is no longer an accurate representation of the actual RTSI. This is due to the fact that (3.14) is not satisfied. 1006 r t 1005'- /' /—— v 1004- .' 1 10034 ,' I I w,(t), rad/s 1002 - I .' 1 1001 7" 1000 0.5 l 1.5 t, s Figure 4.1. Simulation results of RTSl with wn = 1005.04. 51 1060 I I I I I I I I I 1050~ / [I 1 I \ 1040 rad / s 1030 a A I w, (t 1020 1010 _‘-'-r—‘-‘—'-1 . . . \ l 16 18 20 00l- p—e O 1...; t0 p—I A 1000 6 Figure 4.2. Simulation results of RTSl with wn = 1054.1. 4.2.2 RTSZ In this section, the original RTS2, its nonlinear time-varying model and its sim- plified linear time-invariant model are simulated. These simulation results illustrate the performance of RTS2 and verify the accuracy of the proposed simplified model. Assume that the nominal values of R(t), L(t) and C(t) are Re = 0.5 9, Le = 0.01 H and Ge = 0.0001 F, respectively, so that the nominal values of wn(t), C (t) and Q(t) are we = 1000 rad/s, C0 = 0.025 and Qe = 20, respectively. Assume also that V = 2.5 V so that k9 = 1.592 V/rad and let kw = 0.1 V“. Assume further that R(t) = Re, L(t) = Le for all t, but C(t) varies with time as C(t) = 1.000x 10“ F for 0 s S t < 2 s and 8.264 x 10‘5 F for 2 s S t so that the uncontrolled resonant frequency w" (t) of the resonator undergoes a step change from 1000 rad/s to 1100 rad/s. Finally, assume that the excitation frequency w,(t) also undergoes a step change at time t = 4 s from 52 1000 rad/s to 900 rad/s. For each of the following filters 2.827 F(s) : 1.571, F 62.832 F(S)=s+l’ s (3):.m, (4.4) the original resonance tuning system shown in Fig. 2.14, its approximate nonlinear timevarying model given in (3.24) and its simplified linear time-invariant model given in (3.27) are simulated. The simulation results for the resonant frequency wc(t) are shown in Figs. 4.3, 4.4 and 4.5, respectively. In each figure, the solid curve is for the exact system, the dashed curve is for the approximate system and the dashdot curve is for the simplified system. The uncontrolled resonant frequency wn (t) and driving frequency w,(t) are also shown in each of these figures by the solid curves for comparison. These figures indicate that both the approximate nonlinear time-varying model and simplified linear time invariant model predict the behavior of exact system quite accurately. Since the first filter does not contain an integrator, the error between the resonant frequency and driving frequency in the steady-state is not zero as predicted by the previous analysis. This error, on the other hand, is practically zero for the second and third filters as expected since they both contain an integrator. Note also that since the second order filter has better selectivity compared to the other two, its performance in the steady-state is superior to those of the first order filters. These results show that the developed simplified model predicts the performance of the resonant tuning system quite accurately. Therefore, the developed simplified model can be used to facilitate both analysis and design of the resonance tuning system. 53 wc(t), rad/s wc(t), rad/s 1100- e ‘1 1 M 1050- “ - \ \ r\ \\ .\\ 1000 \ - u U l l 950- 1 ‘ I .5;~ 900- 0 l 2 3 4 5 6 t, s Figure 4.3. Simulation results with F(s) = 2.827/(s + 1). llOOr n p w l\ I. \ 1050~ \\ a \ \ \‘\ \ \ \ \ 1000 I“ a 950- - 900* O l 2 3 5 6 Figure 4.4. Simulation results with F (s) = 1.571 / s. 54 1100 I 1050 I 1000 h ‘ 4“ 950* 900~ Figure 4.5. Simulation results with F(s) = 62.832/[s(s + 20)]. 4.2.3 RTS3 In this section, the developed analysis and design methods of RTS3 are applied to two examples. First, a vibrational gyroscope example from [6] is considered. Assume that the parameters of this system are wn(t) = 63881.1 rad/s, C (t) = 0.0005, kg = 0.0666. Moreover, assume w,(t) = 65973.4 rad/s and A1(t) = 1. In the simulations below, the feedback gain k; was selected as k; = 100, the nominal frequency we was taken as we = 65973.4 rad/s and the logic voltage V was chosen as V = 2.5 V. The first simulation of RTS3 uses a multiplication phase detector with the first order low-pass filter -925 F(s) = S , (4.5) The results of this simulation are shown in Figure 4.6. Here, the solid curve is for the actual system shown in Figure 2.15 and the dotted curve is for the nonlinear 55 time-varying model (3.38). The instantaneous input frequency w3(t) is also shown by the solid horizontal line for convenience. 4 x 10 6.65 I l T T I I T I I 6.6 6.55 6.5 wc(t), rad/s 6.45 6.4 l I 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 L 1 6.35 O t,s Figure 4.6. Simulation results of RTSB for vibrational gyroscope example. It should be noted that while the nonlinear model closely approximates the actual system, the corresponding linear model is quite different, and thus not shown in Figure 4.6. This is due to the fact that the deviation of wc(t) from the nominal frequency we is quite large initially. In other words, the system in [6] does not satisfy condition (3.41), and the nonlinearity in (3.38) behaves like saturation due to the extremely small damping ratio. This discrepancy reduces the values of the developed linear time invariant models since their predictions are no longer accurate. Therefore, unless the system is altered to satisfy (3.41), a different design strategy based on the develOped nonlinear time varying models should be used. However, it should also be 56 pointed out that the developed linear model still provides some value in that it acts as starting point in the design process. In order to illustrate the accuracy of the developed linear time invariant model when (3.41) is satisfied, a second example is considered. This simulation considers a hypothetical lightly damped second order system. The parameters of the resonance tuning system are as follows: wn(t) = 1000 rad/s, ((t) = 0.05, kg = 1, w,(t) = 1050 rad/s, A1(t) = 1, k, = 0.5, we = 1000 rad/s and the phase detector is a'multiplication' type with second order filter 10 The results of this simulation are shown in Figure 4.7. Here, the solid, dashed and dashdot curves are for the actual system shown in Figure 2.15, the nonlinear time-varying model (3.38) and the simplified linear time-invariant model (3.40), re- spectively. Again, the instantaneous input frequency w,(t) is shown by the solid horizontal line for convenience. It is evident from Figure 4.7 that the developed models accurately predict the performance of the actual system, which was expected as (5.10) is satisfied in this generic example. Hence, the developed model can be used for both analysis and design of the resonance tuning system. 57 wc(t), rad/s 1060 1050 1040 1030 1020 1010 1000' 990 Figure 4.7. Simulation results of RTS3 with F(s) = l l l 1 013 (l4 (l5 t,s 58 (16 (l7 (l8 (L9 10 RAH—2W CHAPTER 5 Extension - Adaptively Enhanced PLL 5.1 Introduction The following presents an extended use of RTS2 and RTS3, in which these reso- nance tuning systems can be combined with a standard PLL to enhance the perfor- mance of the standard PLL. Much of the derivation behind the equations has been omitted as it is similar to the analysis presented earlier in this thesis. Initial results of the adaptively enhanced PLL are promising, however, more research on this topic still needs to be carried out. 5.2 Modeling In this section, a model of the adaptively enhanced PLL is presented. This model can be seen in Figure 5.1. Here the input to the system is x(t) and the output is y(t). A standard PLL is represented by the components within the dashed box, where PC is a phase comparator, F(s) is a lowpass filter and VCO is a voltage controlled oscillator. FE is a frequency estimator system that provides an estimate, we(t), of the instantaneous frequency of x(t) to the VCO. The purpose of the frequency estimator is to provide the voltage controlled oscillator (VCO) of the PLL with an appropriate free running angular frequency we(t) that closely matches the frequency of the input. 59 we(t) l '_ 7’ 7" —————————— _ "i : PC F(s) Mt) > vco | | 1 ' 1__ _____________ _ __ l " y(t) Figure 5.1. Model of the enhanced PLL system. For this task, a method similar to that of RTSZ is used. In this method, the resonant frequency of a lightly damped second order system is changed using prOportional feedback and the feedback gain is adaptively adjusted to tune the resonant frequency of the resulting closed loop system to its excitation frequency. Hence, at steady state, the resonant frequency of this closed loop system acts as an estimate of the excitation frequency. A block diagram of the frequency estimator system is shown in Figure 5.2. In this figure, H (s) is the transfer function of the lightly damped second order system, PD is a phase detector, ke is the feedback gain and we (t) is the frequency estimate. The external input to the system a:(t) is assumed to be in the form (D(t) = A1 COS [Wot + 61(t)] , (5.1) where A1 > 0 is the amplitude and we is the nominal angular frequency, both of which are assumed to be constant but unknown quantities. The instantaneous phase of :1:(t) 60 I (e(t) I u(t) ' I = + I I I I I I ”e(t) : wni/1+ke(-) _’ I 211 (t) T ”e(t) 1’20?) Figure 5.2. Model of the frequency estimator system. is 01 (t). Thus, the instantaneous frequency of the input is w,(t) = we + 61(t). (5.2) The transfer function of the lightly damped second order system can be expressed as ”'2‘ (5.3) H(s) = 32 + 2(wns + wg’ where w,| is the natural frequency of this system and C is the damping ratio. The dashed box represents the closed loop system, the resonant frequency of which is adjusted to match the instantaneous frequency of the input. The signal ue(t) is the error generated by the phase detector, which is assumed to be an exor type. Following analysis similar to that developed earlier in this thesis, ve(t) is given by 730) = f.(t) * W [920) - 910)] 1 (5-4) 61 where f (t) is the impulse response of the lowpass filter and at is the convolution operation. With ue(t) known, it then follows from Figure 5.2 that u(t) = a:(t) — keve(t)u2(t). (5.5) Thus, the closed loop system within the dashed box is governed by the differential equation 62(t) + 2(wni22(t) + [1 + keue(t)] wiu2(t) = w§x(t), (5.6) where the resonant frequency of this closed loop system is we(t) = tin/1 + 8.64:). (5.7) Hence, the closed loop resonant frequency we(t) can be easily computed from the output of the phase detector ve(t), and provided to the PLL as an estimate of the input frequency. 5.3 Analysis and Design Consider the frequency estimator system shown in Figure 5.2. Here it can be easily seen how the frequency estimate we(t) can be computed when provided the error signal ve(t). However, as good performance of this system requires careful design of the lowpass filter and lightly damped second order system, equations that describe the dynamics of this system are necessary. Following analysis methods similar to those provided earlier in this thesis, the nonlinear equation that describe the dynamics of 62 the frequency estimator system with an exor type phase detector is (.02 _ (1)2 (.02 _ 2 _iliijJ = fe(t)*kek9 arctan[ 82(Ctdinw:(:)t)]' (5.8) Simulations show that (5.8) represents the actual frequency extimator system very accurately. However, a simpler model is still desired for design purposes. After linearization, the the simplified linear time-invariant model of the resonance tuning system becomes 501.0) = fe(t) * k [5ws(t) - 6w.(t)] , (59) where k = keke/(2C). The block diagram of the system described by this equation is as shown in Figure 5.3. This linear model describes the actual system accurately, 6w,(t) ~ 1.1618(3) > 6we(t) Figure 5.3. Simplified model of the resonance tuning system. provide that |6w,(t) - aweltll Cwn 1 < —. . _ 2 (510) However, as the goal of the enhanced PLL is to achieve lock to an input with unknown frequency, whether this condition is satisfied or not, will most likely remain unknown as well. Despite this fact, (5.9) is still a valuable tool in that it provides a starting point for design process. 63 5.4 Simulation In this section, the enhanced PLL system is simulated to demonstrate its lock range capabilities. All simulations were performed in MATLAB/Simulink. In order to determine whether the output signal y(t) has locked to the input signal :c(t), the frequency-phase error between these two signals must be calculated. This error is defined as $0) = («2.0) —w.