AUTO-TUNED NEAR MINIMUM-DEVIATION DIGITAL CONTROLLER FOR HARD SWITCHING DC-DC CONVERTERS

Transcription

1 AUTO-TUNED NEAR MINIMUM-DEVIATION DIGITAL CONTROLLER FOR HARD SWITCHING DC-DC CONVERTERS by Shadi Dashmiz A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright by Shadi Dashmiz 2016

2 AUTO-TUNED NEAR MINIMUM-DEVIATION DIGITAL CONTROLLER FOR HARD SWITCHING DC-DC CONVERTERS Abstract Shadi Dashmiz Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2016 This thesis presents an auto-tuned digital controller applicable to various kinds of hard switching dc-dc converters. Controller consists of three blocks: i) an auto-tuned output current estimator obtaining instantaneous load current value ii) a conventional digital compensator for steady state regulation iii) an auto-tuned transient digital module to minimize voltage deviation. Once a transient happens, the current sensor senses the instantaneous output current and activates the transient controller providing the new value of the load current. The transient nonlinear controller calculates the shortest time needed for the inductor current to ramp up/down to its new steady. Based on the calculated time, the transient controller keeps the main switch on/off. Once the new steady state value of inductor current is achieved, the conventional controller is re-activated through a bump-less mode transition. Experimental results obtained with a boost converter setup confirm accuracy of proposed current estimator and operation of digital controller. ii

3 Acknowledgments I would like to express my gratitude toward Professor Aleksandar Prodic who provided me this amazing opportunity to be part of his research group. Without his none-stop support, insight and encouragement none of these would have been possible. I would like to thank all my friends in the Laboratory for Power Management and Integrated Switch-Mode Power Supplies. I am especially thankful to Behzad Mahdavikhah whose deep knowledge enlightened and made research difficulties look easy. I also want to thank all my lab mates: Nenad, Tim, Maryam, Sam, Tom, Mia, Amr, Ahsan, Parth, Jasmine and Michael for making every moment of work enjoyable. I also would like to thank Texas Instruments for funding my research projects. I also want to thank my committee members Professors Reza Iravani, Josh Taylor and Tung Ng for their feedbacks. I also want to deeply thank my parents Fatemeh and Mahmoud for their faith in me, support and unconditional love throughout my life. I want to thank my brother Shayan for his endless spiritual support and love. At the end I want to express my unfailing gratitude for my husband Alireza whose love and enduring care supported my persistence and encouraged me when all the stress seemed insurmountable. iii

4 Table of Contents Acknowledgments... ii Table of Contents... iv List of Figures... vi Chapter Introduction SMPS Fundamentals Direct vs Indirect Energy Transfer Converters Digital vs Analog Controller Controller for Direct and Indirect Energy Transfer Converters Thesis Overview...4 Chapter Previous Art and Motivation Fast Transient Response and Minimum Deviation Controller for Direct Energy Transfer Converter Output Current Sensor for Direct Energy Transfer Converter Designing Fast Response and Minimum Deviation Controller for Indirect Energy Transfer Hard Switching Converters Boost Converter Characteristics Output Current Estimator Previous Art for Designing Fast Transient Response, Minimum Deviation Controller For Indirect Energy Transfer Converters Time Optimal Controller (TOC) Minimum Deviation Controller for Indirect Energy Transfer Converters...16 Chapter A Hardware Efficient Self-Tuned Output Capacitor and Current Time Estimator for Indirect Energy Transfer Converters...21 iv

5 3.1 Principle of Operation Practical Implementation Slope Identification and Tuning of Estimator Time Constant Load Current Estimation Inductor Current Estimation Transient Detection...30 Chapter Digital Auto-tuned Near Minimum Deviation Controller for Indirect Energy Transfer DC- DC converters Principle of Operation Practical Implementation Simulation Result Simulation Results for Boost Converter Simulation Results for Flyback Converter...48 Chapter Experimental Results Current Sensor Experimental Result Digital Controller Experimental Result...57 Chapter Conclusions and Future Work Future Work...66 A Derivation of Slope Equations for RC Matching Circuit...67 References...70 v

6 List of Figures Fig 1.1. Voltage mode controlled buck converter... 2 Fig 2.1. Boost converter in closed loop system with digital controller Fig 2.2. Key waveforms of a boost converter during steady state and a light-to-heavy load transient under linear controller operation. Top to bottom: Iload is the load current, vout(t) is the output voltage, Vref is the reference voltage, iind(t) is the inductor current, id(t) is diode current and c(t) is the gating signal to the main switch Fig.2.3. Bode blots of boost converter showing the effect of RHPZ..11 Fig.2.4. Conventional method to measure output current...12 Fig 2.5. Digitally measured output current Fig2.6. Transient response of boost converter under TOC. Top to bottom: Iload is the load current, vout(t) is the output voltage, Vref is the reference voltage, iind(t) is the inductor current, id(t) is diode current and c(t) is the gating signal to the main switch. 15 Fig 2.7. Operation of the programmable-deviation controller in a boost converter for light to heavy load transient (maximum switching frequency is limited). Inductor and load current (top), output voltage (middle), and state-plane representation of the voltage and current (bottom). System parameters are: Vin=12 V, Vout=48 V, Iload from 1 A to 4.5 A.[62]..17 Fig 2.8. Schematics of stackable flyback converter in closed loop system Fig 2.9. Flyback main waveforms during transient for the implementation of nonlinear controller. Top to bottom: vout(t) is the output voltage, Vref is the reference voltage, Iload is the load current, iout(t) is the output current, ic(t) is output capacitor current and MS is the gating signal to the main switch, SR is gating signal to secondary switch.. 20 Fig 3.1. Boost converter including conventional PID, adjustment RC filter and current estimator.23 Fig.3.2. Schematics of output filter and output current for indirect energy transfer converters..24 vi

7 Fig 3.3. Comparison of different slope with respect to v esr (t) for diode off time period; (a) Waveform of v esr (t). (b) τ adj (R adj C adj ) > τ esr (R esr C out ), slope of v adj (t) is negative (c) τ adj (R adj C adj ) < τ esr (R esr C out ) slope of v adj (t) is positive during diode off time.. 25 Fig 3.4. Simulation of v esr (t) and v adj (t); (a) Waveforms of v adj (t) and v esr (t) when time constants are matched (b) Waveforms of v esr (t)and v adj (t) when τ adj (R adj C adj ) < τ esr (R esr C out ) (c) Waveforms of v esr (t) and v adj (t) when τ adj (R adj C adj ) > τ esr (R esr C out ) Fig 3.5. Implementation of current sensor circuit..27 Fig 3.6. Implementation of slope identification Fig 3.7. Tuning process to find the best match for R adj..29 Fig 3.8. Boost converter main waveforms during load transient. Top to bottom: Iload is the load current, Vout is the output voltage, v adj (t) is the voltage over adjustment branch resistor and Iind is the inductor current Fig 3.9. Zoomed in version of Fig Fig 4.1. Boost converter including both PID and nonlinear controller...34 Fig 4.2. Boost converter when operated with proposed controller. Top to bottom:load current I Load ; steady-state and transient output voltage v out (t); instantaneous inductor current i ind (t); average inductor currents I ind,avg,h, I ind,avg,l during steady state for heavy and light load conditions respectively; main switch gating signal c(t) Fig 4.3. System implementation including linear and nonlinear controller plus current estimator sensor. 38 Fig 4.4. Flowchart of controller algorithm vii

8 Fig 4.5. Valley point detection for auto-tuning: value of V 2 is always checked against value of V 1. a) KLH= L L b) KLH< v in (t) v in (t) Fig 4.6. Auto-tuning procedure flowchart Fig 4.7. Digital controller (only PID) for boost converter. vout is the output voltage, iind is inductor current, iload is the load current and, c(t) is gating signal to the main switch Fig 4.8. Digital controller (nonlinear with PID) for boost converter. vout is the output voltage, iind is inductor current, iload is the load current, Time_on is the time calculated during transient to keep the main switch on, c(t) is gating signal to the main switch...45 Fig4.9. Addition of auto-tuning system for successive light-to-heavy transitions. vout is the output voltage, K is the auto-tuning parameter Vd the difference of two successive output voltage sample and is equal to V2-V1..46 Fig Heavy to light transition from 0.5 to 0.25 A with PID controller. vout is the output voltage, Iload is the load current, iind is the inductor current, c(t) is gating signal for the main switch Fig Heavy to light transition from 0.5 to 0.25 A including both nonlinear and PID controllers. vout is the output voltage, Iload is the load current, iind is inductor current, c(t) is gating signal for the main switch Fig Schematics of flyback converter in closed loop system Fig Simulation of closed loop flyback with only PID. vout is the output voltage, Iload is the current passing through load resistance and Ilm is the current of magnetizing inductor and, c(t) is gating signal for the main switch 49 viii

9 Fig Simulation of closed loop flyback with PID and nonlinear controller. vout is the output voltage, Iload is the current passing through load resistance and Ilm is the current of magnetizing current inductor and c(t) is gating signal 49 Fig Light-to-heavy transition auto-tuning process for flyback converter. K starts to oscillate between two values when the best match is found. 50 Fig 5.1. Setup including power stage and controller Fig.5.2. v adj (t) waveform for different time constant conditions. Top to bottom: v adj (t) when τ adj > τ esr (R adj = 1 KΩ), (200 mv/div). v adj (t), when τ adj = τ esr (R adj = 500 Ω),( 200 mv/div). v adj (t), when τ adj < τ esr (R adj = 200 Ω), ( 200 mv/div). Timescale is 1 µs/div...52 Fig (a) Key waveforms of the digital controller and the self-tuning estimator. Top-to-bottom: v out (t) output voltage of the converter (500 mv/div), v x (t) switching node of boost converter (5 V/div), i load (t) current of load (500 ma/div), v adj (t) voltage across the estimator resistor (200 mv/div), R adj [n] digital pot code changes, changing resistor and stops when output of comparator is detected zero. (b) zoomed-in version Fig 5.4. Waveforms of boost converter during inductor current reading. Top-to-bottom: v out (t) output voltage of the converter (5V/div),), i lnd (t) current of inductor (500 ma/div), i load (t) current of load (500 ma/div), v adj (t) voltage across the estimator resistor (2 V/div), I lnd -avg [n] digital current reading of inductor in ma. Time-scale is 10 µs/div.. 54 Fig 5.5. Zoomed in version of Fig5.4, (a) current reading in ma during light current steady state. (b) during transient and (c) during heavy load steady state 55 ix

10 Fig Key waveforms of the transient and steady state current estimator. Top-to-bottom: v out (t) output voltage of the converter (500 mv/div), v adj (t) value of adjustment branch voltage after matching process during steady state and transient operation (200 mv/div), i Inductor (t) current of inductor (500 ma/div), I Load value of load current (500 mv/div), digital value of current value of current in ma is updated cycle by cycle during steady state and transient (light to heavy) operation. Time scale is 2 µs/div Fig.5.7. (a) transient response of boost converter to a 0.25 to 0.5A employing conventional converter. Top to bottom: vout(t), is the output voltage 200 mv/div, Iload is step current, 500mv/div, iind (t)is inductor current, 2v/div and, d(t) is the gating signal to main switch,5v/div, time scale is 20µs/div. (b) zoomed in version 58 Fig.5.8. (a) transient response of boost converter to a 0.25 to 0.5 A employing nonlinear converter. vout(t)is output voltage (200mv/div), Iload is step current (500mv/div), iind (t) is inductor current (2v/div), d(t) is gating signal (5v/div), time scale is 10µs/div. (b) zoomed in version...59 Fig.5.9. Transient response of boost converter to a 0.25 to 0.5A employing nonlinear converter without proper re-initialization of PID. Top to bottom: vout (t) is output voltage (500mv/div), iind (t) is inductor current (1v/div), Iload (t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 100µs/div.. 60 Fig Auto-tuning process of K for 0.25 to 0.9A load transition. Top to bottom: v out (t) is output voltage (500mv/div), iload (t) is load current (500mv/div), iind (t) is inductor current (2v/div), code[n] is value of updated K in each cycle, vx(t) is gating signal (5v/div), time scale is 100 ms/div.61 x

11 Fig (a) updating K from 0 to 1, K=0 is equal to PID only, (b) transition K =1 to 2, from figure V2 is smaller than V1 K should be increased.(c) K =2 to 3, still K should be increment (d) K =3 to 4, optimum point, (e) K =4 to 3, K is larger than optimum value, for the next cycle K should be decreased Fig Transient response of boost converter to a 0.5 to 0.25 A step current employing only PID controller. Top to bottom: v out (t) is output voltage (500mv/div), i ind (t) is inductor current (1v/div), Iload(t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 50µs/div.. 63 Fig Transient response of boost converter to a 0.5 to 0.2 A transient employing nonlinear and PID controllers. Top to bottom: v out (t) is output voltage (500mv/div), i ind (t) is inductor current (1v/div), Iload(t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 50µs/div.. 63 Fig A.1. Current Id(t) passing through three parallel branches xi

12 1 1. Introduction The objective of this thesis is to design a practical fast_transient near minimum deviation controller for hard switching dc-dc converters with emphasis on indirect energy transfer converters. In this chapter a review on indirect energy transfer converters is presented. Also, various fast control methods and implementation challenges are investigated SMPS Fundamentals In modern electronics devices, most of power processing is performed through switch mode power supply (SMPS). They have almost completely replaced conventional linear voltage regulator (LVR), which suffer from low power processing efficiency, particularly worsen at high voltage conversion ratios in the targeted devices. SMPS are used in variety of power applications ranging from mili-watts to hundreds of watts. With the increasing energy demand for electronic devices general trend in SMPS research is to provide more efficient solutions while decreasing weight and volume. An SMPS consists of three main components: a switching network, an output filter and, a controller. Switching network usually consists of two or more semiconductor switches [1]. Output filter consists of a capacitor and inductor with corner frequency designed to pass the DC component of the switching signal and filter out switching components at the switching frequency and higher harmonics [1]. The inductor and capacitor also work as energy storing and delivery devices, delivering power from input to the output. Controller is an invariably essential part of

13 2 SMPS, since it is almost always required to maintain a regulated output voltage when a change in the input voltage or the load occurs. As shown in Fig 1.1 in the voltage mode control the output voltage V out (t) is subtracted from a reference V ref and an error signal is generated. Error is fed to a Proportional, Integral, Derivative compensator (PID) which creates a signal for a pulse width modulator (PWM) block that adjusts the on/off times of switch in order to regulate the output voltage. Switching Network Filter L V out (t) v in (t) C(t) D C out R load Pulse Width Modulator (PWM) d(t) Controller PID Compensator e(t) V ref H Fig 1.1. Voltage mode controlled buck converter 1.2. Direct vs Indirect Energy Transfer Converters Various configurations of switching network, and low pass filter give rise to different converter topologies [1], it is possible to categorize dc-dc converters into two groups: direct energy transfer converters and indirect energy transfer converters. In direct energy transfer converters, the inductor is connected to the output capacitor during both on and off time of a switching sequence allowing continuous transfer of energy from input to output. Buck converter is an example of the direct energy transfer converter. In contrary, in indirect energy transfer converters transferring of energy

14 3 to the load occurs in two steps, in the first step, during on time of the main switch, energy is stored in the inductive element while the load and the output capacitor are detached from it. In the second step, during the off time of the main switch, the stored charge in inductor is transferred to the output capacitor and the load. This separation poses challenges for the control of this type of converters compared to direct energy transfer converters because controller face a delay in responding to a change of error signal causing additional voltage drop at the output during transients. Boost converter is an example of an indirect energy transfer converter Digital vs Analog Controller As shown in Fig 1.1, controller consists of a voltage reference, a subtractor, a compensator and, PWM. In analog controller, all the parts are built by analog components. Analog PWM modulator produces a periodic triangular signal, this signal is compared to a control voltage and a square wave is produced. Advantage of analog controller over digital solution is low power consumption and higher frequency of operation [2]-[6]. However, analog controllers are converter specific. Once developed on Integrated Circuit (IC) they cannot be easily transferred from one design or technology to another [7]-[10]. Also for advanced control methods, such as time-optimal and minimum deviation controllers, they do not have sufficient computational power. As a result, digital controllers have gained importance in low power applications. Digital controllers are also gaining popularity due to their portability and transferability from one design to another. They offer possibility of improving dynamic response of converters when load transients are presented in the system [11]-[32]. Digital controllers are easily re-synthesizable in different CMOS technologies. The controller components are essentially similar to that of analog controller but the output voltage value is converted to digital equivalent and the rest of controller is synthetized and implemented using digital blocks.

15 Controller for Direct and Indirect Energy Transfer Converters As was mentioned earlier, in the direct energy transfer converters control output responds directly to the output change. Controller for direct energy transfer converters have been extensively investigated. There are proposed methods to ensure the minimum voltage deviation for buck and buck derived topologies [33]-[38]. In the steady state these controllers act as conventional voltage mode controllers and during transient, the suppression mode reacts to recover the system back to its steady state with lowest possible voltage deviation with no knowledge of converter parameters. On the other hand, for indirect energy transfer converters such as boost and boost derived topologies, implementation of a fast transient and minimum deviation controllers is more challenging due to discontinuity in power transfer. For this reason, there are not many practical methods for a fast transient response control of these converters and, the ones proposed suffer from serious drawbacks. Some of drawbacks include large overshoot, open loop operation during transients, or overly complex calculations to find proper operation point Thesis Overview The main intent of this thesis to showcase the development of a novel self-tuning current sensor applicable to indirect energy transfer converters as well as a practical minimum deviation controller particularly beneficial for indirect energy transfer converters. This thesis is organized into six chapters. Chapter 2 provides a review on nonlinear time-optimal and minimum-deviation control methods previously proposed for various direct and indirect energy transfer hard switching dc-dc converters.

16 5 In Chapter 3, theory, design process and, implementation of a novel hardware efficient self-tuning output current estimator are presented. The estimator can facilitate development of various nonlinear controllers for improving dynamic response, particularly useful for indirect energy transfer converters. Chapter 4 describes a digital controller consisting of linear and nonlinear blocks designed to practically minimize voltage deviation during transients for indirect energy transfer converters. The implementation is elaborated and simulations for two popular indirect energy transfer converters are presented. Chapter 5 describes experimental setups and presents the results that validate the performance of the sensor and the controller introduced in this thesis. Finally, Chapter 6 summarizes the work done in this thesis. Also, possible further improvements are proposed.

17 6 2. Previous Art and Motivation In this chapter various nonlinear controllers providing the optimal dynamic response either in terms of settling time or deviation for both direct and indirect energy transfer converters are reviewed. Challenges involved in designing a nonlinear controller for indirect energy transfer converters are also addressed Fast Transient Response and Minimum Deviation Controller for Direct Energy Transfer Converter Regulating and maintaining a stable output voltage for variety of power converter applications is extremely important [1]. Multiple control methods have been used to minimize output voltage deviation during load transients. This helps to reduce the size of the usually bulky output capacitors [39]-[42]. Among the most efficient methods are the time-optimal and minimum-deviation controllers [43], [44]. Both of these methods activate nonlinear digital controller during transients and linear control during steady state. The time-optimal method [43] performs recovery of output voltage in two phases. For example for light to heavy load transient, in the first phase the inductor current is ramped up to the new load value by keeping the main switch on. In the second phase the lost charge of capacitor is restored through a single on time switch action. After this process is completed, linear controller (usually a PID) takes over the operation. The time that the nonlinear controller keeps the switch on is based on the detection of the valley point in the output voltage. The main challenge in designing a robust

18 7 time-optimal controller is an accurate calculation of the duration of single on/off sequence of the main switch during charge recovery of the output capacitor, imposing a high current stress on the inductor and switches. The calculations are based on the values of the inductor, capacitor and the load. A small inaccuracy in calculating the required time can result in a large mismatch between the inductor and the load currents, making the controller unstable. This sensitivity on very accurate calculations makes the controller extremely parameter dependent. In the minimum deviation controller proposed for buck and buck derived converters [45], the partition of tasks between the linear and nonlinear block is different from the time-optimal controller. The nonlinear controller is activated when a transient, for example light-to-heavy, is detected. The nonlinear controller keeps the main switch on until the inductor current matches the load current, i.e. when a valley point is detected in the output voltage. At this time nonlinear controller is deactivated. The charge recovery phase is handed to the conventional linear controller such as PID. The benefit of this approach is that it reduces computational cost extensively at the expense of a slightly increased voltage recovery time. On the other hand, the switches and the inductor are subject to much lower current stress. Viability of this method depends on determining whether the inductor current has reached to the new load value. This is usually performed by inspecting the instantaneous output capacitor current. When the inductor current matches the load current, the current passing through the output capacitor is zero Output Current Sensor for Direct Energy Transfer Converter As indicated in the previous section, accurate sensing of the output capacitor current and/or load current is a key element for a fast transient response controller. In a realistic SMPS, the output

19 8 capacitor often has non-negligible series resistance referred to as ESR. Together with the output capacitance it forms a RC branch [23]. Various methods have been developed to correctly estimate this RC time constant, which can lead to accurate approximation of the output capacitor and load currents as well as early fault detection of the power supply[46]-[48]. Passive-circuit based, perturbation-based, and self-tuning time constant approximation methods have been proposed to implement RC matching circuit for direct energy transfer converters [49]- [52]. Perturbation-based methods [51],[52] are based on introducing a perturbation at the output node and extracting the output branch capacitor and ESR values using post signal processing of the output waveform. These methods are able to provide a correct estimation of the output capacitor and ESR values i.e time constant, but cannot provide information about the instantaneous output current, making it impractical for fast transient response controllers. Passive-circuit based methods [49],[50] are usually implemented by adding an additional RC filter branch in parallel with the output capacitor. The added RC filter time constant is then matched to output branch and, the output current is estimated by measuring the voltage drop over the RC filter resistor. This method is capable of providing instantaneous current value and time constant but is prone to large errors due to inevitable changes of the output capacitor and ESR values during lifetime of the SMPS. Self-tuning methods have been developed to improve this RC time constant matching technique [53], [54]. These methods go through self-tuning of filter branch to compensate for change of parameters.

20 Challenges in Designing Fast Response and Minimum Deviation Controller for Indirect Energy Transfer Hard Switching Converters The methods mentioned before for the output current estimation are not directly applicable to indirect energy transfer converter topologies, such as boost and flyback. Designing an effective controller for these SMPS is inherently challenging due to the nature of indirect energy transfer converters. In this section, boost converter characteristics and controller design are described. Boost converter characteristics Fig 2.1 shows schematics of a boost converter in a closed loop system. Fig 2.2 shows main converter waveforms during steady state and transient operations employing conventional linear controller. Boost is an indirect energy transfer converter. There is a portion of time where the L i ind (t) i d (t) D v out (t) V in (t) C load C(t) H Pulse Width Modulator (PWM) d[n] PID Compensator Hv out (t) e[n] ADC V ref [n] Fig 2.1. Boost converter in closed loop system with digital controller.

21 10 I load v out (t) V ref i ind (t) i d (t) c(t) Fig 2.2. Key waveforms of a boost converter during steady state and a light-to-heavy load transient under linear controller operation. Top to bottom: I load is the load current, v out(t) is the output voltage, V ref is the reference voltage, i ind(t) is the inductor current, i d(t) is diode current and c(t) is the gating signal to the main switch output capacitor is disconnected from the inductor. This causes a delay in the control response giving rise to a right half plane zero (RHPZ). The averaged small signal transfer function for control to output of the boost converter is shown in Eq 2.1 which mathematically demonstrates the presence of RHPZ.

22 11 40 Bode Diagram Magnitude (db) Phase (deg) Frequency (rad/s) Fig.2.3. Bode blots of boost converter showing the effect of RHPZ v(s) d(s) = V out D (1 s w z ) (1 + s Qw + ( s (2.1) 2 )) 0 w 0 V out is the dc value of output voltage. w z is the frequency of RHPZ in rad/s and is equal to D 2 R, L Q is the quality factor and is equal to D R C L, w 0 is the natural frequency of system equal to D and D, C,L and, R are complement of duty ratio, output capacitor, inductor and load respectively. LC

23 12 The adverse effect of RHPZ can be seen from bode plot of boost converter in Fig 2.3. An additional phase drop of 90 degrees is introduced at high frequencies with an increase in gain. The effect of the RHPZ is initially driving the output voltage opposite to the intended direction. If the duty changes to increase the output, the output voltage initially decreases due to the inductor being disconnected from the output for a longer period of time [1]. The RHPZ location changes with change of duty ratio which can cause significant phase drop leading to potential stability problems. This phenomenon makes design of a robust controller for indirect energy transfer converters challenging. Output Current Estimator A key element to have a precise and robust nonlinear controller is having access to the instantaneous output current. Different RC matching techniques described before [49]-[54] developed for direct energy transfer converters cannot be directly applied to indirect energy transfer converters due to different characteristics of the two categories. Conventional approaches L D v out (t) Vin(t) C(t) C load R sense V sense Controller Fig.2.4. Conventional method to measure output current

24 13 such as estimating the current through adding a series resistance (R sense ) in current path, as shown in Fig.2.4, causes significant loss in the system [55]. Proposed method in [56] is to estimate output current through populating a LUT table for indirect energy transfer converters. At the converter start up calibration is done through measuring the L D v out (t) M(t) V in (t) C(t) C R bleeding load Controller and Current estimator logic Fig 2.5. Digitally measured output current voltage drop in the presence of bleeding resistor. Any further load current is estimated by using this calibration process and added to LUT. The system developed for this purpose is shown in Fig.2.5. First M(t) signal is set to zero, so the voltage drop during switch on time is due to the bleeding resistor only. Current passing through the bleeding resistor I u is equal to: v1 = I u t Cout (2.2)

25 14 t is the time between two consecutive sampling of ADC. Any new load current value, I new, can be estimated using Eq (2.3) by relating v1 to new voltage drop v2 introduced by the adding the load transient: v2 v1 = I new + I u I u = I out,n (2.3) However, the drawback of this method is that it is sensitive to capacitor value and is designed for a specific converter. Due to all these issues design of a robust self-tuning hardware efficient output current estimator is challenging Previous Art for Designing Fast Transient Response, Minimum Deviation Controller for Indirect Energy Transfer Converters. Nonlinear controllers which improve performance during load transients can be generally divided in two groups: time-optimal and minimum-deviation controllers. In this section, these two methods for indirect energy transfer converters are studied. Benefit and drawback of each of them are also discussed. Time-Optimal Controller (TOC) This method utilizes the principal of charge balance of output capacitor to achieve near optimal transient response performance [57]. A digitally implemented algorithm predicts the minimum number of switching cycles and their corresponding duty ratios to recover the output voltage back to its steady value once load transients occur. Main waveforms of a boost converter during operation of the proposed controller for light to heavy transient are shown in Fig 2.6.

26 15 On-off switching sequence is calculated based on the principal of capacitor charge balance which states that the amount of charge taken from the capacitor should be equal to the amount of charge added to the capacitor at the end of transient period. T ON T OFF I load v out (t) V ref I ind (t) I d (t) C(t) Fig 2.6. Transient response of boost converter under TOC. Top to bottom: I load is the load current, v out(t) is the output voltage, V ref is the reference voltage, i ind(t) is the inductor current, i d(t) is diode current and c(t) is the gating signal to the main switch.

27 16 The resulting controller provides rapid transient response in only one on-off switching sequence of the main switch as follows: once transient happens, for example a light-to-heavy transient, inductor starts to ramp up with the highest slew rate possible which is, v in L by keeping the switch on for the calculated TON time and the then decreases by its highest slew rate possible which is v out v in L during calculated TOFF time of the switch. Once this process is completed, linear controller is initialized to the new steady value for the new load condition. At the end of the transient, the charge delivered to capacitor is identical to the charge the capacitor has delivered to the load but at the cost of a large overshoot of the inductor current, imposing stress on the circuit components. This excessive current makes up for the lost charge of capacitor during discharging phase. Also, due to the overcharging of the inductor, the output voltage deviation is not the minimum possible. This method heavily relies on very accurate and complex calculations of the on and off times of switch since there is not access to the output current of the boost converter. Any deviation from the theoretical values causes significant error leading to stability problems. Minimum Deviation Controller for Indirect Energy Transfer Converters. Minimum deviation controllers are designed to minimize voltage deviation during transients leading to possibility of using smaller components and possibly reducing the current stress on the converter components compared to the stress caused by time-optimal method. In the method introduced in [56], minimum deviation of the output voltage is achieved by keeping the main switch on until the inductor current rises to its new steady state value and then the next switching sequence is precisely calculated based on the trajectory of inductor current and output voltage in state-space domain, as shown in Fig.2.7.

28 17 Trajectories are chosen such that a threshold voltage for current and voltage drop are met. In this method minimum voltage deviation is computationally imposed on the converter. The conceptual Fig 2.7 is shown for light to heavy transition. The drawbacks of this method are high computational cost and parameter dependence, which reduce the robustness of the controller. Fig 2.7. Operation of the programmable-deviation controller in a boost converter for light to heavy load transient (maximum switching frequency is limited). Inductor and load current (top), output voltage (middle), and state-plane representation of the voltage and current (bottom). System parameters are: V in=12 V, V out=48 V, I load from 1 A to 4.5 A.[56]

29 18 In [58], a modified near minimum deviation controller is proposed for indirect energy transfer converters, such as flyback. Flyback converter characteristic for stackable applications is shown in fig 2.8. Similarly to boost, Lm is disconnected from output for a part of switching cycle giving rise to the same control challenges. This idea is inspired by the methods developed for minimum deviation controller for direct energy transfer converters [44],[45]. However unlike direct energy transfer converters, it is not possible to leave the switch on or off to reach the valley/ peak point due to lack of information about output current. So, the nonlinear controller is activated during transients and keeps the main switch on with the largest duty ratio. This process of turning the L in i out1 (t) i out (t) v out (t) V in1 L m1 C 1 (t) C' 1 (t) MS 1 SR 1 V in (t) i out2 (t) i c (t) V in2 C out R load L m2 C 2 (t) C' 2 (t) MS 2 SR 2 Controller Fig 2.8. Schematics of stackable flyback converter in closed loop system.

30 19 switch on and off continues and at the discharging phase of inductors the polarity of output capacitor current is monitored by the controller, the key waveforms are shown in Fig 2.9. If polarity of the current is negative it means that the output capacitor is still discharging i.e. the inductor current is still smaller than the load current. This process of checking polarity continues during switch off-time for time period labeled as T off in Fig 2.9. Once a positive polarity of the output capacitor current is detected which means that the inductor current is larger than the load current, the nonlinear controller ceases and, PID controller starts to gradually recover the residual capacitor charge. This exchange of control with PID decreases overshoot of voltage but since the switch is on for a smaller time than switching period, deviation is not the minimum possible. Still it is significantly better than that of a PID controller alone.

31 20 Non-linear controller region v out (t) V ref I load i out (t) i c (t)<0 T on MS 1,2 T off SR 1,2 Fig 2.9. Flyback main waveforms during transient for the implementation of nonlinear controller. Top to bottom: v out(t) is the output voltage, V ref is the reference voltage, I load is the load current, i out(t) is the output current, i c(t) is output capacitor current and MS is the gating signal to the main switch, SR is gating signal to secondary switch.

32 21 3. A Hardware Efficient Self-Tuned Output Capacitor and Current Time Estimator for Indirect Energy Transfer Converters This chapter introduces a novel hardware-efficient auto-tuned estimator for the output capacitor current and time-constant. It is designed for indirect energy transfer converters, and is based on the well-known principle of the utilization of an auxiliary RC circuit for time constant matching. To provide accurate current measurement while avoiding complex calculations/hardware usually existing in other auto-tuned methods, the reconstruction of the time constant is performed through a simple detection of polarity of the slope of the estimator resistor voltage during the main switch on. The effectiveness of the estimator has been verified though simulation and experiment with a 20 W boost-based prototype, demonstrating less than 5% of error in the instantaneous current and time-constant measurements Principal of Operation Self-tuning current estimator is accomplished by addition of an adjustable RC branch in parallel with the output capacitor of boost converter, as shown in Fig Load and inductor currents estimation is achieved using the principle shown in [59]. That is, once the time constant of the RC estimator circuit and output capacitor of Fig3.1 are matched, the voltage waveforms across the resistors of these two branches will have identical wave-shapes. Assuming small output voltage

33 22 ripple and C out >>C adj, the load current can be found based on voltage across R adj during the diode off period, as shown by Eq (3.1)-(3.4): i load (t) = i esr (t) i adj (t) (3.1) i load (t) = v esr(t) R esr v adj(t) R adj (3.2) v esr (t) R esr = v adj(t) R adj C out C adj (3.3) i load (t) = v esr(t) R esr v adj(t) R adj = v adj(t) R adj (1 + C out ) v adj(t) C out (3.4) C adj R adj C adj Where v esr (t) is voltage over R esr and v adj (t) is voltage over R adj. In Eq 3.4, current passing through adjustment branch can be neglected due to the fact that current passing through adjustment branch is much smaller than the one going through the output branch. For a properly matched filter, in steady state, the voltage drop across R adj during the diode turnoff time, is directly proportional to the load current and, has a constant value, i.e. has a zero slope. However, as it will be shown here, when the time constants are mismatched, the estimator resistor voltage during the diode off time is not constant and will have either positive or negative slope, this phenomenon is used in the auto-tuning process. The analysis and implementation below are shown for the boost converter example and can be easily extended to other indirect energy transfer topologies.

34 23 L v x (t) D i d (t) v out (t) V in (t) C(t) i esr (t) i adj (t) i Added RC branch load (t) C out v esr (t) R esr R load C adj v adj (t) R adj Estimator logic and circuitry H DPWM d[n] Compensator e[n] ADC Hv out (t) Vref i load (t) recontruction Fig 3.1. Boost converter including conventional PID, adjustment RC filter and current estimator As will be shown through the analysis, during the time when the output capacitor is not being charged by the inductor, i.e. during the diode off time from Fig. 3.2, the slope of the voltage across R adj, i.e. v adj (t), can be monitored to inspect the matching of the time constants of the two RC branches. In particular, the slope of the v adj (t) will be positive, zero or negative if the time constant of the RC estimator is smaller, equal or greater than that of the output capacitor respectively. To understand how the mismatch of RC estimator time constant affects v adj (t) and to find the matching time constant, current and voltage equations for the circuit of Fig. 3.2 is solved taking

35 24 L D i d (t) v out (t) i esr (t) i adj (t) V in (t) C(t) C out v esr (t) C adj R load R esr Radj Fig.3.2. Schematics of output filter and output current for indirect energy transfer converters into account that the current id(t) is passing through output branches of the converter. Solving the circuit equations for diode off time considering simplifications R esr R load R adj and C adj C out results in: I adj = v out(t) R load R esr (e R adj t C adj R adj (1 C outr esr C adj R adj ) 1 C out R esr + 1) (3.5) Complete derivation of equation can be found in the appendix A. However, in the signal processing part of sensor and during the time when diode is turned off, i.e. during switch on time, the information needed for tuning the resistor is v adj (t) rather than the i adj (t). The reason is that the voltage is a much simpler quantity to measure than the current. Transformation of current to voltage is simply done through (3.6) by multiplying R adj. For the sensor implementation, information about the voltage is not useful alone, to tune the circuit, information about the slope of the voltage is also needed. Slope of the voltage of v adj (t) can be calculated as the derivative of Eq (3.6) shown in Eq (3.7):

36 Voltage (v) Voltage (v) Voltage (v) 25 v adj (t) = R adj i adj (t) v out(t) R load t R esr (e C adj R adj (1 C outr esr C adj R adj ) 1 C out R esr + 1) (3.6) dv adj (t) dt = R adjdi adj (t) dt v out (t) (e t C adj R adj (1 C outr esr R load C adj R adj C adj R adj ) 1 C out ) (3.7) From Eq (3.7) it can be seen, if τ adj (R adj C adj ) > τ esr (R esr C out ) the slope of v adj (t) during switch on time is negative and if τ adj < τ adj the slope is positive, as shown in Fig Accuracy of this analysis is also verified through PSIM simulation shown in Fig In the simulation an RC filter is added to output branch with varying resistor values and, the results matches the prediction of Eq Vesr 0.00 (a) 0.02 Vadj 0.00 (b) 0.02 Vadj (c) 1e-3 Fig 3.3. Comparison of different slope with respect to v esr (t) for diode off time period; (a) Waveform of v esr (t). (b) τ adj (R adj C adj ) > τ esr (R esr C out ), slope of v adj (t) is negative (c) τ adj (R adj C adj ) < τ esr (R esr C out ) slope of v adj (t) is positive during diode off time

37 26 (a) (b) (c) Fig 3.4. Simulation of v esr (t) and v adj (t); (a) Waveforms of v adj (t) and v esr (t) when time constants are matched (b) Waveforms of v esr (t)and v adj (t) when τ adj (R adj C adj ) < τ esr (R esr C out ) (c) Waveforms of v esr (t) and v adj (t) when τ adj (R adj C adj ) > τ esr (R esr C out ) 3.2. Practical Implementation Fig. 3.5 shows a practical realization of the introduced RC estimator for a boost converter. The boost converter is controlled with a mixed signal linear control scheme [12]-[18], where the digital value of error, e[n], is sent to the digitally implemented PID compensator, generating the duty cycle, d[n] for output voltage regulation. The DPWM block then creates the gating signal c(t) for the boost gate driver. The RC estimator circuit is created by adding a small capacitor C adj and an adjustable digital potentiometer, R adj. The circuit for observing the slope of v adj (t) and tuning of R adj is shown in Fig The practical implementation includes the slope identification & tuning circuit which sets the value of R adj and, current estimation logic that estimates the value of the load current based on the value of v adj [n] from ADC2. The RC tuning circuit and current estimator are implemented at the cost of an extra sample and hold block, a comparator and 2 stages of Op- Amps, which amplifies the voltage across R adj, and a simple digital logic. It should be noted that

38 27 the ADC2 shown in Fig3.5 can be eliminated by time-multiplexing the ADC for the output voltage measurement already existing in the circuit. Slope Identification and Tuning of Estimator Time Constant To identify the sign of the slope of v adj (t), a sample and hold circuit (S/H) is used. One point of v adj (t) at the beginning of DT s (T s is switching period) period is sampled and held and then later during the same time interval this sample is compared to the current value of v adj (t) as shown in L v x (t) D i d (t) v out (t) V in (t) C(t) C out i esr (t) v esr (t) R esr R load i adj (t) C adj v adj (t) R adj Op-amp S/H Op-amp Current Estimator ADC 2 Tuning circuit & Slope identification Tuning circuit & Slope identification Comp Current Estimation logic DPWM d[n] Compensator e[n] ADC v out (t) i load (t) V ref Fig 3.5. Implementation of current sensor circuit Fig 3.5. The slope identification circuitry is shown in Fig. 3.6, the slope identification has been shown for the case when τ adj (R adj C adj ) > τ esr (R esr C out ) which translates to a negative slope.

39 28 Based on the v adj (t) waveform of Fig 3.6, for example for the comparator, the output value provided to slope identification block (SI) is high showing that sample and hold output value is bigger than the current voltage value being compared to, which in turns translates to a negative slope. If the output of comparator is zero, it means that the slope is either positive or the matching condition has occurred. L v x (t) D v out (t) v in (t) i esr (t) C out i adj (t) v adj (t) X X v esr (t) R esr C adj v adj (t) R adj S/H Two samples taken from same point during switching cycle Comparator - input PID X Slope identification (SI) Comparator output Comparator + input t Fig 3.6. Implementation of slope identification. Comparator output is used for tuning purposes and if the comparator output is high, a negative slope is detected; thus, digital pot reduces resistance value. Similarly, if the comparator output is low it means that the R adj is either smaller than the desired value or matches the match condition. Since the comparator output zero leaves it undetermined whether the slope is positive or zero, every time that the calibration process starts, digital pot is set to the largest value. So, at the beginning of the calibration process, the output of the comparator is always detected high and pot reduces the resistor value. This process continues until comparator output hits zero for the first

40 29 time during the tuning process. This value is picked as the matching point. Process of decreasing pot resistance to move the slope toward zero is shown in Fig 3.7. L v x(t) D v out(t) v in(t) i esr(t) C out i adj(t) C adj v adj(t) Original negative slope without tuning v esr(t) R esr v adj(t) R adj S/H X X New less negative slope after first round of tuning PID Comparator output Tuning logic t Fig 3.7. Tuning process to find the best match for R adj Load Current Estimation After value of R adj is set following the condition of the time constants match, the voltage over R adj, v adj (t), is sampled during off time of the diode and utilized for estimation of the load current. As described earlier, through Eq.(3.1)-(3.4), the load current is directly proportional to v adj (t) where the constant of proportionality depends on sometimes unknown values of C out and R esr. In some applications, such as utilizing the current estimator for load transient detection and providing load current information for the non-linear transient controller the information about relative (rather than exact) load current value is often sufficient. However, in order to find the exact value of the load current, essential for applications such as on-line efficiency optimization, it is necessary to know the exact value of C out. The output capacitor value can be found by applying a

41 30 known load step and using Eq.4 as described in [10]. Once digital equivalent of v adj (t) is provided by the ADC2, the current estimator digital logic computes the load current. Inductor Current Estimation In addition to estimating the load current, the estimator circuit is capable of estimating the average inductor current. The procedure is similar to load current estimation with one major difference: v adj (t) is ideally sampled in the middle of the diode on time. At this point in time, inductor current is equal to its average value and is charging the output capacitor. The current equation changes from Eq. 3.1 to Eq. 3.7: i load (t) = I L,avg v esr(t) R esr v adj(t) R adj I L,avg v adj(t) R adj C out C adj (3.7) i load (t) is already measured during diode off time and is stored. So, by sampling the v adj (t) in the middle of diode on time and providing the digital sample to logic unit the average inductor current can be found. It is worth mentioning that following the same method and sampling v adj (t) at the end of diode on period, the inductor peak and ripple current values can be found, as well. Transient Detection If any transient occurs during operation of the converter, the estimator is able to detect it instantaneously. This is one of the main advantages of this sensor which can facilitate design of nonlinear controllers for various types of indirect energy transfer converters. In the boost converter, during main switch off times, current passing through diode is equal to instantaneous inductor current. During diode off time the current passing through it is zero. In both cases, there cannot not be a sudden change in the value of diode current when a load transient happens because inductor current cannot change instantaneously. Therefore, any change in the load current is

42 31 detected within the same clock cycle by a voltage drop/peak across v adj (t). This drop or peak is proportional to the new load value. The load transient detection is performed through monitoring v adj (t) value and detecting any sudden voltage drop or overshoot across it. Simulations of Figs 3.8 and 3.9 show the effect of sudden load current change on the v adj (t). An abrupt change in v adj (t) value is observed. Fig 3.8. Boost converter main waveforms during load transient. Top to bottom: I load is the load current, V out is the output voltage, v adj (t) is the voltage over adjustment branch resistor and I ind is the inductor current

43 Fig 3.9. Zoomed in version of Fig

44 33 4. Digital Auto-tuned Near Minimum Deviation Controller for Indirect Energy Transfer DC-DC Converters. In this chapter a robust, hardware-efficient auto-tuned digital controller is introduced applicable to various hard switching dc-dc converters, including indirect energy transfer topologies. Unlike existing fast transient controllers for indirect energy transfer converters, the controller achieves fast transient response and practically minimum deviation of the output voltage without depending on information about converter parameters, i.e. inductor and output capacitor values. This is achieved by utilizing an auto-tuned non-linear controller that, based on the load-step information during a transient, finds the switching sequence for the converter to ramp up/down the inductor current to its new steady state average value in a single on/off switching action Principal of Operation The hardware efficient digital controller introduced here consists of a linear conventional controller for output voltage regulation during steady state and a nonlinear digital controller that takes over the control of switching actions during load transients. As shown in simplified block diagram of Fig4.1 proposed controller consists of two parts: i) one PID controller for steady state regulation and ii) one nonlinear module for transients. The controller has access to the output current through the use of a lossless output current estimator circuit shown in chapter 3. The linear part of the controller operates during steady state and regulates the output voltage using a PID

45 34 compensator. Based on the load current transient measured using the current sensor, the nonlinear controller calculates the switch turn on/off time needed for the inductor current to ramp up/down to its new steady state value. This minimizes the time required to bring inductor current to its new steady state value and allows for recovery from load transients after one switch on or off period, also minimizing output voltage drop/overshoot during load transients. Once the inductor current value matches its new steady state value, the linear PID based controller takes over the converter operation. This makes the capacitor voltage reach its valley/peak point in the following switching cycle as shown in Fig 4.2. After this point the capacitor charge is recovered by the inductor current using linear controller. The linear controller gradually makes up for the lost charge of the capacitor, providing a smooth recovery. In order to have a smooth transition from the non-linear controller L D v out (t) V in (t) c(t) C out R esr Rload C adj R adj Current Estimator H d st [n] PID Compensator Steady State mode Hv out (t) DPWM e[n] ADC d tr [n] Non_linear compensator V ref Transient Mode Fig 4.1: Boost converter including both PID and nonlinear controller

46 35 to PID operation and to avoid any secondary overshoots or voltage drops during the mode transition, the PID integrator is reinitialized based on a self-updating look-up-table right at the point of the control mode transition, to allow for the PID to start operation with a duty cycle required for the steady-state operation with the new load. The key waveforms of a boost converter during operation of the proposed digital controller for light to heavy are shown in Fig Based on the operation of the sensor once a transient is detected, new value of the load current is calculated, while the previous value of the current has already been stored. Eq 4.1 shows the value of i Load. i Load (t) = i Load New (t) i Load old (t) (4.1) The relationship between i Load,L (t) (load current during light load) and average I ind,avg,l (average inductor current for light load) is shown in Eq 4.2 I ind,avg,l = v out(t) R load D = i Load,L(t) D (4.2) Where R load is the load and D is the off time of the switch during the switching cycle. During a light to heavy transient, the relationship between the new average inductor current for heavy load, I ind,avg,h, and the previous average inductor current is shown in Eq 4.3 I ind,avg,h I ind,avg,l = i Load,H(t) D i Load,L(t) D (4.3) So the relationship between i Load (t) and I ind,avg is straight forward and is equal to Eq 4.4 I ind,avg = i Load(t) D (4.4)

47 36 Having known the relationship between the change in load current detected by the sensor and the corresponding change in average inductor current, the next step is to find out the minimum time needed to keep the main switch on to get to the new steady state inductor current. The relationship between the instantaneous inductor current and the switch on time is shown in Eq 4.5 i ind (t) = v in(t) t (4.5) L Where v in (t) is the input voltage value, L is the inductor value. The time needed to charge up the inductor to the new value is combination of Eq (4.4) and (4.5): Linear controller Non-linear controller region Linear controller I load I ind,avg,l v out (t) i ind (t) c(t) ΔI LH Δt LH set by nonlinear controller I ind,avg,h V ref Fig 4.2. Boost converter when operated with proposed controller. Top to bottom:load current I Load ; steady-state and transient output voltage v out (t); instantaneous inductor current i ind (t); average inductor currents I ind,avg,h, I ind,avg,l during steady state for heavy and light load conditions respectively; main switch gating signal c(t).

48 37 i ind (t) = i Load(t) D = v in(t) L t (4.6) From Eq 4.6 it can be observed that i Load (t) and t are related and once the i Load (t) is provided, the time needed to move to next state can be calculated as Eq (4.7) and (4.8): t = i Load(t) D L v in (t) (4.7) t = i Load(t) D K LH (4.8) Similarly, for a heavy to light load transient the time required to bring inductor current to the new steady state value by keeping the main switch off is found as Eq (4.9) and (4.10): t = i Load(t) L D V out v in (t) (4.9) t = i Load(t) D K HL (4.10) In the practical implementation, the required switch on/off period to bring inductor current to its new steady state value, t, is found by measuring I Load, utilizing available information about the duty cycle D from the digital controller, and then by simple multiplication of the load step current by an auto-tuned, self-updating multiplication factor (KLH/HL). Therefore, compared to the existing solutions [62],[63], the introduced controller is significantly less computationally intensive, parameter-insensitive and hardware efficient.

49 Practical Implementation Hardware implementation of the controller is shown in Fig.4.3 and the control algorithm is described through the flowchart of Fig.4.4. In order to produce the gating signal for the switch, a digital pulse width modulator (DPWM) is used. It receives d[n] and produces d(t). This d[n] value can be obtained from either the linear controller d st [n] or the nonlinear controller d tr [n] based on whether it is in steady state or transient condition. During steady state, using ADC, the output is sampled once per switching cycle and the error information is sent to the controller. Steady state duty value is stored each cycle during operation of the system, to update the value of the lookup table. Current information is available to the system through current estimator sensor. Current obtained by the sensor is checked with its previously stored sample and if the difference between the two L v x (t) D id(t) v out (t) V in (t) c(t) SW i esr (t) C out v esr (t) R esr i adj (t) R load C adj v adj (t) R adj H SS LUT for Reinitilization Current estimation and transient detection DPWM Tr Controller Mux d st [n] PID Compensator e[n] Tr ADC Load_current H v out (t) d tr [n] Non_linear compensator V ref Auto-tuning VPD Steady state detection SS Valley point detection VPD Fig 4.3. System implementation including linear and nonlinear controller plus current estimator sensor.

50 39 values is larger or smaller than a threshold, the nonlinear controller is started while the PID compensator goes into a halt condition. During the operation of the nonlinear controller, the switch remains on/off for the duration obtained from Eqns 4.7 and 4.9 for light to heavy or heavy to light transients, respectively, to ramp up/down the inductor current from its previous value to its new steady state value. Once the inductor current reaches its new steady state value, the nonlinear controller stops operation and the PID controller restarts its operation. During steady state operation, the ADC samples the output voltage once per switching cycle and the error information is sent to the PID compensator. Once the steady state is detected, the duty value corresponding to the load current is stored in a lookup table. At each load transient, once the nonlinear controller brings the inductor current to its new steady state value, this look up table is used to re-initialize the PID integrator such that it starts operation with the duty cycle required for the steady-state operation with the new load current. Moreover, at the moment of the mode transition to steady-state operation, the DPWM counter restarts from its previous value before the transient as shown in [18] to avoid any additional secondary voltage over/undershoots. To achieve a simple, parameter insensitive implementation of the non-linear controller, the converter parameter-dependent components that are used to find the optimal value of t LH and t HL once a load transient occurs are lumped into K HL and K LH as shown in Eqns. 4.8 and Here, the auto-tuning process for finding K LH is explained for the case of light-to-heavy load transients. For the heavy to light transients a similar procedure is followed except that the valley point is replaced by the peak point.

51 40 If by the end of non-linear controller operation, inductor current is at its correct new steady-state value, the output voltage will reach its valley point right after the linear controller starts operation as shown in Fig If the inductor current is lower, the output voltage will keep falling until the PID reacts and brings the inductor current up to its new steady state value. Every time a transient happens, after the linear controller takes over the control operation, the Valley point detection block checks whether the valley point condition is met or not in the following switching cycle. If the valley point is not detected at the first cycle, V 2 < V 1, as shown in Fig. 4.5(b), the lumping multiplication factor, K LH of Eq. 4.8, is lower than its correct value and the auto-tuning block of Fig4.1 increments K LH. Otherwise K LH is decremented. The flow diagram of K LH is shown in Fig4.6. It is worth mentioning that the presented auto-tuning process is most suitable for applications of the boost converter where fast variations of input voltage are not expected, such as battery operated devices. If input voltage transients are expected, a modified version of the autotuning process with feed-forward of input voltage can be utilized.

52 41 initiate Current sensor Linear controller Update steady state load current No Yes transient No Enable non-linear controller Sample output in each cycle Get Δi load New duty ratio Calculate Δt on/ off time Update steady state duty ratio Turn on/off switch Re-initialize linear controller Fig 4.4. Flowchart of controller algorithms.

53 42 Linear controller Non-linear controller region Linear controller I load I ind,avg,l i ind (t) I ind,avg,h v out (t) Valley Point V1 V2 (a) Linear controller Non-linear controller region Linear controller I load I ind,avg,h I ind,avg,l i ind (t) v out (t) V1 V2 Valley Point (b) Fig 4.5. Valley point detection for auto-tuning: value of V 2 is always checked against value of V 1. a) KLH= L v in (t) b) KLH< L v in (t)

54 43 initiate Enable current sensor Enable linear controller Update steady state load current No Yes transient No Nonliner Controller ON Yes Nonlinear Controller still on No Measure Two consecutive v out values No V2<=V1 Yes K=K-1 K=K+1 Fig4.6 Auto-tuning procedure flowchart 4.3. Simulation Results In this section to verify system operation, combination of current sensor and digital controller has been simulated. Two different indirect energy transfer topologies have been chosen for the simulation. First, a boost converter and then a flyback converter.

55 44 Simulation Result for Boost Converter Fig 4.7 shows simulation result for 1.5 V to 3.3 V boost converter having 5 µh inductor and 20 µf capacitor. The converter is controlled with two different control methods. First a PID controller only which is well-tuned for light-to-heavy transitions and, second a mixed linear/nonlinear controller designed for light to heavy transition. Fig 4.7 shows boost converter when is run only with linear controller for a load change from 0.25 A to 0.5 A. Fig 4.7. Digital controller (only PID) for boost converter. v out is the output voltage, i ind is inductor current, i load is the load current and, c(t) is gating signal to the main switch. Fig 4.8 shows response when the proposed nonlinear controller is added. The results confirm the operation described in previous section. As it can be seen once a transient occurs the nonlinear controller calculates the on time of switch and keeps the main switch on causing inductor current to ramp up. Once this process is finished operation is back to PID for a smooth charge recovery

56 45 eliminating potentially secondary transients. Comparing Figs 4.7 and 4.8 shows that the voltage drop is reduced by half with introduced controller, allowing for similar reduction of output capacitance and the response time is also reduced by 30% Fig 4.8. Digital controller (nonlinear with PID) for boost converter. v out is the output voltage, i ind is inductor current, i load is the load current, time_on is the time calculated during transient to keep the main switch on, c(t) is gating signal to the main switch.

57 46 Fig 4.9 shows the addition of auto-tuning process for successive light to heavy transients for the same converter parameters. Initial value of parameter K has been set to a different value than the one resulting in optimal response to trigger the auto-tuning block action. As it can be seen from Fig 4.9, voltage drops are more prominent at the beginning of the tuning process for light to heavy transitions due to the deviation of converter parameters from their nominal values. Initial conditions are set such that the auto-tuning process reduces K. Once the best match for the valley point is found, K starts to oscillate between the two successive values and the voltage drop is Fig4.9. Addition of auto-tuning system for successive light-to-heavy transitions. v out is the output voltage, K is the auto-tuning parameter V d the difference of two successive output voltage sample and is equal to V2-V1. 6 s.v minimized. The best match according to simulation is K= which results in a switch H on time of 1.6 µs matches the theoretical calculation of 1.66 µs. The last set of simulations shown in Figs 4.10 and 4.11 is for a nonlinear controller designed to handle heavy to light as well as light to heavy load transients. The converter parameters have remained the same and the load current changes from 0.5 to 0.25 A.

58 47 Fig 4.10 shows a heavy to light transition when only the PID controller is used.fig 4.11 shows Fig Heavy to light transition from 0.5 to 0.25 A with PID controller. v out is the output voltage, I load is the load current, i ind is the inductor current, c(t) is gating signal for the main switch. same converter in closed loop when nonlinear controller is activated. Overshoot is reduced by 30% Fig Heavy to light transition from 0.5 to 0.25 A including both nonlinear and PID controllers. v out is the output voltage, I load is the load current, i ind is inductor current, c(t) is gating signal for the main switch.

59 48 and response time is reduced by 20 µs Simulation Result for Flyback Converter To further verify applicability of the proposed method to other types of indirect energy transfer dc-dc converters, simulations are also performed for a flyback converter. Flyback converter is a very popular topology, possessing galvanic isolation, which is capable of stepping down or up the input voltage. The simulation is carried for a 48 V to 12 V flyback with 40 µf output capacitor and magnetizing inductance 200 µh and a transformer ratio of 4:1. A schematic of the flyback is shown in Fig It is similar to the boost converter in terms that the inductor and output capacitor are not charged simultaneously. The magnetizing inductor is charged up during the first portion of the switching cycle when the main switch is on. Its energy is transferred to the output capacitor through secondary widening in second part of switching cycle. i out (t) i c (t) v out (t) I load V in (t) C out L m R load C(t) DPWM PID & nonlinear V ref H Hv out Fig Schematics of flyback converter in closed loop system.

60 49 Fig Simulation of closed loop flyback with only PID. v out is the output voltage, I load is the current passing through load resistance and I lm is the current of magnetizing inductor and, c(t) is gating signal for the main switch Fig Simulation of closed loop flyback with PID and nonlinear controller. v out is the output voltage, I load is the current passing through load resistance and i lm is the current of magnetizing current inductor and c(t) is gating signal Simulations have been carried for a nonlinear controller designed the for light-to-heavy transitions

61 50 (load changes from 2A to 4A). It can be seen that the addition of the nonlinear digital controller reduces voltage drop and response time. First simulations show transient response when only PID is used and the results are illustrated in Fig The improved response obtained with addition of the proposed controller to PID is shown in Fig The output voltage drop is reduced by 80% and response time is reduced by around 35%. The final set of simulations is the demonstration of auto-tuning for the flyback converter for successive light-to-heavy transitions and results are shown in Fig Fig Light-to-heavy transition auto-tuning process for flyback converter. K starts to oscillate between two values when the best match is found

62 51 5. Experimental Results To verify the performance of the introduced sensor and digital controller, boost power stage and controller board have been designed and tested on the setup shown in Fig 5.1.The digital controller is implemented with an FPGA development board. Also, a custom printed circuit board (PCB) consisting of boost power stage and current estimator has been designed. Boost Power Stage Current Estimator and Tuning Circuits Controller To FPGA Fig 5.1. Setup including power stage and controller

63 Current Sensor: Experimental Results The performance and functionality of the introduced RC matching method are verified with a 500 khz, 2.5 V to 5 V, 20 W boost experimental prototype. The prototype was build based on the diagram of Fig.3.5. The boost output capacitor has capacitance of 20 µf and an ESR of 6 mω, Cadj is 220 pf, and an 8-bit digital potentiometer of 1 kω max resistance is used. The digital controller and estimator logic are created using an FPGA based development board. To verify that equations correctly describe the slope of v adj based on the relation of the time constants, the waveform of v adj for three different scenarios where τ adj > τ esr, τ adj = τ esr and Fig.5.2. v adj (t) waveform for different time constant conditions. Top to bottom: v adj (t) when τ adj > τ esr (R adj = 1 KΩ), (200 mv/div). v adj (t), when τ adj = τ esr (R adj = 500 Ω),( 200 mv/ div). v adj (t), when τ adj < τ esr (R adj = 200 Ω), ( 200 mv/div). Time-scale is 1 µs/div

64 53 τ adj < τ esr are measured and shown in Fig 5.2 Different slope obtained with three different conditions perfectly match the simulations results of Chapter3. The waveforms of Fig 5.2 are =5 V Code[n] (a) =5 V Code[n] (b) Fig (a) Key waveforms of the digital controller and the self-tuning estimator. Top-to-bottom: v out (t) output voltage of the converter (500 mv/div), v x (t) switching node of boost converter (5 V/div), i load (t) current of load (500 ma/div), v adj (t) voltage across the estimator resistor (200 mv/div), R adj [n] digital pot code changes, changing resistor and stops when output of comparator is detected zero. (b) zoomed-in version

65 54 magnified using op-amps for better visualization. Fig 5.3 shows self-tuning process of R adj. Results show the effort of the auto-tuning circuit, i.e. the process of deciding what is the best match value of R adj. It can be seen that the tuning process is completed when the digital pot code remains constant, in this case at code 128 which translates to the resistance value of 550 Ω. =5 V Code[n] Fig 5.4. Waveforms of boost converter during inductor current reading. Top-to-bottom: v out (t) output voltage of the converter (5V/div),), i lnd (t) current of inductor (500 ma/div), i load (t) current of load (500 ma/div), v adj (t) voltage across the estimator resistor (2 V/div), I lnd -avg [n] digital current reading of inductor in ma. Time-scale is 10 µs/div Experimental result of Fig 5.4 shows the sensor reading for the inductor current. As was described in Chapter 3. By sampling in the middle of diode on-time, it is possible to find the value of average inductor current. The experimental results of Fig 5.4 for inductor current reading show the reading during steady state and light-to-heavy transition from A to 1.1 A. it can be seen that sensor

66 55 =5 V =0.8 A V Code[n] (a) =5 V Code[n] (b) =5 V =2 A 3V Code[n] (c) Fig 5.5. Zoomed in version of Fig5.4, (a) current reading in ma during light current steady state. (b) during transient and (c) during heavy load steady state is able to accurately measure the value of the inductor current both in steady state and during

67 56 transients as shown in Fig 5.5. Adjustment voltage is magnified through op-amps for more accurate measurement Experimental results of Fig.5. 6 further verify the transient performance of the estimator. It can be seen that the current estimator constantly reads current both during steady state and transient. Once the light to heavy load transient happens, v adj drops, and the value of this drop is proportional to Fig Key waveforms of the transient and steady state current estimator. Top-to-bottom: v out (t) output voltage of the converter (500 mv/div), v adj (t) value of adjustment branch voltage after matching process during steady state and transient operation (200 mv/div), i Inductor (t) current of inductor (500 ma/div), I Load value of load current (500 mv/div), digital value of current value of current in ma is updated cycle by cycle during steady state and transient (light to heavy) operation. Time scale is 2 µs/div

68 57 new value of current. This value is fed to ADC 2 of Fig. 3.5 and converted to the equivalent current value Digital Controller Experimental Results The performance and functionality of the introduced fast response near minimum deviation controller are verified with a 500 khz, 1.5 V to 3.3 V, 6 W boost experimental prototype. The prototype was build based on the diagram of Fig.4.3. The boost output capacitor has capacitance of 20 µf and an ESR of 6 mω, Cadj is 1nF, and an 8-bit digital potentiometer of 1 kω max resistance is used. Inductor value is 5 µh. The digital controller and estimator logic are created using the same FPGA based development board as for the previously described setup. Fig 5.7 shows result of load transient from 0.25 A to 0.5 A, when only PID is present. It can be seen that the total voltage drop is 420 mv and transient recovery time is about 100 µs. 420m V 3.3 V 250m A 750m A 500m A 1.5 A c(t) (a)

69 58 420m V 3.3 V 250m A 750m A 500m A 1.5 A c(t) (b) Fig.5.7. (a) transient response of boost converter to a 0.25 to 0.5A employing conventional converter. Top to bottom: v out(t), is the output voltage 200 mv/div, I load is step current, 500mv/div, i ind (t) is inductor current, 2v/div and, d(t) is the gating signal to main switch,5v/div, time scale is 20µs/div. (b) zoomed in version Fig.5.8 shows the response of system when the nonlinear controller is added to the system and the conventional controller is reinitialized. The same load step is applied to the converter and it can be seen that the voltage drop is reduced by half and recovery time is less than 50 µs. This is significant improvement compared to only PID controller. Experimental results shown in Fig.5.9 demonstrates the importance of the PID re-initialization process. They show response of the controller without proper re-initialization of the controller. It can be seen that a secondary transient happens.

70 59 200m V 3.3 V 250m A 750m A 500m A 1.5 A c(t) (a) 200m V 3.3 V 250m A 750m A 500m A 1.5 A c(t) (b) Fig.5.8. (a) transient response of boost converter to a 0.25 to 0.5 A employing nonlinear converter. v out(t)is output voltage (200mv/div), I load is step current (500mv/div), i ind (t) is inductor current (2v/div), d(t) is gating signal (5v/div), time scale is 10µs/div. (b) zoomed in version

71 60 200m V 3.3 V c(t) 750m A 250m A 1.5 A 500m A Fig.5.9. Transient response of boost converter to a 0.25 to 0.5A employing nonlinear converter without proper re-initialization of PID. Top to bottom: v out (t) is output voltage (500mv/div), i ind (t) is inductor current (1v/div), I load (t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 100µs/div. A set of experimental results verifies auto-tuning function of the controller as shown in Fig For the auto-tuning process, as described in section 4.2 earlier, valley point is detected and base on the algorithm of Fig 4.6 the constant K is calculated. In the experimental result shown in Fig 5.10 the auto-tuning algorithm gradually increases tuning constant K which was intentionally set to a lower value than the nominal. Experimental results are obtained for a light-to-heavy transient of 0.25 A to 0.9 A.

72 61 Code[n] Fig Auto-tuning process of K for 0.25 to 0.9A load transition. Top to bottom: v out (t) is output voltage (500mv/div), i load (t) is load current (500mv/div), i ind (t) is inductor current (2v/div), code[n] is value of updated K in each cycle, v x(t) is gating signal (5v/div), time scale is 100 ms/div. Fig 5.11 shows zoomed-in waveforms of Fig 5.10 for successive light-to-heavy transition of 0.25 A to 0.9 A during K adjustment.

73 62 (a) (b) (c) (d) (e) Fig (a) updating K from 0 to 1, K=0 is equal to PID only, (b) transition K =1 to 2, from figure V2 is smaller than V1 K should be increased.(c) K =2 to 3, still K should be increment (d) K =3 to 4, optimum point, (e) K =4 to 3, K is larger than optimum value, for the next cycle K should be decreased

74 63 The result for the heavy to light transition for both conventional PID and the introduced controller are shown in Figs 5.12 and 5.13 respectively. Fig 5.12 shows the result when only PID is present 450m V 3.3 V c(t) 1.5 A 500m A 750m A 250m A Fig Transient response of boost converter to a 0.5 to 0.25 A step current employing only PID controller. Top to bottom: v out (t) is output voltage (500mv/div), i ind (t) is inductor current (1v/div), I load(t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 50µs/div. 100m V 3.3 V c 1.5 A 500m A 750m A 250m A Fig Transient response of boost convertr to a 0.5 to 0.2 A transient employing nonlinear and PID controllers. Top to bottom: v out (t) is output voltage (500mv/div), i ind (t) is inductor current (1v/div), I load(t) is step current (500mv/div), d(t) is gating signal to main switch (5v/div), time scale is 50µs/div.