Evaluating the switching frequency of a buck converter with a sliding mode control by a hysteresis band controller

Evaluating the switching frequency of a buck converter with a sliding mode control by a hysteresis band controller Bachelor’s Degree Thesis Submitted to the Faculty at the Escola Tècnica Superior d'Enginyeria de Telecomunicació de Barcelona de la Universitat Politècnica de Catalunya by Alex Nova Bellavista In partial fulfillment of the requirements for the DEGREE IN ELECTRONIC ENGINEERING Tutors: Domingo Biel Solé and Víctor Repecho del Corral Barcelona, June 2024 ETSETB – Degree in Electronic Engineering -2- -3Summary Resum______________________________ Aquest projecte se centra en l’estudi de diferents mètodes per calcular la freqüència de commutació de convertidors de potència controlats en mode lliscant. Per això, s'ha dissenyat un control en mode lliscant per a diferents sistemas, garantitzant una regulació robusta de la tensió de sortida. S'han considerat dues superfícies diferents i s'han comparat mitjançant simulacions. S'ha implementat experimentalment un convertidor buck amb control en mode lliscant per contrastar els resultats obtinguts per simulació. Finalment, s'ha trobat que el mètode piece-wise approximation és l'únic capaç d'aproximar amb precisió i simplicitat a tot el rang d'operació del convertidor. ____________________________Resumen Este proyecto se centra en el estudio de diferentes métodos para calcular la frecuencia de conmutación de convertidores de potencia controlados en modo deslizante. Para ello, se ha diseñado un control en modo deslizante para diferentes sistemas, garantizando una regulación robusta de la tensión de salida. Se han considerado dos superficies diferentes y se han comparado mediante simulación. Se ha implementado experimentalmente un convertidor buck con control en modo deslizante para contrastar los resultados obtenidos por simulación. Finalmente, se ha encontrado que el método piece-wise approximation es el único capaz de aproximar con precisión y simplicidad en todo el rango de operación del convertidor. Summary____________________________ This project focuses on the study of different methods for calculating the switching frequency of sliding mode controlled power converters. For this purpose, a sliding mode control has been designed for different systems, ensuring a robust output voltage regulation. Two different surfaces have been considered and compared by simulation. A buck converter with sliding mode control has been implemented experimentally to contrast the results obtained by simulation. Finally, the piece-wise approximation method was found to be the only one capable of approximating with accuracy and simplicity over the entire operating range of the converter. -4Acknowledgements First of all, I would like to express my gratitude to my TFG tutors, Domingo Biel and Victor Repecho, for their invaluable help and guidance throughout the project. Their knowledge and constant support, insisting until the last day so that everything went as planned, have been fundamental for the development of the work. On the other hand, I would like to thank my family, especially my mother, for putting up with me and helping me even in the worst moments. Finally, I would like to thank my girlfriend for her support not only during this project but also throughout my degree. -5Revision history and approval record Revision Date Description 1.0 02/05/2024 Document creation 1.1 18/06/2024 Error correction 2.0 20/06/2024 Revised after review 3.0 20/06/2024 Final version DOCUMENT DISTRIBUTION LIST Role Surname(s) and Name Student Alexandre Nova Bellavista Project Supervisor 1 Domingo Biel Solé Project Supervisor 2 Victor Repecho del Corral Written by: Reviewed and approved by: Date 02/05/2024 Date 20/06/2024 Name Alexandre Nova Bellavista Nome Domingo Biel Solé Role Project Author Role Project Supervisor -6Contents Summary ........................................................................................................................................................................ 3 Acknowledgements ................................................................................................................................................... 4 Revision history and approval record ............................................................................................................... 5 Contents ......................................................................................................................................................................... 6 List of Figures............................................................................................................................................................... 8 List of Tables ............................................................................................................................................................. 10 Abbreviations ............................................................................................................................................................ 11 1. Introduction ..................................................................................................................................................... 12 1.1. Project Overview and Work Goals ................................................................................................ 12 1.2. Requirements and Specifications .................................................................................................. 13 1.3. Methods and Procedures .................................................................................................................. 13 1.4. Work plan ................................................................................................................................................ 14 2. State of the art of the technology used or applied in this thesis ................................................ 18 2.1. What is Sliding Mode Control.......................................................................................................... 18 2.2. Applications of SMC in Power Electronics ................................................................................. 19 2.3. Problems with the Implementation of SMC .............................................................................. 19 3. Methodology / project development ..................................................................................................... 21 3.1. Design of the SMC ................................................................................................................................ 21 3.1.1. Control Design Procedure ....................................................................................................... 21 3.1.2. First Design Example: Simple Mathematical Model ..................................................... 24 3.1.3. Second Design Example: Buck Converter Model ........................................................... 28 3.2. Analysis of the Switching Frequency in Sliding Mode Control .......................................... 38 3.2.1. Methods .......................................................................................................................................... 38 4. Results ................................................................................................................................................................ 46 4.1. Example of Analysis ............................................................................................................................ 46 4.1.1. Case 1: Limit of the operating range................................................................................... 47 4.1.2. Case 2: Typical Operating Range .......................................................................................... 56 4.1.3. Conclusions ................................................................................................................................... 62 4.2. Experimental Part ................................................................................................................................ 63 4.2.1. Design of the Controller ........................................................................................................... 64 -74.2.2. Design the Circuit to Set the Hysteresis Value ............................................................... 69 4.2.3. Functionality of the SMC ......................................................................................................... 70 4.2.4. Results ............................................................................................................................................. 73 4.2.5. Conclusions ................................................................................................................................... 83 5. Sustainability Analysis and Ethical Implications .............................................................................. 86 5.1. Environmental Impact ....................................................................................................................... 86 5.2. Economic Impact .................................................................................................................................. 87 5.3. Social impact .......................................................................................................................................... 88 6. Conclusions and Future Work .................................................................................................................. 90 6.1. Conclusions ............................................................................................................................................. 90 6.2. Future Directions ................................................................................................................................. 93 Bibliography .............................................................................................................................................................. 94 Appendix A ................................................................................................................................................................. 95 -8List of Figures Figure 1. Work Breakdown Structure ............................................................................................................. 14 Figure 2. Example of the sliding surface. ....................................................................................................... 18 Figure 3. Control of the simple mathematical model. .............................................................................. 27 Figure 4. Behaviour of the signals of the simple mathematical model. ............................................ 27 Figure 5. Buck converter circuit. ....................................................................................................................... 29 Figure 6. Control of the buck converter model. .......................................................................................... 34 Figure 7. Behaviour of the buck converter model signals with voltage control............................ 35 Figure 8. Control of the buck converter model with current control. ............................................... 37 Figure 9. Behaviour of the buck converter model signals with current control. .......................... 37 Figure 10. Switching function behaviour. ..................................................................................................... 39 Figure 11. Plot of G(jω) and W(jΩ). ................................................................................................................. 42 Figure 12. Plot of G(jω) and W(jΩ) zoomed in. ........................................................................................... 42 Figure 13. Plot of J(ω) and the straight line -πb/(4c). ............................................................................. 44 Figure 14. Switching function for ∆ = 1.3. ..................................................................................................... 47 Figure 15. Error vs hysteresis value graph. .................................................................................................. 48 Figure 16. Error vs switching frequency graph. ......................................................................................... 48 Figure 17. Switching function for ∆ = 2.33.................................................................................................... 50 Figure 18. Error vs hysteresis value graph. .................................................................................................. 50 Figure 19. Error vs switching frequency graph. ......................................................................................... 51 Figure 20. Switching function for ∆ = 5.85.................................................................................................... 52 Figure 21. Error vs hysteresis value graph. .................................................................................................. 52 Figure 22. Error vs switching frequency graph. ......................................................................................... 53 Figure 23. Error vs hysteresis value graph. .................................................................................................. 55 Figure 24. Error vs switching frequency graph. ......................................................................................... 55 Figure 25. Switching function for ∆ = 11.8.................................................................................................... 57 Figure 26. Error vs hysteresis value graph. .................................................................................................. 57 Figure 27. Error vs switching frequency graph. ......................................................................................... 58 Figure 28. Error vs hysteresis value graph. .................................................................................................. 60 Figure 29. Error vs switching frequency graph. ......................................................................................... 61 Figure 30. Multiphase prototype board. ........................................................................................................ 63 Figure 31. Controller PCB assembled. ............................................................................................................ 65 -9Figure 32. Main schematic of the SMC. ........................................................................................................... 65 Figure 33. Input connector schematic of the SMC. .................................................................................... 66 Figure 34. Enable drivers and conditioning schematic of the SMC. ................................................... 66 Figure 35. Surface creation schematic of the SMC. .................................................................................... 67 Figure 36. Comparator schematic of the SMC. ............................................................................................ 67 Figure 37. Simulation of the SMC with LTspice. ......................................................................................... 68 Figure 38. Circuit to set the hysteresis value. .............................................................................................. 70 Figure 39. Schematic of the hysteresis circuit. ............................................................................................ 70 Figure 40. Behaviour of the buck converter signals on the oscilloscope. ........................................ 71 Figure 41. Behaviour of the buck converter signals on the oscilloscope zoomed in. .................. 72 Figure 42. Line Regulation. .................................................................................................................................. 73 Figure 43. Load Regulation. ................................................................................................................................ 73 Figure 44. Equipment set up. .............................................................................................................................. 75 Figure 45. Behaviour of the signals with Vout =12 V and a measured switching frequency of 39.97 kHz. ................................................................................................................................................................... 75 Figure 46. Error vs hysteresis value graph. .................................................................................................. 76 Figure 47. Error vs switching frequency graph. ......................................................................................... 77 Figure 48. Error vs hysteresis value graph. .................................................................................................. 78 Figure 49. Error vs switching frequency graph. ......................................................................................... 78 Figure 50. Behaviour of the signals with Vout = 21 V and a measured switching frequency of 41.03 kHz. ................................................................................................................................................................... 79 Figure 51. Error vs hysteresis value graph. .................................................................................................. 80 Figure 52. Error vs switching frequency graph. ......................................................................................... 80 Figure 53. Behaviour of the signals with Vout = 3 V and a measured switching frequency of 40.22 kHz. ................................................................................................................................................................... 81 Figure 54. Error vs hysteresis value graph. .................................................................................................. 82 Figure 55. Error vs switching frequency graph. ......................................................................................... 83 Figure 56. Behaviour of the signals with Vout = 21 V and a 5 Ω load connected. The measured switching frequency was 32.78 kHz. ............................................................................................................... 84 Figure 57. Behaviour of the signals with Vout = 12 V and a 5 Ω load connected. The measured switching frequency was 40 kHz. ..................................................................................................................... 85 -16Table 3. Work Package 3. Project: Evaluation of switching frequency in hysteresis control WP ref: WP3 Major constituent: HW Sheet 3 of 4 Short description: Experimental implementation of the proposed control system. Planned start date: 21/03/2024 Planned end date: 01/06/2024 Start event: End of simulations End event: Obtained experimental results Internal task T1: Implementation of the buck converter. Internal task T2: Measuring the real switching frequency for different hysteresis values. Deliverables: Dates: Table 4. Work Package 4. Project: Evaluation of switching frequency in hysteresis control WP ref: WP4 Major constituent: Documentation Sheet 4 of 4 Short description: Conclusions Planned start date: 08/02/2024 Planned end date: 21/03/2024 Start event: Obtained experimental results End event: End of the project Internal task T1: Document conclusions drawn during the project Deliverables: Dates: -17Gantt Diagram TFG Alex Nova Time (h) 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th 16th 17th 18th #WP1 105 Internal Task T1 40 30 10 Internal Task T2 35 5 5 5 5 5 5 5 Internal Task T3 30 5 5 5 5 5 5 #WP2 175 Internal Task T1 30 20 10 Internal Task T2 60 15 25 20 Internal Task T3 65 5 25 25 10 Internal Task T4 20 20 #WP3 200 Internal Task T1 125 25 25 25 25 25 Internal Task T2 75 25 25 25 #WP4 60 Internal Task T1 60 30 30 Total 540 -182. State of the art of the technology used or applied in this thesis ____________________________________________________________ CHAPTER 2 State of the Art of the Technology Used or Applied in this Thesis 2.1. What is Sliding Mode Control Sliding Mode Control (SMC) is a robust control technique that alters the dynamics of a nonlinear system by applying a discontinuous control signal. This forces the system state to "slide" along a predetermined surface, known as the sliding surface. SMC is particularly useful for systems with uncertainties and disturbances due to its inherent robustness. The concept of SMC was introduced as part of variable structure systems (VSS) and has been widely studied for its effectiveness in handling nonlinearities and parameter variations (Utkin, Guldner, & Shi, 2009). There are listed key concepts of sliding mode control: - The sliding surface is a manifold within the state space of the system. When the system's state is limited to this surface, it exhibits the desired dynamic behavior. Creating a control strategy that directs the system state to this surface and keeps it there is the main objective of SMC. Figure 2. Example of the sliding surface. - The control signal in SMC is discontinuous, meaning it can rapidly switch between different values. This feature is essential because it allows the -19controller to respond quickly to any deviations from the sliding surface, thereby correcting the system's state and ensuring it follows the intended path. - The robustness of SMC is a key advantage. A wide range of uncertainties and external disturbances can be handled without significantly deviating from the desired performance by the discontinuous control signal. The high-gain nature of the control law, which can effectively counteract disturbances, is attributed to this robustness. 2.2. Applications of SMC in Power Electronics Power electronics systems are significantly enhanced by sliding mode control. The robustness and adaptability of SMC can be used to effectively manage nonlinearities, parameter variations, and external disturbances. The versatility and effectiveness of SMC in improving efficiency, stability, and overall system performance in power electronics can be seen in applications ranging from DC-DC converters to motor drives and PFC circuits. The primary advantages of using SMC in power electronics include: • Robustness: SMC can handle system parameter variations and external disturbances effectively. • Fast dynamic response: The control law in SMC ensures a quick response to changes in system states. • Reduced steady-state error: SMC can achieve accurate tracking of reference signals. In conclusion, sliding mode control is a powerful tool in the control engineer's arsenal, providing a robust solution for managing nonlinear systems with uncertainties and disturbances. Its application across various industries underscores its versatility and effectiveness in achieving desired system performance. According to (Utkin, Guldner, & Shi, 2009), the application of SMC in power electronics also faces several challenges: variable switching frequency, chattering and implementation, which are described in the next section. 2.3. Problems with the Implementation of SMC Despite its advantages, the implementation of SMC in power electronics faces several challenges, such as: -20- • Variable Switching Frequency: SMC typically results in a variable switching frequency, which can complicate the design of the power stage and filter components. • Chattering: The high-frequency switching can cause chattering, leading to potential wear and tear of electronic components. Chattering occurs due to the high and variable switching frequency, which results in increased electromagnetic interference and higher switching losses. • Complexity in Practical Implementation: Implementing SMC requires precise measurement and control, which can be complex and costly. The design of the hysteresis band to limit the switching frequency is a common approach to mitigate some of these issues. Chattering, discrete-time implementation issues, and the need for accurate parameter tuning and system identification are some of the challenges faced by the implementation of sliding mode control in power electronics. Advanced control techniques, careful system design and high-performance hardware and algorithms are needed to address these challenges. The performance and reliability of power electronics systems can be enhanced by overcoming these hurdles. These problems need careful consideration and design to fully leverage the benefits of SMC while minimizing its drawbacks. -213. Methodology / project development ____________________________________________________________ CHAPTER 3 Methodology / Project Development 3.1. Design of the SMC 3.1.1. Control Design Procedure The design of a Sliding Mode Control (SMC) system involves several critical steps to ensure robust and stable control of nonlinear systems. The following procedure, extracted from (Biel & Dòria), outlines the key steps required to design an effective SMC system: 1. Select the Proper Sliding Surface 2. Analyse the Sliding Motion 3. Determine the Equivalent Control ueq 4. Control Law: 𝒔∙𝒔󰇗<𝟎 5. Determine the Sliding Domain Each of these steps is crucial in establishing the sliding mode and achieving desired control performance. 1. Select the Proper Sliding Surface The sliding surface is a key component in SMC design. It defines the manifold on which the system's state should ideally remain. The sliding surface is typically chosen as a linear combination of system states. For a system described by the state vector x: 𝑥󰇗=𝑓(𝑥)+𝑔(𝑥)𝑢 (1) -22where x ∈ U, an open set of ℝ𝑛, f and g are smooth vector fields on U with g(x) ≠ everywhere, and u ∶ U → ℝ is the control input. Let S be a submanifold in U defined by a smooth function s ∶ U → ℝ, namely: 𝑆={𝑥 ∈ 𝑈 | 𝑠(𝑥)=0} (2) where 𝜕𝑠 𝜕𝑥≠0,∀ 𝑥∈U and 𝑆∩𝑈≠0 are assumed. As for the input, let u be defined as: 𝑢={𝑢+(𝑥), 𝑖𝑓 𝑠(𝑥)>0 𝑢−(𝑥), 𝑖𝑓 𝑠(𝑥)<0 (3) where 𝑢+ and 𝑢− are smooth functions of x and 𝑢+(𝑥)≠𝑢−(𝑥). 2. Analyse the Sliding Motion The next step is to analyse the sliding motion, which refers to the system's behaviour when it is constrained to move along the sliding surface. The sliding condition ensures that the system's state remains on the sliding surface for all future times once it reaches this surface. S is a sliding surface for the dynamical system if and only if there exists an open neighbourhood N of S, N ∩ S ≠ 0 such that ∀x ∈ N: 𝑠(𝑥)·𝜕𝑠(𝑥) 𝜕𝑡 <0 (4) Deriving with respect to x: 𝑠(𝑥)·𝜕𝑠(𝑥) 𝜕𝑥 ·(𝑓(𝑥)+𝑔(𝑥)𝑢)<0 (5) The objective is to ensure that the system dynamics lead the state to converge to and remain on the sliding surface. 3. Determine the Equivalent Control ueq The equivalent control ueq is the continuous control input that would keep the system state on the sliding surface if applied alone. It is derived from the condition that the sliding surface derivative  equals zero: 𝜕𝑠(𝑥) 𝜕𝑡 =𝜕𝑠(𝑥) 𝜕𝑥 ·(𝑓(𝑥)+𝑔(𝑥)𝑢𝑒𝑞)=0 (6) -23By solving this equation, we obtain the equivalent control ueq: 𝑢𝑒𝑞=−𝜕𝑠(𝑥) 𝜕𝑥 ·𝑓(𝑥) 𝜕𝑠(𝑥) 𝜕𝑥 ·𝑔(𝑥) (7) Notice that a necessary condition for the existence of equivalent control is: 𝜕𝑠(𝑥) 𝜕𝑥 ·𝑔(𝑥)≠0 (8) This is known as the transversality equation. This control represents the ideal control action and the Ideal Sliding Motion (ISM), which is governed by: 𝑥󰇗=𝑓(𝑥)+𝑔(𝑥)𝑢𝑒𝑞=0 (9) 4. Control Law: 𝒔∙𝒔󰇗<𝟎 The final step is to establish the control law, ensuring that the system's state reaches and stays on the sliding surface. This is achieved by enforcing the condition 𝒔∙𝒔󰇗<𝟎, where s is the derivative with respect time of the sliding surface. This condition guarantees that the sliding surface s and its derivative s always have opposite signs, driving the system state towards the sliding surface. 𝑠∙𝑠󰇗=𝑠·𝜕𝑠 𝜕𝑥·𝑥󰇗<0 (10) Replacing 𝑥󰇗 from Equation (1): 𝑠·𝜕𝑠 𝜕𝑥·(𝑓(𝑥)+𝑔(𝑥)𝑢)<0 (11) which can be rewritten as: 𝑠·𝜕𝑠 𝜕𝑥·(𝑓(𝑥)+𝑔(𝑥)𝑢−𝑔(𝑥)𝑢𝑒𝑞+𝑔(𝑥)𝑢𝑒𝑞)<0 (12) By definition: 𝑠󰇗|𝑢=𝑢𝑒𝑞=0⟹𝜕𝑠 𝜕𝑥·(𝑓(𝑥)+𝑔(𝑥)𝑢𝑒𝑞)=0 (13) Therefore, -24𝑠·𝜕𝑠 𝜕𝑥·𝑔(𝑥)·(𝑢−𝑢𝑒𝑞)<0 (14) Defining the control by: {𝑠·𝜕𝑠 𝜕𝑥·𝑔(𝑥)>0⟹𝑢<𝑢𝑒𝑞⟹𝑢=min {𝑢−,𝑢+} 𝑠·𝜕𝑠 𝜕𝑥·𝑔(𝑥)<0⟹𝑢>𝑢𝑒𝑞⟹𝑢=max {𝑢−,𝑢+} (15) This control law ensures that the system state is driven towards the sliding surface and remains there, achieving the desired sliding mode control. 5. Determine the Sliding Domain Sliding mode exists on the submanifold of s = 0 defined by: min{𝑢−,𝑢+}<𝑢𝑒𝑞<max {𝑢−,𝑢+} (16) which is known as the sliding domain. Note that sliding modes will be observed only if the switching surface, s = 0, is reached in finite time. By following these steps, an effective SMC system can be designed to achieve robust and stable control performance for nonlinear systems. 3.1.2. First Design Example: Simple Mathematical Model Let us introduce a first example with simple values to familiarize with the design of the sliding and ensure that the system has the desired behaviour. This model was extracted from (Repecho, Biel, Olm, & Fossas, 2017) and some parameters were adapted. The system is described by: 𝑥1󰇗=−𝑥1+𝑥2 (17) 𝑥2󰇗=−𝑥1+𝑀𝑢 (18) where 𝑀∈ℝ. Following the steps described in the previous section: -251. Select the Proper Sliding Surface The system is forced to slide over the switching surface: 𝑠(𝑥,𝑡)=𝑥2−𝑥2∗(𝑡)=0 (19) where 𝑥2∗(𝑡)=𝐴 (20) with 𝐴∈ℝ, by the control law Equation (3), where 𝑢+=1 and 𝑢−=−1. Indeed, identifying f(x) and g(x) from Equation (1) in the system described by Equation (17) and Equation (18), one gets that: 𝑓(𝑥)=(−𝑥1+𝑥2,−𝑥1)𝑇, 𝑔(𝑥)=(0,𝑀)𝑇 (21) The condition s = 0 implies: 𝑥2=𝑥2∗ (22) 2. Analyse the Sliding Motion 𝑠󰇗=𝑥2󰇗−𝑥2∗󰇗 (23) Replacing 𝑥2󰇗 from Equation (18), and being 𝑥2∗󰇗=0: 𝑠󰇗=−𝑥1+𝑀𝑢 (24) What is the control u? 3. Determine the Equivalent Control ueq 𝑠󰇗=0⟹−𝑥1+𝑀𝑢𝑒𝑞=0 (25) By solving this equation, we obtain the equivalent control ueq: 𝑢𝑒𝑞=1 𝑀𝑥1 (26) In accordance with the transversality equation, sliding motion exists when: 𝑀>|𝑥1| (27) 4. Control Law: 𝒔∙𝒔󰇗<𝟎 -323. Determine the Equivalent Control ueq 𝑠󰇗=0⟹𝜆1 C(−𝑣𝐶 R+𝑖𝐿)+𝜆2(1 𝑅𝐶(−𝑣𝐶 R+𝑖𝐿)−𝑣𝐶 𝐿+𝐸𝐿𝑢𝑒𝑞)=0 (51) By solving this equation, we obtain the equivalent control ueq: 𝑢𝑒𝑞=1𝐸(−𝛼𝜆1 𝜆2𝑖𝐿+(1+𝛼𝜆1 𝑅𝜆2)𝑣𝑐) (52) with 𝛼=𝐿𝐶(1−𝜆2 𝑅𝜆1) (53) 4. Control Law: 𝒔∙𝒔󰇗<𝟎 𝑠∙𝑠󰇗<0 (54) Replacing 𝑠󰇗 from Equation (50): 𝑠·(𝜆1 𝐶(−𝑣𝐶 𝑅+𝑖𝐿)+𝜆2(1 𝑅𝐶(−𝑣𝐶 𝑅+𝑖𝐿)−𝑣𝐶 𝐿+𝐸𝐿𝑢))<0 (55) which can be rewritten as: 𝑠·(𝜆1 C(−𝑣𝐶 R+𝑖𝐿)+𝜆2(1 𝑅𝐶(−𝑣𝐶 R+𝑖𝐿)−𝑣𝐶 𝐿+𝐸𝐿𝑢)−𝜆2𝐸𝐿𝑢𝑒𝑞+𝜆2𝐸𝐿𝑢𝑒𝑞)<0 (56) By definition: 𝑠󰇗|𝑢=𝑢𝑒𝑞=0⟹𝜆1 C(−𝑣𝐶 R+𝑖𝐿)+𝜆2(1 𝑅𝐶(−𝑣𝐶 R+𝑖𝐿)−𝑣𝐶 𝐿+𝐸𝐿𝑢𝑒𝑞)=0 (57) Therefore, 𝑠·𝜆2𝐸 𝐿·(𝑢−𝑢𝑒𝑞)<0 (58) Defining the control by: { 𝑠·𝜆2𝐸 𝐿>0⟹𝑢<𝑢𝑒𝑞⟹𝑢=0 𝑠·𝜆2𝐸 𝐿<0⟹𝑢>𝑢𝑒𝑞⟹𝑢=1 (59) -33As 𝐸𝜆2>0, the control law forces sliding motion over s = 0. 5. Determine the Sliding Domain Sliding mode exists on the submanifold of s = 0 defined by: 0<𝑢𝑒𝑞<1 (60) Substituting the ueq from Equation (52): 0<1𝐸(−𝛼𝜆1 𝜆2𝑖𝐿+(1+𝛼𝜆1 𝑅𝜆2)𝑣𝑐)<1 (61) With the ideal sliding dynamics given by: 𝐶𝜕𝑣𝐶 𝜕𝑡=−𝜆1 𝜆2𝑣𝐶+𝜆1 𝜆2𝑣𝐶∗ (62) 𝐿𝜕𝑖𝐿 𝜕𝑡=𝛼𝜆1 𝑅𝜆2𝑣𝐶−𝛼𝜆1 𝜆2𝑖𝐿 (63) 𝑠󰇗=𝜆2 𝐿·(−𝑣𝐶+𝐸𝑢) (64) It is then immediate that the steady-state solution is asymptotically stable if and only if: 0<𝜆2 𝜆1<𝑅 (65) As in the previous example, this model was implemented and simulated in the environment Matlab-Simulink. Parameters of the circuit were the following: Input voltage (E) 24 V Inductance (L) 22 µH Capacitor (C) 50 µF Output load (R) 2 Ω Output voltage (vC) 12V -34Output current (iL) 6 A The surface parameters have been set to 𝜆1=0.2 and 𝜆2=0.38, providing sliding mode transient response for the desired output voltage with a time constant of 95 µs, as it is determined in (Repecho, Biel, Olm, & Fossas, 2017). The fundamental sample time for the simulation with MATLAB-Simulink was 10-8, in order to have enough points for cases with elevated frequency. The implementation of the SMC is shown in Figure (6) and the behaviour of the buck converter signals is shown in Figure (7) where the control signal u is in yellow (raised up to 15 V just to see it properly), the surface s is in red, the output voltage vC is in green and the output current iL is in blue. Figure 6. Control of the buck converter model. -35Figure 7. Behaviour of the buck converter model signals with voltage control. It is easy to see that the system worked as expected. The steady state was reached in about 0.5 ms, when the output voltage reached the desired value of 12 V. In this case, the hysteresis value was set at ∆ = 0.466, ensuring a switching frequency of 110 kHz. The surface was centred on 0, going from -∆ to +∆ and vice versa. As it was defined, iL = vC/R = 6 A in equilibrium, which can be observed in the graph. In the case, the duty cycle of the buck converter is 𝐷=𝑣𝐶∗ 𝐸=50%, as it was seen in the introduction of this section. Considering that the filter resonance frequency is 𝑓0=1 2𝜋√𝐿𝐶=4.8 𝑘𝐻𝑧, the system is working at a switching frequency fsw > 20·f0. SMC with current control This model was also implemented with a surface controlled by current to have the same control as the simple mathematical model. The SMC was designed but it was easy this time since we chose the same surface than the simple mathematical model, so the calculations were basically the same. Following the procedure from the previous case, these results were obtained: The new surface would be: 𝑠(𝑖𝐿,𝑡)=𝑖𝐿−𝑖𝐿∗(𝑡)=0 (66) where -36𝑖𝐿∗(𝑡)=𝑖𝐿0 (67) The equivalent control ueq will be: 𝑢𝑒𝑞=𝑣𝑐 𝐸 (68) Defining the control by: { 𝑠·𝐸𝐿>0⟹𝑢<𝑢𝑒𝑞⟹𝑢=0 𝑠·𝐸𝐿<0⟹𝑢>𝑢𝑒𝑞⟹𝑢=1 (69) Sliding mode exists on the submanifold of s = 0 defined by: 0<𝑣𝑐 𝐸<1 (70) With the ideal sliding dynamics given by: 𝑖𝐿=𝑖𝐿∗ (71) 𝑣𝐶󰇗 =−𝑣𝐶 𝑅+𝑖𝐿∗ (72) 𝑠󰇗=1𝐿·(−𝑣𝐶+𝐸𝑢) (73) This buck converter model with current control was also implemented and simulated. Figure (8) and Figure (9) show the implementation of the SMC and the behaviour of the buck converter signals. The signals are the same than in the previous case with voltage control. -37Figure 8. Control of the buck converter model with current control. Figure 9. Behaviour of the buck converter model signals with current control. As in the previous models, the system worked as expected. This time, the switching frequency of 101 kHz with a hysteresis value of ∆ = 0.585. The output current reached the desired value of 10.5 A and the steady state in about 0.5 ms. This time the output voltage was vC = iL·R = 21 V in equilibrium. The duty cycle is 𝐷=𝑖𝐿∗·𝑅 𝐸=87.5%. The filter resonance frequency is the same as in the voltage control model, 𝑓0= 4.8 𝑘𝐻𝑧. Therefore, the system is also working at a switching frequency fsw > 20·f0. -383.2. Analysis of the Switching Frequency in Sliding Mode Control The analysis of the switching frequency in sliding mode control (SMC) is crucial to understand the performance and stability of the system. The switching frequency can vary due to several factors, including system parameters and external disturbances. The aim of this section is to evaluate different methods to analyse the switching frequency in SMC to determine the best approach for designing the control system. Specifically, we seek to find the hysteresis value that achieves the desired switching frequency of the converter. The best method will be identified as the one that best meets the following requirements: 1. Accuracy 2. Simplicity of use and implementation 3. Applicability across different operating conditions If one method does not meet one of the listed requirements, it will be discarded. To assess the accuracy of the methods, the error that each one makes when estimating in respect to the real frequency will be checked through different cases. Having into account that this study is about SMC in power converters, the filter resonance frequency f0, will be a reference. Typically, the switching frequency of power converters is, at least, 10 times the filter resonance frequency and, in some cases, even 20 times. Working below 5·f0 is not a realistic operation point. Therefore, even if we test all the possible frequencies, the most important results for determining the accuracy of the methods will be those obtained when evaluating switching frequency for values higher than 5·f0. 3.2.1. Methods Time-based methods • Piece-wise approximation The piece-wise approximation method involves analysing the time-domain behaviour of the SMC system to determine the switching frequency. This approach typically includes: -39- - Measuring the time intervals between consecutive switching. - Calculating the switching frequency as the inverse of the average time interval. According to (Repecho, Biel, Olm, & Fossas, 2017), this method is straightforward and useful for systems where the switching intervals can be easily measured. The hysteresis band controller plays a significant role in regulating the switching frequency, by adjusting the hysteresis band to maintain a desired frequency. In this method, the switching frequency is calculated assuming that the system is fast enough, so that the input may be considered constant during the period of self-excited oscillations. With this assumption the sliding function can be approximated by straight lines. Figure 10. Switching function behaviour. Figure (10) shows the behaviour of s when evolves from s = -∆ to s = ∆ and vice versa. The required time to reach s = ∆ from s = -∆ is denoted with Tk+, whereas Tkis the time needed to recover s = -∆ from s = ∆. The switching frequency would be calculated as the inverse of the period, Tk, calculated as: 𝑇𝑘=𝑇𝑘++𝑇𝑘−=2𝛥(1 𝑠𝑘+󰇗−1 𝑠𝑘−󰇗) (74) This equation shows that the period Tk is directly proportional to the hysteresis band value. By adjusting it, the switching frequency can be regulated to achieve the desired performance. In order to obtain the switching frequency in an easy way, Equation (74) was implemented in a MATLAB script. To do so, the formula was computed taking as 𝑠𝑘+󰇗, -40the expression of 𝑠󰇗 from Equation (38), Equation (64) or Equation (73), depending on the surface definition, and evaluating it when 𝑢=𝑢+. The same happens with 𝑠𝑘−󰇗, which was evaluated when 𝑢=𝑢−. We just had to introduce the corresponding parameters of the system and run the script, which will return us the calculated frequency for the parameters introduced. Note that for one specific system described, to obtain all the calculated frequencies for different hysteresis values required only the creation of a vector ∆ with all the values. The script will then return a vector with all the calculated frequencies. • Poincare Map The Poincare Map method investigates the behaviour of the system by projecting its trajectories onto a low-dimensional subspace defined by a specific event or condition. The method includes: - Transforming continuous time into discrete time by selecting a specific crosssection on the phase space of the system to observe the evolution of state points on that cross-section (Sun & Wang, 2024). - Solving an equations system In linear systems, the switching frequency can be found by solving these two equations with two unknown variables θ1 and θ2 presented by Boiko in (Boiko, 2016): {𝑟𝑠−𝛥𝑟𝑠−𝑪(𝑰−𝑒𝑨𝑇)−1𝑨−1[𝑒𝑨𝑇−2𝑒𝑨𝜃2+𝑰]𝑩=𝑏 𝑟𝑠−𝛥𝑟𝑠−𝑪(𝑰−𝑒𝑨𝑇)−1𝑨−1[2𝑒𝑨𝜃1−𝑒𝑨𝑇−𝑰]𝑩=−𝑏 (75) where rs is the surface of the SMC with the variables equal to 0, just with the constants and the reference values, b is the hysteresis value, T = θ1 + θ2 = 2πΩ, where Ω is the switching frequency and A, B and C are the matrix that define the following system: {𝒙󰇗=𝑨𝒙+𝑩𝑢 𝑦=𝑪𝒙 (76) θ1 and θ2 are lengths of the positive and negative control pulses, respectively. This method will be discarded due to its complexity, solving the equations system described in Equation (75) is not trivial. Additionally, this method involves evaluating the switching function evaluating it point by point, which does not make sense for this study, because one of the requirements for the methods in this context is simplicity. -41Frequency-based methods • Describing Function The Describing Function (DF) method is used to approximate the nonlinear characteristics of the SMC system by representing it as a frequency-dependent function. This method involves: - Replacing the nonlinear elements of the system around a specific operating point by an equivalent frequency response called Describing Function. - Analysing the resulting linear system to determine its frequency response. - Estimating the switching frequency based on the frequency response characteristics. The DF of the relay function is given as follows: 𝑁(𝑎)=4𝑐 𝜋𝑎√1−(𝑏𝑎)2−𝑗4𝑐𝑏 𝜋𝑎2 (77) where a is the amplitude of the symmetric oscillations, b is the hysteresis value, and c is the output level (amplitude) of the relay. This method calculates the switching frequency by obtaining the intersection point Ω of the harmonic balance equation, defined as W(jΩ) = -1/N(a), with the transfer function of the system G(jω). The switching frequency would be the value of ω from G(jω) in the intersection point Ω, as it is explained in (Biel, Dòria-Cerezo, & Olm, 2023). The simple mathematical model presented in Section (3.1.2) was used to familiarize ourselves with the method and obtain initial results. During this process, some limitations were found. First of all, the transfer function G(s) of the system was obtained. The system described by Equation (17) and Equation (18) was rewritten using Laplace transform: 𝑠·𝑋1(𝑠)=−𝑋1(𝑠)+𝑋2(𝑠) (78) 𝑠·𝑋2(𝑠)=−𝑋1(𝑠)+𝑀·𝑈(𝑠) (79) From this point, the transfer function can be easily obtained: 𝐺(𝑠)=𝑋2(𝑠) 𝑈(𝑠)=𝑀·(𝑠+1) 𝑠2+𝑠+1 (80) -48Figure 15. Error vs hysteresis value graph. Figure 16. Error vs switching frequency graph. It can be easily observed that, as expected, the method works almost perfectly for higher frequencies and starts to increase the error made for lower frequencies. The error made in absolute value for different frequencies multiples of the filter resonance frequency f0 = 0.16 Hz is shown in Table (5). -49Table 5. Error made respect f0 with piece-wise approximation method with the simple mathematical model. In both cases the error for fsw > 5·f0 is below 1%, for fsw > 10·f0 is about 0.2% and for fsw > 20·f0 is below 0.1%. The difference between both control laws is that with u ∊ {0, 1}, we have more error in the intermediate frequencies, but less in higher and lower frequencies. Obtaining more error in higher frequencies make sense because the system is slower compared to the control law u ∊ {-1, 1}. Buck converter model On the other hand, the buck converter model presented in the Chapter 3 was also used, with both designed surfaces, the first one with voltage control and the second one with current control. For the voltage control, the reference value of the output voltage vC* was changed to 21 V, to simulate the model in the limit of its operating range. - Voltage control: Following the steps done with the simple mathematical example, the hysteresis was increased, and, for the higher values, the deformation of the switching function can be observed, as shown in Figure (17), with ∆ = 2.33. The frequency in this case was fsw = 9.003 kHz, which was less than 2 times f0. -50Figure 17. Switching function for ∆ = 2.33. The results of approximating the switching frequency with the piece-wise approximation method are shown in Figure (18) and Figure (19), representing the error in % versus the hysteresis value and the frequency, respectively. Figure 18. Error vs hysteresis value graph. -51Figure 19. Error vs switching frequency graph. It can be easily observed that, as expected and as in the previous case with the simple model, the method works almost perfectly for higher frequencies and starts to increase the error made for lower frequencies. The error made in absolute value for different frequencies multiples of the filter resonance frequency f0 = 4.8 kHz is shown in Table (6). Table 6. Error made respect f0 with piece-wise approximation method with voltage control (Vout = 21 V). The error for fsw > 5·f0 is below 2%, for fsw > 10·f0 is about 0.3% and for fsw > 20·f0 is about 0.2%. -52- - Current control: Here, the same process was done for the buck converter model with current control. The hysteresis was increased, and for the higher values, the deformation of the switching function can be observed, as shown in Figure (20). The frequency in this case was fsw = 10.198 kHz, with ∆ = 5.85, which was about 2 times f0. Figure 20. Switching function for ∆ = 5.85. Figure (21) and Figure (22) represents the error in % versus the hysteresis value and the frequency from approximating with the piece-wise approximation method, respectively. Figure 21. Error vs hysteresis value graph. -53Figure 22. Error vs switching frequency graph. Following the same behaviour as in the previous cases, so the conclusions are the same. The error made in absolute value for different frequencies multiples of the filter resonance frequency f0 = 4.8 kHz is shown in Table (7). Table 7. Error made respect f0 with piece-wise approximation method with current control (Vout = 21 V). The error for fsw > 5·f0 is below 2%, for fsw > 10·f0 is below 0.3% and for fsw > 20·f0 is about 0.2%. Comparing these results with those obtained with the voltage control model, we can say that the defined surface does not modify the error made directly. The results are very similar, following the same behaviour. This model was also useful to compare the buck model with the simple mathematical model. The surface of the SMC was the same, and the only change between the definition of the systems was the value of the parameters: {𝑥1󰇗=−𝑥1+𝑥2 𝑥2󰇗=−𝑥1+𝑀𝑢 s=𝑥2−𝑥2∗ {𝐶·𝑣𝐶󰇗 =−𝑣𝐶/𝑅+𝑖𝐿 𝐿·𝑖𝐿󰇗=−𝑣𝐶+𝐸𝑢 s=𝑖𝐿−𝑖𝐿∗ (84) -54By normalizing the system with a variable change, the buck converter model was transformed into the simple mathematical model. The expected result of this comparation was that the error made for equivalent frequencies respect f0, would be the same. Normalizing the system: 𝑥1=√𝐶·𝑣𝐶 (85) 𝑥2=√𝐿·𝑖𝐿 (86) Replacing (85) and (86) in (84) we obtain: { 𝐶·𝑥1󰇗 √𝐶=− 𝑥1 √𝐶·𝑅+𝑥2 √𝐿 𝐿·𝑥2󰇗 √𝐿=−𝑥1 √𝐶+𝐸𝑢 (87) Simplifying: {√𝐿𝐶·𝑥1󰇗=−𝑥1√𝐿/𝐶·1𝑅+𝑥2 √𝐿𝐶·𝑥2󰇗=−𝑥1+𝐸√𝐶·𝑢 (88) Defining the variable change: 𝑡=√𝐿𝐶·𝜏 (89) where τ is the time in the simple mathematical model and t is the time in the buck converter model, therefore: {𝜕𝑥1 𝜕𝜏=−𝛾·𝑥1+𝑥2 𝜕𝑥2 𝜕𝜏=−𝑥1+𝑀𝑢 (90) with 𝛾=√𝐿/𝐶·1𝑅 and 𝑀=𝐸√𝐶. We needed γ = 1 to have the same system, it was fulfilled for a specific value of 𝑅=√𝐿/𝐶=√22 𝜇𝐻/50 𝜇𝐹=0.663 Ω. Hence, we needed to simulate the system for a different value of R, so we decided to simulate it for that specific value and also for a bigger R to see the load effect in the approximation of the frequency (keeping the desired output voltage vC* = iL*/R constant). Note that neither the calculation of 𝑠󰇗 in Equation (73) nor the calculation of the switching period in Equation (74) depends on R. We noticed that the duty cycle of the simple mathematical model and the buck converter model were not the same, so we changed the parameters from the simple -55mathematical model to ensure that both duty cycles were the same before comparing them. 𝐷𝑏𝑢𝑐𝑘=𝐷𝑠⟹𝑅·𝑖𝐿∗ 𝐸=0.663·21 0.663 24 =21 24=𝑥2∗ 𝑀⟹𝑀=12,𝑥2∗=10.5 (91) Dbuck is the duty cycle in the buck model while Ds is the duty cycle in the simple mathematical model. The results of using the piece-wise approximation method with different load values in the buck converter model with current control are shown in Figure (23) and Figure (24). These figures illustrate the error percentage in relation to the hysteresis value and the frequency, respectively. Figure 23. Error vs hysteresis value graph. Figure 24. Error vs switching frequency graph. On the one hand, results demonstrated that the load effect acts as an offset value for the error. Higher loads tend to scroll down all the waveform, keeping the same -56behaviour. It is true that for the limit case, when the switching frequency is very low, the graphic tends to increase again. On the other hand, Table (8) shows the comparation between the simple mathematical model (with the control law u ∊ {0, 1}) and the buck converter model with current control with R = 0.663 Ω. As can be observed, the error made in each case meets, confirming that the models are equivalent. Table 8. Comparative table of the errors made respect f0 between models. 4.1.2. Case 2: Typical Operating Range Let us define the typical operating range as the case when the duty cycle is 50%, which is the same as saying that Vout = Vin/2. For this case, the buck converter model with voltage control presented in the Chapter 3 was the only one analysed, keeping all the parameters as described previously, with the desired Vout = 12 V. This case was especially interesting for two main reasons: - Analyse the performance of the methods in a typical operating range of the converter. - Use the LPRS method, since it is the only case where it can be implemented. The procedure will be the same as in case 1: increase the hysteresis in little steps and compare the simulation values with the calculated values, this time, with two different methods. For the higher hysteresis values, the deformation of the switching function can be observed as shown in Figure (25), with ∆ = 11.8. The frequency in this case was fsw = 7.047 kHz, which was less than 2 times f0. -57Figure 25. Switching function for ∆ = 11.8. - Piece-wise approximation method The results of approximating with the piece-wise approximation method are shown in Figure (26) and Figure (27), representing the error in % versus the hysteresis value and the frequency, respectively. Figure 26. Error vs hysteresis value graph. -64The controller part was redesigned and replaced to implement the SMC described in previously chapters. However, the PCB used was the same as a sample without the assembled components was already manufactured, which simplify and accelerated the process. Furthermore, to set the hysteresis value, a through-hole board was designed. The procedure of this part can be summarised in: 1) Design and simulate the controller part. 2) Design the circuit to set the hysteresis value. 3) Connect the control PCB into the multiphase prototype board. 4) Check the SMC functionality in open-loop control. 5) Check the SMC functionality in close-loop control. 6) Take measures with the oscilloscope. 7) Compare the real switching frequency with theoretical predictions according to the method used. The equipment used is listed below: - Power Supply EA-PS 2384-05 B - Oscilloscope Yokogawa DLM2024 - Programmable Function Generator HAMEG HM8131-2 - Programmable AC/DC Electronic Load Chroma Model 63804 4.2.1. Design of the Controller As said before, the controller was based on an already designed board from Adrià Arroyo, but it has been modified. Since the sample used ad no assembled components, all of them were hand-welded. Figure (31) shows the PCB with mounted components. Figure (32), Figure (33), Figure (34), Figure (36) and Figure (36) shows the schematics of the PCB. -65Figure 31. Controller PCB assembled. Figure 32. Main schematic of the SMC. -66Figure 33. Input connector schematic of the SMC. Figure 34. Enable drivers and conditioning schematic of the SMC. -67Figure 35. Surface creation schematic of the SMC. Figure 36. Comparator schematic of the SMC. The functionality of the controller was tested via simulation using LTspice. Figure (37) shows the surface s and control signal u. -68Figure 37. Simulation of the SMC with LTspice. Main modifications from the original design: - Switching surface The implementation of the surface can be observed on Figure (x). By changing the resistors value, the surface parameters can be modified. The chosen ones were to implement the described surface: 𝑠(𝑣𝑐,𝑣𝐶󰇗,𝑡)=𝜆1(𝑣𝐶−𝑣𝐶∗)+𝜆2𝐶𝑣𝐶󰇗 =0 (105) which can be rewritten in terms of iC as: 𝑠(𝑣𝑐,𝑣𝐶󰇗,𝑡)=𝜆1(𝑣𝐶−𝑣𝐶∗)+𝜆2𝑖𝐶=0 (106) The desired output voltage vc* was set by the value of the potentiometer R47. As determined in the simulations, 𝜆1=0.2 and 𝜆2=0.38. To implement the parameters as desired it was needed to considerate the sense of every signal: The capacitor current iC was sensed by a 47/250 factor (a resistor of 47 Ω parallel to an electromagnetic coil with 250 turns, in order to transform the current into voltage) while the capacitor voltage vc was sensed by a 1/12 factor (with a voltage divider). Note that the reference output voltage had to be set with a 1/12 factor as well, i.e. if the desired output voltage is 12 V, the reference voltage vref must be 1 V. Moreover, the surface had an offset of 5 V, in order to always have a positive signal. The hysteresis value, which will be compared with surface, was also calculated respect 5 V. When it was said that the hysteresis value was 0.5, that meant that hys+ = 5.5 V and hys- = 4.5 V. -69By popping the input resistors of the amplifier IC2A, the model with current control can be implemented. Meanwhile, this amplifier is working as a voltage follower. - Hysteresis comparator The comparator used to compare the hysteresis value and the surface was the MCP6567 from Microchip Technology. This comparator ensures its functionality only when working below VDD + 0.3 V. Since VDD was 5 V and our surface had an offset level of 5 V, for hysteresis values bigger than 0.3 the comparator did not operate correctly, switching before the surface reaches the hysteresis level. The comparator was replaced for the dual comparator LM319A from TI, which allows higher operating voltages. The disadvantage was that this comparator was about 25 ns slower than the MCP6567 and was not pin compatible, so a rework was needed, connecting each pin of the component with a wire. - Mounted components and reworks This PCB was designed to control all the four phases and, in our case, only one phase shall be controlled, so it was necessary to select which components were needed for the project. The capacitor voltage vC input pin was wrongly designed, so it was fixed putting a wire from one input pin to the other. The hysteresis value input was inverted, being hys+ = 5V - ∆ and hys- = 5V + ∆. It was supposed to be hys+ = 5V + ∆ and hys- = 5V - ∆, so it was modified in the circuit that allows us to set the hysteresis value. Lastly, we did not have SMD 10 kΩ potentiometers left, so another one which was not pin compatible was used. 4.2.2. Design the Circuit to Set the Hysteresis Value In order to set the desired hysteresis value, it was needed to design a little throughhold board that allows us to change the value with a potentiometer. Figure (38) shows the mounted circuit and Figure (39) shows the schematic of the circuit. This circuit allows us to reach hysteresis values ∆ of 5 V. The circuit was designed with the operational amplifier TLC2272. -70Figure 38. Circuit to set the hysteresis value. Figure 39. Schematic of the hysteresis circuit. 4.2.3. Functionality of the SMC After designing the SMC and the circuit to set the hysteresis value, it was time to check that it works as expected. The first step was to connect all together and check that the open-loop control was working correctly. A square signal with a frequency of 50 kHz and a duty cycle of 50% was generated with the function generator and was introduced in the circuit as the control signal u. Following the circuit signals it was observed that the surface was the expected and the output voltage was 12 V. The next step was to disconnect the function generator and close the loop. It worked, controlling correctly the output voltage, which could be regulated by changing the reference value, and the switching frequency, which can be regulated by changing the -71hysteresis value. The output load was connected, and the output current was monitored. Figure (40) shows the control signal u in Channel 1, the output voltage vC in Channel 2, the surface s in Channel 3 and the output current iL in Channel 4. Figure (41) is the same screenshot but zooming in when the output load was changed. Figure 40. Behaviour of the buck converter signals on the oscilloscope. -72Figure 41. Behaviour of the buck converter signals on the oscilloscope zoomed in. It is easy to see that the system worked as expected. The output voltage was regulated at the desired 12 V. The surface going from -∆ to +∆ and vice versa. The system was working at a switching frequency of 100 kHz. It is worth comparing Figure (40) with Figure (7), concluding that the simulated system meets with the experimental one. The output load change was from 100 Ω to 2 Ω, and the control was capable of continue regulating the output voltage without any distortion. This change in the load increased the output current and the consumption from 120 mA and 1.44 W to 6 A and 72 W. It was really good to see that the control worked so well with that heavy change. Moreover, line and load regulation were checked, obtaining fair values. The line regulation is the ability of the converter to maintain the desired output voltage despite variations of the input voltage, while the load regulation is the ability to maintain the desired output voltage despite variations in the output load. Figure (42) shows the line regulation variating the input voltage from 16 to 32 V, and Figure (43) shows the load regulation variating the output load from 2 to 1000 Ω. -73Figure 42. Line Regulation. Figure 43. Load Regulation. 4.2.4. Results Once the controller was verified to be working properly, we were able to start the analysis of the switching frequency for different cases. Following the examples done via simulations, the cases when the desired output voltage was 12 and 21 V were analysed. Additionally, the case when the desired output voltage was 3 V was also tested. In this part all the parameters maintained the specified value except the load, which was not connected. This change was because with the load, especially with a low value like 2 Ω, some components were heating up too quickly, which did not allow us to make measurements in ease, we had to turn off the power supply all the time to let it cool down. In addition, we realised that with the load connected there were important losses in the cases where the output voltage was high, due to the high output current. Since in the simulations we did not consider losses, with the load connected there were a big error between simulation and experimental part. -80The results of approximating with the piece-wise approximation method are shown in Figure (51) and Figure (52), representing the error in % versus the hysteresis value and the frequency, respectively. Figure 51. Error vs hysteresis value graph. Figure 52. Error vs switching frequency graph. The error made in absolute value for different frequencies multiples of the filter resonance frequency f0 = 4.8 kHz is shown in Table (15). -81Table 15. Error made respect f0 with piece-wise approximation method (Vout = 21 V). From the table, we can see that the error for fsw = 5·f0 is about 9%, for fsw = 10·f0 is about 4.2% and for fsw = 20·f0 is about 4%. Case 3: Vout = 3 V The last case analysed was when the desired output voltage was 3 V. In this case only the piece-wise approximation method will be used. Figure 53. Behaviour of the signals with Vout = 3 V and a measured switching frequency of 40.22 kHz. As in the other cases, the deviation between the simulation and the experimental measures were compared in Table (16). -82Table 16. Comparative table between the simulated and the experimental switching frequency for Vout = 3 V. The results of approximating with the piece-wise approximation method are shown in Figure (54) and Figure (55), representing the error in % versus the hysteresis value and the frequency, respectively. Figure 54. Error vs hysteresis value graph. -83Figure 55. Error vs switching frequency graph. The error made in absolute value for different frequencies multiples of the filter resonance frequency f0 = 4.8 kHz is shown in Table (17). Table 17. Error made respect f0 with piece-wise approximation method (Vout = 3 V). From the table, we can see that the error for fsw = 5·f0 is about 19.4%, for fsw = 10·f0 is about 2.8% and for fsw = 20·f0 is about 0.2%. 4.2.5. Conclusions Taking all the results from the experimental part into consideration, we can say that, in exception of the lower frequency case with an output voltage of 3 V, the experimental measures meet with the simulated ones with a deviation between 0 and 7%, which is quite accurate. It is worth saying that part of the deviation can be caused by the value of the resistors mounted to create the surface signal. These resistors are R20, R27, R30 and R32. In the calculations of the value of these resistors, there were ideally R27 = R32 = 4167 Ω, and R20 = R30 = 4947 Ω. Since that specific values do not exist, the mounted resistors were of 4300 Ω and 4700 Ω, respectively. A way to reduce this effect would be using high precision resistors. This validates the design of the controller done. -84When we decided not to connect the load to take the experimental measurements was because in the case of 21 V, when the output current was 4.2 A with a 5 Ω load, the losses caused by the system components (MOSFETs, inductor serial losses resistor, etc.) were bigger than expected, altering the measurements done. Figure (56) shows exactly the same case of Figure (50), but with the load connected. The frequency had a drop from 41 kHz to 33 kHz. Furthermore, the delay of the comparator was accentuated, resulting in the surface not switching at the moment it reaches the hysteresis value. This behaviour meets with the load effect previously viewed with the simulations of the buck converter model with current control, when lower the load, lower the frequency. Figure 56. Behaviour of the signals with Vout = 21 V and a 5 Ω load connected. The measured switching frequency was 32.78 kHz. On the other hand, in the cases with lower output voltage, where the output current was also lower, this phenomenon did not affect the switching frequency. This was because the losses in these cases were lower. Figure (57) shows the same case of Figure (45), but with a load connected. Note that the switching frequency in this case remains the same. -85Figure 57. Behaviour of the signals with Vout = 12 V and a 5 Ω load connected. The measured switching frequency was 40 kHz. As commented before, the cases when the frequency was lower than 5 time the filter resonance frequency could not be validated because when we were trying to work under that frequency, the buck stopped switching due to a high value of the hysteresis. This confirms that working under a frequency of 5·f0 is not a realistic case in power converters, as it was said at Section 3.2. Furthermore, working near that frequency can lead to fails in the regulation of the output voltage, as can be observed in Figure (56), where the output voltage is 20.6 V instead of 21 V. To conclude, looking the error made by approximating the measured switching frequency with both the piece-wise approximation and LPRS methods, we can affirm that both works perfectly, with low error, especially in most interesting operating ranges with high frequency. Recalling the conclusions from the simulation part, it is confirmed that with low frequencies, the error made is higher with the piece-wise approximation while the LPRS method remains being very accurate. -865. Sustainability Analysis and Ethical Implications ____________________________________________________________ CHAPTER 5 Sustainability Analysis and Ethical Implications 5.1. Environmental Impact The development of this project involved various stages, including research, simulations, and hardware testing. The material used was not required to be bought since it was already in stock in the laboratory where the project took place, situated in the IOC at UPC Campus Sud. Even for the implementation of the controller PCB designed, was not needed to manufacture a new PCB because in the IOC they got a sample without components assembled, so we decided to use it, mounting the needed components. This decision, added to the fact that buying components was not necessary, allowed to reduce the environmental impact of the project since the manufacturing and transportation of new material was avoided, reducing the carbon footprint. Furthermore, about the execution of the project we can determine that the implementation of the SMC improves the efficiency of the buck converter, reducing energy losses and lowering overall energy consumption. This has a positive environment impact by decreasing the electricity demand and reducing greenhouse gas emissions. However, laboratory equipment and a personal computer was used to develop and execute the project. It is important to estimate the environment impact associated to the utilization of this equipment. Taking into account that electronic devices with a life span of 5 years emit about 180 kg CO2, it can be calculated that the emissions would be 4 g CO2 every hour. The emissions from welding can be estimated as 200 g CO2 every hour of use. Since the worst case must be calculated, let us say that all the hours spent to develop and execute the project the personal computer used. The equipment used in the laboratory to carry out the experimental part was a power supply, an oscilloscope, an electronic load, a function generator, and a soldering station. Let us say that from the 200 hours spent -87to the experimental part, 150 hours were spent to measurements and testing of the buck, while the other 50 hours were spent welding. Additionally, we shall calculate the pollution generated using transport to go to the lab. I used a motorbike to go to the laboratory, which have an emissions of 53 g CO2 every km at an average speed of 80 km/h. The route from my house to the laboratory is about 35 km, so every I have done 70 km. Estimating that I went to the laboratory a total of 40 days. The following table shows the total pollution emitted taking into account these approximations. Table 18. Environmental impact associated to the project. Equipment Emissions (g CO2/h) Total Hours Total Emissions (g CO2) PC 4 g CO2/h 540 2160 g CO2 Power Supply 4 g CO2/h 150 600 g CO2 Oscilloscope 4 g CO2/h 150 600 g CO2 Electronic Load 4 g CO2/h 150 600 g CO2 Function Generator 4 g CO2/h 150 600 g CO2 Soldering Station 200 g CO2/h 50 10000 g CO2 Motorbike 54 g CO2/km* 2800 km* 151200 g CO2 Total 165760 g CO2 We have determined that the total environmental emission of the project is 165.760 g CO2. Looking at the calculations, we can easily observe that most of these emissions are from the use of the motorbike. In order to take measures with the objective of reducing the environmental impact for future projects, the best option will be using another mean of transport. 5.2. Economic Impact The cost associated with the development of the project include human resources, material and equipment costs and software costs. -88The human costs can be easily calculated since a total of 540 hours were spent by one member. The price that the UPC fix for students is 10€/hour. This is a total cost of 5400€ due to human resources. The costs associated to material and equipment used is not that easy to estimate. The estimated cost of the materials is 200€, including the PCB, components, etc. The cost of the equipment used is divided in two parts, the cost of the equipment and the cost associated with the consumption of the equipment. For the first part, we will add up what each equipment cost: - Computer LENOVO: 400€ - Power Supply EA-PS 2384-05 B: 1027€ - Oscilloscope Yokogawa DLM2024: 3570€ (the price is not exactly because it is discontinued) - Programmable Function Generator HAMEG HM8131-2: 603€ - Programmable AC/DC Electronic Load Chroma Model 63804: 18640€ - Soldering Station JBC: 100€ The total acquisition cost of the equipment is 24340€. For the second part, we will estimate that the use of the computer have a consumption of 200 W and the laboratory equipment have a total consumption of 150 W. Being the mean prince of the electricity 0.1270€/kWh we can calculate: 𝐶𝑜𝑠𝑡=0.1270 € 𝑘𝑊ℎ·(200 𝑊·540 ℎ+150 𝑊·150 ℎ) 1000 =16.57€ Considering all the costs calculated we obtain a total cost of 29956.57€. 5400€+200€+24340€+16.57€=29956.57€ Obviously, this is not the real cost of the project, since most part of the cost is from the acquisition of the laboratory equipment, which it is used for lots of projects. Taking into account that the equipment used is already amortised we can underestimate the cost of acquisition, rising a total of 5616.57€. 5.3. Social impact Since the project is oriented towards research purposes and aimed at researchers, engineers and professionals in the field of power electronics, the social impact it may have is minimal. There is no direct relationship with people outside the area of study of the project. -89Obviously, the development of the project may have some social impact, but only in the personal sphere of the people who have carried it out. In addition, this project could lead to other projects, so that other students could expand the research, thus having a certain impact. -96Case 1: Vout = 12 V fsw = 124.91 kHz, ∆ = 0.405 fsw = 102.45 kHz, ∆ = 0.511 fsw = 80.84 kHz, ∆ = 0.624 fsw = 60.76 kHz, ∆ = 0.823 fsw = 39.97 kHz, ∆ = 1.24 fsw = 24.16 kHz, ∆ = 2.195 -97Case 2: Vout = 21 V fsw = 124.01 kHz, ∆ = 0.187 fsw = 99.66 kHz, ∆ = 0.239 fsw = 82.2 kHz, ∆ = 0.285 fsw = 59 kHz, ∆ = 0.399 fsw = 41.03 kHz, ∆ = 0.585 fsw = 23.87 kHz, ∆ = 1.05 -98Case 3: Vout = 3 V fsw = 122 kHz, ∆ = 0.185 fsw = 101.36 kHz, ∆ = 0.224 fsw = 81.89 kHz, ∆ = 0.279 fsw = 59.8 kHz, ∆ = 0.378 fsw = 40.22 kHz, ∆ = 0.539 fsw = 24.73 kHz, ∆ = 0.739