Chapter 9 Sinusoids and Phasors
sinusoid医学翻译
sinusoid医学翻译sinusoid的医学翻译为"窦状血窦"，它是指一种血管结构，典型地表现为一条长而细的、波浪状的管道，类似于正弦曲线。
窦状血窦广泛分布于许多组织和器官中，包括肝脏、脾脏和骨髓等。
以下是10个关于窦状血窦的双语例句：1. The liver is rich in sinusoids, which allow the exchange of nutrients and waste products between blood and liver cells.肝脏富含窦状血窦，可以在血液和肝细胞之间进行营养和废物的交换。
2. The spleen is a major site for the destruction of old red blood cells, and sinusoids in the spleen play a role in this process.脾脏是旧红细胞破坏的主要场所，脾脏中的窦状血窦在这一过程中起着作用。
3. Sinusoids in the bone marrow are responsible for the production and release of blood cells into the bloodstream.骨髓中的窦状血窦负责产生和释放血液细胞到血液中。
4. The structure of sinusoids allows for the efficient exchange of substances between blood and surrounding tissues.窦状血窦的结构使得血液与周围组织之间的物质交流更加高效。
5. Diseases such as cirrhosis can disrupt the normal functioning of sinusoids in the liver.像肝硬化这样的疾病可能会破坏肝脏中窦状血窦的正常功能。
小王子第七、八、九章读后感英文
小王子第七、八、九章读后感英文After reading Chapters 7, 8, and 9 of "The Little Prince", I was struck by the deep, philosophical musings of the young prince as he encounters various characters on his journey through the universe. These chapters delve into themes of love, loneliness, and the nature of human relationships, offering poignant insights into the complexity of human emotions.In Chapter 7, the Little Prince meets the Fox, who imparts to him the wisdom of building meaningful relationships through time and effort. The Fox teaches the Little Prince that relationships cannot be formed instantaneously but require the investment of time and care. This lesson resonated with me as it reminded me of the importance of patience and perseverance in cultivating relationships with others.Chapter 8 introduces the Little Prince to the character of the Railway Switchman, who laments the monotony of his existence and the lack of meaning in his job. This chapter serves as a critique of modern society and the dehumanizing effects of industrialization. Through the Switchman's story, the Little Prince learns that true happiness cannot be found in material possessions or societal expectations but in pursuing one's passions and connecting with others on a deeper level.In Chapter 9, the Little Prince encounters the Geographer, who represents a stark contrast to the imaginative and curious nature of the Little Prince. The Geographer is preoccupied with collecting data and mapping out the universe, but he lacks a true understanding of the worlds he studies. This chapter explores the limitations of knowledge and the importance of viewing the world with an open mind and a sense of wonder.Overall, these chapters of "The Little Prince" invite readers to reflect on the complexities of human relationships, the pursuit of happiness, and the nature of knowledge. Through the character of the Little Prince, Antoine de Saint-Exupéry offers timeless lessons on love, loneliness, and the meaning of life that continue to resonate with readers of all ages. I was moved by the profound wisdom contained in these chapters and inspired to approach life with a sense of curiosity, empathy, and humility.。
Alice's Adventures in Wonderland(《爱丽丝漫游奇境》英文原版)第九章
Alice's Adventures in Wonderland爱丽丝漫游奇境by Lewis Carroll（刘易斯·卡罗尔）CHAPTER IX: THE MOCK TURTLE’S STORY'You can't think how glad I am to see you again, you dear old thing!' said the Duchess, as she tucked her arm affectionately into Alice's, and they walked off together.Alice was very glad to find her in such a pleasant temper, and thought to herself that perhaps it was only the pepper that had made her so savage when they met in the kitchen.'When I'M a Duchess,' she said to herself, (not in a very hopeful tone though), 'I won't have any pepper in my kitchen AT ALL. Soup does very well without—Maybe it's always pepper that makes people hot–tempered,' she went on, very much pleased at having found out a new kind of rule, 'and vinegar that makes them sour—and camomile that makes them bitter—and—and barley–sugar and such things that make children sweet–tempered. I only wish people knew that: then they wouldn't be so stingy about it, you know—'She had quite forgotten the Duchess by this time, and was a little startled when she heard her voice close to her ear. 'You're thinking about something, my dear, and that makes you forget to talk. I can't tell you just now what the moral of that is, but I shall remember it in a bit.''Perhaps it hasn't one,' Alice ventured to remark.'Tut, tut, child!' said the Duchess. 'Everything's got a moral, if only you can find it.' And she squeezed herself up closer to Alice's side as she spoke.Alice did not much like keeping so close to her: first, because the Duchess was VERY ugly; and secondly, because she was exactly the right height to rest her chin upon Alice's shoulder, and it was an uncomfortably sharp chin. However, she did not like to be rude, so she bore it as well as she could.'The game's going on rather better now,' she said, by way of keeping up the conversation a little.''Tis so,' said the Duchess: 'and the moral of that is—"Oh, 'tis love, 'tis love, that makes the world go round!"''Somebody said,' Alice whispered, 'that it's done by everybody minding their own business!''Ah, well! It means much the same thing,' said the Duchess, digging her sharp little chin into Alice's shoulder as she added, 'and the moral of THAT is—"Take care of the sense, and the sounds will take care of themselves."''How fond she is of finding morals in things!' Alice thought to herself.'I dare say you're wondering why I don't put my arm round your waist,' the Duchess said after a pause: 'the reason is, that I'm doubtful about the temper of your flamingo. Shall I try the experiment?''HE might bite,' Alice cautiously replied, not feeling at all anxious to have the experiment tried.'Very true,' said the Duchess: 'flamingoes and mustard both bite. And the moral of that is—"Birds of a feather flock together."''Only mustard isn't a bird,' Alice remarked.'Right, as usual,' said the Duchess: 'what a clear way you have of putting things!''It's a mineral, I THINK,' said Alice.'Of course it is,' said the Duchess, who seemed ready to agree to everything that Alice said; 'there's a large mustard–mine near here. And the moral of that is—"The more there is of mine, the less there is of yours."''Oh, I know!' exclaimed Alice, who had not attended to this last remark, 'it's a vegetable. It doesn't look like one, but it is.''I quite agree with you,' said the Duchess; 'and the moral of that is—"Be what you would seem to be"—or if you'd like it put more simply—"Never imagine yourself not to be otherwise than what it might appear to others that what you were or might have been was not otherwise than what you had been would have appeared to them to be otherwise."''I think I should understand that better,' Alice said very politely, 'if I had it written down: but I can't quite follow it as you say it.''That's nothing to what I could say if I chose,' the Duchess replied, in a pleased tone.'Pray don't trouble yourself to say it any longer than that,' said Alice.'Oh, don't talk about trouble!' said the Duchess. 'I make you a present of everything I've said as yet.''A cheap sort of present!' thought Alice. 'I'm glad they don't give birthday presents like that!' But she did not venture to say it out loud.'Thinking again?' the Duchess asked, with another dig of her sharp little chin.'I've a right to think,' said Alice sharply, for she was beginning to feel a little worried.'Just about as much right,' said the Duchess, 'as pigs have to fly; and the m—'But here, to Alice's great surprise, the Duchess's voice died away, even in the middle of her favourite word 'moral,' and the arm that was linked into hers began to tremble. Alice looked up, and there stood the Queen in front of them, with her arms folded, frowning like a thunderstorm.'A fine day, your Majesty!' the Duchess began in a low, weak voice.'Now, I give you fair warning,' shouted the Queen, stamping on the ground as she spoke; 'either you or your head must be off, and that in about half no time! Take your choice!'The Duchess took her choice, and was gone in a moment.'Let's go on with the game,' the Queen said to Alice; and Alice was too much frightened to say a word, but slowly followed her back to the croquet–ground.The other guests had taken advantage of the Queen's absence, and were resting in the shade: however, the moment they saw her, they hurried back to the game, the Queen merely remarking that a moment's delay would cost them their lives.All the time they were playing the Queen never left off quarrelling with the other players, and shouting 'Off with his head!' or 'Off with her head!' Those whom she sentenced were taken into custody by the soldiers, who of course had to leave off being arches to do this, so that by the end of half an hour or so there were no archesleft, and all the players, except the King, the Queen, and Alice, were in custody and under sentence of execution.Then the Queen left off, quite out of breath, and said to Alice, 'Have you seen the Mock Turtle yet?''No,' said Alice. 'I don't even know what a Mock Turtle is.''It's the thing Mock Turtle Soup is made from,' said the Queen.'I never saw one, or heard of one,' said Alice.'Come on, then,' said the Queen, 'and he shall tell you his history,'As they walked off together, Alice heard the King say in a low voice, to the company generally, 'You are all pardoned.' 'Come, THAT'S a good thing!' she said to herself, for she had felt quite unhappy at the number of executions the Queen had ordered.They very soon came upon a Gryphon, lying fast asleep in the sun. (IF you don't know what a Gryphon is, look at the picture.) 'Up, lazy thing!' said the Queen, 'and take this young lady to see the Mock Turtle, and to hear his history. I must go back and see after some executions I have ordered'; and she walked off, leaving Alice alone with the Gryphon. Alice did not quite like the look of the creature, but on the whole she thought it would be quite as safe to stay with it as to go after that savage Queen: so she waited.The Gryphon sat up and rubbed its eyes: then it watched the Queen till she was out of sight: then it chuckled. 'What fun!' said the Gryphon, half to itself, half to Alice.'What IS the fun?' said Alice.'Why, SHE,' said the Gryphon. 'It's all her fancy, that: they never executes nobody, you know. Come on!''Everybody says "come on!" here,' thought Alice, as she went slowly after it: 'I never was so ordered about in all my life, never!'They had not gone far before they saw the Mock Turtle in the distance, sitting sad and lonely on a little ledge of rock, and, as they came nearer, Alice could hear him sighing as if his heart would break. She pitied him deeply. 'What is his sorrow?' she asked the Gryphon, and the Gryphon answered, very nearly in the same words as before, 'It's all his fancy, that: he hasn't got no sorrow, you know. Come on!'So they went up to the Mock Turtle, who looked at them with large eyes full of tears, but said nothing.'This here young lady,' said the Gryphon, 'she wants for to know your history, she do.''I'll tell it her,' said the Mock Turtle in a deep, hollow tone: 'sit down, both of you, and don't speak a word till I've finished.'So they sat down, and nobody spoke for some minutes. Alice thought to herself, 'I don't see how he can EVEN finish, if he doesn't begin.' But she waited patiently.'Once,' said the Mock Turtle at last, with a deep sigh, 'I was a real Turtle.'These words were followed by a very long silence, broken only by an occasional exclamation of 'Hjckrrh!' from the Gryphon, and the constant heavy sobbing of the Mock Turtle. Alice was very nearly getting up and saying, 'Thank you, sir, for your interesting story,' but she could not help thinking there MUST be more to come, so she sat still and said nothing.'When we were little,' the Mock Turtle went on at last, more calmly, though still sobbing a little now and then, 'we went to school in the sea. The master was an old Turtle—we used to call him Tortoise—''Why did you call him Tortoise, if he wasn't one?' Alice asked.'We called him Tortoise because he taught us,' said the Mock Turtle angrily: 'really you are very dull!''You ought to be ashamed of yourself for asking such a simple question,' added the Gryphon; and then they both sat silent and looked at poor Alice, who felt ready to sink into the earth. At last the Gryphon said to the Mock Turtle, 'Drive on, old fellow! Don't be all day about it!' and he went on in these words:'Yes, we went to school in the sea, though you mayn't believe it—''I never said I didn't!' interrupted Alice.'You did,' said the Mock Turtle.'Hold your tongue!' added the Gryphon, before Alice could speak again. The Mock Turtle went on.'We had the best of educations—in fact, we went to school every day—''I'VE been to a day–school, too,' said Alice; 'you needn't be so proud as all that.''With extras?' asked the Mock Turtle a little anxiously.'Yes,' said Alice, 'we learned French and music.''And washing?' said the Mock Turtle.'Certainly not!' said Alice indignantly.'Ah! then yours wasn't a really good school,' said the Mock Turtle in a tone of great relief. 'Now at OURS they had at the end of the bill, "French, music, AND WASHING—extra."''You couldn't have wanted it much,' said Alice; 'living at the bottom of the sea.''I couldn't afford to learn it.' said the Mock Turtle with a sigh. 'I only took the regular course.''What was that?' inquired Alice.'Reeling and Writhing, of course, to begin with,' the Mock Turtle replied; 'and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.''I never heard of "Uglification,"' Alice ventured to say. 'What is it?'The Gryphon lifted up both its paws in surprise. 'What! Never heard of uglifying!' it exclaimed. 'You know what to beautify is, I suppose?''Yes,' said Alice doubtfully: 'it means—to—make—anything—prettier.''Well, then,' the Gryphon went on, 'if you don't know what to uglify is, you ARE a simpleton.'Alice did not feel encouraged to ask any more questions about it, so she turned to the Mock Turtle, and said 'What else had you to learn?''Well, there was Mystery,' the Mock Turtle replied, counting off the subjects on his flappers, '—Mystery, ancient and modern, with Seaography: then Drawling—the Drawling–master was an old conger–eel, that used to come once a week: HE taught us Drawling, Stretching, and Fainting in Coils.''What was THAT like?' said Alice.'Well, I can't show it you myself,' the Mock Turtle said: 'I'm too stiff. And the Gryphon never learnt it.''Hadn't time,' said the Gryphon: 'I went to the Classics master, though. He was an old crab, HE was.''I never went to him,' the Mock Turtle said with a sigh: 'he taught Laughing and Grief, they used to say.''So he did, so he did,' said the Gryphon, sighing in his turn; and both creatures hid their faces in their paws.'And how many hours a day did you do lessons?' said Alice, in a hurry to change the subject.'Ten hours the first day,' said the Mock Turtle: 'nine the next, and so on.''What a curious plan!' exclaimed Alice.'That's the reason they're called lessons,' the Gryphon remarked: 'because they lessen from day to day.'This was quite a new idea to Alice, and she thought it over a little before she made her next remark. 'Then the eleventh day must have been a holiday?''Of course it was,' said the Mock Turtle.'And how did you manage on the twelfth?' Alice went on eagerly.'That's enough about lessons,' the Gryphon interrupted in a very decided tone: 'tell her something about the games now.'。
TrendReader Standard 2说明书
COMPREHENSIVE Excel Tutorial 8 Developing an Excel Application.\n \n \n \n \n "," \n \n \n \n \n \n TrendReader Standard 2 This generation of TrendReader Standard software utilizes the more familiar Windows format (\u201ctree\u201d) views of functions and file.\n \n \n \n \n "," \n \n \n \n \n \n E-LABORATORY PRACTICAL TEACHING FOR APPLIED ENGINEERING SCIENCES W O R K S H O P University of Oradea, Romania February 6, 2012 G E N E R A L P R E S E.\n \n \n \n \n "," \n \n \n \n \n \n Microsoft Office 2007 Access 2007 Chapter 9 Administering a Database System.\n \n \n \n \n "," \n \n \n \n \n \n Hands-On Virtual Computing\n \n \n \n \n "," \n \n \n \n \n \n Understand the concept to improve parameter setting\n \n \n \n \n "," \n \n \n \n \n \n SFT 2841 IN CONNECTED MODE Prepare setting files.\n \n \n \n \n "," \n \n \n \n \n \n Electrical Distribution Training\n \n \n \n \n "," \n \n \n \n \n \n 1 Fluke DAQ Software Visual Demo Guide 2680A-APSW.\n \n \n \n \n "," \n \n \n \n \n \n | | Tel: | | Computer Training & Personal Development Outlook Express Complete.\n \n \n \n \n "," \n \n \n \n \n \n 4 May I\/O Control. 4 May What is I\/O Control? A Doors feature that allows you to program input events to drive output responses \u2013 for example:\n \n \n \n \n "," \n \n \n \n \n \n Copyright \u00a9 2007, Oracle. All rights reserved. Managing Concurrent Requests.\n \n \n \n \n "," \n \n \n \n \n \n All rights reserved, property and \u00a9 CAD Computer GmbH & Co.KG 2009 Cover page.\n \n \n \n \n "," \n \n \n \n \n \n Electrical Distribution Training\n \n \n \n \n "," \n \n \n \n \n \n Current operation (black keys) Descriptions Parameter \/ protection setting (blue keys) Main points to remember A B C D ? SEPAM UMIs SEPAM UMI OPERATION.\n \n \n \n \n "," \n \n \n \n \n \n All rights reserved, property and \u00a9 CAD Computer GmbH & Co.KG 2009 Cover page.\n \n \n \n \n "," \n \n \n \n \n \n Specview 32 Release 2.5 Enhancements\n \n \n \n \n "," \n \n \n \n \n \n VistA Imaging Capture via Scanning. October VistA Imaging Capture via Scanning The information in this documentation includes only new and updated.\n \n \n \n \n "," \n \n \n \n \n \n Local Alarm Station Data Acquisition, Storage and Visualization for Radiation Portal Monitor (RPM).\n \n \n \n \n "," \n \n \n \n \n \n Electrical Distribution Training Logic Equation Editor.\n \n \n \n \n "," \n \n \n \n \n \n VistA Imaging Workstation Configuration. October The information in this documentation includes functionality of the software after the installation.\n \n \n \n \n "," \n \n \n \n \n \n Programmable Logic Training Course HDL Editor\n \n \n \n \n "," \n \n \n \n \n \n Chapter 10 Function Block Diagram\n \n \n \n \n "," \n \n \n \n \n \n 11 INSTALLING AND MANAGING HARDWARE Chapter 6. Chapter 6: Installing and Managing Hardware2 INSTALLING AND MANAGING HARDWARE \uf0a7 Install hardware in a Microsoft.\n \n \n \n \n "," \n \n \n \n \n \n Overview of MSWS Control of 212-LC July 15, 2006.\n \n \n \n \n "," \n \n \n \n \n \n VApp Product Support Engineering Rev E VMware Confidential.\n \n \n \n \n "," \n \n \n \n \n \n SmartMQn Motor Horner APG, LLC September 9, 2008.\n \n \n \n \n "," \n \n \n \n \n \n JDS5 Training Guide. On Start Up you will see this screen click the OK button Click OK.\n \n \n \n \n "," \n \n \n \n \n \n Quick guide to ASIMON configuration For version 3.0 or greater SAFETY AT WORK Date: 3\/18\/2009.\n \n \n \n \n "," \n \n \n \n \n \n Chapter \u2013 8 Software Tools.\n \n \n \n \n "," \n \n \n \n \n \n Chapter 5 Introduction To Form Builder. Lesson C Objectives \uf07d Use sequences to automatically generate primary key values in a form\uf07d Create lists of values.\n \n \n \n \n "," \n \n \n \n \n \n \u00a9 2013 Eaton Corporation. All rights reserved. easy Programmable Relay Training Program Exercise 3.\n \n \n \n \n "," \n \n \n \n \n \n Chapter 10 Using Macros, Controls and Visual Basic for Applications (VBA) with Excel Microsoft Excel 2013.\n \n \n \n \n "," \n \n \n \n \n \n Chapter 5.\n \n \n \n \n "," \n \n \n \n \n \n PROPRIETARY INFORMATION - \u00a9 2015 WOODWARD, INC. PAGE 1 HighPROTEC-2 MRI4-2 Feeder Relay Sales presentation 2015\/08\/10.\n \n \n \n \n "," \n \n \n \n \n \n HighPROTEC-2 MRA4-2 Directional Feeder Relay Sales presentation\n \n \n \n \n "," \n \n \n \n \n \n Local Alarm Station Data Acquisition, Storage and Visualization for Radiation Portal Monitor (RPM).\n \n \n \n \n "," \n \n \n \n \n \n DIGITAL MULTIMETER DMG\n \n \n \n \n "," \n \n \n \n \n \n Excel Tutorial 8 Developing an Excel Application\n \n \n \n \n "," \n \n \n \n \n \n HighPROTEC-2 MRM4-2 Motor Protection Relay Sales presentation\n \n \n \n \n "," \n \n \n \n \n \n Monitoring systems COMET types MS55 & MS6\n \n \n \n \n "," \n \n \n \n \n \n HighPROTEC-2 MCA4-2 Directional Feeder Relay Sales presentation\n \n \n \n \n "," \n \n \n \n \n \n Haiwell PLC Modules Introduce\n \n \n \n \n "," \n \n \n \n \n \n CONFIGURING HARDWARE DEVICE & START UP PROCESS\n \n \n \n \n "," \n \n \n \n \n \nMcCrometer Nano Board Test Procedure\n \n \n \n \n "," \n \n \n \n \n \n HighPROTEC-2 MRDT4-2 Transformer Protection with differential Sales presentation 2015\/08\/10.\n \n \n \n \n "," \n \n \n \n \n \n Training Module Introduction to the TB9100\/P25 CG\/P25 TAG Customer Service Software (CSS) Describes Release 3.95 for Trunked TB9100 and P25 TAG Release.\n \n \n \n \n "," \n \n \n \n \n \n easYgen-3000XT Series Training\n \n \n \n \n "]; Similar presentationssft 2841 authorization codeDownload Zip ☆☆☆☆☆ https://t.co/LJGzQQSy0JMonitoring systems COMET types MS55 & MS6 MS55\/MS6 Configuration.\n \n \n \n \n "," \n \n \n \n \n \n Fundamentals of Electric Circuits Chapter 11\n \n \n \n \n "," \n \n \n \n \n \n DEMONSTRATION FOR SIGMA DATA ACQUISITION MODULES Tempatron Ltd Data Measurements Division Darwin Close Reading RG2 0TB UK T : +44 (0) F :\n \n \n \n \n "," \n \n \n \n \n \n Workflow Based Tools Simplifying All Job Functions.\n \n \n \n \n "," \n \n \n \n \n \n SCADA and Telemetry Presented By:.\n \n \n \n \n "," \n \n \n \n \n \n G650 Generator Protection & Control System\n \n \n \n \n "," \n \n \n \n \n \n E-LABORATORY PRACTICAL TEACHING FOR APPLIED ENGINEERING SCIENCES W O R K S H O P University of Oradea, Romania February 6, 2012 G E N E R A L P R E S E.\n \n \n \n \n "," \n \n \n \n \n \n ECE Electric Drives Topic 10: Cycloconverters Spring 2004.\n \n \n \n \n "," \n \n \n \n \n \n Average Power and Power Factor ET 242 Circuit Analysis II Electrical and Telecommunication Engineering Technology Professor Jang.\n \n \n \n \n "," \n \n \n \n \n \n Fundamentals of Electric Circuits Chapter 11\n \n \n \n \n "," \n \n \n \n \n \n SunGuide SM Software Development Project Ramp Meter Firmware Discussions August 8, 2006.\n \n \n \n \n "," \n \n \n \n \n \n POWER PLANT USED IN TELECOM\n \n \n \n \n "," \n \n \n \n \n \n Single Loop Analogue addressable fire control Panel\n \n \n \n \n "," \n \n \n \n \n \n Electrical Distribution Training\n \n \n \n \n "," \n \n \n \n \n \n DLS Digital Controller Tony Dobbing Head of Power Supplies Group.\n \n \n \n \n "," \n \n \n \n \n \n A new class in PC Data Acquisition Delphin Technology AG Felix Ottofuelling V5.0.\n \n \n \n \n "," \n \n \n \n \n \n Chapter 7 AC Power Analysis\n \n \n \n \n "," \n \n \n \n \n \n Electrical Distribution Training\n \n \n \n \n "," \n \n \n \n \n \n Current operation (black keys) Descriptions Parameter \/ protection setting (blue keys) Main points to remember A B C D ? 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西塞罗作品英文版
西塞罗作品英文版Cicero's Works in English Translation: A Timeless LegacyCicero, the renowned Roman statesman, philosopher, and orator, has left an indelible mark on the Western intellectual tradition. His extensive body of work, spanning a wide range of topics, from politics and rhetoric to philosophy and ethics, has been the subject of intense scholarly interest and admiration for centuries. The enduring relevance of Cicero's ideas and the profound impact they have had on the development of Western thought have led to a rich tradition of translating his works into various languages, including English.The English translations of Cicero's works have played a crucial role in disseminating his ideas and making them accessible to a global audience. One of the earliest and most influential English translations of Cicero's works was that of the 16th-century scholar, Thomas Bentley. Bentley's translations of Cicero's philosophical treatises, such as "On the Nature of the Gods" and "On Duties," helped to introduce Cicero's thought to a wider English-speaking readership and laid thefoundation for the subsequent proliferation of Cicero's works in the English language.Over the centuries, numerous scholars and translators have contributed to the English translation of Cicero's vast literary corpus. In the 17th and 18th centuries, figures such as Meric Casaubon, Walter Moyle, and Charles Yonge produced new translations of Cicero's works, further expanding the reach of his ideas in the English-speaking world. These translations not only conveyed the content of Cicero's writings but also sought to capture the elegance and rhetorical power of his original Latin prose.One of the most significant and comprehensive English translations of Cicero's works was undertaken in the late 19th and early 20th centuries by the renowned scholar, Walter Ker. Ker's translations, published in a multi-volume series, covered a vast array of Cicero's writings, including his political speeches, philosophical treatises, and letters. Ker's meticulous and scholarly approach to translation ensured that the nuances and complexities of Cicero's thought were faithfully rendered in English, making his works more accessible to a modern audience.In more recent decades, the translation of Cicero's works has continued, with new generations of scholars and translators bringing fresh perspectives and approaches to the task. The Oxford UniversityPress, for example, has published a series of high-quality translations of Cicero's works, including "On the Commonwealth" and "On the Laws," which have been widely acclaimed for their accuracy and readability.The enduring popularity and relevance of Cicero's works in the English-speaking world can be attributed to several factors. Firstly, Cicero's writings touch upon timeless themes and questions that continue to resonate with readers today. His insights into the nature of politics, the role of the state, and the pursuit of virtue and justice remain highly relevant in the modern world. Moreover, Cicero's mastery of rhetoric and his ability to articulate complex ideas with clarity and eloquence have ensured that his works continue to be studied and admired by students of language, literature, and philosophy.Furthermore, the English translations of Cicero's works have played a crucial role in preserving and disseminating his legacy. By making his writings accessible to a wider audience, these translations have facilitated a deeper understanding and appreciation of Cicero's contributions to Western thought. They have also inspired new generations of scholars and thinkers to engage with Cicero's ideas and to explore their relevance in the contemporary world.In conclusion, the English translations of Cicero's works represent asignificant and enduring aspect of the Western intellectual tradition. From the pioneering efforts of 16th-century scholars to the latest scholarly endeavors, the translation of Cicero's writings has ensured that his ideas and his influence continue to be felt across the globe. As we navigate the complexities of the modern world, the timeless wisdom and insights found in Cicero's works remain a valuable resource, guiding us in our pursuit of justice, virtue, and the betterment of society.。
江苏省南京市2025届高三英语下学期第三次模拟试题含解析
江苏省南京市2025届高三英语下学期第三次模拟试题（含解析）第一部分听力（共两节，满分30分）做题时，先将答案标在试卷上。
录音内容结束后。
你将有两分钟的时间将试卷上的答案转涂到答题卡上。
第一节（共5小题；每小题1.5分，满分7.5分）听下面5段对话。
每段对话后有一个小题。
从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。
听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题，每段对话仅读一遍。
1. 【此处可播放相关音频，请去附件查看】When does the conversation probably take place?A. In the morning.B. In the afternoon.C. In the evening.【答案】B【解析】【原文】M: I’m so hungry, Mom. Is dinner going to be ready soon?W: We just had lunch an hour ago! And you had two full plates of breakfast when you woke up.2. 【此处可播放相关音频，请去附件查看】What is the full price of the man’s jacket？A. $15.B. $30.C. $50.【答案】B【解析】【原文】W: So, what did you buy?M: A jacket. It was a real bargain. I got it for half price, so I saved 15 dollars.W: That’s very cheap. I bought a similar o ne for 50 dollars last year.3. 【此处可播放相关音频，请去附件查看】Who is the woman?A. A passenger.B. A health worker.C. A customs officer.【答案】C【解析】【原文】M: Is the customs examination here?W: That’s right. Your passport and health certificate, please.M: Here you are.4. 【此处可播放相关音频，请去附件查看】What does the woman know about?A. Major rivers.B. Famous mountains.C. Capital cities.【答案】C【解析】【原文】M: Do you know the major rivers and famous mountains in Europe?W: Hm. Ask me about capital cities instead.5. 【此处可播放相关音频，请去附件查看】Why does the man meet the woman？A. To apply for a job.B. To sell her something.C. To reserve a seat.【答案】A【解析】【原文】W: Take a seat, Mr. Black. Could you tell me which position interests you most?M: The sales manager position.W: OK. But do you have any relevant experience?其次节（共15小题；每小题1.5分，满分22.5分）听下面5段对话或独白。
OSHA现场作业手册说明书
DIRECTIVE NUMBER: CPL 02-00-150 EFFECTIVE DATE: April 22, 2011 SUBJECT: Field Operations Manual (FOM)ABSTRACTPurpose: This instruction cancels and replaces OSHA Instruction CPL 02-00-148,Field Operations Manual (FOM), issued November 9, 2009, whichreplaced the September 26, 1994 Instruction that implemented the FieldInspection Reference Manual (FIRM). The FOM is a revision of OSHA’senforcement policies and procedures manual that provides the field officesa reference document for identifying the responsibilities associated withthe majority of their inspection duties. This Instruction also cancels OSHAInstruction FAP 01-00-003 Federal Agency Safety and Health Programs,May 17, 1996 and Chapter 13 of OSHA Instruction CPL 02-00-045,Revised Field Operations Manual, June 15, 1989.Scope: OSHA-wide.References: Title 29 Code of Federal Regulations §1903.6, Advance Notice ofInspections; 29 Code of Federal Regulations §1903.14, Policy RegardingEmployee Rescue Activities; 29 Code of Federal Regulations §1903.19,Abatement Verification; 29 Code of Federal Regulations §1904.39,Reporting Fatalities and Multiple Hospitalizations to OSHA; and Housingfor Agricultural Workers: Final Rule, Federal Register, March 4, 1980 (45FR 14180).Cancellations: OSHA Instruction CPL 02-00-148, Field Operations Manual, November9, 2009.OSHA Instruction FAP 01-00-003, Federal Agency Safety and HealthPrograms, May 17, 1996.Chapter 13 of OSHA Instruction CPL 02-00-045, Revised FieldOperations Manual, June 15, 1989.State Impact: Notice of Intent and Adoption required. See paragraph VI.Action Offices: National, Regional, and Area OfficesOriginating Office: Directorate of Enforcement Programs Contact: Directorate of Enforcement ProgramsOffice of General Industry Enforcement200 Constitution Avenue, NW, N3 119Washington, DC 20210202-693-1850By and Under the Authority ofDavid Michaels, PhD, MPHAssistant SecretaryExecutive SummaryThis instruction cancels and replaces OSHA Instruction CPL 02-00-148, Field Operations Manual (FOM), issued November 9, 2009. The one remaining part of the prior Field Operations Manual, the chapter on Disclosure, will be added at a later date. This Instruction also cancels OSHA Instruction FAP 01-00-003 Federal Agency Safety and Health Programs, May 17, 1996 and Chapter 13 of OSHA Instruction CPL 02-00-045, Revised Field Operations Manual, June 15, 1989. This Instruction constitutes OSHA’s general enforcement policies and procedures manual for use by the field offices in conducting inspections, issuing citations and proposing penalties.Significant Changes∙A new Table of Contents for the entire FOM is added.∙ A new References section for the entire FOM is added∙ A new Cancellations section for the entire FOM is added.∙Adds a Maritime Industry Sector to Section III of Chapter 10, Industry Sectors.∙Revises sections referring to the Enhanced Enforcement Program (EEP) replacing the information with the Severe Violator Enforcement Program (SVEP).∙Adds Chapter 13, Federal Agency Field Activities.∙Cancels OSHA Instruction FAP 01-00-003, Federal Agency Safety and Health Programs, May 17, 1996.DisclaimerThis manual is intended to provide instruction regarding some of the internal operations of the Occupational Safety and Health Administration (OSHA), and is solely for the benefit of the Government. No duties, rights, or benefits, substantive or procedural, are created or implied by this manual. The contents of this manual are not enforceable by any person or entity against the Department of Labor or the United States. Statements which reflect current Occupational Safety and Health Review Commission or court precedents do not necessarily indicate acquiescence with those precedents.Table of ContentsCHAPTER 1INTRODUCTIONI.PURPOSE. ........................................................................................................... 1-1 II.SCOPE. ................................................................................................................ 1-1 III.REFERENCES .................................................................................................... 1-1 IV.CANCELLATIONS............................................................................................. 1-8 V. ACTION INFORMATION ................................................................................. 1-8A.R ESPONSIBLE O FFICE.......................................................................................................................................... 1-8B.A CTION O FFICES. .................................................................................................................... 1-8C. I NFORMATION O FFICES............................................................................................................ 1-8 VI. STATE IMPACT. ................................................................................................ 1-8 VII.SIGNIFICANT CHANGES. ............................................................................... 1-9 VIII.BACKGROUND. ................................................................................................. 1-9 IX. DEFINITIONS AND TERMINOLOGY. ........................................................ 1-10A.T HE A CT................................................................................................................................................................. 1-10B. C OMPLIANCE S AFETY AND H EALTH O FFICER (CSHO). ...........................................................1-10B.H E/S HE AND H IS/H ERS ..................................................................................................................................... 1-10C.P ROFESSIONAL J UDGMENT............................................................................................................................... 1-10E. W ORKPLACE AND W ORKSITE ......................................................................................................................... 1-10CHAPTER 2PROGRAM PLANNINGI.INTRODUCTION ............................................................................................... 2-1 II.AREA OFFICE RESPONSIBILITIES. .............................................................. 2-1A.P ROVIDING A SSISTANCE TO S MALL E MPLOYERS. ...................................................................................... 2-1B.A REA O FFICE O UTREACH P ROGRAM. ............................................................................................................. 2-1C. R ESPONDING TO R EQUESTS FOR A SSISTANCE. ............................................................................................ 2-2 III. OSHA COOPERATIVE PROGRAMS OVERVIEW. ...................................... 2-2A.V OLUNTARY P ROTECTION P ROGRAM (VPP). ........................................................................... 2-2B.O NSITE C ONSULTATION P ROGRAM. ................................................................................................................ 2-2C.S TRATEGIC P ARTNERSHIPS................................................................................................................................. 2-3D.A LLIANCE P ROGRAM ........................................................................................................................................... 2-3 IV. ENFORCEMENT PROGRAM SCHEDULING. ................................................ 2-4A.G ENERAL ................................................................................................................................................................. 2-4B.I NSPECTION P RIORITY C RITERIA. ..................................................................................................................... 2-4C.E FFECT OF C ONTEST ............................................................................................................................................ 2-5D.E NFORCEMENT E XEMPTIONS AND L IMITATIONS. ....................................................................................... 2-6E.P REEMPTION BY A NOTHER F EDERAL A GENCY ........................................................................................... 2-6F.U NITED S TATES P OSTAL S ERVICE. .................................................................................................................. 2-7G.H OME-B ASED W ORKSITES. ................................................................................................................................ 2-8H.I NSPECTION/I NVESTIGATION T YPES. ............................................................................................................... 2-8 V.UNPROGRAMMED ACTIVITY – HAZARD EVALUATION AND INSPECTION SCHEDULING ............................................................................ 2-9 VI.PROGRAMMED INSPECTIONS. ................................................................... 2-10A.S ITE-S PECIFIC T ARGETING (SST) P ROGRAM. ............................................................................................. 2-10B.S CHEDULING FOR C ONSTRUCTION I NSPECTIONS. ..................................................................................... 2-10C.S CHEDULING FOR M ARITIME I NSPECTIONS. ............................................................................. 2-11D.S PECIAL E MPHASIS P ROGRAMS (SEP S). ................................................................................... 2-12E.N ATIONAL E MPHASIS P ROGRAMS (NEP S) ............................................................................... 2-13F.L OCAL E MPHASIS P ROGRAMS (LEP S) AND R EGIONAL E MPHASIS P ROGRAMS (REP S) ............ 2-13G.O THER S PECIAL P ROGRAMS. ............................................................................................................................ 2-13H.I NSPECTION S CHEDULING AND I NTERFACE WITH C OOPERATIVE P ROGRAM P ARTICIPANTS ....... 2-13CHAPTER 3INSPECTION PROCEDURESI.INSPECTION PREPARATION. .......................................................................... 3-1 II.INSPECTION PLANNING. .................................................................................. 3-1A.R EVIEW OF I NSPECTION H ISTORY .................................................................................................................... 3-1B.R EVIEW OF C OOPERATIVE P ROGRAM P ARTICIPATION .............................................................................. 3-1C.OSHA D ATA I NITIATIVE (ODI) D ATA R EVIEW .......................................................................................... 3-2D.S AFETY AND H EALTH I SSUES R ELATING TO CSHO S.................................................................. 3-2E.A DVANCE N OTICE. ................................................................................................................................................ 3-3F.P RE-I NSPECTION C OMPULSORY P ROCESS ...................................................................................................... 3-5G.P ERSONAL S ECURITY C LEARANCE. ................................................................................................................. 3-5H.E XPERT A SSISTANCE. ........................................................................................................................................... 3-5 III. INSPECTION SCOPE. ......................................................................................... 3-6A.C OMPREHENSIVE ................................................................................................................................................... 3-6B.P ARTIAL. ................................................................................................................................................................... 3-6 IV. CONDUCT OF INSPECTION .............................................................................. 3-6A.T IME OF I NSPECTION............................................................................................................................................. 3-6B.P RESENTING C REDENTIALS. ............................................................................................................................... 3-6C.R EFUSAL TO P ERMIT I NSPECTION AND I NTERFERENCE ............................................................................. 3-7D.E MPLOYEE P ARTICIPATION. ............................................................................................................................... 3-9E.R ELEASE FOR E NTRY ............................................................................................................................................ 3-9F.B ANKRUPT OR O UT OF B USINESS. .................................................................................................................... 3-9G.E MPLOYEE R ESPONSIBILITIES. ................................................................................................. 3-10H.S TRIKE OR L ABOR D ISPUTE ............................................................................................................................. 3-10I. V ARIANCES. .......................................................................................................................................................... 3-11 V. OPENING CONFERENCE. ................................................................................ 3-11A.G ENERAL ................................................................................................................................................................ 3-11B.R EVIEW OF A PPROPRIATION A CT E XEMPTIONS AND L IMITATION. ..................................................... 3-13C.R EVIEW S CREENING FOR P ROCESS S AFETY M ANAGEMENT (PSM) C OVERAGE............................. 3-13D.R EVIEW OF V OLUNTARY C OMPLIANCE P ROGRAMS. ................................................................................ 3-14E.D ISRUPTIVE C ONDUCT. ...................................................................................................................................... 3-15F.C LASSIFIED A REAS ............................................................................................................................................. 3-16VI. REVIEW OF RECORDS. ................................................................................... 3-16A.I NJURY AND I LLNESS R ECORDS...................................................................................................................... 3-16B.R ECORDING C RITERIA. ...................................................................................................................................... 3-18C. R ECORDKEEPING D EFICIENCIES. .................................................................................................................. 3-18 VII. WALKAROUND INSPECTION. ....................................................................... 3-19A.W ALKAROUND R EPRESENTATIVES ............................................................................................................... 3-19B.E VALUATION OF S AFETY AND H EALTH M ANAGEMENT S YSTEM. ....................................................... 3-20C.R ECORD A LL F ACTS P ERTINENT TO A V IOLATION. ................................................................................. 3-20D.T ESTIFYING IN H EARINGS ................................................................................................................................ 3-21E.T RADE S ECRETS. ................................................................................................................................................. 3-21F.C OLLECTING S AMPLES. ..................................................................................................................................... 3-22G.P HOTOGRAPHS AND V IDEOTAPES.................................................................................................................. 3-22H.V IOLATIONS OF O THER L AWS. ....................................................................................................................... 3-23I.I NTERVIEWS OF N ON-M ANAGERIAL E MPLOYEES .................................................................................... 3-23J.M ULTI-E MPLOYER W ORKSITES ..................................................................................................................... 3-27 K.A DMINISTRATIVE S UBPOENA.......................................................................................................................... 3-27 L.E MPLOYER A BATEMENT A SSISTANCE. ........................................................................................................ 3-27 VIII. CLOSING CONFERENCE. .............................................................................. 3-28A.P ARTICIPANTS. ..................................................................................................................................................... 3-28B.D ISCUSSION I TEMS. ............................................................................................................................................ 3-28C.A DVICE TO A TTENDEES .................................................................................................................................... 3-29D.P ENALTIES............................................................................................................................................................. 3-30E.F EASIBLE A DMINISTRATIVE, W ORK P RACTICE AND E NGINEERING C ONTROLS. ............................ 3-30F.R EDUCING E MPLOYEE E XPOSURE. ................................................................................................................ 3-32G.A BATEMENT V ERIFICATION. ........................................................................................................................... 3-32H.E MPLOYEE D ISCRIMINATION .......................................................................................................................... 3-33 IX. SPECIAL INSPECTION PROCEDURES. ...................................................... 3-33A.F OLLOW-UP AND M ONITORING I NSPECTIONS............................................................................................ 3-33B.C ONSTRUCTION I NSPECTIONS ......................................................................................................................... 3-34C. F EDERAL A GENCY I NSPECTIONS. ................................................................................................................. 3-35CHAPTER 4VIOLATIONSI. BASIS OF VIOLATIONS ..................................................................................... 4-1A.S TANDARDS AND R EGULATIONS. .................................................................................................................... 4-1B.E MPLOYEE E XPOSURE. ........................................................................................................................................ 4-3C.R EGULATORY R EQUIREMENTS. ........................................................................................................................ 4-6D.H AZARD C OMMUNICATION. .............................................................................................................................. 4-6E. E MPLOYER/E MPLOYEE R ESPONSIBILITIES ................................................................................................... 4-6 II. SERIOUS VIOLATIONS. .................................................................................... 4-8A.S ECTION 17(K). ......................................................................................................................... 4-8B.E STABLISHING S ERIOUS V IOLATIONS ............................................................................................................ 4-8C. F OUR S TEPS TO BE D OCUMENTED. ................................................................................................................... 4-8 III. GENERAL DUTY REQUIREMENTS ............................................................. 4-14A.E VALUATION OF G ENERAL D UTY R EQUIREMENTS ................................................................................. 4-14B.E LEMENTS OF A G ENERAL D UTY R EQUIREMENT V IOLATION.............................................................. 4-14C. U SE OF THE G ENERAL D UTY C LAUSE ........................................................................................................ 4-23D.L IMITATIONS OF U SE OF THE G ENERAL D UTY C LAUSE. ..............................................................E.C LASSIFICATION OF V IOLATIONS C ITED U NDER THE G ENERAL D UTY C LAUSE. ..................F. P ROCEDURES FOR I MPLEMENTATION OF S ECTION 5(A)(1) E NFORCEMENT ............................ 4-25 4-27 4-27IV.OTHER-THAN-SERIOUS VIOLATIONS ............................................... 4-28 V.WILLFUL VIOLATIONS. ......................................................................... 4-28A.I NTENTIONAL D ISREGARD V IOLATIONS. ..........................................................................................4-28B.P LAIN I NDIFFERENCE V IOLATIONS. ...................................................................................................4-29 VI. CRIMINAL/WILLFUL VIOLATIONS. ................................................... 4-30A.A REA D IRECTOR C OORDINATION ....................................................................................................... 4-31B.C RITERIA FOR I NVESTIGATING P OSSIBLE C RIMINAL/W ILLFUL V IOLATIONS ........................ 4-31C. W ILLFUL V IOLATIONS R ELATED TO A F ATALITY .......................................................................... 4-32 VII. REPEATED VIOLATIONS. ...................................................................... 4-32A.F EDERAL AND S TATE P LAN V IOLATIONS. ........................................................................................4-32B.I DENTICAL S TANDARDS. .......................................................................................................................4-32C.D IFFERENT S TANDARDS. .......................................................................................................................4-33D.O BTAINING I NSPECTION H ISTORY. .....................................................................................................4-33E.T IME L IMITATIONS..................................................................................................................................4-34F.R EPEATED V. F AILURE TO A BATE....................................................................................................... 4-34G. A REA D IRECTOR R ESPONSIBILITIES. .............................................................................. 4-35 VIII. DE MINIMIS CONDITIONS. ................................................................... 4-36A.C RITERIA ................................................................................................................................................... 4-36B.P ROFESSIONAL J UDGMENT. ..................................................................................................................4-37C. A REA D IRECTOR R ESPONSIBILITIES. .............................................................................. 4-37 IX. CITING IN THE ALTERNATIVE ............................................................ 4-37 X. COMBINING AND GROUPING VIOLATIONS. ................................... 4-37A.C OMBINING. ..............................................................................................................................................4-37B.G ROUPING. ................................................................................................................................................4-38C. W HEN N OT TO G ROUP OR C OMBINE. ................................................................................................4-38 XI. HEALTH STANDARD VIOLATIONS ....................................................... 4-39A.C ITATION OF V ENTILATION S TANDARDS ......................................................................................... 4-39B.V IOLATIONS OF THE N OISE S TANDARD. ...........................................................................................4-40 XII. VIOLATIONS OF THE RESPIRATORY PROTECTION STANDARD(§1910.134). ....................................................................................................... XIII. VIOLATIONS OF AIR CONTAMINANT STANDARDS (§1910.1000) ... 4-43 4-43A.R EQUIREMENTS UNDER THE STANDARD: .................................................................................................. 4-43B.C LASSIFICATION OF V IOLATIONS OF A IR C ONTAMINANT S TANDARDS. ......................................... 4-43 XIV. CITING IMPROPER PERSONAL HYGIENE PRACTICES. ................... 4-45A.I NGESTION H AZARDS. .................................................................................................................................... 4-45B.A BSORPTION H AZARDS. ................................................................................................................................ 4-46C.W IPE S AMPLING. ............................................................................................................................................. 4-46D.C ITATION P OLICY ............................................................................................................................................ 4-46 XV. BIOLOGICAL MONITORING. ...................................................................... 4-47CHAPTER 5CASE FILE PREPARATION AND DOCUMENTATIONI.INTRODUCTION ............................................................................................... 5-1 II.INSPECTION CONDUCTED, CITATIONS BEING ISSUED. .................... 5-1A.OSHA-1 ................................................................................................................................... 5-1B.OSHA-1A. ............................................................................................................................... 5-1C. OSHA-1B. ................................................................................................................................ 5-2 III.INSPECTION CONDUCTED BUT NO CITATIONS ISSUED .................... 5-5 IV.NO INSPECTION ............................................................................................... 5-5 V. HEALTH INSPECTIONS. ................................................................................. 5-6A.D OCUMENT P OTENTIAL E XPOSURE. ............................................................................................................... 5-6B.E MPLOYER’S O CCUPATIONAL S AFETY AND H EALTH S YSTEM. ............................................................. 5-6 VI. AFFIRMATIVE DEFENSES............................................................................. 5-8A.B URDEN OF P ROOF. .............................................................................................................................................. 5-8B.E XPLANATIONS. ..................................................................................................................................................... 5-8 VII. INTERVIEW STATEMENTS. ........................................................................ 5-10A.G ENERALLY. ......................................................................................................................................................... 5-10B.CSHO S SHALL OBTAIN WRITTEN STATEMENTS WHEN: .......................................................................... 5-10C.L ANGUAGE AND W ORDING OF S TATEMENT. ............................................................................................. 5-11D.R EFUSAL TO S IGN S TATEMENT ...................................................................................................................... 5-11E.V IDEO AND A UDIOTAPED S TATEMENTS. ..................................................................................................... 5-11F.A DMINISTRATIVE D EPOSITIONS. .............................................................................................5-11 VIII. PAPERWORK AND WRITTEN PROGRAM REQUIREMENTS. .......... 5-12 IX.GUIDELINES FOR CASE FILE DOCUMENTATION FOR USE WITH VIDEOTAPES AND AUDIOTAPES .............................................................. 5-12 X.CASE FILE ACTIVITY DIARY SHEET. ..................................................... 5-12 XI. CITATIONS. ..................................................................................................... 5-12A.S TATUTE OF L IMITATIONS. .............................................................................................................................. 5-13B.I SSUING C ITATIONS. ........................................................................................................................................... 5-13C.A MENDING/W ITHDRAWING C ITATIONS AND N OTIFICATION OF P ENALTIES. .................................. 5-13D.P ROCEDURES FOR A MENDING OR W ITHDRAWING C ITATIONS ............................................................ 5-14 XII. INSPECTION RECORDS. ............................................................................... 5-15A.G ENERALLY. ......................................................................................................................................................... 5-15B.R ELEASE OF I NSPECTION I NFORMATION ..................................................................................................... 5-15C. C LASSIFIED AND T RADE S ECRET I NFORMATION ...................................................................................... 5-16。
信号与系统奥本海姆英文版课后答案chapter9
Chapter 9 Answers9.1 （a ）The given integral may be written as(5)0t j t e e dt σω∞-+⎰If σ<-5 ,then the function (5)te σ-+ grows towards ∞ with increasing t and the given integral does not converge .but if >-5,then the integral does converge (b) The given integral may be written as0(5)t j t e e d σω-+-∞⎰tIf σ>-5 ,then the function (5)te σ-+ grows towards ∞as t decreases towards -∞and the given integral does not converge .but if σ<-5,then the integral does converge (c) The given integral may be written as5(5)5t j t e e d σω-+-⎰tClearly this integral has a finite value for all finite values of σ. (d) The given integral may be written as(5)t j t e e d σω∞-+-∞⎰tIf σ>-5 ,then the function (5)t e σ-+ grows towards ∞as t decreases towards -∞and the given integral does not converge If σ<-5, ,then function (5)te σ-+ grows towards ∞ with increasing t and the given integral does not converge If σ=5, then the integral stilldoes not have a finite value. therefore, the integral does not converge for any value of σ. (e) The given integral may be written as 0(5)t j t e e d σω-+-∞⎰t+ (5)0t j t e e d σω∞-+⎰tThe first integral converges for σ<-5, the second internal converges if σ>-5,therefore, the given internal converges when σ<5. (f) The given integral may be written as0(5)t j t e e d σω-+-∞⎰tIf σ>5 ,then the function (5)te σ--+grows towards ∞ as t decrease towards -∞ and the given integral does not converge .but if σ<5,then the integral does converge. 9.2 (a)X(s)= 5(1)t dt eu t e dt ∞---∞⎰- =(5)0s tedt ∞-+⎰ =(5)5s es -++As shown in Example 9.1 the ROC will be {}Re s >-5. (b) By using eg.(9.3), we can easily show that g(t)=A 5te -u(-t-0t ) has the LaplacetransformG(s)= 0(5)5s t Ae s ++The ROC is specified as {}Re s <-5 . Therefore ,A=1 and 0t =-19.3 Using an analysis similar to that used in Example 9.3 we known that given signal has a Laplace transform of the formX(s)115s s β+++The corresponding ROC is {}Re s >max(-5,Re{β}). Since we are given that the ROCis Re{s}>-3, we know that Re{β}=3 . there are no constraints on the imaginary part of β. 9.4 We know form Table 9.2 thatWe also know form Table 9.1 thatx(t)= 1()Lx t -←−→X(s)= 1()X s - The ROC of X(s) is such that if 0s was in the ROC of 1()X s , then -0s will be in the ROC of X(s). Putting the two above equations together ,we havex(t)= 1x (-t) =sin(2)()t e t u t --L ←−→X(s)= 1()X s -=-222(1)2s -+, {}Re s <1the denominator of the form 2s -2s+5. Therefore, the poles of X(s) are 1+2j and 1-2j.9.5 (a) the given Laplace transform may be written as ()X s =24(1)(3)s s s +++.Clearly ,X(s) has a zero at s=-2 .since in X(s) the order of the denominator polynomial exceeds the order of the numerator polynomial by 1 ,X(s) has a zero at ∞. Therefore ,X(s) has one zero in finite s-plane and one zero at infinity. (b) The given Laplance transform may be written asX(s)=1(1)(1)s s s +-+= 11s -Clearly ,X(s) has no zero in the finite s-plane .Since in X(s) the of the denominatorpolynomial exceeds the order the numerator polynomial by 1,X(s) has a zero at ∞.therefore X(s) has no zero in the finite s-plane and one zero at infinity. (c) The given Laplace transform may be written as22(1)(1)()1(1)s s s X s s s s -++==-++Clearly ，X （s ）has a zero at s=1.since in X(s) the order of the numerator polynomial exceeds the order of the denominator polynomial by 1,X(s) has zeros at ∞ .therefore , X(s) has one zero in the s-plane and no zero at infinity .9.6 (a) No. From property 3 in Section 9.2 we know that for a finite-length signal .the ROC is the entire s-plane .therefore .there can be no poles in the finite s-plane for a finite length signal . Clearly in this problem this not the case.(b) Yes. Since the signal is absolutely integrable, The ROC must include, the j ω-axis . Furthermore ,X(s) has a pole at s=2 .therefore, one valid ROC for the signal would be Re{s}<2. From property 5 in section 9.2 we know that this would correspond to a left-sided signal (C) No . Since the signal is absolutely integrable, The ROC must include , the j ω-axis . Furthermore ,X(s) has a pole at s=2. therefore ,we can never have an ROC of the form Re{s}> α. From property 5 in section 9.2 we knew that x(t) can not be a right-side signal(d) Yes . Since the signal is absolutely integrable, The ROC must include , the j ω-axis . Furthermore ,X(s) has a pole at s=2 .therefore, one valid ROC for the signal could be α<Re{s}<2 such that α<0 .From property 6 in section 9.2 ,we know that this would correspond to a two side signal9.7 We may find different signal with the given Laplace transform by choosing different regions of convergence , the poles of the given Laplace transform are02s =- 13s =- 212s j =- 312s j =-Based on the locations of the locations of these poles , we my choose form the following regions of convergence: (i) Re{s}>- 12(ii)-2< Re{s}<- 12(iii)-3<Re{s}<-2 (iv)Re{s}<-39.8 From Table 9.1,we know thatG(t)= 2()()(2)Lt e x t G s X s ←−→=-. The ROC of G(s) is the ROC of X(s) shifted to the right by 2We are also given that X(s) has exactly 2 poles at s=-1 and s=-3. since G(s)=X(s-2), G(s)also has exactly two poles ,located at s=-1+2=1 and s=-3+2=-1 since we are given G(j ω) exists , we may infer that j ω-axis lies in the ROC of G(s). Given this fact and the locations of thepoles ,we may conclude that g(t) is a two side sequence .Obviously x(t)= 2te g(t) will also be two sided9.9 Using partial fraction expansion X(s)= 4243s s -++Taking the inverse Laplace transform, X(t)=443()2()t t e u t e u t ---9.10 The pole-zero plots for each of the three Laplace transforms is as shown in Figure S9.10(a) form Section 9.4 we knew that the magnitude of the Fourier transform may be expressed aswe se that the right-hand side of the above expression is maximum for ω=0 and decreases asω becomes increasing more positive or more negative . Therefore 1()H j ω is approximatelylowpass(b) From Section 9.4 we know that the magnitude of the Fourier transform may be expressaswe see that the right-hand side of the above expression is zero for ω=0.It thenincreams with increasing |ω| until |ω| reach 1/2. Then it starts decreasing as |ω| increase even further.Therefore | 2H (j )ω| is approximately bandpass.(c) From Section 9.4 we know that the magnitude of the Fourier transform may be expressas2We see that the right-hand side of the above expression is zero for ω=0. It thenincreases with increasing |ω| until |ω| reaches 21. Then |ω| increases,| 3()H j ω|decreases towards a value of 1(because all the vector lengths became almost identical and the ratio become 1) .Therefore |3()H j ω| is approximately highpass.9.11 X(s) has poles ats=1-2and 1-2.X(s) has zeros ats=12and1ω)| is(Length of vector form ω to -1)(Length of vector form ω to 111(Length of vector from toto 22ωωThe terms in the numerator and denominator of the right-band side of above expressioncancel our giving us |X(j ω)|=1.9.12 (a) If X(s) has only one pole, then x(t) would be of the form A ate -.Clearly such a signal violates condition 2. Therefore , this statement is inconsistent with the given information.(b) If X(s) has only two poles, then x(t) would be of the form A 0sin()atet ω- .Clearlysuch a signal could be made to satisfy all three conditions(Example:0ω=80π,α=19200).Therefore, this statement is consistent with the given information. (c) If X(s) has more than two poles (say 4 poles), then x(t) could be assumed to be of the form 00sin()sin()atbt Aet Be t ωω--+. Clearly such a signal could still be made tosatisfy all three conditions. Therefore, this statement is consistent with the given information.9.13 We have1}Re{,1)(->+=s s s X β.Also,1}Re{1),()()(<<--+=s s X s X s G αTherefore, ].11[)(2s s s s G -++-=ααβComparing with the given equation for G(s), ,1-=α .21=β9.14. Since X(s) has 4 poles and no zero in the finite s-plane, we many assume that X(s) is of the form.))()()(()(d s c s b s a s As X ----=Since x(t) is real ,the poles of X(s) must occur in conjugate reciprocal pairs. Therefore, we may assume that b=*a and d=*c . This result in .))()()(()(**c s c s a s a s As X ----=Since the signal x (t) is also even , the Laplace transform X(s) must also be even . This implies that the poles have to be symmetric about the j ω-axis. Therefore, we may assume that c=*a -. This results in .))()()(()(**a S a s a s a s As X ++--=We are given that the location of one of the poles is (1/2)4πj e . If we assume that this pole is a, we have 4444AX(s)=.1111(s-)(s-)(s+)(s+)2222j j j j e e e e ππππ--This gives us22().11()()44AX s s s s s =Also ,we are give that ()(0)4x t dt X ∞-∞==⎰Substituting in the above expression for X(s), we have A=1/4. Therefore,221/4().11()()44X s s s s s =9.15. Taking the Laplace transform of both sides of the two differential equations, wes X(s)=1)(2+-s Y and s Y(s)=2X(s) . Solving for X(s) and Y(s), we obtain 4)(2+=s s s X and Y(s)= 22s 4+.The region of convergence for both X(s) and Y(s) is Re{s}>0 because both are right-handsignals.9.16. Taking the Laplace transform of both sides of the given differential equations ,we obtain).(])1()1()[(223s X s s s s Y =+++++αααα therefore,.)1()1(1)()()(223αααα+++++==s s s s X s Y s H(a) Taking the Laplace transform of both sides of the given equation, we haveG(s) = s H(s)+ H(s). Substituting for H(s) from above,.1)1()1()1()(22223αααααα++=++++++=s s s s s s s GTherefore, G(s) has 2 poles.(b) we know thatH(s) =.))(1(122αα+++s s s Therefore, H(s) has poles at and j ),2321(,1+--α ).2321(j --α If the system has to bestable, then the real part of the poles has to be less than zero. For this to be true, we require that ,02/<-α i.e.,0>α.9.17 The overall system show in Figure 9.17 may be treated as two feedback system of the form shown in figure 9.31 connected in parallel. By carrying out an analysis similar to that described in Section 9.8.1, we find the system function of the upper feedback system to be.82)/2(41/2)(1+=+=s s s s HSimilarly, the system function of the lower feedback system is .21)2/1(21/1)(2+=+=s s s HThe system function of the overall system is now .1610123)()()(221+++=+=s s s s H s H s HSince H(s)=Y(s)/X(s), we may write]123)[(]1610)[(2+=++s s X s s s Y . Taking the inverse Laplace transform, we obtaindtt dx t x t y dt t dy dt t y d )(3)(12)(16)(10)(2+=++9.18. ( a) From problem 3.20, we know that differential equation relating the input and output of the RLC circuit is2()()()().d y t dy t y t x t dtdt++=Taking the Laplace transform of this (while nothing that the system is causal and stable), we obtain2()[1]().Y s s s X s ++= Therefore ,2()1(),()1Y s H s X s s s ==++ 1{}.e s ℜ>-(b) We note that H(s) has twopoles at 12s =--12s =-+zeros in thefinite s-plane. From Section 9.4 we know that the magnitude of the Fourier transform maybe expressed asWe see that the right hand side of the above expression Increases with increasing|ω| until |ω| reaches 12. Then it starts decreasing as |ω| increasing even further. It finally reaches 0 for |ω|=∞. Therefore 2|()|H j ω is approximately lowpass. (c) By repeating the analysis carried out in Problem 3.20 and part (a) of this problem with R =310-Ω, we can show that2()1(),()1Y s H s X s s s ==++ {}0.0005.e s ℜ>-(d) We haveWe see that when |ω| is in he vicinity 0.0005, the right-hand side of the aboveequation takes on extremely large value. On either side of this value of |ω| the value of |H (j ω)| rolls off rapidly. Therefore, H(s) may be considered to be approximately bandpass.9.19. (a) The unilateral Laplace transform isX(s) = 20(1)t st e u t e dt -∞--+⎰= 20t st e e dt -∞--⎰=21+s {} 2.e s ℜ>- (b) The unilateral Laplace transform is2(3)0()[(1)()(1)]t st X s t t e u t e dt δδ-∞-+-=++++⎰2(3)0[()]t st t e e dt δ-∞-+-=+⎰612e s -=++ {} 2.e s ℜ>- (c) The unilateral Laplace transform is240()[()()]t t st X s e u t e u t e dt -∞---=⎰240[]t t st e e e dt -∞---=+⎰1124s s =+++ {} 2.e s ℜ>- 9.20. In Problem 3.29, we know that the input of the RL circuit are related by ).()()(t x t y dtt dy =+Applying the unilateral Laplace transform to this equation, we have ).()()0()(s x s y y s sy =+--(a) For the zero-state response, set (0)0y -=.Also we have u s x =)(L{)(2t u et-}=21+s .Therefore,y(s)(s+1)=.21+sComputing the partial fraction expansion of the right-hand side of the above equation and then taking its inverse unilateral Laplace transform, we have(b) For the zero-state response, assume that x(t) = 0.Since we are given that (0)1y -=,.11)(0)(1)(+=⇒=+-s s y s y s syTaking the inverse unilateral Laplace transform, we have ()().t y t e u t -=Figure S9.21(c) The total response is the sum of the zero-state and zero-input response. This is 2()2()().t t y t e u t e u t --=-9.21. The pole zero plots for all the subparts are shown in figure S9.21. (a) The Laplace transform of x(t) is X(s)= 230()t t st e e e dt ∞---+⎰= (2)(3)00[/(2)]|[/(3)]|s t s te s e s -+∞-+∞-++-+ = 211252356s s s s s ++=++++(b) Using an approach similar to that show in part (a), we have41(),4L t e u t s -←−→+ {} 4.e s ℜ>-Also,551(),55L t j t e e u t s j -←−→+-{}5e s ℜ>-. and(){}551,555LT t j t e e u t e s s j --←−→ℜ>-++.From this we obtain()()()()55555215sin 52525LTt t j t t j t e t u t e e e e u t js ----⎡⎤=-←−→⎣⎦++ ,where {}5e s ℜ>- .Therefore, ()()(){}245321570sin 5,51490100LTt t s s e u t e t u t e s s s s --+++←−→ℜ>-+++. (c)The Laplace transform of ()x t is ()()023t t st X s e e e dt --∞=+⎰()()()()2300/2|/3|s t s t e s e s ----∞-∞⎡⎤⎡⎤=--+--⎣⎦⎣⎦ 211252356s s s s s -=+=---+.The region of convergence (ROC) is {}2e s ℜ<.(d)Using an approach along the lines of part (a),we obtain(){}21,22LT t e u t e s s -←−→ℜ>-+. (S9.21-1) Using an approach along the lines of part (c) ,we obtain(){}21,22LT t e u t e s s -←−→ℜ<-. (S9.21-2)R b Im()()222224t LT t t s e e u t e u t s --=+-←−→-, {}22e s -<ℜ<. Using the differentiation in the s-domain property , we obtain(){}22222228,2244t LT d s s te e s ds s s -+⎡⎤←−→-=--<ℜ<⎢⎥-⎣⎦-. (e)Using the differentiation in the s-domain property on eq.(S9.21-1),we get()(){}2211,222LT t d te u t e s ds s s -⎡⎤←−→-=ℜ>-⎢⎥+⎣⎦+.Using the differentiation in the s-domain property on eq (S9.21-2),we get ()(){}2211,222LT t d te u t e s ds s s ⎡⎤--←−→=-ℜ<⎢⎥-⎣⎦-.Therefore,()()()(){}222224,2222t LT t t st e te u t te u t e s s s ---=--←−→-<ℜ<+-.(f)From the previous part ,we have()()(){}2221,22LT t t t e u t te u t e s s -=--←−→-ℜ<-.(g)Note that the given signal may be written as ()()()1x t u t u t =-- .Note that (){}1,0LTu t e s s←−→ℜ>.Using the time shifting property ,we get(){}1,0s LT e u t e s s--←−→ℜ>.Therefore , ()1x t()()11,sLT e u t u t s----←−→ All s . Note that in this case ,since the signal is finite duration ,the ROC is the entire s-plane. (h)Consider the signal ()()()11x t t u t u t =--⎡⎤⎣⎦.Note that the signal ()x t may beexpressed as ()()()112x t x t x t =+-+ . We have from the previous part()()11sLT e u t u t s----←−→, All s . Using the differentiation in s-domain property ,we have()()()12111s s s LT d e se e x t t u t u t ds s s---⎡⎤--+=--←−→=⎡⎤⎢⎥⎣⎦⎣⎦, All s . Using the time-scaling property ,we obtain()121s s LT se e x t s --+-←−→, All s . Then ,using the shift property ,we have()21212s s LT sse e x t e s ---+-+←−→ ,All s . Therefore ,()()()21122112s s s sLTs se e se e x t x t x t es s ----+--+=+-+←−→+, All s. (i) The Laplace transform of ()()()x t t u t δ=+ is (){}11/,0X s s e s =+ℜ>.(j) Note that ()()()()33t u t t u t δδ+=+.Therefore ,the Laplace transform is the same as the result of the previous part. 9.22 (a)From Table 9.2,we have ()()()1sin 3x t t u t =.(b)From Table 9.2 we know that()(){}2cos 3,09LT st u t e s s ←−→ℜ>+. Using the time scaling property ,we obtain()(){}2cos 3,09LT s t u t e s s -←−→-ℜ<+Therefore ,the inverse Laplace transform of ()X s is ()()()cos 3x t t u t =--.(c)From Table 9.2 we know that()()(){}21cos 3,119LTt s e t u t e s s -←−→ℜ>-+. Using the time scaling property ,we obtain ()()(){}21cos 3,119LTt s e t u t e s s -+-←−→-ℜ<-++. Therefore ,the inverse Laplace transform of ()X s is ()()()cos 3t x t e t u t -=--.(d)Using partial fraction expansion on ()X s ,we obtain ()2143X s s s =-++ . From the given ROC ,we know that ()x t must be a two-sided signal .Therefore ()()()432t t x t e u t e u t --=+-.(e)Using partial fraction expansion on ()X s ,we obtain ()2132X s s s =-++. From the given ROC ,we know that ()x t must be a two-sided signal ,Therefore,()()()332ttx t e u t e u t --=+-.(f)We may rewrite ()X s as ()2311s X s s s =+-+ ()()22311/22ss =+-+1=+Using Table 9.2 ,we obtain()())())()/2/23cos/2sin/2t t x t t e u t u t δ--=+.(g)We may rewrite ()X s as()()2311sX s s =-+.From Table 9.2,we know that(){}21,0LT tu t e s s ←−→ℜ>.Using the shifting property ,we obtain()(){}21,11LT t e tu t e s s -←−→ℜ>-+.Using the differentiation property ,()()()(){}2,11LT t t t d s e tu t e u t te u t e s dt s ---⎡⎤=-←−→ℜ>-⎣⎦+.()()()()33t t x t t e u t te u t δ--=--.9.23.The four pole-zero plots shown may have the following possible ROCs:·Plot (a): {}2e s ℜ<- or {}22e s -<ℜ< or {}2e s ℜ>. ·Plot (b): {}2e s ℜ<- or {}2e s ℜ>-. ·Plot (c): {}2e s ℜ< or {}2e s ℜ>. ·Plot (d): Entire s-plane.Also, suppose that the signal ()x t has a Laplace transform ()X s with ROC R . (1).We know from Table 9.1 that ()()33LTt e x t X s -←−→+.The ROC 1R of this new Laplace transform is R shifted by 3 to the left .If ()3t x t e - is absolutely integrable, then 1R must include the jw -axis. ·For plot (a), this is possible only if R was {}2e s ℜ> . ·For plot (b), this is possible only if R was {}2e s ℜ>-. ·For plot (c), this is possible only if R was {}2e s ℜ> . ·For plot (d),R is the entire s-plane. (2)We know from Table 9.2 that(){}1,11LT t e u t e s s -←−→ℜ>-+.Also ,from Table 9.1 we obtain()()(){}2,11LT t X s x t e u t R R e s s -⎡⎤*←−→=ℜ>-⎡⎤⎣⎦⎣⎦+I If ()()t e u t x t -*is absolutely integrable, then 2R must include the jw -axis. ·For plot (a), this is possible only if R was {}22e s -<ℜ<. ·For plot (b), this is possible only if R was {}2e s ℜ>-. ·For plot (c), this is possible only if R was {}2e s ℜ< . ·For plot (d),R is the entire s-plane.(3)If ()0x t = for 1t > ,then the signal is a left-sided signal or a finite-duration signal .·For plot (a), this is possible only if R was {}2e s ℜ<-. ·For plot (b), this is possible only if R was {}2e s ℜ<-. ·For plot (c), this is possible only if R was {}2e s ℜ< . ·For plot (d),R is the entire s-plane.(4)If ()0x t =for 1t <-,then the signal is a right-sided signal or a finite-duration signal·For plot (a), this is possible only if R was {}2e s ℜ>. ·For plot (b), this is possible only if R was {}2e s ℜ>- . ·For plot (c), this is possible only if R was {}2e s ℜ>.·For plot (d),R is the entire s-plane.9.24.(a)The pole-zero diagram with the appropriate markings is shown Figure S9.24. (b)By inspecting the pole-zero diagram of part (a), it is clear that the pole-zero diagram shown in Figure S9.24 will also result in the same ()X jw .This would correspond to the Laplace()112X s s =-, {}12e s ℜ<.(c)≮()X jw π=-≮()1X jw .(d)()2X s with the pole-zero diagram shown below in Figure S9.24 would have the property that ≮()2X jw =≮()X jw .Here , ()211/2X s s -=-. (e) ()()21/X jw X jw =.(f)From the result of part (b),it is clear that ()1X s may be obtained by reflecting the poles and zeros in the right-half of the s-plane to the left-half of the s-plane .Therefore,()11/22s X s s +=+.From part (d),it is clear that ()2X s may be obtained by reflecting the poles (zeros) in the right-half of the s-plane to the left-half and simultaneously changing them to zeros (poles).Therefore, ()()()()2211/22s X s s s +=++ 9.25.The plots are as shown in Figure S9.25. 9.26.From Table 9.2 we have()()(){}2111,22LT t x t e u t X s e s s -=←−→=ℜ>-+and()()(){}3111,33LTt x t e u t X s e s s -=←−→=ℜ>-+.Using the time-shifting time-scaling properties from Table 9.1,we obtain()(){}22112,22s LT s e x t e X s e s s ---←−→=ℜ>-+and()(){}33223,33s LT s e x t e X s e s s---+←−→-=ℜ>--.Therefore, using the convolution property we obtain()()()()23122323s s LTe e y t x t x t Y s s s --⎡⎤⎡⎤=-*-+←−→=⎢⎥⎢⎥+-⎣⎦⎣⎦. 9.27.From clues 1 and 2,we know that ()X s is of the form()()()AX s s a s b =++.Furthermore , we are given that one of the poles of ()X s is 1j -+.Since ()x t is real, the poles of ()X s must occur in conjugate reciprocal pairs .Therefore, 1a j =-and 1b j =+ and()()()11AH s s j s j =+-++.From clue 5,we know that ()08X =.Therefore, we may deduce that 16A = and ()21622H s s s =++ .Let R denote the ROC of ()X s .From the pole locations we know that there are two possible choices of R .R may either be {}1e s ℜ<-or {}1e s ℜ>-.We will now use clue 4 to pick one .Note that()()()()22LTt y t e x t Y s X s =←−→=-.The ROC of ()Y s is R shifted by 2 to the right .Since it is given that ()y t is not absolutelyintegrable ,the ROC of ()Y s should not include the jw axis -.This is possible only ofR is {}1e s ℜ>-.9.28.(a) The possible ROCs are(i) {}2e s ℜ<-.(ii) {}21e s -<ℜ<-. (iii) {}11e s -<ℜ<. ( iv) {}1e s ℜ>.(b)(i)Unstable and anticausal. (ii) Unstable and non causal. （iii ）Stable and non causal. (iv) Unstable and causal. 9.29.(a)Using Table 9.2,we obtain (){}1,11X s e s s =ℜ>-+and(){}1, 2.2H s e s s =ℜ>-+(b) Since ()()()y t x t h t =*,we may use the convolution property to obtain()()()()()112Y s X s H s s s ==++.The ROC of ()Y s is {}1e s ℜ>-.(c) Performing partial fraction expansion on ()Y s ,we obtain . ()1112Y s s s =-++.Taking the inverse Laplace transform, we get()()()2t t y t e u t e u t --=-. (d)Explicit convolution of ()x t and ()h t gives us()()()y t h x t d τττ∞-∞=-⎰()()20t e e u t d ττττ∞---=-⎰t t e e d ττ--=⎰ for 0t>()2.t t e e u t --⎡⎤=-⎣⎦ 9.30.For the input ()()x t u t =, the Laplace transform is (){}1,0.X s e s s=ℜ>The corresponding output ()()1t t y t e te u t --⎡⎤=--⎣⎦has the Laplace transform()()(){}221111,0111Y s e s s s s s s =--=ℜ>+++. Therefore,()()()(){}21,0.1Y s H s e s X s s ==ℜ>+ Now ,the output ()()3123t t y t e e u t --⎡⎤=-+⎣⎦has the Laplace transform ()()(){}12316,0.1313Y s e s ss s s s s =-+=ℜ>++++ Therefore , the Laplace transform of the corresponding input will be()()()()(){}1161,0.3Y s s X s e s H s s s +==ℜ>+Taking the inverse Laplace transform of the partial fraction expansion of ()1,X s we obtain()()()3124.t x t u t e u t -=+9.31.(a).Taking the Laplace transform of both sides of the given differential equation andsimplifying, we obtain()()()212Y s H s X s s s ==--.b).The partial fraction expansion of ()H s is()1/31/321H s s s =--+. (i).If the system is stable ,the ROC for ()H s has to be {}12e s -<ℜ< . Therefore ()()()21133t t h t e u t e u t -=---.(ii).If the system is causal, the ROC for ()H s has to be {}2e s ℜ> .Therefore()()()21133t t h t e u t e u t -=-.(iii)If the system is neither stable nor causal ,the ROC for ()H s has to be {}1e s ℜ<-. Therefore ,()()()21133t t h t e u t e u t -=--+-9.32. If ()2t x t e =produces ()()21/6t y t e =,then ()()21/6H =. Also, by taking the Laplace transform of both sides of the given differential equation we get ()()()()442s b s H s s s s ++=++.Since ()21/6H = ,we may deduce that 1b = .Therefore()()()()()222424s H s s s s s s +==+++. 9.33.Since ()()()t t t x t e e u t e u t --==+-, ()()(){}112,111111Xs e s s s s s -=-=-<ℜ<+-+-. We are also given that ()2122s H s s s +=++.Since the poles of ()H s are at 1j -±, and since ()h t is causal ,we may conclude that the ROC of ()H s is {}1e s ℜ>-.Now()()()()()22221Y s H s X s s s s -==++-. The ROC of ()Y s will be the intersection of the ROCs of ()X s and ()H s .This is {}11e s -<ℜ<.We may obtain the following partial fraction expansion for ()Y s : ()22/52/56/5122s Ys s s s +=-+-++. We may rewrite this as()()()222/521411551111s Y s s s s ⎡⎤⎡⎤+=-++⎢⎥⎢⎥-++++⎢⎥⎢⎥⎣⎦⎣⎦. Nothing that the ROC of ()Y s is {}11e s -<ℜ<and using Table9.2,we obtain()()()()224cos sin 555t t t y t e u t e tu t e tu t --=-++9.34.We know that()()(){}111,0LTx t u t X s e s s=←−→=ℜ> Therefore, ()1X s has a pole at 0s =.Now ,the Laplace transform of the output ()1y t of thesystem with ()1x t as the input is ()()()11Y s H s X s =Since in clue 2, ()1Y s is given to be absolutely integrable ,()H s must have a zero at0s =which cancels out the pole of ()1X s at 0s =.We also know that()()(){}2221,0LT x t tu t X s e s s=←−→=ℜ> Therefore , ()2x s has two poles at 0s =.Now ,the Laplace transform of the output ()2y t of the system with ()2x t as the input is()()()22Y s H s X s =Since in clue 3, ()2Y s is given to be not absolutely integrable ,()H s does not have two zerosat 0s =.Therefore ,we conclude that ()H s has exactly one zero at 0s =. From clue 4 we know that the signal ()()()()2222d h t dh t p t h t dt dt=++is finite duration .Taking the Laplace transform of both sides of the above equation ,weget()()()()222P s s H s sH s H s =++. Therefore,()()222P s H s s s =++.Since ()p t is of finite duration, we know that ()P s will have no poles in the finite s-plane.Therefore, ()H s is of the form ()()1222Ni i A s z H s s s =-=++∏,where i z ,1,2,....,i N =represent the zeros of ()P s .Here ,A is some constant.From clue 5 we know that the denominator polynomial of ()H s has to have a degree which is exactly one greater than the degree of the numerator polynomial .Therefore, ()()1222A s s H s s s -=++.Since we already know that ()H s has a zero at 0s = ,we may rewrite this as()222As H s s s =++ From clue 1 we know that ()1H is 0.2.From this ,we may easily show that 1A = .Therefore, ()222s H s s s =++.Since the poles of ()H s are at 1j -± and since ()h t is causal and stable ,the ROC of()H s is {}1e s ℜ>-.。
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Chapter 9 Sinusoids and PhasorsSinusoidsA sinusoid is a signal that has the form of the sine or cosine function.anglephase um ent t frequencyangular am plitudeVm where t V v m ==+==+=φφϖϖφϖarg )cos()cos(φϖ+=t V v m φωtfTππω22==radians/second (rad/s)f is in hertz(Hz))cos()()cos()(222111φωφω+=+=t V t v t V t v m m Phase difference:φθφθθθφφφωφωθby v lags v by v leads v phase in are v and v phaseof out are v and v if t t 210210210210)()(2121<>=≠-=+-+=Complex Numberforml exponentia form sinusoidal formpolar form r rectangula φφφφj rez jrsin rcos z r z jy x z =+=∠=+=φPhasora phasor is a complex number representing the amplitude and phase angle of a sinusoidal voltage or current.Eq.(8-1)and Eq. (8-2) Eq.(8-3)When Eq.(8-2) is applied to the general sinusoid we obtainE q.(8-4)The phasor V is written asEq.(8-5)Fig. 8-1 shows a graphical representation commonly calleda phasor diagram.Fig. 8-1: Phasor diagram Two features of the phasor concept need emphasis:1.Phasors are written in boldfacetype like V or I1 to distinguishthem from signal waveformssuch as v(t)and i1(t).2. A phasor is determined byamplitude and phase angle anddoes not contain anyinformation about the frequencyof the sinusoid.In summary, given a sinusoidal signal , the corresponding phasor representation is . Conversely, given the phasor , the corresponding sinusoid is found by multiplying the phasor by and reversing the steps in Eq.(8-4) as follows:E q.(8-6))cos()(φϖ+=t V t v m φ∠=Vm V Time domainrepresentationPhase-domain representationProperties of Phasors•additive propertyEq.(8-7)Eq.(8-8)Eq.(8-9)•derivative propertyEq.(8-10)Vj dtdv ω⇔∴Time domain representationPhase-domain representation•Integral propertyTime domain representationPhase-domainrepresentation⎰⇔ωj Vvdt The differences between v(t) and V:V(t) is the instantaneous or time-domain representation, while V is the frequency or phasor-domain representation.2.V(t) is a real signal which is time dependent, while V is just a supposed value to simplify the analysisThe complex exponential is sometimes called a rotating phasor, and the phasor V is viewed as a snapshot of the situation at t=0.Fig. 8-2: Complex exponential+ j + real-real-j ωtV mθ= 0θ= 90 or π/2θ= -90 or -π/2θ= 180 or π151050510151513.51210.597.564.531.5010V r ms ac signal at 0.5 Hzvoltage in voltsa n g u l a r f r e q u e n c y t i m e s t i m e i n r a d i a n s12.566-ω-t n⋅14.14214.142-v rea l t ()n5101515129630369121510V rms ac signal at 0.5 Hzangular frequency times time in radiansV o l t a g e i n v o l t s14.14214.142-v im a g t n ()12.5660ωt ⋅()n)sin(is axis )(imaginary j on the phasor rotating the of projection The t V v m ima g ω⋅=)cos(is axis real on the phasor rotating the of projection The t V v m rea l ω⋅=()()caseparticular In this 5.02cos 102cos t t f V v m ⋅⋅=⋅⋅=ππEXAMPLE 8-1(a)Construct the phasors for the following signals:(b) Use the additive property of phasors and the phasorsfound in (a) to find v(t)=v1(t)+v2(t).SOLUTION(a) The phasor representations of v(t)=v1(t)+ v2(t) are(b) The two sinusoids have the same frequent so the additive property of phasors can be used to obtain their sum:The waveform corresponding to this phasor sum isV1V21jVEXAMPLE 8-2(a)Construct the phasors representing the following signals:(b) Use the additive property of phasors and the phasors foundin (a) to find the sum of these waveforms.SOLUTION:(a) The phasor representation of the three sinusoidal currents are(b) The currents have the same frequency, so the additive property of phasors applies. The phasor representing the sum of these current isFig. 8-4EXAMPLE 8-3Use the derivative property of phasors to find the time derivative of v(t)=15cos(200t-30°).The phasor for the sinusoid is V=15∠-30 °.According tothe derivative property, the phasor representing the dv/dt isfound by multiplying V by jω.SOLUTION:The sinusoid corresponding to the phasor jωV isDevice Constraints in Phasor FormV oltage-current relations for a resistor in the: (a) time domain, (b) frequency domain.Resistor:RejImI VIV m m RI Vφφ==Device Constraints in Phasor FormInductor:V oltage-current relations for an inductor in the: (a) time domain, (b) frequency domain.ω︒+==90I V mm LI V φφωDevice Constraints in Phasor Form Capacitor:ωV oltage-current relations for a capacitor in the: (a) time domain, (b) frequency domain.︒+==90VImmCVIφφωConnection Constraints in Phasor Form KVL in time domainKirchhoff's laws in phasor form (in frequency domain)KVL: The algebraic sum of phasor voltages around a loop iszero.KCL: The algebraic sum of phasor currents at a node is zero.The IV constraints are all of the formV=ZI or Z= V/IEq.(8-16)where Z is called the impedance of the elementThe impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms(Ω)reactance. the is Z Im X and resistance the is Z Re R where ==+=jXR Z The impedance is inductive when X is positiveis capacitive when X is negativeθθθθsin,cos tan, where 122Z X Z R and RXX R Z Z Z ===+=∠=-EXAMPLE 8-5Fig. 8-5The circuit in Fig. 8-5 is operating in the sinusoidal steady state with and . Find the impedance of the elements in the rectangular box.SOLUTION:︒VI0.278/R=37.9-∠=3L2The Admittance ConceptThe admittance Y is the reciprocal of impedance, measured in siemens (S)VI Z Y ==1Y=G+jBWhere G=Re Y is called conductance and B=Im Y is called the susceptance 2222,1XR XB X R R G jX R jB G +-=+=+=+How get Y=G+jB from Z=R+jX ?Cj Y capacitor Lj Y inductor GR Y resistor C L R ωω====:1:1:Basic Circuit Analysis with PhasorsStep 1: The circuit is transformed intothe phasor domain by representing theinput and response sinusoids as phasorand the passive circuit elements bytheir impedances.Step 2: Standard algebraic circuittechniques are applied to solve thephasor domain circuit for the desiredunknown phasor responses.Step 3: The phasor responses areinverse transformed back into time-domain sinusoids to obtain theresponse waveforms.Series Equivalence And Voltage Divisionwhere R is the real part and X is the imaginary partEXAMPLE 8-6Fig. 8-8The circuit in Fig. 8 -8 is operating in the sinusoidal steady state with(a) Transform the circuit into the phasor domain.(b) Solve for the phasor current I.(c) Solve for the phasor voltage across each element.(d) Construct the waveforms corresponding to the phasors found in (b) and (c)SOLUTION:PARALLEL EQUIVALENCE AND CURRENT DIVISIONRest ofthecircuitY1Y1Y2Y NIVI1I2I3 phasor version of the current division principleEXAMPLE 8-9Fig. 8-13The circuit in Fig. 8-13 is operating in the sinusoidal steady state with i S(t)=50cos2000t mA.(a) Transform the circuit into the phasor domain.(b) Solve for the phasor voltage V.(c) Solve for the phasor current through each element.(d) Construct the waveforms corresponding to the phasors found in (b) and (c).SOLUTION:(a) The phasor representing the input source current isIs=0.05∠0°A. The impedances of the three passive elements areFig. 8-14And the voltage across the parallel circuit isThe sinusoidal steady-state waveforms corresponding to thephasors in (b) and (c) areThe current through each parallel branch isEXAMPLE 8-10Fig. 8-15Find the steady-state currents i(t), and i C(t)in the circuit of Fig. 8-15 (for Vs=100cos2000t V, L=250mH, C=0.5 μF, and R=3kΩ).SOLUTION:Vs=100∠0°Y←→△TRANSFORMATIONSThe equations for the △to Ytransformation areThe equations for a Y-to-△transformation arewhen Z1=Z2=Z3=Z Y or Z A=Z B=Z C=Z N.Z Y=Z N/3 and Z N=3Z Y balanced conditionsEXAMPLE 8-12Use a △to Y transformation to solve for the phasor current I X in Fig. 8-18.Fig. 8-18SOLUTION:ABC△to Y。