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《自动控制原理》+胡寿松+习题答案(附带例题课件

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uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ      ሱ‫׮‬॥ᇅჰ৘   ‫׈‬ሰ࢝σ        uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ uሱ‫׮‬॥ᇅჰ৘vज़ӱ࢝࿐ն‫۔‬ ज़ӱщ‫ݼ‬ğ ज़ӱ଀ӫğሱ‫׮‬॥ᇅჰ৘ ႇ໓଀ӫğ"VUPNBUJD$POUSPM5IFPSZ ज़ӱো྘:ğህြࠎԤсྩज़ ሹ ࿐ ൈğ**** ࢃज़࿐ൈğ56 ഈࠏ࿐ൈğ8 ࿐  ൈğ**** ࿐  ‫ٳ‬ğ4 ൡႨؓའğ‫׈‬గ‫۽‬ӱࠣఃሱ‫߄׮‬ህြč‫׈‬৯༢๤ࠣሱ‫߄׮‬a‫׈‬৯༢๤࠿‫׈‬Ќ޹a‫׈‬ຩࡓ॥࠯ඌa‫܂‬ Ⴈ‫࠯׈‬ඌህြٚཟĎ ༵ྩज़ӱğۚ֩ඔ࿐aն࿐໾৘aࠒ‫ٳ‬эߐa‫׈‬ਫ਼aඔሳ‫׈‬ሰ࠯ඌaଆ୅‫׈‬ሰ࠯ඌ ၂aज़ӱྟᇉaଢ֥‫ބ‬಩ༀ Чज़ӱູ‫׈‬గ‫۽‬ӱࠣఃሱ‫߄׮‬ህြ֥ᇶေህြࠎԤज़ӱᆭ၂đଢ֥൞൐࿐ളᅧ໤‫ّڵ‬ঌ॥ᇅჰ৘a॥ ᇅ༢๤ඔ࿐ଆ྘֥ࡹ৫‫ބ‬༢๤ྟି‫ٳ‬༅aഡ࠹֥ࠎЧٚ‫م‬đ஡အ࿐ള‫ٳ‬༅‫ބ‬ഡ࠹ሱ‫׮‬॥ᇅ༢๤ྟି֥ࠎЧ ି৯ѩିડቀః෱ު࿃ህြज़ӱؓሱ‫׮‬॥ᇅ৘ંᆩ്֥ླေb ‫ؽ‬a࢝࿐ࠎЧေ౰ Чज़ӱҐႨൈთ‫م‬a۴݅ࠖ‫ބم‬௔ੱหྟ‫ؓم‬ሱ‫׮‬॥ᇅ༢๤֥ྟିࣉྛ‫ٳ‬༅‫ބ‬ഡ࠹đ࿐ປЧज़ӱႋղ ֞ၛ༯ࠎЧေ౰b 1ēᅧ໤‫ّڵ‬ঌ॥ᇅჰ৘ ᅧ໤‫ّڵ‬ঌ॥ᇅჰ৘đି‫ٳܔ‬༅‫ّڵ‬ঌ॥ᇅ༢๤֥‫ݖࢫט‬ӱѩ߂ԛཌྷႋ֥॥ᇅ༢๤ٚॿ๭bਔࢳ॥ᇅ ༢๤֥ࠎЧ‫ܒ‬Ӯ‫ٳބ‬োb 2ēඃ༑ࡹ৫॥ᇅ༢๤ඔ࿐ଆ྘֥ٚ‫م‬ ඃ༑Ⴈঘ൦эߐ‫م‬౰ࢳཌྟ༢๤ັ‫ٳ‬ٚӱ֥ࠎЧٚ‫م‬bᅧ໤॥ᇅ༢๤Ԯ‫ݦ־‬ඔa‫׮‬෿ࢲ‫ܒ‬๭ࡹ৫‫ࡥބ‬ ߄ٚ‫م‬b 3ēඃ༑ᄎႨൈთ‫ٳ‬༅‫ٳم‬༅༢๤ྟି֥ٚ‫م‬ ᅧ໤‫ࢨؽ྘ׅ‬༢๤֥ֆ໊ࢨᄁཙႋၛࠣྟିᆷѓ֥౰౼bᅧ໤Ⴈসථսඔ໗‫ק‬஑ऌ஑؎༢๤֥໗‫ྟק‬ ֥ٚ‫م‬bᅧ໤౰༢๤֥໗෿༂ҵࠣ༂ҵ༢ඔ֥ٚ‫م‬b 4ēඃ༑Ⴈ۴݅ࠖ‫ٳ‬༅‫ٳم‬༅॥ᇅ༢๤ྟି֥ٚ‫م‬ ᅧ໤۴ऌ༢๤षߌԮ‫ݦ־‬ඔ֥ਬaࠞׄ‫߻҃ٳ‬ᇅоߌ༢๤۴݅ࠖ๭֥ࠎЧٚ‫م‬b۴ऌ۴݅ࠖ๭‫ٳ‬༅॥ ᇅ༢๤֥ྟିbਔࢳषߌਬaࠞׄؓ༢๤ྟି֥႕ཙb 5ēඃ༑௔ੱ‫ٳ‬༅‫ٳم‬༅॥ᇅ༢๤ྟି֥ٚ‫م‬ ඃ༑‫ࢫߌ྘ׅ‬௔ੱหྟ֥౰౼ၛࠣ௔ੱหྟ౷ཌđᅧ໤༢๤षߌؓඔ௔ੱหྟ౷ཌaࠞቕѓ౷ཌ߻ᇅ ֥ࠎЧٚ‫م‬bਔࢳ۴ऌषߌؓඔ௔ੱหྟ౷ཌ‫ٳ‬༅оߌ༢๤ྟି֥ٚ‫م‬bඃ༑Ⴈ଱উථห໗‫ק‬஑ऌ஑؎༢ 1 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ๤໗‫֥ྟק‬ٚ‫م‬bᅧ໤໗‫ק‬ღ؇֥࠹ෘٚ‫م‬b 6ēඃ༑॥ᇅ༢๤཮ᆞ֥ٚ‫م‬ ਔࢳԱ৳ӑభ཮ᆞaԱ৳ᇌު཮ᆞ֥཮ᆞልᇂഡ࠹֥ࠎЧჰ৘‫ބ‬ٚ‫م‬b 7ēඃ༑٤ཌྟ॥ᇅ༢๤֥‫ٳ‬༅ٚ‫م‬ ਔࢳ٤ཌྟ॥ᇅ༢๤֥หׄ‫ބ‬ӈ࡮٤ཌྟหྟbඃ༑٤ཌྟ॥ᇅ༢๤֥૭ඍ‫ݦ‬ඔ‫م‬b ೘a࢝࿐ଽಸࠣေ౰ č၂Ďሱ‫׮‬॥ᇅ༢๤֥ࠎЧ‫୑ۀ‬ ਔࢳሱ‫׮‬॥ᇅ৘ં֥ᇶေ಩ༀၛࠣ࿹࣮ؓའđᅧ໤‫ّڵ‬ঌ॥ᇅჰ৘ѩ‫ٳ‬༅॥ᇅ༢๤֥ሱ‫׮‬॥ᇅ‫ݖ‬ӱđ ඃ༑ሱ‫׮‬॥ᇅ༢๤֥ࠎЧ‫ܒ‬Ӯѩି߻ᇅ॥ᇅ༢๤ٚॿ๭đਔࢳሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬োٚ‫ࠎބم‬Чေ౰b ᇶေଽಸ 1ē ሱ‫׮‬॥ᇅაሱ‫׮‬॥ᇅ༢๤ 2ē ‫ّڵ‬ঌ‫ࢫט‬ჰ৘ 3ē ሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬ো 3ēؓ॥ᇅ༢๤֥ྟିေ౰ 4ēሱ‫׮‬॥ᇅ৘ં‫ؿ‬ᅚࡥൎ č‫ؽ‬Ď ሱ‫׮‬॥ᇅ༢๤֥ඔ࿐ଆ྘ ඃ༑༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫đঘ൦эߐࠣఃႋႨbᅧ໤༢๤Ԯ‫ݦ־‬ඔ֥‫ק‬ၬࠣ౰౼đ༢๤‫׮‬෿ࢲ‫ܒ‬๭ ֥ࡹ৫ࠣఃࡥ߄ၛࠣ༢๤҂๝Ԯ‫ݦ־‬ඔ֥‫ק‬ၬࠣ౰౼b 1ē॥ᇅ༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫ 2ē٤ཌྟඔ࿐ଆ྘֥ཌྟ߄ 3ē॥ᇅ༢๤֥Ԯ‫ݦ־‬ඔ 4ē‫֥ࢫߌ྘ׅ‬Ԯ‫ݦ־‬ඔ 5ē॥ᇅ֥‫׮‬෿ࢲ‫ܒ‬๭ࠣэߐ 6ēྐ‫ݼ‬ੀ๭઼ࠣ࿠‫܄‬ൔ 7ēّঌ॥ᇅ༢๤֥Ԯ‫ݦ־‬ඔ č೘Ďሱ‫׮‬॥ᇅ༢๤֥ൈთ‫ٳ‬༅‫م‬ ඃ༑॥ᇅ༢๤֥ൈთᆷѓđ၂ࢨ༢๤֥ֆ໊ࢨᄁཙႋaོ௡ཙႋၛࠣྟିᆷѓ֥౰౼bᅧ໤‫ࢨؽ྘ׅ‬ ༢๤֥ֆ໊ࢨᄁཙႋၛࠣྟିᆷѓ֥౰౼bᅧ໤সථ໗‫ק‬஑ऌ‫ٳ‬༅༢๤֥໗‫ྟק‬ٚ‫م‬bඃ༑॥ᇅ༢๤໗෿ ༂ҵ‫ٳ‬༅ၛࠣ໗෿༂ҵa༂ҵ༢ඔ֥౰౼b 1ē ॥ᇅ༢๤ྟିᆷѓ֥‫ק‬ၬ 2ē၂ࢨ༢๤ྟି‫ٳ‬༅ 3ē‫ࢨؽ‬༢๤ྟି‫ٳ‬༅ 4ē ఴቅୄ‫ࢨؽ‬༢๤֥ൈთ‫ٳ‬༅‫ބ‬ᆷѓ࠹ෘ 5ē ۚࢨ༢๤֥ൈთ‫ٳ‬༅aоߌᇶ֝ࠞׄ‫ࢨۚބ‬༢๤֥ࢆࢨ 2 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 6ē ॥ᇅ༢๤֥໗‫ٳྟק‬༅ 7ē ॥ᇅ༢๤֥໗෿༂ҵ‫ٳ‬༅‫ࣉڿބ‬ծീ 8ē ๝҄‫ৣࠏ׈ؿ‬Ո‫ࢫט‬č࿊ࢃĎ čඹĎ۴݅ࠖ‫ٳ‬༅‫م‬ ਔࢳ۴֥݅ࠖࠎЧ‫୑ۀ‬đඃ༑۴֥݅ࠖ߻ᇅܿᄵđᅧ໤ቋཬཌྷ໊༢๤֥۴݅ࠖ๭߻ᇅđਔࢳ٤ቋཬཌྷ ໊༢๤۴݅ࠖ๭֥߻ᇅbᄎႨ۴݅ࠖ‫ٳم‬༅༢๤֥ᄠ෿หྟb 1ē۴֥݅ࠖࠎЧ‫୑ۀ‬č۴݅ࠖa۴݅ࠖٚӱĎ 2ē߻ᇅ1800 ۴֥݅ࠖࠎЧ‫م‬ᄵ 3ē߻ᇅ 00 ۴֥݅ࠖࠎЧ‫م‬ᄵ 4ēܼၬ۴݅ࠖ 5ē٤ቋཬཌྷ໊༢๤֥۴݅ࠖ 6ēႨ۴݅ࠖ‫ٳم‬༅༢๤ྟି č໴Ď௔ੱ‫م‬ ਔࢳ௔ੱหྟ֥ࠎЧ‫୑ۀ‬đ௔ੱหྟ֥ࠫ‫ޅ‬іൕٚ‫م‬đඃ༑‫֥ؓࢫߌ྘ׅ‬ඔ௔ੱหྟ౷ཌčBode ๭Ď ߻ᇅ‫ࠞބ‬ቕѓ౷ཌčNyquist ౷ཌĎđᅧ໤༢๤षߌؓඔ௔ੱหྟ౷ཌ֥߻ᇅđਔࢳ༢๤षߌࠞቕѓ౷ཌ߻ ᇅ֥၂Ϯٚ‫م‬đඃ༑षߌؓඔ௔ੱหྟ֮௔‫؍‬aᇏ௔‫؍‬aۚ௔‫֥؍‬หᆘđ࿐߶ᄎႨ଱উථห໗‫ק‬஑ऌ஑؎ оߌ༢๤֥໗‫ྟק‬đᅧ໤༢๤໗‫ק‬ღ؇֥ࠎЧ‫࠹ބ୑ۀ‬ෘٚ‫م‬đਔࢳ༢๤ྟି‫ބ‬षߌ௔ੱหྟ֥ܱ༢b 1ē௔ੱหྟ֥ࠎЧ‫ޅࠫބ୑ۀ‬іൕ 2ē‫֥ࢫߌ྘ׅ‬௔ੱหྟ 3ē॥ᇅ༢๤षߌؓඔ௔ੱหྟ‫ࠞބ‬ቕѓ౷ཌ֥߻ᇅ 4ēቋཬཌྷ໊༢๤Ԯ‫ݦ־‬ඔ֥ಒ‫ק‬ 5ē଱উථห໗‫ק‬஑ऌ‫ ބ‬Bode ๭ഈ֥໗‫ק‬஑ऌ 6ē໗‫ק‬ღ؇֥ࠎЧ‫࠹ބ୑ۀ‬ෘٚ‫م‬ 7ē௔ੱหྟა༢๤ྟି֥ࠎЧܱ༢ čੂĎ॥ᇅ༢๤ྟି֥཮ᆞ ਔࢳ཮ᆞልᇂ‫཮ބ‬ᆞٚ‫م‬đඃ༑Ա৳ӑభ཮ᆞaԱ৳ᇌު཮ᆞ֥ࠎЧჰ৘‫ބ‬ٚ‫م‬bਔࢳ௔ੱ‫ّم‬ঌ཮ ᆞ֥ࠎЧჰ৘‫ބ‬ٚ‫م‬č࿊ࢃĎb 1ē॥ᇅ༢๤཮ᆞ֥ࠎЧ‫ބ୑ۀ‬၂Ϯٚ‫م‬ 2ē௔ੱ‫م‬Ա৳ӑభ཮ᆞ֥ࠎЧჰ৘‫ބ‬ٚ‫م‬ 3ē௔ੱ‫م‬Ա৳ᇌު཮ᆞ֥ࠎЧ‫ބ୑ۀ‬ٚ‫م‬ ē௔ੱ‫ّم‬ঌ཮ᆞ֥ࠎЧჰ৘‫ބ‬ٚ‫م‬č࿊ࢃĎ č௾Ď٤ཌྟ॥ᇅ༢๤ ਔࢳ٤ཌྟ༢๤აཌྟ༢๤֥౵љđਔࢳ٤ཌྟหྟ‫ބ‬٤ཌྟ༢๤֥ᇶေหᆘđ࿐߶٤ཌྟ༢๤֥૭ ඍ‫ݦ‬ඔ‫ٳ‬༅ٚ‫م‬đਔࢳ٤ཌྟ༢๤֥ཌྷ௜૫‫ٳ‬༅‫م‬č࿊ࢃĎb 3 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ē ٤ཌྟ༢๤֥ࠎЧ‫୑ۀ‬ ē ‫྘ׅ‬٤ཌྟหྟa٤ཌྟ༢๤֥ᇶေหᆘ ē ૭ඍ‫ݦ‬ඔ‫ק‬ၬaႋႨ่ࡱ‫ބ‬౰౼ٚ‫م‬ ē ႋႨ૭ඍ‫ݦ‬ඔ‫ٳ‬༅٤ཌྟ༢๤֥໗‫ྟק‬ ē ٤ཌྟ༢๤ሱࠗᆒ֕‫ٳ‬༅‫࠹ބ‬ෘ ē ࢺക٤ཌྟ༢๤ཌྷ௜૫‫ٳ‬༅‫م‬č࿊ࢃĎ ඹa෮‫ݣ‬ൌ࡬ߌࢫ ๙‫ ؓݖ‬."5-"# ٟᆇೈࡱ֥࿐༝đᅧ໤ ."5-"# ೈࡱ֥ࠎЧႋႨٚ‫م‬đି‫ܔ‬࿐߶ᄎႨ ."5-"# ೈࡱ‫ٳ‬༅ ॥ᇅ༢๤֥ྟି‫ࠎބ‬Чഡ࠹ٚ‫م‬b ሱ‫׮‬॥ᇅ৘ં֥ൌဒνஆᄝज़ӱଽ࠹ෘࠏഈປӮđषഡ 4 ۱ൌဒğ 1ēඃ༑ MATLAB ೈࡱ֥ࠎЧ൐Ⴈٚ‫م‬đѩ০Ⴈ MATLAB ӱ྽ൌགྷ॥ᇅ༢๤‫ିྟ֥ࢫߌ྘ׅ‬ٟᆇb čဒᆣྟൌဒĎ 2 ࿐ൈ 2ē০Ⴈ MATLAB ӱ྽߻ᇅ॥ᇅ༢๤ࢨᄁཙႋ౷ཌa࠹ෘྟିᆷѓđษંषߌ٢նПඔؓоߌ༢๤ཙ ႋ෎؇a໗‫ބྟק‬໗෿༂ҵ֥႕ཙ bčဒᆣྟൌဒĎ 2 ࿐ൈ 3ē০Ⴈ MATLAB ӱ྽߻ᇅ॥ᇅ༢๤֥ Nyquist ౷ཌaBode ๭đ࠹ෘ॥ᇅ༢๤֥‫ږ‬ᆴღ؇‫ބ‬ཌྷ໊ღ؇b čဒᆣྟൌဒĎ 2 ࿐ൈ 4ē০Ⴈ MATLAB ೈࡱഡ࠹॥ᇅ༢๤čഡ࠹ྟൌဒĎ 2 ࿐ൈ ໴aज़ຓ༝ีࠣज़ӱษં ູղ֞Чज़ӱ֥࢝࿐ࠎЧေ౰đज़ຓ༝ี၂Ϯ҂ႋഒႿ 50 ีb ੂa࢝࿐ٚ‫م‬ა൭‫؍‬ Чज़ӱҐႨၛज़ถϰ඀ູᇶൡ֒ࢲ‫ุૂ؟ކ‬ज़ࡱѩ‫ڣ‬ၛഈࠏൌဒ֥ٚൔࣉྛ࢝࿐b ௾a۲࢝࿐ߌࢫ࿐ൈ‫ٳ‬஥ ᅣࢫčࠇଽಸĎ ࢃज़ ༝ีज़ ษંज़ ഈࠏ ః෱ ‫࠹ކ‬ ሱ‫׮‬॥ᇅ༢๤֥ࠎЧ‫୑ۀ‬       ॥ᇅ༢๤֥ඔ࿐ଆ྘       ཌྟ༢๤֥ൈთ‫ٳ‬༅‫م‬       ۴݅ࠖ‫ٳ‬༅‫م‬       ཌྟ༢๤֥௔თ‫ٳ‬༅‫م‬       ॥ᇅ༢๤֥཮ᆞ       ٤ཌྟ༢๤       ‫࠹ކ‬       ϖaॉ‫ނ‬ٚൔ Чज़ӱॉ‫ູނ‬௹ଌоजг൫‫ބ‬௜ൈॉ‫ނ‬ཌྷࢲ‫ކ‬b࿐ള֥ज़ӱሹ௟ӮࠛႮ௜ൈӮࠛčᅝ 30%Ď‫ބ‬௹ଌॉ 4 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ൫Ӯࠛčᅝ 70%Ďਆ҆‫ܒٳ‬Ӯđ௜ൈӮࠛᇏഈࠏൌဒӮࠛᅝ 20%đԛ౗aቔြaज़ถิ໙a࿐༝ᇶ‫֩ྟ׮‬ ᅝ 10%b ࣴa๷ࡩ࢝ҋ‫࢝ބ‬࿐ҕॉ඀ ࢝ ҋğ uሱ‫׮‬॥ᇅჰ৘vđ‫ݓ‬ٝ‫۽‬ြԛϱഠđ຦߃၂ᇶщđ2001 ୍ ҕॉ඀ğ uሱ‫׮‬॥ᇅჰ৘vđ‫ݓ‬ٝ‫۽‬ြԛϱഠđ‫ޱ‬൰඾ᇶщđ ୍ uሱ‫׮‬॥ᇅჰ৘vđౢ޿ն࿐ԛϱഠđ໱Ḁᇶщđ ୍ uགྷս॥ᇅ‫۽‬ӱvđ‫׈‬ሰ‫۽‬ြԛϱഠđ,BUTVIJLP0HBUB ᇷđ੓ѵႇaႿ‫ݚ‬࿔֩ၲđ ୍ ն‫۔‬ᇅ‫ר‬ದğဗᆽӑ ն‫۔‬ബ‫ק‬ದğ৙༵ᄍ ᇅ‫ר‬ರ௹ğ2005 ୍ 6 ᄅ 5 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ uሱ‫׮‬॥ᇅჰ৘vज़ӱൌဒ࢝࿐ն‫۔‬  ၂aൌဒ࢝࿐ଢѓაࠎЧေ౰ uሱ‫׮‬॥ᇅჰ৘vज़ӱൌဒ๙‫ݖ‬ഈࠏ൐Ⴈ ."5-"# ೈࡱđ൐࿐ളԚ҄ᅧ໤ ."5-"# ೈࡱᄝ॥ᇅ৘ંᇏ֥ ࠎЧႋႨđ࿐߶০Ⴈ ."5-"# ೈࡱ‫ٳ‬༅॥ᇅ༢๤đՖ‫ࡆط‬ധؓሱ‫׮‬॥ᇅ༢๤֥ಪ്đϺᇹ৘ࢳࣜ‫ׅ‬ሱ‫׮‬॥ ᇅ֥ཌྷܱ৘ં‫ٳބ‬༅ٚ‫م‬b๙‫ݖ‬Чज़ӱഈࠏൌဒđေ౰࿐ളؓ ."5-"# ೈࡱႵ၂۱ࠎЧ֥ਔࢳđᅧ໤ ."5-"# ೈࡱᇏࠎЧඔቆ‫ބ‬इᆔ֥іൕٚ‫م‬đᅧ໤ ."5-"# ೈࡱ֥ࠎЧ߻๭‫ିۿ‬đ࿐߶ ."5-"# ೈࡱᇏሱ‫׮‬॥ᇅ৘ં ӈႨ‫ݦ‬ඔ֥൐Ⴈđ࿐߶ᄝ ."5-"# ೈࡱ‫۽‬ቔԳ१ࣉྛࢌ޺ൔٟᆇ‫ބ‬൐Ⴈ .@'JMF ۬ൔ֥ࠎЧщӱٚ‫م‬đԚ҄ ᅧ໤০Ⴈ ."5-"# ೈࡱࣉྛ॥ᇅ༢๤ഡ࠹đಞ࿐ള֤֞ለཿБ֥ۡࠎЧ࿞਀b ‫ؽ‬aЧज़ӱൌဒ֥ࠎЧ৘ંაൌဒ࠯ඌᆩ് ҐႨ ."5-"# ೈࡱഈࠏࣉྛൌဒđࣼ൞০Ⴈགྷս࠹ෘࠏ႗ࡱ‫࠹ބ‬ෘࠏೈࡱ࠯ඌđၛඔሳٟᆇ࠯ඌູ‫ނ‬ ྏđൌགྷؓሱ‫׮‬॥ᇅ༢๤ࠎЧ৘ં‫ٳބ‬༅ٚ‫֥م‬ဒᆣၛࠣ॥ᇅ༢๤ഡ࠹b ๙‫ݖ‬ഈࠏൌဒđ൐࿐ളᄝ ."5-"# ೈࡱ֥ࠎЧ൐Ⴈaщӱ‫ט‬൫aٟᆇൌဒඔऌ֥ࠆ౼aᆜ৘a‫ٳ‬༅ၛ ࠣൌဒБ֥ۡለཿ֩ࠎЧ࠯ି֤֞࿞਀b ೘aൌဒٚ‫م‬aหׄაࠎЧေ౰ Чज़ӱൌဒҐႨ࠹ෘࠏ ."5-"# ೈࡱٟᆇٚ‫م‬đఃหׄ൞০Ⴈ ."5-"# ೈࡱ‫ݦିۿ֥ڶپ‬ඔaਲࠃ֥щ ӱ‫טބ‬൫൭‫؍‬ၛ఼ࠣն֥ದࠏࢌ޺‫ބ‬๭ྙൻԛ‫ିۿ‬đॖၛൌགྷؓ॥ᇅ༢๤ᆰܴ‫ބ‬ٚь֥‫ٳ‬༅‫ބ‬ഡ࠹b Чज़ӱൌဒ֥ࠎЧေ౰൞đ൐࿐ളؓ ."5-"# ೈࡱႵ၂۱ࠎЧ֥ਔࢳđᅧ໤ ."5-"# ೈࡱᇏࠎЧඔቆ‫ބ‬ इᆔ֥іൕٚ‫م‬đᅧ໤ ."5-"# ೈࡱ֥ࠎЧ߻๭‫ିۿ‬đ࿐߶ ."5-"# ೈࡱᇏሱ‫׮‬॥ᇅ৘ંӈႨ‫ݦ‬ඔ֥൐Ⴈđ ࿐߶ᄝ ."5-"# ೈࡱ‫۽‬ቔԳ१ࣉྛࢌ޺ൔٟᆇ‫ބ‬൐Ⴈ .@'JMF ۬ൔ֥ࠎЧщӱٚ‫م‬đԚ҄ᅧ໤০Ⴈ ."5-"# ೈࡱࣉྛ॥ᇅ༢๤ഡ࠹đಞ࿐ള֤֞ለཿБ֥ۡࠎЧ࿞਀b ඹaൌဒᇶေ၎ఖഡС ஥СႵ  ෻࠹ෘࠏđѩνል ."5-"#9 ೈࡱb ໴aൌဒཛଢ֥ഡᇂაଽಸิေ ൌൌ ྽ ൌဒཛଢ ଽಸิေ ဒ ဒ ૄቆದ ൌဒေ౰ ‫ݼ‬ ࿐ ো ඔ ൈ ྘ ඃ༑ MATLAB ೈࡱ֥ࠎЧ൐Ⴈٚ ဒ  ॥ᇅ༢๤‫ٳିྟࢫߌ྘ׅ‬༅ ‫م‬đѩ০Ⴈ MATLAB ൌགྷ॥ᇅ༢๤  ᆣ  сቓ ‫֥ିྟࢫߌ྘ׅ‬ٟᆇ‫ٳ‬༅ ০Ⴈ MATLAB ӱ྽߻ᇅ॥ᇅ༢๤  ሱ‫׮‬॥ᇅ༢๤֥໗‫ބྟק‬໗ ࢨᄁཙႋ౷ཌa࠹ෘྟିᆷѓđษ  ဒ  сቓ ෿༂ҵ‫ٳ‬༅ ંषߌ٢նПඔؓоߌ༢๤ཙႋ෎ ᆣ ؇a໗‫ބྟק‬໗෿༂ҵ֥႕ཙ 6 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ০Ⴈ MATLAB ӱ྽߻ᇅ॥ᇅ༢๤  ሱ‫׮‬॥ᇅ༢๤௔თ‫ٳ‬༅ ֥ Nyquist ๭aBode ๭đ஑љ໗‫  ק‬ဒ  сቓ ྟđ࠹ෘ॥ᇅ༢๤֥‫ږ‬ᆴღ؇‫ބ‬ཌྷ ᆣ сቓ  ॥ᇅ༢๤཮ᆞࠣഡ࠹ ໊ღ؇ ੂaൌဒБۡေ౰ ০Ⴈ MATLAB ೈࡱഡ࠹॥ᇅ༢๤ ഡ   ࠹ ૄՑഈࠏൌဒсྶิࢌൌဒБۡbൌဒБۡႮൌဒჰ৘aൌဒଽಸaٟᆇӱ྽aൌဒඔऌ࠺੣ࠣ‫ٳ‬༅ ԩ৘֩ଽಸቆӮb ௾aॉ‫ނ‬ٚൔაӮࠛ௟‫ק‬ѓሙ ൌဒӮࠛğყ༝aഈࠏҠቔaБۡ ϖa࢝ҋࠣᇶေҕॉሧਘ ࢝ ҋğ uሱ‫׮‬॥ᇅ৘ંൌဒᆷ֝඀vđ຦ٙaဗᆽӑщཿđ2007 ୍ ҕॉ඀ğuሱ‫׮‬॥ᇅჰ৘vđ‫ݓ‬ٝ‫۽‬ြԛϱഠđ຦߃၂ᇶщđ ୍ uࠎႿ ."5-"# ֥༢๤‫ٳ‬༅აഡ࠹v॥ᇅ༢๤đੌඨ฿aႿ໏щᇷđ༆ν‫׈‬ሰ॓࠯ն࿐ԛ ϱഠđ ୍ u."5-"# ॥ᇅ༢๤ഡ࠹აٟᆇvđᅵ໓‫ڂ‬щᇷđ༆ν‫׈‬ሰ॓࠯ն࿐ԛϱഠđ ୍    ն‫۔‬ᇅ‫ר‬ದğဗᆽӑ ն‫۔‬ബ‫ק‬ದğ৙༵ᄍ ᇅ‫ר‬ರ௹ğ ୍  ᄅ         7 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ሱ‫׮‬॥ᇅჰ৘൱ज़࠹߃č ࿐ൈĎ ླ൱ ൱ज़ࠇൌဒଽಸᅋေ ज़ຓቔြ ྽ᇛႨ ज़ ‫ݼ‬ൈൈ ྟ 1-2a1-6 ඔᇉ 2-4(a) 2-5(b) 1 1 4 ৘ં ࿂ં ࢝࿐ 2-2(1)(2) 2-8 2 2 ৘ં ༢๤ඔ࿐ଆ྘֥หׄaো྘‫ࡹބ‬ଆჰᄵ 2-9 2 ࢝࿐ ༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫ 2-6 2-10(c) 32 ৘ં ٤ཌྟඔ࿐ଆ྘ཌྟ߄ 2-11(a) 43 2 ࢝࿐ ཌྟ༢๤֥Ԯ‫ݦ־‬ඔ 2-12(b) 53 2-13(a) **** ৘ં ཌྟ༢๤֥Ԯ‫ݦ־‬ඔ 2-14(c) 74 2 ࢝࿐ ‫ࠣࢫߌ྘ׅ‬ఃԮ‫ݦ־‬ඔ ປӮൌဒБ 85 ۡ 10 5 2 ৘ં ༢๤ࢲ‫ܒ‬๭ 11 6 ࢝࿐ 3-2 12 6 13 7 2 ৘ં ྐ‫ݼ‬ੀ๭઼ࠣ࿠‫܄‬ൔ 3-3 14 7 ࢝࿐ 3-4 2 ഈࠏ ॥ᇅ༢๤‫ٳିྟࢫߌ྘ׅ‬༅ഈࠏൌဒ ൌဒ 3-7 3-11 ৘ં ཌྟ༢๤ൈࡗཌྷႋ֥ྟିᆷѓ ປӮൌဒБ 2 ࢝࿐ ၂ࢨ༢๤֥ൈთ‫ٳ‬༅ ۡ 2 ৘ં ‫ࢨؽ‬༢๤֥ൈთ‫ٳ‬༅ 4-2 ࢝࿐ 4-12 ৘ં ‫ࢨؽ‬༢๤֥ൈთ‫ٳ‬༅ 4-13 2 ࢝࿐ ۚࢨ༢๤֥ൈთ‫ٳ‬༅ ৘ં ཌྟ༢๤໗‫୑ۀྟק‬a‫ק‬ၬ‫ࡱ่ބ‬ 2 ࢝࿐ ཌྟ༢๤֥սඔ໗‫ק‬஑ऌ 2 ৘ં ཌྟ༢๤֥༂ҵ‫ٳ‬༅ ࢝࿐ 2 ഈࠏ ሱ‫׮‬॥ᇅ༢๤໗‫ބྟק‬໗෿ྟି‫ٳ‬༅ ൌဒ 15 8 1 ৘ં ۴݅ࠖ‫ࠎ֥م‬Ч‫୑ۀ‬ ࢝࿐ 16 8 3 ৘ં ߻ᇅ༢๤ 180 ۴֥݅ࠖࠎЧ‫م‬ᄵ ࢝࿐ 17 9 2 ৘ં 0° ۴݅ࠖ߻ᇅࠣҕэਈ۴݅ࠖ߻ᇅ ࢝࿐ 8 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ླ൱ ൱ज़ࠇൌဒଽಸᅋေ ज़ຓቔြ ྽ᇛႨ ज़ ‫ݼ‬ൈൈ ྟ 4-4a4-6 4-8a4-15 ඔᇉ 5-4(2)a(4) 18 9 2 ৘ં ॥ᇅ༢๤֥۴݅ࠖ‫ٳ‬༅ٚ‫م‬ 5-3(4) ࢝࿐ 5-5(2)(3) 5-10 ৘ં ௔ੱหྟࠎЧ‫୑ۀ‬a‫ק‬ၬࠣࠫ‫ޅ‬іൕ‫م‬ 5-7 19 10 2 ࢝࿐ ‫֥ࢫߌ྘ׅ‬௔ੱหྟ 5-9 ৘ં ‫֥ࢫߌ྘ׅ‬௔ੱหྟ ປӮൌဒБ 20 10 2 ࢝࿐ षߌࠞቕѓ๭֥߻ᇅ ۡ ৘ં षߌѵ֣๭֥߻ᇅ 6-1 21 11 2 ࢝࿐ ቋཬཌྷ໊༢๤Ԯ‫ݦ־‬ඔ֥ಒ‫ק‬ 6-2 22 11 2 ৘ં ଱উථห໗‫ק‬஑ऌ ࢝࿐ 23 12 2 ৘ં ໗‫ק‬ღ؇ ࢝࿐ ৘ં оߌ௔ੱหྟ 24 12 2 ࢝࿐ ௔ੱหྟ‫ٳ‬༅ 25 13 2 ഈࠏ ሱ‫׮‬॥ᇅ༢๤௔თ‫ٳ‬༅ഈࠏൌဒ ൌဒ ৘ં ༢๤֥ഡ࠹ࠣ཮ᆞ໙ี 26 13 2 ࢝࿐ ௔ੱ‫م‬Ա৳཮ᆞ 27 14 2 ৘ં ௔ੱ‫م‬Ա৳ӑభ཮ᆞ ࢝࿐ 28 14 2 ৘ં ௔ੱ‫م‬Ա৳Ӿު཮ᆞ ࢝࿐ 29 15 2 ഈࠏ ॥ᇅ༢๤֥཮ᆞࠣഡ࠹ഈࠏൌဒ ປӮൌဒБ ൌဒ ۡ ৘ં ‫྘ׅ‬٤ཌྟหྟa٤ཌྟหᆘa٤ཌྟ༢๤‫ٳ‬༅ٚ‫م‬ 30 15 2 ࢝࿐ ૭ඍ‫ݦ‬ඔ ৘ં ૭ඍ‫ݦ‬ඔ 31 16 2 ࢝࿐ ૭ඍ‫ݦ‬ඔ‫ٳ‬༅‫م‬ 7-2 32 16 2 ৘ં ૭ඍ‫ݦ‬ඔ‫ٳ‬༅‫م‬ 7-7 ࢝࿐ 7-8    9 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 1 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ၂ᅣ ࿂ંč1-3 ࢫĎ 1ēሱ‫׮‬॥ᇅᄝ۲ਵთ֥ႋႨ 2ēሱ‫׮‬॥ᇅ֥ቔႨ ᇶေଽಸ 3ēሱ‫׮‬॥ᇅ‫ק‬ၬğሱ‫׮‬॥ᇅࣼ൞ᄝીႵದᆰࢤҕა֥౦ঃ༯đ০Ⴈ॥ᇅఖ൐Ф॥ؓའčࠇ ‫ݖ‬ӱĎ֥ଖུ໾৘ਈሱ‫ֹ׮‬οყ༵۳‫ੰ֥ܿק‬ಀᄎྛb 4ēሱ‫׮‬॥ᇅ༢๤֥ࠎЧᆯିჭࡱࠣࠎЧॿ๭֩ 5ēषߌ॥ᇅაоߌ॥ᇅ ଢ֥აေ ਔࢳሱ‫׮‬॥ᇅ༢๤֥ࠎЧᆯିჭࡱaࠎЧඌეࠣٚॿ๭ ౰ ᅧ໤ሱ‫׮‬॥ᇅ‫ק‬ၬ ᅧ໤षߌaоߌ॥ᇅ֥‫ק‬ၬaࠎЧॿ๭ ᇗ ׄ ა ଴ ᇗׄğሱ‫׮‬॥ᇅ֥‫ק‬ၬaषߌ॥ᇅაоߌ॥ᇅ֥‫ק‬ၬࠣॿ๭ ׄ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 1-2  ႄ࿽ ໭ં൞ದૌ֥ರӈളࠃa‫۽‬ြളӁđߎ൞ॢࡗฐ෬a֝֐ᇅ֝֩ࡕ؊॓࠯ਵთᇏđሱ‫׮‬॥ᇅ࠯ඌ ໭෮҂ᄝa໭෮҂ିbሱ‫׮‬॥ᇅ৘ં‫࠯ބ‬ඌၘࣜറ๩֞ഠ߶aࣜ࠶‫॓ބ‬࿐࿹࣮֥۲۱ٚ૫b ሱ‫׮‬॥ᇅ࠯ඌ൞ࡹ৫ᄝ॥ᇅંࠎԤഈ֥đ‫ط‬॥ᇅં࿹࣮֥൞॥ᇅ֥၂Ϯྟ৘ંđ෱҂ऎุ૫ؓଖ ၂ো॥ᇅ༢๤֥đၹՎ෱൞၂૊ၛ৘ંູᇶ֥ज़ӱb ሱ‫׮‬॥ᇅ৘ં൞၂૊৘ંྟ‫۽ބ‬ӱྟ֥ሸ‫॓ކ‬࿐b ē॥ᇅ৘ં֥ࠎԤܴ୑ ॥ᇅ৘ં൞ࡹ৫ᄝႵॖି‫ؿ‬ᅚ၂ᇕٚ‫م‬ট࿹࣮۲ൔ۲ဢ༢๤ᇏ॥ᇅ‫ݖ‬ӱᆃ၂ࠎԤഈ֥৘ંč္ࠧđ ෱൞࿹࣮༢๤‫֥ྟ܋‬॥ᇅ‫ݖ‬ӱ֥৘ંđॖၛϜൌ࠽ؓའ֥໾৘‫ݤ‬ၬԎའԛটđၹՎđ෱၂‫ק‬൞ၛඔ࿐ ‫۽‬ऎቔູᇶေ࿹࣮൭‫֥؍‬Ďb ē॥ᇅ৘ં֥࿹࣮ؓའ ॥ᇅં֥࿹࣮൞૫ཟ༢๤֥b ܼၬֹࢃğ॥ᇅં൞࿹࣮ྐ༏֥ӁളaሇߐaԮ‫־‬a॥ᇅ‫ބ‬ყБ֥॓࿐Ġ ༮ၬֹࢃğ۴ऌ௹ຬ֥ൻԛট‫ڿ‬э॥ᇅൻೆđ൐༢๤֥ൻԛିղ֞ଖᇏყ௹ི֥‫ݔ‬b ē॥ᇅંაඔ࿐ࠣሱ‫࠯߄׮‬ඌ֥ܱ༢ ॥ᇅં൞ႋႨඔ࿐֥၂۱‫ٳ‬ᆦđ෱֥ଖུ৘ં֥࿹࣮ߎေࢹᇹႿԎའඔ࿐b‫ط‬॥ᇅં֥࿹࣮Ӯ‫ݔ‬ ೏ေႋႨႿൌ࠽‫۽‬ӱᇏđࣼсྶᄝ৘ં‫୑ۀ‬აႨটࢳथᆃུൌ࠽໙ี֥ൌႨٚ‫م‬ᆭࡗࡏఏ၂ቖూਃb ሱ‫׮‬॥ᇅ‫ބ‬ሱ‫׮‬॥ᇅ༢๤  ሱ‫׮‬॥ᇅ໙ี֥ิԛ ದૌթᄝሢ၂ᇕ௴ђ֥ေ౰ࠇ༐ຬđࠧေ౰ଖུ໾৘ਈົӻᄝଖᇕห‫֥ק‬čೂ‫קޚ‬҂эࠇοଖᇕ ܿੰэ߄ࠇ۵ሶଖ۱э߄֥ਈ֩֩Ďѓሙഈb ২ೂğඣད֥၁໊ۚ؇ H ֥‫קޚ‬॥ᇅࣼ൞၂۱൅‫֥྘ׅٳ‬২ሰb ದ‫۽‬॥ᇅ֥༢๤đหљ൞ᄝ༢๤ࢠູ‫گ‬ᄖൈđః॥ᇅ֥ॹ෎ྟaሙಒྟ‫ބ‬໗‫ࣼྟק‬҂ಸၞ֤֞Ќ ᆣđമᇀ൞҂ॖିղ֞ყ௹֥॥ᇅི‫ݔ‬đ๝ൈđ္҂০Ⴟิۚস‫׮‬ളӁ৯‫ࢳބ‬٢ದোሱദb ሱ‫׮‬॥ᇅ֥‫ק‬ၬࠣࠎЧᆯିჭࡱ ēሱ‫׮‬॥ᇅ֥‫ק‬ၬ ሱ‫׮‬॥ᇅࣼ൞ᄝીႵದᆰࢤҕა֥౦ঃ༯đ০Ⴈ॥ᇅఖ൐Ф॥ؓའčࠇ‫ݖ‬ӱĎ֥ଖུ໾৘ਈčФ॥ਈĎ ሱ‫ֹ׮‬οყ༵۳‫ੰ֥ܿק‬ಀᄎྛb ēࠎЧᆯିჭࡱ 10 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ бࢠሱ‫׮‬॥ᇅაದ‫۽‬॥ᇅ༢๤đॖ࡮ሱ‫׮‬॥ᇅ༢๤թᄝሢ೘۱ቋࠎЧ֥ᆯିჭࡱb čĎҩਈჭࡱაэෂఖğսูದ֥ငࣕđປӮྐ‫֥ݼ‬Ґࠢҩਈ‫ބ‬эෂĠ čĎ॥ᇅఖğսูದ֥նଷđປӮбࢠa࠹ෘa஑؎đѩ‫ؿ‬ԛ‫ࢫט‬ᆷ਷Ġ čĎᆳྛჭࡱğսูದ֥ࠔಽ‫ބ‬൭đປӮࠇൌགྷؓФ॥ؓའ֥‫ࢫט‬ቔႨb ಩‫ޅ‬ൌ࠽֥ሱ‫׮‬॥ᇅ༢๤đ‫׻‬ഒ҂ਔഈඍ೘۱֥ᆯିჭࡱč҆ࡱa҆‫ٳ‬Ďb ēሱ‫׮‬॥ᇅᇏ֥ӈႨඌე z ॥ᇅ༢๤č$POUSPMTZTUFNĎğູਔղ֞ყ௹֥ଢ֥čཙႋĎ‫ط‬ഡ࠹ԛট֥༢๤đ෱Ⴎཌྷ޺ܱ৳ ֥҆ࡱቆ‫طކ‬Ӯb z ॥ᇅؓའč$POUSPMQMBOUĎğᆷФ॥ഡСࠇ‫ݖ‬ӱb z ॥ᇅఖčControllerĎğ൐Ф॥ؓའղ֞෮ေ౰֥ྟିࠇሑ෿֥॥ᇅഡСb෱ࢤ൳ൻೆྐ‫ࠇݼ‬ொ ҵྐ‫ݼ‬đοყ‫֥ק‬॥ᇅܿੰ۳ԛ॥ᇅྐ‫ݼ‬čҠቔਈĎđෂ֞ᆳྛჭࡱč٢նఖĎࠇФ॥ؓའb z ༢๤č4ZTUFNĎğູൌགྷყ௹֥ଢѓ‫ࡼط‬Ⴕܱ҆ࡱč҆‫ٳ‬Ď޺৳ᄝ၂ఏ֥ᆜุb z ༢๤ൻԛđ္ӫФ॥ਈč4ZTUFNPVUQVUĎğᆷФ॥ᇅ֥ਈb෱іᆘФ॥ؓའࠇ‫ݖ‬ӱ֥ሑ෿‫ྟބ‬ ିđ෱ႻӈӈФӫູ༢๤ؓൻೆ֥ཙႋč3FTQPOTFĎb z ॥ᇅਈčҠቔਈ$POUSPMTJHOBMĎğ൞Ⴎ॥ᇅఖ۳ԛ֥ቔႨႿᆳྛࠏ‫ࠇܒ‬Ф॥ؓའ֥ྐ‫ݼ‬đ෱ ุགྷਔؓФ॥ؓའ֥‫ࢫט‬ቔႨb z ҕॉൻೆࠇ۳‫ק‬ൻೆࠇ༐ຬൻೆč%FTJSFE*OQVUĎğ൞ದູ۳‫֥ק‬༢๤ყ௹ൻԛ֥༐ຬᆴb z ಠ‫׮‬č*OUFSBDUJPOĎğ‫ۄ‬ಠ‫ބ‬௥ߊ༢๤ყ௹ྟି‫ބ‬ൻԛ֥‫ۄ‬ಠྐ‫ݼ‬čቔႨĎbႮ༢๤ଽ҆Ӂള֥ ӫູଽ҆ಠ‫׮‬đႮ༢๤ຓ҆Ӂള֥ӫູຓ҆ಠ‫׮‬đ౏ຓ҆ಠ‫ؓ׮‬༢๤‫ط‬࿽൞၂ᇕൻೆਈb z ொҵྐ‫ݼ‬č&SSPSTJHOBMĎğҕॉൻೆაൌ࠽ൻԛ֥ҵӫູொҵྐ‫ݼ‬đொҵྐ‫ݼ‬၂Ϯቔູ॥ᇅఖ ֥ൻೆྐ‫ݼ‬b ēٚॿ๭č#MPDLĎ ࡼ༢๤۲҆‫ٳ‬Ⴈٚॿѩᇿၛ໓ሳࠇ‫ݼژ‬οྐ༏Ԯ‫ܱ־‬༢৳ࢲఏট֥၂ᇕ๭ྙіൕb ٚॿ๭ૼಒֹіൕਔ༢๤ଽ۲҆‫ྐؓٳ‬༏ࡆ‫֥۽‬ଽಸ‫ྐބ‬༏ࡗ֥ܱ༢đၛࠣྐ༏֥Ԯ‫־‬ਫ਼ࣥđ൞၂ ᇕᆰܴ֥๭ྙіൕđᄝ‫۽‬ӱ۲ਵთႨႿࣉྛ‫קބྟק‬ਈ‫ٳ‬༅đၹՎ֤֞ࠞఃܼ֥ٗႋႨb षߌ॥ᇅაоߌ॥ᇅ षߌ॥ᇅ षߌ॥ᇅ൞॥ᇅਈაФ॥ؓའᆭࡗᆺႵ၂่๙ਫ਼‫ط‬ીႵّঌ๙ਫ਼đ္ࠧ॥ᇅቔႨ֥Ԯ‫־‬ਫ਼ࣥ҂൞о ‫֥ކ‬đࠇᆀඪൻԛྐ‫ݼ‬҂ّঌቔႨႿൻೆྐ‫ݼ‬b षߌ॥ᇅႻॖၛ‫ູٳ‬ο۳‫ק‬॥ᇅ‫ބ‬οಠ‫׮‬॥ᇅ ēο۳‫ק‬॥ᇅ ᆃᇕषߌ॥ᇅ༢๤֥ࢲ‫ࡥܒ‬ֆđ‫ט‬ᆜٚьđӮЧ္ࢠ֮đః॥ᇅ֥ྟିčೂࣚ؇Ďᇶေ౼थႿ‫ܒ‬Ӯ ༢๤෮Ⴈჭࡱ֥ྟିႪਜ‫ބ‬ຓࢸߌ࣢b षߌ॥ᇅ֥ಌׄ൞ğ čĎ॥ᇅࣚ؇ࢠҵĠ čĎ༢๤֥ॆ‫ۄ‬ಠྟିࢠҵčਗ਼ϿྟࢠҵĎb ෮ၛ‫گ‬ᄖ֥॥ᇅ༢๤‫ࣚބ‬؇ေ౰ࢠ֥ۚӆ‫ކ‬၂Ϯ҂ൡ‫ކ‬ႋႨषߌ॥ᇅb ēοಠ‫׮‬॥ᇅ Ֆο۳‫֥ק‬षߌ॥ᇅ‫ٳ‬༅ॖᆩđ႕ཙ॥ᇅ༢๤ࣚ؇֥ᇶေၹ෍൞ಠ‫׮‬đؓႿପུყ༵ૼಒఃؓ༢๤ ႕ཙ֥ಠ‫׮‬đॖၛ۴ऌҩ֤֥ಠ‫׮‬ਈնཬđؓ༢๤Ґ౼၂ᇕҀӊ‫ྩބ‬ᆞԩ৘đၛַཨࠇࡨཬಠ‫ؓ׮‬༢๤ ൻԛ֥႕ཙđ෱ॖၛิۚ॥ᇅ༢๤֥ࣚ؇đࡨཬಠ‫ؓ׮‬ൻԛ֥႕ཙđิۚ༢๤֥ॆ‫ۄ‬ಠି৯b ֒ಖοಠ‫֥׮‬षߌ॥ᇅсྶႵਆ۱భิ่ࡱđࠧ čĎಠ‫ؓ׮‬ൻԛ֥႕ཙหྟсྶ൞ყᆩ֥Ġ  čĎಠ‫׮‬сྶ൞ॖҩਈ֥b ēषߌ॥ᇅ֥‫ק‬ၬ ೏༢๤֥ൻԛਈؓ༢๤֥॥ᇅቔႨીႵ႕ཙ֥༢๤ӫູषߌ॥ᇅb หׄğ čĎൻԛਈӁള॥ᇅቔႨᆰࢤ႕ཙൻԛਈĠ čĎൻԛਈؓൻೆӁള֥॥ᇅቔႨીႵ႕ཙč໭ّঌቔႨĎb оߌ॥ᇅč$MPTFE-PPQ$POUSPMĎ 11 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ᄝ॥ᇅ༢๤ᇏđ۴ऌൌ࠽ൻԛটྩᆞ॥ᇅቔႨđൌགྷؓФ॥ؓའࣉྛ॥ᇅ֥಩ༀđᆃᇕ॥ᇅჰ৘ӫ ّູঌ॥ᇅჰ৘bႮႿႄೆਔൻԛਈّ֥ঌྐ༏đ൐ᆜ۱༢๤Ӯູо‫֥ކ‬đၹՎđοّঌჰ৘ࡹ৫ఏট ֥॥ᇅ༢๤đࢡቓоߌ॥ᇅ༢๤bၹູ൞οொҵ֥նཬࣉྛ‫ࢫט‬॥ᇅ֥đ෮ၛоߌّঌ॥ᇅႻӫູοொ ҵ॥ᇅb оߌ॥ᇅ֥‫ק‬ၬğ ༢๤֥ൻԛྐ‫ؓݼ‬॥ᇅቔႨିႵᆰࢤ႕ཙ֥༢๤đࢡቓоߌ॥ᇅ༢๤b หׄ‫ބ‬Ⴊׄğ čĎ༢๤֥॥ᇅఖ۴ऌொҵ֥նཬট‫ؿ‬ԛ‫֥ݼྐࢫט‬đ෮ၛđႻӫоߌ॥ᇅູοொҵ॥ᇅb čĎྐ‫֥ݼ‬Ԯ‫־‬๯ࣥྙӮ၂۱о‫ߌ֥ކ‬ਫ਼đӫູоߌbоߌ༢๤ॖၛ൞ᆞّঌ֥đ္ॖၛ൞‫ّڵ‬ঌ ֥đൌ࠽ॖႨ֥ሱ‫׮‬॥ᇅ༢๤၂Ϯ൞‫ّڵ‬ঌ॥ᇅ༢๤b čĎॖၛႨࣚ؇҂֥ۚჭఖࡱđ‫ܒ‬Ӯࣚ؇ࢠ֥ۚ॥ᇅ༢๤b čĎ༢๤ऎႵࢠ఼֥ॆ‫ۄ‬ಠି৯b ಌׄğčĎᄹࡆਔቆӮ༢๤֥ჭఖࡱඔਈđՖ‫ط‬ᄹࡆਔ༢๤֥ӮЧb čĎᄹࡆਔ༢๤֥‫گ‬ᄖྟb čĎ༢๤֥ᄹၭč٢նПඔĎ෥ാđᆃၩ໅ሢ॥ᇅ༢๤֥‫ିࢫט‬৯Фཤ೐đཙႋ෎؇эતb čĎႮႿႄೆਔّঌđ෮ၛоߌ༢๤ࣼթᄝሢ໗‫ྟק‬໙ีđ‫ط‬໗‫ྟק‬൞॥ᇅ༢๤‫۽‬ቔ֥൮ေ่ࡱb ః෱॥ᇅٚൔ ᄝൌ࠽֥‫۽‬ӱ॥ᇅൌ࡬ᇏđߎॖၛࡼഈඍਆᇕ॥ᇅٚൔࣉྛൡ֥֒ቆ‫ކ‬đ‫ܒ‬Ӯ‫ކگ‬॥ᇅٚൔb ē ‫ࡆڸ‬۳‫ק‬ൻೆ֥Ҁӊ॥ᇅٚൔ ē‫ࡆڸ‬ಠ‫׮‬ൻೆ֥Ҁӊ॥ᇅٚൔ 12 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 2 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ၂ᅣ ࿂ંč4-6 ࢫĎ ᇶေଽಸ ሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬োࠣቆӮ ؓ॥ᇅ༢๤֥ေ౰ ॥ᇅ༢๤֥‫ٳ‬༅აഡ࠹ ሱ‫׮‬॥ᇅ৘ં֥‫ؿ‬ᅚࡥൎ ّঌ॥ᇅई২ ଢ֥აေ ᅧ໤ሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬োၛࠣؓ༢๤֥ေ౰ ౰ ਔࢳ॥ᇅ༢๤֥ࠎЧቆӮa॥ᇅ༢๤֥‫ٳ‬༅აഡ࠹ ਔࢳሱ‫׮‬॥ᇅ৘ં֥‫ؿ‬ᅚࡥൎ ᇗׄა଴ ᅧ໤ሱ‫׮‬॥ᇅ༢๤֥‫۽‬ቔჰ৘‫ބ‬ٚॿ๭֥߻ᇅ ׄ ᇗׄğሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬ো ؓ॥ᇅ༢๤֥ေ౰ ଴ׄğሱ‫׮‬॥ᇅ༢๤֥‫۽‬ቔჰ৘‫ބ‬ٚॿ๭֥߻ᇅ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 1-6  ሱ‫׮‬॥ᇅ༢๤֥‫ٳ‬োࠣࠎЧቆӮ ο۳‫֥ݼྐק‬หᆘ‫ٳ‬ো ē‫ޚ‬ᆴ॥ᇅ༢๤ หׄğ༐ຬ༢๤֥ൻԛົӻᄝ۳‫ק‬ᆴഈ҂эࠇэ߄‫ཬޓ‬đᆃো॥ᇅ༢๤൞ቋӈ࡮֥đӈӈ္ӫູ ሱᆓ‫ק‬༢๤đཞ࿢৯aੀਈa໑؇a෎؇a‫׈‬࿢a‫׈‬ੀ֩‫קޚ‬॥ᇅ༢๤b ēෛ‫׮‬॥ᇅ༢๤ หׄğᆃো॥ᇅ༢๤֥ᇶေหׄ൞۳‫֥ݼྐק‬э߄ܿੰ൞൙༵҂ಒ‫֥ק‬ෛࠏྐ‫ݼ‬đ॥ᇅ༢๤֥ᇶ ေ಩ༀ൞൐༢๤֥ൻԛିॹ෎aሙಒֹ۵ෛൻೆ֥э߄‫ط‬э߄đ‫ܣ‬ᆃো༢๤ӈӈႻӫູ۵ሶ॥ᇅ༢๤b ӈ࡮֥২ሰೂࠅ஛aূղa֝֐ᇅ֝֩॥ᇅ༢๤b ēӱ྽॥ᇅ༢๤ หׄğӱ྽॥ᇅ༢๤აෛࠏ॥ᇅ༢๤֥҂๝ᄝႿ༢๤֥۳‫ק‬ൻೆ҂൞ෛࠏ֥đ‫ط‬൞ಒ‫֥ק‬aοყ ༵֥ܿੰэ߄b෱ေ౰༢๤֥ൻԛି࿸۬οൻೆэ߄‫ط‬э߄đѩऎႵቀ‫֥ࣚܔ‬؇đӈ࡮֥২ሰೂඔ॥ ࡆ‫۽‬aሱ‫׮‬ੀඣളӁཌ༢๤֩b ēēο༢๤֥ඔ࿐ଆ྘‫ٳ‬ো ᆃᇕ‫ٳ‬োٚ‫م‬൞οᅶჭࡱࠇ༢๤֥ඔ࿐ଆ྘čٚӱࠇඔ࿐૭ඍĎ֥หᆘđ၇ऌఃൻೆൻԛᆭࡗ֥ ܱ༢টࣉྛ‫ٳ‬োđӈॖၛ‫ູٳ‬ཌྟ༢๤‫ބ‬٤ཌྟ༢๤ਆնোb ēཌྟ༢๤ ؓႿ၂۱༢๤đ֒ఃൻೆčࠗৣĎ‫ބ‬ൻԛčཙႋĎ๝ൈડቀ‫ބྟࡆן‬ఊՑྟൈӫఃູཌྟ༢๤b ۴ऌཌྟ༢๤֥‫ק‬ၬđડቀཌྟหྟ֥ჭࡱӫູཌྟჭࡱđ‫ܒط‬Ӯ༢๤֥෮Ⴕჭࡱनູཌྟჭࡱ ֥đсູཌྟ༢๤b ෮໌ཌྟหྟđՖࠫ‫ޅ‬ഈটुđ൞ᆷჭࡱ֥࣡෿หྟູ၂่๙‫ݖ‬ቕѓჰ֥ׄᆰཌb ཌྟ༢๤ӈॖၛႨັ‫ٳ‬ٚӱটіൕđ೏ັ‫ٳ‬ٚӱ֥༢ඔनູӈඔđᄵӫູཌྟ‫ק‬ӈ༢๤b ২ ğ஑؎༯ਙൻԛཙႋؓႋ֥༢๤൞‫ູڎ‬ཌྟ༢๤b ∫ ༢๤ ğ = + t x(τ )dτđt >  y1 (t) 3q(0) 5 0 0 ༢๤ ğ y2 (t) = 3q(0) + 5x 2 (t)đt > 0  ༢๤ ğ y3 (t) = 3q 2 (0) + 5x(t)đt > 0  ༢๤ ğ y4 (t) = 3q 2 (0) + lg x(t)đt > 0 13 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ <ࢳ>ğ ∫ ༢๤ ğਬൻೆཙႋ[3q(0)] ‫ބ‬ਬሑ෿ཙႋ[5 t x(τ )dτ ] नऎႵཌྟྟđ‫ູܣ‬ཌྟ༢๤b 0 ༢๤ ğࣇਬൻೆཙႋ[3q(0)] ऎႵཌྟྟĠ ༢๤ ğࣇਬሑ෿ཙႋ[5x(t)]ऎႵཌྟྟĠ ༢๤ ğਬൻೆཙႋ‫ބ‬ਬሑ෿ཙႋन҂ऎႵཌྟྟĠ ē٤ཌྟ༢๤ ُ൞҂ડቀཌྟ༢๤หྟ֥༢๤đ๤ӫູ٤ཌྟ༢๤bऎֹุࢃđᆺေ༢๤ᇏթᄝ၂۱ࠇ၂۱ၛഈ ֥٤ཌྟჭࡱđପહđᆃ۱༢๤ࣼ൞٤ཌྟ༢๤b٤ཌྟ༢๤Ⴈ٤ཌྟٚӱটіൕb ॖၛࡼ٤ཌྟหྟ‫ູٳ‬ਆնোđࠧ٤Чᇉ٤ཌྟ‫ބ‬Чᇉ٤ཌྟb čĎ٤Чᇉ٤ཌྟğؓႿଖ၂ো٤ཌྟหྟđᄝଖ၂౵თଽॖၛ࣍රູཌྟܱ༢đ‫ط‬ᄝնٓຶ‫۽‬ ቔ౵თൈđᆃᇕ࣍ර֥ཌྟܱ༢ࣼ҂թᄝਔb čĎЧᇉ٤ཌྟğؓႿ಩ၩնཬ֥ൻೆྐ‫ݼ‬đनӯགྷ٤ཌྟหྟ֥ᆃো٤ཌྟหྟb ‫֥྘ׅ‬Чᇉ٤ཌྟೂ༯ğ ٤Чᇉ٤ཌྟ༢๤ॖၛ๙‫ؓݖ‬٤Чᇉ٤ཌྟᄝ‫۽‬ቔׄ‫ྛࣉ࣍ڸ‬ཌྟ߄ԩ৘‫֤֞ط‬ཌྟ߄ު֥༢๤ ඔ࿐ଆ྘đಯॖοཌྟ༢๤֥৘ંࣉྛ‫ٳ‬༅‫ބ‬ഡ࠹b‫ط‬Чᇉ٤ཌྟหྟđᆺିοᅶ٤ཌྟ༢๤֥ٚ‫ࣉم‬ ྛ‫ٳ‬༅‫ބ‬ഡ࠹b οྐ‫ݼ‬Ԯ‫֥־‬৵࿃ྟ߃‫ٳ‬ ē৵࿃༢๤ ᆃো༢๤ᇏ֥෮Ⴕჭࡱ֥ൻೆൻԛྐ‫ݼ‬नູൈࡗ֥৵࿃‫ݦ‬ඔđ෮ၛႻӈӫູଆ୅༢๤b ē৖೛༢๤ ༢๤ᇏᆺေႵ၂ԩ֥ྐ‫ݼ‬൞ઝԊ྽ਙࠇඔሳྐ‫ݼ‬ൈđ‫ھ‬༢๤ࣼ൞৖೛༢๤bᆃো༢๤ӈႨҵ‫ٳ‬ٚӱ টіൕb৖೛༢๤ൌགྷഈ൞ࡼ৵࿃ྐ‫ݖࣜݼ‬Ґဢު৖೛߄ູઝԊࠇඔሳྐ‫ުݼ‬ෂೆ࠹ෘࠏࣉྛ‫ٳ‬༅aԩ ৘aथҦުđྙӮઝԊࠇඔሳൔ॥ᇅྐ‫ݼ‬đѩߎჰູཌྷႋ֥ଆ୅ਈ॥ᇅྐ‫ؓݼ‬Ф॥ؓའൌགྷ॥ᇅb ο༢๤֥ൻೆൻԛྐ‫֥ݼ‬ඔਈ‫ٳ‬ো ēֆэਈ༢๤č4*40Ď ෮໌ֆэਈ༢๤൞ᆷ༢๤ᆺႵ၂۱ൻೆ‫ބ‬၂۱ൻԛđ෱ᆺᇿᇗ༢๤֥ຓ҆ൻೆ‫ބ‬ൻԛđ‫ط‬҂ܱྏ༢ ๤ଽ֥҆ሑ෿э߄đ෮ၛֆൻೆֆൻԛ༢๤ॖၛϜ༢๤ुӮູ၂۱‫ޑ‬༨ሰb ࣜ‫ׅ‬॥ᇅ৘ં࿹࣮֥ؓའᇶေ൞ֆൻೆֆൻԛ֥ཌྟ‫ק‬ӈ༢๤b ē‫؟‬эਈ༢๤č.*.0Ď ෮໌‫؟‬эਈ༢๤൞ᆷ༢๤Ⴕ‫؟‬۱ൻೆࠇֆ۱ൻԛࠇ‫؟‬۱ൻԛđ෱҂ࣇࣇᇿᇗ༢๤֥ൻೆ‫ބ‬ൻԛэ ਈđߎ۷‫ྏֹܱ؟‬༢๤ࢲ‫ܒ‬ଽ҆۲ሑ෿эਈ֥э߄‫ބ‬۱ሑ෿эਈᆭࡗ֥ᯒ‫ܱކ‬༢b ‫؟‬эਈ༢๤൞གྷս॥ᇅ৘ં࿹࣮֥ᇶေؓའđᄝඔ࿐ഈၛሑ෿ॢࡗэਈ‫ބم‬इᆔ৘ંູᇶေ࿹࣮‫۽‬ ऎb ሱ‫׮‬॥ᇅ༢๤֥ࠎЧቆӮ ē۳‫ק‬ჭࡱğఃᆯି൞۳ԛა௹ຬ֥ൻԛཌྷؓႋ֥༢๤ൻೆਈđ൞၂োӁള༢๤॥ᇅᆷ਷֥ልᇂb ēҩਈჭࡱğఃᆯି൞࡟ҩФ॥ਈč༢๤ൻԛĎđѩࣉྛྐ‫֥ݼ‬эߐčೂ٤‫׈‬ਈሇߐĎ‫ބ‬ԮൻđႨ ႿّঌФ॥ਈ֞бࢠჭࡱაൻೆࣉྛбࢠčྙӮொҵྐ‫ݼ‬Ďb ēбࢠჭࡱğఃᆯି൞Ϝҩਈჭࡱ࡟ҩ֥֞ൌ࠽ൻԛਈა۳‫ק‬ჭࡱ۳ԛ֥ൻೆਈࣉྛбࢠđ֤֞ ொҵྐ‫ݼ‬b ē٢նჭࡱğఃᆯି൞ࡼັ೐֥ொҵྐ‫ྛࣉݼ‬٢նđၛቀ‫ੱۿ֥ܔ‬ট๷‫׮‬ᆳྛࠏ‫ࠇܒ‬Ф॥ؓའb ēᆳྛჭࡱğఃᆯି൞ᆰࢤ॥ᇅФ॥ؓའđ൐ఃФ॥ਈ‫ؿ‬ളэ߄đ২ೂ‫ل‬૊aය‫֩ܒࠏڛ‬b ē཮ᆞჭࡱğఃᆯି൞ູਔ‫ڿ‬೿ࠇิۚ॥ᇅ༢๤֥ྟିčೂ໗‫ྟק‬a໗෿ࣚ؇aཙႋ෎؇֩Ďđᄝ ॥ᇅ༢๤֥ࠎЧࢲ‫ܒ‬ഈ‫ࡆڸ‬၂‫֥ק‬ልᇂčჭࡱĎđᆃᇕ‫཮֥ࡆڸ‬ᆞልᇂčჭࡱĎॖၛႵ‫؟‬ᇕྙൔđೂԱ ৳཮ᆞaѩ৳཮ᆞaّঌ཮ᆞ֩b  ؓ॥ᇅ༢๤֥ေ౰‫ٳބ‬༅ഡ࠹ ؓ༢๤֥ေ౰ ৘མ֥॥ᇅ༢๤đсྶऎСਆٚ૫֥ྟିđࠧ čĎ൐༢๤֥ൻԛॹ෎aሙಒֹοൻೆྐ‫ݼ‬ေ౰֥௹ຬൻԛᆴэ߄Ġ čĎ൐༢๤֥ൻԛ࣐ਈ҂൳಩‫ޅ‬ಠ‫֥׮‬႕ཙĠ 14 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ؓሱ‫׮‬॥ᇅ༢๤ྟି֥ᇶေေ౰ູğ čĎ໗‫ྟק‬ğေ౰༢๤໗‫ק‬ѩऎႵ၂‫֥ק‬໗‫ק‬ღ؇b čĎට෿ᇉਈğေ౰༢๤֥ට෿ཙႋॹ෎౏э߄௜໗b čĎ໗෿ࣚ؇ğေ౰༢๤֥໗෿༂ҵડቀഡ࠹֥ေ౰b ഈඍ೘۱ေ౰ສສ‫ޓ‬଴๝ൈડቀđѩ౏ཌྷ޺ᆭࡗႵ၂‫֥ק‬ᇅჿܱ༢đ২ೂđູЌᆣ༢๤Ⴕቀ‫֥ࣚܔ‬ ؇đေ౰༢๤֥षߌ٢նПඔᄀնᄀ‫ݺ‬đ֌षߌ٢նПඔ֥նཬđಏ൳ᇅაоߌ༢๤֥໗‫ྟק‬đၹՎᆃ ུ၂్ᆭࡗླေࣉྛᅼᇏ࿊ᄴb ॥ᇅ༢๤֥‫ٳ‬༅‫ބ‬ഡ࠹ ē༢๤‫ٳ‬༅ ၂Ϯ҄ᇧğčĎࡹ৫ඔ࿐ଆ྘ĠčĎ‫ٳ‬༅༢๤֥ྟିđ࠹ෘఏऎุ֥ྟିᆷѓĠčĎ‫ٳ‬༅༢๤ҕඔ э߄ؓ༢๤ྟି֥႕ཙđѩथ‫ק‬࿊ᄴ‫ކ‬৘֥‫ٳ‬༅ٚ‫م‬b ༢๤֥‫ٳ‬༅ٚ‫م‬ສສෛሢඔ࿐ଆ྘֥҂๝‫ط‬҂๝đᄝࣜ‫ׅ‬॥ᇅ৘ંᇏđӈႨ֥‫ٳ‬༅ᇶေႵൈთ‫ٳ‬༅ ‫م‬a‫گ‬௔თ‫ٳ‬༅‫م‬a۴݅ࠖ‫ٳ‬༅‫֩م‬b ༢๤ഡ࠹  15 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 3 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘č1a2 ࢫĎ ᇶေଽಸ ඔ࿐ଆ྘֥ࠎЧ‫୑ۀ‬aหׄaো྘ ༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫ ଢ֥აေ ਔࢳࡹ৫ඔ࿐ଆ྘֥ၩၬ‫ބ‬ඔ࿐ଆ྘֥หׄaো྘ ౰ ᅧ໤ࡹ৫ັ‫ٳ‬ٚӱඔ࿐ଆ྘֥ٚ‫҄ބم‬ᇧđ ࠏྀ௜၍a࿈ሇ༢๤a‫׈‬࿐༢๤‫گބ‬ᄖ༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫ ᇗ ׄ ა ଴ ᇗׄğັ‫ٳ‬ٚӱ֥ࡹ৫ ଴ׄğ۴ऌ໾৘ࠏ৘ࡹ৫۲ো҂๝༢๤֥ັ‫ٳ‬ٚӱ ׄ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 2-4čaĎa2-5čbĎ ቔြี 2.1 ႄ࿽ ඔ࿐ଆ྘ğ૭ඍ༢๤‫׮‬෿หྟࠣఃэਈᆭࡗܱ༢֥ඔ࿐іղൔࠇః෱ྙൔ֥іൕӫູඔ࿐ଆ྘b 2.1.1 ඔ࿐ଆ྘֥หׄ 1ēཌྷරྟ‫ބ‬Ԏའ߄ ऎႵཌྷ๝ඔ࿐ଆ྘֥҂๝֥ऎุ༢๤ᆭࡗࣼ൞ཌྷර༢๤b 2ēࡥ߄ྟ‫ࣚބ‬ಒྟ ᄝࡹଆ֥ൈީđေᄝࡥ߄‫ࣚބ‬ಒᆭࡗቔᅼᇏ࿊ᄴđఃჰᄵ൞ࡥ߄ު֥ඔ࿐ٚӱ֥ࢳ֥ࢲ‫ݔ‬сྶડ ቀ‫۽‬ӱൌ࠽֥ေ౰ѩ਽Ⴕ၂‫֥ק‬Ⴥֹb 3ē‫׮‬෿ଆ྘ ෮໌‫׮‬෿ଆ྘൞ᆷ૭ඍ༢๤эਈ֥۲ࢨ֝ඔᆭࡗܱ༢֥ັ‫ٳ‬ٚӱӫູ༢๤֥‫׮‬෿ଆ྘b 4ē࣡෿ଆ྘ ෮໌࣡෿ଆ྘൞ᆷᄝ࣡෿่ࡱ༯đࠧ૭ඍ༢๤эਈ֥۲ࢨ֝ඔູਬđ૭ඍэਈᆭࡗܱ༢֥սඔٚ ӱӫູ࣡෿ଆ྘b 2.1.2 ඔ࿐ଆ྘֥ᇕো ඔ࿐ଆ྘Ⴕ‫؟‬ᇕྙൔđ২ೂັ‫ٳ‬ٚӱaҵ‫ٳ‬ٚӱaሑ෿ٚӱ‫ބ‬Ԯ‫ݦ־‬ඔaࢲ‫ܒ‬๭a௔ੱหྟ֩֩đ ࣮ࣨ࿊Ⴈଧᇕଆ྘đ၂Ϯေ൪ҐႨ֥‫ٳ‬༅ٚ‫ބم‬༢๤֥ো྘‫קط‬đ২ೂğ ৵࿃༢๤֥ֆൻೆ/ֆൻԛ༢๤֥ൈთ‫ٳ‬༅‫م‬đॖҐႨັ‫ٳ‬ٚӱb ৵࿃‫؟‬ൻೆ‫؟‬ൻԛ༢๤֥ൈთ‫ٳ‬༅‫م‬ॖၛҐႨሑ෿ٚӱb ‫ٳ‬༅௔თ‫م‬ॖၛҐႨ௔ੱหྟb৖೛༢๤ॖၛҐႨҵ‫ٳ‬ٚӱđ֩֩b 2.2 ༢๤ັ‫ٳ‬ٚӱ֥ࡹ৫ 2.2.1 ၂Ϯ҄ᇧ č1Ďಒ‫ק‬༢๤֥ൻೆਈaൻԛਈࠣᇏࡗэਈđ୫ౢ۲эਈᆭࡗ֥ܱ༢Ġ č2Ď၇ऌ‫ކ‬৘֥ࡌഡđޭ੻၂ུՑေၹ෍đ൐໙ีࡥ߄Ġ č3Ď۴ऌᆦ஥༢๤۲҆‫׮ٳ‬෿หྟ֥ࠎЧ‫ੰק‬đਙཿԛ۲҆‫֥ٳ‬ჰ൓ٚӱđః၂Ϯჰᄵ൞ğ AēՖ༢๤ൻೆ؊ष൓đ၇ՑਙཿቆӮ༢๤۲҆‫֥ٳ‬ᄎ‫׮‬ٚӱb Bēཌྷਣჭ҆ࡱᆭࡗު၂ࠩ೏ቔູభ၂ࠩ‫ڵ‬ᄛ֥đေॉ੮ᆃᇕ‫ڵ‬ᄛིႋb Cēӈ࡮֥ࠎЧ‫ੰק‬ᇶေႵđ୤‫ؘ‬೘ն‫ੰק‬čܸྟ‫ੰק‬aࡆ෎؇‫ੰק‬aቔႨ‫ّބ‬ቔႨ‫ੰק‬Ďa ିਈ൯‫ੰקޚ‬a‫׮‬ਈ൯‫ੰקޚ‬a॓༐ࠉ‫׈ڏ‬࿢a‫׈‬ੀ‫ੰק‬a໾ᇉ൯‫ࠣੰקޚ‬۲࿐॓Ⴕܱ֝ԛ‫֩֩ੰק‬b č4Ďਙཿᇏࡗэਈაః෱эਈ֥ၹ‫ܱݔ‬༢ൔčӫູ‫ڣ‬ᇹٚӱൔĎb č5Ď৳৫ഈඍٚӱቆđཨಀᇏࡗэਈđቋᇔ֤֞༢๤ܱႿൻೆൻԛэਈ֥ັ‫ٳ‬ٚӱb č6Ďѓሙ߄đࠧࡼაൻೆэਈႵܱ֥۲ཛ٢֞ٚӱൔ֩‫֥ݼ‬ႷҧđࡼაൻԛэਈႵܱ֥۲ཛ٢֞ ٚӱൔ֩‫֥ݼ‬ቐҧđ౏۲ࢨ֝ඔοࢆૢஆਙb 2.2.2 ৘མჭࡱ֥ັ‫ٳ‬ٚӱ૭ඍ ᄝ‫׈‬గ‫ྀࠏބ‬༢๤ᇏࠫᇕቋӈ࡮֥৘མჭࡱႵğ 1ē‫׈‬ಸ 16 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ∫ ‫׈‬ಸਆ؊‫׈‬࿢ა‫׈‬ੀ֥ܱ༢ູğ i(t) = C du c (t) ࠇ uc (t) = 1 i(t)dt C dt 2ē‫ۋ׈‬ = diL ࠇ 1 ∫ ੀ‫׈ۋ׈ݖ‬ੀაਆ؊֥‫׈‬࿢֥ܱ༢ູğ (t) L dt = L u L iL (t) uL (t)dt 3ē֐ྟ৯෱൞၂ᇕ֐ߞ֥֐ྟ߫‫گ‬৯đఃնཬაࠏྀэྙӮᆞбđ֐ྟ৯‫ٳ‬௜‫ބ׮‬࿈ሇਆᇕb ∫ F = ky = k vdt ࠇ v = 1 dF đൔᇏ k ູ֐ߞ֥֐ྟ༢ඔđ F ູቔႨႿ֐ߞ֥ຓ৯đ y ູᆰཌ໊ k dt ၍ਈđ v ູᆰཌ໊၍෎؇b 4ēቅୄఖ ௜‫׮‬ቅୄఖቅୄ৯ğ F = fv = f dy đൔᇏ f ູቅୄ༢ඔđ F ູቅୄ৯b dt ࿈ሇቅୄఖቅୄ৯इğ T = fω = f dθ đൔᇏω ູ࿈ሇ࢘෎؇đθ ູ࿈ሇ࢘؇đ T ູቅୄ৯इb dt ቅୄఖЧദ҂ԥթିਈđ෱་൬ିਈѩၛಣ֥ྙൔ‫ݻ‬೛‫ו‬b 2.2.3 ඔ࿐ࡹଆई২ ২ 1ğ֐ߞ-ᇉਈ-ቅୄఖԱ৳༢๤đ൫ਙཿၛຓ৯ F(t) ູൻೆđၛᇉਈ໊၍ y(t) ູൻԛ֥ັ‫ٳ‬ٚӱ ൔb [ࢳ]ğᆃ൞၂۱ࣜ‫֥ׅ‬ᆰཌࠏ໊ྀ၍‫׮‬৯࿐༢๤đॖၛࡌ‫ק‬༢๤ҐႨࠢᇏҕඔđm ູᇉׄ č1Ď ༢๤֥ൻೆູ FčtĎđൻԛູ yčtĎđ֐ߞ֥֐ྟቅ৯ູ Fk (t) đቅୄఖ ֥ቅୄ৯ູ Ff (t) नູᇏࡗэਈb č2Ď ߂ԛ m ֥൳৯๭ č3Ď Ⴎ୤‫ੰקؽֻؘ‬čࠧࡆ෎؇‫ੰק‬Ďğ ∑ Fi = **** = d2y m dt 2 Fk (t) F F(t) - Fk (t) − Ff (t) = m d2 y dt 2 m č4Ďਙཿᇏࡗэਈa Ff (t) іղൔ Ff (t) Fk (t) = ky(t), Ff (t) = f dy dt č5Ďࡼഈඍᇏࡗэਈ֥‫ڣ‬ᇹٚӱսೆჰ൓ٚӱđཨಀᇏࡗэਈ Fk (t) ‫ ބ‬Ff (t) F(t) - ky(t) − f dy = m d 2 y dt dt 2 č6Ďѓሙ߄đ֤֞ğ m d 2 y + f dy + ky(t) = F(t) dt 2 dt ೏਷ Tm2 =m k , Tf =f k đᄵഈඍٚӱॖၛіൕູğ m d 2 y + f dy + y = 1 k dt 2 k dt F(t) kb ࣡෿ٚӱູğ y(t) = 1 F(t) đၹՎ1 k Ⴛӫູ༢๤֥࣡෿٢նПඔb k ২ 2ğR-L-C Ա৳‫׈‬ਫ਼đൻೆູ‫׈‬࿢ u r (t) đൻԛູ‫׈‬ಸ‫׈‬࿢ u c (t) đ൫౰ൻೆൻԛັ‫ٳ‬ٚӱb [ࢳ]ğč1Ďಒ‫ק‬༢๤֥ൻೆູ‫׈‬࿢ u r (t) đൻԛູ‫׈‬ಸ‫׈‬࿢ u c (t) đᇏࡗэਈູ‫׈‬ੀ i(t) 17 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ďຩ઎οࠢᇏҕඔॉ੮đ౏ൻԛູषਫ਼‫׈‬࿢đࠧ໭ުࠩ‫ڵ‬ᄛb č3ĎႮक़༐ࠉ‫ੰקڏ‬ཿԛჰ൓ٚӱğ L di + Ri + u c = ur dt č4Ďਙཿᇏࡗэਈ i(t) ა u c (t) ֥ܱ༢ൔğ i = C du c dt č5Ďࡼ i(t) սೆჰ൓ٚӱđཨಀᇏࡗэਈ i( t ) đ֤֞ğ LC d2u c + RC du c + uc = ur dt 2 dt ২ 3ğ‫׈‬൷॥ᇅ֥ᆰੀ‫ࠏ׈‬đᄝ༢๤ᇏđൻೆ‫׈‬൷‫׈‬࿢ u a ᄝ‫׈‬൷߭ਫ਼ᇏӁള‫׈‬൷‫׈‬ੀ i a đᄜႮ i a ა ৣՈՈ๙ཌྷ޺ቔႨӁള‫׈‬Ոሇइ M D đՖ‫ط‬൐‫׈‬൷࿈ሇđ຀‫ڵ׮‬ᄛđປӮਔႮ‫ି׈‬ᄝՈӆቔႨ༯ཟࠏྀ ିሇߐ֥‫ݖ‬ӱb č1Ďಒ‫ק‬ൻೆaൻԛਈູğ u a ູൻೆ‫׈‬࿢đ ω ູൻԛ࢘௔ੱđ M L ູ‫ڵ‬ᄛಠ‫׮‬৯इĠ č2Ďޭ੻‫׈‬൷ّႋaՈᇌ֩႕ཙđ֒ If = C ൈđৣՈՈ๙҂эđэਈܱ༢ॖ൪ູཌྟܱ༢Ġ č3Ďਙཿჰ൓ٚӱ La di a + Raia + Ea = ua dt Ⴎ‫ุې‬࿈ሇ‫ੰק‬ཿԛ‫ࠏ׈‬ᇠഈ֥ࠏྀᄎ‫׮‬ٚӱ d 2θ = J dω = MD − ML č4Ďཿԛ‫ڣ‬ᇹٚӱൔ J dt dt 2 ‫׈‬൷ّႋّ֥‫׈‬൝ğ E a = k eω đ k e ູ‫׈‬൝༢ඔđႮ‫ܒࢲࠏ׈‬ҕඔथ‫ק‬b ‫׈‬Ոሇइğ M D = k mia đ k m ູሇइ༢ඔđႮ‫ܒࢲࠏ׈‬ҕඔथ‫ק‬b č5Ďཨಀᇏࡗэਈ iaaEaaM D đ M L ູ‫ڵ‬ᄛಠ‫׮‬ൻೆ৯इ LaJ d 2ω + RaJ dω +ω = 1 ua − La dM L − Ra ML kekm dt 2 kekm dt ke kekm dt kekm ೏਷ Tm = R a J k e k m ູࠏ‫׈‬ൈࡗӈඔđ Ta = La R a ູ‫׈‬Ոൈࡗӈඔđᄵഈൔॖၛཿູğ Ta Tm d 2ω + Tm dω +ω = 1 ua − Ta Tm dM L − Tm ML dt 2 dt ke J dt J ೏ޭ੻‫ڵו‬ᄛሇइ M L đࠧॢᄛൈđᄵႵğ Ta Tm d 2ω + Tm dω +ω = 1 ua dt 2 dt ke 2.2.4 ັ‫ٳ‬ٚӱ֥၂Ϯหᆘ ັ‫ٳ‬ٚӱ֥၂Ϯྙൔğ a0 d nc + a1 d n-1c + a2 d n-2c + + an-1 dc + anc = b0 d mr + b1 d m-1r + b2 d m-2r + + bm-1 dr + bmr dt n dt n-1 dt n-2 dt dt m dt m-1 dt m-2 dt č1Ďఃᇏ a i đ b j ູൌӈඔđႮ໾৘༢๤֥ҕඔथ‫ק‬Ġ č2Ďٚӱൻԛэਈ֥ັ‫ࢨٳ‬ՑۚႿൻೆэਈ֥ັ‫ࢨٳ‬Ցđၹູ໾৘༢๤‫ݣ‬Ⴕԥିჭࡱđࠧ n > m b 18 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 4 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘č3a4 ࢫĎ ᇶေଽಸ ٤ཌྟଆ྘֥ཌྟ߄ ཌྟӈ༢ඔັ‫ٳ‬ٚӱ֥౰ࢳ Ԯ‫ݦ־‬ඔ֥‫ק‬ၬaൌ࠽ၩၬaྟᇉࠣັܴࢲ‫ܒ‬ ଢ֥აေ ᅧ໤٤ཌྟଆ྘֥ཌྟ߄֥ٚ‫م‬a่ࡱ‫҄ބ‬ᇧ ౰ ᅧ໤ཌྟӈ༢ඔັ‫ٳ‬ٚӱ֥ঘ൦эߐ‫م‬౰ࢳ ᅧ໤ཌྟ‫ק‬ӈ༢๤Ԯ‫ݦ־‬ඔ֥‫ק‬ၬaൌ࠽ၩၬaྟᇉࠣັܴࢲ‫ܒ‬ ᇗ ׄ ა ଴ ᇗׄğ Ԯ‫ݦ־‬ඔ֥‫ק‬ၬaྟᇉࠣັܴࢲ‫ܒ‬ ଴ׄğཌྟӈ༢ඔັ‫ٳ‬ٚӱ֥౰ࢳ ׄ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 2-2č1Ďč2Ď 2-8a2-9 ቔြี 2.3 ٤ཌྟඔ࿐ଆ྘ཌྟ߄ ൌ࠽ၩၬഈՂե֥ཌྟ༢๤൞҂թᄝ֥đቆӮ༢๤֥ჭࡱࠇ‫ࠇ؟‬ഒֹթᄝሢ٤ཌྟหྟđؓ٤Ч ᇉ֥٤ཌྟหྟ໡ૌေࣉྛཌྟ߄ԩ৘đ࠻ཌྟ࣍රb 2.3.1 ཬொҵཌྟ߄֥‫୑ۀ‬ ২ೂೂ༯֥٤ཌྟหྟđؓႿ‫۽‬ቔׄ A(x 0 , y0 ) đ೏ᄝ‫۽‬ቔׄ A ‫ຶ֥ٓཬޓ࣍ڸ‬ଽ‫۽‬ቔđࠧэਈ xa y ཌྷؓႿ (x 0 , y0 ) ቔັཬ֥ᄹਈ ∆x a ∆y ֥э߄đၛ A ׄ (x 0 , y0 ) ԩ్֥ཌটսูᄝ (∆x, ∆y) ٓຶଽ ‫ཬޓ‬၂؊౷ཌđၹູᆃᇕཌྟ߄ԩ৘Фཋᇅᄝ‫۽‬ቔׄ (x 0 , y0 ) ‫∆ ຶ֥ٓཬޓ࣍ڸ‬x a ∆y ଽҌ֤ၛӮ৫đ ၹՎӫູ“ཬொҵ‫”م‬b 2.3.2 ཌྟ߄֥ඔ࿐ၩၬ‫҄ބ‬ᇧ 1aഡ y = f (x) ູ٤ཌྟหྟٚӱđ໗‫۽ק‬ቔׄ (x 0 , y0 ) ࣍රൔ ∆y = f ′(x 0 )∆x = K∆x đൔᇏ༢ ඔ K = f ′(x 0 ) ູ‫۽‬ቔׄ (x 0 , y0 ) ԩ్ཌོ֥ੱđࠧ K = f ′(x 0 ) = tgα đᆃဢࣼࡼ٤ཌྟหྟ y = f (x) ࣍රູཌྟหྟ ∆y = f ′(x 0 )∆x = K∆x b ২ğၘᆩ٤ཌྟ‫ݦ‬ඔ y = sinθ đ൫ᄝθ0 = 0 ‫ބ‬θ0 = π ਆׄԩቔཌྟ߄ԩ৘b ࢳğཁಖ y = sinθ ൞၂۱٤ཌྟ‫ݦ‬ඔđ໡ૌՖθ0 = 0 ‫ބ‬θ0 = π ਆׄԩॖၛᆰֹܴुԛđᄝఃਵთ ଽӯགྷԛ‫֥ۚޓ‬ཌྟ؇b č1Ďθ0 = 0 ൈđ y0 = sinθ0 θ0 =0 = 0 đ‫( ܣ‬θ0 , y0 ) = (0,0) đ ∆y = y′(θ 0 )∆θ = cos θ |θ0 =0 ∆θ = ∆θ ֌໡ૌӈӈ༝ܸಯࡼ ∆y ཿӮ y đ ∆θ ཿӮθ đ҂‫ݖ‬Վൈ֥θ ‫ ބ‬y न൞ᆌؓ (θ0 , y0 ) = (0,0) ‫۽‬ቔׄ ԩ֥ᄹਈ ∆θ a ∆y đႿ൞Ⴕ࣍ර֥ཌྟ‫ݦ‬ඔູ y = θ č2Ďθ0 = π ൈđ y0 = sinθ0 θ0 =π = 0 đ‫( ܣ‬θ0 , y0 ) = (π ,0) đ ∆y = y′(θ 0 )∆θ = cosθ |θ0 =π ∆θ = −∆θ 19 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Ⴟ൞Ⴕ࣍ර֥ཌྟ‫ݦ‬ඔູ y = −θ b ཁಖ‫۽‬ቔׄ҂๝ൈđཌྟ߄֥༢ඔ൞҂๝֥b 2aؓ‫؟‬эਈ٤ཌྟ‫ݦ‬ඔđ๝ဢॖၛႵোර֥ཌྟ߄ԩ৘đഡႵ٤ཌྟ‫ݦ‬ඔ y = f (x1, x 2 , , x n ) đ ൔᇏ֥ x1, x 2 , , x n ູ༢๤֥ n ۱ൻೆčࠗৣĎđ೏༢๤֥‫۽‬ቔູׄ y = f (x10 , x 20 , , x n0 ) đ࣍රіղ ൔ ∆y = K1∆x1 + K2∆x2 + + Kn∆xn b ༢ඔ K1 = ∂f đK 2 = ∂f đ đK n = ∂f ູ۲‫۽‬ቔ֥ׄொ֝ඔđหљ൞֒ n=2 ൈႵğ ∂x1 ∂x2 ∂xn X0 X0 X0 ∆y = K1∆x1 + K 2∆x 2 đ K 1 = ∂f đK2 = ∂f ∂x1 ∂x2 x 1 = x 10 x 1 = x 10 x 2 = x 20 x 2 = x 20 ܱႿཌྟ߄֥ࠫׄඪૼğ č1Ďཌྟ߄сྶᆌؓଖ၂۱‫۽‬ቔׄԩࣉྛđ‫۽‬ቔׄ҂๝ᄵཌྟ߄֥ࢲ‫္ݔ‬҂၂ဢĠ č2Ďཌྟ߄่֥ࡱ൞ᄝ‫۽‬ቔׄ‫ຶٓཬ֥࣍ڸ‬ଽđડቀཬொҵ่֥ࡱĠ č3Ďཌྟ߄ᆺିᆌؓ٤Чᇉ٤ཌྟหྟࣉྛđՖඔ࿐ၩၬഈࢃđࣼ൞٤ཌྟ‫ݦ‬ඔсྶ൞ֆᆴa৵࿃a ܻ߁‫ބ‬ॖ֥֝bĠ č4Ďཌྟ߄֥ࢲ‫ݔ‬൞֤֞ࠎႿ‫۽‬ቔׄ‫࣍ڸ‬čਣთĎэਈᄹਈ (∆x, ∆y) ֥ཌྟٚӱൔđ༝ܸഈ໡ૌಯ ࡼ ∆x, ∆y ཿӮ xđyb 2.4 ཌྟ༢๤֥Ԯ‫ݦ־‬ඔ 2.4.1 ັ‫ٳ‬ٚӱ֥౰ࢳ ັ‫ٳ‬ٚӱ֥౰ࢳ‫ູٳ‬ൈთ‫ބم‬эߐთ‫م‬đ෱ૌᆭࡗ֥ܱ༢ॖၛႨ༯๭টіൕğ rčtĎ ັ‫ٳ‬ٚӱൔ ౰ࢳັ‫ٳ‬ٚ ൈთࢳ C Laplace эߐ Laplace ّэߐ RčsĎ ౰սඔٚӱ S თࢳ CčSĎ S ֥սඔٚӱ CčsĎ Ֆ๭ᇏॖᆩđ๙‫ ݖ‬Laplace эߐđॖၛࡼັ‫ٳ‬ٚӱ֥ࢳࡥ߄ູ‫گ‬эთᇏܱႿ s ֥սඔٚӱđѩ֤֞ൻ ԛ֥ Laplace эߐ C(s)ުđّэߐ֤֞ັ‫ٳ‬ٚӱ֥ൈࡗთࢳ c(t)b 2.4.2 Ԯ‫ݦ־‬ඔčTransfer FunctionĎ 1ēԮ‫ݦ־‬ඔ‫ק‬ၬ ᄝཌྟ༢๤ᇏđ֒Ԛ൓่ࡱູਬൈđ༢๤ൻԛ֥ Laplace эߐའ‫ݦ‬ඔ C(s) აൻೆ֥ Laplace эߐའ ‫ݦ‬ඔ R(s) ᆭбđӫູ༢๤֥Ԯ‫ݦ־‬ඔb ഡཌྟൈ҂э༢๤֥ັ‫ٳ‬ٚӱູğ a0 d nc + a1 d n-1c + a2 d n-2c + + an-1 dc + anc = b0 d mr + b1 d m-1r + b2 d m-2r + + bm-1 dr + bmr dt n dt n-1 dt n-2 dt dt m dt m-1 dt m-2 dt ᄝਬԚ൓่ࡱ༯đؓഈඍັ‫ٳ‬ٚӱਆш๝ൈ౰ Laplace эߐđѩ਷ൻԛ c(t) ֥ Laplace эߐູ C(s) đ ൻೆ r(t) ֥ Laplace эߐູ R(s) đ০Ⴈ Laplace эߐ֥ັ‫ྟٳ‬ᇉđ֤֞ğ (a 0s n + a1s n-1 + + a n-1s + a n )C(s) = (b 0s m + b1s m-1 + + b m-1s + b m )R(s) ౰֤Ԯ‫ݦ־‬ඔğ G(s) = C(s) = b0sm + b1sm-1 + + b m-1s + b m R(s) a 0s n + a1s n-1 + + a n-1s + a n ২ğ൫౰ RLC Ա৳‫׈‬ਫ਼֥Ԯ‫ݦ־‬ඔb 20 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ࢳğ‫׈ھ‬ਫ਼֥ັ‫ٳ‬ٚӱభ૫ၘࣜ౰֤ LC d 2u c + RC du c + uc = ur dt 2 dt ਷Ԛ൓่ࡱູਬđٚӱਆш౰ Laplace эߐđ֤֞ğ LCs 2 U c (s) + RCsU c (s) + U c (s) = U r (s) (LCs2 + RCs + 1)Uc (s) = U r (s) ⇒ G(s) = Uc (s) = LCs2 1 +1 U r (s) + RCs ২ğ‫׮‬৯࿐༢๤ೂ༯๭đ൫౰Ԯ‫ݦ־‬ඔb ࢳğ‫ھ‬༢๤֥ັ‫ٳ‬ٚӱູ my′′ + (f1 + f2 )y′ + ky = F(t) ਷Ԛ൓่ࡱູਬđؓٚӱਆш๝ൈ౰ Laplace эߐđ֤֞ğ (ms 2 + (f1 + f 2 )s + k)Y(s) = F(s) ⇒ G (s) = Y(s) = ms 2 + 1 f 2 )s + k F(s) (f1 + 2ēԮ‫ݦ־‬ඔ֥ࢲ‫ܒ‬หᆘğ č1ĎԮ‫ݦ־‬ඔ൞Ֆັ‫ٳ‬ٚӱဆэ‫ݖ‬ট֥đၹՎԮ‫ݦ־‬ඔ๝ဢіᆘਔ༢๤֥‫ܥ‬Ⴕหྟđ෱൞༢๤ᄝ‫گ‬ эთᇏ֥၂ᇕඔ࿐ଆ྘Ġ č2ĎႮႿԮ‫ݦ־‬ඔൡႨႿཌྟ༢๤đ෮ၛԮ‫ݦ־‬ඔ҂߶ၹູൻೆਈࠇൻԛਈ‫ݦ‬ඔ‫ط‬ၳĠ č3ĎԮ‫ݦ־‬ඔЇ‫ݣ‬ਔັ‫ٳ‬ٚӱ֥ಆ҆༢ඔđ෮ၛ෱აັ‫ٳ‬ٚӱ൞ປಆཌྷ๙֥đೂ‫ݔ‬Ԯ‫ݦ־‬ඔᇏ҂թ ᄝ‫ٳ‬ሰ‫ٳ‬ଛؓཨ֥ၹሰđପહԮ‫ݦ־‬ඔაັ‫ٳ‬ٚӱ၂ဢЇ‫ݣ‬ਔ༢๤֥ಆ҆ྐ༏Ġ č4ĎԮ‫ݦ־‬ඔ֥‫ٳ‬ଛ‫؟‬ཛൔࣼ൞ັ‫ٳ‬ٚӱቐ؊‫ݦ‬ඔ֥ັ‫ٳ‬ෘሰ‫؟ژ‬ཛൔđ္ࣼ൞༢๤֥หᆘ‫؟‬ཛൔđ ҂‫ݖ‬෱൞‫گ‬эთ৚֥іགྷྙൔbԮ‫ݦ־‬ඔ֥‫ٳ‬ሰ‫؟‬ཛൔࣼ൞ັ‫ٳ‬ٚӱႷ؊‫ݦ‬ඔ֥ັ‫ٳ‬ෘሰ‫؟ژ‬ཛൔĠ č5ĎෙಖԮ‫ݦ־‬ඔაັ‫ٳ‬ٚӱ൞ཌྷ๙֥đ֌ՖྙൔഈඪđԮ‫ݦ־‬ඔ൞၂۱‫ݦ‬ඔđ‫ٳັط‬ٚӱ൞၂۱ ٚӱđၹՎԮ‫ݦ־‬ඔᄝᄎෘ‫ބ‬ቔ๭ٚ૫൞бࢠٚь֥đ֌෱္ջটਔ‫ٳ‬ሰ‫ٳ‬ଛཌྷཨ֩໙ีĠ č6ĎႮႿԮ‫ݦ־‬ඔაັ‫ٳ‬ٚӱ֥ཌྷ๙ྟđၹՎᆺေࡼັ‫ٳ‬ٚӱᇏ֥ັ‫ٳ‬ෘሰ‫ ژ‬P = d dt ߐӮ‫گ‬эਈ s đࠧॖ֤֞Ԯ‫ݦ־‬ඔĠ ਷ P = d dt → s đ c(t) → C(s), r(t) → R(s) đᄵॖ֤֞Ԯ‫ݦ־‬ඔ G(s) b č7ĎԮ‫ݦ־‬ඔ G(s) ა༢๤֥Ԋࠌཙႋ g(t) ູ၂ؓэߐؓđࠧ G(s) ↔ g(t) ູ ໡ ૌ ิ ‫ ܂‬ਔ Ԯ ‫ ݦ ־‬ඔ ֥ ၂ ᇕ ౰ ౼ ٚ ‫ م‬đ ࠧ ೏ ၘ ᆩ ༢ ๤ ֥ Ԋ ࠌ ཙ ႋ g(t) đ ᄵ ః Ԯ ‫ ݦ ־‬ඔ G(s) = L[g(t)] b č8ĎؓႿ၂۱໾৘ഈॖൌགྷ֥ཌྟࠢᇏҕඔؓའđఃԮ‫ݦ־‬ඔс‫ק‬൞࿸۬ᆇႵ৘‫ݦ‬ඔđ္ࠧԮ‫ݦ־‬ ඔ֥‫ٳ‬ሰ‫؟‬ཛൔ֥ࢨՑ m ሹ൞ཬႿ‫ٳ‬ଛ‫؟‬ཛൔ֥ࢨՑ n đ m < n b ২ğഡ༢๤֥ֆ໊ࢨᄁཙႋູ c(t) = 1 − e-2t + e−t đ൫౰༢๤֥Ԯ‫ݦ־‬ඔ‫ބ‬ઝԊཙႋ‫ݦ‬ඔb ࢳğٚ‫م‬၂ğ۴ऌԮ‫ݦ־‬ඔ֥‫ק‬ၬđ༢๤֥Ԯ‫ݦ־‬ඔູğ C (s ) L (1 − e − 2t + e−t ) 1 − 1 + 1 s 2 + 4s + 2 s s+2 s +1 G (s) = = = = R (s) L[l( t )] 1 (s + 1)( s + 2) s ઝԊཙႋ‫ݦ‬ඔູğ g(t) = L−1[G(s)] = δ (t) + 2e−2t − e−t ٚ‫ؽم‬ğ۴ऌཌྟ༢๤֥ྟᇉđ༢๤֥ઝԊཙႋ‫ݦ‬ඔູğ g(t) = dc(t) = δ (t) + 2e−2t − e−t dt ༢๤֥Ԯ‫ݦ־‬ඔູğ G(s) = L[g(t)] = L[δ (t) + 2e−2t − e−t ] = 1 + 2 − 1 = s2 + 4s + 2 s + 2 s + 1 (s + 1)(s + 2)  21 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 5 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘č4a6 ࢫĎ ᇶေଽಸ ਬࠞׄ‫҃ٳ‬ა༢๤ཙႋ֥ܱ༢ ‫ࠣࢫߌ྘ׅ‬Ԯ‫ݦ־‬ඔ ଢ֥აေ ᅧ໤ਬࠞׄ‫҃ٳ‬ა༢๤ཙႋ֥ܱ༢ ౰ ᅧ໤۲‫֥ࢫߌ྘ׅ‬Ԯ‫ݦ־‬ඔaᄎ‫׮‬ٚӱaࢨᄁཙႋࠣหׄ ᇗ ׄ ა ଴ ᇗׄğਬࠞׄ‫҃ٳ‬ა༢๤ཙႋ֥ܱ༢a‫֥ࢫߌ྘ׅ‬Ԯ‫ݦ־‬ඔ ׄ ଴ׄğਬࠞׄ‫҃ٳ‬ა༢๤ཙႋ֥ܱ༢ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 3ēԮ‫ݦ־‬ඔ֥ਬaࠞׄ č1ĎԮ‫ݦ־‬ඔ֥ਬׄაࠞׄ G(s) = C(s) = b0sm + b1sm-1 + + bm-1s + b m = M(s) R(s) a 0s n + a1sn-1 + + a n-1s + a n N(s) ‫ק‬ၬԮ‫ݦ־‬ඔ‫ٳ‬ሰ‫؟‬ཛൔ M(s) = 0 ֥ m ۱۴ z1,z2 , ,zm ູਬׄđ‫ٳ‬ଛ‫؟‬ཛൔ N(s) = 0 ֥ n ۱۴ p1, p2 , , pn ູࠞׄđൌ࠽ഈ္ࠞׄࣼ൞หᆘٚӱൔ֥۴đࠧหᆘ۴bਬׄ‫ׄࠞބ‬ॖၛ൞ൌඔࠇ‫܋‬ᣢ ‫گ‬ඔb č2ĎԮ‫ݦ־‬ඔ֥іൕྙൔ ®ਬࠞׄіղൔ G(s) = b0sm + b1sm-1 + + b m-1s + b m = Kg (s − z1 )((s − z 2 ) (s − zm ) a 0s n + a1s n-1 + + a n-1s + a n (s − p1 )(s − p2 ) (s − pn ) ൔᇏ K g = b0 a 0 ӫູԮ‫־‬༢ඔࠇ۴݅ࠖᄹၭb ®݂၂߄čൈࡗӈඔĎіղൔ G(s) = b0sm + b1s m-1 + + b m-1s + b m (τ 1s + 1)(τ 2 s 2 + 2τ 2ς 2s + 1) 2 =K a 0s n + a1s n-1 + + a n-1s + a n (T1s + 1)(T22s2 + 2T2ξ2s + 1) ൔᇏ K = bm a n ӫູ༢๤Ԯ‫ݦ־‬ඔ֥࣡෿č໗෿Ď٢ն༢ඔb m K ა Kg ֥ܱ༢ğ K = Kg (−z1)(−z2 ) ∏ (−zm) = Kg (−1)m zi (−p1)(−p2 ) ∏ (−pn) i=1 n (−1)n p j j=1 č3Ďਬׄaࠞׄ๭ ༝ܸഈᄝ s ௜૫ഈၛ“○”іൕਬׄđၛ“×”іൕ֥ࠞׄ๭ӫູਬࠞׄ๭b ২ೂğ༢๤֥Ԯ‫ݦ־‬ඔູ G(s) = (s + 2) (s + 3)(s2 + 2s + 2) đᄵԮ‫ݦ־‬ඔॖіൕູ G(s) = (s + 2) (s + 3)(s +1 + j)(s +1 − j) 4ēਬׄaࠞׄaԮ‫־‬༢ඔა༢๤ཙႋ֥ܱ༢ č1Ďࠞׄथ‫ק‬ਔ༢๤ሱႮč‫ܥ‬ႵĎᄎ‫׮‬උྟ ༢๤֥ሱႮᄎ‫׮‬൞༢๤֥‫ܥ‬Ⴕᄎ‫׮‬උྟđ‫ط‬აຓ҆ൻೆྐ‫ݼ‬໭ܱb ၂Ϯ‫ط‬࿽đԮ‫ݦ־‬ඔ G(s) ֥ࠞׄྙൔđथ‫ק‬ਔ༢๤ሱႮᄎ‫׮‬ଆ෿֥ऎุྙൔb ®ູ֒ࠞׄ޺҂ཌྷ֥֩ൌඔ۴ p1đp đ đp n ൈđሱႮᄎ‫֥׮‬ଆ෿ྙൔູğ 2 e đ p1t e đ p2t đe pnt 22 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ®֒ࠞׄႵ‫܋‬ᣢ‫گ‬ඔ۴ೂ pi = σ i ± jωi ൈđሱႮᄎ‫֥׮‬ଆ෿ྙൔࡼԛགྷğ eσit cosωi t ࠇ eσitsinωi t ®֒ࠞׄԛགྷൌඔᇗ۴đೂ m ᇗൌඔ۴ pi ൈđሱႮᄎ‫֥׮‬ଆ෿ྙൔࡼԛགྷğ e pitđte đ pit đt m−1e pit ®֒ࠞׄԛགྷ‫گ‬ඔᇗ۴đೂ m ᇗ‫گ‬ඔ۴ pi = σ i ± jωi ൈđሱႮᄎ‫֥׮‬ଆ෿ྙൔࡼԛགྷğ eσit cosωi tđteσit cosωi tđ đt m-1eσitcosωi t ࠇ eσitsinωi tđteσitsinωi tđ đt m-1eσitsinωi t ®֒ࠞׄ࠻Ⴕ޺ၳ֥ൌඔ۴a‫܋‬ᣢ‫گ‬ඔ۴đႻႵᇗൌඔ۴aᇗ‫گ‬ඔ۴ൈđሱႮᄎ‫֥׮‬ଆ෿ྙൔࡼ൞ ഈඍࠫᇕྙൔ֥ཌྟቆ‫ކ‬b č2Ď໊ࠞׄᇂथ‫ק‬ਔ༢๤ཙႋ֥໗‫ॹބྟק‬෎ྟ ●༢๤Ԯ‫ݦ־‬ඔ֥षߌ֥ࠞׄൌ҆नཬႿਬđՖ s ௜૫টुđ෮Ⴕࠞׄन໊Ⴟఃቐ϶௜૫đᄵఃଆ෿ ࣼ߶ෛሢൈࡗ t ֥ᄹӉ‫ط‬කࡨđቋᇔཨാb༢๤ཙႋ֥ሱႮᄎ‫ٳ׮‬ਈčࠧି֤֞໗෿ཙႋĎି‫ܔ‬ཨാ֥ӫ ູ໗‫ק‬༢๤đၹՎ༢๤֥໗‫ྟק‬Ⴎఃಆ໊֥҆ࠞׄᇂটथ‫ק‬b ●ؓႿ໗‫֥ק‬༢๤đࠧ෮ၛࠞׄन໊Ⴟ S ቐ϶௜૫đૄ۱ࠞׄ෮ؓႋ֥ᄎ‫׮‬ଆ෿đෛሢൈࡗ t කࡨ֥ ॹતđᄵႮ‫ׄࠞھ‬৖षྴᇠ֥ए৖টथ‫ק‬đཁಖ৖षྴᇠᄀჹđᄵකࡨ֤ᄀॹđ৖षྴᇠᄀ࣍đᄵකࡨ ᄀતb‫ط‬༢๤ཙႋཙႋ֥ॹ෎ྟđࠧᄠ෿ཙႋකࡨ֥ॹતđࣼ൞Ⴎࠞׄथ‫֥ק‬ሱႮᄎ‫׮‬ଆ෿කࡨ֥ॹતđ ၹՎ༢๤ཙႋ֥ॹ෎ྟđႮఃಆ҆ࠞׄᄝ S ቐ϶௜૫ഈ֥‫҃ٳ‬थ‫ק‬b č3Ďਬׄथ‫ק‬ਔᄎ‫׮‬ଆ෿֥бᇗ ਬׄथ‫ק‬ਔ۲ଆ෿ᄝཙႋᇏ෮ᅝ֥“бᇗ”đၹ‫္ࣼط‬႕ཙ༢๤ཙႋ֥౷ཌྙሑđၹՎ္ࣼ߶႕ཙ༢ ๤ཙႋ֥ॹ෎ྟb ਬׄ৖ࠞׄࢠჹൈđཌྷႋႿ‫ׄࠞھ‬ଆ෿෮ᅝ֥бᇗࢠնđ৖ࠞׄࢠ࣍ൈđཌྷႋႿ‫ׄࠞھ‬ଆ෿෮ᅝ֥ бᇗࢠཬb֒ਬׄაࠞׄᇗ‫ކ‬đԛགྷਬࠞׄؓཨགྷའđՎൈཌྷႋႿ‫֥ׄࠞھ‬ଆ෿္ࣼཨാਔčൌ࠽ഈ൞ ‫ھ‬ଆ෿֥бᇗູਬĎbၹՎਬׄႵቅ؎ࠞׄଆ෿“Ӂള”ࠇ“ളӮ”֥ቔႨb č4ĎԮ‫־‬༢ඔथ‫ק‬ਔ༢๤໗෿Ԯ‫ିྟ־‬ K थ‫ק‬ਔ໗෿ཙႋ֥٢նПඔܱ༢b 2.5 ‫ࠣࢫߌ྘ׅ‬ఃԮ‫ݦ־‬ඔ 1ēб২ߌࢫč٢նߌࢫĎ ᄎ‫׮‬ٚӱğ c(t) = Kr(t) đԮ‫ݦ־‬ඔູğ G(s) = K đ б২ߌࢫ֥‫྘ׅ‬২ሰႵᄎෘ٢նఖaԂ੽э෎དa‫໊׈‬ఖaҩ෎‫֩ࠏ׈ؿ‬b 2ēܸྟߌࢫ ᄎ‫׮‬ٚӱğ T dc(t) + c(t) = r(t) đԮ‫ݦ־‬ඔğ G(s) = 1 = s ω0 đω0 = 1 dt Ts + 1 + ω0 T T ӫູܸྟൈࡗӈඔđᆺႵ၂۱ൌඔࠞׄ −1 T đીႵਬׄb ೏ൻೆູࢨᄁ‫ݦ‬ඔ r(t) = ul(t) đᄵ‫֥ࢫߌྟܸھ‬ൻԛູğ c(t) = u + [c(0) − u]e−t T đ c(0) ູ c ֥Ԛᆴb ‫֥྘ׅ‬ൌ২Ⴕ၂ࢨ RC ‫׈‬ਫ਼b 3ē ࠒ‫ࢫߌٳ‬ 23 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ∫ ᄎ‫׮‬ٚӱğT dc(t) = r(t) ࠇ c(t) = 1 t r(t)dt đࠧߌࢫൻԛູൻೆྐ‫ٳࠒ֥ݼ‬đߌࢫႮՎ֤଀b dt T0 Ԯ‫ݦ־‬ඔğ G(s) = 1 s ࠒ‫֥ࢫߌٳ‬หׄ൞Ԣ٤ൻೆྐ‫ ݼ‬r(t) ‫ ູޚ‬0đ‫ڎ‬ᄵࠒ‫֥ࢫߌٳ‬ൻԛਈ c(t) ҂ॖିົӻູӈඔ҂эb 4ēັ‫ࢫߌٳ‬ ᄎ‫׮‬ٚӱğ c(t) = T dr(t) đࠧߌࢫൻԛਈູൻೆྐ‫ٳັ֥ݼ‬đߌࢫႮՎ֤଀b dt Ԯ‫ݦ־‬ඔğ G(s) = Ts đႵൈ G(s) = Ts + 1္ӫູັ‫ࢫߌٳ‬b 5ēᆒ֕ߌࢫ ᄎ‫׮‬ٚӱğT 2 d 2c(t) + 2ξT dc(t) + c(t) = r(t) dt 2 dt Ԯ‫ݦ־‬ඔğ G(s) = 1 = 1T2 = ω 2 n T 2 s 2 + 2ξTs + 1 s 2 + 2ξs T + 1 T 2 s2 + 2ξωn s + ω 2 n ఃᇏ ωn = 1 T đ 0 ≤ ξ < 1b‫ࢫߌھ‬ऎႵ၂ؓ‫܋‬ᣢ‫گ‬ඔࠞׄđ໭ਬׄbఃֆ໊ࢨᄁཙႋӯ‫֥྘ׅ‬ᆒ֕කࡨ ྙൔb 6ē࿼ൈߌࢫ ᄎ‫׮‬ٚӱğ c(t) = r(t − τ ) đᆃ۱ٚӱൌ࠽ഈ҂൞ັ‫ٳ‬ٚӱ‫ط‬൞ҵ‫ٳ‬ٚӱb Ԯ‫ݦ־‬ඔğ G(s) = e−τs đ൞ s ֥໭৘‫ݦ‬ඔđ‫ݦ‬ඔ e−τs ᄝ s = ∞ ׄႵ໭౫‫؟‬۱ࠞׄ‫ބ‬ਬׄb 24 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 5 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘č7 ࢫĎ ᇶေଽಸ ॥ᇅ༢๤֥‫׮‬෿ࢲ‫ܒ‬๭ ଢ֥აေ ᅧ໤༢๤ࢲ‫ܒ‬๭֥‫ק‬ၬaࠎЧቆӮa߻ᇅaࠎЧ৵ࢤྙൔaི֩эߐࠣ߄ࡥ ౰ ᅧ໤оߌ༢๤ӈႨ֥Ԯ‫ݦ־‬ඔ ᇗׄა଴ ᇗׄğ༢๤ࢲ‫ܒ‬๭֥߻ᇅ‫ࡥ߄ބ‬aоߌ༢๤ӈႨ֥Ԯ‫ݦ־‬ඔ ׄ ଴ׄğ༢๤ࢲ‫ܒ‬๭֥߻ᇅa߄ࡥ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 2-6 2-10čcĎ2-11čaĎ ቔြี 2.7 ༢๤֥‫׮‬෿ࢲ‫ܒ‬๭ 1ē‫ק‬ၬ ॥ᇅ༢๤֥‫׮‬෿ࢲ‫ܒ‬๭൞૭ඍ༢๤۲ჭࡱᆭࡗྐ‫ݼ‬ੀཟ‫ބ‬Ԯ‫ܱ־‬༢֥ඔ࿐๭ൕଆ྘đ෱іൕ༢๤۲ эਈᆭࡗ֥ၹ‫ܱݔ‬༢ၛࠣؓ۲эਈ෮ࣉྛ֥ᄎෘđ൞॥ᇅ৘ંᇏ૭ඍ‫گ‬ᄖ༢๤֥၂ᇕࡥьٚ‫م‬đ෱ൡ ႨႿཌྟ‫ބ‬٤ཌྟ༢๤b 2ēࢲ‫ܒ‬๭֥ቆӮ ࢲ‫ܒ‬๭Ⴎྐ‫ݼ‬ཌa‫ٳ‬ᆦׄaཌྷࡆčሸ‫ކ‬Ďׄ‫ބ‬ٚॿ֩ֆჭb ई২ğᆰੀ‫ࠏ׈‬ሇ෎॥ᇅ༢๤čॢᄛĎb ൳॥ؓའԮ‫ݦ־‬ඔğ G1 (s) = Ω(s) = 1 Ke +1 Ua (s) TmTa s 2 + Tm s ॥ᇅఖč٢նఖĎԮ‫ݦ־‬ඔğ G2 (s) = Ua (s) = Ka E(s) бࢠߌࢫԮ‫ݦ־‬ඔğ E(s) = U r (s) − UT (s) ّঌҩਈčҩ෎‫ࠏ׈ؿ‬ĎԮ‫ݦ־‬ඔğ G3 (s) = UT (s) = KT Ω(s) ۴ऌഈඍ۲ߌࢫ֥ྐ‫ݼ‬Ԯ‫ބ־‬эߐܱ༢đ֤֞ఃࢲ‫ܒ‬๭ೂ༯ğ 3ēࢲ‫ܒ‬๭֥߻ᇅٚ‫م‬ ҄ᇧğč1Ďਙཿૄ۱ჭࡱ֥ჰ൓ັ‫ٳ‬ٚӱđѩᇿၩ‫ڵ‬ᄛིႋĠ č2Ďࡼჰ൓ັ‫ٳ‬ٚӱ౰ Laplace эߐđѩࡼ֤֥֞Ԯ‫ݦ־‬ඔཿೆٚॿᇏĠ č3Ďࡼᆃུٚॿοྐ‫֥ݼ‬ੀཟ‫ބ‬Ԯ‫ܱ־‬༢Ⴈྐ‫ݼ‬ཌaཌྷࡆׄ‫ٳބ‬ᆦׄ৵ࢤఏটđ֤ࠧ֞ᆜ۱༢๤ ֥ࢲ‫ܒ‬๭b ২ี 1ğ߂ԛ RC ‫׈‬ຩ઎֥ࢲ‫ܒ‬๭ 25 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ਙཿჰ൓ັ‫ٳ‬ٚӱğ ౰ Laplace эߐğ ⎧ ⎪u ⎪ R (t ) = u1 (t) − u2 (t) ⎪⎨⎧UI (Rs)( s) = U1(s) − U 2 (s) uR (t) R = U R (s) R ⎨i(t) = ⇒ ⎪ 1 ⎪⎩U 2 (s) = I (s) sc ⎪u C ⎩ 2 (t) = ∫ i(t)dt ߂ԛࢲ‫ܒ‬๭ğ ‫ھ‬ຩ઎֥Ԯ‫ݦ־‬ඔູğ G(s) = U 2 (s) U1(s) = 1 (RCs + 1) ২ี 2ğ߂ԛਆࠩ RC ‫׈‬ຩ઎֥ࢲ‫ܒ‬๭ ਙཿჰ൓ٚӱğ ౰ Laplace эߐğ ⎧uR1 (t) = u1 (t) − uc1 (t) ⎧U R1 (s) = U1 (s) − U c1 (s) ⎪⎪i1 (t) = uR1 (t) R1 ⎪ ⎪ I1 (s) = U R1 (s) R1 ∫ ⎪ = 1 [i1 (t) − i2 (t)]dt ⇒ ⎪ = I1(s) − I2 (s) C1 ⎨U c1 (s) sC1 ⎨uc1 (t) ⎪ ⎪ ∫ ⎪ ⎪ (t) = 1 i2 (t)dt ⎪U 2 (s) = I2 (s) ⎪u2 C2 ⎩ sC2 ⎩ ߂ԛࢲ‫ܒ‬๭ğ ຩ઎Ԯ‫ݦ־‬ඔູğ G(s) = U 2 (s) U1 (s) = 1 [R1C1R2C2 s 2 + (R1C1 + R2C2 + R1C2 )s + 1] 4ēࢲ‫ܒ‬๭֥ࠎЧ৵ࢤྙൔ č1ĎԱ৳čseriesĎ G(s) = C(s) R(s) = G1 (s)G2 (s) n č2Ďѩ৳čparallelĎ ∏ Gn (s) = Gi (s) i =1 26 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ G(s) = C(s) R(s) = G1 (s) + G2 (s) + n č3Ďّঌ৵ࢤčFeedbackĎ ∑ + Gn (s) = Gi (s) i =1 ΦB (s) = C(s) = 1 ± G(s) (s) đΦ E ( s) = E(s) = 1 R(s) G(s)H R(s) 1+ G(s)H (s) 5ēࢲ‫ܒ‬๭ི֥֩эߐ ࢲ‫ܒ‬๭ི֩эߐ‫م‬ᄵᇶေႵğ‫ٳ‬ᆦׄభ၍a‫ٳ‬ᆦׄު၍aཌྷࡆׄభ၍aཌྷࡆׄު၍b ई২ğ༢๤ࢲ‫ܒ‬๭ೂ༯đ౰Ԯ‫ݦ־‬ඔ b G(s) = C(s) R(s) ࢳğ ∴G(s) = G3(s)(G1(s)G2 (s) + G4 (s)) 1+ G2 (s)G3 (s)H (s) 6ē༢๤֥ӈႨԮ‫ݦ־‬ඔ ‫྘ׅ‬॥ᇅ༢๤ࢲ‫ܒ‬๭ğ 27 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č1Ďభཟ๙֡Ԯ‫ݦ־‬ඔğ C(s) E(s) = G1 (s)G2 (s) č2Ďّঌ๙֡Ԯ‫ݦ־‬ඔğ B(s) C(s) = H (s) č3ĎषߌԮ‫ݦ־‬ඔğ G0 (s) = B(s) E(s) = G1 (s)G2 (s)H (s) č4ĎоߌԮ‫ݦ־‬ඔğ Φc (s) = C(s) R(s) = G1 (s)G2 (s [1 + G1 (s)G2 (s)H (s)] č5Ď༂ҵԮ‫ݦ־‬ඔğ ⎧Φ E (s) = E(s) R(s) = 1 [1 + G1 (s)G2 (s)H (s)] (s)H (s)] ⎩⎨Φ N (s) = N (s) R(s) = −G2 (s)H (s) [1 + G1 (s)G2 č6Ďоߌหᆘٚӱൔğ1 + G1 (s)G2 (s)H (s) = 1 + G0 (s) = 0 ഈൔіૼğ č1Ď༢๤֥оߌหᆘٚӱൔູषߌԮ‫ݦ־‬ඔႵ৘‫ٳ‬ൔ֥‫ٳ‬ଛ‫؟‬ཛൔა‫ٳ‬ሰ‫؟‬ཛൔᆭ‫ބ‬b č2ĎᆺေषߌԮ‫ݦ־‬ඔ൞࿸۬ᆇႵ৘‫ݦ‬ඔđᄵоߌ༢๤აषߌ༢๤֥หᆘ‫؟‬ཛൔࣼऎႵཌྷ๝֥ࢨՑb č3Ďоߌ༢๤აषߌ༢๤֥Ԯ‫ݦ־‬ඔીႵ‫ׄࠞ܋܄‬b č4Ďоߌ༢๤აषߌ༢๤֥Ԯ‫ݦ־‬ඔऎႵཌྷ๝֥ਬׄb 28 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 7 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘č7 ࢫĎ ᇶေଽಸ ྐ‫ݼ‬ੀ๭઼ࠣ࿠‫܄‬ൔ ଢ֥აေ ᅧ໤ྐ‫ݼ‬ੀ๭֥‫ק‬ၬaቆӮჭ෍ ౰ ᅧ໤ࢲ‫ܒ‬๭აྐ‫ݼ‬ੀ๭֥ؓႋܱ༢aྐ‫ݼ‬ੀ๭֥߻ᇅ ᅧ໤઼࿠‫܄‬ൔ‫ބ‬ႋႨ ᇗ ׄ ა ଴ ᇗׄğࢲ‫ܒ‬๭აྐ‫ݼ‬ੀ๭֥ؓႋܱ༢aྐ‫ݼ‬ੀ๭֥߻ᇅa઼࿠‫܄‬ൔ‫ބ‬ႋႨ ׄ ଴ׄğྐ‫ݼ‬ੀ๭֥߻ᇅa઼࿠‫܄‬ൔ‫ބ‬ႋႨ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 2-12čbĎ2-13čaĎ2-14čcĎ ቔြี 2.8 ྐ‫ݼ‬ੀ๭ ྐ‫ݼ‬ੀ๭൞၂ᇕіൕ၂ቆཌྟսඔٚӱ֥๭ൕٚ‫م‬bཞࢲ‫ܒ‬๭၂ဢđ෱္൞၂ᇕ૭ඍ༢๤ଽ҆ྐ ‫ݼ‬Ԯ‫ܱ־‬༢֥ඔ࿐ଆ྘bྐ‫ݼ‬ੀ๭бࢲ‫ܒ‬๭۷ࡥьૼਔđ҂Ⴈࣉྛ߄ࡥđࣼॖ০Ⴈ઼࿠čMasonĎ‫܄‬ൔ ౰ԛ಩ၩਆׄᆭࡗ֥Ԯ‫ݦ־‬ඔđ֌෱აࢲ‫ܒ‬๭҂๝֥൞ᆺିႨট૭ඍཌྟ༢๤b 1ēྐ‫ݼ‬ੀ๭֥ቆӮ ྐ‫ݼ‬ੀ๭Ⴎࢫׄ‫ބ‬Ⴕཟཌ‫؍‬čᆦਫ਼ĎቆӮđࢫׄčႨჵಁіൕĎіൕ༢๤ᇏ֥эਈčЇওൻೆa ൻԛэਈĎđႵཟཌ‫؍‬іൕਆ۱ࢫׄᆭࡗ֥Ԯ‫־‬ٚཟ‫ݼྐބ‬čэਈĎ֥эߐܱ༢đᄝႵཟཌ‫؍‬ഈٚѓᇿ ᄹၭđᄹၭ൞ਆ۱эਈᆭࡗ֥ၹ‫ܱݔ‬༢ൔđൌ࠽ഈᆃ৚֥ᄹၭࣼ൞ਆ۱эਈᆭࡗ֥Ԯ‫ݦ־‬ඔіղൔb ༯๭൞ྐ‫ݼ‬ੀ๭ඌეൕၩ๭ğ a12 a23a34 a46 ӫູՖ x1 ֞ x6 ֥๙֡Ԯൻđ a23a32 ӫູ߭ߌԮൻđ a22 ӫູሱ߭ߌԮൻb 2ēࢲ‫ܒ‬๭აྐ‫ݼ‬ੀ๭֥ؓႋܱ༢ 2 ่ჰᄵğ č1Ďࢫׄ෮іൕ֥эਈ֩Ⴟੀೆ‫ݼྐ֥ׄࢫھ‬ᆭ‫ބ‬Ġ č2ĎՖࢫׄੀԛ֥ૄ၂ᆦਫ਼ྐ‫֩׻ݼ‬Ⴟ‫ׄࢫھ‬෮іൕ֥эਈĠ 6 ่ؓႋܱ༢‫ބ‬ᇿၩ໙ีğ č1Ďࢲ‫ܒ‬๭ᇏ֥ཌྷࡆׄ‫ٳބ‬ᆦׄؓႋႿྐ‫ݼ‬ੀ๭ᇏ֥ࠁ‫ׄࢫކ‬Ġ č2Ďࢲ‫ܒ‬๭ᇏ֥ൻೆྐ‫ބݼ‬ൻԛྐ‫ؓݼ‬ႋႿྐ‫ݼ‬ੀ๭ᇏ֥ჷࢫׄ‫ׄࢫ߸ބ‬Ġ č3Ďࢲ‫ܒ‬๭ᇏ֥ٚॿؓႋႿྐ‫ݼ‬ੀ๭ᇏ֥ᆦਫ਼đॿᇏ֥Ԯ‫ݦ־‬ඔؓႋႿᆦਫ਼ԮൻĠ č4Ďࢲ‫ܒ‬๭ᇏ֥‫ّڵ‬ঌ‫“ݼژ‬-”сྶ࠹ೆཌྷႋ֥ᆦਫ਼ᇏčԮ‫ݦ־‬ඔĎĠ č5Ďࢲ‫ܒ‬ᇏმ֞ཌྷਢ֥ཌྷࡆׄ‫ٳބ‬ᆦׄൈđؓႋ֞ྐ‫ݼ‬ੀ๭ᇏൈđсྶࡼཌྷਢ֥ཌྷࡆׄ‫ٳބ‬ᆦׄ ൪ູ 2 ۱ࢫׄđᆭࡗ๙‫ݖ‬Ԯൻູ 1 ֥ᆦਫ਼৵ࢤĠ č6Ďᄝࣉྛࢲ‫ܒ‬๭აྐ‫ݼ‬ੀ๭֥ؓႋ‫ݖ‬ӱᇏđ໭ྶؓჰࢲ‫ܒ‬๭༵ࣉྛ߄ࡥಖުᄜؓႋ֞ཌྷႋ֥ྐ‫ݼ‬ ੀ๭đႋ‫ھ‬ҐႨ“ᆰၲ”֥ٚ‫م‬b 29 ई২ğ G2 (s) uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ G2 (s) C(s) G1 (s) G1 (s) R(s) − H (s) C(s) R(s) H (s) 3ē઼࿠‫܄‬ൔ ∑ P = 1 n ∆ PK ∆ K K =1 ൔᇏğ P ູሹᄹၭđᄝ॥ᇅ৘ંᇏॖၛ৘ࢳູଖ 2 ׄᆭࡗ֥Ԯ‫ݦ־‬ඔ G(s) Ġ PK ູՖჷࢫׄ֞߸ࢫֻ֥ׄ K ่భཟ๙ਫ਼֥ᄹၭčԮ‫ݦ־‬ඔ֥ӰࠒĎĠ ∆ ູྐ‫ݼ‬ੀ๭֥หᆘൔđऎุູğ ∆ = 1 − (෮Ⴕ҂๝߭ਫ਼ᄹၭᆭ‫ )ބ‬+čૄਆ۱޺҂ࢤԨ߭ਫ਼ᄹၭ֥Ӱࠏᆭ‫ބ‬Ď− čૄ೘۱޺҂ࢤԨ߭ਫ਼ᄹၭ֥Ӱࠏᆭ‫ބ‬Ď+ ∑ ∑ ∑ = 1 − La + Lb Lc − Ld Le L f + a bac daeaf ∆ K ູಀԢაֻ K ่భཟ๙ਫ਼ཌྷࢤԨ֥߭ਫ਼ުഺჅ֥ྐ‫ݼ‬ੀ๭֥หᆘൔđӫֻູ K ่భཟ๙ਫ਼หᆘ ൔ֥Ⴥሰൔđ෱ॖၛՖ ∆ ᇏԢಀა๙ਫ਼ PK ཌྷࢤԨ֥߭ਫ਼ᄹၭު֤֞b ᇿğሹᄹၭ P ൌ࠽ഈࣼ൞༢๤ଖਆׄᆭࡗ֥Ԯ‫ݦ־‬ඔđหᆘൔ ∆ ൌ࠽ഈࣼ൞оߌ༢๤֥หᆘ‫؟‬ཛൔb 4ē২ี ২ 1ğ ∆ = 1 − (bi + dj + fk + bcdefgm) + (bidj + bifk + djfk ) − (bidjfk ) [ࢳ] P1 = abcdefghđ∆1 = 1 ২ 2ğ x8 = P1∆1 = abcdefgh x0 ∆ 1 − (bi + dj + fk + bcdefgm) + (bidj + bifk + djfk) − (bidjfk) [ࢳ] ∆ =1− (G2H2 + G3H3 + G6H6 + G7H7 ) + (G2H2G6H6 + G2H2G7H7 + G3H3G6H6 + G3H3G7H7 ) P1 = G1G2G3G4đ∆1 = 1−čG6 H6 + G7 H Ď 7 P2 = G5G6G7G8đ∆ 2 = 1 −čG2 H 2 + G3 H Ď 3 G(s) = G1G2G3G4[1−čG6H6 + G7 H7Ď] + G5G6G7G8[1−čG2H2 + G3H3 )] 1− (G2H2 + G3H3 + G6H6 + G7 H7 ) + (G2H2G6H6 + G2H2G7 H7 + G3H3G6H6 + G3H3G7 H7 ) 30 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 8 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ‫ؽ‬ᅣ ॥ᇅ༢๤֥ඔ࿐ଆ྘ ᇶေଽಸ ॥ᇅ༢๤‫ٳିྟࢫߌ྘ׅ‬༅ഈࠏൌဒ ଢ֥აေ ඃ༑ MATLAB ೈࡱ֥൐Ⴈߌ࣢ ౰ ඃ༑ MATLAB ೈࡱ֥ Simulink ֥ࠎЧ൐Ⴈٚ‫م‬ ๙‫ ݖ‬Simulink ೈࡱٟᆇđ‫ٳ‬༅۲‫ିྟ֥ࢫߌ྘ׅ‬ ᇗׄა଴ ᇗׄğඃ༑ MATLAB ೈࡱ֥ Simulink ֥ࠎЧ൐Ⴈٚ‫م‬ ׄ ๙‫ ݖ‬Simulink ೈࡱٟᆇđ‫ٳ‬༅۲‫ିྟ֥ࢫߌ྘ׅ‬ ࢝࿐൭‫ ؍‬ഈࠏൌဒ නॉีࠇ ປӮൌဒБۡ ቔြี ၂aൌဒଢ֥ 1aඃ༑۲ᇕ‫ࢨ֥ࢫߌ྘ׅ‬ᄁཙႋ౷ཌĠ 2aਔࢳҕඔэ߄ؓ‫׮ࢫߌ྘ׅ‬෿หྟ֥႕ཙb ‫ؽ‬aൌဒ಩ༀ 1aб২ߌࢫč K Ď Ֆ๭ྙ९ᛍফఖᇏ຀ှ StepčࢨᄁൻೆĎaGainčᄹၭଆॶĎaScopečൕѯఖĎଆॶٟ֞ᆇҠቔ߂ ૫đ৵ࢤӮٟᆇॿ๭b‫ڿ‬эᄹၭଆॶ֥ҕඔđՖ‫ڿط‬эб২ߌࢫ֥٢նПඔ K đܴҳ෱ૌ֥ֆ໊ࢨᄁ ཙႋ౷ཌэ߄౦ঃbॖၛ๝ൈཁൕ೘่ཙႋ౷ཌđٟᆇॿ๭ೂ๭෮ൕğ 2aࠒ‫ࢫߌٳ‬č 1 Ď Ts ࡼഈ๭ٟᆇॿ๭ᇏ֥ GainčᄹၭଆॶĎߐӮ Transfer FcnčԮ‫ݦ־‬ඔĎଆॶđഡᇂ Transfer FcnčԮ ‫ݦ־‬ඔĎଆॶ֥ҕඔđ൐ఃԮ‫ݦ־‬ඔэӮ 1 ྘b‫ڿ‬э Transfer FcnčԮ‫ݦ־‬ඔĎଆॶ֥ҕඔđՖ‫ڿط‬ Ts эࠒ‫ ֥ࢫߌٳ‬T đܴҳ෱ૌ֥ֆ໊ࢨᄁཙႋ౷ཌэ߄౦ঃbٟᆇॿ๭ೂഈႷ๭෮ൕb 3a၂ࢨܸྟߌࢫč 1 Ď Ts +1 ࡼഈ๭ᇏ Transfer FcnčԮ‫ݦ־‬ඔĎଆॶ֥ҕඔᇗྍഡᇂđ൐ఃԮ‫ݦ־‬ඔэӮ 1 ྘đ‫ڿ‬эܸྟ Ts +1 ߌࢫ֥ൈࡗӈඔ T đܴҳ෱ૌ֥ֆ໊ࢨᄁཙႋ౷ཌэ߄౦ঃbٟᆇॿ๭ೂ๭෮ൕğ 31 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 4aൌ࠽ັ‫ࢫߌٳ‬č Ks Ď Ts +1 ࡼ Transfer FcnčԮ‫ݦ־‬ඔĎଆॶ֥ҕඔᇗྍഡᇂđ൐ఃԮ‫ݦ־‬ඔэӮ Ks ྘đčҕඔഡᇂൈႋᇿၩ Ts +1 T 1 Ďb ਷ K ҂эđ‫ڿ‬э Transfer FcnčԮ‫ݦ־‬ඔĎଆॶ֥ҕඔđՖ‫ڿط‬э T đܴҳ෱ૌ֥ֆ໊ࢨᄁཙ ႋ౷ཌэ߄౦ঃbٟᆇॿ๭ೂഈႷ๭෮ൕb 5a‫ࢨؽ‬ᆒ֕ߌࢫč s2 ωn2 Ď + 2ξωns + ωn2 ࡼ Transfer FcnčԮ‫ݦ־‬ඔĎଆॶ֥ҕඔᇗྍഡᇂđ൐ఃԮ‫ݦ־‬ඔэӮ s2 ωn2 ྘čҕඔഡ + 2ξωns + ωn2 ᇂൈႋᇿၩ 0 < ξ < 1 Ďđٟᆇॿ๭ೂ๭෮ൕğ č1Ď਷ωn ҂эđ ξ ౼҂๝ᆴč 0 < ξ < 1 Ďđܴҳఃֆ໊ࢨᄁཙႋ౷ཌэ߄౦ঃĠ č2Ď਷ξ =0.2 ҂эđ ωn ౼҂๝ᆴđܴҳఃֆ໊ࢨᄁཙႋ౷ཌэ߄౦ঃb 6a࿼Ӿߌࢫč e−τ s Ď ࡼٟᆇॿ๭ᇏ֥ Transfer FcnčԮ‫ݦ־‬ඔĎଆॶߐӮ Transport Delayčൈࡗ࿼ӾĎଆॶđ‫ڿ‬э࿼Ӿൈ ࡗτ đܴҳֆ໊ࢨᄁཙႋ౷ཌэ߄౦ঃbٟᆇॿ๭ೂഈႷ๭෮ൕb ೘aൌဒඔऌ 1a б২ߌࢫč K Ď 2aࠒ‫ࢫߌٳ‬č 1 Ď Ts 3a၂ࢨܸྟߌࢫč 1 Ď 4aൌ࠽ັ‫ࢫߌٳ‬č Ks Ď Ts +1 Ts +1 32 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 5a‫ࢨؽ‬ᆒ֕ߌࢫč s2 ωn2 Ď + 2ξωns + ωn2 č1Ď਷ωn =5 ҂эđ ξ ౼ 0.5 ‫ ބ‬0.2 č2Ď਷ξ =0.2 ҂эđωn ౼ 5 ‫ ބ‬10 6a࿼Ӿߌࢫč e−τ s Ď ඹaൌဒࢲં 1a Ֆֆ໊ࢨᄁཙႋটुđб২ߌࢫି‫ܔ‬҂ാᆇ֥Ӯб২֥‫گ‬གྷൻೆđб২ߌࢫ K ‫ڿ‬эđᄵ٢նПඔ‫ڿ‬эb 2a Ֆֆ໊ࢨᄁཙႋটुđࠒ‫֥ࢫߌٳ‬ൻԛ൞ൻೆؓൈࡗഈ֥ࠒ‫ٳ‬đ‫ڿ‬эൈࡗӈඔ T đT ᄀնđᄵࠒ‫ٳ‬ᄀ તb 3a Ֆֆ໊ࢨᄁཙႋটुđܸྟߌࢫି࿼ߏ֥‫گ‬གྷൻೆđ T ᄀնđܸྟᄀնđ‫گ‬གྷൻೆᄀતb 4a Ֆֆ໊ࢨᄁཙႋটुđັ‫ࢫߌٳ‬ൻԛିّႋൻೆؓൈࡗ֥э߄ੱb 5a Ֆֆ໊ࢨᄁཙႋটुđ೏‫ࢨؽ‬ᆒ֕ߌࢫ֥ ωn ཌྷ๝đξ ᄀնđӑ‫ט‬ਈᄀཬđཙႋᄀॹĠ೏ξ ཌྷ๝đωn ‫ڿ‬ эđ҂႕ཙӑ‫ט‬ਈđ ωn ᄀնđཙႋᄀॹb 6a Ֆֆ໊ࢨᄁཙႋটुđ࿼Ӿߌࢫൻԛᄝτ ൈख़ުି‫گ‬གྷൻೆđτ ֥նཬđ႕ཙൻԛᇌު֥ൈࡗb ๙‫ݖ‬ᆃՑൌဒđඃ༑ਔ۲‫ࢫߌ྘ׅ‬ၛࠣఃֆ໊ࢨᄁཙႋđਔࢳ֥۲‫֥ࢫߌ྘ׅ‬หׄđၛࠣཌྷႋҕඔ э߄ؓཙႋ֥႕ཙb  33 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 9 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅č1a2 ࢫĎ ᇶေଽಸ ‫྘ׅ‬ൻೆྐ‫ݼ‬ ༢๤ཙႋ֥ൈთྟିᆷѓ ၂ࢨ༢๤֥ൈთ‫ٳ‬༅ ਔࢳሱ‫׮‬॥ᇅ༢๤֥‫྘ׅ‬ൻೆྐ‫ݼ‬đᅧ໤॥ᇅ༢๤֥‫׮‬෿ྟିᆷѓ‫ބ‬໗෿ྟିᆷѓ ଢ֥აေ ᅧ໤၂ࢨ༢๤֥ඔ࿐ଆ྘aࢨᄁཙႋࠣఃྟିᆷѓ ౰ ਔࢳ၂ࢨ༢๤֥ઝԊཙႋ‫ོބ‬௡ཙႋ ᇗ ׄ ა ଴ ᇗׄğ॥ᇅ༢๤֥‫׮‬෿ྟିᆷѓ‫ބ‬໗෿ྟିᆷѓa၂ࢨ༢๤֥ࢨᄁཙႋࠣఃྟିᆷѓ ׄ ଴ׄğ॥ᇅ༢๤֥‫׮‬෿ྟିᆷѓ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี  ႄ࿽ ؓႿཌྟ॥ᇅ༢๤‫۽‬ӱഈӈႨ֥‫ٳ‬༅ٚ‫م‬Ⴕൈთ‫ٳ‬༅‫م‬a۴݅ࠖ‫ٳ‬༅‫م‬a௔თ‫ٳ‬༅‫֩م‬bЧᅣᇗ ׄࢺകൈთ‫ٳ‬༅‫م‬đ෮໌ൈთ‫ٳ‬༅‫م‬đ൞ᆷ॥ᇅ༢๤ᄝ၂‫֥ק‬ൻೆྐ‫ݼ‬ቔႨ༯đ۴ऌ༢๤ൻԛਈ֥ൈ თіղൔđ‫ٳ‬༅༢๤֥໗‫ྟק‬aᄠ෿‫ބ‬໗෿ྟିb ē‫྘ׅ‬ൻೆྐ‫ݼ‬ čĎֆ໊ԊࠌčઝԊĎྐ‫ ݼ‬δ (t)  δ ⎧0 t≠0 δ0+ (t)dt = 1 đ-BQMBDF эߐğ L[δ (t)] = 1  ⎩⎨∞ đ 0− t =0 ∫ ‫ק‬ၬğ (t ) = čĎֆ໊ࢨᄁྐ‫ ݼ‬1(t)  ‫ק‬ၬğ 1(t ) = ⎧0 t < 0 đ-BQMBDF эߐğ L[1(t)] = 1 s ⎨ t≥0 ⎩1 čĎֆ໊ོ௡č෎؇Ďྐ‫ݼ‬ ‫ק‬ၬğ rv (t ) = ⎧0 t<0 s2  ⎩⎨t t ≥ 0 đ-BQMBDF эߐğ L[rv (t)] = 1 čĎֆ໊ࡆ෎؇č஘໾ཌĎྐ‫ݼ‬ ⎧0 t <0 ⎪ t≥ ‫ק‬ၬğ ra (t) = ⎨ 1 đ-BQMBDF эߐğ L[ra (t)] =1 s3  ⎪⎩ 2 2 t 0 čĎᆞ༿čჅ༿Ďྐ‫ݼ‬ ‫ק‬ၬğ r(t) = ⎧0 t < 0 đ-BQMBDF эߐğ L[r(t)] = ω  ⎨ t +ω2 ⎩sin ωt ≥ 0 s2 ‫ק‬ၬğ r(t) = ⎧0 t < 0 đ-BQMBDF эߐğ L[r(t)] = s  ⎨ +ω2 ⎩cos ωt t ≥ 0 s2 ēᄠ෿ཙႋ‫ބ‬໗෿ཙႋ ಩‫ޅ‬၂۱॥ᇅ༢๤֥ൈࡗཙႋ‫׻‬ॖၛ‫ູٳ‬ᄠ෿ཙႋ‫ބ‬໗෿ཙႋਆ҆‫ٳ‬b čĎᄠ෿ཙႋ ᄠ෿ཙႋႻӫູᄠ෿‫ݖ‬ӱࠇ‫ݖ‬؈‫ݖ‬ӱb෱൞ᆷᄝൻೆྐ‫ࠇݼ‬ಠ‫ݼྐ׮‬ቔႨ༯đ༢๤ֹൻԛਈՖԚ ൓ሑ෿֞ቋᇔሑ෿֥ཙႋ‫ݖ‬ӱb 34 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  ‫ݖ‬ӱॖၛіགྷູཙႋ౷ཌ֥ֆ‫ࡨט‬aֆ‫ט‬ᄹaකࡨᆒ֕a֩‫ږ‬ᆒ֕a‫ؿ‬೛֩‫؟‬ᇕྙൔđఃྙൔႮ༢ ๤֥ࠎЧᄎ‫׮‬ଆ෿थ‫ק‬bཁಖ၂۱ൌ࠽ॖྛ֥॥ᇅ༢๤ఃཙႋֹᄠ෿‫ݖ‬ӱсྶ൞ֆ‫ࠇࡨט‬කࡨᆒ֥֕đ ္ࠧіགྷູ༢๤֥ᄠ෿‫ݖ‬ӱ൞ෛൈࡗᇯࡶཨാ֥đᆃဢ֥༢๤Ҍ൞໗‫֥ק‬b čĎ໗෿ཙႋ ໗෿ཙႋႮӫູ໗෿‫ݖ‬ӱb෱൞ᆷ༢๤ᄝൻೆྐ‫ࠇݼ‬ಠ‫ݼྐ׮‬ቔႨ༯đ֒ൈࡗ t ౴࣍Ⴟ໭౫նൈđ༢ ๤֥ൻԛཙႋ֥ሑ෿b໗෿‫ݖ‬ӱّ႘ਔ༢๤ൻԛਈ‫گ‬གྷൻೆਈ֥ӱ؇đӈӈႨ໗෿ࣚ؇ࠇ໗෿༂ҵট‫ޙ‬ ਈb ēᄠ෿‫ބ‬໗෿ྟିᆷѓ ູਔ‫ק‬ਈіൕ॥ᇅ༢๤ᄠ෿‫ބ‬໗෿ཙႋ֥ྟିđᄝ‫۽‬ӱഈ၂Ϯၛֆ໊ࢨᄁྐ‫ݼ‬ቔູൻೆ൫ဒྐ‫ݼ‬ট‫ק‬ ၬ༢๤֥ᄠ෿‫ބ‬໗෿ྟିᆷѓb c(t)   c(tp ) 5%× c(∞)  c(∞) ࠇ2%× c(∞)  1 c(∞)  2  0 td tr tp ts t  čĎഈശൈࡗ tr ğ༢๤ࢨᄁཙႋՖਬष൓ֻ၂Ցഈശ֞໗෿ᆴ֥ൈࡗčႵൈॖ౼ཙႋ֥໗෿ᆴ֤  ֞ ෮ؓႋ֥ൈࡗĎb čĎ࿼Ӿൈࡗ td ğ༢๤ࢨᄁཙႋՖਬष൓ֻ၂Ցഈശ֞໗෿ᆴ ֥ൈࡗb čĎ‫ڂ‬ᆴൈࡗ t p ğ༢๤ࢨᄁཙႋՖਬष൓ֻ၂Ցӑ‫ݖ‬໗෿ᆴղֻ֞၂۱‫ڂ‬ᆴ֥ൈࡗb čĎ‫ࢫט‬ൈࡗ ts ğ༢๤ࢨᄁཙႋ౷ཌࣉೆܿ‫ק‬ᄍྸ֥༂ҵջ c(∞) × ∆% ٓຶđѩ౏ၛު҂ᄜӑԛᆃ ۱༂ҵջ෮ླ֥ൈࡗb čĎӑ‫ט‬ਈ M p % ༢๤ࢨᄁཙႋ֥ቋն‫ڂ‬ᆴ c(t p ) ა໗෿ᆴ c(∞) ֥ҵᆴა໗෿ᆴ c(∞) ᆭб֥Ϥ‫ٳ‬ ඔđࠧ M p % = c(t p ) − c(∞) ×100% b༂ҵջॖ౼ğ ∆ = ±2% ࠇ ∆ = ±5% b c(∞) čĎᆒ֕Ցඔ /ğᄝ 0 ≤ t ≤ ts ൈࡗଽđ༢๤ࢨᄁཙႋ౷ཌԬᄀః໗෿ᆴ c(∞) Ցඔ֥၂϶čԬᄀ  Ցཌྷ֒Ⴟᆒ֕  ՑĎb čĎ໗෿༂ҵ ess ğ֒ൈࡗ t → ∞ ൈđ༢๤௹ຬൻԛაൌ࠽ൻԛᆭҵđࠧ ess = lim e(t) = lim[r(t) − c(t)] t→∞ t→∞ ཬࢲğഈඍྟିᆷѓᇏ tratdat pats ّႋਔ༢๤ᄠ෿ཙႋ֥ॹ෎ྟđఃᇏ ts ّႋਔ༢๤ཙႋ֥ሹ ุॹ෎ྟđ෮ၛ၂Ϯಪູᄝ ts ᆭభູᄠ෿ཙႋđ ts ᆭުູ໗෿ཙႋb M p % ّႋਔ༢๤ᄠ෿‫ݖ‬ӱ֥ᆒ֕ ྟđఃЧᇉّႋਔ༢๤֥ཌྷؓ໗‫ྟק‬b ess ّႋਔ༢๤֥໗෿ࣚ؇b ၂ࢨ༢๤֥ൈთ‫ٳ‬༅ R(s) 1 C(s) ē၂ࢨ༢๤֥ඔ࿐ଆ྘ ၂ࢨ༢๤֥ັ‫ٳ‬ٚӱğT dc(t) + c(t) = r(t)  t ≥ 0 Ġ Ts dt ၂ࢨ༢๤֥Ԯ‫ݦ־‬ඔğ G(s) = 1 đൔᇏT ູൈࡗӈඔ Ts + 1 ༢๤ࢲ‫ܒ‬๭ğ ২၂ࢨ 3$ ‫׈‬ਫ਼ğ ē၂ࢨ༢๤֥ֆ໊ࢨᄁཙႋ ൻೆ r(t) = 1(t) ⇔ R(s) = 1 s  35 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ൻԛ c(t) = L−1[C(s)] = L−1[Φ(s)R(s)] = L−1[ 1 × 1 = 1− e − t đt ≥ 0  ] T Ts + 1 s čĎ࿼Ӿൈࡗğ td = 0.69T ĠčĎഈശൈࡗğ tr = 2.20T ĠčĎ‫ࢫט‬ൈࡗğ ts = ⎧3Tđ∆ = ±5% Ġ ⎩⎨4Tđ∆ = ±2% čĎཙႋ౷ཌӯֆ‫ט‬ഈശđ໭ӑ‫ט‬đ໭ᆒ֕ĠčĎ໗෿༂ҵğ ess = 0 b ཙႋ෎؇აൈࡗӈඔ T Ӯّбđࠧ T ᄀཬđཙႋ෎؇ᄀॹđᄝ t = 0 ԩቋնđѩෛൈࡗᄹն‫ط‬эཬđ ᆰᇀູਬb ē၂ࢨ༢๤ֆ໊ઝԊཙႋ ൻೆ r(t) = δ (t) ⇔ R(s) = 1  ൻԛ c(t) = L−1[C(s)] = L−1[Φ(s)R(s)] = L−1[ 1 ×1] = 1 e − t đt ≥ 0 T Ts + 1 T čĎཙႋ౷ཌӯֆ‫ט‬༯ࢆđ໭ӑ‫ט‬đ໭ᆒ֕đᄝ t = 0 ԩ༯ࢆ෎ੱቋնđᆭު෎ੱэཬđ౏༯ࢆ෎ੱ აൈࡗӈඔ T Ӯّбđࠧ T ᄀཬđ༯ࢆ෎ੱᄀॹb ēֆ໊ོ௡ཙႋ ൻೆğ r(t) = t •1(t) ⇔ R(s) = 1 s 2  ൻԛğ c(t) = L−1[C(s)] = L−1[Φ(s)R(s)] = L−1[ 1 × 1 ] = L−1[ 1 − T + T ]  Ts + 1 s 2 s2 s s +1 T  = t − T (1 − −t )đ t ≥ 0 eT −t ༂ҵ‫ݦ‬ඔğ e(t) = r(t) − c(t) = T (1 − e T )  ໗෿༂ҵğ ess = T  ࢲંğ čĎ၂ࢨ༢๤ᄝ۵ሶֆ໊ོ௡ྐ‫ݼ‬ൈđሹթᄝ໊ᇂ༂ҵđ౏໊ᇂ༂ҵ֥նཬෛൈࡗ‫ط‬ᄹնđቋᇔ౴ Ⴟൈࡗӈඔ T đၹՎ T ᄀնᄵ༂ҵ္ᄀնb −t čĎཙႋ෎؇čོࠧੱĎູ c′(t) = 1 − e T đ෮ၛཙႋ෎؇ᄝ t = 0 ԩቋཬđt ‫ޓ‬նč৘ંഈ t → ∞ Ď ཙႋ෎؇ቋն č౷ཌ࣍ެԼᆰႿൈࡗᇠĎb 36 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 10 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅č3 ࢫĎ ᇶေଽಸ ‫ࢨؽ‬༢๤֥၂Ϯඔ࿐ଆ྘‫ބ‬ҕඔ ‫ࢨؽ‬༢๤֥оߌࠞׄ‫ބ҃ٳ‬ҕඔ֥ܱ༢ ‫ࢨؽ‬༢๤֥ֆ໊ࢨᄁཙႋ ఴቅୄ‫ࢨؽ‬༢๤‫׮‬෿ྟିᆷѓ֥࠹ෘ ଢ֥აေ ਔࢳ‫ࢨؽ‬༢๤оߌ֥ࠞׄ‫ބ҃ٳ‬ҕඔ֥ܱ༢ ౰ ᅧ໤‫ࢨؽ‬༢๤֥҂๝ቅୄб༯֥ֆ໊ࢨᄁཙႋ ᅧ໤ఴቅୄ‫ࢨؽ‬༢๤‫׮‬෿ྟିᆷѓ֥࠹ෘ ᇗׄა଴ ᇗׄğ‫ࢨؽ‬༢๤֥҂๝ቅୄб༯֥ֆ໊ࢨᄁཙႋaఴቅୄ‫ࢨؽ‬༢๤‫׮‬෿ྟିᆷѓ֥࠹ෘ ׄ ଴ׄğఴቅୄ‫ࢨؽ‬༢๤‫׮‬෿ྟିᆷѓ֥࠹ෘ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 3-2 3-3  ē‫ࢨؽ‬༢๤֥ൈთ‫ٳ‬༅ R(s) ω 2 C(s) ē‫ࢨؽ྘ׅ‬༢๤֥ඔ࿐ଆ྘ n षߌԮ‫ݦ־‬ඔğ GC (s) = ω 2  s ( s + 2ξω n s ) n s(s + 2ξωn ) оߌԮ‫ݦ־‬ඔğ Φ(s) = ω 2 = 1  n s2 + 2ξωn s + ω 2 T 2 s 2 + 2ξTs + 1 n ൔᇏğ ωn ӫູ‫ࢨؽ‬༢๤֥໭ቅୄᆒ֕௔ੱࠇሱಖᆒ֕௔ੱđֆ໊൞ rad / s đ T = 1 ωn ӫູ‫ࢨؽ‬ ༢๤֥ൈࡗӈඔđξ ӫູቅୄбđ၂Ϯ൞໭ਈ‫֥۔‬b ২ğ3-$ Ա৳‫׈‬ਫ਼֥ັ‫ٳ‬ٚӱູ LC d 2u0 (t) dt 2 + RC du0 dt + u0 (t) = ur (t) đ ఃԮ‫ݦ־‬ඔູ G(s) = U 0 (s) = 1 = 1 LC  U r (s) Lcs2 + Rcs + 1 s 2 + (R C)s + 1 LC ॖၛ֤֞ğ ω 2 =1 LC ⇒ ωn =1 LC đ 2ξωn = R C ⇒ ξ = R C n 2 L ē‫ࢨؽ‬༢๤֥оߌਬࠞׄ ‫ࢨؽ‬༢๤֥оߌหᆘٚӱൔğ s 2 + 2ξωn s + ω 2 = 0 n หᆘ۴čоߌࠞׄĎູğ s1,2 = −ξωn ± ωn ξ 2 −1  ཁಖෛሢቅୄбξ ౼ᆴ֥҂๝đоߌ֥ࠞׄ‫္҃ٳ‬҂๝b ē‫ࢨؽ‬༢๤֥ࢨᄁཙႋ čĎఴቅୄč ξ < 1 Ďཙႋ s1,2 = −ξω n ± jω n 1 − ξ 2 = −ξω n ± jω d đωd = ωn 1 − ξ 2 ӫູቅୄᆒ֕௔ੱđ β = arctg 1− ξ 2 = arccosξ = arcsin 1− ξ 2  ξ ࢲંğႮഈ๭ॖ࡮đᄝሱಖᆒ֕௔ੱωn Ќӻ҂э֥౦ঃ༯đቅୄ࢘ β ᄀնđᄵቅୄбξ ᄀཬĠቅ ୄ࢘ β ᄀཬđᄵቅୄбξ ᄀնb 37 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  ఴቅୄ‫ࢨؽ‬༢๤֥ֆ໊ࢨᄁཙႋູğ c(t) = 1− e−ξωnt sin(ωd t + β) (t ≥ 0)  1−ξ 2   ‫ٳ‬༅ఴቅୄ‫ࢨؽ‬༢๤ֆ໊ࢨᄁཙႋіղൔđॖၛ֤ԛೂ༯ࢲંğ ֻ၂đᄠ෿ཙႋູ ct (t) = e −ξωnt sin(ωd t + β ) đ໗෿ཙႋູ css (t) = 1 Ġ 1−ξ 2 ֻ‫ؽ‬đᄠ෿ཙႋູοᆷඔකࡨ֥ᆞ༿ᆒ֕ྙ෿Ġ ֻ೘đቅୄб ξ ֥ࡨཬࡼ֝ᇁ༢๤ཙႋ֥ᆒ֕ࡆखđ౏කࡨ෎؇эતĠ čĎ໭ቅୄč ξ = 0 Ďཙႋ หᆘ۴ğ s1,2 = ± jωn đֆ໊ࢨᄁཙႋğ c(t) = 1 − cosωnt (t ≥ 0)  ‫ٳ‬༅ࢲંğ໭ቅୄ‫ࢨؽ‬༢๤֥ֆ໊ࢨᄁཙႋ౷ཌӯ֩‫ږ‬ᆒ֕ྙൔđఃᆒ֕௔ੱູωn đ‫ږ‬ᆴູ b  čĎਢࢸቅୄč ξ = 1 Ďཙႋ หᆘ۴ğ s1,2 = −ωn đֆ໊ࢨᄁཙႋğ c(t) = 1− e−ξωnt (1+ ωnt) (t ≥ 0)  ‫ٳ‬༅ࢲંğਢࢸቅୄ‫ࢨؽ‬༢๤֥ֆ໊ࢨᄁཙႋ౷ཌοᆷඔֆ‫ט‬ഈശྙൔđ֌ႮႿ൞ᇗࠞׄđ෮ၛཙႋ ᇏᄹࡆਔ ωnte−ξωnt ၂ཛb čĎ‫ݖ‬ቅୄč ξ > 1 Ďཙႋ หᆘ۴ğ s1,2 = −ξωn ± ωn ξ 2 −1 đ c(t) = 1+ T1 −t T2 −t T2 − T1 e T1 + T1 − T2 e T2 , (t ≥ 0)  ൔᇏğT1 = 1 ωn (ξ − ξ 2 − 1) = −1 s1 đT2 =1 ωn (ξ + ξ 2 − 1) = −1 s2  ‫ٳ‬༅ࢲંğ ֻ၂đႮႿ‫ݖ‬ቅୄሑ෿֥ਆ۱หᆘ۴नູ‫ڵ‬ൌඔđ෮ၛఃཙႋൌ࠽ഈࣼ‫ູࢳٳ‬ਔਆ۱කࡨ֥ᆷඔཛ e s1tae s2t ֥ཌྟቆ‫ކ‬đቆ‫֥ުކ‬ཙႋ౷ཌ൞٤ᆒ֥֕ֆ‫ט‬ഈശྙൔb ֻ‫ؽ‬đཙႋ෎؇Ⴎਆ۱ൈࡗӈඔ T1aT2 ‫܋‬๝थ‫ק‬b ēఴቅୄ‫ࢨؽ‬༢๤ྟିᆷѓ čĎഈശൈࡗ tr  38 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ tr =π −β = π −β  ωd ωn 1−ξ 2 čĎ‫ڂ‬ᆴൈࡗ t p  tp =π = π čĎӑ‫ט‬ਈ M p %  ωd ωn 1−ξ 2 Mp% = c(t p ) − c(∞) ×100% = e−ξπ 1−ξ 2 ×100%  c(∞) Ֆഈൔॖᆩӑ‫ט‬ਈ M p % ࣇაቅୄбξ Ⴕܱđ‫ط‬აሱಖᆒ֕௔ੱ໭ܱb ࢲંğቅୄбξ ᄀཬđᄵӑ‫ט‬ਈ M p % ᄀնĠቅୄбξ ᄀնđᄵӑ‫ט‬ਈ M p % ᄀཬb čĎ‫ࢫט‬ൈࡗ t s  ֒ 0.1 < ξ < 0.9 ൈđ 1 − ξ 2 ≈ 1đႿ൞‫ࢫט‬ൈࡗ֥࣍ර࠹ෘ‫܄‬ൔູğ ts = 3 đ౼ ∆ =5 ξω n ts = 4 đ౼ ∆ = 2  ξω n ࢲંğ ֻ၂đ‫ࢫט‬ൈࡗ t s ა ξωn Ӯّбđࠧა֥ࠞׄൌ҆ඔᆴӮّбbᆃඪૼࠞׄए৖ྴᇠᄀჹđ༢๤֥ ‫ࢫט‬ൈࡗᄀ؋b ֻ‫ؽ‬đႮႿቅୄб t s ֥࿊౼ᇶေ൞၇ऌ༢๤ؓӑ‫ט‬ਈ M p % ֥ေ౰টಒ‫֥ק‬đ‫ط‬ӑ‫ט‬ਈ M p % აሱ ಖᆒ֕௔ੱ ωn ໭ܱđ‫ܣ‬ॖၛᄝЌӻቅୄб t s ҂э֥భิ༯đൡ֒ᄹնሱಖᆒ֕௔ੱ ωn đՖ‫࠻ط‬ॖЌᆣ ӑ‫ט‬ਈ M p % ֥҂эđႻି൐‫ࢫט‬ൈࡗ t s ෪؋b ‫ٳ‬༅ğ๙‫ࢫטؓݖ‬ൈࡗaӑ‫ט‬ਈაቅୄбᆭࡗܱ༢֤бࢠđॖၛ֤ԛೂ༯֥ࠎЧࢲંğ‫ࢫט‬ൈࡗ t s a ӑ‫ט‬ਈ M p % ؓቅୄб ξ ֥ေ౰൞ཌྷ޺઱‫֥؛‬đࠧቅୄб ξ ֥࿊ᄴđ໭‫م‬๝ൈડቀ‫ࢫט‬ൈࡗ t s aӑ‫ט‬ਈ M p % бࢠཬ֥ေ౰b ‫۽‬ӱഈ౼ξ = 2 2 = 0.707 ቔູ༢๤ྟିቋࡄ֥ഡ࠹၇ऌđՎൈđ༢๤ྟିሹุࠆ֤oቋࡄpb ২ ğෛ‫׮‬॥ᇅ༢๤ೂ༯๭෮ൕđൻೆྐ‫ ݼ‬r(t) = l(t) đ൫ čĎ,đ࠹ෘ‫׮‬෿ྟିᆷѓĠ čĎ, ‫ ބ‬,đ‫ٳ‬љ࠹ෘษંభᇂ٢նఖؓ༢๤‫׮‬෿ྟି֥႕ཙb R(s) K 5 C(s) C(s) s (s + 34 .5)  ࢳğčĎ, ൈ षߌԮ‫ݦ־‬ඔູğ G c (s) = 5K s(s + 34.5)  оߌԮ‫ݦ־‬ඔູğ Φ(s) = 5K (s2 + 34.5s + 5K)  бᅶ‫ࢨؽ‬༢๤ѓሙྙൔđ౰֤ğωn = 5K = 1000 = 31.č6 rad / sĎđξ = 34b5 2ωn = 0.545 b ႮՎ֤֞ྟିᆷѓğ t p = π ωd = π ωn 1 − ξ 2 = 0.12s đ M p % = e−ξπ 1−ξ 2 ×100% = 13% đ 39 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ts = 3 ξωn = 0.17s ࠇ ts = 4 ξωn = 0.23s  čĎษં , ‫ ބ‬, ൈ֥౦ঃ K = 1500 K = 13.5 ξ = 0.2,ωn = 86.2(rad / s) ξ = 2.1,ωn = 8.22(rad / s)  t p = 0.037s, M p = 52.7%, t s = 0.17s T1 = 0.5s, T2 = 0.05s, T1 >> T2 , t s = 3T1 = 1.46s K ↑→ ξ ↓→ Mp ↑ K ↓→ ξ ↑→ Mp ↓  Ⴎ‫ھ‬২ีॖᆩđ֒ ‫ط‬ →ts,tp ↓ →ts,tp ↑ ২ ഡ‫ࢨؽ‬༢๤֥ֆ໊ࢨᄁཙႋ౷ཌೂ༯๭෮ൕđ൫ಒ‫ק‬༢๤֥Ԯ‫ݦ־‬ඔb  ࢳğՖཙႋ౷ཌૼཁॖၛुԛđᄝֆ໊ࢨᄁ‫ݦ‬ඔቔႨ༯đ༢๤ཙႋ֥໗෿ᆴູ đ‫ܣ‬Վ༢๤ֹᄹၭ҂ ൞ đ‫ط‬൞ đၹՎ༢๤֥Ԯ‫ݦ־‬ඔྙൔႋູğ Φ (s) = 3ω 2  n s2 + 2 ξω ns + ω 2 n ⎧ c(t p ) − c(∞) ×100% 4 − 3 ×100% − ξπ ⎪⎪M c(∞) 3 1−ξ 2 ⎨ % = = = 33% =e ×100% p  ⎪ ωn 1−ξ 2 ⎪⎩t p = 0.1(s) = π ࢳ֤ğξ = 0.33 đωn = 33.2(rad / s) đԮ‫ݦ־‬ඔູğ Φ ( s ) = 3306 .72  s 2 + 22s + 1102 .4 ২ ഡ༢๤ೂ༯๭෮ൕđೂ‫ݔ‬ေ౰༢๤ֹӑ‫ט‬ਈ֩Ⴟ đ‫ڂ‬ᆴൈࡗ֩Ⴟ Tđ൫ಒ‫ק‬ᄹၭ K1 ‫ބ‬෎ ؇ّঌ༢ඔ K f đ๝ൈ࠹ෘᄝՎ K1 ‫ ބ‬K f ඔᆴ༯༢๤֥ഈശൈࡗ‫ࢫטބ‬ൈࡗb ⎧ − ξπ ⎧ξ = 0.517 1−ξ 2 ⎨⎩ωn = 4.588 ࢳğ ⎪⎨Mp % = e ×100% = 15% ⇒  1− ξ 2 = 0.8 ⎪ t p = π ωn ⎩ оߌԮ‫ݦ־‬ඔູğ Φ(s) = K1 [s2 + (1 + K1K f )s + K1 ] Ֆ‫ط‬Ⴕğ ω 2 = K1,2ξωn = 1+ K1Kf ⇒ K1 = 21.05, K f = 0.178  n ᄝՎ K1 = 21.05, Kf = 0.178 ඔᆴ༯đॖ౰֤ഈശൈࡗ‫ڂބ‬ᆴൈࡗ ⎧ t r = π −β 2 = 0 . 538 ⎪⎪ ωn 1− ξ ⎨ ⎪ 3 ⎪⎩ t s = ξω n = 1 . 234s  ২ ğႵ၂໊ᇂෛ‫׮‬༢๤đఃࢲ‫ܒ‬๭ೂ༯๭෮ൕbఃᇏ ,đ5b൫౰ğ čĎ‫ھ‬༢๤֥໭ቅୄᆒ֕௔ੱωn ĠčĎ༢๤ֹቅୄбξ ĠčĎ༢๤֥ӑ‫ט‬ਈ M p % ‫ࢫטބ‬ൈࡗ t s ĠčĎ 40 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ༢๤֥ഈശൈࡗ t r ĠčĎೂ‫ݔ‬ေ౰ ξ = 2 2 đᄝ҂‫ڿ‬эൈࡗӈඔ 5 ֥౦ঃ༯đႋᄸဢ‫ڿ‬э༢๤षߌ٢ ն༢ඔ ,Ĥ    ࢳğ༢๤ֹоߌԮ‫ݦ־‬ඔູğ Φ(s) = K = KT = 4 Ts 2 + s + K s2 + 1 T s + K T s2 + s + 4 čĎωn = K T = 2 ĠčĎξ = 1 2ωn = 0.25 Ġ čĎ M p % = e−ξπ 1−ξ ×100% = 44.4% đ t s = 3 ξωn = 6s Ġ čĎ t r = (π − β ) ωn 1 − ξ 2 = 0.94s Ġ čĎ֒ξ = 2 2 ൈđωn = 1 2ωn = 0.707 đᄵ K = ω 2 = 0.5 đॖ࡮ေડቀ‫ࢨؽ‬༢๤ቋࡄቅୄб֥ေ n ౰đсྶࢆ֮षߌ٢նПඔ , ֥ඔᆴb  41 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 11 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅č3a4 ࢫĎ ‫ࢨؽ‬༢๤֥ֆ໊Ԋࠌཙႋ‫ބ‬ֆ໊ོ௡ཙႋ ᇶေଽಸ ‫ࢨؽ‬༢๤֥ྟି‫ڿ‬೿ ӈܿ॥ᇅᄝሱ‫׮‬॥ᇅ༢๤ᇏ֥ႋႨ ۚࢨ༢๤֥ൈთ‫ٳ‬༅ ଢ֥აေ ਔࢳ‫ࢨؽ‬༢๤֥ֆ໊Ԋࠌཙႋ‫ބ‬ֆ໊ོ௡ཙႋၛࠣ‫ڿ‬೿‫ࢨؽ‬༢๤ྟି֥ਆᇕٚ‫م‬ ౰ ਔࢳӈܿ॥ᇅᄝሱ‫׮‬॥ᇅ༢๤ᇏ֥ႋႨ ਔࢳۚࢨ༢๤֥ࢨᄁཙႋaਬࠞׄ‫ࢨۚؓ҃ٳ‬༢๤ཙႋ֥႕ཙ ᅧ໤оߌᇶ֥֝ࠞׄၩၬa‫ק‬ၛ่ࠣࡱ ᅧ໤ۚࢨ༢๤֥ࢆࢨ ᇗ ׄ ა ଴ ᇗׄğۚࢨ༢๤֥ࢨᄁཙႋaоߌᇶ่֥֝ࠞׄࡱaۚࢨ༢๤֥ࢆࢨ ׄ ଴ׄğ‫ڿ‬೿‫ࢨؽ‬༢๤ྟି֥ਆᇕٚ‫م‬aਬࠞׄ‫ࢨۚؓ҃ٳ‬༢๤ཙႋ֥႕ཙ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี ē‫ࢨؽ‬༢๤֥ֆ໊ઝԊཙႋ  r(t) = δ (t) ↔ R(s) = 1đ C(s) = G(s) * R(s) = ωn2 (s 2 + 2ξω n s + ω 2 )  n čĎ໭ቅୄξ = 0  g(t) = L−1[C(s)] = L−1[ω 2 (s 2 + ω 2 )] = ωn sinωn t đ t ≥ 0  n n čĎఴቅୄ 0 ≤ ξ ≤ 1 g(t) = L−1[C(s)] = L−1[ω 2 (s2 + 2ξωns + ωn2 )] = ωn e-ξωnt sinωn 1−ξ2t đt ≥ 0 n 1−ξ 2 čĎਢࢸቅୄξ = 1  g(t) = L−1 [ C (s )] = L− 1 [ω 2 (s 2 + 2ω n s + ω 2 ) ] = ω 2 te -ω n t đt ≥ 0 n n n čĎ‫ݖ‬ቅୄξ > 1  g(t) = L−1[C(s)] = L−1[ω 2 (s2 + 2ξωns + ωn2 )] = ωn [e-(ξ − ξ 2 −1)t - e-(ξ + ξ 2 −1)t ] đ t ≥ 0  n 1−ξ 2 ē‫ࢨؽ‬ఴቅୄ༢๤ོ֥௡ཙႋ r(t) = δ (t) ↔ R(s) = 1 s2  C(s) = G(s) * R(s) = ω 2 ×1 n s2 + 2ξω n s + ω 2 s2 n c(t) = t − 2ξ +1 e−ξωntsin(ωd t + 2β ) đ t ≥ 0đωd = ωn 1 − ξ 2 đ β = arccosξ đ ωn ωd ໗෿༂ҵğ ess = 2ξ  ωn ཁಖ‫ࢨؽ‬ఴቅୄ༢๤ॖၛ۵ሶൻೆྐ‫ݼ‬э߄đ֌թᄝሢ‫֥קܥ‬༂ҵđᆃᇕ༂ҵ൓ᇔթᄝđᆺି๙ ‫טݖ‬ᆜ༢๤֥ҕඔ‫ܒࢲބ‬টࡨཬđ֌҂ିཨԢbೂ‫ݔ‬ေ൐༂ҵࡨཬđᄵсྶᄹնωn ࠇࡨཬξ đ֌ᆃဢ ቓ֥ࢲ‫ݔ‬ॖି߶൐༢๤֥‫׮‬෿ྟିᆷѓэҵđၹՎေ‫ڿ‬೿‫ࢨؽ‬༢๤֥ྟିđສສေ๙‫ݖ‬ᄹࡆ཮ᆞልᇂ ֩Ϸ‫م‬টൌགྷb ē ‫ࢨؽ‬༢๤ྟି֥‫ڿ‬೿ ‫ڿ‬೿‫ࢨؽ‬༢๤ྟି֥ӈႨ཮ᆞٚ‫م‬ᇶေႵၛ༯  ᇕb 42 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  čĎ༂ҵྐ‫֥ݼ‬б২ັ‫ٳ‬॥ᇅč1* ॥ᇅĎ  б২ັ‫ٳ‬॥ᇅđൌ࠽ഈᄝభཟ๙֡ഈࡆೆ༂ҵྐ‫֥ݼ‬б২ ັ‫֥ٳ‬॥ᇅఖđࡥӫູ 1* ॥ᇅđ॥ᇅఖ ֥Ԯ‫ݦ־‬ඔູğ G c (s) = U a (s) = Kp + Td s  E(s) ൈთіղൔູğ u a ( t ) = K pe(t) + Td de(t)  dt Φ (s) = ω 2 ( K p + Td s)  n s2 + ( 2 ξω n + T d ω 2 ) s + K pω 2 n n ࢲંğֻ၂đ༢๤ི֥֩ቅୄб‫ބ‬໭ቅୄᆒ֕௔ੱ‫׻‬ᄹࡆਔđᄝ‫ކ‬৘࿊ᄴ K p ,Td ުđི֩ቅୄб֥ ᄹࡆđࡼ߶Ⴕֹིၝᇅ༢๤֥ᆒ֕đࡨཬӑ‫ט‬ਈĠ ֻ‫ؽ‬đ༢๤Ⴎ‫ࢨؽ֥྘ׅ‬༢๤đэӮູ၂۱‫ࡆڸ‬Ⴕ၂۱ਬ֥ׄ‫ࢨؽ‬༢๤bᆃ۱‫֥ࡆڸ‬ਬׄđऎႵັ ‫ٳ‬ቔႨđॖၛ൐༢๤֥ᄠ෿ཙႋ෎؇ࡆॹb čĎൻԛਈ֥෎؇ّঌ॥ᇅč෎؇ّঌ཮ᆞĎ ॥ᇅ֥оߌԮ‫ݦ־‬ඔູğ R(s) E(s) ωn2 C(s) Φ(s) = ω 2  s(s + 2ξωn ) n s2 + (2ξωn + K f ωn2 )s + ω 2 n ི֩ቅୄб ξ ′ = ξ + 1 ωn  Kf s 2 K f ࢲંğֻ၂đջ෎؇ّঌ֥‫ࢨؽ‬༢๤ಯಖ൞‫ࢨؽ྘ׅ‬༢๤đః໭ቅୄᆒ֕௔ੱીႵ‫ڿ‬эĠ ֻ‫ؽ‬đႵֹིิۚਔ༢๤֥ቅୄбđ༢๤֥ӑ‫ט‬ਈॖၛૼཁࡨཬĠ ֻ೘đႮႿ ωn Ќӻ҂эđ‫ط‬ቅୄбᄹնđՖ‫ط‬༢๤֥‫ࢫט‬ൈࡗ ts эཬđᄵ༢๤֥ཙႋ෎؇֤֞ࡆ ॹb ২ ğၘᆩ෎؇ّঌ༢๤đေ౰༢๤֥ӑ‫ט‬ਈູ đt p = 1૰đ൫౰ K ‫ ބ‬K f ֥ᆴđ೏Ќӻ , ҂эđ ‫ط‬౼ཨ෎؇ّঌč K f = 0 Ďđᄜ౰ M p % ֥ᆴb ࢳğоߌԮ‫ݦ־‬ඔູğ Φ(s) = K  s 2 + (1 + KK f )s + K ཁಖႵğ R(s) E(s) K C(s) s(s + 1) ⎪⎧2ξωn =1+ KK f  K K fs ⎪⎩⎨ω 2 = n  Ⴎğ ⎪⎪⎧M p% = e −ξπ 1−ξ 2 ×100% đ౰֤ğ ⎧ξ = 0.456 (rad / s)  ⎨ = π 1 ⎪⎨ωn = 3.53 ⎪t p 1− ξ = ⎪⎩ts = 1.875s ⎪⎩ ωn 2 Ⴟ൞Ⴕğ K = ω 2 = 12.5đK f = (2ξωn − 1) K = 0.178  n ၇ऌีଢေ౰Ќӻ , ҂эđ਷ K f = 0 đࠧ໭෎؇ّঌൈđႵğ 43 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Φ(s) = s2 12.5 ⎧ξ = 0.14 (rad / s) đ ⎧M p % = ****%  + s + 12.5 đ ⎩⎨ωn = 3.53 ⎨ = 6.07s ⎩t s бࢠॖ࡮đҐႨ෎؇ّঌުđႮႿቅୄб֥ᄹࡆđ൐֤༢๤֥ӑ‫ט‬ਈննཬႿીႵ෎؇ّঌ֥౦ঃĠ ๝ൈ‫ࢫט‬ൈࡗ္ࡨഒđՖ‫ط‬൐༢๤֥ཙႋ෎؇ࡆॹb ۚࢨ༢๤֥ൈთ‫ٳ‬༅ ēۚࢨ༢๤֥ֆ໊ࢨᄁཙႋ ഡ၂Ϯ॥ᇅ֥Ԯ‫ݦ־‬ඔູğ G(s) = b0sm + b1sm-1 + + b m-1s + b m = Kg (s − z1 )((s − z 2 ) (s − zm )  a 0s n + a1s n-1 + + a n-1s + a n (s − p1 )(s − p2 ) (s − pn ) ᄝԚ൓่ࡱູਬ‫ބ‬ֆ໊ࢨᄁൻೆྐ‫ݼ‬ቔႨ༯đ౰ఃൻԛཙႋđ֤֞ğ (s − z1)((s − z2 ) (s − zm ) × 1 = A0 + n Ai  (s − p1)(s − p2 ) (s − pn ) s s i=1 s − pi ∑ C(s)= = G(s)R(s) Kg n ∑ Ⴟ൞༢๤֥ֆ໊ࢨᄁཙႋູğ c ( t ) = A 0 + A i e pit  i =1 ‫ٳ‬༅ఃཙႋॖ࡮đ༢๤֥ཙႋ൞Ⴎ၂༢ਙᆷඔ‫ݦ‬ඔčᄎ‫׮‬ଆ෿Ď֥‫ܒބ‬Ӯđఃᇏૄ၂ཛ෮ᅝ֥oб ᇗpࣼႮ਽ඔ Ai ֥նཬथ‫ק‬đ‫ ط‬Ai ֥նཬႮਬׄ‫܋ׄࠞބ‬๝টथ‫ק‬đ ପུჹ৖ቕѓჰ֥ׄࠞׄ෮ؓႋ֥ᄎ‫ྙ׮‬෿ؓࢨᄁཙႋ֥႕ཙ‫ཬޓ‬b ࢲંğᄝ‫ٳ‬༅ۚࢨ༢๤ൈđؓႿ‫ކژ‬ഈඍਆᇕ่ࡱ֥ࠞׄॖၛቔູՑေၹ෍‫੻ޭط‬đՖ‫ط‬Ϝۚࢨ༢ ๤ࢆࢨ໊֮ࢨ༢๤ট࣍රb ēоߌᇶ֝ࠞׄ‫ࢨۚބ‬༢๤֥࣍ර čĎᄝ໗‫ࢨ֥ۚק‬༢๤ᇏđؓఃൈࡗཙႋఏᇶ֝ቔႨ֥оߌࠞׄӫູоߌᇶ֝ࠞׄb čĎоߌᇶ่֥֝ࠞׄࡱğ ֻ၂ğᄝ 4 ቐ϶௜૫ഈ৖ྴᇠቋ࣍đ౏ఃᇛຶીႵਬ֥ׄࠞׄĠ ֻ‫ؽ‬ğఃൌ҆৖ྴᇠ֥ए৖൞ః෱ࠞׄ৖ྴᇠ֥ए৖  Пၛഈb čĎۚࢨ༢๤֥࣍ර ۚࢨ༢๤ࡥ߄ࢆࢨູ֮ࢨ༢๤֥ჰᄵ‫ބ‬ऎุ҄ᇧğ ჰᄵᄝ‫҃ٳׄࠞކژ‬ေ౰֥భิ༯đсྶЌᆣࡥ߄భުԮ‫ݦ־‬ඔ֥໗෿ᆴေཌྷ֩b ҄ᇧֻ၂đಒ‫ק‬༢๤֥оߌᇶ֝ࠞׄĠֻ‫ؽ‬đࡼۚࢨ༢๤֥षߌԮ‫ݦ־‬ඔࠇоߌԮ‫ݦ־‬ඔཿູൈ ࡗӈඔіղྙൔĠֻ೘đޭ੻ཬൈࡗӈඔཛb ২ğଖۚࢨ༢๤֥оߌԮ‫ݦ־‬ඔູğ Φ(s) = (s2 + 10ωn2 + p)  2ξωns + ωn2 )(s ೂ‫໊֥ׄࠞݔ‬ᇂડቀ p ξωn > 5  ᄵ‫ھ‬༢๤֥оߌᇶູ֝ࠞׄğ s1,2 = −ξωn ± jωn 1 − ξ 2 đႿ൞༢๤ॖၛࢆࢨູğ Φ(s) = 10ωn2 = 10ωn2 ≈− 10ωn2  + 2ξωns + ωn2 )(s + + 2ξωns + ωn2 ) (s2 p) p(s2 + 2ξωn s + ωn2 )( 1 s + 1) p(s2 p ࡥ߄భ֥໗෿ᆴčֆ໊ࢨᄁྐ‫ݼ‬Ď c(∞) = lim sC(s) = lim sΦ(s)R(s) = lim s • 1 • (s2 + 10ωn2 + p) = 10  s 2ξωns + ωn2 )(s p s→0 s→0 s→0 ࡥ߄ު֥໗෿ᆴčֆ໊ࢨᄁྐ‫ݼ‬Ď c(∞) = lim sC(s) = lim sΦ(s)R(s) = lim s • 1 • p(s2 10ωn2 = 10  s + 2ξωns + ωn2 ) p s→0 s→0 s→0 44 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 12 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅č5 ࢫĎ ᇶေଽಸ ༢๤໗‫ٳྟק‬༅ ଢ֥აေ ᅧ໤໗‫קބ୑ۀ֥ק‬ၬa༢๤໗‫֥ק‬ԉေ่ࡱ ౰ ᅧ໤ཌྟ༢๤֥সථ஑ऌࠣႋႨᇏ֥ห൹໙ี ᇗ ׄ ა ଴ ᇗׄğ༢๤໗‫֥ק‬ԉေ่ࡱaসථ஑ऌ ׄ ଴ׄğসථ஑ऌࠣᄝ༢๤໗‫ٳྟק‬༅ᇏ֥ႋႨ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 3-4 3-5 ཌྟ༢๤֥໗‫ٳྟק‬༅ ၂۱ཌྟ॥ᇅ༢๤ି‫ܔ‬ᆞӈ‫۽‬ቔ֥൮ေ่ࡱđࣼ൞෱сྶ൞໗‫֥ק‬b ē༢๤ᄎ‫֥׮‬໗‫ྟק‬ ໗‫ྟק‬૭ඍğೂ‫ݔ‬ཌྟ༢๤൳֞ಠ‫֥׮‬ቔႨ‫ط‬൐Ф॥ਈӁളொҵđ֒ಠ‫׮‬ཨാުđෛሢൈࡗ֥๷ ၍đ‫ھ‬ொҵᇯࡶࡨཬѩ౴ႿਬđࠧФ॥ਈ߭֞ჰট֥௜‫۽ޙ‬ቔሑ෿đᄵӫ‫ھ‬༢๤໗‫ק‬bّᆭđ೏ᄝಠ ‫֥׮‬႕ཙ༯đ༢๤֥Ф॥ਈෛሢൈࡗ֥๷၍‫ؿط‬೛đᄵӫ༢๤҂໗‫ק‬b ೏ᄠ෿ཙႋିཨാ֥đᄵ༢๤൞໗‫֥ק‬đ೏ᄠ෿ཙႋ҂ିཨാđᄵ༢๤҂໗‫ק‬bཌྟ༢๤֥໗‫ק‬ ྟđა༢๤֥ൻೆྐ‫ݼ‬aԚ൓ሑ෿न໭ܱđ෱൞༢๤֥‫ܥ‬ႵЧᇉඋྟđປಆ౼थႿ༢๤֥ࢲ‫ބܒ‬ҕඔb ēཌྟ॥ᇅ༢๤໗‫֥ྟק‬ԉ‫ٳ‬сေ่ࡱ ࡌഡ༢๤֥Ԛ൓่ࡱູਬđຓ҆ࠗৣູൻೆྐ‫ ݼ‬r(t) = δ (t) ğ ྽‫ݼ‬ ઝԊ‫ݦ‬ඔࠞཋᆴ ઝԊཙႋකࡨ౦ঃ ໗‫ק‬ሑ෿  lim g(t) = 0  කࡨ ༢๤໗‫ק‬ t→∞  lim g(t) = ∞  ‫ؿ‬೛ ༢๤҂໗‫ק‬ t→∞  lim g(t) = C  ֩‫ږ‬ᆒ֕ ༢๤ਢࢸ໗‫ק‬ t→∞ ཌྟ༢๤໗‫ק‬ေડቀ lim g(t) = 0 ่֥ࡱđൌ࠽ഈ౼थႿఃหᆘ۴đ္ࠧ༢๤оߌԮ‫ݦ־‬ඔ֥ࠞׄb t→∞ ཌྟ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱູğ ༢๤ັ‫ٳ‬ٚӱ֥หᆘ۴֥ಆ҆۴‫׻‬൞‫ڵ׻‬ൌඔࠇൌູ҆‫گ֥ڵ‬ඔđ္ࠧđ༢๤оߌԮ‫ݦ־‬ඔ֥ࠞ ׄन໊Ⴟ 4 ௜૫֥ቐ϶௜૫b ֒หᆘ۴ԛགྷᆞൌඔࠇൌູ҆ᆞ֥‫گ‬ඔࠇႵࠞׄ‫҃ٳ‬Ⴟ 4 ௜૫֥Ⴗ϶௜૫ൈđཌྟ༢๤ູ҂໗‫ק‬Ġ ֒หᆘ۴ԛགྷՂྴඔࠇႵ໊ࠞׄႿ 4 ௜૫֥ྴᇠൈđཌྟ༢๤ູਢࢸ໗‫ק‬b ২ ğ༢๤֥оߌԮ‫ݦ־‬ඔູğ Φ(s) = 2(s −1) (s + 1)(s + 2) ஑љ༢๤໗‫ྟק‬b ࢳğ༢๤֥оߌࠞׄ‫ٳ‬љູ p1 = −2 a p2 = −1đ෮ၛ༢๤໗‫ק‬b ২ ğၘᆩཌྟ༢๤֥оߌหᆘٚӱູ DB (s) = (s + 10)(s2 + 16) = 0 đ൫஑љ༢๤֥໗‫ྟק‬b ࢳğႮ۳‫֥ק‬оߌหᆘٚӱ DB (s) = (s + 10)(s2 + 16) = 0 đॖၛ౰֤หᆘ۴ູğ s1 = −10 đ s2 = ± j4 đ၇ऌཌྟ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱॖᆩ༢๤ູਢࢸ໗‫ק‬b ২ ğֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູğ G0 (s) = 2 s(s + 3) đ൫஑љоߌ༢๤֥໗‫ྟק‬b ࢳğоߌ༢๤֥Ԯ‫ݦ־‬ඔູ Φ(s) = 2 (s2 + 3s + 2) đఃоߌູࠞׄ p1 = −2 a p2 = −1 ෮ၛ༢๤ ໗‫ק‬b ēսඔ໗‫ק‬஑ऌ ༢๤หᆘٚӱ֥۲ཛ༢ඔა༢๤֥໗‫ྟק‬ᆭࡗ၂‫ק‬թᄝሢଖᇕଽᄝ֥৳༢b 45 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  + an−1s + an = 0 đѩ၇ᅶၛ༯֥ٚ‫ܒم‬ᄯ সථ໗‫ק‬஑ऌ ഡ࿹࣮֥ཌྟ༢๤֥หᆘٚӱູğ a0sn + a1sn−1 + ၂۱і۬đᆃ۱і۬ӫູসථіđ‫ܒ‬ᄯٚ‫م‬ೂ༯b sn a0 a2 a4 s n−1 a1 a3 a5 s n−2 a2a1 − a0a3 b1 = b2 = a4a1 − a0a5 b3 a6 a1 − a0a7 a1 a1 a1 s n−3 c1 = b1a3 − a1b2 c2 = b1a5 − a1b3 c3 = b1a7 − a1b4  b1 b1 b1 s2 d1 d2 d3 e1 e2 s1 f1 s0 ****VUI ஑ऌğཌྟ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱ൞ ****VUI іֻ၂ਙ֥෮Ⴕჭ෍‫ݼژ‬҂‫ڿ‬эđ౏‫ڿݼژ‬э֥ Ցඔູหᆘ۴໊Ⴟ 4 Ⴗ϶௜૫֥۱ඔb ২ ğษં‫ࢨؽ‬a೘ࢨ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱb ‫ࢨؽ‬༢๤ğ a0s2 + a1s + a2 = 0 đ‫ܒ‬ᄯ ****VUI іğ s2 a0 a2 s1 a1 0 Ⴎ ****VUI іѩ၇ऌসථ஑ऌॖᆩđ‫ࢨؽ‬༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱູğ s0 a1a2 − a0 × 0 = a2 a1 a0 > 0, a1 > 0, a2 > 0 b ೘ࢨ༢๤ğ a0s3 + a1s2 + a2s + a3 = 0 đ‫ܒ‬ᄯ ****VUI іğ s3 a0 a2 s2 a1 a3 s1 a1a 2 − a0a3 đႮ ****VUI іѩ၇ऌসථ஑ऌॖᆩđ೘ࢨ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱູğ s0 a1 0 a3 a0 > 0, a1 > 0, a2 > 0, a3 > 0 đ౏ a1a2 > a0a3 b ২ ğഡоߌ༢๤֥หᆘٚӱູ DB (s) = s4 + 2s3 + 3s2 + 4s + 5 = 0 đ൫஑љః໗‫ྟק‬b ࢳğ‫ܒ‬ᄯসථі s4 1 35 s3 2 s2 1 4 5 Ⴎসථіॖ࡮đఃֻ၂ਙჭ෍֥‫ؿݼژ‬ളਔ  Ց‫ڿ‬эđ෮ၛ‫ھ‬༢๤൞҂໗‫֥ק‬đ౏Ⴕ  s1 − 6 s0 5 ۱หᆘ۴໊Ⴟ 4 Ⴗ϶௜૫b ࠫᇕห൹౦ঃ ֻ၂ᇕห൹౦ঃğসථіᇏֻ၂ਙ֥ଖ၂ྛჭ෍ԛགྷਬჭ෍đ‫֥ྛھط‬ఃჅჭ෍҂ಆູਬb ࢲંğ֒ԛགྷᆃᇕ౦ঃൈđඪૼ༢๤หᆘٚӱऎႵᆞൌඔ۴ࠇՂྴ۴đ༢๤҂໗‫ࠇק‬ਢࢸ໗‫ק‬b ԩ৘ٚ‫م‬ğॖၛႨ၂۱ཬᆞඔ ε টսูପ۱ਬჭ෍đಖު࠿࿃‫ܒ‬ᄯ༯ಀđѩ਷ ε → 0+ đ஑љֻ၂ਙ ჭ෍‫ڿݼژ‬э֥Ցඔb 46 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ২ ğഡ༢๤֥หᆘٚӱູ s3 − 3s + 2 = 0  ࢳğ‫ܒ‬ᄯসථіೂ༯đѩቔห൹ԩ৘b s3 1 −3 s2 0 = ε 2 đႮসථіॖ࡮đఃֻ၂ਙჭ෍֥‫ڿݼژ‬эਔ  Ցđ‫ܣ‬༢๤҂໗‫ק‬đ౏ऎႵ  ۱ห − 3ε − 2 0 s1 ε s0 2 ᆘ۴໊Ⴟ 4 Ⴗ϶௜૫b ֻ‫ؽ‬ᇕห൹౦ঃğসථіᇏԛགྷଖ၂ྛჭ෍ಆູਬb ࢲંğԛགྷᆃᇕห൹౦ঃđඪૼթᄝሢ֩ᆴّ‫֥ݼ‬ൌඔ۴ࠇӮؓԛགྷ֥Ղྴ۴ࠇؓӫႿ 4 ௜૫ቕѓ ᇠჰ֥ׄ୽ඔؓ‫܋‬ᣢ‫گ‬ඔ۴b༢๤൞҂໗‫ࠇ֥ק‬ਢࢸ໗‫ק‬b ԩ৘ٚ‫م‬ğ০Ⴈಆਬྛഈ၂ྛ֥ჭ෍ࠣཌྷႋ֥ࢨՑ‫ܒ‬ᄯ‫ڣ‬ᇹ‫؟‬ཛൔ F (s) đѩၛ dF(s) ds ۲༢ඔս ูಆਬྛჭ෍đಖު࠿࿃‫ܒ‬ᄯসථі֥ఃჅ҆‫ٳ‬b ২ ğഡ༢๤֥หᆘٚӱູ s5 + 2s4 + 24s3 + 48s2 − 25s − 50 = 0  ࢳğ‫ܒ‬ᄯসථіೂ༯đѩቔห൹ԩ৘b s5 1 24 − 25 s4 2 48 s3 0 → 8 0 → 96 − 50 ⎧ F(s) = 2s4 + 48s2 − − 50 ⎪ dF (s) = 8s 3 + 96s s2 24  ⎨ 50  s1 112.7 ⎪⎩ ds s0 − 50 ⇐ ႮႿসථіֻ၂ਙჭ෍֥‫ؿݼژ‬ള‫ڿ‬эđ෮ၛ༢๤сಖ҂໗‫ק‬bൌ࠽ഈॖၛ౰֤‫ھ‬༢๤֥  ۱หᆘ ۴‫ٳ‬љູ s1,2 = ±1, s3 = −2, s4,5 = ± j5 b ēսඔ஑ऌ֥ႋႨ ২ ğഡֆّ໊ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G0 (s) = K s(s + 1)(s + 5) đ൫‫ٳ‬༅оߌ༢๤໗‫ק‬ൈ٢ նПඔ , ֥౼ᆴٓຶb ࢳğоߌหᆘٚӱູ s(s + 1)(s + 5) + K = 0 ⇒ s3 + 6s2 + 5s + K = 0  Ⴎ೘ࢨ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱॖၛ֤֞đ֒ 0 < K < 30 ൈ‫ھ‬оߌ༢๤൞໗‫֥ק‬b ২ ğഡֆّ໊ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G0 (s) = K s(Tms + 1)(T f s + 1) đ౏Tm > 0,Tf > 0 đ൫ ಒ‫ק‬оߌ༢๤໗‫ק‬ൈ٢նПඔ , ֥౼ᆴٓຶb ࢳğ༢๤֥оߌԮ‫ݦ־‬ඔູ Φ(s) = s3 K )s2  + Tf TmT f + (Tm + s + K ༢๤֥оߌหᆘٚӱູğTmTf s3 + (Tm + Tf )s2 + s + K = 0  ೘ࢨ༢๤໗‫ູࡱ่֥ק‬ğTm + Tf > KTmT f ,K > 0 ⇒ 0 < K < Tm + Tf =1 +1  TmT f Tm Tf ২ ğഡֆّ໊ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G0 (s) = K s(s + 1)(s + 5) đ೏ေ౰оߌหᆘ۴न໊Ⴟ 4 ௜૫ s = −0.1Լཌ֥ቐҧđ൫‫ٳ‬༅оߌ༢๤໗‫ק‬ൈ , ֥౼ᆴٓຶb ࢳğоߌหᆘٚӱູ s(s + 1)(s + 5) + K = 0 ⇒ s3 + 6s2 + 5s + K = 0  ቔཌྟэߐđ਷ s = z − 0.1đѩսೆഈඍоߌหᆘٚӱđᆜ৘֤֞ DB (s) = s3 + 5.7s2 + 3.82s + (K − 0.441) = 0  ۴ऌসථ஑ऌđॖၛ֤֞ડቀ่ࡱൈ , ֥౼ᆴٓຶđ 0.441< K < 21.39 b 47 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 13 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅č6 ࢫĎ ᇶေଽಸ ཌྟ༢๤֥༂ҵ‫ٳ‬༅ ଢ֥აေ ᅧ໤༂ҵ֥‫קބ୑ۀ‬ၬa۳‫ק‬ቔႨ༯໗෿༂ҵ֥࠹ෘࠣؓႋ۲ᇕൻೆ༯༂ҵ༢ඔ֥࠹ෘ ౰ ᅧ໤ಠ‫׮‬ቔႨ༯໗෿༂ҵ֥࠹ෘ ਔࢳิۚ༢๤॥ᇅࣚ؇֥ծീ ᇗ ׄ ა ଴ ᇗׄğ۳‫ק‬ቔႨ༯໗෿༂ҵ֥࠹ෘࠣؓႋ۲ᇕൻೆ༯༂ҵ༢ඔ֥࠹ෘ ׄ ଴ׄğಠ‫׮‬ቔႨ༯໗෿༂ҵ֥࠹ෘ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 3-7 3-11 ॥ᇅ༢๤֥໗෿༂ҵ‫ٳ‬༅ ē༂ҵ֥‫ק‬ၬ e(t) = r(t) − b(t) ====⇒ E(s) = R(s) − B(s)  e(t) = r(t) − c(t) =====⇒ E(s) = R(s) − C(s)  ē໗෿༂ҵ ॥ ᇅ ༢ ๤ ֥ ༂ ҵ ‫ ݦ‬ඔ e(t) ္ ॖ ၛ ‫  ູ ٳ‬۱ ‫ ٳ‬ਈ đ ࠧ ᄠ ෿ ‫ ٳ‬ਈ ets (t) ‫ ބ‬໗ ෿ ‫ ٳ‬ਈ ess (t) e(t) = ets (t) + ess (t) ؓႿ໗‫֥ק‬༢๤đ֒ൈࡗ t ౴ཟ໭౫նൈđᄠ෿҆‫ ٳ‬ets (t) ҂‫ק‬౴Ⴟਬđ‫ط‬ ᆺ਽༯໗෿҆‫ ٳ‬ess (t) đ‫ط‬༢๤֥໗෿༂ҵॖၛཿູ ess = lim e(t)  t→∞ ē༂ҵԮ‫ݦ־‬ඔ  čĎႮ r(t) ቔႨ༯֥༂ҵ‫ݦ‬ඔ ਷ n(t) = 0 đ Φ ER (s) = 1+ 1 = 1+ 1  G0 (s) G1(s)G2 (s)H (s) čĎႮ n(t) ቔႨ༯֥༂ҵ‫ݦ‬ඔ ਷ r(t) = 0 đ Φ EN (s) = − + G2 (s)H (s) (s)  G1(s)G2 (s)H 1 čĎ༢๤ሹ༂ҵ E(s) = ER (s) + EN (s) = Φ ER (s)R(s) + ΦEN (s)N (s)  ໗෿༂ҵ࠹ෘ ླေᆷԛ֥൞đᆺႵ֒༢๤໗‫ק‬ൈđ࿹࣮໗෿༂ҵҌႵၩၬđၹՎđᄝ࠹ෘ༢๤֥໗෿༂ҵᆭభđ сྶ஑؎༢๤൞໗‫֥ק‬b Ⴎ -BQMBDF эߐ֥ᇔᆴ‫ק‬৘đॖ֤֞໗෿༂ҵ֥࠹ෘ ess = lim e(t) = lim sE(s)  t→∞ s→0 ২ ğഡ॥ᇅ༢๤֥໗෿༂ҵೂ༯๭෮ൕđ֒ൻೆྐ‫ູݼ‬ֆ໊ོ௡‫ݦ‬ඔൈđ൫౰༢๤֥໗෿༂ҵĠ ູ൐໗෿༂ҵཬႿ  ႋ‫ھ‬ೂ‫טޅ‬ᆜ , ᆴb 48 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  R(s) E(s) K (0.5 s + 1) C(s) ࢳğ஑љ༢๤֥໗‫ྟק‬đоߌ หᆘٚӱູ s ( s + 1)( 2 s + 1) 2s3 + 3s2 + (1 + 0.5K )s + k = 0  ༢๤໗‫ק‬ൈ , ֥౼ᆴٓຶ൞ğ 0 < K < 6  ༢๤֥༂ҵ‫ݦ‬ඔູğ Φ ER (s) = E(s) = 1 = s(s + 1)(2s + 1)  R(s) K (0.5s + 1) 1+ s(s + 1)(2s + 1) + K (0.5s + 1) s(s + 1)(2s + 1) ༢๤֥໗෿༂ҵູğ ess = lim sE(s) = lim sΦ ER (s) R(s) = lim s • s(s + 1)(2s + 1) • 1 = 1  s→0 s(s + 1)(2s + 1) + K (0.5s + 1) s2 K s→0 s→0 ູ൐໗෿༂ҵཬႿ đᄵᆺေડቀ ess = 1 < 0.1đࠧ K > 10  K ֌Ⴎ༢๤໗‫ࡱ่֥ק‬ᆩ֡đ֒ K > 10 ൈđ༢๤ࡼэູ҂໗‫ק‬đ‫ܣ‬໭‫م‬๙‫ݖ‬࿊ᄴ , ᆴটղ֞໗෿༂ ҵཬႿ ֥ေ౰b ໗෿༂ҵ‫ٳ‬༅ ഡ༢๤оߌԮ‫ݦ־‬ඔೂ༯đѩіൕູ݂၂߄čൈࡗӈඔĎྙൔ G(s) = b0sm + b1sm-1 + + bm-1s + bm = K (τ1s + 1)(τ 22s2 + 2τ 2ς 2s + 1) = K G0 (s)  a 0s n + a1sn-1 + + a n-1s + a n sγ (T1s + 1)(T22s2 + 2T2ξ2s + 1) sγ ఃᇏğ G0 ( s) (τ1s + 1)(τ 22s2 + 2τ 2ς 2s + 1) đ౏ཁಖႵ lim G0 (s) = G0 (0) = 1 đK ູ༢๤षߌ٢ (T1s + 1)(T22s2 + 2T2ξ2s + 1) s→0 նПඔđ τ1aaτ a đT1aT2a ູൈࡗӈඔđ γ ູࠒ‫ࢫߌٳ‬ඔଢb 2 ๙ӈ۴ऌ༢๤षߌԮ‫ݦ־‬ඔ෮‫֥ࢫߌٳࠒݣ‬ඔଢ γ টؓ༢๤ࣉྛ‫ٳ‬োđ֒ γ ູ 0đ1đ2đ3đlൈđ ‫ٳ‬љ‫ק‬ၬ༢๤֥྘љູ 0 ྘aú྘aû྘aü྘đl/ ྘đ γ ္ӫູ༢๤֤໭ҵ؇ࢨඔb ⎧ൻೆྐ‫ݼ‬r(t)ྙൔ ܱ༢֞༢๤໗෿༂ҵ֥ၹ෍Ⴕğ༢๤ ⎪⎨षߌ٢նПඔK  ⎪⎩षߌԮ‫ݦ־‬ඔᇏࠒ‫֥ࢫߌٳ‬ඔଢγ čĎൻೆྐ‫ູݼ‬ֆ໊ࢨᄁ‫ݦ‬ඔ‫࣡ބ‬෿໊ᇂ༂ҵ༢ඔ ഡൻೆྐ‫ ູݼ‬r(t) = 1(t) ⇔ R(s) = 1 s  ໗෿༂ҵğ ess = lim sE(s) = lim s • 1 • 1 = 1  1+ GK (s) s s→0 s→0 1+ lim GK (s) s→0 ਷ğ K p = lim GK (s)  s→0 Kp ູ༢๤࣡෿໊ᇂ༂ҵ༢ඔđ༢๤ᄝֆ໊ࢨᄁ‫ݦ‬ඔቔႨ༯֥໗෿༂ҵູğ ess = 1  1+ Kp ໗෿༂ҵູਬ֥༢๤ӫູ໭ҵ༢๤đ໗෿༂ҵູႵཋᆴ֥༢๤ӫູႵҵ༢๤b čĎൻೆྐ‫ູݼ‬ֆ໊ོ௡‫ݦ‬ඔ‫࣡ބ‬෿෎؇༂ҵ༢ඔ ഡൻೆྐ‫ ູݼ‬r(t) = t1(t) ⇔ R(s) = 1 s2  ໗෿༂ҵğ ess = lim sE(s) = lim s • 1 • 1 = 1 s2 s→0 s→0 1 + GK (s) lim sGK (s) s→0 ਷ğ Kv = lim sGK (s)  s→0 Kv ູ༢๤֥࣡෿෎؇༂ҵ༢ඔđ༢๤ᄝֆ໊ོ௡‫ݦ‬ඔቔႨ༯֥໗෿༂ҵູğ ess = 1  Kv 49 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  čĎൻೆྐ‫ູݼ‬ֆ໊ࡆ෎؇‫ݦ‬ඔ‫࣡ބ‬෿ࡆ෎؇༂ҵ༢ඔ ഡൻೆྐ‫ ູݼ‬r(t) = 1 t 2 ⇔ R(s) = 1 s3  2 ໗෿༂ҵğ ess = lim sE(s) = lim s • 1 • 1 = 1 s→0 1 + GK (s) s3 s→0 lim s 2GK ( s) s→0 ਷ğ Ka = lim s 2GK (s)  s→0 Ka ູ༢๤֥࣡෿ࡆ෎؇༂ҵ༢ඔđ༢๤ᄝֆ໊ࡆ෎؇‫ݦ‬ඔቔႨ༯֥໗෿༂ҵູğ ess = 1  Ka čĎൻೆྐ‫ູݼ‬ֆ໊ࢨᄁaོ௡aࡆ෎؇ྐ‫ݼ‬ൈ֥໗෿༂ҵ ഡൻೆྐ‫ ູݼ‬r(t) = 1 + t + 1 t 2 ⇔ R(s) = 1 + 1 + 1  2 s s2 s3 ০Ⴈཌྟ༢๤֥‫ࡆן‬ჰ৘đॖ֤༢๤֥໗෿༂ҵູ ess =1 + 1 + 1  1+ Kp Kv Ka ༢ ࣡෿༂ҵ༢ඔ ࢨᄁྐ‫ݼ‬ ོ௡ྐ‫ݼ‬ ࡆ෎؇ྐ‫ݼ‬ ๤ r(t) = A1(t)  r(t) = Bt1(t)  r(t) = Ct 2 2  ྘ Kp  ໊ᇂ༂ҵ ෎؇༂ҵ ࡆ෎؇༂ҵ љ Kv  Ka  ess = A (1 + K p )  ess = B Kv  ess = C Ka   ,   A (1+ K)  ∞ ∞  ∞  ,   B K   ∞  ∞  ,   C K  ∞  ∞  ∞     ‫ٳ‬༅ࢲંğčĎ༢๤֥໗෿༂ҵაൻೆྐ‫ݼ‬ႵܱĠ čĎ༢๤֥໗෿༂ҵაषߌ٢նПඔ , ࠎЧӮّбܱ༢bؓႿႵҵ֥༢๤đ, ᆴᄀնđ໗෿༂ҵ ᄀཬđ֌๝ൈ༢๤֥໗‫ྟק‬эҵĠ čĎ༢๤֥໗෿༂ҵაषߌԮ‫ݦ־‬ඔ֥ࠒ‫ࢫߌٳ‬ඔν Ⴕܱbࠒ‫ࢫߌٳ‬ඔᄹࡆđ໗෿༂ҵࡨཬđ֌ ๝ൈ༢๤֥໗‫ྟק‬эҵb ২ ğ1% ॥ᇅ༢๤ೂ༯෮ൕđൻೆྐ‫ ູݼ‬r(t) = 1 + t + 1 t 2 đ൫ቔ໗‫ٳྟק‬༅‫ބ‬໗෿༂ҵ‫ٳ‬༅b 2  R(s) K 1 (τs + 1) Km C(s)  s 2 (Tm s + 1)   ࢳčĎ໗‫ٳྟק‬༅ğ ༢๤оߌหᆘٚӱ Tms3 + s2 + Km K1τs + Km K = 0  ۴ऌসථ஑ऌđູ൐оߌ༢๤໗‫ק‬đсྶડቀTm > 0, Km > 0, K1 > 0,τ > 0,τ > Tm  čĎ໗෿༂ҵ‫ٳ‬༅ ༢๤षߌԮ‫ݦ־‬ඔູ GK (s) = K1Km (τs + 1) s2 (Tms + 1)  ཁಖູ  ྘༢๤đ‫ܣ‬ః࣡෿༂ҵ༢ඔ‫ٳ‬љູ K p = ∞, Kv = ∞, Ka = K1Km  ༢๤໗෿༂ҵູ ess = ess1 + ess2 + ess3 = 1 K1Km  ২  ğ ၘ ᆩ ֆ ໊ ّ ঌ ॥ ᇅ ༢ ๤ ֥ ष ߌ Ԯ ‫ ݦ ־‬ඔ ູ GK (s) = 10 s(T1s +1)(T2s +1) đ ൔ ᇏ T1 = 0.1s,T2 = 0.5s đ೏ൻೆྐ‫ ູݼ‬r(t) = 2 + 0.5t đ൫౰༢๤֥໗෿༂ҵb 50 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  ࢳğоߌหᆘٚӱT1T2s3 + (T1 + T2 )s2 + s + 10 = 0 ۴ऌসථ஑ऌđ༢๤໗‫ູࡱ่֥ק‬T1 + T2 > 10T1T2 ⇒ 0.1+ 0.5 = 0.6 > 10× 0.1× 0.5 = 0.5 ཁಖđડቀ༢๤໗‫ࡱ่֥ק‬đ෮ၛоߌ༢๤໗‫ק‬b षߌԮ‫ݦ־‬ඔ GK (s) = 10 s(T1s +1)(T2s +1)  ཁಖ൞  ྘༢๤đః࣡෿໊ᇂ༂ҵ༢ඔa࣡෿෎؇༂ҵ༢ඔ‫ٳ‬љູ K p = ∞, Kv = 10  ෮ၛđ༢๤໗෿༂ҵູ ess = ess1 + ess2 = 0 + 0.5 10 = 0.05  ēಠ‫ݼྐ׮‬༂ҵ‫ٳ‬༅ ಠ‫ݼྐ׮‬ቔႨ༯֥໗෿༂ҵđّ႘ਔ༢๤ॆ‫ۄ‬ಠ֥ି৯b৘མ౦ঃ༯đಠ‫׮‬Ӂള֥༂ҵᄀཬᄀ‫ݺ‬b  Ⴎಠ‫׮‬Ӂള֥༂ҵđॖၛіൕູ ਷ R(s) = 0  EN (s) = −CN (s)H (s) = − + G2 (s)H (s) N (s)  G1(s)G2 (s)H 1 (s) Ⴎಠ‫׮‬ႄఏ֥໗෿༂ҵູ essn = lim sE(s) = − lim s • G2 (s)H (s) • N (s)  s→0 1+ G1(s)G2 (s)H (s) s→0 ēࡨཬࠇཨԢ໗෿༂ҵ֥ծീ ࡨཬࠇཨԢ໗෿༂ҵ֥ծീᇶေႵ čĎб২ࠒ‫ٳ‬č1*Ď॥ᇅ   ∫ ॥ᇅఖඔ࿐ଆ྘U a (s) = (K p + Ki ⇒ = + t đ Ki ࠒ‫ٳ‬ൈࡗӈඔb s )E(s) ua (t) K p e(t ) Ki 0 e(t)dt ཁಖ॥ᇅఖ֥ൻԛྐ‫ݼ‬൞༂ҵྐ‫ ݼ‬e(t) б২ ࠒ‫ٳ‬đ‫طܣ‬ӫູб২ࠒ‫ٳ‬॥ᇅb षߌԮ‫ݦ־‬ඔ GK (s) = ωn2 (K ps + Ki ) đν = 2 đоߌԮ‫ݦ־‬ඔ Φ(s) = ωn2 (K ps + Ki ) s2 (s + 2ξωn ) s3 + 2ξωn s 2 + K ω2 s + K ω2 pn in Ⴎসථ஑ऌॖᆩđ֒ 0 < Ki < 2ξωn đоߌ༢๤໗‫ק‬b ⎧ ⎧ ⎪⎨⎪⎪eessssrr ࣡෿༂ҵ༢ඔ ⎪ Kp = ∞ ⇒ = 0, r(t) = 1(t)  ⎪⎪ Kv = ∞ = 0, r(t) = t1(t) ⎨ ⎪ ⎪ ⎪ Kiωn ⎪⎪⎩essr 2ξ ⎪⎩ K a = 2ξ = Kiωn ཁಖđҐႨб২ࠒ‫ٳ‬॥ᇅުđ༢๤ᄝಠ‫׮‬ቔႨ༯֥༂ҵࡨཬਔb č2Ď‫ކگ‬॥ᇅ ‫ކگ‬॥ᇅ‫ູٳ‬ൻೆҀӊ॥ᇅ‫ބ‬ಠ‫׮‬Ҁӊ॥ᇅਆᇕྙൔb 51 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 14 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ೘ᅣ ཌྟ༢๤֥ൈთ‫ٳ‬༅ ᇶေଽಸ ሱ‫׮‬॥ᇅ༢๤໗‫ބྟק‬໗෿ྟି‫ٳ‬༅ഈࠏൌဒ ଢ֥აေ ඃ༑ MATLAB ೈࡱؓ༢๤໗‫ٳྟק‬༅֥ࠎЧଁ਷ეओ ౰ ඃ༑ MATLAB ೈࡱؓ༢๤༂ҵ‫ٳ‬༅֥ Simuink ٟᆇ ๙‫ݖ‬щӱࠇ Simuink ٟᆇປӮ༢๤໗‫ބྟק‬໗෿ྟି‫ٳ‬༅ ᇗ ׄ ა ଴ ᇗׄğ๙‫ݖ‬щӱࠇ Simuink ٟᆇປӮ༢๤໗‫ބྟק‬໗෿ྟି‫ٳ‬༅ ׄ ଴ׄğ༢๤໗‫ބྟק‬໗෿ྟି‫ٳ‬༅ ࢝࿐൭‫ ؍‬ഈࠏ නॉีࠇ ປӮൌဒБۡ ቔြี ၂aൌဒଢ֥ 1a࿹࣮ۚࢨ༢๤֥໗‫ྟק‬đဒᆣ໗‫ק‬஑ऌ֥ᆞಒྟĠ 2aਔࢳ༢๤ᄹၭэ߄ؓ༢๤໗‫֥ྟק‬႕ཙĠ 3aܴҳ༢๤ࢲ‫ބܒ‬໗෿༂ҵᆭࡗ֥ܱ༢b ‫ؽ‬aൌဒ಩ༀ 1a໗‫ٳྟק‬༅ რ஑؎༢๤֥໗‫ྟק‬đᆺေ౰ԛ༢๤֥оߌࠞׄࠧॖđ‫ط‬༢๤֥оߌࠞׄࣼ൞оߌԮ‫ݦ־‬ඔ֥‫ٳ‬ ଛ‫؟‬ཛൔ֥۴đॖၛ০Ⴈ MATLAB ᇏ֥ tf2zp ‫ݦ‬ඔ౰ԛ༢๤֥ਬࠞׄđࠇᆀ০Ⴈ root ‫ݦ‬ඔ౰‫ٳ‬ଛ‫؟‬ཛ ൔ֥۴টಒ‫ק‬༢๤֥оߌࠞׄđՖ‫ط‬஑؎༢๤֥໗‫ྟק‬b č1Ďၘᆩֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = 0.2(s + 2.5) đႨ MATLAB щ s(s + 0.5)(s + 0.7)(s + 3) ཿӱ྽ট஑؎оߌ༢๤֥໗‫ྟק‬đѩ߻ᇅоߌ༢๤֥ਬࠞׄ๭b č2Ďၘᆩֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = k(s + 2.5) đ֒౼ k =1đ10đ100 s(s + 0.5)(s + 0.7)(s + 3) Ⴈ MATLAB щཿӱ྽ট஑؎оߌ༢๤֥໗‫ྟק‬b ᆺေࡼč1Ďս઒ᇏ֥ k ᆴэູ 1đ10đ100đࠧॖ֤֞༢๤֥оߌࠞׄđՖ‫ط‬஑؎༢๤֥໗‫ྟק‬đ ѩษં༢๤ᄹၭ k э߄ؓ༢๤໗‫֥ྟק‬႕ཙb 2a໗෿༂ҵ‫ٳ‬༅ č1Ďၘᆩೂ๭෮ൕ֥॥ᇅ༢๤bఃᇏ G(s) = s+5 đ൫࠹ෘ֒ൻೆູֆ໊ࢨᄁྐ‫ݼ‬aֆ໊ོ s2 (s +10) ௡ྐ‫ބݼ‬ֆ໊ࡆ෎؇ྐ‫ݼ‬ൈ֥໗෿༂ҵb Ֆ Simulink ๭ྙ९ᛍফఖᇏ຀ှ Sumč౰‫ބ‬ଆॶĎaPole-ZeročਬࠞׄĎଆॶaScopečൕѯఖĎ ଆॶٟ֞ᆇҠቔ߂૫đ৵ࢤӮٟᆇॿ๭ೂႷഈ๭෮ൕğ č2Ď೏ࡼ༢๤эູ I ྘༢๤đG(s) = 5 đᄝࢨᄁൻೆaོ௡ൻೆ‫ࡆބ‬෎؇ྐ‫ݼ‬ൻೆቔႨ༯đ s(s +10) ๙‫ݖ‬ٟᆇট‫ٳ‬༅༢๤֥໗෿༂ҵb 52 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ೘aൌဒඔऌ 1a ໗‫ٳྟק‬༅ č1Ďၘᆩֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = 0.2(s + 2.5) đႨ MATLAB щཿ s(s + 0.5)(s + 0.7)(s + 3) ӱ྽ট஑؎оߌ༢๤֥໗‫ྟק‬đѩ߻ᇅоߌ༢๤֥ਬࠞׄ๭b ӱ྽ս઒aᄎྛࢲ‫ݔ‬ၛࠣਬࠞׄ‫҃ٳ‬๭࡮ᆷ֝඀b Ⴎᄎྛࢲ‫ݔ‬ॖᆩđk=0.2 ൈđ༢๤൞໗‫֥ק‬b č2Ďၘᆩֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = k(s + 2.5) đ֒౼ k =1đ10đ100 s(s + 0.5)(s + 0.7)(s + 3) Ⴈ MATLAB щཿӱ྽ট஑؎оߌ༢๤֥໗‫ྟק‬b æ k =1 ൈđMATLAB ӱ྽ս઒ೂ༯ğ z=-2.5;p=[0,-0.5,-0.7,-3];k=1;Go=zpk(z,p,k);Gc=feedback(Go,1);Gctf=tf(Gc);[z,p,k]=zpkdata(Gctf,'v'); pz****p(Gctf);grid; ᄎྛࢲ‫ݔ‬ೂ༯ğ z = -2.5000Ġp = -3.0297 a1.3319 a0.0808 + 0.7829ia0.0808 - 0.7829iĠk = 1 ൻԛਬࠞׄ‫҃ٳ‬๭ೂ༯๭෮ൕğ Ⴎఃоߌ֥ࠞׄᆴࠇᆀਬࠞׄ‫҃ٳ‬๭ॖᆩ k = 1 ൈđ༢๤҂໗‫ק‬b ç k =10 ൈđᆺླࡼഈ૫֥ӱ྽ս઒ k ֥ᆴэູ 10đࣼ҂ᄜઅਙb ᄎྛࢲ‫ݔ‬ೂ༯ğ z = -2.5000Ġp = 0.6086 + 1.7971ia0.6086 - 1.7971ia-3.3352 a-2.0821Ġk = 10 ൻԛਬࠞׄ‫҃ٳ‬๭ೂ༯๭෮ൕğ Ⴎఃоߌ֥ࠞׄᆴࠇᆀਬࠞׄ‫҃ٳ‬๭ॖᆩ k = 10 ൈđ༢๤҂໗‫ק‬b è k =100 ൈđᄎྛࢲ‫ݔ‬ೂ༯ğ z = -2.5000Ġp = 1.8058 + 3.9691ia1.8058 - 3.9691ia-5.3575a-2.4541Ġk = 100 ൻԛਬࠞׄ‫҃ٳ‬๭ೂഈ๭b Ⴎఃоߌ֥ࠞׄᆴࠇᆀਬࠞׄ‫҃ٳ‬๭ॖᆩ k = 10 ൈđ༢๤҂໗‫ק‬b 2a໗෿༂ҵ‫ٳ‬༅ č1Ďၘᆩೂ๭෮ൕ֥॥ᇅ༢๤đఃᇏ G(s) = s+5 đ൫࠹ෘ֒ൻೆູֆ໊ࢨᄁྐ‫ݼ‬aֆ໊ོ s2 (s +10) ௡ྐ‫ބݼ‬ֆ໊ࡆ෎؇ྐ‫ݼ‬ൈ֥໗෿༂ҵb ٟᆇॿ๭ၛࠣൻԛѯྙ࡮ᆷ֝඀đՖൻԛѯྙॖၛुԛ҂๝ൻೆቔႨ༯֥༢๤֥໗෿༂ҵđ༢๤൞ II ྘༢๤đၹՎᄝࢨᄁൻೆ‫ོބ‬௡ൻೆ༯đ༢๤໗෿༂ҵູਬđᄝࡆ෎؇ྐ‫ݼ‬ൻೆ༯đթᄝ໗෿༂ҵb 53 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ď೏ࡼ༢๤эູ I ྘༢๤đ G(s) = 5 đᄝࢨᄁൻೆaོ௡ൻೆ‫ࡆބ‬෎؇ྐ‫ݼ‬ൻೆቔႨ༯đ s(s +10) ๙‫ݖ‬ٟᆇট‫ٳ‬༅༢๤֥໗෿༂ҵb ࢨᄁൻೆቔႨ༯֥ٟᆇॿ๭ၛࠣൻԛѯྙೂ༯๭෮ൕğ ོ௡ൻೆቔႨ༯֥ٟᆇॿ๭ၛࠣൻԛѯྙೂ༯๭෮ൕğ ࡆ෎؇ྐ‫ݼ‬ൻೆቔႨ༯֥ٟᆇॿ๭ၛࠣൻԛѯྙೂ༯๭෮ൕğ Ⴎٟᆇࢲ‫ݔ‬ॖᆩđ༢๤൞ I ྘༢๤đၹՎᄝࢨᄁൻೆቔႨ༯đ༢๤໗෿༂ҵູਬĠᄝོ௡ൻೆቔႨ༯ թᄝ໗෿༂ҵđ൞Ⴕҵ۵ሶĠᄝࡆ෎؇ྐ‫ݼ‬ൻೆ༯đ໗෿༂ҵ౴Ⴟ໭౫նđၹՎ I ྘༢๤҂ି۵ሶࡆ෎؇ ྐ‫ݼ‬b ඹaൌဒࢲં 1aษં༢๤ᄹၭ k э߄ؓ༢๤໗‫֥ྟק‬႕ཙb Ֆࢲ‫ݔ‬ॖᆩđᄹၭ k эնđ༢๤֥໗‫ିྟק‬эҵb 2aษં༢๤྘ඔၛࠣ༢๤ൻೆؓ༢๤໗෿༂ҵ֥႕ཙb Ֆࢲ‫ݔ‬ॖᆩđ༢๤྘ඔᄀۚđି໭ҵ۵ሶ֥ൻೆྐ‫ࢨ֥ݼ‬ՑᄀۚĠ҂๝ൻೆቔႨ༯đ๝၂۱༢๤ؓ ఃཙႋ֥໗෿༂ҵ҂๝đൻೆࢨՑᄀۚđေ౰༢๤֥྘ඔᄀۚđҌିൌགྷ໭ҵ۵ሶb 54 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 15 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻඹᅣ ۴݅ࠖ‫م‬č1a2 ࢫĎ ۴݅ࠖ‫ࠎ֥م‬Ч‫୑ۀ‬ ᇶေଽಸ ߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵ ଢ֥აေ ਔࢳ۴݅ࠖ‫م‬a۴֥݅ࠖ‫ק‬ၬ ౰ ᅧ໤۴݅ࠖٚӱč‫ږ‬ᆴٚӱ‫ބ‬ཌྷ࢘ٚӱĎ ᅧ໤߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵčఏׄ‫ބ‬ᇔׄa৵࿃ྟ‫ؓބ‬ӫྟa‫ٳ‬ᆦඔaࡶ࣍ཌaൌ ᇠഈ֥۴݅ࠖ‫҃ٳ‬Ď ᇗ ׄ ა ଴ ᇗׄğ۴֥݅ࠖ‫ק‬ၬa۴݅ࠖٚӱa߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵ ׄ ଴ׄğ۴݅ࠖٚӱa߻ᇅ۴֥݅ࠖࠎЧ‫م‬ᄵ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 4.1 ႄ࿽ ۴݅ࠖ‫ٳ‬༅‫م‬ğ֒षߌ༢๤֥၂۱ࠇ‫؟‬۱ҕඔ‫ؿ‬ളэ߄ൈđ۴ऌ༢๤֥षߌਬׄ‫ׄࠞބ‬đࢹᇹႿ ೏‫߻่ۄ‬๭‫م‬ᄵđ߻ᇅԛоߌหᆘ۴э߄֥݅ࠖb০Ⴈ۴݅ࠖ‫م‬ॖၛ‫ٳ‬༅оߌ༢๤֥໗‫ྟק‬đ࠹ෘčࠇ ‫ܙ‬ෘĎоߌ༢๤֥ᄠ෿‫ބ‬໗෿ྟିᆷѓđಒ‫ק‬оߌ༢๤֥ଖུҕඔؓႿ༢๤ྟି֥႕ཙၛࠣؓоߌ༢ ๤ࣉྛ཮ᆞ֩b 1ē۴݅ࠖ ۴݅ࠖğ֒༢๤षߌԮ‫ݦ־‬ඔᇕଖ۱ҕඔčೂЧ২ᇏ֥۴݅ࠖᄹၭ K g Ďᄝଖ၂ٓຶଽčೂ 0 → ∞ Ď ৵࿃э߄ൈđоߌหᆘ۴ᄝ S ௜૫ഈ၍‫֥ࠖ݅׮‬đӫູ۴݅ࠖb ॖၛႮ۴݅ࠖ๭ট‫ٳ‬༅༢๤֥ྟିb R(s) E(s) C(s) 2ē۴݅ࠖٚӱ č1Ď‫ّڵ‬ঌ༢๤֥۴݅ࠖٚӱ G (s) ‫ّڵ྘ׅ‬ঌ॥ᇅ༢๤֥ࢲ‫ܒ‬๭ ೂႷ๭෮ൕb H (s) ۴݅ࠖٚӱ൞ܱႿ‫گ‬эਈ s ٚӱđཿӮࠞቕѓྙൔೂ༯ m m ∏ ∏ Kg ⎡m ⎤ (s − zi ) s − zi n e ∑ ∑ = −1 = e j i=1 ⎢ ( s− zi )− ( s− p j ) ⎥ n ⎢⎣ ⎥⎦ i=1 = Kg i =1 j =1 j (2k +1)π n ∏(s − pj) ∏ s− pj j =1 j=1 Ⴟ൞đ۴݅ࠖٚӱႻॖၛ‫ږູࢳٳ‬ᆴٚӱ‫ބ‬ཌྷ࢘ٚӱೂ༯ m n ∏ s − zi m n ∏ s− pj j =1 ‫ږ‬ᆴٚӱğ i =1 ∏ ∏ = 1 ⇒ Kg m Kg n s − zi = s− pj ⇒ Kg = ∏ s− pj i =1 j =1 ∏ s − zi j =1 i =1 ∑ ∑ ⎡m n ⎤ ⎥ ⎢ ⎥ = + 1)π = 2 kπ +π ⎢ ཌྷ࢘ٚӱğ (s− zi )− ( s − p j ) (2k đ k = 0,±1,±2, ⎣ i=1 j =1 ⎦ č2Ď‫ږ‬ᆴٚӱaཌྷ࢘ٚӱ֥ࠫ‫ޅ‬ၩၬ Ֆ߻ᇅ۴݅ࠖ๭֥࢘؇টुđ۴݅ࠖഈ֥಩ၩ၂ׄᆺေડቀཌྷ࢘ٚӱđࠧॖ߂ԛ۴݅ࠖਔđॖၛ ඪཌྷ࢘ٚӱ൞۴֥݅ࠖԉ‫ٳ‬сေ่ࡱb‫ږط‬ᆴٚӱ֥ቔႨᇶေႨটಒ‫ၘק‬ᆩׄؓႋ֥ᄹၭb 55 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  čĎᆞّঌ༢๤֥۴݅ࠖٚӱ ೏༢๤ູᆞّঌൈđః۴݅ࠖٚӱູ m ∏ s − zi i =1 ‫ږ‬ᆴٚӱູğ Kg n =1 ∏ s− pj j =1 ∑ ∑ ⎡m n ⎤ 0 ⎢ ⎥ ཌྷ࢘ٚӱູğ ⎢ (s− zi )− ( s − p j ) ⎥ = 2 kπ = 0 đ k = 0,±1,±2, ⎣ i=1 j =1 ⎦ ਸ਼ຓđ 0 ≤ K g < +∞ ൈđ‫ّڵ‬ঌ༢๤֥۴݅ࠖӫູ1800 ۴݅ࠖđᆞّঌ༢๤֥۴݅ࠖࣼӫູ 00 ۴ ݅ࠖb ߻ᇅ۴֥݅ࠖࠎЧ‫م‬ᄵ ē1800 ۴݅ࠖቔ๭‫م‬ᄵ ‫م‬ᄵ ğ۴֥݅ࠖఏׄ‫ބ‬ᇔׄ ۴֥݅ࠖఏׄ൞ᆷ۴݅ࠖᄹၭ Kg = 0 ൈđоߌࠞׄᄝ 4 ௜૫ഈ໊֥ᇂđ‫ط‬۴֥݅ࠖᇔׄᄵ൞ᆷ K g = ∞ ൈоߌࠞׄᄝ 4 ௜૫ഈ໊֥ᇂb ۴݅ࠖఏ൓Ⴟ༢๤֥षߌࠞׄčЇওᇗࠞׄĎđ‫ط‬ᇔᆸႿषߌਬׄb ಖ‫ط‬ൌ࠽֥॥ᇅ༢๤ᇏđषߌԮ‫ݦ־‬ඔ֥‫ٳ‬ሰ‫؟‬ཛൔࢨՑ m ა‫ٳ‬ଛ‫؟‬ཛൔࢨՑ n ᆭࡗđડቀ n ≥ m ֥ ܱ༢bೂ‫ ݔ‬n > m đପહഺჅ֥ n − m ่۴݅ࠖ‫ٳ‬ᆦᇔᆸႿ໭౫ჹԩb ‫م‬ᄵ ğ۴֥݅ࠖ৵࿃ྟ‫ٳބ‬ᆦඔ ۴݅ࠖऎႵ৵࿃ྟđ౏ؓӫႿൌᇠb ‫م‬ᄵ ğ۴֥݅ࠖ‫ٳ‬ᆦඔ ۴֥݅ࠖ‫ٳ‬ᆦඔაषߌႵཋਬׄඔ m ‫ބ‬Ⴕཋࠞׄඔ n ᇏ֥նᆀཌྷ֩b֒ n ≥ m ൈđ‫ٳ‬ᆦඔ֩Ⴟ n đ ࠧ༢๤֥ࢨඔb ‫م‬ᄵ ğ۴֥݅ࠖࡶ࣍ཌ ෮໌۴֥݅ࠖࡶ࣍ཌđ൞ᆷ֒ n > m ൈđႋႵ n − m ่۴݅ࠖ‫ٳ‬ᆦ֥ᇔׄᄝ໭౫ჹԩđ෮໌ࡶ࣍ཌđ ॖၛಪູ֒ K g → ∞ ൈđ۴݅ࠖაࡶ࣍ཌ൞ᇗ‫֥ކ‬b ࡶ࣍ཌაൌᇠᆞٚཟ֥ࡃູ࢘ğ ϕa = (2 k + 1)π đ k = 0,1,2, ,n − m −1 n−m ࡶ࣍ཌაൌᇠ֥ࢌູׄğ n m ∑ p j − ∑ zi j =1 i =1 σa =  n−m ‫م‬ᄵ 5ğൌᇠഈ۴֥݅ࠖ‫҃ٳ‬ ൌᇠഈଖ౵თđ೏ఃႷш֥षߌਬׄ‫ބ‬षߌࠞׄ۱ඔᆭ‫ູބ‬అඔđᄵ‫ھ‬౵თс൞۴݅ࠖb 56 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 16 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻඹᅣ ۴݅ࠖ‫م‬č2 ࢫĎ ᇶေଽಸ ߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵ ଢ֥აေ ᅧ໤߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵğ‫ٳ‬৖ׄa߶‫֥ׄކ‬ಒ‫ק‬aაྴᇠ֥ࢌׄaԛഝ࢘აೆ ౰ ഝ֥࢘࠹ෘa֥ࠞׄ‫ބ‬აࠒb ᇗ ׄ ა ଴ ᇗׄğ߻ᇅ 180¢۴֥݅ࠖࠎЧ‫م‬ᄵ ׄ ଴ׄğ‫ٳ‬৖ׄa߶‫֥ׄކ‬ಒ‫ק‬aԛഝ࢘აೆഝ֥࢘࠹ෘ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 4-1a4-2a 4-9č3Ď ቔြี ‫م‬ᄵ 6ğ۴֥݅ࠖ‫ٳ‬৖č߶‫ކ‬Ďׄ ਆ่ࠇਆ่ၛഈ۴݅ࠖ‫ٳ‬ᆦᄝ‫گ‬௜૫ഈཌྷმުႻ‫ٳ‬৖֥ׄӫູ‫ٳ‬৖č߶‫ކ‬Ďׄb ۴֥݅ࠖ‫ٳ‬৖č߶‫ކ‬Ďׄൌᇉഈоߌหᆘٚӱ֥ᇗ۴đၹ‫ط‬ॖၛႨ౰ࢳٚӱൔᇗ۴֥ٚ‫م‬টಒ‫ק‬ ఃᄝ‫گ‬௜૫ഈ໊֥ᇂb ഡ༢๤षߌԮ‫ݦ־‬ඔູ m ∏∏ GK (s) = K (s − zi ) = KN (s)  D(s) i=1 n (s − pj ) j =1 ఃоߌหᆘٚӱູğ1+ KN (s) = 0 ⇒ f (s) = D(s) + KN (s) = 0 D(s) ડቀၛ༯಩‫ޅ‬၂۱ٚӱđ౏Ќᆣ K ູᆞൌඔ֥ࢳđࠧ൞۴֥݅ࠖ‫ٳ‬৖č߶‫ކ‬Ďׄb ⎧ df (s) = d[D(s) + KN (s)] = 0 ⎪ ds ds ⎪ ⎪ dK ⎨ = 0 ⎪ ds ⎪ d [ D(s) ] = 0 ⎪⎩ ds N (s) ‫م‬ᄵ ğ۴݅ࠖაྴᇠ֥ࢌׄ ۴݅ࠖაྴᇠ֥ࢌׄđൌᇉഈࣼ൞оߌ༢๤֥ਢࢸ໗‫۽ק‬ቔׄbၹՎਢࢸ‫۽‬ቔ֥ׄ౰‫م‬Ⴕೂ༯ਆ ᇕٚ‫م‬b ٚ‫م‬၂ğᄝоߌหᆘٚӱ DB (s) = 0 ᇏđ਷ s = jω đ֤֞ DB ( jω) = 0 đࡼ DB ( jω) ‫ູٳ‬ൌ҆‫ބ‬ ྴ҆đࠧ Re[DB ( jω)] + j Im[DB ( jω)] = 0  Ⴟ൞Ⴕ ⎧⎨⎩RIme[[DDBB ( jω)] = 0 đ౰ࢳ֤֞ ω ᆴđູࠧ۴݅ࠖაྴᇠ֥ࢌׄቕѓ௔ੱb ( jω)] = 0 ٚ‫ؽم‬ğႮসථ໗‫ק‬஑ऌđ਷সථіᇏԛགྷಆਬྛđ֌ֻ၂ਙჭ෍‫ݼژ‬Ќӻ҂эđՎൈ༢๤ԩႿ ਢࢸ໗‫ק‬ሑ෿đѩॖ౰֤۴݅ࠖაྴᇠ֥ࢌׄb ২ ğഡֆ໊‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = K g s(s + 1)(s + 5) đ൫߻ᇅ༢๤֥оߌ ۴݅ࠖđѩ౰۴֥݅ࠖ‫ٳ‬৖č߶‫ކ‬Ďׄၛࠣაྴᇠ֥ࢌׄb 57 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ࢳğषߌࠞׄğ p1 = 0, p2 = −1, p3 = −5 đ໭षߌਬׄđ n = 3, m = 0  Ⴕ  ่۴݅ࠖ‫ٳ‬ᆦđ‫ٳ‬љఏ൓Ⴟ  ۱षߌࠞׄ p1 = 0, p2 = −1, p3 = −5 đᇔᆸႿ໭౫ჹԩb ࡶ࣍ཌࡃ࢘‫ބ‬ൌᇠ֥ࢌׄቕѓ‫ٳ‬љູ ⎧π 3 = 600, k = 0 (2K +1)π (2K +1)π = ⎪⎨π = 1800, k = 1  ϕa = n−m = 3 ⎪⎩5π 3 = 3000, k = 2 ∑ ∑ σ a = p j − zi = p1 + p2 + p3 = −2  n−m 3 ൌᇠഈ۴݅ࠖ‫ູ҃ٳ‬čđĎđčđ ∞ Ďb ۴݅ࠖ‫ٳ‬৖č߶‫ކ‬Ďׄb ႮषߌԮ‫ݦ־‬ඔđॖၛ֤֞оߌหᆘٚӱൔ K g + s(s + 1)(s + 5) = 0 đᄵ Kg = −s(s +1)(s + 5) đ਷ dK g = − ds(s +1)(s + 5) = 3s 2 + 12s + 5 = 0 đ౰֤ğ ds ds s1 = −0.473, s2 = −3.52 đႮႿ s2 ҂ᄝ۴݅ࠖഈđ෮ၛ‫ٳ‬৖č߶‫ކ‬Ďູׄ s1 = −0.473b ۴֥݅ࠖ‫ٳ‬৖č߶‫ކ‬Ď࢘đθd = 1800 k = 1800 2 = 900 b ۴݅ࠖაྴᇠ֥ࢌׄğ ٚ‫م‬၂ğоߌหᆘٚӱğ s3 + 6s 2 + 5s + K g = 0  ਷ s = jω սೆоߌหᆘٚӱ ( jω)3 + 6( jω)2 + 5( jω) + K g = 0  ‫ູࢳٳ‬ൌ҆‫҆ྴބ‬ğ (K g − 6ω 2 ) + j(5ω − ω 3 ) = 0  Ⴟ൞Ⴕğ ⎪⎧K g − 6ω 2 = 0 ⇒ ⎪⎧ω = 1,± 5 đཁಖࢌູׄ ⎪⎧ω =± 5  ⎨ −ω3 0 ⎪⎩⎨K = 0,30 ⎪⎩⎨K g = 30 ⎪⎩5ω = g ٚ‫ؽم‬ğ‫ܒ‬ᄯসථі s3 1 5 s2 6 K g ֒ K g = 30 ൈđ s1 ྛູಆਬđসථіֻ၂ਙ‫ݼژ‬҂эđ༢๤หᆘ۴ູՂྴඔđॖ s1 30 − K g 0 Im 6 s0 Kg ۴ऌ s 2 ྛ֥‫ڣ‬ᇹٚӱğ F (s) = 6s 2 + K g = 0 đ ౰֤ s = ± j 5 đၹՎ۴݅ࠖაྴᇠ֥ࢌׄቕѓູğ Re (K g = 30, s = ± j 5) b -4 -3 -2 -1 0 ۴݅ࠖҤ๭ೂ༯b  ‫ٳ‬৖ׄ  ‫م‬ᄵ ğ۴֥݅ࠖԛഝ࢘‫ބ‬ೆഝ࢘ ֒षߌਬׄ‫ބ‬षߌࠞׄԩႿ‫گ‬௜૫ൈđ۴݅ࠖ৖षषߌࠞׄԩ్֥ཌაᆞൌᇠ֥ٚཟࡃ࢘đӫູ۴ ֥݅ࠖԛഝ࢘čԛ‫࢘ؿ‬Ďb๝ဢđ۴݅ࠖࣉೆषߌਬׄԩ్֥ཌაᆞൌᇠ֥ٚཟࡃ࢘đӫູ۴֥݅ࠖೆഝ ࢘čᇔᆸ࢘ĎbႨ‫܄‬ൔіൕູğ m n ∑ ∑ θ px = (2k + 1)π + ∠( p x − zi ) − ∠( p x − p j ) i =1 j =1 j≠x Ⴎ px ა px+1 ֥‫܋‬ᣢྟđ θ p x +1 = − θ p x ๝৘ॖ֤đ‫گ‬ඔਬ֥ׄೆഝ࢘Ⴈ‫܄‬ൔіൕູ 58 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ m n ∑ ∑ ϕ zx = (2k + 1)π − ∠( z x − zi ) + ∠( z x − p j ) i =1 j =1 i≠x Ⴎ z x ა z x+1 ֥‫܋‬ᣢྟđ ϕ z x +1 = − ϕ z x b s(s 2 + 2s + 2) đ൫߻ᇅ༢๤֥ປᆜ۴݅ ২ğഡֆّ໊ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູğ GK (s) = K g ࠖđѩေ౰࠹ෘԛഝ࢘b ࢳğषߌູࠞׄ p1 = 0, p2 = −1 + j, p3 = −1 − j đ໭षߌਬׄđ n = 3, m = 0 č1ĎႮႿ n = 3, m = 0 đ෮ၛ۴݅ࠖႵ 3 ่‫ٳ‬ᆦĠ č2Ď۴݅ࠖఏ൓Ⴟषߌࠞׄ p1 = 0, p2 = −1 + j, p3 = −1 − j đᇔᆸႿ໭౫ჹԩb č3Ď3 ่۴֥݅ࠖࡶ࣍ཌࡃ࢘‫ׄࢌބ‬ቕѓ (2k + 1)π (2k + 1)π ⎧π 3 (60 0 ), k = 0 = = = ⎪⎨π (180 0 ), k = 1 ϕa ⎪⎩5π 3 (−60 0 ,300 0 ), k = 2 n−m 3 σa = p1 + p2 + p3 = 0−1+ j −1− j =−2 3 3 3 č4Ďൌᇠഈ֥۴ູ݅ࠖč- ∞đ0Ď,ࠧᆜ۱‫ڵ‬ൌᇠĠ č5Ď۴݅ࠖ໭‫ٳ‬৖č߶‫ކ‬ĎׄĠ č6Ďఏ൓Ⴟ p2 = −1 + j, p3 = −1 − j ۴݅ࠖ‫ٳ‬ᆦཟሢϕa = π 3,5π 3 ֥ਆ่ࡶ࣍ཌЯ࣍Ġ č7Ď۴݅ࠖაྴᇠ֥ࢌׄ оߌหᆘٚӱğ s3 + 2s2 + 2s + Kg = 0 ਷ s = jω սೆหᆘٚӱđ ( jω)3 + 2( jω)2 + 2( jω) + K g = 0 ࠇ ⎪⎧K g − 2ω 2 = 0 ⇒ ⎪⎧ω = ± 2 ⎨ −ω3 0 ⎪⎩⎨K = ⎪⎩2ω = g 4 čĎ۴݅ࠖԛഝ࢘ θ p2 = 1800 − ∠( p2 − p1 ) − ∠( p2 − p3 ) = 1800 − ∠(−1 + j) − ∠[(−1 + j − (−1 − j)] = −450 θ p3 = 450 č9Ď߻ᇅ۴݅ࠖೂ༯ ‫م‬ᄵ 9ğ༢๤оߌ֥ࠞׄ‫ބ‬აࠒ ༢๤षߌԮ‫ݦ־‬ඔ 59 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ G(s) = Kg (s − z1)((s − z2 ) (s − zm ) = Kg sm + b1sm-1 + + bm-1s + bm (s − p1)(s − p2 ) (s − pn ) sn + a1sn-1 + + an-1s + an  = Kg sm − (z1 + z2 + + zm )sm−1 + + (−1)m z1z2 zm sn − ( p1 + p2 + + pm )sn−1 + + (−1)n p1 p2 pn षߌਬ֥ׄ‫ބ‬აࠒğ m m zm = (−1) m bm ∑ zi =z1 + z2 + ∏ + zm = −b1 đ zi =z1 z2 i =1 i =1 षߌ֥ࠞׄ‫ބ‬აࠒğ n n ∑ ∏ pj =p1 + p2 + + pn = −a1đ p j = p1 p2 pn = (−1)n an j =1 j =1 оߌหᆘٚӱğ DB (s) = sn + a1sn-1 + + an-1s + an + K g (sm + b1sm-1 + + bm-1s + bm ) = 0 ೏ഡ༢๤֥оߌູࠞׄ si č i = 1,2, , n ĎđᄵႵ n ∏ DB (s) = (s - si ) = sn + C1sn−1 + C2sn−1 + + Cn−1sn−1 + Cn = 0 i =1 č1Ď༢๤оߌ֥ࠞׄ‫ބ‬ n ∑ si = − c1 i =1 ֒ n − m ≥ 2 ൈđоߌࠞׄᆭ‫֩ބ‬Ⴟषߌࠞׄᆭ‫ބ‬đ౏ູӈඔđᆃ۱ӈඔ္ӫູоߌ֥ࠞׄᇗྏb ᆃіૼğ֒ K g Ⴎ 0 → ∞ э߄ൈđоߌࠞׄᆭ‫ބ‬Ќӻ҂эđ౏֩Ⴟ n ۱षߌࠞׄᆭ‫ބ‬bᆃၩ໅ሢ၂ ҆‫ٳ‬оߌࠞׄᄹնൈđਸ਼ຓ၂҆‫ٳ‬оߌࠞׄсಖэཬb္ࠧđೂ‫ݔ‬၂҆‫ٳ‬оߌ۴݅ࠖෛሢ K g ֥ᄹࡆ‫ط‬ཟ Ⴗ၍‫׮‬ൈđਸ਼ຓ၂҆‫ٳ‬۴݅ࠖсࡼෛሢ K g ֥ᄹࡆ‫ط‬ཟቐ၍‫׮‬đ൓ᇔЌӻоߌ֥ࠞׄᇗྏ҂эb č2Ď༢๤оߌ֥ࠞׄࠒ n ∏ si =(−1)n cn = (−1)n (an + K gbm ) i =1 = (−1)n an + (−1)n K gbm n m ∏ ∏ = pi +(−1)n−m K g z j i =1 j =1 ೏༢๤թᄝਬᆴषߌࠞׄčࠧ an = 0 ĎđႿ൞༢๤оߌ֥ࠞׄࠒູ n m m ∏ ∏ ∏ si =(−1)n−m Kg z j = Kg[(−1)n−m z j ] i =1 j =1 j =1 ᆃіૼ༢๤֥оߌ֥ࠞׄࠒა༢๤֥षߌ۴݅ࠖᄹၭӮᆞбb  60 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 17 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻඹᅣ ۴݅ࠖ‫م‬č2 ࢫĎ ᇶေଽಸ ߻ᇅ 0¢۴֥݅ࠖࠎЧ‫م‬ᄵ ҕэਈ༢๤֥۴݅ࠖ ٤ቋཬཌྷ໊༢๤֥۴݅ࠖ ଢ֥აေ ᅧ໤߻ᇅ 0¢۴֥݅ࠖࠎЧ‫م‬ᄵđѩა 180¢۴݅ࠖ߻ᇅ‫م‬ᄵཌྷбࢠđᅧ໤ః҂๝ᆭԩ ౰ ᅧ໤ҕэਈ༢๤֥۴݅ࠖ߻ᇅ֥҄ᇧ ᅧ໤٤ቋཬཌྷ໊༢๤֥‫ק‬ၬၛࠣ 180¢ა 0¢۴֥݅ࠖ஑؎ ᇗׄა଴ ᇗׄğ߻ᇅ 0¢۴֥݅ࠖࠎЧ‫م‬ᄵaҕэਈ༢๤֥۴݅ࠖ߻ᇅ ׄ ଴ׄğҕэਈ༢๤֥۴݅ࠖ߻ᇅ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 4-8a4-12a4-13 ቔြี 2ē 00 ۴݅ࠖቔ๭‫م‬ᄵ 00 ֩ཌྷ࢘۴݅ࠖቔ๭‫م‬ᄵđა1800 ֩ཌྷ࢘۴݅ࠖቔ๭‫م‬ᄵ෮҂๝֥൞đေྩ‫ڿ‬აཌྷ่࢘ࡱႵܱ֥ ܿᄵđऎุႵğ č1Ď۴֥݅ࠖࡶ࣍ཌ ࡶ࣍ཌ֥ࢌׄቕѓ҂эđౠ࢘‫ູڿ‬ ϕa = 2kπ đ k = 0,1,2, ,n − m −1 n−m č2Ďൌᇠഈ֥۴݅ࠖ‫҃ٳ‬ ൌᇠഈଖ౵თđ೏ఃႷш֥षߌਬׄ‫ބ‬षߌࠞׄ۱ඔᆭ‫୽ູބ‬ඔčЇও 0Ďđᄵ‫ھ‬౵თс൞۴݅ࠖb č3Ď۴֥݅ࠖԛഝ࢘‫ބ‬ೆഝ࢘ m n ∑ ∑ ԛഝ࢘ğθ p x = ∠ ( px − zi) − ∠ ( px − p j) i=1 j =1 j≠ x m n ∑ ∑ ೆഝ࢘ğϕzx = − ∠(zx − zi ) + ∠(zx − p j ) i=1 j =1 i≠x 3ēҕэਈ۴݅ࠖ ၛഈษં֥൞༢๤۴݅ࠖᄹၭ K g ቔູҕэਈൈоߌ۴֥݅ࠖቔ๭ܿᄵđ֌ൌ࠽౦ঃ҂ປಆ൞ᆃဢ ֥b֒ၛ༢๤ᇏః෱ҕඔቔູэਈčൈࡗӈඔaّঌ༢ඔaषߌਬׄ‫֩ׄࠞބ‬Ďൈ֥۴݅ࠖӫູҕэ ਈ۴݅ࠖࠇܼၬ۴݅ࠖb Ֆ৘ં֥࢘؇টुđҕਈ۴݅ࠖѩ໭ห൹ᆭԩđఃԩ৘ٚ‫م‬൞đࡼჰট֥षߌԮ‫ݦ־‬ඔࣜ‫ݖ‬ඔ࿐ эߐӮၛҕэਈቔູo۴݅ࠖᄹၭpི֥֩षߌԮ‫ݦ־‬ඔྙൔđಖު၇ഈඍ֥ቔ๭ܿᄵ߻ᇅ۴݅ࠖ๭b ২ğၘᆩ‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = G(s)H (s) = 1 (s + a) / s2 (s + 1) 4 ൫߻ᇅҕඔ a Ֆ 0 → ∞ э߄ൈđоߌ༢๤֥۴݅ࠖb ࢳğоߌหᆘٚӱ DB (s) =1+ GK (s) =1+ 1 (s + a) = 0 đࠧğ s3 + s2 + 1 s + 1 a = 0 b 4 4 4 s2 (s +1) 61 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 1a a a 4 4s2 + 4s ࡼٚӱэྙູğ 1+ s2 1 = 0 ⇒ 1+ 4s3 = 0 ⇒ 1+ s(4s2 =0 s3 + + s + + s + 1) 4 ི֥֩षߌԮ‫ݦ־‬ඔູğ GK′ (s) = s(4s2 a + 1) đቔԛҕඔ a э߄ൈ֥۴݅ࠖ๭ೂ༯ğ + 4s 4ēܱႿ1800 ‫ ބ‬00 ֩ཌྷ࢘۴֥݅ࠖࠫ۱໙ี ۴ऌ۴݅ࠖᄹၭ֥ᆞ‫ڵ‬aّঌࢲ‫ܒ‬čᆞّঌa‫ّڵ‬ঌĎđॖၛ‫ٳ‬љ֤֞۴݅ࠖٚӱđ‫ٳ‬༅ॖᆩ໭ં൞ 1800 ۴݅ࠖߎ൞ 00 ۴݅ࠖđ෱ૌ෮҂๝֥൞ཌྷ࢘ٚӱđ‫ږ‬ᆴٚӱ൞၂ဢ֥đऎุ֥ؓႋܱ༢ሹࢲೂ༯іb ྽‫ݼ‬ Kg ‫ّڵ‬ঌ ᆞّঌ 1 0 ≤ K g < +∞ 1800 ۴݅ࠖ 00 ۴݅ࠖ 2 −∞ < Kg ≤ 0 00 ۴݅ࠖ 1800 ۴݅ࠖ ֌ေᇿၩ֥൞đᄝ۴֥݅ࠖ߻๭ܿᄵᇏđषߌԮ‫ݦ־‬ඔ GK (s) ‫ٳ‬ሰa‫ٳ‬ଛ֥ྙൔູ s − zi a s − p j đ ၹՎႵུ॥ᇅ֥ෙಖ൞‫ّڵ‬ঌࢲ‫ܒ‬đ֌ᄝఃषߌԮ‫ݦ־‬ඔ֥‫ٳ‬ሰࠇ‫ٳ‬ଛ‫؟‬ཛൔᇏđ s ֥ቋۚՑૢ֥༢ඔ ູ‫ڵ‬đ൐༢๤ऎႵᆞّঌ֥ྟᇉđೂğ GK1 (s) = K (1− 2s) đ GK 2 (s) = K (−s2 + 3s − 2) đ౏ K > 0 (s + 2)(s + 3) s(s2 + 2s + 3) ࣼඋՎ౦ঃđმ֞ᆃᇕ౦ঃൈđᄝ߻ᇅ۴݅ࠖభđႋ༵ࡼ෱ૌሇߐູਬaࠞׄྙൔb GK 1 ( s) = − 2K (s − 0.5) = Kg1(s − 0.5) , K g1 < 0 (s + 2)(s + 3) (s + 2)(s + 3) GK 2 (s) = − K (s2 − 3s + 2) = K g2 (s2 − 3s + 2) , Kg2 < 0 s(s2 + 2s + 3) s(s2 + 2s + 3)   ႮႿ K g1 < 0 a K g2 < 0 đ෮ၛ෱ૌၹ‫ھ‬ҐႨ 00 ۴݅ࠖ‫م‬ᄵ߻ᇅb ᄝ߻ᇅ۴݅ࠖൈđ೏ીႵห൹ᆷૼđ၂Ϯಪູ൞߻ᇅ‫ّڵ‬ঌ॥ᇅ༢๤֥۴݅ࠖđ۴݅ࠖᄹၭ K g ቔ ູҕэਈđ౏նႿਬđࠧ߻ᇅ1800 ۴݅ࠖb 62 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 18 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻඹᅣ ۴݅ࠖ‫م‬č3 ࢫĎ ᇶေଽಸ ༢๤֥۴݅ࠖ‫ٳ‬༅ٚ‫م‬ ଢ֥აေ ᅧ໤оߌਬ֥ࠞׄಒ‫ק‬ ౰ ਔࢳоߌਬ֥ࠞׄ‫ؓ҃ٳ‬༢๤ྟି֥႕ཙ ਔࢳषߌਬ֥ࠞׄ‫ؓ҃ٳ‬༢๤ྟି֥႕ཙ ᅧ໤۴ऌ߻ᇅ֥۴݅ࠖ‫ٳ‬༅༢๤֥໗‫ྟק‬aֆ‫ྟט‬ ᇗ ׄ ა ଴ ᇗׄğ༢๤֥໗‫ྟק‬aֆ‫ٳྟט‬༅ ׄ ଴ׄğоߌਬ֥ࠞׄಒ‫ק‬a ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 4-4a4-6a4-15 ቔြี ۴݅ࠖ‫ٳ‬༅‫م‬ ႋႨ۴݅ࠖ‫ٳم‬༅༢๤đॖၛࣉྛ༢๤ྟି֥‫ٳྟק‬༅‫ܙބ‬ෘđၛࠣ‫࠹ܙ‬ҕඔэ߄ؓ༢๤ྟି֥ ႕ཙđѩิԛऎุ֥‫ڿ‬೿č཮ᆞĎٚ‫م‬b ēоߌਬׄaࠞׄ‫ބ‬षߌ۴݅ࠖᄹၭ֥ಒ‫ק‬ čĎоߌਬׄ оߌԮ‫ݦ־‬ඔਬ֥ׄಒ‫ק‬൞൅‫ٳ‬ಸၞ֥đ෱ൌ࠽ഈ൞భཟ๙֥֡ਬׄ‫ّބ‬ঌ๙֥֡ࠞׄቆӮđ֒ ູֆّ໊ঌൈđоߌਬׄࣼ൞षߌਬׄb čĎоߌࠞׄ оߌ֥ࠞׄಒ‫ק‬ٚ‫م‬ğ ၂Ϯ‫ط‬࿽đؓႿбࢠࡥֆ֥༢๤ॖ༵൐Ⴈ‫ږ‬ᆴٚӱࣉྛ൫ฐಒ‫ٳ҆ק‬оߌൌඔࠞׄđಖުႨሸ‫ކ‬ ӉԢ‫م‬౰ఃჅ֥оߌࠞׄđࠇҐႨоߌ֥ࠞׄ‫ބ‬აࠒ֥ྟᇉটಒ‫ק‬ఃჅ֥оߌࠞׄb č3Ď۴݅ࠖᄹၭ ೏ၘᆩ༢๤֥оߌਬׄ‫ބ‬оߌࠞׄđᄵॖ০Ⴈ‫ږ‬ᆴٚӱটಒ‫ؓק‬ႋ֥۴݅ࠖᄹၭ K g b ഡषߌਬׄaࠞׄ‫ٳ‬љູ zi č i = 1,2, , m Ďa p j č j = 1,2, , n Ďđ౰оߌ۴݅ࠖഈଖ၂ׄ sl ؓ ႋ֥۴݅ࠖᄹၭ K g đᄵႮ‫ږ‬ᆴٚӱ֤֞ğ m n ∏ (sl − zi ) ∏ (sl − p j ) j =1 K i=1 =1⇒ Kg = m  gn ∏ (sl − p j ) ∏ (sl − zi ) j =1 i=1 ēоߌਬׄaࠞׄ‫ؓ҃ٳ‬༢๤ྟି֥႕ཙ čĎ໗‫ྟק‬đေ౰оߌ༢๤໗‫ק‬đః۴݅ࠖсྶಆ໊҆Ⴟ 4 ቐ϶௜૫bೂ‫ݔ‬༢๤թᄝ೘่ࠇ೘่ ၛഈ֥ࡶ࣍ཌđᄵсႵ၂۱ K g ᆴđ൐༢๤ԩႿਢࢸ໗‫ק‬ሑ෿b čĎᄎ‫ྙ׮‬෿đ༢๤֥ࠎЧᄎ‫ྙ׮‬෿Ⴎоߌ໊֥ࠞׄᇂथ‫ק‬b֒Ⴕ၂оߌਬׄ‫ބ‬оߌࠞׄᇗ‫ކ‬ൈđ ‫ؽ‬ᆀ‫ܒ‬Ӯ၂ؓ୽ࠞሰb਽ඔ֥࠹ෘॖᆩđؓႋႿ୽ࠞሰ֥ପ۱֥ࠞׄᄎ‫ྙ׮‬෿‫ٳ‬ਈࡼཨാb֒оजࠞ ׄಆ໊҆Ⴟ‫ڵ‬ൌᇠൈđཙႋӯֆ‫ט‬ഈശሑ෿b֒оߌࠞׄԛགྷ‫گ‬ඔൈđཙႋӯකࡨᆒ֕ྙൔb čĎ௜໗ྟđቅୄ࢘ β ᄀն ቅୄб ξ ཬđ༢๤֥ᆒ֕௔ੱᄀۚđᆒ֕ᄀखਛbေ൐༢๤֥ᄠ෿ ཙႋ௜໗đ๝ൈႻႵбࢠ‫ॹ֥ݺ‬෎ྟđ༢๤֥ቅୄб҂ି෾նđ္҂ି෾ཬđ৘ંഈࢃ β = 450 đቅ ୄбξ = 0.707 ൈđ༢๤֥ሹุྟିቋ‫ݺ‬b čĎॹ෎ྟđေ൐༢๤ऎႵࢠ‫ॹ֥ݺ‬෎ྟđԢоߌᇶ֝ࠞׄၛຓđఃჅоߌࠞׄႋ‫ھ‬ჹ৖ྴᇠđ൐ః ᄠ෿ཙႋ‫ٳ‬ਈකࡨࡆॹđ༢๤‫ࢫט‬ൈࡗࡨཬđՖ‫ۚิط‬༢๤֥ཙႋ෎؇b 63 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ  ē০Ⴈ۴݅ࠖ‫ܙ‬ෘ༢๤ྟି ۴݅ࠖ‫ܙم‬ෘоߌ༢๤֥ྟି֥ٚ‫م‬đᇶေ൞০Ⴈ۴݅ࠖࣉྛ๭ࢳb ২ğഡֆّ໊ঌ༢๤֥षߌԮ‫ݦ־‬ඔູğ K s(s +1)(0.5s +1) đ൫ႋႨ۴݅ࠖ‫م‬౰༢๤֥ֆ໊ࢨᄁ ཙႋđѩ‫ܙ‬ෘ༢๤֥ྟିᆷѓb ࢳğࡼषߌԮ‫ݦ־‬ඔ‫ھ‬ཿູਬࠞׄіൕྙൔ GK (s) = K s(s +1)(0.5s +1) = 2K s(s +1)(s + 2) = K g s(s +1)(s + 2)  K g = 2K  ࠹ෘ֤֞۴֥݅ࠖ‫ٳ‬৖߶‫ׄކ‬ቕѓູğ d = −0.423, K g = 0.385 Ġ ۴݅ࠖაྴᇠ֥ࢌູׄ s = ± j1.414, K g = 6 b ߻ᇅ۴݅ࠖ๭ೂ༯ğ  οีଢ۳‫ק‬ေ౰ğ K = 0.525  ᄵ K g = 2K = 1.05 đཁಖؓႋ ֥оߌࠞׄႋ‫໊ھ‬Ⴟ‫ٳ‬৖ č߶‫ކ‬Ďׄၛުč K gd = 0.385 Ď ֥‫گ‬௜૫ഈđ֌҂߶ӑԛቐ϶ ௜૫čაྴᇠࢌׄԩ֥۴݅ࠖᄹၭູ K gs = 6 Ďb ࣜ൫ฐ֒ K g = 2K = 1.05 ൈ֥  ۱оߌູࠞׄ s1,2 = −0.33 ± j0.58 b ཿԛоߌหᆘٚӱğ DB (s) = s 3 + 3s 2 + 2s + K g = s 3 + 3s 2 + 2s + 1.05 = 0  ႮႿ n − m = 3 − 0 = 3 > 2 đ෮ၛႮหᆘ۴֥‫ܱ֥ࠒࠇބ‬༢ॖၛ౰౼ֻ೘۱оߌࠞׄğ s1 + s2 + s3 = −3 ⇒ s3 = −3 − s1 − s2 = −2.43b ႮႿ൞ֆّ໊ঌ॥ᇅ༢๤đ౏໭षߌਬׄđ෮ၛ֤֞༢๤֥оߌԮ‫ݦ־‬ඔູğ Φ(s) = 1.05  (s + 2.43)(s + 0.33 + j0.58)(s + 0.33 − j0.58) ༢๤֥ֆ໊ࢨᄁཙႋູğ C(s) = Φ(s)R(s) = 1.05 ×1  (s + 2.43)(s + 0.33 + j0.58)(s + 0.33 − j0.58) s c(t) = L−1[C(s)] = 1 − 0.125e−2.34t −1.297e−0.33t sin(0.58t + 44.30 ) č t ≥ 0 Ď ྟିᆷѓ‫ܙ‬ෘğ Ⴎ Ⴟ ‫ گ‬ඔ ࠞ ׄ s1,2 = −0.33 ± j0.58 ᇛ ຶ ી Ⴕ ਬ ׄ đ ౏ s1,2 << s3 đ ෮ ၛ ॖ ၛ ಒ ‫ק‬ s1,2 = −0.33 ± j0.58 ູоߌᇶ֝ࠞׄđႿ൞оߌԮ‫ݦ־‬ඔॖၛࡥ߄ູ‫ࢨؽ‬༢๤b Φ(s) = 2.43(s + 0.33 + 1.05 + 0.33 − j0.58) = s2 + 0.448  j0.58)(s 0.66s + 0.448 ౰֤ğωn = 0.67(rad / s) đξ = 0.49  ࠹ෘྟିᆷѓູğ M p % = eξπ 1−ξ 2 ×100% = 16.4% đ ts = 3 ξωn = 12.1s č ∆ = 0.02 Ď ēषߌਬaࠞׄ‫ؓ҃ٳ‬༢๤ྟି֥႕ཙ čĎᄹࡆषߌਬׄؓ۴֥݅ࠖ႕ཙ ֻ၂đࡆೆषߌਬׄđ‫ڿ‬эࡶ࣍ཌ่֥ඔ‫࣍ࡶބ‬ཌ֥ౠ࢘Ġ ֻ‫ؽ‬đᄹࡆषߌਬׄđཌྷ֒Ⴟᄹࡆັ‫ٳ‬ቔႨđ൐۴݅ࠖཟቐ၍‫ࠇ׮‬ຕ౷đՖ‫ۚิط‬ਔ༢๤֥ཌྷؓ໗ ‫ྟק‬b༢๤ቅୄᄹࡆđ‫ݖ‬؈‫ݖ‬ӱൈࡗ෪؋Ġ ֻ೘đᄹࡆ֥षߌਬׄᄀࢤ࣍ቕѓჰׄđັ‫ٳ‬ቔႨᄀ఼đ༢๤֥ཌྷؓ໗‫ྟק‬ᄀ‫ݺ‬b  čĎᄹࡆषߌࠞׄؓ۴֥݅ࠖ႕ཙ **** uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ၂đࡆೆषߌࠞׄđ‫ڿ‬эࡶ࣍ཌ่֥ඔ‫࣍ࡶބ‬ཌ֥ౠ࢘Ġ ֻ‫ؽ‬đᄹࡆषߌࠞׄđཌྷ֒Ⴟᄹࡆࠒ‫ٳ‬ቔႨđ൐۴݅ࠖཟႷ၍‫ࠇ׮‬ຕ౷đՖ‫֮ࢆط‬ਔ༢๤֥ཌྷؓ໗ ‫ྟק‬b༢๤ቅୄࡨཬđ‫ݖ‬؈‫ݖ‬ӱൈࡗࡆӉĠ ֻ೘đᄹࡆ֥षߌࠞׄᄀࢤ࣍ቕѓჰׄđࠒ‫ٳ‬ቔႨᄀ఼đ༢๤֥ཌྷؓ໗‫ྟק‬ᄀҵb čĎᄹࡆषߌ୽ࠞሰ֥ቔႨ ᄹࡆ၂ؓषߌ୽ࠞሰđॖၛ‫ڿ‬೿༢๤֥໗෿ྟିb ഡषߌ୽ࠞሰ֥ਬູׄ zc aູࠞׄ pc đႮႿ୽ࠞሰડቀ zc ≈ pc đࠧ ∠(s − zc ) = ∠(s − pc ), s − zc = s − pc  ၹՎ෱ૌؓ۴݅ࠖࠫެીႵ႕ཙb ഡ༢๤ᄝીႵᄹࡆषߌ୽ࠞሰൈ֥षߌ٢նПඔູ K  m m ∏ ∏ Kg (s − zi ) Kg (−zi ) i=1 i=1 n ∏ ∏ K = lim = n  s→0 (s − pj ) (− p j ) j=1 j=1 ᄹࡆषߌ୽ࠞሰު֥षߌ٢նПඔູ Kc  m ∏∏ Kc=Kg (−zi ) zc =K× zc  pc pc i=1 n (− p j ) j=1 ೏౼ zc = −0.5 đ pc = −0.05 đᄵႵ Kc = K× zc = K × 0.5 = 10K  pc 0.05 ཁಖषߌ٢նПඔᄹࡆਔ  Пđ༢๤֥໗෿ࣚ؇֤֞ਔႵི֥ิۚb 65 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 19 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č1a2 ࢫĎ ௔ੱหྟ֥ࠎЧ‫୑ۀ‬a‫ק‬ၬၛࠣࠫ‫ޅ‬іൕ‫م‬ ᇶေଽಸ ‫֥ࢫߌ྘ׅ‬௔ੱหྟ ଢ֥აေ ਔࢳ௔ੱหྟ֥ࠎЧ‫୑ۀ‬ ౰ ᅧ໤௔ੱหྟ֥‫ק‬ၬ ᅧ໤ӈႨ௔ੱหྟč‫ږ‬௔หྟaཌྷ௔หྟaൌ௔หྟ‫ྴބ‬௔หྟĎ ᅧ໤௔ੱหྟ֥ਆᇕࠫ‫ޅ‬іൕٚ‫م‬ ਔࢳ۱‫֥ࢫߌ྘ׅ‬௔ੱหྟ ᇗ ׄ ა ଴ ᇗׄğ௔ੱหྟ֥‫ק‬ၬaӈႨ௔ੱหྟၛࠣਆᇕࠫ‫ޅ‬іൕٚ‫م‬ ׄ ଴ׄğ௔ੱหྟ֥ਆᇕࠫ‫ޅ‬іൕٚ‫م‬ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 5-2 5.1 ௔ੱหྟ ၂aࠎЧ‫୑ۀ‬ ༯૫ၛ RC ‫׈‬ਫ਼ູ২đඪૼ௔ੱหྟ֥ࠎЧ‫୑ۀ‬b ഈ๭ຩ઎֥Ԯ‫ݦ־‬ඔູ C(s) = 1 đఃᇏ T = RC b R(s) Ts +1 ೏ຩ઎ൻೆູᆞ༿ྐ‫ݼ‬đࠧ r(t) = R sin ωt ൈđॖ֤ğlim c(t) = R sin(ωt − arctan ωT ) đ x→∞ 1+ ω 2T 2 ॖ࡮ຩ઎֥໗෿ൻԛಯಖ൞ᆞ༿‫׈‬࿢đః௔ੱ‫ބ‬ൻೆ‫׈‬࿢௔ੱཌྷ๝đ‫ږ‬ᆴ൞ൻೆ֥ 1 Пđཌྷ 1+ ω 2T 2 ࢘бൻೆᇌު arctan ωT đਆᆀ‫׻‬൞ω ֥‫ݦ‬ඔđ 1 ӫູ RC ຩ઎֥‫ږ‬௔หྟđ − arctan ωT ӫ 1+ ω 2T 2 ູ RC ຩ઎֥ཌྷ௔หྟb ᄵ 1 ie− j arctanωT = 1 j∠ 1 1 = G( jω) ӫູຩ઎֥௔ੱหྟb 1+ ω2T 2 1+ jωT ie = 1+ jωT 1+ jωT G( jω) = 1 1 bᆃ၂ࢲંؓ಩‫ޅ‬໗‫֥ק‬ཌྟ‫ק‬ӈ༢๤‫׻‬൞ᆞಒ֥b = 1+ jωT 1+ Ts s= jω ‫ؽ‬a௔ੱหྟ֥‫ק‬ၬ ௔ੱหྟğൻԛྐ‫֥ݼ‬ Fourier эߐაൻೆྐ‫֥ݼ‬ Fourier эߐᆭбbG( jω) = C( jω) = G(s) s= jω b R( jω) ః໾৘ၩၬّࠧ႘ਔ༢๤ؓᆞ༿ྐ‫֥ݼ‬೘նԮ‫ି־‬৯ğ๝௔aэ‫ږ‬a၍ཌྷb ‫ږ‬௔หྟ‫ݦ‬ඔğ A(ω) = G( jω) ཌྷ௔หྟ‫ݦ‬ඔğϕ(ω) = ∠G( jω) ൌ௔หྟ‫ݦ‬ඔğ Re(ω) = Re[G( jω)] ྴ௔หྟ‫ݦ‬ඔğ Im(ω) = Im[G( jω)] 66 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ೘a௔ੱหྟ֥ࠫ‫ޅ‬іൕٚ‫م‬ 1aࠞቕѓ๭ࠇ‫ږ‬ཌྷ௔ੱหྟ౷ཌčNyquist ౷ཌĎ ᆃᇕ౷ཌ֥หׄ൞Ϝ௔ੱ ω ुӮҕэਈđ֒ ω Ֆ 0 → ∞ э߄ൈđࡼ௔ੱหྟ֥‫ږ‬௔‫ބ‬ཌྷ௔หྟࠇൌ ௔‫ྴބ‬௔หྟ๝ൈіൕᄝ‫گ‬ඔ௜૫ഈb ࠞቕѓ๭߻ᇅൈႋᇿၩၛ༯ࠫׄğ æܿ‫ק‬ൌᇠᆞٚཟູཌྷ࢘ਬ؇ཌđ౷ཌ୉ൈᆌٚཟູᆞđඨൈᆌٚཟູ‫ڵ‬Ġ çؓႿ၂۱۳‫֥ק‬ω ᆴđсႵ၂۱‫ږ‬ᆴູ G( jω) aཌྷູ࢘ ∠G( jω) ֥ཟਈაᆭؓႋb è‫ږ‬ཌྷ౷ཌൌ࠽ഈ൞ G( jω) ֒ ω : 0 → ∞ э߄ൈđ൏ਈ G( jω) ֥൏؊ᄝ S ௜૫ഈ૭߻ԛ֥၂่౷ ཌĠ 2aؓඔቕѓ๭čBode ๭Ď ؓඔ௔ੱหྟ๭္ӫູѵ֣๭čBodeĎđ൞Ⴎؓඔ‫ږ‬௔౷ཌ‫ؓބ‬ඔཌྷ௔౷ཌ 2 ่౷ཌቆӮbؓඔ௔ੱ หྟ๭֥‫ޘ‬ቕѓ൞ω đҐႨၛ 10 ָູ֥ؓඔ lg ω ‫ٳ‬؇čؓႿ௔ੱω ൞٤ཌྟ‫ٳ‬؇Ďđֆ໊൞ rad s bؓ ඔ‫ږ‬௔౷ཌ֥ሺቕѓіൕؓඔ‫ږ‬௔หྟ֥‫ݦ‬ඔᆴđο 20 lg G( jω) नᄋཌྟ‫ٳ‬؇đֆ໊൞‫ٳ‬Н‫غ‬čdBĎđ ӈႨ‫ ݼژ‬L(ω) іൕđࠧ L(ω) = 20 lg G( jω) (dB)Ġؓඔཌྷ௔หྟ֥ሺቕѓູϕ(ω) = ∠G( jω) đο؇ č 0 Ďࠇ޶؇č rad Ďཌྟ‫ٳ‬؇b ᆴ֤ᇿၩ֥൞đ‫ޘ‬ቕѓၛ lg ω ཌྟ‫ٳ‬؇đ֌ؓႿ ω ಏ൞٤ཌྟ‫ٳ‬؇֥b ؓඔ௔ੱหྟ๭֥Ⴊׄğ æۚ௔҆‫ޘٳ‬ቕѓ֤֞ਔ࿢෪đ‫֮ط‬௔҆‫ٳ‬ཌྷؓᅚॺbՖ‫ط‬ॖၛঔն௔ੱ֥ܴҳٓຶđ๝ൈႻॖၛ ᄝ๝၂‫ږ‬๭ഈܴҳ֮௔‫֥؍‬༥ཬэ߄Ġ çॖၛࡥ߄ᄎෘ ॖၛࡼؓඔ‫ږ‬௔֥ཌྷӰᄎෘđэӮູਔ౼ؓඔၛު֥ࡆ‫م‬ᄎෘđՖ‫ط‬൐ᄎෘ֤֞ਔࡥ߄b èٚь‫ࡆן‬ቔ๭ ᄝؓඔቕѓ๭ഈđؓඔ௔ੱหྟॖၛႨ‫؍ٳ‬ᆰཌ֥ࡶ࣍ཌটіൕđၹՎᄝࣉྛ‫ࡆן‬ቔ๭ൈđᆺླေ ᄝᆰཌོੱэ߄ൈđྩᆞᆰཌོ֥ੱࠧॖb 5.2 ‫֥ࢫߌ྘ׅ‬௔ੱหྟ 1ēб২ߌࢫ Ԯ‫ݦ־‬ඔğ G(s) = K đ௔ੱหྟğ G( jω) = K đ ‫ږ‬௔หྟğ A(ω) = K đཌྷ௔หྟğϕ(ω) = 00 ؓඔ‫ږ‬௔หྟğ L(ω) = 20 lg K đؓඔཌྷ௔หྟğϕ(ω) = 00 ᄵࠞቕѓ๭ğ ѵ֣๭ğ 2ēࠒ‫ࢫߌٳ‬ Ԯ‫ݦ־‬ඔğ G(s) = 1 s đ௔ੱหྟğ G(s) = 1 jω đ‫ږ‬௔หྟğ A(ω) = 1 ω đ‫ږ‬ᆴ֥նཬა ω Ӯّ бđཌྷ௔หྟğϕ(ω) = −900 đၹՎ౷ཌᄝ‫ྴڵ‬ᇠഈb ؓඔ‫ږ‬௔หྟğ L(ω) = 20 lg 1 = −20 lgω đؓඔཌྷ௔หྟğϕ(ω) = −900 ω 67 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ࠞቕѓ๭ğ ѵ֣๭ğ 3ēັ‫ࢫߌٳ‬ Ԯ‫ݦ־‬ඔğ G(s) = s đ௔ੱหྟğ G( jω) = jω đ ‫ږ‬௔หྟğ A(ω) = ω đཌྷ௔หྟğϕ(ω) = 900 đ‫ږ‬ᆴնཬა ω ӮᆞбđၹՎ౷ཌᄝᆞྴᇠഈbؓ ඔ‫ږ‬௔หྟğ L(ω) = 20 lgω đؓඔཌྷ௔หྟğϕ(ω) = 900 ࠞቕѓ๭ğ ѵ֣๭ğ ັ‫ࢫߌٳ‬აࠒ‫֥ࢫߌٳ‬Ԯ‫ݦ־‬ඔ޺ູ֚ඔđؓඔ‫ږ‬௔หྟ‫ؓބ‬ඔཌྷ௔หྟࣇҵ၂۱‫ݼژ‬đၹՎ෰ૌ ֥ѵ֣๭ܱႿ‫ޘ‬ᇠؓӫb 4ēܸྟߌࢫ Ԯ‫ݦ־‬ඔğ G(s) = 1 (1+ Ts) đ௔ੱหྟ G( jω) = 1 (1+ jωT ) ğđ ‫ږ‬௔หྟğ A(ω) = 1 1 + (ωT )2 đཌྷ௔หྟğϕ(ω) = −arctgωT G( jω) = 1 (1+ jωT ) = 1 − j ωT 1+ (ωT )2 1+ (ωT )2 ؓඔ‫ږ‬௔หྟğ L(ω) = 20 lg1 1+ (ωT )2 = −20 lg 1+ (ωT )2 = ⎪⎧− 20 lg 1+ 0 = 0,ω << 1 T ⎨ ⎪⎩− 20 lg (ωT )2 = −20 lgω − 20 lgT ,ω >> 1 T 68 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ؓඔཌྷ௔หྟğϕ(ω) = −arctgωT ܸྟߌࢫ֥ؓඔཌྷ௔หྟ౷ཌູğ 5ē၂ࢨັ‫ࢫߌٳ‬ Ԯ‫ݦ־‬ඔğ G(s) = 1+ Ts đ௔ੱหྟ G( jω) = 1 + jωT ğđ ‫ږ‬௔หྟğ A(ω) = 1 + (ωT )2 đཌྷ௔หྟğϕ(ω) = arctgωT ؓඔ‫ږ‬௔หྟğ L(ω) = 20 lg 1+ (ωT )2 = ⎪⎧20 lg 1+ 0 = 0,ω << 1 T ⎨ ⎪⎩20 lg (ωT )2 = 20 lgω + 20 lgT ,ω >> 1 T ၂ࢨັ‫֥ؓࢫߌٳ‬ඔ‫ږ‬௔หྟ౷ཌაܸྟߌࢫ֥ؓඔ‫ږ‬௔หྟ൞၇‫ޘ‬ᇠӮࣤཞؓӫ֥b ؓඔཌྷ௔หྟğϕ(ω) = arctgωT ၂ࢨັ‫֥ؓࢫߌٳ‬ඔཌྷ௔หྟ౷ཌაܸྟߌࢫ֥ؓඔཌྷ௔หྟ൞၇‫ޘ‬ᇠӮࣤཞؓӫ֥b 69 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 20 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č2a3 ࢫĎ ᇶေଽಸ ‫֥ࢫߌ྘ׅ‬௔ੱหྟ षߌࠞቕѓ๭֥߻ᇅ ଢ֥აေ ᅧ໤۲ᇕ‫֥ࢫߌ྘ׅ‬௔ੱหྟࠣఃѵ֣๭‫ࠞބ‬ቕѓ๭֥หׄ ౰ ᅧ໤षߌࠞቕѓ๭֥߻ᇅ ᇗ ׄ ა ଴ ᇗׄğ۲ᇕ‫֥ࢫߌ྘ׅ‬ѵ֣๭aषߌࠞቕѓ๭֥߻ᇅ ׄ ଴ׄğषߌࠞቕѓ๭֥߻ᇅ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 5-4č2Ďač4Ď ቔြี 6ēᆒ֕ߌࢫ Ԯ‫ݦ־‬ඔğ G(s) = ωn2 Im s2 + 2ξωns + ωn2 ω=∞ ω =0 ௔ੱหྟ G( jω) = ωn2 Re ( jω)2 + 2ξωn ( jω) + ωn2 ξ1 ‫ږ‬௔หྟğ A(ω) = 1 ξ2 ξ1 > ξ2 > ξ3 (1− T 2ω 2 )2 + (2ξTω)2 ཌྷ௔หྟğ ϕ (ω ) = −arctg 2ξTω ξ3 − T 2ω 1 2 ൌ௔หྟğ Re(ω) = 1− T 2ω2 ྴ௔หྟğ Im(ω) = − 2ξTω (1− T 2ω2 )2 + (2ξTω)2 (1− T 2ω2 )2 + (2ξTω)2 ൔᇏ T = 1 ωn đູᆒ֕ߌࢫ֥ሇᅼ௔ੱb ᄝቅୄб ξ < 0.707 đ౏ ω = ωr ൈđᆒ֕ߌࢫ‫ؿ‬ളਔཾᆒbॖၛ౰֤ཾᆒൈ֥ ωr ‫ ބ‬Mr b d A(ω) = 0 ⇒ ωr = 1 1− 2ξ 2 dω T ω =ωr M r = A(ωr ) = 2ξ 1 1−ξ 2 ⎧ξ > 0.707ൈđ໭ཾᆒ‫ڂ‬ᆴđAčωĎֆ‫ט‬කࡨ ⎪⎨⎪⎪ξξ = 0.707ൈđਢࢸཾᆒđωr = 0đMr = 1 < 0.707ൈđႵཾᆒ‫ڂ‬ᆴđωr > 0đMr > 1 ⎪⎪ξ = 0ൈđωr = ωnđMr = ∞đᆃඪૼຓࡆ֥ྐ‫ݼ‬௔ੱӮ‫ٳ‬ა ⎪⎩༢๤֥ሱಖᆒ֕௔ੱཌྷ֩ൈđࡼႄఏ༢๤Ӂള‫܋‬ᆒགྷའ ؓඔ‫ږ‬௔หྟğ ⎧20 lg 1+ 0 = 0dB,ω << 1 T L(ω) = −20 lg (1− T 2ω 2 )2 + (2ξTω)2 = ⎪⎨− 20 lg 2ξ = −6.02 − 20 lgξ (dB),ω = 1 T ⎪⎩20 lg(Tω)2 = 40 lgω + 40 lgT (dB),ω >> 1 T ഈൔω = ωn = 1 T ӫູᆒ֕ߌࢫ֥ሇᅼ௔ੱb 70 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ᆒ֕ߌࢫ֥ؓඔ‫ږ‬௔หྟ౷ཌູğ ϕ(ω)(0 ) ؓඔཌྷ௔หྟğ ϕ (ω ) = − arctg 2ξTω 00 0.1 T 1 T ω − T 2ω 1 2 10 T ⎧00 ,ω << 1 T − 900 2ξTω ⎪ − 1800 − 90 0 dec − T 2ω ⎨− ϕ (ω ) = −arctg 1 2 = 900 ,ω = 1 T ⎪⎩−1800,ω >> 1 T 7ē‫ࢫߌٳັࢨؽ‬ Ԯ‫ݦ־‬ඔğ G(s) = s2 ωn2 = + 1 + 1 , T= 1 + 2ξωns + ωn2 Ts 2 2ξTs ωn ௔ੱหྟğ G( jω) = T 2 ( jω)2 + 2ξT ( jω) +1 ‫ږ‬௔หྟğ A(ω) = (1 − T 2ω 2 )2 + (2ξTω)2 = (1− T 2ω2 ) + j(2ξTω) ཌྷ௔หྟğϕ(ω) = arctg 2ξTω 1− (Tω)2 ൌ௔หྟğ Re(ω) = 1− T 2ω2 đྴ௔หྟğ Im(ω) = 2ξTω ؓඔ‫ږ‬௔หྟğ ‫֥ؓࢫߌٳັࢨؽ‬ඔ௔ੱหྟაᆒ֕ߌࢫ൞၇‫ޘ‬ᇠӮࣤཞؓӫ֥đႿ൞Ⴕ ω = ωn = 1 T ູ‫ࢨؽ‬ ັ‫֥ࢫߌٳ‬ሇᅼ௔ੱb ؓඔཌྷ௔หྟğϕ(ω) = arctg 2ξTω 1 − T 2ω 2 ‫֥ؓࢫߌٳັࢨؽ‬ඔཌྷੱหྟაᆒ֕ߌࢫ൞၇‫ޘ‬ᇠӮࣤཞؓӫ֥ğ 8ē࿼Ӿߌࢫ 71 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Ԯ‫ݦ־‬ඔğ G(s) = e−τs ,௔ੱหྟğ G( jω) = e− jωτ ‫ږ‬௔หྟğ A(ω) = 1,ཌྷ௔หྟğϕ(ω) = −ωτ ‫ږ‬ཌྷ௔ੱหྟğ ؓඔ‫ږ‬௔หྟğ L(ω) = 20 lg A(ω) = 20 lg1 = 0 ؓඔཌྷ௔หྟğϕ(ω) = −ωτ 5.3 ༢๤षߌ௔ੱหྟ֥߻ᇅ ၂a षߌࠣቕѓ๭ ࡼषߌԮ‫ݦ־‬ඔіൕູൈࡗӈඔіղྙൔ m1 m2 ∏ ∏ + bm-1s + bm ∏ ∏ + an-1s + an (τ k s +1) (τ 2 s 2 + 2τ lς s + 1) l b0 s m + b1s m-1 + l K a0 s n + a1s n-1 + sν G(s) = = K k =1 l =1 = G0 (s) n1 n2 sν (Tis +1) (T 2 s 2 + 2Tjξ js + 1) j i =1 j =1 ൔᇏğ m1 + 2m2 = m đν + n1 + 2n2 = n č1Ďಒ‫ࠞק‬ቕѓ๭֥ఏׄ ࠞቕѓ๭֥ఏׄ൞ ω → 0 ൈ Gk ( j0+ ) ᄝ‫گ‬௜૫ഈ໊֥ᇂb Gk ( j0+ ) = ( K G0 ( jω) = ( K jω )ν jω )ν ω→0 ⎧⎪⎪νࠧ=ࠞ0ቕൈѓđ๭GKఏ( j൓ω)Ⴟ=ᆞK∠ൌ0ᇠ0 ,ഈ֥ଖ၂ׄ ⎪⎨ν ≠ 0ൈđGK ( jω) = ∞∠ −ν × 900 , ⎪⎩ࠧࠞቕѓ๭ఏ൓Ⴟ໭౫ჹԩ č2Ďಒ‫ࠞק‬ቕѓ֥ᇔׄč n > m Ď ࠞቕѓ๭֥ఏׄ൞ω → +∞ ൈ Gk (+ j∞) ᄝ‫گ‬௜૫ഈ໊֥ᇂb 72 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Gk (+ j∞) = b0sm + b1sm-1 + + bm-1s + bm = b0 × ( 1 n−m = b0 a0 a0sn + a1sn-1 + + an-1s + an a0 jω) ( jω)n−m ω→∞ ⎧n − m = 1ൈđGK ( jω) = 0∠ − 900đՖ‫ྴڵ‬ᇠ֥ٚཟࣉೆቕѓჰׄ ⎪ − m = 2ൈđGK ( jω) = 0∠ −1800đՖ‫ڵ‬ൌᇠ֥ٚཟࣉೆቕѓჰׄ ⎨n ⎪⎩n − m = 3ൈđGK ( jω) = 0∠ − 2700đՖᆞྴᇠ֥ٚཟࣉೆቕѓჰׄ č3Ďࠞቕѓ๭Ԭᄀൌᇠ໊֥ᇂ ਷௔ੱหྟ G( jω) ֥ྴູ҆ਬđࠧ Im[G( jω)] = 0 đѩ౰֤ཌྷႋ֥௔ੱωx đಖުࡼՎ௔ੱ ωx սೆ ௔ੱหྟ G( jω) ֥ൌ҆đᄵ Re[G( jωx )]ࣼ൞ࠞቕѓ๭აൌᇠ֥ࢌׄb ২ğഡ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = k (T1s +1)(T2s +1) đ൫߻ᇅః‫ږ‬ཌྷ౷ཌb ࢳğ༢๤֥षߌ௔ੱหྟğ GK ( jω ) = k 1 1 T2 T1T2 ( jω + T1 )( jω + ) Ⴎषߌ௔ੱหྟॖᆩđ༢๤ູ 0 ྘đࠧν = 0 b ‫ږ‬ཌྷ౷ཌ֥ఏູׄğ GK ( j0) = k∠00 đ‫ږ‬ཌྷ౷ཌ֥ᇔູׄğ GK ( j∞) = 0∠ −1800 b ՙ੻߂ԛ‫ږ‬ཌྷ౷ཌೂ༯ğ ২ğၘᆩֆ໊‫ّڵ‬ঌ༢๤षߌԮ‫ݦ־‬ඔູ Gk (s) = 1 đ൫߻ᇅ༢๤‫ږ‬ཌྷ౷ཌb s(s +1) ࢳğ๤֥षߌ௔ੱหྟğ GK ( jω) = 1 =− 1 − j ω (1 1 ω 2 ) jω( jω +1) 1+ω2 + Ⴎषߌ௔ੱหྟॖᆩđ༢๤ູ I ྘đࠧν = 1 b Ⴟ൞‫ږ‬ཌྷ౷ཌ֥ఏູׄğ GK ( j0) = ∞∠ − 900 đ֒ω = 0 ൈđൌ҆‫ݦ‬ඔႵࡶ࣍ཌ-1b Ⴎ n − m = 2 ॖ֤‫ږ‬ཌྷ౷ཌ֥ᇔູׄğ GK ( j∞) = 0∠ −1800 ๙‫ٳݖ‬༅ൌ҆‫ݦ҆ྴބ‬ඔॖᆩაቕѓᇠ໭ࢌׄbႮഈ‫ٳ‬༅ࢲંቔԛ༢๤֥षߌࠞቕѓ๭ೂ༯ğ 73 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ২ğഡ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = k(2s +1) s2 (0.5s +1)(s +1) đ൫ՙ੻߻ᇅః‫ږ‬ཌྷ౷ཌb ࢳğ༢๤֥षߌ௔ੱหྟğ GK ( jω) = ( jω)2 ( k( j2ω +1) jω + 1) j0.5ω +1)( Ⴎषߌ௔ੱหྟॖᆩđ༢๤ູ 2 ྘đࠧν = 2 b Ⴟ൞‫ږ‬ཌྷ౷ཌ֥ఏູׄğ GK ( j0) = ∞∠ −1800 ‫ږ‬ཌྷ౷ཌ֥ᇔູׄğ GK ( j∞) = 0∠ − 2700 ‫ږ‬ཌྷ౷ཌაൌᇠ֥ࢌׄğ k( j2ω +1) k ( jω)2 ( j0.5ω +1)( jω ω 2 (1+ 0.25ω 2 )(1+ ω 2 ) [ ] GK ( jω) = + 1) = − (1+ 2.5ω2 ) − j(0.5 − ω 2 ) Im[GK ( jω)] = 0.5 − ω 2 = 0 đ౰֤ ω 2 = 0.5 đսೆ౰֤‫ږ‬ཌྷ౷ཌაൌᇠ֥ࢌູׄ x Re[GK ( jωx )] = −2.67k ՙ੻߂ԛ‫ږ‬ཌྷ౷ཌೂ༯  74 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 21 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č3 ࢫĎ ᇶေଽಸ ॥ᇅ༢๤֥षߌ௔ੱหྟ ଢ֥აေ ᅧ໤षߌ Bode ๭֥߻ᇅ ౰ ᅧ໤۴ऌ Bode ๭ಒ‫ק‬ቋཬཌྷ໊༢๤֥Ԯ‫ݦ־‬ඔ ᇗ ׄ ა ଴ ᇗׄğषߌ Bode ๭֥߻ᇅa۴ऌ Bode ๭ಒ‫ק‬ቋཬཌྷ໊༢๤֥Ԯ‫ݦ־‬ඔ ׄ ଴ׄğ๝ഈ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 5-3č4Ďa5-5č2Ďč3Ď ቔြี ‫ؽ‬aषߌѵ֣๭ ൭‫۽‬ቔ๭֥၂Ϯ҄ᇧğ č1ĎࡼषߌԮ‫ݦ־‬ඔіൕູൈࡗӈඔіղྙൔ m1 m2 ∏ ∏ + bm-1s + bm ∏ ∏ + an-1s + an (τ k s +1) (τ 2 s 2 + 2τlς l s + 1) l b0 s m + b1sm-1 + K G(s) = a0 s n + a1sn-1 + = K k =1 l =1 = sν G0 (s) n1 n2 sν (Tis +1) (T 2 s 2 + 2Tjξ s + 1) j j i =1 j =1 ൔᇏğ m1 + 2m2 = m đν + n1 + 2n2 = n č2Ď౰ 20 lg K ֥ᆴđѩૼಒࠒ‫֥ࢫߌٳ‬۱ඔν č3Ďಒ‫ק‬۲‫֥ࢫߌ྘ׅ‬ሇᅼ௔ੱđѩοႮཬ֞նஆ྽ č4Ďಒ‫֮ק‬௔‫࣍ࡶ؍‬ཌb ֮௔‫؍‬௔ੱหྟູ GK ( jω) = ( K G0 ( jω) = ( K jω )ν jω )ν ω→0 ؓඔ‫ږ‬௔หྟູğ L(ω ) = 20 lg ( K )ν = 20 lg K − 20 ×ν × lgω jω ؓඔཌྷ௔หྟູğϕ(ω) = −ν × 900 ഈඍіૼğ æ֮௔‫֥ؓ؍‬ඔ‫ږ‬௔หྟᆰཌོ֥ੱູ − 20×νdB dec đཌྷ௔࢘؇ູ −ν × 900 Ġ ç֒ ω = 1 ൈđ֮௔‫؍‬ᆰཌࠇః࿼Ӊཌčᄝ ω < 1֥ٓຶଽႵሇᅼ௔ੱĎ֥‫ٳ‬Нᆴູ 20 lg K đᆃ ൞ၹູႮ֮௔‫ږ֥؍‬௔ٚӱđॖ֤֞ L(ω) = 20 lg K − 20 ×ν × lgω ω=1 = 20lg K 1 è֮௔‫؍‬ᆰཌčࠇః࿼ӉཌĎაਬ‫ٳ‬Нཌč‫ޘ‬ᇠĎ֥ࢌׄ௔ੱູ ω0 = Kν đؓႿ I ྘༢๤ࢌׄ௔ ੱູ ω0 = K đII ྘༢๤ࢌׄ௔ੱູ ω0 = K Ġᆃ൞ၹູႮ֮௔‫ږ֥؍‬௔ٚӱđॖ֤֞ L(ω) = 20 lg K − 20 ×ν × lgω = 0 ⇒ 20 lg K = 20 ×ν × lgω = 20 lgων 1 Ⴟ൞Ⴕğ ων = K ⇒ ω0 = K ν 75 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č5Ď߻ᇅᇏ௔‫؍‬ ൮༵ᄝ‫ޘ‬ቕѓᇠഈࡼሇᅼ௔ੱοՖ֥֮֞ۚඨ྽ѓԛ۲ሇᅼ௔ੱbಖުđ၇Ցᄝ۲ሇᅼ௔ੱԩ‫ڿ‬э ᆰཌོ֥ੱđ‫ڿ‬э֥‫؟‬ഒ౼थႿሇᅼԩߌࢫ֥ྟᇉđೂܸྟߌࢫོ֥ੱູ − 20dB dec đᆒ֕ߌࢫູ − 40dB dec đ၂ࢨັ‫ ູࢫߌٳ‬+ 20dB dec đ‫ ູࢫߌٳັࢨؽ‬+ 40dB dec ֩֩b ২ğၘᆩֆّ໊ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = 100(s + 2) s(s +1)(s + 20) đ൫߻ᇅఃषߌ ༢๤֥ Bode ๭b ࢳğႮ۳‫֥ק‬༢๤षߌԮ‫ݦ־‬ඔ֤֞༢๤֥षߌ௔ੱหྟđѩࡼః݂၂߄ GK ( jω) = 100( jω + 2) jω( jω +1)( jω + 20) = 100 × 2( j0.5ω +1) jω( jω +1) × 20 × ( j0.05ω +1) ႮՎॖ࡮đ༢๤ֹषߌ௔ੱหྟႵ 5 ۱‫ܒࢫߌ྘ׅ‬Ӯđ‫ٳ‬љູ č1Ďб২ߌࢫğ G1( jω) = 10 đL L1(ω) = 20 lg10 = 20dB,ϕ1(ω) = 00 č2Ď၂ࢨັ‫ࢫߌٳ‬ğ G2 ( jω) = j0.5ω +1đሇᅼ௔ੱω = 1 T = 1 0.5 = 2 č3Ďࠒ‫ࢫߌٳ‬ğ G3 ( jω) = 1 jω č4Ď၂ࢨܸྟߌࢫğ G2 ( jω) = 1 ( jω +1) đሇᅼ௔ੱω = 1 T = 1 1 = 1 č5Ď၂ࢨܸྟߌࢫğ G2 ( jω) = 1 ( j0.05ω +1) đሇᅼ௔ੱω = 1 T = 1 0.05 = 20 ‫ކ‬Ӯު֥༢๤षߌؓඔ‫ږ‬௔หྟğ L(ω) = L1(ω) + L2 (ω) + L3(ω) + L4 (ω) + L5 (ω) ‫ކ‬Ӯު֥༢๤षߌؓඔཌྷ௔หྟğϕ(ω) = ϕ1(ω) + ϕ2 (ω) + ϕ3 (ω) + ϕ4 (ω) + ϕ5 (ω) ೘aቋཬཌྷ໊༢๤ ቋཬཌྷ໊༢๤‫ק‬ၬğ༢๤षߌԮ‫ݦ־‬ඔ֥ਬׄaࠞׄಆ໊҆Ⴟ S ቐ϶௜૫đ๝ൈႻ໭Ղᇌުߌࢫ֥ ༢๤ӫູቋཬཌྷ໊༢๤b‫ڎ‬ᄵࣼ൞٤ቋཬཌྷ໊༢๤b 76 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ൫߻ᇅ G1(s) = 1+ s ‫ ބ‬G2 (s) = 1− s ֥ؓඔ௔ੱหྟೂ༯ѩбࢠೂ༯ 1+ 2s 1+ 2s ⎪⎨⎧L1(ω) = −20 lg 1 + ω 2 − 20 lg lg 1+ 4ω 2 đ ⎩⎨⎧ϕϕ12((ωω)) = arctgω − arctg 2ω ⎪⎩L2 (ω) = −20 lg 1 + ω 2 − 20 lg lg 1+ 4ω 2 = −arctgω − arctg2ω Ⴎഈඍбࢠđॖၛ֤ԛೂ༯ࢲં č1Ď֒ω = 0 → ∞ э߄ൈđቋཬཌྷ໊༢๤֥ཌྷ࢘э߄ቋཬđ‫ط‬٤ቋཬཌྷ໊༢๤֥ཌྷ࢘э߄၂ϮࢠնĠ č2Ďቋཬཌྷ໊༢๤֥ؓඔ‫ږ‬௔ L(ω) ོ֥ੱэ߄౴൝აؓඔཌྷ௔ϕ(ω) ֥э߄౴൝၂ᇁđ‫ط‬٤ቋཬཌྷ ໊༢๤ᄵ҂ಖb ႮႿቋཬཌྷ໊༢๤֥‫ږ‬௔აཌྷ௔֥၂၂ؓႋܱ༢đၹՎॖၛࣇႮ༢๤֥षߌ‫ږ‬௔หྟটಒ‫ק‬༢๤֥ ௔ੱหྟčࠇԮ‫ݦ־‬ඔĎđ‫ط‬҂߶ႄఏఅၳb ২ğၘᆩ༢๤֥षߌؓඔ‫ږ‬௔หྟೂ༯đ൫ಒ‫ק‬༢๤֥षߌԮ‫ݦ־‬ඔb ࢳğႮ๭ॖ࡮đ֮௔‫ ູੱོ֥؍‬− 20dB dec đ෮ၛषߌԮ‫ݦ־‬ඔႵ၂۱ࠒ‫ࢫߌٳ‬b ႮႿᄝ֮௔‫ ؍‬ω = 1 ൈđ L(ω) = 15dB đ෮ၛ༢๤֥षߌ٢նПඔູડቀ 20 lg K = 15 đՖ‫ط‬౰֤ K = 1015 20 = 100.75 = 5.6 b ႮՎॖၛཿԛ༢๤֥षߌԮ‫ݦ־‬ඔູ 5.6( 1 s + 1) 5.6(0.14s + 1) 7 G(s) = = s(1 s +1) s(0.5s +1) 2 ௔ੱหྟູğ G( jω) = 5.6( j ω + 1) đཌྷ௔หྟູğ ϕ (ω ) = −900 + arctg ω − arctg ω  jω( j + 1) 7 2 7 ω 2 77 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 22 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ᇶေଽಸ ଱൦໗‫ק‬஑ऌ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č4 ࢫĎ ଢ֥აေ ਔࢳ‫ڣ‬ᇹ‫ݦ‬ඔ֥‫ܒ‬Ӯၛࠣ଱൦஑ऌ֥๷֝‫ݖ‬ӱ ౰ ᅧ໤଱൦໗‫ק‬஑ऌࠣᄹҀཌ֥߻ᇅ ᇗ ׄ ა ଴ ᇗׄğ ଱൦໗‫ק‬஑ऌ ׄ ଴ׄğ ᄹҀཌ֥߻ᇅa໗‫֥ק‬஑؎ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 5-10 5.4 ଱উථห໗‫ק‬஑ऌ č1ĎषߌԮ‫ݦ־‬ඔીႵࠒ‫ࢫߌٳ‬ ၘᆩषߌ༢๤หᆘٚӱൔᄝ S Ⴗ϶௜૫֥۴֥۱ඔູ Pđ֒ ω Ֆ 0 → ∞ э߄ൈđषߌ௔ੱหྟ֥݅ ࠖᄝ G( jω)H ( jω) ௜૫ຶಡ (−1, j0) ֥ׄಁඔູ R čܿ‫ק‬ඨൈᆌ࿈ሇູ‫ڵ‬đ୉ൈᆌ࿈ሇູᆞĎđᄵоߌ ༢๤หᆘٚӱൔᄝ S Ⴗ϶௜૫֥۱ඔູ Z đ౏ Z = P − 2R b ೏ Z = 0 đᄵоߌ༢๤หᆘ۴न໊Ⴟ S ቐ϶௜૫đоߌ༢๤໗‫ק‬Ġ ೏ Z ≠ 0 đᄵоߌ༢๤Ⴕ Z ۱หᆘ۴ᄝ S Ⴗ϶௜૫đоߌ༢๤҂໗‫ק‬b ၛഈ֥֩ࡎіൕ ؓႿषߌ໗‫֥ק‬༢๤čࠧ P = 0 Ďđ GK (s) ᄝ S Ⴗ϶௜૫ഈ໭ࠞׄđоߌ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱ ൞ఃषߌࠞቕѓ౷ཌ҂Їຶ (−1, j0) ׄĠ ؓႿषߌ҂໗‫֥ק‬༢๤čࠧ P ≠ 0 ĎđGK (s) ᄝ S Ⴗ϶௜૫ഈႵ P ۱ࠞׄđоߌ༢๤໗‫֥ק‬ԉ‫ٳ‬с ေ่ࡱ൞ఃषߌࠞቕѓ౷ཌ֒ω Ֆ 0 → +∞ э߄ൈđၛ୉ൈᆌٚཟЇຶ (−1, j0) ׄ P 2 ಁđࠧ R = P 2 Ġ ೏оߌ༢๤҂໗‫ק‬đᄵ‫ھ‬༢๤ᄝ s Ⴗ϶௜૫֥ࠞׄ۱ඔູ Z = P − 2R đ R ູࠞቕѓ౷ཌຶಡ (−1, j0) ֥ׄಁඔčܿ‫ק‬ඨൈᆌ࿈ሇູ‫ڵ‬đ୉ൈᆌ࿈ሇູᆞĎb ๷ંğ೏ࠞቕѓ౷ཌඨൈᆌٚཟЇຶ (−1, j0) ׄ (R < 0) đᄵ҂ંषߌ༢๤໗‫ק‬ა‫ڎ‬đоߌ༢๤ሹ ൞҂໗‫֥ק‬b ২ğၘᆩ۲༢๤֥षߌ‫ږ‬ཌྷ௔ੱ౷ཌೂ༯đ൫஑љఃоߌ༢๤֥໗‫ྟק‬b ࢳğ๭čaĎđၘᆩ p = 0 đ౏ R = 0 đࠧषߌ‫ږ‬ཌྷ౷ཌીႵЇຶč-1đj0Ďׄđ෮ၛоߌ༢๤໗‫ק‬Ġ ๭ č b Ďđ ၘ ᆩ p = 0 đ ౏ R = −1 đ ࠧ ष ߌ ‫ ږ‬ཌྷ ౷ ཌ ඨ ൈ ᆌ Ї ຶ č -1 đ j0 Ď 1 ಁ đ Ⴟ ൞ Z = P − 2R = 0 − 2 × (−1) = 2 ≠ 0 đႵ 2 ۱оߌหᆘ۴໊Ⴟ S Ⴗ϶௜૫đ෮ၛоߌ༢๤໗‫ק‬Ġ ๭čcĎđၘᆩ p = 0 đ౏ R = −1 2 +1 2 = 0 đࠧषߌ‫ږ‬ཌྷ౷ཌીႵЇຶč-1đj0Ďׄđ෮ၛоߌ༢ ๤໗‫ק‬b ২ğֆ໊‫ّڵ‬ঌ༢๤षߌԮ‫ݦ־‬ඔູ GK (s) = K Ts −1đ൫஑؎оߌ༢๤֥໗‫ྟק‬b ࢳğ߻ᇅ‫ږ‬ཌྷ౷ཌ๭ 78 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ GK ( jω) = K jTω −1 = −k − j 1 kT ω 2 1+ T 2ω2 +T 2ω ఏׄğ GK ( j0) = k∠ −1800 đᇔׄğ GK ( j∞) = 0∠ − 900 đ‫ٳ‬༅ൌ҆‫ݦ҆ྴބ‬ඔॖᆩაൌᇠ໭ࢌׄ čԢఏׄĎbቔࠞቕѓ๭ೂ༯ğ ༢๤Ⴕ၂۱षߌ໊ࠞׄႿႷ϶௜૫đ෮ၛ P=1b ֒ −k < −1ࠧ k > 1ൈđ R = 1/ 2 đᄵ Z = P − 2R = 0 đ༢๤໗‫ק‬Ġ ֒ −k > −1 ࠧ 0 < k < 1ൈđ R = 0 đᄵ Z = P − 2R = 1 ≠ 0 đ༢๤҂໗‫ק‬b č2Ď೏षߌԮ‫ݦ־‬ඔႵࠒ‫ࢫߌٳ‬ ֒षߌԮ‫ݦ־‬ඔႵࠒ‫ࢫߌٳ‬ൈđԩ৘֥ٚ‫م‬൞Ֆ‫ږ‬ཌྷ౷ཌ ω = 0+ ໊֥ᇂष൓đခ୉ൈᆌٚཟ߂ ν × 900 ֥ჵ޶ᄹҀཌč৘ંഈ϶ູࣥ໭౫նĎđᄝ࠹ෘЇຶ (−1, j0) ֥ׄಁඔൈđေ৵๝෮߂֥ჵ޶ᄹҀ ཌᄝଽb ২ğၘᆩ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = K s(T1s +1)(T2s +1) đ൫߂ԛ‫ږ‬ཌྷ౷ཌ๭đѩ஑љоߌ༢ ๤֥໗‫ྟק‬b ࢳğ߻ᇅ‫ږ‬ཌྷ౷ཌ๭ ఏׄğ GK ( j0) = ∞∠ − 900 đᇔׄğ GK ( j∞) = 0∠ − 3× 900 = ∠ − 2700 აൌᇠ֥ࢌׄğ GK ( jω) = K s(T1 jω + 1)(T2 jω + 1) = 1+ ω2 (T12 + 1 ) + ω 4T12T22 ⎢⎣⎡− K (T1 + T2 ) − j K (1 − ω 2T1T2 )⎥⎦⎤ ਷ T22 ω ྴ҆֩Ⴟਬđ֤֞ğ1 − ω 2 T T đ౰֤ğ ω 2 =1 T1T2 x 1 2 Ⴟ൞‫ږ‬ཌྷ౷ཌაൌᇠ֥ࢌׄቕѓູ − K (T1 + T2 ) = − KT1T2 1 + ω 2 (T12 + T22 ) + ω 4T12T22 ω 2 =1 T1T2 T1 + T2 x ႮՎ߂ԛ‫ږ‬ཌྷ౷ཌೂ༯ ႮႿν = 1 đ෮ၛླՖω = 0+ ໊֥ ᇂष൓୉ൈᆌ߂ 900 ֥ᄹҀཌđೂ๭ᇏྴཌ෮ൕđ࠹ෘ‫ږ‬ཌྷ౷ཌЇຶ (−1, j0) ֥ׄಁඔb ֒ − KT1T2 < −1 đࠧ K > T1 + T2 ൈđ‫ږ‬ཌྷ౷ཌඨൈᆌЇຶ (−1, j0) ׄ 1 ಁđࠧ R = −1 đႿ൞ T1 + T2 T1T2 Z = P − 2R = 2 đ෮ၛоߌ༢๤҂໗‫ק‬Ġ ֒ − KT1T2 > −1 đࠧ K < T1 + T2 ൈđ‫ږ‬ཌྷ౷ཌ҂Їຶ (−1, j0) ׄđࠧ R = 0 đႿ൞ Z = P − 2R = 0 đ T1 + T2 T1T2 ෮ၛоߌ༢๤໗‫ק‬b 79 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ২ğၘᆩ༢๤षߌԮ‫ݦ־‬ඔູ GK (s) = k(0.1s +1) đႮ଱൦໗‫ק‬஑ऌ஑؎оߌ༢๤֥໗‫ྟק‬b s(s −1) ࢳğ߻ᇅ‫ږ‬ཌྷ౷ཌ๭ ༢๤֥௔ੱหྟູ GK ( jω) = k( j0.1ω +1) = −1.1ωk + j(1− 0.1ω2 )k jω( jω −1) ω(1+ ω2 ) ఏׄğ GK ( j0) = ∞∠ − 2700 đ౏Ⴕࡶ࣍ཌ Re(0+ ) = −k Ġ ᇔׄğ GK ( j∞) = 0∠ − 3× 900 = ∠ − 2700 აൌᇠ֥ࢌׄğ֒ Im(ω) = 0 ൈđω = 10 սೆൌ҆đ֤ Re(ω) = −0.1k b ॖቔԛ༢๤ࠞቕѓ๭ೂ༯ğ ႮႿν = 1 đ෮ၛླՖ ω = 0+ ໊֥ᇂष൓୉ൈᆌ߂ 900 ֥ᄹҀཌđೂ๭ᇏྴཌ෮ൕb ႮषߌԮ‫ݦ־‬ඔॖᆩ P=1b ֒֒ −0.1k < −1ࠧ k > 10 ൈđ R = 1−1/ 2 = 1/ 2 đᄵ Z = P − 2R = 0 đ༢๤໗‫ק‬Ġ ֒ −0.1k > −1 ࠧ 0 < k < 10 ൈđ R = −1/ 2 đᄵ Z = P − 2R = 2 ≠ 0 đ༢๤҂໗‫ק‬b  80 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 23 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č4a5 ࢫĎ ᇶေଽಸ ѵ֣๭ഈ֥໗‫ק‬஑ऌ ໗‫ק‬ღ؇ ଢ֥აေ ਔࢳࠞቕѓ๭აѵ֣๭֥ؓႋ ౰ ᅧ໤ѵ֣๭ഈ֥໗‫ק‬஑ऌ ᅧ໤໗‫ק‬ღ؇֥࠹ෘ ᇗ ׄ ა ଴ ᇗׄğ ѵ֣๭ഈ֥໗‫ק‬஑ऌa໗‫ק‬ღ؇֥࠹ෘ ׄ ଴ׄğ໗‫ק‬ღ؇֥࠹ෘ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 5-7a5-9 ቔြี ඹa଱উථห໗‫ק‬஑ऌᄝѵ֣๭ᇏ֥ႋႨ č1ĎܱႿ‫ږ‬ཌྷ౷ཌຶಡ (−1, j0) ׄಁඔ R ֥࠹ෘ ‫ڵ‬Ԭᄀğ‫ږ‬ཌྷ౷ཌႮ༯ཟഈč‫ཬࡨ࢘ږ‬ĎԬᄀ (−∞,−1) ౵ࡗ֥Ԭᄀӫູ‫ڵ‬Ԭᄀđ࠹ູ N− = 1b‫ط‬ Ֆ (−∞,−1) ౵ࡗൌᇠष൓֥‫ڵ‬Ԭᄀӫູ϶Ց‫ڵ‬Ԭᄀđ࠹ູ N− = 1 2 ĠᆞԬᄀğ‫ږ‬ཌྷ౷ཌႮഈཟ༯č‫ږ‬ ࢘ᄹնĎԬᄀ (−∞,−1) ౵ࡗ֥ԬᄀӫູᆞԬᄀđ࠹ູ N+ = 1b‫ط‬Ֆ (−∞,−1) ౵ࡗൌᇠष൓֥ᆞԬᄀӫ ູ϶ՑᆞԬᄀđ࠹ູ N+ = 1 2 bႿ൞đ‫ږ‬ཌྷ౷ཌຶಡ (−1, j0) ֥ׄಁඔູğ R = N+ − N− č2Ďषߌ༢๤֥ࠞቕѓ๭აཌྷႋ֥ؓඔቕѓ๭֥ؓႋܱ༢ æࠞቕѓ๭ഈ֥ֆ໊ჵؓႋႿؓඔቕѓ๭ഈ֥ਬ‫ٳ‬НཌĠ çࠞቕѓ๭ഈ֥‫ڵ‬ൌᇠؓႋႿؓඔቕѓ๭ഈ֥ −1800 ཌྷ໊ཌb č3Ďؓඔ௔ੱหྟ๭ഈ֥ᆞ‫ڵ‬Ԭᄀ Ⴎᆞ‫ڵ‬Ԭᄀ֥‫ק‬ၬ‫ބ‬षߌ༢๤֥ࠞቕѓ๭აཌྷႋ֥ؓඔቕѓ๭֥ؓႋܱ༢ॖ࡮đؓඔ௔ੱ หྟ๭ഈ֥ᆞ‫ڵ‬Ԭᄀູğᄝؓඔቕѓ๭ഈ L(ω) > 0č A(ω) > 1Ď֥ٓຶଽđ֒ω ᄹࡆൈđཌྷ௔หྟ౷ ཌϕ(ω) ՖഈԬ‫ ݖ‬−1800 ཌྷ໊ཌčཌྷ໊ࡨཬĎ֥ӫູ‫ڵ‬ԬᄀĠ‫ط‬ཌྷ௔หྟ౷ཌϕ(ω) Ֆ༯Ԭ‫ ݖ‬−1800 ཌྷ ໊ཌčཌྷ໊ᄹնĎ֥ӫູᆞԬᄀb 81 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č4Ďؓඔ௔ੱ໗‫ק‬஑ऌ ೏༢๤Ї‫ࢫߌٳࠒݣ‬đᄝؓඔཌྷ௔౷ཌ ω ູ 0+ֹ֥ٚҀ߂၂่Ֆ GK ( j0+ ) +ν × 90° ֞ GK ( j0+ ) ֥ྴ ཌđ࠹ෘᆞ‫ڵ‬ԬᄀՑඔൈႋࡼҀ߂֥ྴཌुӮؓඔཌྷ௔หྟ౷ཌ֥၂҆‫ٳ‬b ഡ P ູषߌԮ‫ݦ־‬ඔ GK (s) ᄝ S Ⴗ϶௜૫֥ࠞׄඔđоߌ༢๤໗‫֥ק‬ԉ‫ٳ‬сေ่ࡱ൞đؓඔቕѓ๭ ഈ‫ږ‬௔หྟ L(ω) > 0 ֥෮Ⴕ௔‫؍‬ଽđ֒௔ੱω ᄹࡆൈđؓඔཌྷ௔หྟؓ −1800 ཌྷ໊ཌ֥ᆞ‫ڵ‬ԬᄀՑඔҵ ູP 2b ২ğၘᆩଖ‫ّڵ‬ঌ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = K s2 (Ts + 1) đ൫஑љоߌ༢๤֥໗‫ྟק‬b ࢳğč1Ď߂ԛषߌؓඔ௔ੱหྟ౷ཌೂ༯๭Ġč2Ď P = 0,ν = 2, m = 0, n = 3 Ġ č3Ď߂ԛᄹҀཌč๭ᇏྴཌĎĠč4Ď N+ = 0, N− = 1đᄵ R = N+ − N− = −1 č5Ď Z = P − 2R = 2 ≠ 0 đоߌ༢๤҂໗‫ק‬b 5.5 ໗‫ק‬ღ؇ č1Ď༢๤໗‫ܱ֥ྟק‬༢ ॖၛࡼቋཬཌྷ໊༢๤֥໗‫ۀྟק‬ওູğ ⎧҂ຶಡ(−1, j0)ׄൈ,оߌ༢๤໗‫ק‬ ‫ږ‬ཌྷ௔ੱ౷ཌ⎪⎨‫(ݖ‬−1, j0)ׄൈ,оߌ༢๤ਢࢸ໗‫ק‬ ⎪⎩ຶಡ(−1, j0)ׄൈ,оߌ༢๤҂໗‫ק‬ ⎧L(ω)‫ݖ‬0dBཌު,ϕ(ω) Ҍ‫ ݖ‬-1800ཌྷ໊ཌൈ,оߌ༢๤໗‫ק‬ ؓඔ௔ੱ౷ཌ⎪⎨L(ω)‫ݖ‬0dBཌ,ϕ(ω)္‫ ݖ‬-1800ཌྷ໊ཌൈ,оߌ༢๤ਢࢸ໗‫ק‬ ⎪⎩L(ω)‫ݖ‬0dBཌభ,ϕ(ω)ၘ‫ ݖ‬-1800ཌྷ໊ཌൈ,оߌ༢๤҂໗‫ק‬ č2Ď‫ږ‬ᆴ໗‫ק‬ღ؇ h (Lg ) ਷‫ږ‬ཌྷ౷ཌԬᄀ −1800 ཌྷ໊ཌ෮ؓႋ֥௔ੱູωg đᆃ۱௔ੱӫູཌྷ࢘Ԭᄀ௔ੱđՎ௔ੱ෮ؓႋ֥‫ږ‬ ᆴູ A(ωg ) b ‫ږ‬ᆴ໗‫ק‬ღ؇֥‫ק‬ၬğཌྷ࢘Ԭᄀ௔ੱൈ֥‫ږ‬௔หྟ֥֚ඔӫູ‫ږ‬ᆴ໗‫ק‬ღ؇đࡥӫ‫ږ‬ᆴღ؇đࠧ h = 1 A(ωg ) ࠇ h × A(ωg ) = 1 ᄝؓඔቕѓ๭ഈđҐႨ Lg іൕ h ֥‫ٳ‬Нᆴđࠧ Lg = 20 lg h = 20 lg[1 A(ωg )] = −20 lg A(ωg ) dB ‫ږ‬ᆴღ؇֥໾৘ၩၬğ ໗‫֥ק‬༢๤đ೏༢๤֥षߌ٢նПඔᄜᄹնູჰট֥ h Пčࠇؓඔ‫ږ‬௔หྟ౷ཌཟഈ၍‫ ׮‬Lg ‫ٳ‬НĎđ ᄵ༢๤ࡼэູਢࢸ໗‫ק‬ሑ෿Ġ೏षߌ٢նПඔࣉ၂҄ᄹնđᄵ༢๤ࡼэູ҂໗‫ק‬Ġّᆭၧಖb č3Ďཌྷ໊໗‫ק‬ღ؇ γ ਷‫ږ‬ཌྷ౷ཌԬᄀ 0dB ཌ෮ؓႋ֥௔ੱູ ωc đᆃ۱௔ੱӫູ‫ږ‬ᆴԬᄀ௔ੱđՎ௔ੱ෮ؓႋ֥ཌྷູ໊ ϕ(ωc ) b 82 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ཌྷ໊໗‫ק‬ღ؇֥‫ק‬ၬğ‫ږ‬ᆴԬᄀ௔ੱ ωc ൈ֥ཌྷ௔หྟა −1800 ᆭҵӫູཌྷ໊໗‫ק‬ღ؇đࡥӫཌྷ໊ღ ؇đࠧ γ = ϕ(ωc ) − (−1800 ) = 1800 + ϕ(ωc ) ཌྷ໊ღ؇֥໾৘ၩၬğ ೂ‫ݔ‬༢๤൞໗‫֥ק‬đᄵཌྷ௔หྟϕ (ωc ) ᄜᇌު γ ࢘؇čࠇؓඔཌྷ௔หྟϕ(ω) ᄜ༯၍ γ ࢘؇Ďđᄵо ߌ༢๤ࡼэູਢࢸ໗‫ק‬ሑ෿Ġ೏ؓඔཌྷ௔หྟϕ(ω) ᄜࣉ၂҄ࡨཬđᄵ༢๤ࡼэູ҂໗‫ק‬Ġّᆭၧಖb č4Ď০Ⴈ໗‫ק‬ღ؇஑љ༢๤໗‫ྟק‬ ؓႿቋཬཌྷ໊༢๤টඪđLg > 0 ‫ ބ‬γ > 0 ሹ൞๝ൈ‫ؿ‬ളࠇ๝ൈ҂‫ؿ‬ളđၹՎ‫۽‬ӱഈӈᆺႨཌྷ໊໗‫ק‬ღ ؇ γ টіൕbཁಖđ༢๤໗‫ק‬ൈđсႵ Lg > 0 ‫ ބ‬γ > 0 b ২ğၘᆩֆّ໊ঌ֥ቋཬཌྷ໊༢๤đఃषߌؓඔ‫ږ‬௔หྟೂ༯๭෮ൕđ൫౰षߌԮ‫ݦ־‬ඔđѩ࠹ෘ ༢๤֥໗‫ק‬ღ؇b ࢳğč1ĎႮ۳‫֥ؓק‬ඔ‫ږ‬௔หྟॖၛ౰֤षߌԮ‫ݦ־‬ඔູ GK (s) = K (s + 1) s 2 (0.1s + 1)2 č2Ď࠹ෘ٢նПඔ K A(ω c ) = ωc2 K ωc2 + 1 = 1 (0.1ωc )2 +1 ॉ੮֞ ωc = 3.16 > 1 đ෮ၛ ω 2 >> 1, (0.1ωc )2 << 1 đႿ൞đഈൔॖၛࡥ߄ູ c A(ωc ) = Kωc =1⇒ K = ωc = 3.16 ωc2 ×1 č3Ď࠹ෘ໗‫ק‬ღ؇ ཌྷ໊໗‫ק‬ღ؇ğ γ = 1800 + ϕ(ωc ) = 1800 + arctgωc − 2 × 900 − 2 × arctg0.1ωc = 1800 + arctg3.16 − 2 × 900 − 2 × arctg0.1× 3.16 = 1800 + 72.400 −1800 − 2 ×17.50 = 37.40 ‫ږ‬ᆴ໗‫ק‬ღ؇ğ Ⴎϕ(ωg ) = −1800 ॖ֤ arctgωg − 2 × 900 − 2 × arctg0.1ωg = −1800 ߄ࡥ֤֞ğ arctgωg = 2 × arctg 0.1ωg ਷ğ ϕ = arctg 0.1ωg đᄵğ tg[arctgωg ] = tg[2 × arctg 0.1ωg ] Ⴎ೘࢘‫܄‬ൔॖ֤ğ ωg = tg2ϕ = sin 2ϕ cos 2ϕ = 2sinϕ cosϕ = 2tgϕ = 2 × 0.1ωg 2 cos2 ϕ − sin 2 ϕ 1 − tg 2ϕ 1 − 0.01ω g ࢳ֤ğωg = 8.94(rad s) 83 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Ⴟ൞‫ږ‬ᆴ໗‫ק‬ღ؇ູğ Lg = −20 lg A(ω ) = −20 lg K ×ω g = −20 lg 3.16 = 9.03dB 8.49 g ω 2 g ཁಖğႮႿ Lg > 0,γ > 0 ෮ၛоߌ༢๤໗‫ק‬b ২ğၘᆩֆّ໊ঌ֥ቋཬཌྷ໊༢๤đఃषߌؓඔ‫ږ‬௔หྟೂ༯๭෮ൕđ൫౰č1ĎषߌԮ‫ݦ־‬ඔĠč2Ď ࠹ෘ༢๤֥ཌྷ໊ღ؇đѩ஑؎༢๤໗‫ྟק‬Ġč3Ďࡼఃؓඔ‫ږ‬௔หྟཟႷ௜၍൅П௔ӱđ൫ษંؓ༢๤ྟ ି֥႕ཙb ࢳğč1Ď༢๤թᄝ 2 ۱ሇᅼ௔ੱ 0.1 ‫ ބ‬20đ‫ܣ‬ఃषߌԮ‫ݦ־‬ඔູ G(s) = k s(s / 0.1+1)(s / 20 +1) ౏ 20 lg k = 0 10 ֤ k = 10 đ ෮ၛ G(s) = 10 b s(s / 0.1+1)(s / 20 +1) č2ĎႮ༢๤षߌؓඔ‫ږ‬௔หྟॖᆩ A(ωc ) ≈ 10 = 1 i ωc i1 ωc 0.1 ֤ ωc = 1 ཌྷ௔หྟູ ϕ (ω ) = −90° − arctan ω− arctan ω đࡼ ωc = 1 սೆ֥ϕ(ωc ) = −177.15° 0.1 20 γ = 180° + ϕ(ωc ) = 2.85° đ‫ܣ‬༢๤໗‫ק‬b (3) ࡼؓඔ‫ږ‬௔หྟཟႷ௜၍൅П௔ӱđॖ֤ྍ֥षߌԮ‫ݦ־‬ඔ G(s) = 100 s(s /1+1)(s / 200 +1) ఃࢩᆸ௔ੱ ωc1 = 10ωc = 10 đ‫ط‬ ϕ (ωc1 ) = −90° − arctan ωc1 − arctan ωc1 = −177.15° đᄵ 200 γ1 = γ = 2.85° đ༢๤໗‫ିྟק‬҂эb 84 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 24 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬č6a7 ࢫĎ ᇶေଽಸ оߌ௔ੱหྟa௔ੱหྟ‫ٳ‬༅ ਔࢳоߌ௔ੱหྟაषߌ௔ੱหྟ֥ܱ༢ ଢ֥აေ ਔࢳоߌ௔ੱหྟ֥ྟିᆷѓ ౰ ᅧ໤༢๤໗෿ྟିa‫׮‬෿ྟିࠣॆ‫ۄ‬ಠྟି‫ބ‬षߌ௔ੱหྟ֥ܱ༢ ᇗ ׄ ა ଴ ᇗׄğषߌ௔ੱหྟ‫ٳ‬༅ ׄ ଴ׄğоߌ௔ੱหྟ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 5.6 оߌ༢๤֥௔ੱหྟ 1ēႨཟਈ‫م‬౰౼оߌ௔ੱหྟ Im ༢๤֥оߌ௔ੱหྟູ Φ( jω) = 1 G( jω) P + G( jω) Re → → → → -1 θ ϕ − θ 0 G( jω1) = OA = OA e jϕđ1+ G( jω1) = PA = PA e jθ G( jω1 ) ϕ Ⴟ൞ᄝ‫ھ‬௔ੱᆴൈ֥оߌ௔ੱหྟᆴູ → → A OA e jϕ Φ( jω1 ) = 1 G( jω1 ) ) = = OA + G( jω1 → e j(ϕ −θ ) PA e jθ → PA → OA → ‫ܣ‬ğ Φ( jω1) = M (ω)∠α (ω) đ M (ω1 ) = → ,α (ω1 ) = ϕ − θ = ∠ PAO PA бࢠषߌؓඔ‫ږ‬௔‫ބ‬оߌؓඔ‫ږ‬௔ॖၛ‫ؿ‬གྷğ č1Ď֮௔‫؍‬оߌؓඔ‫ږ‬௔ა 0dB ཌᇗ‫ކ‬aཌྷ௔ა 00 ཌᇗ‫ކ‬Ġ č2Ďۚ௔‫؍‬оߌؓඔ‫ږ‬௔౴Ⴟषߌؓඔ‫ږ‬௔đоߌؓඔཌྷ௔౴Ⴟоߌؓඔཌྷ௔Ġ č3Ďоߌؓඔ‫ږ‬௔Ӂളཾᆒ‫ڂ‬ᆴb 2ē༢๤ջॺ‫ބ‬ջॺ௔ੱ ༯๭൞оߌ༢๤֥‫ږ྘ׅ‬௔หྟ ‫ק‬ၬğоߌ‫ږ‬௔หྟ֥‫ږ‬ᆴႮ M (0) කࡨ֞ 0.707M (0) ൈ֥௔ੱđӫູоߌ༢๤֥ջॺ௔ੱđႨωb টіൕđ 0jωb ູࠧ௔ջॺ؇đࡥӫջॺb ջॺč௔ջॺ؇Ď֥ၩၬğՖջॺ֥‫ק‬ၬॖᆩđ֒ൻೆྐ‫֥ݼ‬௔ੱۚႿջॺ௔ੱωb ൞đ༢๤ൻԛ ֥‫ږ‬ᆴකࡨ‫ޓ‬նđ҂ି‫ֹّݺޓ‬႘ൻೆྐ‫ݼ‬đၹՎ༢๤֥ջॺđൌ࠽ഈّ႘ਔ༢๤ؓൻೆྐ‫گ֥ݼ‬གྷ ି৯đ֒ൻೆྐ‫֥ݼ‬௔ੱ֮Ⴟջॺ௔ੱൈđ༢๤ॖၛӁളቀ‫఼ܔ‬؇֥௔ੱཙႋb 85 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 5.7 ௔ੱหྟ‫ٳ‬༅ 1ē০Ⴈषߌ௔ੱหྟ‫ٳ‬༅༢๤֥໗෿ྟି č1Ďषߌ௔ੱหྟ֥֮௔‫؍‬थ‫ק‬ਔ༢๤֥໗෿ྟି ֮௔‫؍‬௔ੱหྟູ GK ( jω) = ( K G0 ( jω) = ( K jω )ν jω )ν ω→0 ؓඔ‫ږ‬௔หྟູğ L (ω ) = 20 lg ( K )ν = 20 lg K − 20 ×ν × lgω jω ؓඔཌྷ௔หྟູğϕ(ω) = −ν × 900 æ֮௔‫֥ؓ؍‬ඔ‫ږ‬௔หྟᆰཌོ֥ੱູ − 20×νdB dec đཌྷ௔࢘؇ູ −ν × 900 Ġ ç֒ω = 1 ൈđ֮௔‫؍‬ᆰཌࠇః࿼Ӊཌčᄝω < 1֥ٓຶଽႵሇᅼ௔ੱĎ֥‫ٳ‬Нᆴູ 20 lg K Ġ 1 è֮௔‫؍‬ᆰཌčࠇః࿼ӉཌĎაਬ‫ٳ‬Нཌč‫ޘ‬ᇠĎ֥ࢌׄ௔ੱູ ω0 = Kν đؓႿ I ྘༢๤ࢌׄ௔ੱ ູ ω0 = K đII ྘༢๤ࢌׄ௔ੱູ ω0 = K b 2ē௔თྟିᆷѓაൈთྟିᆷѓ֥ܱ༢ č1Ď‫ࢨؽ྘ׅ‬༢๤֥षߌ௔თᆷѓაᄠ෿ྟିᆷѓ֥ܱ༢ ‫ࢨؽ྘ׅ‬༢๤֥षߌ௔ੱหྟğ GK ( jω) = ( jω)( ωn2 2ξωn ) jω + ‫ږ‬ᆴԬᄀ௔ੱ ωc ğ ω c = ω n 4ξ 4 + 1 − 2ξ 2 ཌྷ໊ღ؇ γ ğ γ = 1800 + ϕ(ωc ) = arctg 2ξ 4ξ 4 + 1 − 2ξ 2 ֒ቅୄб 0 < ξ < 0.707 ൈđॖ࣍රູཌྟܱ༢ğ ξ = 0.01γ b − ξπ ႮႿӑ‫ט‬ਈ M p % = e 1−ξ 2 ×100% đࠧӑ‫ט‬ਈ M p % ൞ቅୄб ξ ֥ֆᆴ‫ݦ‬ඔb ‫ٳ‬༅ॖ֤ğཌྷ໊ღ؇ γ ᄀཬđ༢๤֥ֆ໊ࢨᄁཙႋӑ‫ט‬ਈ M p % ьᄀնb ‫ࢫט‬ൈࡗູ ts ⎧3 = 6 ×1 ,∆ =5 = ⎪⎪⎪⎨ξω3n = ωc tgγ ,∆ = 2 8 ⎪⎩ξωn ωc ×1 tgγ ‫ٳ‬༅ࢲંğ֒ཌྷ໊ღ؇ γ ҂эൈđ‫ࢫט‬ൈࡗ ts აࢩᆸ௔ੱč‫ږ‬ᆴԬᄀ௔ੱĎωc Ӯّбܱ༢bࠧ ωc ᄀ նđ‫ࢫט‬ൈࡗ ts ᄀཬđᄠ෿ཙႋ෎؇ᄀॹđၹՎࢩᆸ௔ੱ ωc іᆘਔ༢๤֥ᄠ෿ཙႋ֥ॹ෎ྟb č2Ďఴቅୄ‫ࢨؽ‬༢๤оߌ௔თྟିᆷѓაൈთᄠ෿ྟିᆷѓ֥ܱ༢ Φ( jω) = ωn2 ఴቅୄ‫ࢨؽ‬༢๤оߌ௔ੱหྟູ ( jω)2 + 2ξωn ( jω) + ωn2 ωn2 M (ω) = (ωn2 − ω 2 )2 + (2ξωnω)2 ωr = ωn 1 − 2ξ 2 ֒ 0 < ξ < 0.707 ൈđ 1 Mr = 1−ξ2 2ξ ႮႿ M (0) = 1đ෮ၛ֒ M (ω) = 0.707M (0) ൈđॖ֤֞ջॺ௔ੱ ωb 86 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ωb = ωn 1 − 2ξ 2 + 2 − 4ξ 2 + 4ξ 4 æ M P % ა M r ֥ܱ༢ğ M r ᄀཬđ༢๤֥ቅୄྟିᄀ‫ݺ‬đՖ‫ط‬༢๤֥ӑ‫ט‬ਈ M P % ္ᄀཬĠ M r ࢠ նൈđ༢๤֥ӑ‫ט‬ਈ M P % ࢠնđ༢๤ཙႋ֥௜໗ྟࢠҵb၂Ϯ‫ط‬࿽đ֒ M r = 1.2j15 ൈđ༢๤֥ӑ‫ט‬ ਈ M P % = 20%j30% đՎൈ༢๤Ⴕൡ؇֥ᆒ֕đ௜໗ྟࢠ‫ݺ‬b ç ts ა M r a ωb ֥ܱ༢ğ Ⴎğ ωb = ωn 1 − 2ξ 2 + 2 − 4ξ 2 + 4ξ 4 ॖ֤ğ ωbts ⎧3 = 3 1 − 2ξ 2 + 2 − 4ξ 2 + 4ξ 4 = ⎨⎪⎪ξωn = ξ 1 − 2ξ 2 + 2 − 4ξ 2 + 4ξ 4 4 ⎪4 ⎪⎩ξωn ξ ωbts ෛ M r ֥ᄹࡆ‫ط‬ֆ‫ט‬ᄹࡆb֒ M r ҂эൈđ‫ࢫט‬ൈࡗ ts აջॺ௔ੱ ωb Ӯّбܱ༢b č3Ďۚࢨ༢๤֥௔თᆷѓაൈთᄠ෿ྟିᆷѓ֥ܱ༢ æ ӑ‫ט‬ਈ M P % ෛཾᆒ‫ڂ‬ᆴ M r ֥ᄹն‫ط‬ᄹնĠ ç ‫ࢫט‬ൈࡗ ts ෛཾᆒ‫ڂ‬ᆴ M r ֥ᄹն‫ࡆط‬Ӊđѩა ωc ӮّбĠ è ཾᆒ‫ڂ‬ᆴ M r აཌྷ໊ღ؇ γ ֥࣍රܱ༢ູğ M r = 1 sin γ đᆃіૼđཌྷ໊ღ؇ γ ᄀཬđᄵཾᆒ‫ڂ‬ ᆴ M r ᄀնđ༢๤ᄀಸၞᆒ֕b֒ M r → ∞ ൈđཌྷ֒Ⴟཌྷ໊ღ؇ γ = 00 đ༢๤ԩႿ҂໗‫֥ק‬шჸčਢࢸ໗ ‫ק‬ሑ෿Ďb é༢๤֥ཌྷ໊ღ؇ γ ҂ࣇّ႘ਔ༢๤֥ཌྷؓ໗‫ྟק‬đߎ႕ཙሢ༢๤֥ᄠ෿ཙႋ෎؇b‫ط‬ཌྷ໊ღ؇ γ ֥ նཬᇶေ౼थႿषߌؓඔ‫ږ‬௔หྟᄝ ωc ‫࣍ڸ‬čᇏ௔‫؍‬Ď֥ྙሑđ၂Ϯေ౰༢๤֥ཌྷ໊ღ؇ γ = 300j600 đ ᆃࣼေ౰षߌؓඔ‫ږ‬௔หྟ L(ω) ‫ ݖ‬0dB ཌོ֥ੱູ − 20dB / dec đ౏ᅝऌ၂‫֥ק‬௔ջॺ؇b 3ēषߌ௔ੱหྟ֥ۚ௔‫ؓ؍‬༢๤ྟି֥႕ཙ ؓႿֆّ໊ঌ༢๤đषߌ‫ބ‬оߌԮ‫ݦ־‬ඔ֥ܱ༢ູ Φ(s) = G(s) Ġ 1+ G(s) ᄵ௔ੱหྟᆭࡗ֥ܱ༢ູ Φ( jω) = G( jω) Ġ 1+ G( jω) ᄝۚ௔‫؍‬၂Ϯ 20 lg G( jω) 0 đࠧ G( jω) 1đ‫ܣ‬Ⴕ Φ( jω) = G( jω) ≈ G( jω) 1+ G( jω) ࠧоߌ‫ږ‬௔֩Ⴟषߌ‫ږ‬௔bၹՎđषߌؓඔ‫ږ‬௔หྟۚ௔‫ږ֥؍‬ᆴđᆰࢤّ႘ਔ༢๤ؓൻೆ؊ۚ௔ ྐ‫֥ݼ‬ၝᇅି৯đۚ௔‫ٳ؍‬Нᆴᄀ֮đ༢๤ॆ‫ۄ‬ಠି৯ᄀ఼b  87 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 25 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ໴ᅣ ཌྟ༢๤֥௔თ‫ٳ‬༅ٚ‫م‬ ᇶေଽಸ ሱ‫׮‬॥ᇅ༢๤௔თ‫ٳ‬༅ഈࠏൌဒ ଢ֥აေ ඃ༑ MATLAB ೈࡱᄝ௔თ‫ٳ‬༅ᇏ֥ࠎЧႋႨ ౰ ඃ༑ MATLAB ೈࡱ߻ᇅ Bode ๭aNyquist ౷ཌ Ⴎ MATLAB ೈࡱ߻ᇅ֥ Bode ๭஑љоߌ༢๤֥໗‫ྟק‬ ᇗׄა଴ ᇗׄğ."5-"# ೈࡱ߻ᇅ #PEF ๭a/ZRVJTU ౷ཌࠣ໗‫ྟק‬஑؎ ׄ ଴ׄğMATLAB ೈࡱ߻ᇅ Bode ๭aNyquist ౷ཌ ࢝࿐൭‫ ؍‬ഈࠏ නॉีࠇ ປӮൌဒБۡ ቔြี ၂aൌဒଢ֥ 1a০Ⴈ MATLAB ߻ᇅ༢๤֥௔ੱหྟ๭Ġ 2a۴ऌ Nyquist ๭஑؎༢๤֥໗‫ྟק‬Ġ 3a۴ऌ Bode ๭࠹ෘ༢๤֥໗‫ק‬ღ؇b ‫ؽ‬aൌဒ಩ༀ ০Ⴈ MATLAB ߻ᇅ༢๤֥௔ੱหྟ๭đ൞ᆷ߻ᇅ Nyquist ๭aBode ๭đ෮Ⴈ֥֞‫ݦ‬ඔᇶေ൞ nyquista ngridabode ‫ ބ‬****rgin ֩b 1aNyquist ๭֥߻ᇅࠣ໗‫ྟק‬஑؎ nyquist ‫ݦ‬ඔॖၛ࠹ෘ৵࿃ཌྟ‫ק‬ӈ༢๤֥௔ੱཙႋđ֒ଁ਷ᇏ҂Ї‫ݣ‬ቐ؊эਈൈđࣇӁള Nyquist ๭b ଁ਷ nyquist(num,den)ࡼ߂ԛ༯ਙԮ‫ݦ־‬ඔ֥ Nyquist ๭ğ GH (s) = bm s m + bm−1sm−1 +… + b1s + b0 an s n + an−1sn−1 +… + a1s + a0 ఃᇏ num = [bm bm−1 b1 b0 ] đ den = [an an−1 a1 a0 ] b č1Ďၘᆩଖ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = 50 đႨ MATLAB ߻ᇅ༢๤֥ Nyquist (s + 5)(s − 2) ๭đѩ஑؎༢๤֥໗‫ྟק‬b MATLAB ӱ྽ս઒ೂ༯ğ num=[50]Ġden=[1,3,-10]Ġnyquist(num,den)Ġaxis([-6 2 -2 0])Ġtitle('Nyquist ๭') ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Nyquist ๭ೂ๭෮ൕğ Ⴎഈ๭ॖᆩ Nyquist ౷ཌ୉ൈᆌЇຶ(-1đj0)ׄ϶ಁđ‫ط‬षߌ༢๤ᄝႷ϶௜૫Ⴕ၂۱ࠞׄđ‫ܣ‬༢๤ ໗‫ק‬b 88 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ďၘᆩ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = 100k đႨ MATLAB ‫ٳ‬љ߻ᇅ k = 1,8, 20 ൈ༢๤ s(s + 5)(s +10) ֥ Nyquist ๭đѩ஑؎༢๤֥໗‫ྟק‬b 2aBode ๭֥߻ᇅࠣ໗‫ק‬ღ؇֥࠹ෘ MATLAB ิ‫߻܂‬ᇅ༢๤ Bode ๭‫ݦ‬ඔ bode( )đbode( num,den)߻ᇅၛ‫؟‬ཛൔ‫ݦ‬ඔіൕ֥༢๤ Bode ๭b č1Ďၘᆩ‫֥ࢫߌࢨؽ྘ׅ‬Ԯ‫ݦ־‬ඔູ G(s) = s2 ωn2 đ ః ᇏ ωn = 0.7 đ‫ٳ‬љ߻ᇅ + 2ξωns + ωn2 ξ = 0.1, 0.4,1,1.6, 2 ൈ֤ Bode ๭b MATLAB ӱ྽ս઒ೂ༯ğ w=[0,logspace(-2,2,200)]Ġwn=0.7Ġtou=[0.1,0.4,1,1.6,2]Ġfor j=1:5Ġ sys=tf([wn*wn],[1,2*tou(j)*wn,wn*wn])Ġbode(sys,w)Ġhold on ĠendĠgtext('tou=0.1')Ġ gtext('tou=0.4')Ġgtext('tou=1')Ġgtext('tou=1.6')Ġgtext('tou=2') ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Bode ๭ೂ๭෮ൕğ č2Ďၘᆩଖۚࢨ༢๤֥Ԯ‫ݦ־‬ඔູ G(s) = 5(0.0167s +1) đ߻ᇅ༢๤֥ Bode s(0.03s +1)(0.0025s +1)(0.001s +1) ๭đѩ࠹ෘ༢๤֥ཌྷ࢘ღ؇‫ږބ‬ᆴღ؇b MATLAB ӱ྽ս઒ೂ༯ğ num=5*[0.0167,1]Ġden=conv(conv([1,0],[0.03,1]),conv([0.0025,1],[0.001,1]))Ġ sys=tf(num,den)Ġw=logspace(0,4,50)Ġbode(sys,w)ĠgridĠ[Gm,Pm,Wg,Wc]=****rgin(sys) ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Bode ๭ೂ๭෮ൕğ ᄎྛࢲ‫ݔ‬ೂ༯ğ Gm = 455.2548ĠPm = 85.2751ĠWg = 602.4232ĠWc = 4.9620 Ⴎ ᄎ ྛ ࢲ ‫ ݔ‬ॖ ᆩ đ ༢ ๤ ֥ ‫ ږ‬ᆴ ღ ؇ Ag = 455.2548 đ ཌྷ ࢘ ღ ؇ γ = 85.2751° đ ཌྷ ࢘ Ԭ ᄀ ௔ ੱ ωg = 602.4262rad/s đࢩᆸ௔ੱωc = 4.962rad/s b 89 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č3Ďၘᆩଖۚࢨ༢๤֥Ԯ‫ݦ־‬ඔູ G(s) = 100(0.5s +1) đ߻ᇅ༢๤֥ Bode ๭đѩ࠹ s(s +1)(0.1s +1)(0.05s +1) ෘ༢๤֥ཌྷ࢘ღ؇‫ږބ‬ᆴღ؇b ೘aൌဒඔऌࠣࢲં 1aNyquist ๭֥߻ᇅࠣ໗‫ྟק‬஑؎ č2Ďၘᆩ༢๤֥षߌԮ‫ݦ־‬ඔູ G(s) = 100k đႨ MATLAB ‫ٳ‬љ߻ᇅ k = 1,8, 20 ൈ༢๤ s(s + 5)(s +10) ֥ Nyquist ๭đѩ஑؎༢๤֥໗‫ྟק‬b æ k=1 ൈ MATLAB ӱ྽ս઒ೂ༯ğ num=[100];den=[1,15,50,0];nyquist(num,den);axis([-3 2 -1 1]);title('Nyquist ๭'); ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Nyquist ๭ೂ๭෮ൕğ ༢๤ູ I ྘༢๤đᄝഈ๭ቔᄹҀཌđNyquist ౷ཌ҂Їຶ(-1đj0)ׄđ‫ط‬षߌ༢๤ᄝႷ϶௜૫ીႵࠞׄđ ‫ܣ‬༢๤໗‫ק‬b ç k=8 ൈ MATLAB ӱ྽ս઒ೂ༯ğ num=[800];den=[1,15,50,0];nyquist(num,den);axis([-3 2 -1 1]);title('Nyquist ๭'); ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Nyquist ๭ೂ๭෮ൕğ ቔᄹҀཌު Nyquist ౷ཌඨൈᆌЇຶ(-1đj0)ׄ၂ಁđ‫ط‬षߌ༢๤ᄝႷ϶௜૫ીႵࠞׄđ‫ܣ‬༢๤҂໗ ‫ק‬b è k=20 ൈ MATLAB ӱ྽ս઒ೂ༯ğ num=[2000];den=[1,15,50,0];nyquist(num,den);axis([-4 2 -1 1]);title('Nyquist ๭'); ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Nyquist ๭ೂ๭෮ൕğ 90 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ቔᄹҀཌު Nyquist ౷ཌඨൈᆌЇຶ(-1đj0)ׄ၂ಁđ‫ط‬षߌ༢๤ᄝႷ϶௜૫ીႵࠞׄđ‫ܣ‬༢๤҂໗ ‫ק‬b Ⴎၛഈࢲંॖᆩđk ᄹնđ༢๤֥໗‫ିྟק‬эҵb 2a Bode ๭֥߻ᇅࠣ໗‫ק‬ღ؇֥࠹ෘ č3Ďၘᆩଖۚࢨ༢๤֥Ԯ‫ݦ־‬ඔູ G(s) = 100(0.5s +1) đ߻ᇅ༢๤֥ Bode ๭đѩ s(s +1)(0.1s +1)(0.05s +1) ࠹ෘ༢๤֥ཌྷ࢘ღ؇‫ږބ‬ᆴღ؇b MATLAB ӱ྽ս઒ೂ༯ğ num=100*[0.5,1];den=conv(conv([1,0],[1,1]),conv([0.1,1],[0.05,1]));sys=tf(num,den); w=logspace(0,4,50);bode(sys,w);grid;[Gm,Pm,Wg,Wc]=****rgin(sys); ᆳྛ‫ھ‬ӱ྽ުđ༢๤֥ Bode ๭ೂ๭෮ൕğ ᄎྛࢲ‫ݔ‬ೂ༯ğ Gm = 0.5080ĠPm = -16.2505ĠWg = 13.0505ĠWc = 18.0572 Ⴎ ᄎ ྛ ࢲ ‫ ݔ‬ॖ ᆩ đ ༢ ๤ ֥ ‫ ږ‬ᆴ ღ ؇ Ag = 0.5080 đ ཌྷ ࢘ ღ ؇ γ = −16.2505° đ ཌྷ ࢘ Ԭ ᄀ ௔ ੱ ωg = 13.0505rad/s đࢩᆸ௔ੱωc = 18.0572rad/s đႮࢲ‫ݔ‬ॖᆩđ༢๤҂໗‫ק‬b 91 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 26 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻੂᅣ ॥ᇅ༢๤֥཮ᆞč1a2 ࢫĎ ᇶေଽಸ ༢๤֥ഡ࠹ࠣ཮ᆞa௔ੱ‫م‬Ա৳ӑభ཮ᆞ ଢ֥აေ౰ ਔࢳൈთa௔თᇏ֥ྟିᆷѓ ਔࢳࠫᇕ཮ᆞࢲ‫ܒ‬ ᅧ໤௔ੱ‫م‬Ա৳ӑభ཮ᆞຩ઎ࠣԮ‫ݦ־‬ඔ ᇗׄა଴ׄ ᇗׄğൈთa௔თᇏ֥ྟିᆷѓa௔ੱ‫م‬Ա৳ӑభ཮ᆞຩ઎ ଴ׄğ௔ੱ‫م‬Ա৳ӑభ཮ᆞ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇቔြ ี 6.1 ႄ࿽ ؓႿྸ‫؟‬ൌ࠽֥॥ᇅ༢๤đࣇ๙‫ࢫטݖ‬༢๤֥ଖ၂۱ҕඔđ໭‫ۚิܤ࡙م‬༢๤֥۲ٚ૫ྟିđൌ࠽ഈ ༢๤֥ྟିᆷѓؓႿଖ၂ҕඔ֥၇ঠ൞ཌྷ઱‫֥؛‬bၹՎູ‫ڿ‬೿༢๤֥ሹุྟିđ໡ૌສສေᄝ༢๤ᇏ‫ࡆڸ‬ ልᇂđটؓ༢๤ࣉྛࢲ‫ܒ‬ഈ֥཮ᆞđՖ‫ط‬o௧൐p༢๤֥ྟିડቀഡ࠹ေ౰b 1ē཮ᆞ ູਔൌགྷყ௹ྟି‫ؓط‬॥ᇅ༢๤ࢲ‫ט֥ྛࣉܒ‬ᆜӫູ཮ᆞbၹՎđ཮ᆞࣼ൞ູ૛Ҁ༢๤֥ྟି҂ቀ‫ط‬ ࣉྛ֥ࢲ‫טܒ‬ᆜb 2 ē཮ᆞልᇂ ູൌགྷؓ༢๤ࣉྛ֥཮ᆞđ‫ط‬ᄝჰ༢๤ᇏ‫֥ࡆڸ‬ልᇂӫູ཮ᆞልᇂb཮ᆞልᇂ֥ྙൔॖၛ‫؟‬ᇕ‫؟‬ဢđ ॖၛ൞‫׈‬ਫ਼aࠏྀልᇂa၁࿢ልᇂaగ‫׮‬ልᇂđമᇀႿॖି൞၂۱ೈࡱ֩֩b 3ēྟିᆷѓ ॥ᇅ༢๤ሸ‫ކ‬ഡ࠹ൈ࠻ေॉ੮ൈთ֥ྟିᆷѓđႻေॉ੮௔თ֥ྟିᆷѓb๝ൈߎေᇿၩ໗෿a‫׮‬෿ ‫ބ‬໗‫ିྟק‬đၛࠣॆ‫ۄ‬ಠି৯b 4ē཮ᆞࢲ‫ܒ‬ ॥ᇅ༢๤֥཮ᆞࢲ‫ܒ‬ᆺေႵԱ৳཮ᆞaّঌ཮ᆞaభঌ཮ᆞ‫཮ކگބ‬ᆞ֩ඹᇕٚൔb 5ē཮ᆞٚ‫م‬ ཌྟ॥ᇅ཮ᆞഡ࠹֥ᇶေٚ‫م‬Ⴕğ č1Ď‫ٳ‬༅‫م‬ğ‫ٳ‬༅‫္م‬ӫູ൫ฐ‫م‬đ෱бࢠᆰܴđ໾৘ഈၞႿൌགྷb֌ေ౰ഡ࠹ᆀऎႵ၂‫۽֥ק‬ӱ ࣜဒĠ 92 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ďሸ‫مކ‬ğሸ‫္مކ‬ӫູ௹ຬหྟ‫م‬bᆃᇕഡ࠹ٚ‫م‬൞۴ऌоߌ༢๤ྟିაषߌ༢๤หྟૡ్ཌྷ ܱᆃ၂‫୑ۀ‬ԛ‫ؿ‬đಖު๙‫ିྟݖ‬ᆷѓಒ‫֥ק‬௹ຬหྟაჰႵčໃ཮ᆞ༢๤Ď༢๤หྟཌྷбࢠđՖ‫ط‬ಒ‫ק‬ ཮ᆞልᇂ֥ྙൔ‫ބ‬ҕඔbᆃᇕٚ‫م‬ऎႵܼ֥ٗ৘ંၩၬđ֌ႮႿ཮ᆞልᇂ֥Ԯ‫ݦ־‬ඔбࢠ‫گ‬ᄖđ໾৘ൌ གྷбࢠ঒଴b ໭ં൞ҐႨ‫ٳ‬༅‫م‬đߎ൞ሸ‫مކ‬ഡ࠹཮ᆞልᇂđӈӈॖၛᄝ௔თᇏࣉྛđࠇҐႨ۴݅ࠖ‫م‬b‫ط‬ᄝ௔ თᇏࣉྛ༢๤ഡ࠹đ൞၂ᇕбࢠࡗࢤ֥ٚ‫م‬đᆃ൞ၹູ௔თᇏഡ࠹ડቀ֥ᆷѓ൞௔თᆷѓđ෱აൈთᆷ ѓᆺႵࡗࢤܱ༢b֌ᆃᇕٚ‫م‬бࢠьࢮđၞႿᅧ໤b ᄝ௔თᇏഡ࠹॥ᇅ༢๤đေ቎࿖֥ჰᄵ൞ğ č1Ďषߌ௔ੱหྟ֥֮௔‫؍‬іᆘਔоߌ༢๤֥໗෿ྟିĠ č2Ďषߌ௔ੱหྟ֥ᇏ௔‫؍‬іᆘਔоߌ༢๤֥‫׮‬෿ྟିĠ č3Ďषߌ௔ੱหྟ֥ۚ௔‫؍‬іᆘਔоߌ༢๤֥‫گ‬ᄖྟ‫ބ‬ᄮലၝᇅି৯b  ၹՎؓ༢๤཮ᆞ֥ଢ֥൞൐཮ᆞުषߌؓඔ‫ږ‬௔หྟऎႵ௹ຬ֥ྙሑđࠧ č1Ď֮௔‫؍‬Ⴕቀ‫ܔ‬ն֥ᄹၭđၛЌᆣ༢๤֥໗෿ࣚ؇ေ౰Ġ č2Ďᇏ௔‫ؓ؍‬ඔ‫ږ‬௔หྟ‫ ݖ‬0dB ཌོ֥ੱູ − 20dB dec đѩᅝऌ၂‫֥ק‬௔ջॺ؇Ġ č3Ďۚ௔‫؍‬ᄹၭ࣐ॹࡨཬđࠧۚ௔‫ੱོ֥؍‬бࢠնđ၂ϮေཬႿ − 40dB dec đཤ೐ᄮല֥႕ཙb 6.2 ௔ੱ‫م‬Ա৳཮ᆞ R1 ၂aԱ৳ӑభ཮ᆞ 1aӑభ཮ᆞຩ઎ࠣหྟ ຩ઎Ԯ‫ݦ־‬ඔğ C G(s) = R2 = R2 1 + αTs = 1 × 1 + αTs U1 R2 U2 R1 // 1 + R2 R1 + R2 1 + Ts α 1+ Ts sc ൔᇏğα = (R1 + R1) R2 > 1 ӫູකࡨၹሰđT = R1R2c (R1 + R2 ) ႮႿ1 α < 1 đ෮ၛ཮ᆞຩ઎ቔႨႿ༢๤ުđࡼ߶൐ᆜ۱༢๤č཮ᆞުĎ֥षߌᄹၭ༯ࢆα ПđՖ‫ط‬ ࢆ֮༢๤֥໗෿ྟିđູՎđႋ‫ھ‬ᄝ཮ᆞު֥༢๤ᇏđᄹն༢๤֥षߌ٢նПඔα ПđၛҀӊႮႿ཮ᆞ ຩ઎֥ቔႨᄯӮ֥٢նПඔ༯ࢆb ၹՎđ࿹࣮ӑభ཮ᆞຩ઎หྟൈđᆺླေ࿹࣮ G(s) = 1 + αTs ࠧॖb 1+ Ts ӑభ཮ᆞຩ઎֥ؓඔ‫ږ‬௔‫ؓބ‬ඔཌྷ௔หྟ౷ཌđೂ༯෮ൕb ӑభ཮ᆞຩ઎หྟ æᄝሇᅼ௔ੱω1 = 1 αT ‫ބ‬ω2 = 1 T ᆭࡗđຩ઎ऎႵૼཁ֥ັ‫ٳ‬ቔႨĠ çᄝሇᅼ௔ੱ ω1 = 1 αT ‫ ބ‬ω2 = 1 T ᆭࡗđຩ઎ऎႵཌྷ໊ӑభቔႨđ‫ܣ‬ӑభຩ઎္ႮՎ֤଀Ġ èᄝ ω = ωm ԩႵቋն֥ӑభཌྷ໊࢘ϕm đ౏ ωm ໊Ⴟω1 = 1 αT ‫ ބ‬ω2 = 1 T ᆭࡗ֥ࠫ‫ޅ‬ᇏྏĠ 93 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ωm = T 1 = 1 ×1 = ω1ω 2 đ lgωm = lg ω1ω2 = 1 (lg ω1 + lgω2 ) α αT T 2 éᄝ ω = ωm ԩđࠆ֤֥ӑభཌྷູ໊࢘ ϕm = arctg α −1 = arcsin α −1 2 α α +1 ‫ھ‬ൔіૼđකࡨ༢ඔα ᄀնđӑభཌྷ໊࢘ϕm ᄀնđՖ‫ٳັط‬ቔႨᄀ఼b êᄝ ω = ωm ԩđؓඔ‫ږ‬௔ᆴູğ L(ωm ) = 10 lgα = 1 (0 + 20 lgα ) Ġ 2 ëӑభ཮ᆞຩ઎ॖၛुቔູ൞၂۱ۚ๙ੲѯఖčྐ‫֥ۚݼ‬௔҆‫ٳ‬๙‫ݖ‬đ֮௔҆‫ٳ‬ФකࡨĎb 94 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 27 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻੂᅣ ॥ᇅ༢๤֥཮ᆞč2 ࢫĎ ᇶေଽಸ ௔ੱ‫م‬Ա৳ӑభ཮ᆞ֥ٚ‫҄ࠣم‬ᇧ ௔ੱ‫م‬Ա৳Ӿު཮ᆞຩ઎ ଢ֥აေ ᅧ໤ӑభ཮ᆞຩ઎֥ҕඔಒ‫֥ק‬ٚ‫҄ࠣم‬ᇧ ౰ ਔࢳӾު཮ᆞຩ઎ ᇗׄა଴ ᇗׄğԱ৳ӑభ཮ᆞ֥ٚ‫҄ࠣم‬ᇧ ׄ ଴ׄğԱ৳ӑభ཮ᆞ֥ٚ‫҄ࠣم‬ᇧ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 6-1 2aӑభ཮ᆞ ູਔࠆ֤ቋն֥ཌྷ໊ӑభਈđႋ൐֤ӑభຩ઎֥ቋնӑభཌྷ໊‫ؿ‬ളᄝ཮ᆞު༢๤֥‫ږ‬ᆴԬᄀ௔ੱ ωc ԩđࠧ཮ᆞު֥‫ږ‬ᆴԬᄀ௔ੱωc′ = ωm đՖ‫ط‬൐཮ᆞຩ઎ᄝ ωc′ = ωm ԩӁള֥ӑభཌྷ໊ϕm ૛Ҁ཮ᆞ భ༢๤ཌྷ໊໗‫ק‬ღ؇֥҂ቀđᆃࣼ൞ӑభ཮ᆞ֥ჰ৘b ၇ऌӑభ཮ᆞ֥ჰ৘đ཮ᆞ֥ऎุ҄ᇧ൞ğ æቔԛໃ཮ᆞ༢๤֥षߌؓඔ‫ږ‬௔หྟ L(ω) ‫ބ‬ཌྷ௔หྟϕ(ω) ऎุቓ‫م‬ğ۴ऌषߌؓඔ‫ږ‬௔หྟ L(ω) đ࡟ဒ༢๤֥໗෿ྟିb೏҂ડቀ ໗෿༂ҵေ౰đॖο ᅶေ౰ᄹնषߌ٢նПඔ K ᆴđࠧࡼषߌؓඔ‫ږ‬௔หྟ L(ω) ཟഈ௜၍đՖ‫ط‬ॖၛಒ‫ק‬ડቀ໗෿ྟିေ ౰֥षߌ٢նПඔ K ᆴb ç০Ⴈഈ၂҄ᇧಒ‫֥ק‬षߌ٢նПඔ K ᆴđࢲ‫ކ‬षߌؓඔཌྷ௔หྟϕ(ω) đ࠹ෘࠇႮ๭ࠆ֤཮ᆞ భ֥ࢩᆸ௔ੱ ωc ‫ބ‬ཌྷ໊໗‫ק‬ღ؇ γ Ġ ⎧ໃ཮ᆞγ < ௹ຬ֥γ ′ ⎪⎨ໃ཮ᆞωc < ௹ຬ֥ωc′ ⎪⎩ໃ཮ᆞ༢๤‫ݖ‬0dBཌོ֥ੱཬႿࠇ֩Ⴟ − 40dB dec è࠹ෘླေҀӊ֥ӑభཌྷ໊࢘ϕm ğ ऎุቓ‫م‬ğ਷ ϕm = γ ′ − γ + (5j120 ) bൔᇏ ∆γ = γ ′ − γ ູླေҀӊ֥ཌྷ໊࢘đ (5j120 ) ູᄹࡆ ֥ღਈ࢘b é࠹ෘකࡨၹሰ α ऎุቓ‫م‬ğႮഈ၂҄ᇧಒ‫֥ק‬ӑభຩ઎Ҁӊ࢘ ϕ m đ၇ऌ ϕ m = arctg α −1 = α −1 đ࠹ෘක 2 α arcsin α +1 ࡨၹሰα ֤֞ğ α = 1+ sin ϕm 1 − sin ϕm ֌൞đೂ‫཮ؓݔ‬ᆞު༢๤֥ࢩᆸ௔ੱ ωc′ ၘࣜิԛေ౰đᄵॖၛ࿊ᄴ௹ຬ֥ ωc′ ቔູ཮ᆞު֥ࢩᆸ௔ ੱđᄝؓඔ‫ږ‬௔หྟ๭ഈҰᅳ֞ໃ཮ᆞ༢๤֥ᄝ ωc′ ԩ֥‫ږ‬ᆴ L(ωc′ ) đ౼ ωm = ωc′ đѩ਷ L(ωc′ ) +10 lgα = 0 ႮՎॖၛ౰֤කࡨၹሰα b 95 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ êಒ‫཮ק‬ᆞު༢๤֥ࢩᆸ௔ੱ ωc′ ऎุቓ‫م‬ğಒ‫཮ק‬ᆞభ༢๤֥षߌؓඔ‫ږ‬௔หྟ֥‫ږ‬ᆴ֩Ⴟ − 10 lgα ൈ֥௔ੱđ࿊ᄴՎ௔ੱቔູ཮ ᆞު༢๤֥‫ږ‬ᆴԬᄀ௔ੱ ωc′ čࠧࢩᇀ௔ੱĎđ‫ھ‬௔ੱູࠧ཮ᆞຩ઎Ӂളቋնӑభཌྷ໊࢘ϕm ෮ؓႋ֥௔ੱ ωm đࠧ ωm =1 = ω1ω2 b Tα ëಒ‫཮ק‬ᆞຩ઎Ԯ‫ݦ־‬ඔ ऎุቓ‫م‬ğႮႿ ωm =1 = ω1ω2 đ෮ၛ T =1 =1 Tα ωm α ωc′ α Ⴟ൞Ⴕğ ω1 =1 = ωc′ đ ω2 =1 = aωc′ αT α T ཮ᆞልᇂ֥Ԯ‫ݦ־‬ඔູğ Gc (s) = 1+ s ω1 1+ s ω2 ཮ᆞު༢๤֥षߌԮ‫ݦ־‬ඔູğ G′(s) = G(s)Gc (s) b ì཮ဒྟିᆷѓ ऎุቓ‫م‬ğႮ཮ᆞު༢๤֥ؓඔ௔ੱหྟ L′(ω) aϕ′(ω) ཮ဒ༢๤֥ྟିᆷѓ൞‫ڎ‬ડቀေ౰đ೏҂ડ ቀđᄵေᇗ‫گ‬ഈඍ֥҄ᇧđᆰᇀડቀေ౰ູᆸb ২ğଖ॥ᇅ༢๤֥षߌԮ‫ݦ־‬ඔູğ GK (s) = K s(0.1s + 1)(0.001s + 1Ďđؓ‫ھ‬༢๤֥ေ౰൞ğč1Ď༢ ๤֥ཌྷ໊ღ؇ γ ′ ≥ 450 Ġč2Ď࣡෿෎؇༂ҵ༢ඔ Kv = 1000(s−1) b౰཮ᆞ֥Ԯ‫ݦ־‬ඔb ࢳğč1ĎႮႿ༢๤ູ 1 ྘đ෮ၛ࣡෿෎؇༂ҵ༢ඔࣼ൞༢๤֥षߌ٢նПඔđࠧ K = Kv = 1000 Ġ č2Ďໃ཮ᆞ༢๤֥षߌԮ‫ݦ־‬ඔູğ GK (s) = 1000 s(0.1s + 1)(0.001s + 1Ď ߂ԛໃ཮ᆞ༢๤֥ Bode ๭đႮ๭ॖ౰֤đ ωc = 100,γ = 00 đ༢๤ԩႿਢࢸ໗‫ק‬ሑ෿đ‫ܣ‬ॖҐႨӑభ Ա৳཮ᆞb ҐႨ࠹ෘ֥ٚ‫م‬๝ဢॖၛ֤ԛ၂ဢ֥ࢲંğ Ⴎࢩᆸ௔ੱ֥‫ק‬ၬॖ֤ 20 lg1000 − 20 lg ωc − 20 lg (0.1ωc )2 + 1 − 20 lg (0.001ωc )2 + 1 = 0 ႮႿ10 < ωc < 1000 đ෮ၛഈൔॖၛ࣍රູğ 20 lg1000 − 20 lgωc − 20 lg 0.1ωc = 20 lg103 − 20 lg 0.1ωc2 = 0 ࠧğ 20 lg104 = 20 lg ωc2 đႿ൞ ωc = 100 ཌྷ໊ღ؇ğ [ ] γ = 1800 + − 900 − arctg0.1ωc − arctg0.001ωc = 1800 − 900 − arctg0.1×100 − arctg0.001×100 = 00 q3rಒ‫ק‬ӑభҀӊཌྷ࢘ ϕ m ϕm = γ ′ − γ + (50j120 ) = 450 − 00 + 50 = 500 č4Ďಒ‫ק‬කࡨ༢ඔ α α = 1 + sin ϕm = 7.5 1 − sin ϕm č5Ďಒ‫཮ק‬ᆞު֥ࢩᆸ௔ੱ ωc′ ႮႿğ10 lgα = 10 lg 7.5 = 8.75dB đູՎᄝໃ཮ᆞ༢๤षߌؓඔ‫ږ‬௔ L(ω) ഈᅳ֞‫ږ‬ᆴູ − 8.75dB ؓ ႋ֥௔ੱđູࠧ཮ᆞު֥ࢩᆸ௔ੱ ωc′ ωc′ = 1****.5(rad / s) Ⴈ࠹ෘ֥ٚ‫م‬౰ࢳೂ༯ğ 96 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 20 lg1000 − 20 lg ωc′ − 20 lg 0.1ωc′ = 20 lg103 − 20 lg 0.1ωc′2 = −8.75 ౰֤཮ᆞު֥ࢩᆸ௔ੱ ωc′ ğ ωc′ = 1****.19(rad / s) č6Ď౰཮ᆞልᇂԮ‫ݦ־‬ඔ ಒ‫ק‬ሇᅼ௔ੱ ω1,ω2 b ω1 = ωc = 1****.5 = 60(rad / s) α 7.5 ω2 = αωc = 7.5 ×1****.5 = 450(rad / s) ཮ᆞልᇂԮ‫ݦ־‬ඔູğ Gc (s) = 1+ s ω1 = 1+ 0.0167s 1+ s ω2 1 + 0.00222s č7Ď཮ဒࢲ‫ݔ‬ ཮ᆞު༢๤֥षߌԮ‫ݦ־‬ඔູğ G′(s) = Gc (s)G(s) = 1 + 0.0167s × s(0.1s 1000 + 1Ď= s(1 + 1000(1 + 0.0167s) + 1Ď 1 + 0.00222s + 1)(0.001s 0.00222s)(0.1s + 1)(0.001s ཮ဒğ æ཮ᆞ༢๤֥षߌ٢նПඔ K = 1000 = Kv đડቀ༢๤֥໗෿ྟିေ౰Ġ ç཮ᆞު֥ཌྷ໊ღ؇ γ ′ = 450 đડቀ໗‫֥ିྟק‬ေ౰Ġ èᇏ௔‫ ݖ؍‬0dB ཌൈོ֥ੱູ − 20dB / dec đ౏ᅝऌ 390(rad / s) ֥௔ջॺ؇đ෮ၛоߌ༢๤֥ӑ‫ט‬ ਈ༯ࢆĠ éႮႿ཮ᆞު ωc′ = 1****.5 ჹჹնႿໃ཮ᆞൈ֥ ωc = 100 đ෮ၛоߌ༢๤֥௔ջॺ؇ႵིᄹࡆđՖ‫ط‬ ൐ཙႋ෎؇ࡆॹĠ 3aԱ৳ӑభ཮ᆞ֥൐Ⴈ൳ཋ֥ 2 ۱ᇶေၹ෍ æ೏ໃ཮ᆞ༢๤҂໗‫ק‬đູਔ֤֞ေ౰֥ཌྷ໊ღ؇đླေӑభ཮ᆞຩ઎ิ‫ޓ܂‬ն֥ཌྷ࢘ღਈđᆃဢ ၂টđӑభຩ઎֥කࡨ༢ඔα ࣼ߶‫ޓ‬նđ၂ٚ૫൐໾৘ൌགྷбࢠ঒଴đਸ਼၂ٚ૫đႻ߶ᄯӮၘ཮ᆞ༢๤ ֥௔ջॺ؇‫ݖ‬նđ൐֤๙‫ݖ‬༢๤֥ۚ௔ᄮല‫׈‬௜‫ۚݖ‬đ‫ޓ‬ॖି൐༢๤ാ॥Ġ çؓႿᄝࢩᆸ௔ੱ‫࣍ڸ‬ཌྷ࢘࿡෎ࡨཬ֥ໃ཮ᆞ༢๤đ၂Ϯ҂ൡၒҐႨӑభ཮ᆞbᆃ൞ၹູෛሢࢩᆸ ௔ੱ֥ᄹնđໃ཮ᆞ༢๤֥ཌྷ࢘࿡෎ࡨཬđᇁ൐཮ᆞު֥༢๤ཌྷ໊ღ؇‫ڿ‬೿҂նb ‫ؽ‬aԱ৳Ӿު཮ᆞ R1 1aӾު཮ᆞຩ઎ࠣหྟ ຩ઎Ԯ‫ݦ־‬ඔğ R2 + 1 1 + scR2 = 1 + bTs U1 R2 sc U2 G(s) = = 1 1 + sc(R1 + R2 ) 1 + Ts C R1 + sc + R2 ൔᇏğ b = R2 (R1 + R2 ) < 1 đ T = (R1 + R2 )c b ӫູӾުຩ઎֥‫ٳ‬؇༢ඔđіൕӾު֥ധ؇b Ӿު཮ᆞຩ઎֥ؓඔ‫ږ‬௔‫ؓބ‬ඔཌྷ௔หྟ౷ཌđೂ༯෮ൕb  97 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 28 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻੂᅣ ॥ᇅ༢๤֥཮ᆞč2 ࢫĎ ᇶေଽಸ Ӿު཮ᆞຩ઎֥หׄ Ӿު཮ᆞ֥ٚ‫҄ބم‬ᇧ ଢ֥აေ ᅧ໤Ӿު཮ᆞຩ઎֥ҕඔಒ‫֥ק‬ٚ‫҄ࠣم‬ᇧ ౰ ਔࢳّঌ཮ᆞ֥ࠎЧ‫୑ۀ‬aّঌ཮ᆞ֥ࠎЧჰ৘ّࠣঌ཮ᆞ֥ࠎЧࢲ‫ބܒ‬ٚ‫م‬ ᇗ ׄ ა ଴ ᇗׄğӾު཮ᆞ֥ٚ‫҄ބم‬ᇧ ׄ ଴ׄğ๝ഈ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 6-2 Ӿު཮ᆞຩ઎หྟğ æ ᄝሇᅼ௔ੱ ω1 = 1 T ‫ ބ‬ω2 = 1 bT ᆭࡗđຩ઎ऎႵૼཁ֥ࠒ‫ٳ‬ቔႨĠ ç ᄝሇᅼ௔ੱ ω1 = 1 T ‫ ބ‬ω2 = 1 bT ᆭࡗđຩ઎ऎႵཌྷ໊ӾުቔႨĠ èᄝ ω = ωm ԩđऎႵቋն֥Ӿުཌྷ໊࢘ϕm đ౏ ωm ໊Ⴟ ω1 = 1 T ‫ ބ‬ω2 = 1 bT ᆭࡗ֥ࠫ‫ޅ‬ᇏྏĠ ωm = 1 = 1× 1 = ω1ω2 đ lgωm = lg ω1ω2 = 1 (lg ω1 + lg ω2 ) Tb T bT 2 éᄝ ω = ωm ԩđࠆ֤֥Ӿުཌྷູ໊࢘ ϕ m = arctg b −1 = arcsin b −1 2 b b +1 ‫ھ‬ൔіૼđ‫ٳ‬؇༢ඔ b ᄀնđӾުཌྷ໊࢘ϕm ᄀնđՖ‫ٳࠒط‬ቔႨᄀ఼b֌Ӿުభཌྷ໊ϕm ֥ࠞཋᆴ ູ − 900 đՎൈ‫ٳ‬؇༢ඔ b ູ໭౫նđՖ‫ط‬൐໾৘ൌགྷ‫ؿ‬ള঒଴b êᄝ ω = ωm ԩđؓඔ‫ږ‬௔ᆴູğ L(ωm ) = 10 lg b = 1 (0 + 20 lg b) Ġ 2 ëӾު཮ᆞຩ઎ॖၛुቔູ൞၂۱֮๙ੲѯఖčྐ‫֥֮ݼ‬௔҆‫ٳ‬๙‫ݖ‬đۚ௔҆‫ٳ‬ФකࡨĎb ᄝഈඍ཮ᆞຩ઎หྟᇏđؓ༢๤ఏ཮ᆞቔႨ֥൞ğ æႮႿ֮௔‫؍‬ა 0dB ཌᇗ‫ކ‬đ‫֮ܣ‬௔‫ؓ؍‬ႵႨྐ‫ݼ‬҂ӁളකࡨቔႨđၹ‫ط‬҂߶႕ཙ༢๤֥໗෿ྟ ିĠ çႮႿۚ௔‫ږ֥؍‬௔ູ 20 lg b < 0dB đࠧऎႵۚ௔කࡨหྟđ෮ၛᆃ۱หྟቔႨႿФ཮ᆞؓའުđ ॖၛ൐཮ᆞު֥༢๤ۚ௔‫ॹ࣐؍‬කࡨđࢩᆸ௔ੱ ωc′ భ၍đՖ‫ڿط‬೿༢๤֥ཌྷ໊ღ؇ γ Ġ èႮႿཌྷ໊Ӿުหྟđหљ൞ᄝ ωm ԩđཌྷ໊Ӿު࢘ቋնđູਔх૧ཌྷ໊Ӿުؓ༢๤ཌྷ໊ღ؇ γ ֥ ႕ཙđႋ‫ھ‬൐཮ᆞຩ઎ӁളቋնӾުཌྷ࢘ϕm ֥௔ੱׄ ωm ჹჹཬႿ༢๤֥ྍࢩᆸ௔ੱ ωc′ b ๙ӈॖ౼ğ ω2 = 1 = ωc′ = 0.1ωc′ bT 10 ՎൈđӾުຩ઎ᄝ ωc′ ԩӁള֥ཌྷ࢘Ӿުॖο༯ൔಒ‫ק‬ğϕc (ωc′ ) ≈ arctg0.1(b −1) éԱ৳Ӿު཮ᆞ֥൐Ⴈӆ‫ ކ‬၂Ϯ ωc′ < ωc đࠧ཮ᆞު௹ຬ֥ࢩᆸ௔ੱཬႿໃ཮ᆞ༢๤֥षߌࢩᇀ ௔ੱb‫ط‬ႮႿྍ֥ࢩᆸ௔ੱ ωc′ ֥эཬđၹՎӾު཮ᆞ൞๙‫ݖ‬࿢෪௔ջđ།വ༢๤֥ཙႋ෎؇đটิۚ ༢๤֥໗‫ק‬ღ؇֥b 2aӾު཮ᆞ Ӿު཮ᆞ֥ࠎЧჰ৘൞০ႨӾު཮ᆞຩ઎֥ۚ௔‫ږ‬ᆴකࡨหྟđ൐཮ᆞު༢๤֥‫ږ‬ᆴԬᄀ௔ੱčࢩ ᆸ௔ੱĎ༯ࢆđࢹᇹႿ཮ᆞభ༢๤ᄝ‫ږھ‬ᆴԬᄀ௔ੱԩ֥ཌྷ໊đ൐༢๤ࠆ֤ቀ‫֥ܔ‬ཌྷ໊ღ؇b 98 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ၇ऌӾު཮ᆞ֥ჰ৘đ཮ᆞ֥ऎุ҄ᇧ൞ğ æቔԛໃ཮ᆞ༢๤֥षߌؓඔ‫ږ‬௔หྟ L(ω) ‫ބ‬ཌྷ௔หྟϕ(ω) ऎุቓ‫م‬ğ۴ऌषߌؓඔ‫ږ‬௔หྟ L(ω) đ࡟ဒ༢๤֥໗෿ྟିb೏҂ડቀ໗෿༂ҵေ౰đॖοᅶေ ౰ᄹնषߌ٢նПඔ K ᆴđࠧࡼषߌؓඔ‫ږ‬௔หྟ L(ω) ཟഈ௜၍đՖ‫ط‬ॖၛಒ‫ק‬ડቀ໗෿ྟିေ౰֥ष ߌ٢նПඔ K ᆴb ç০Ⴈഈ၂҄ᇧಒ‫֥ק‬षߌ٢նПඔ K ᆴđࢲ‫ކ‬षߌؓඔཌྷ௔หྟϕ(ω) đ࠹ෘࠇႮ๭ࠆ֤ໃ཮ᆞ భ֥ࢩᆸ௔ੱ ωc ‫ބ‬ཌྷ໊໗‫ק‬ღ؇ γ Ġ ⎧ໃ཮ᆞγ < ௹ຬ֥γ ′ ⎩⎨ໃ཮ᆞωc > ௹ຬ֥ωc′ è۴ऌཌྷ໊ღ؇֥ေ౰đ࿊ᄴѩಒ‫཮ၘק‬ᆞ༢๤֥ࢩᆸ௔ੱ ωc′ Ġ ऎ ุ ቓ ‫ م‬ğ ॉ ੮ ֞ Ӿ ު ཮ ᆞ ຩ ઎ ᄝ ωc′ ԩ ߶ Ӂ ള ၂ ‫ ֥ ק‬ཌྷ ࢘ Ӿ ު ϕ(ωc′ ) đ ၹ Վ ༯ ൔ Ӯ ৫ ğ γ ′ = γčωc′Ď+ ϕ(ωc′ ) ൔᇏ γčωc′Ďູໃ཮ᆞ༢๤ᄝྍࢩᆸ௔ੱ ωc′ ԩ֥ཌྷ࢘đϕ(ωc′ ) ູ཮ᆞຩ઎ᄝ ωc′ ԩ֥Ӿުཌྷ࢘đᄝ ωc′ ໃ ಒ‫ק‬ᆭభđॖ౼ϕ(ωc′ ) = −60 bᆃဢႮ‫܄‬ൔ γ ′ = γčωc′Ď+ ϕ(ωc′ ) ॖၛ౰֤ྍࢩᆸ௔ੱ ωc′ b éಒ‫ק‬Ӿުຩ઎ҕඔ b ‫ ބ‬T ⎪⎧L(ωc′ ) + 20 lg b = 0 ⎨1 ⎪⎩bT = 0.1ωc′ ᇿၩğ 1 ॖᄝ (0.1j0.25)ωc′ ᆭࡗ࿊౼đϕ(ωc′ ) ॖᄝ (−60j − 140 ) ᆭࡗ࿊౼b bT êಒ‫ק‬Ӿު཮ᆞຩ઎Ԯ‫ݦ־‬ඔ ω1 = 1 T ,ω2 = 1 bT Gc (s) = 1+ s ω2 1+ s ω1 ë཮ဒࢲ‫ݔ‬ ऎุቓ‫م‬ğ཮ဒࢲ‫ݔ‬đ೏҂ડቀေ౰đᄵေᇗ‫گ‬ഈඍ҄ᇧb ২ğഡֆّ໊ঌ༢๤֥षߌԮ‫ݦ־‬ඔູ GK (s) = K s(s + 25) đေ౰༢๤֥࣡෿෎؇༂ҵ༢ඔູ Kv = 100 đཌྷ࢘ღ؇ γ ≥ 450 đҐႨԱ৳Ӿު཮ᆞđ൫ಒ‫཮ק‬ᆞልᇂ֥Ԯ‫ݦ־‬ඔb ࢳğ č1Ď۴ऌ Kv = 100 ֥ေ౰đॖಒ‫ק‬षߌ٢նПඔđၹ༢๤ູ྘đ‫ طܣ‬Kv = K 25 = 100 đ෮ၛषߌ ٢նПඔ K = 25Kv = 25 ×100 = 2500 b षߌԮ‫ݦ־‬ඔູğ GK (s) = 2500 s(s + 25) = 100 s(0.04s + 1) č2Ď߻ᇅໃ཮ᆞ༢๤֥ѵ֣๭đՖ๭ᇏॖ֤ğ ωc = 50(rad /) đ γ = 270 < 450 ္ॖ࠹ෘ֤֞ğ 20 lg100 − 20 lgωc − 20 lg (0.04ωc )2 + 1 = 0dB ၹ ωc > 25 ,‫( ܣ‬0.04ωc )2 >> 1 đ෮ၛഈൔॖၛ࣍රູ 20 lg100 − 20 lg ωc − 20 lg 0.04ω = 0 ౰֤ğ ωc = 50(rad /) ཌྷ໊ღ؇ğ γ = 1800 + ϕ(ωc ) = 1800 + [−900 − arctg0.04 × 50] = 260 č3Ď۴ऌ γ ′ ≥ 450 ֥ေ౰đ౼ γ ′ = 450 đϕc (ωc′ ) = −60 đᄵႵ γ ′ = γčωc′Ď+ ϕ(ωc′ ) ֤֞ğ 450 = ϕ(ωc′ ) − 60 ϕ(ωc′ ) = 510 = −900 − arctg0.04ω′ 99 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ౰֤ğ ωc′ = 20.2(rad / s) č4Ďໃ཮ᆞ༢๤ᄝ ωc′ = 20.2(rad / s) ԩ֥‫ږ‬ᆴູ L(ωc′ ) = 14dB đᆃ۱ᆴॖၛႮ๭ഈਈ֤đ္ॖ๙‫ݖ‬ ࠹ෘࠆ֤ ࣚಒ࠹ෘğ L(ωc′ ) = 20 lg100 − 20 lg ωc′ − 20 lg (0.04ωc′ )2 + 1 = 40 − 20 lg 20.2 − lg (0.04 × 20.2)2 + 1 = 11.8dB ࣍ර࠹ෘ ğ L(ωc′ ) = 20 lg100 − 20 lgωc′ − 20 lg (0.04ωc′ )2 +1 ≈ 40 − 20 lg 20.2 = 14dB č5Ďಒ‫཮ק‬ᆞልᇂ֥ሇᅼ௔ੱ Ⴎ L(ωc′ ) + 20 lg b = 0 đॖၛ֤֞ 20 lg b = −L(ωc′ ) = −14dB đ౰֤ b = 0.2 Ⴟ൞ğ 1 = 0.1ωc′ = 0.1× 20.2 đ౰֤ T = 2.5(s) bT č6Ďಒ‫཮ק‬ᆞልᇂ֥Ԯ‫ݦ־‬ඔ Gc (s) = 1 + bTs = 1+ 0.5s 1 + Ts 1+ 2.5s č7Ď཮ဒࢲ‫ݔ‬ ཮ᆞު༢๤֥षߌԮ‫ݦ־‬ඔູ: G′(s) = Gc (s)G(s) = 100(1 + 0.5s) s(1 + 2.5s)(1 + 0.04s) ཮ᆞު༢๤֥ྟିᆷѓğ ⎧γ ′ = 1800 + [−900 + arctg0.5 × 20 − arctg 2.5 × 20 − arctg0.04 × 20] = 46.80 > 450 ⎨ ⎩Kv = 100 ‫ކژ‬ഡ࠹ေ౰b 100 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 29 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻੂᅣ ॥ᇅ༢๤֥཮ᆞ ᇶေଽಸ ॥ᇅ༢๤֥཮ᆞࠣഡ࠹ഈࠏൌဒ ଢ֥აေ ඃ༑ႋႨ MATLAB ೈࡱഡ࠹༢๤֥ࠎЧٚ‫م‬ ౰ ඃ༑ႋႨ SISO Design Tool ࣉྛ༢๤ഡ࠹֥ࠎЧٚ‫م‬ ๙‫ݖ‬࿐༝ሱྛഡ࠹ປӮ၂۱‫ࢨؽ‬༢๤Ա৳཮ᆞഡ࠹಩ༀ ᇗ ׄ ა ଴ ᇗׄğሱྛഡ࠹ປӮ၂۱‫ࢨؽ‬༢๤Ա৳཮ᆞഡ࠹಩ༀ ׄ ଴ׄğሱྛഡ࠹ປӮ၂۱‫ࢨؽ‬༢๤Ա৳཮ᆞഡ࠹಩ༀ ࢝࿐൭‫ ؍‬ഈࠏ නॉีࠇ ປӮൌဒБۡ ቔြี ၂aൌဒଢ֥ 1aᅧ໤Ա৳཮ᆞߌࢫؓ༢๤໗‫֥ྟק‬႕ཙĠ 2aਔࢳ൐Ⴈ SISO ༢๤ഡ࠹‫۽‬ऎčSISO Design ToolĎࣉྛ༢๤ഡ࠹b ‫ؽ‬aഡ࠹಩ༀ Ա৳཮ᆞ൞ᆷ཮ᆞჭࡱა༢๤֥ჰট҆‫ٳ‬Ա৳đೂ๭ 1 ෮ൕb ๭ ᇏ đ Gc (s) і ൕ ཮ ᆞ ҆ ‫ ֥ ٳ‬Ԯ ‫ ݦ ־‬ඔ đ Go (s) і ൕ ༢ ๤ ჰ ট భ ཟ ๙ ֡ ֥ Ԯ ‫ ݦ ־‬ඔ b ֒ Gc (s) = 1+ aTs (a > 1) đູԱ৳ӑభ཮ᆞĠ֒ Gc (s) = 1+ aTs (a < 1) đູԱ৳Ӿު཮ᆞb 1+ Ts 1+ Ts ໡ૌॖၛ൐Ⴈ SISO ༢๤ഡ࠹Ա৳཮ᆞߌࢫ֥ҕඔđSISO ༢๤ഡ࠹‫۽‬ऎčSISO Design ToolĎ൞Ⴈ Ⴟֆൻೆֆൻԛّঌ॥ᇅ༢๤Ҁӊఖഡ࠹֥๭ྙഡ࠹ߌ࣢b๙‫۽ھݖ‬ऎđႨ޼ॖၛॹ෎ປӮၛ༯‫۽‬ቔğ ০Ⴈ۴݅ࠖٚ‫࠹م‬ෘ༢๤֥оߌหྟaᆌؓषߌ༢๤ Bode ๭֥༢๤ഡ࠹aเࡆҀӊఖ֥ਬࠞׄaഡ࠹ ӑభ/ᇌުຩ઎‫ބ‬ੲѯఖa‫ٳ‬༅оߌ༢๤ཙႋa‫ט‬ᆜ༢๤‫ږ‬ᆴࠇཌྷ໊ღ؇֩b č1Ďյष SISO ༢๤ഡ࠹‫۽‬ऎ ᄝ MATLAB ଁ਷Գ१ᇏൻೆ sisotool ଁ਷đॖၛյष၂۱ॢ֥ SISO Design Toolđ္ॖၛᄝ sisotool ଁ਷֥ൻೆҕඔᇏᆷ‫ ק‬SISO Design Tool ఓ‫׮‬ൈಌസյष֥ଆ྘bᇿၩ༵ᄝ MATLAB ֥֒భ‫۽‬ቔॢࡗ ᇏ‫ק‬ၬ‫ھݺ‬ଆ྘bೂ๭ 2 ູ၂۱ DC ‫֥ࠏ׈‬ഡ࠹ߌ࣢b č2Ďࡼଆ྘ᄛೆ SISO ഡ࠹‫۽‬ऎ ๙‫ ݖ‬file/import ଁ਷đॖၛࡼ෮ေ࿹࣮֥ଆ྘ᄛೆ SISO ഡ࠹‫۽‬ऎᇏbׄࠌ‫ھ‬Ғֆཛުđࡼ֐ԛ Import System Data ؓ߅ॿđೂ๭ 3 ෮ൕb č3Ď֒భ֥ҀӊఖčCurrent CompensatorĎ ๭ 2 ᇏ֒భ֥ҀӊఖčCurrent CompensatorĎ၂ণཁൕ֥൞ଢభഡ࠹֥༢๤Ҁӊఖ֥ࢲ‫ܒ‬bಌസ֥ Ҁӊఖᄹၭ൞၂۱ીႵ಩‫׮ޅ‬෿උྟ֥ֆ໊ᄹၭđ၂֊ᄝ۵݅ࠖ๭‫ ބ‬Bode ๭ᇏเࡆਬࠞׄࠇ၍‫׮‬౷ཌđ ‫ھ‬ণࡼሱ‫׮‬ཁൕҀӊఖࢲ‫ܒ‬b č4Ďّঌࢲ‫ܒ‬ SISO Design Tool ᄝಌസ่ࡱ༯ࡼҀӊఖ٢ᄝ༢๤֥భཟ๙֡ᇏđႨ޼ॖၛ๙‫ݖ‬o+/-pο୦࿊ᄴ ᆞ‫ّڵ‬ঌđ๙‫ݖ‬oFSpο୦ᄝೂ༯๭ 4 ࠫᇕࢲ‫ܒ‬ᆭࡗࣉྛ్ߐb 101 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ๭ 2 SISO ༢๤֥๭ྙഡ࠹ߌ࣢ ๭ 3 Import System Data ؓ߅ॿ ๭ 4 SISO Design Tool ᇏّ֥ঌ॥ᇅࢲ‫ܒ‬ 1a๭ 1 ෮ൕ֥॥ᇅ༢๤đჰषߌԮ‫ݦ־‬ඔູ Go (s) = 2 đ൫Ⴈ SISO ༢๤ഡ࠹‫۽‬ऎčSISO + 1)(0.3s s(0.1s + 1) Design ToolĎഡ࠹ӑభ཮ᆞߌࢫđ൐ః཮ᆞު༢๤֥࣡෿෎؇༂ҵ༢ඔ Kv ≤ 6 đཌྷ࢘ღ؇ູ 45° đѩ߻ ᇅ཮ᆞభު֥ Bode ๭đѩ࠹ෘ཮ᆞభު֥ཌྷ࢘ღ؇b č1Ďࡼଆ྘ᄛೆ SISO ഡ࠹‫۽‬ऎ ᄝ MATLAB ଁ਷Գ१༵‫ק‬ၬ‫ݺ‬ଆ྘ Go (s) = s(0.1s 2 + 1) đս઒ೂ༯ğ + 1)(0.3s num=2Ġden=conv([0.1,1,0],[0.3,1])ĠG=tf(num,den) ֤֞ࢲ‫ݔ‬ೂ༯ğ 2 ---------------------- 0.03 s^3 + 0.4 s^2 + s ൻೆ sisotool ଁ਷đॖၛյष၂۱ॢ֥ SISO Design Toolđ๙‫ ݖ‬file/import ଁ਷đॖၛࡼଆ྘ G ᄛೆ SISO ഡ࠹‫۽‬ऎᇏđೂ๭ 5 ෮ൕb č2Ď‫ט‬ᆜᄹၭ ۴ऌေ౰༢๤֥࣡෿෎؇༂ҵ༢ඔ Kv ≤ 6 đҀӊఖ֥ᄹၭႋູ 3đࡼ๭ 5 ᇏ֥ C(s)=1 ‫ ູڿ‬3đೂ๭ 5 ෮ൕbՖ๭ᇏ Bode ཌྷ௔๭ቐ༯࢘ॖၛुԛཌྷ໊ღ؇ γ = 21.2° đ҂ડቀေ౰b č3Ďࡆೆӑభ཮ᆞຩ઎ ᄝषߌ Bode ๭ᇏׄࠌඊѓႷ࡯đ࿊ᄴoAdd Pole/Zerop༯֥oLeadpҒֆđ‫ଁھ‬਷ࡼᄝ॥ᇅఖᇏเ ࡆ၂۱ӑభ཮ᆞຩ઎bᆃൈඊѓ֥ܻѓࡼэӮoXpྙሑđࡼඊѓ၍֞ Bode ๭‫ږ‬௔౷ཌഈࢤ࣍ቋႷ؊ࠞ ໊֥ׄᇂο༯ඊѓđ֤֞ೂ๭ 6 ෮ൕ֥༢๤b Ֆ๭ᇏ Bode ཌྷ௔๭ቐ༯࢘ॖၛुԛཌྷ໊ღ؇ γ = 28.4° đಯ҂ડቀေ౰đླࣉ၂҄‫ט‬ᆜӑభߌࢫ֥ ҕඔb č4Ď‫ט‬ᆜӑభຩ઎֥ਬࠞׄ ࡼӑభຩ઎֥ਬׄ၍‫֞׮‬ौ࣍ჰটቋቐш໊֥ࠞׄᇂđࢤ༯টࡼӑభຩ઎֥ࠞׄཟႷ၍‫׮‬đѩᇿၩ ၍‫ݖ׮‬ӱᇏཌྷ࢘ღ؇֥ᄹӉđ၂ᆰ֞ཌྷ࢘ღ؇ղ֞ 45° đՎൈӑభຩ઎ડቀഡ࠹ေ౰bೂ๭ 7 ෮ൕb 102 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ๭ 5 ‫ڿ‬эᄹၭު֥༢๤ ๭ 6 ᄹࡆӑభຩ઎ު֥༢๤ ๭ 7 ቋުડቀေ౰֥༢๤ Ֆ๭ᇏॖၛुԛটđӑభຩ઎֥Ԯ‫ݦ־‬ඔູ 3(1+ 0.26s) đቋު༢๤֥ Kv = 6đγ = 45.9° b (1+ 0.054s) 2a๭ 1 ෮ൕ֥॥ᇅ༢๤đჰषߌԮ‫ݦ־‬ඔູ Go (s) = k + 1) đ൫Ⴈ SISO ༢๤ഡ࠹‫۽‬ऎčSISO s(0.2s Design ToolĎഡ࠹ӑభ཮ᆞߌࢫđ൐ః཮ᆞު༢๤֥࣡෿෎؇༂ҵ༢ඔ Kv ≤ 100 đཌྷ࢘ღ؇ູ 30° đѩ ߻ᇅ཮ᆞభު֥ Bode ๭đѩ࠹ෘ཮ᆞభު֥ཌྷ࢘ღ؇b 3a൐Ⴈ SISO Design Tool ഡ࠹ᆰੀ‫טࠏ׈‬෎༢๤b‫ܒࢲࠏ׈྘ׅ‬ൕၩ๭ೂ๭ 8 ෮ൕđ॥ᇅ༢๤֥ൻ ೆэਈູൻೆ‫׈‬࿢Ua (t) đ༢๤ൻԛ൞‫ڵࠏ׈‬ᄛ่ࡱ༯֥ሇ‫࢘׮‬෎؇ ω(t) bགྷഡ࠹Ҁӊఖ֥ଢ֥൞๙‫ݖ‬ ؓ༢๤ൻೆ၂‫׈֥ק‬࿢đ൐‫ࠏ׈‬ջ‫ڵ׮‬ᄛၛ௹ຬ֥࢘෎؇ሇ‫׮‬đѩေ౰༢๤ऎႵ၂‫֥ק‬໗‫ק‬ღ؇b ๭ 8 ᆰੀ‫טࠏ׮׈‬෎༢๤ ᆰੀ‫׮ࠏ׈‬෿ଆ྘Чᇉഈॖၛ൪ູ‫ࢨؽ྘ׅ‬༢๤đഡଖᆰੀ‫֥ࠏ׈‬Ԯ‫ݦ־‬ඔູ G(s) = s2 1.5 40.02 +14s + 103 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ༢๤֥ഡ࠹ᆷѓູğഈശൈࡗ tr < 0.5s đ໗෿༂ҵ ess < 5% đቋնӑ‫ט‬ਈ M P % < 10% đ‫ږ‬ᆴღ؇ Lg > 20dB đཌྷ࢘ღ؇ γ > 40° b ༢๤ഡ࠹҄ᇧğ č1Ď‫ט‬ᆜҀӊఖ֥ᄹၭ ೂ‫ھؓݔ‬༢๤ࣉྛൈთٟᆇđॖ‫ؿ‬གྷఃࢨᄁཙႋൈࡗ‫ޓ‬նđิۚ༢๤ཙႋ෎؇֥ቋࡥֆٚ‫ࣼم‬൞ᄹ ࡆҀӊఖᄹၭ֥նཬbᄝ SISO ֥ഡ࠹‫۽‬ऎᇏॖၛ‫ޓ‬ٚь֥ൌགྷҀӊఖᄹၭ֥‫ࢫט‬ğඊѓ၍‫ ֞׮‬Bode ‫ږ‬ ᆴཌഈđο༯ඊѓቐ࡯ሂ౼ Bode ‫ږ‬ᆴཌđཟഈ຀‫׮‬đ൤٢ඊѓđ༢๤ሱ‫࠹׮‬ෘ‫ڿ‬э֥༢๤ᄹၭ‫ׄࠞބ‬b ࠻ಖ༢๤ေ౰ഈശൈࡗ tr < 0.5s đႋ‫ט‬ᆜ༢๤ᄹၭđ൐֤༢๤֥Ԭᄀ௔ੱ ωc ໊Ⴟ 3rad/s ‫࣍ڸ‬bᆃ൞ ၹູ 3rad/s ֥௔ੱ໊ᇂ࣍රؓႋႿ 0.33s ֥ഈശൈࡗb ູਔ۷ౢԣ֥Ұᅳ༢๤֥Ԭᄀ௔ੱđׄࠌඊѓႷ࡯đᄝॹࢮҒֆᇏ࿊ᄴoGridpଁ਷đࡼᄝ Bode ๭ ᇏ߻ᇅຩ۬ཌb ܴҳ༢๤֥ࢨᄁཙႋđॖၛु֞༢๤֥໗෿༂ҵ‫ބ‬ഈശൈࡗၘ֤֞‫ڿ‬೿đ֌ေડቀ෮Ⴕ֥ഡ࠹ᆷѓđ ߎႋࡆೆ۷‫گ‬ᄖ֥॥ᇅఖb č2Ďࡆೆࠒ‫ٳ‬ఖ ׄࠌඊѓႷ࡯đᄝ֐ԛ֥ॹࢮҒֆᇏ࿊ᄴoAdd Pole/Zerop༯֥oIntegratorpҒֆđᆃൈ༢๤ࡼࡆ ೆ၂۱ࠒ‫ٳ‬ఖđ༢๤֥Ԭᄀ௔ੱෛᆭ‫ڿ‬эđႋ‫ט‬ᆜҀӊఖ֥ᄹၭࡼԬᄀ௔ੱ‫ט‬ᆜ߭ 3rad/s ໊֥ᇂb č3Ďࡆೆӑభ཮ᆞຩ઎ ູਔเࡆ၂۱ӑభ཮ᆞຩ઎đᄝषߌ Bode ๭ᇏׄࠌඊѓႷ࡯đ࿊ᄴoAdd Pole/Zerop༯֥oLeadp Ғֆđ‫ଁھ‬਷ࡼᄝ॥ᇅఖᇏเࡆ၂۱ӑభ཮ᆞຩ઎bᆃൈඊѓ֥ܻѓࡼэӮoXpྙሑđࡼඊѓ၍֞ Bode ๭‫ږ‬௔౷ཌഈࢤ࣍ቋႷ؊໊֥ࠞׄᇂο༯ඊѓb Ֆ Bode ๭ᇏॖၛुԛ‫ږ‬ᆴღ؇ߎીႵղ֞ေ౰đߎླࣉ၂҄‫ט‬ᆜӑభߌࢫ֥ҕඔb č4Ď၍‫׮‬Ҁӊఖ֥ਬࠞׄ ູਔิۚ༢๤֥ཙႋ෎؇đࡼӑభຩ઎֥ਬׄ၍‫֞׮‬ौ࣍‫ࠏ׈‬ჰটቋቐш໊֥ࠞׄᇂđࢤ༯টࡼӑ భຩ઎֥ࠞׄཟႷ၍‫׮‬đѩᇿၩ၍‫ݖ׮‬ӱᇏ‫ږ‬ᆴღ؇֥ᄹӉb္ॖၛ๙‫ࢫטݖ‬ᄹၭটᄹࡆ༢๤֥‫ږ‬ᆴღ ؇b ൫οᅶഈඍٚ‫טم‬ᆜӑభຩ઎ҕඔ‫ބ‬ᄹၭđቋᇔડቀഡ࠹֥ေ౰b ೘aൌဒ҄ᇧࠣࢲ‫ݔ‬ 2a๭ 1 ෮ൕ֥॥ᇅ༢๤đჰषߌԮ‫ݦ־‬ඔູ Go (s) = k + 1) đ൫Ⴈ SISO ༢๤ഡ࠹‫۽‬ऎčSISO s(0.2s Design ToolĎഡ࠹ӑభ཮ᆞߌࢫđ൐ః཮ᆞު༢๤֥࣡෿෎؇༂ҵ༢ඔ Kv ≤ 100 đཌྷ࢘ღ؇ູ 30° đѩ ߻ᇅ཮ᆞభު֥ Bode ๭đѩ࠹ෘ཮ᆞభު֥ཌྷ࢘ღ؇b č1Ďࡼଆ྘ᄛೆ SISO ഡ࠹‫۽‬ऎ ᄝ MATLAB ଁ਷Գ१༵‫ק‬ၬ‫ݺ‬ଆ྘ Go (s) = k + 1) đս઒ೂ༯ğ s(0.2s num=1Ġden=conv([1,0],[0.2,1])ĠG=tf(num,den) ൻೆ sisotool ଁ਷đॖၛյष၂۱ॢ֥ SISO Design Toolđ๙‫ ݖ‬file/import ଁ਷đॖၛࡼଆ྘ G ᄛೆ SISO ഡ࠹‫۽‬ऎᇏđೂ๭෮ൕğ č2Ď‫ט‬ᆜᄹၭ ۴ऌေ౰༢๤֥࣡෿෎؇༂ҵ༢ඔ Kv ≤ 100 đҀӊఖ֥ᄹၭႋູ 100đࡼഈ๭ᇏ֥ C(s)=1 ‫ ູڿ‬100đ ೂ๭෮ൕbՖ๭ᇏ Bode ཌྷ௔๭ቐ༯࢘ॖၛुԛཌྷ໊ღ؇ γ = 12.8° đ҂ડቀေ౰b 104 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č3Ďࡆೆӑభ཮ᆞຩ઎ ᄝषߌ Bode ๭ᇏׄࠌඊѓႷ࡯đ࿊ᄴoAdd Pole/Zerop༯֥oLeadpҒֆđ‫ଁھ‬਷ࡼᄝ॥ᇅఖᇏเ ࡆ၂۱ӑభ཮ᆞຩ઎bᆃൈඊѓ֥ܻѓࡼэӮoXpྙሑđࡼඊѓ၍֞ Bode ๭‫ږ‬௔౷ཌഈࢤ࣍ቋႷ؊ࠞ ໊֥ׄᇂο༯ඊѓđ֤֞ೂ༯๭෮ൕ֥༢๤ğ Ֆ๭ᇏ Bode ཌྷ௔๭ቐ༯࢘ॖၛुԛཌྷ໊ღ؇ γ = 12.9° đಯ҂ડቀေ౰đླࣉ၂҄‫ט‬ᆜӑభߌࢫ֥ ҕඔb č4Ď‫ט‬ᆜӑభຩ઎֥ਬࠞׄ ࡼӑభຩ઎֥ਬׄ၍‫֞׮‬ौ࣍ჰটቋቐш໊֥ࠞׄᇂđࢤ༯টࡼӑభຩ઎֥ࠞׄཟႷ၍‫׮‬đѩᇿၩ ၍‫ݖ׮‬ӱᇏཌྷ࢘ღ؇֥ᄹӉđ၂ᆰ֞ཌྷ࢘ღ؇ղ֞ 30° đՎൈӑభຩ઎ડቀഡ࠹ေ౰đೂ๭෮ൕb Ֆ๭ᇏॖၛुԛটđӑభຩ઎֥Ԯ‫ݦ־‬ඔູ 100(1+ 0.21s) đቋު༢๤֥ Kv = 100 đ γ = 30.5° b (1+ 0.033s) 3a൐Ⴈ SISO Design Tool ഡ࠹ᆰੀ‫טࠏ׈‬෎༢๤b ᆰੀ‫׮ࠏ׈‬෿ଆ྘Чᇉഈॖၛ൪ູ‫ࢨؽ྘ׅ‬༢๤đഡଖᆰੀ‫֥ࠏ׈‬Ԯ‫ݦ־‬ඔູ G(s) = s2 1.5 đ༢๤֥ഡ࠹ᆷѓູğഈശൈࡗ tr < 0.5s đ໗෿༂ҵ ess < 5% đቋնӑ‫ט‬ਈ +14s + 40.02 M P % < 10% đ‫ږ‬ᆴღ؇ Lg > 20dB đཌྷ࢘ღ؇ γ > 40° b ༢๤ഡ࠹҄ᇧğ č1Ďࡼଆ྘ᄛೆ SISO ഡ࠹‫۽‬ऎ ᄝ MATLAB ଁ਷Գ१༵‫ק‬ၬ‫ݺ‬ଆ྘ G(s) = s2 1.5 đս઒ೂ༯ğ +14s + 40.02 num=1.5Ġden=[1 14 40.02]ĠG=tf(num,den) ൻೆ sisotool ଁ਷đ๙‫ ݖ‬file/import ଁ਷đࡼଆ྘ G ᄛೆ SISO ‫۽‬ऎᇏđೂ๭෮ൕğ 105 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ď‫ט‬ᆜҀӊఖ֥ᄹၭ ඊѓ၍‫ ֞׮‬Bode ‫ږ‬ᆴཌഈđο༯ඊѓቐ࡯ሂ౼ Bode ‫ږ‬ᆴཌđཟഈ຀‫׮‬đ൤٢ඊѓđ༢๤ሱ‫࠹׮‬ෘ ‫ڿ‬э֥༢๤ᄹၭ‫ׄࠞބ‬b࠻ಖ༢๤ေ౰ഈശൈࡗ tr < 0.5s đႋ‫ט‬ᆜ༢๤ᄹၭđ൐֤༢๤֥Ԭᄀ௔ੱ ωc ໊ Ⴟ 3rad/s ‫࣍ڸ‬bᆃ൞ၹູ 3rad/s ֥௔ੱ໊ᇂ࣍රؓႋႿ 0.33s ֥ഈശൈࡗbՎൈđ༢๤ᄹၭູ 34.8đೂ༯ ๭෮ൕb ܴҳ༢๤֥ࢨᄁཙႋđॖၛु֞༢๤֥໗෿༂ҵ‫ބ‬ഈശൈࡗၘ֤֞‫ڿ‬೿đ֌ေડቀ෮Ⴕ֥ഡ࠹ᆷѓđ ߎႋࡆೆ۷‫گ‬ᄖ֥॥ᇅఖb č2Ďࡆೆࠒ‫ٳ‬ఖ ׄࠌඊѓႷ࡯đᄝ֐ԛ֥ॹࢮҒֆᇏ࿊ᄴoAdd Pole/Zerop༯֥oIntegratorpҒֆđ༢๤ࡆೆ၂۱ ࠒ‫ٳ‬ఖđ༢๤֥Ԭᄀ௔ੱෛᆭ‫ڿ‬эđႋ‫ט‬ᆜҀӊఖ֥ᄹၭࡼԬᄀ௔ੱ‫ט‬ᆜ߭ 3rad/s ໊֥ᇂđՎൈ༢๤ᄹၭ ູ 108đೂ༯๭෮ൕğ č3Ďࡆೆӑభ཮ᆞຩ઎ ᄝषߌ Bode ๭ᇏׄࠌඊѓႷ࡯đ࿊ᄴoAdd Pole/Zerop༯֥oLeadpҒֆđ‫ଁھ‬਷ࡼᄝ॥ᇅఖᇏเ ࡆ၂۱ӑభ཮ᆞຩ઎bᆃൈඊѓ֥ܻѓࡼэӮoXpྙሑđࡼඊѓ၍֞ Bode ๭‫ږ‬௔౷ཌഈࢤ࣍ቋႷ؊ࠞ ໊֥ׄᇂο༯ඊѓbೂ๭෮ൕğ 106 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ Ֆ Bode ๭ᇏॖၛुԛ‫ږ‬ᆴღ؇ߎીႵղ֞ေ౰đߎླࣉ၂҄‫ט‬ᆜӑభߌࢫ֥ҕඔb č4Ď၍‫׮‬Ҁӊఖ֥ਬࠞׄ ູਔิۚ༢๤֥ཙႋ෎؇đࡼӑభຩ઎֥ਬׄ၍‫֞׮‬ौ࣍‫ࠏ׈‬ჰটቋቐш໊֥ࠞׄᇂđࢤ༯টࡼӑ భຩ઎֥ࠞׄཟႷ၍‫׮‬đѩᇿၩ၍‫ݖ׮‬ӱᇏ‫ږ‬ᆴღ؇֥ᄹӉbೂ๭෮ൕğ Ֆ๭ᇏॖၛुԛđՎൈ‫ږ‬ᆴღ؇ Lg = 20.5dB đཌྷ࢘ღ؇ γ = 65.1° đડቀေ౰đԬᄀ௔ੱ ωc =3.98rad/s ໊Ⴟ 3rad/s ‫࣍ڸ‬đܴҳఃࢨᄁཙႋđॖၛु֞༢๤֥໗෿༂ҵ‫ބ‬ӑ‫ט‬ਈनડቀေ౰đՎൈҀӊఖ֥Ԯ‫ݦ־‬ ඔູ C(s) = 108(1+ 0.28s) b s(1+ 0.028s)  107 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 30 Ցज़ ൱ज़ൈࡗ 2 ࿐ൈ ൱ज़ีଢčᅣaࢫĎ ֻ௾ᅣ ٤ཌྟ༢๤‫ٳ‬༅č1a2 ࢫĎ ᇶေଽಸ ‫྘ׅ‬٤ཌྟหྟa٤ཌྟหᆘࠣ٤ཌྟ༢๤‫ٳ‬༅ٚ‫م‬ ૭ඍ‫ݦ‬ඔ֥‫ק‬ၬ ਔࢳ‫྘ׅ‬٤ཌྟหྟčЎ‫ބ‬aඵಀaᇌߌa࠿‫׈‬หྟĎ ଢ֥აေ ਔࢳ٤ཌྟหᆘčԚᆴૹ‫ྟۋ‬a໗‫ྟק‬a௔ੱཙႋaሱࠗᆒ֕aࠞཋߌ֩Ď ౰ ਔࢳ٤ཌྟ༢๤‫ٳ‬༅ٚ‫م‬č૭ඍ‫ݦ‬ඔ‫م‬aཌྷ௜૫‫م‬Ď ᅧ໤૭ඍ‫ݦ‬ඔ‫ק‬ၬčႋႨ่ࡱa૭ඍ‫ݦ‬ඔ‫ק‬ၬĎ ᇗ ׄ ა ଴ ᇗׄğ‫྘ׅ‬٤ཌྟหྟa૭ඍ‫ݦ‬ඔ‫ק‬ၬ ׄ ଴ׄğ٤ཌྟหᆘ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 7.1 ႄ࿽ ؓႿ٤ཌྟ༢๤đ૭ඍఃᄎ‫׮‬ሑ෿֥ඔ࿐ଆ྘൞٤ཌྟٚӱđ෱აཌྟ༢๤֥ቋնҵљ൞҂ି൐ Ⴈ‫ࡆן‬ჰ৘b ၂a٤ཌྟ༢๤֥หׄ č1Ďඔ࿐ଆ྘ ‫ࡆן‬ჰ৘໭‫م‬ႋႨႿ٤ཌྟັ‫ٳ‬ٚӱᇏb č2Ď໗‫ྟק‬ ٤ཌྟ༢๤֥໗‫ྟק‬҂ࣇა༢๤֥ࢲ‫ބܒ‬ҕඔႵܱđ‫౏ط‬ა༢๤֥ൻೆྐ‫ބݼ‬Ԛ൓่ࡱႵܱb࿹ ࣮٤ཌྟ༢๤֥໗‫ྟק‬đсྶૼಒਆׄğ၂൞ᆷૼ۳‫ק‬༢๤֥Ԛ൓ሑ෿đ‫ؽ‬൞ᆷૼཌྷؓႿଧ၂۱௜‫ޙ‬ ሑ෿ট‫ٳ‬༅໗‫ྟק‬b č3Ď༢๤֥ਬൻೆཙႋ ‫ط‬٤ཌྟ༢๤֥ਬൻೆཙႋྙൔა༢๤֥Ԛ൓ሑ෿ಏႵܱb֒Ԛ൓ሑ෿҂๝ൈđ๝၂۱٤ཌྟ༢ ๤ॖႵ҂๝֥ਬൻೆཙႋྙൔb č4Ďሱࠗᆒ֕ࠇࠞཋߌ Ⴕུ٤ཌྟ༢๤đᄝԚ൓ሑ෿֥ࠗৣ༯đॖၛӁള‫קܥ‬ᆒ‫קܥބږ‬௔ੱ֥ᇛ௹ᆒ֕đᆃᇕᇛ௹ᆒ ֕ӫູ٤ཌྟ༢๤֥ሱࠗᆒ֕ࠇࠞཋߌbೂ‫ݔ‬٤ཌྟႵ၂۱໗‫֥ࠞק‬ཋߌđᄵ෱֥ᆒ‫ބږ‬௔ੱ҂൳ಠ ‫ބ׮‬Ԛ൓ሑ෿֥႕ཙb ‫ؽ‬a٤ཌྟ༢๤֥࿹࣮ٚ‫م‬ ଢభ‫۽‬ӱഈӈႨ֥‫ٳ‬༅٤ཌྟ༢๤֥ٚ‫م‬Ⴕ૭ඍ‫ݦ‬ඔ‫م‬aཌྷ௜૫‫م‬đၛࠣ‫ٳ‬༅٤ཌྟ༢๤໗‫ྟק‬ ֥۷၂Ϯ֥ٚ‫م‬đࠧ৙࿮௴୶‫ڏ‬ᆰࢤ‫م‬b֒ಖđ୍࣍ࠫ‫ؿ‬ᅚఏট֥၂ུ٤ཌྟ༢๤‫ٳ‬༅‫م‬Ⴕğപࣜຩ ઎a‫ྙٳ‬৘ંaህࡅ༢๤֩֩b y ೘a‫྘ׅ‬٤ཌྟหྟ č1ĎЎ‫ބ‬หྟ M k Ў‫ބ‬٤ཌྟหྟ֥ඔ࿐૭ඍ -a x ⎧M x > a y = ⎪⎨kx x < a a ⎪⎩− M x < −a Ў‫ބ‬٤ཌྟหྟ֥หׄ൞ğ֒ൻೆྐ‫ཬࢠݼ‬ൈđ‫۽‬ቔᄝཌྟ౵თĠ֒ൻೆྐ‫ࢠݼ‬նൈđൻԛӯЎ ‫ބ‬ሑ෿b 108 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ č2Ďඵ౵หྟ y ඵ౵٤ཌྟหྟ֥ඔ࿐૭ඍ ⎧0 x ≤a k ⎪ x>a y = ⎨k (x − a) x < −a - a x ⎪⎩k(x + a) ඵ౵٤ཌྟԛགྷᄝ၂ུؓཬྐ‫ݼ‬҂ਲૹ֥ልᇂᇏđ ೂҩਈჭࡱaᆳྛࠏ‫֩ܒ‬bఃหׄ൞ğ֒ൻೆྐ‫ཬࢠݼ‬ൈđ໭ൻԛྐ‫ݼ‬Ġ֒ྐ‫ݼ‬նႿඵ౵ުđൻԛྐ‫ݼ‬ Ҍෛሢൻೆྐ‫ݼ‬э߄b y č3Ďᇌߌหྟ b ᇌߌ٤ཌྟ္ӫູࡗ༣٤ཌྟđఃඔ࿐૭ඍູ y = ⎧k(x − asignx) y≠0 - x ⎩⎨b ⋅ signx y=0 a - ᇌߌ٤ཌྟᇶေթᄝႿࠏྀࡆ‫۽‬ഡСႮႿል஥ջট֥ࡗ༣đఃหྟ൞ğ֒ൻೆྐ‫ཬࢠݼ‬ൈđൻԛູ ਬĠ֒ൻೆྐ‫ݼ‬նႿ a ൈđൻԛӯཌྟэ߄Ġ֒ൻೆྐ‫ّݼ‬ཟൈđൻԛЌӻ҂эđᆰ֞ൻೆཬႿ-a ൈđൻ ԛҌႻӯཌྟэ߄b č4Ď࠿‫׈‬ఖหྟ ࠿‫׈‬ఖ٤ཌྟႵච໊࠿‫׈‬ఖหྟaऎႵඵ౵֥࠿‫׈‬ఖหྟaऎႵᇌߌ֥࠿‫׈‬ఖหྟaऎႵඵ౵‫ބ‬ᇌ ߌ࠿‫׈‬ఖหྟb 7.2 ૭ඍ‫ݦ‬ඔ ٤ཌྟ༢๤֥૭ඍ‫ݦ‬ඔіൕđ൞ཌྟ҆‫ٳ‬௔ੱหྟіൕ‫֥م‬၂ᇕ๷ܼb‫ھ‬ٚ‫م‬൮༵๙‫ݖ‬૭ඍ‫ݦ‬ඔࡼ ٤ཌྟหྟཌྟ߄đಖުႋႨཌྟ༢๤֥௔ੱ‫ؓم‬༢๤ࣉྛ‫ٳ‬༅b ၂a૭ඍ‫ݦ‬ඔ֥‫ק‬ၬ č1Ď૭ඍ‫ݦ‬ඔ֥ࠎЧ‫୑ۀ‬ ഡ٤ཌྟߌࢫ֥ൻೆྐ‫ູݼ‬ᆞ༿ྐ‫ ݼ‬x(t) = Asinωt ఃൻԛ y(t) ၂Ϯູ٤ᇛ௹ᆞ༿ྐ‫ݼ‬đॖၛᅚषູ‫ڰ‬൦ࠩඔ ∞ ∑ y(t) = A0 + ( An cos nωt + Bn sin nωt) n=1 ೏٤ཌྟߌࢫ֥ൻೆൻԛ҆‫֥࣡ٳ‬෿หྟ౷ཌ൞అؓӫ֥đࠧ y(x) = − y(−x) đႿ൞ൻԛᇏࡼ҂߶ԛ གྷᆰੀ‫ٳ‬ਈđՖ‫ ط‬A0 = 0 b 1 2π (ω t ) đ B n 1 2π π π y (t ) cos y (t ) sin n ω td (ω t ) 0 0 ∫ ∫ ൔᇏğ An = n ω td = ๝ൈđ೏ཌྟ҆‫ ֥ٳ‬G(s) ऎႵ֮๙ੲѯఖ֥หྟđՖ‫ط‬٤ཌྟൻԛᇏ֥ۚ௔‫ٳ‬ਈ҆‫ٳ‬Фཌྟ҆‫ٳ‬ն նཤ೐đॖၛ࣍රಪູ٤ཌྟߌࢫ֥໗෿ൻԛᇏᆺЇ‫ݣ‬Ⴕࠎѯ‫ٳ‬ਈđࠧ y(t) = A1 cos nωt + B1 sin nωt = Y1 sin(ωt + ϕ1) 109 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 1 2π 1 2π π π y (t ) cos y (t ) sin 0 ∫ ∫ ൔᇏğ A1 = ω td (ω t ) đ B1 = ω td (ω t ) đ 0 Y1 = A 2 + B 2 đ ϕ1 = arctg A1 1 1 B1 č2Ď૭ඍ‫ݦ‬ඔ֥‫ק‬ၬ োරႿཌྟ༢๤ᇏ֥௔ੱหྟ‫ק‬ၬğ٤ཌྟჭࡱ໗෿ൻԛ֥ࠎѯ‫ٳ‬ਈაൻೆᆞ༿ྐ‫گ֥ݼ‬ඔᆭбӫ ູ٤ཌྟߌࢫ֥૭ඍ‫ݦ‬ඔđႨ N ( A) টіൕb N ( A) = Y1 e jϕ1 = A12 + B12 ∠arctg A1 A A B1 ཁಖđϕ1 ≠ 0 ൈđ N ( A) ູ‫گ‬ඔb č3Ď૭ඍ‫ݦ‬ඔ֥ႋႨ่ࡱ æ٤ཌྟ༢๤֥ࢲ‫ܒ‬๭ॖၛࡥ߄ູᆺႵ၂۱٤ཌྟߌࢫ N ‫ބ‬၂۱ཌྟߌࢫ G(s) Ա৳֥оߌࢲ‫ܒ‬b ç٤ཌྟหྟ֥࣡෿ൻೆൻԛܱ༢൞అؓӫ֥đࠧ y(x) = − y(−x) đၛЌᆣ٤ཌྟߌࢫᄝᆞ༿ྐ‫ݼ‬ቔ Ⴈ༯֥ൻԛᇏ҂Ї‫ݣ‬ᆰੀ‫ٳ‬ਈb è༢๤֥ཌྟ҆‫ ٳ‬G(s) ऎႵਅ‫֥֮ݺ‬๙ੲѯหྟđၛЌᆣ٤ཌྟߌࢫᄝᆞ༿ൻೆቔႨ༯֥ൻԛᇏ֥ ۚ௔‫ٳ‬ਈФննཤ೐b 110 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 31 Ցज़U U ൱ज़ൈࡗ 2 ࿐ൈ U U ൱ज़ีଢčᅣaࢫĎ ֻ௾ᅣ ٤ཌྟ༢๤‫ٳ‬༅č2 ࢫĎ ᇶေଽಸ ૭ඍ‫ݦ‬ඔ ଢ֥აေ ᅧ໤૭ඍ‫ݦ‬ඔč࠿‫׈‬หྟaЎ‫ބ‬หྟ૭ඍ‫ݦ‬ඔ౰‫م‬aӈႨ٤ཌྟหྟ૭ඍ‫ݦ‬ඔіĎ ᅧ໤ቆ‫ކ‬٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔčѩ৳aԱ৳Ď ౰ ᇗ ׄ ა ଴ ᇗׄğ࠿‫׈‬หྟaЎ‫ބ‬หྟ૭ඍ‫ݦ‬ඔ౰‫م‬aӈႨ٤ཌྟหྟ૭ඍ‫ݦ‬ඔ ׄ ଴ׄğቆ‫ކ‬٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ ቔြี 7-2 ‫ؽ‬a૭ඍ‫ݦ‬ඔ֥౰‫م‬ ૭ඍ‫ݦ‬ඔ౰ࢳ֥၂Ϯ҄ᇧ൞ğ æ൮༵Ⴎ٤ཌྟหྟ౷ཌđ߂ԛᆞ༿ྐ‫ݼ‬ൻೆ༯֥ൻԛѯྙđѩཿԛൻԛѯྙ֥ y(t) ֥ඔ࿐іղ ൔb ç০Ⴈ‫ڰ‬൦ࠩඔ౰ԛ y(t) ֥ࠎѯ‫ٳ‬ਈb èࡼࠎѯ‫ٳ‬ਈսೆ૭ඍ‫ݦ‬ඔ‫ק‬ၬđࠧॖ౰֤ཌྷႋ֥૭ඍ‫ݦ‬ඔ N ( A) b ၛ࠿‫׈‬ఖ٤ཌྟหྟູ২đඪૼ૭ඍ‫ݦ‬ඔ֥౰ࢳٚ‫م‬b ႮႿ٤ཌྟູච໊࠿‫׈‬ఖđࠧᄝൻೆնႿਬൈđൻԛ֩Ⴟ‫ק‬ᆴ M đ‫ط‬ൻೆཬႿਬൈđൻԛູ‫ק‬ᆴ − M đ‫طܣ‬đᄝᆞ༿ൻೆྐ‫֥ݼ‬ቔႨ༯đ٤ཌྟ҆‫֥ٳ‬ൻԛѯྙູٚѯᇛ௹ྐ‫ݼ‬đ౏ᇛ௹๝ൻೆ֥ᆞ ༿ྐ‫ ݼ‬2π bఃѯྙೂ༯๭෮ൕb Ⴎѯྙ๭ॖ࡮đൻԛ֥ٚѯᇛ௹ྐ‫ູݼ‬అ‫ݦ‬ඔđᄵః‫ڰ‬൦ࠩඔᇏ֥ᆰੀ‫ٳ‬ਈაࠎѯ֥୽‫ݦ‬ඔ‫ٳ‬ਈ ༢ඔनູਬđࠧ A0 = 0, An = 0(n = 1,2,3, ,), Bn = 0(n = 2,4,6, ) Ⴟ൞đൻԛྐ‫ ݼ‬y(t) ॖіൕູ y(t ) = 4M (sin ωt + 1 sin 3ωt + 1 sin 5ωt + 1 sin 7ωt + ∑ ) = 4M ∞ sin(2n + 1)ωt π 3 5 7 π n=0 2n + 1 ౼ൻԛ֥ࠎѯ‫ٳ‬ਈđࠧ y1 (t ) = 4M sin ωt π Ⴟ൞đ࠿‫׈‬ఖ٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔູ N ( A) = Y1 ∠ϕ1 = 4M A πA ཁಖđ N ( A) ֥ཌྷູ໊࢘ਬ؇đః‫ږ‬ᆴ൞ൻೆᆞ༿ྐ‫ږݼ‬ᆴ A ֥‫ݦ‬ඔb 111 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ӈ࡮٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔğ æ࠿‫׈‬ఖ٤ཌྟ૭ඍ‫ݦ‬ඔ N ( A) = Y1 ∠ϕ1 = 4M A πA çЎ‫ބ‬٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔ N ( A) = B1 = 2k ⎡ a + a 1 − ( a )2 ⎤ (A ≥ a) A π ⎢arcsin A A ⎥ ⎢⎣ A ⎥⎦ èః෱ӈ࡮٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔđ࡮ Page291 і 7-1 ২ğ࿹࣮٤ཌྟ‫ݦ‬ඔ y = 1 x + 1 x 3 ֥૭ඍ‫ݦ‬ඔb 2 4 ࢳğ߂ԛ۳‫ק‬٤ཌྟหྟ౷ཌb ཁಖđ٤ཌྟหྟູֆᆴఅ‫ݦ‬ඔđ෮ၛ A0 = A1 = 0 đ ∫ B 1=1 2π (1 x + 1 x3 )sinωtd (ωt) π 02 4 ࡼ x = Asin ωt սೆഈൔđ֤֞ 1 2π ( 1 x + 1 x3 )sinωtd (ωt) = 1 2π ( 1 Asinωt + 1 A3 sin3 ωt)sinωtd (ωt) ∫ ∫ B1=π π 02 4 02 4 = 1 A + 3 A3 2 16 Ⴟ൞đ૭ඍ‫ݦ‬ඔູ N ( A) = B1 = 1 + 3 A2 A 2 16 ೘aቆ‫ކ‬٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔ ࡥֆ٤ཌྟࠎЧ৵ࢤྙൔႵԱ৳aѩ৳b č1Ď٤ཌྟหྟ֥ѩ৳࠹ෘ ሹ֥૭ඍ‫ݦ‬ඔູ N ( A) = N1( A) + N2 ( A) ႮՎॖ࡮đ೏‫ۄ‬۱٤ཌྟߌࢫѩ৳ުሹ֥૭ඍ‫ݦ‬ඔđ֩Ⴟ۱ѩ৳ߌࢫ૭ඍ‫ݦ‬ඔᆭ‫ބ‬b č2Ď٤ཌྟߌࢫ֥Ա৳ 112 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ਆ۱٤ཌྟߌࢫཌྷԱ৳đԱ৳ުሹ֥٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔѩ҂֩Ⴟ۱Ա৳ߌࢫ૭ඍ‫ݦ‬ඔ֥Ӱࠒb ‫ط‬൞ႋ‫༵ھ‬౰ԛᆃਆ۱Ա৳٤ཌྟหྟི֥֩٤ཌྟหྟđಖުᄜ౰ᆃ۱ི֩٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔb ২ğೂ༯ਆ۱٤ཌྟหྟཌྷԱ৳ ႮԱ৳ުི֥֩٤ཌྟหྟđؓᅶі 7-1 ֥ඵ౵ࡆЎ‫ބ‬٤ཌྟหྟđॖ࡮đk = 2,a = 2, ∆ = 1 đႿ൞đ ི֩٤ཌྟหྟ֥૭ඍ‫ݦ‬ඔູ N ( A) = 2k ⎡ a − arcsin ∆ + a 1− ( a )2 − ∆ 1 − ( ∆ ) 2 ⎤ π ⎢arcsin A A A AA ⎥ ⎢⎣ A ⎥⎦ = 4 ⎡ 2 − arcsin 1 + 2 1− ( 2 )2 − 1 1 − ( 1 )2 ⎤ π ⎢arcsin A A A AA ⎥ ⎢⎣ A ⎥⎦  113 uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ֻ 32 Ցज़UU ൱ज़ൈࡗ 2 ࿐ൈU U ൱ज़ีଢčᅣaࢫĎ ֻ௾ᅣ ٤ཌྟ༢๤‫ٳ‬༅č3 ࢫĎ ᇶေଽಸ ૭ඍ‫ݦ‬ඔ‫ٳ‬༅‫م‬ ଢ֥აေ ᅧ໤٤ཌྟ༢๤໗‫ٳྟק‬༅č‫֚ڵ‬૭ඍ‫ݦ‬ඔđ૭ඍ‫ݦ‬ඔ஑љ٤ཌྟ༢๤໗‫֥ྟק‬଱উථห ౰ ஑ऌĎ ᅧ໤ሱࠗᆒ֕‫ٳ‬༅čሱࠗᆒ่֕ࡱaሱࠗᆒ֕໗‫ྟק‬ ᅧ໤ሱࠗᆒ֕࠹ෘٚ‫م‬ ᇗ ׄ ა ଴ ᇗׄğ٤ཌྟ༢๤໗‫ٳྟק‬༅aሱࠗᆒ֕‫ٳ‬༅ ׄ ଴ׄğሱࠗᆒ֕‫ٳ‬༅ ࢝࿐൭‫ ؍‬൱ज़a২ีࢃࢳ නॉีࠇ 7-6a7-7a7-8 ቔြี 7.3 ٤ཌྟ༢๤֥૭ඍ‫ݦ‬ඔ‫م‬ ҐႨ૭ඍ‫ݦ‬ඔ‫م‬࿹࣮֥٤ཌྟ༢๤ႋ‫ھ‬൞ೂ༯ࢲ‫ܒ‬b ၂a٤ཌྟ༢๤֥໗‫ྟק‬ ᄝഈඍ෮ൕ֥٤ཌྟ༢๤ࢲ‫ܒ‬ᇏđ٤ཌྟ҆‫ ٳ‬N ॖၛႨ૭ඍ‫ݦ‬ඔ N ( A) іൕđཌྟ҆‫ ٳ‬G(s) ᄵႨ ௔ੱหྟ G( jω) іൕb Ⴎоߌ༢๤֥ࢲ‫ܒ‬๭đॖ֤֞༢๤֥оߌ௔ੱหྟ Φ( jω) ೂ༯ Φ( jω) = C( jω) = N ( A)G( jω) R( jω) 1 + N ( A)G( jω) ఃоߌหᆘٚӱູ1 + N ( A)G( jω) = 0 đՖ‫ط‬Ⴕ G( jω) = − 1 N ( A) ഈൔ − 1 N ( A) ӫູ٤ཌྟหྟ֥‫֚ڵ‬૭ඍ‫ݦ‬ඔb ০Ⴈ૭ඍ‫ݦ‬ඔ஑љ٤ཌྟ༢๤໗‫֥ྟק‬଱উථห஑ऌ൞ğ æ೏ G( jω) ౷ཌ҂Їຶ − 1 N ( A) ౷ཌđᄵ٤ཌྟ༢๤൞໗‫֥ק‬Ġ ç೏ G( jω) ౷ཌЇຶ − 1 N ( A) ౷ཌđᄵ٤ཌྟ༢๤൞҂໗‫֥ק‬Ġ è೏ G( jω) ౷ཌა − 1 N ( A) ౷ཌཌྷࢌđ৘ંഈࡼӁളᆒ֕đࠇӫູሱࠗᆒ֕b uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ ‫ؽ‬aሱࠗᆒ֥֕‫ٳ‬༅ა࠹ෘ Ⴎഈඍ‫ٳ‬༅ॖᆩđ֒ཌྟ҆‫֥ٳ‬௔ੱหྟ G( jω) ა‫֚ڵ‬૭ඍ‫ݦ‬ඔ౷ཌ − 1 N ( A) ཌྷࢌൈđ٤ཌྟ༢๤ Ӂളሱࠗᆒ֕b༯૫ࣉ၂҄‫ٳ‬༅ሱࠗᆒ่֥֕ࡱ‫ބ‬ሱࠗᆒ֥֕໗‫ྟק‬b æሱࠗᆒ่֕ࡱ G ( jω ) = − 1 đࠧ ⎧ G( jω)N ( A) = 1 ( A) ⎨ + ∠N ( A) N ⎩∠G( jω) = −π çሱࠗᆒ֥֕໗‫ྟק‬ ෮໌ሱࠗᆒ֥֕໗‫ྟק‬൞ᆷđ֒٤ཌྟ༢๤൳֞ಠ‫׮‬ቔႨ‫ط‬ொ৖ჰট֥ᇛ௹ᄎ‫׮‬ሑ෿đ֒ಠ‫׮‬ཨാ ުđ༢๤ି‫֞߭ܔ‬ჰট֥֩‫ږ‬ᆒ֕ሑ෿֥đӫູ໗‫֥ק‬ሱࠗᆒ֕bّᆭđӫູ҂໗‫֥ק‬ሱࠗᆒ֕b ೂႷ๭෮ൕđཌྟ҆‫֥ٳ‬௔ੱหྟ G( jω) ა Im ‫֚ڵ‬૭ඍ‫ݦ‬ඔ౷ཌ − 1 N ( A) Ⴕਆ۱ཌྷࢌׄ M1 a M 2 đ ᆃඪૼ༢๤Ⴕਆ۱ሱࠗᆒ֕ׄb ؓႿ M1 ׄđ೏൳֞ಠ‫׮‬൐‫ږ‬ᆴ A ᄹնđᄵ‫۽‬ቔ a M1 Re ׄࡼႮ M1 ׄ၍ᇀ a ׄbႮႿ a ׄ҂Ф G( jω) Їຶđ ༢๤൞໗‫֥ק‬đ‫ܣ‬ᆒ֕කࡨđᆒ‫ ږ‬A ሱ‫ཬࡨ׮‬đ ᆒ֕ b ‫۽‬ቔׄࡼခ − 1 N ( A) ౷ཌႻ߭֞ M1 ׄb ّᆭၧಖb෮ၛ M1 ׄ൞໗‫֥ק‬ሱࠗᆒ֕b c d M2 ๝৘‫ٳ‬༅ॖᆩđ M 2 ׄ൞҂໗‫֥ק‬ሱࠗᆒ֕b G( jω) ஑љሱࠗᆒ֕໗‫֥ק‬ٚ‫م‬൞ğᄝ‫گ‬௜૫ሱࠗᆒ֕‫࣍ڸ‬đ֒ο‫ږ‬ᆴ A ᄹն֥ٚཟခ − 1 N ( A) ౷ཌ၍‫׮‬ ൈđ೏༢๤Ֆ҂໗‫ק‬౵თࣉೆ໗‫ק‬౵თ֥đᄵ‫ׄࢌھ‬սі֥ሱࠗᆒ֕൞໗‫֥ק‬bّᆭđ֒ο‫ږ‬ᆴ A ᄹն ֥ٚཟခ − 1 N ( A) ౷ཌ၍‫׮‬൞Ֆ໗‫ק‬౵თࣉೆ҂໗‫ק‬౵თ֥đᄵ‫ׄࢌھ‬սі֥ሱࠗᆒ֕൞҂໗‫֥ק‬b èሱࠗᆒ֥֕࠹ෘ ؓႿ໗‫֥ק‬ሱࠗᆒ֕đఃᆒ‫ބږ‬௔ੱ൞ಒ‫ק‬ѩ౏൞ॖၛҩਈ֥đऎุ֥࠹ෘٚ‫م‬൞ğᆒ‫ږ‬ॖႮ −1 N ( A) ౷ཌ֥ሱэਈ A টಒ‫ק‬đᆒ֕௔ੱ ω Ⴎ G( jω) ౷ཌ֥ሱэਈ ω টಒ‫ק‬bླေᇿၩ֥൞đ࠹ෘ֤ ֥֞ᆒ‫ބږ‬௔ੱđ൞٤ཌྟߌࢫ֥ൻೆྐ‫ ݼ‬x(t) = Asinωt ֥ᆒ‫ބږ‬௔ੱđ‫ط‬҂൞༢๤֥ൻԛྐ‫ ݼ‬c(t) b ২ğऎႵ৘མ࠿‫׈‬ఖหྟ֥٤ཌྟ༢๤ೂ༯෮ൕđఃᇏཌྟ҆‫֥ٳ‬Ԯ‫ݦ־‬ඔູ G(s) = 10 s(s + 1)(s + 2) đ൫ಒ‫ק‬ఃሱࠗᆒ֥֕‫ږ‬ᆴ‫ބ‬௔ੱb ࢳğ࠿‫׈‬ఖ٤ཌྟ֥૭ඍ‫ݦ‬ඔູ N (A) = 4M = 4 πA πA ‫֚ڵ‬૭ඍ‫ݦ‬ඔູ − 1 = − πA N ( A) 4 ֒ A = 0 → ∞ ൈđ −1 N ( A) ౷ཌູᆜ۱‫ڵ‬ൌᇠb ཌྟ҆‫֥ٳ‬௔ੱหྟູ G( jω) = 10 =− 30 − j 10(2 − ω 2 ) jω( jω + 1)( jω + 2) ω 4 + 5ω 2 + 4 ω(ω 4 + 5ω 2 + 4) ߂ԛ G( jω) ‫ ބ‬−1 N ( A) ౷ཌೂ༯đႮ๭ॖᆩđਆ౷ཌᄝ‫ڵ‬ൌᇠഈႵ၂۱ࢌׄđ౏‫ھ‬ሱࠗᆒ֕ׄ൞໗ ‫֥ק‬b ਷ Im[G( jω)] = 0 đࠧ Im 115 − 1 N ( A) -2 -1 Re uሱ‫׮‬॥ᇅჰ৘v‫׈‬ሰ࢝σ 10(2 −ω2) = 0 ⇒ 2 − ω 2 = 0 ω(ω 4 + 5ω 2 + 4) ࡼ ω = 2 սೆ G( jω) ֥ൌ҆đ֤֞ Re[G( jω)] ω= =− 30 = −1.66 2 ω 4 + 5ω 2 + 4 ω= 2 Ⴎ G( jω)N (A) = −1 đॖ֤֞ − 1 = G( jω) N ( A) ࠧႵ − 1 = − πA = −1.66 N ( A) 4 Ⴟ൞౰֤ሱࠗᆒ֥֕‫ږ‬ᆴູ A = 2.1 Ġሱࠗᆒ֕௔ੱູ ω = 2(rad / s) b 116

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