印刷電路板最短抓取時間零件槽幾何形狀之研究

(1)

Оה࿪ྮڕ౵ൺԹפॡม࿬Іᇿೀңԛې̝ࡁտ

ෲ˧Җ

̚ර̂ጯგநጯੰ

ૺ ࿭ Ղ঍൞

̚ර̂ጯϹ఼ᄃۏ߹გநጯր/̚ර̂ጯࡊԫგநࡁտٙ

ၡ! ࢋ

ώࡁտ೩΍˘౵ൺԹפॡม۞͞ڱĂӀϡ࿬Іᇿ̚Տ˘̮І˭˘ѨజԹפ

̝፟த۞̙Т઄నĂֽࢍზОה࿪ྮڕ௡྅ሀ௡̚࿬ІᇿπӮொજ෼ᗓ۞˯ࢨ ࣃᄃ˭ࢨࣃĂ֭੫၆็௚ܜ୧ݭă็௚ಏ˘๪ሹݭăፖШଵЕ๪ሹݭ̈́Т͕๪

ᒖېଵЕඈα჌ೀңԛې࿬ІᇿซҖෞҤĄࡁտඕڍពϯТ͕๪ଵЕёᒖݭ࿬

Іᇿ౵ָĂТॡ༊̮І჌ᙷᇴດкॡĂԼචड़தດព඾Ą ᙯᔣෟĈОה࿪ྮڕă࿬Іᇿೀңԛېă೧פІॡมĄ

MINIMIZING PICK-UP-TIME FOR GEOMETRIC CONFIGURATION OF PRINTED CIRCUIT BOARD FEEDER

Li-Hsing Ho

Collegel of Management Chung Hua University Hsinchu, Taiwan 300, R.O.C.

Ching Chang Tai-lin Li

Department of Transportation and Logistics Management / Institute of Management of Technology Chung Hua University

Hsinchu, Taiwan 300, R.O.C.

Key Words: PCB, feeder configuration, pick up time.

ABSTRACT

We propose a simple method to compute the upper and lower bounds of the mean pick up time in which the rotary head has to move from one position to the other position of the PCB feeder. Four different PCB feeder configurations of line, circle, line-circle, and concentric are evaluated. The upper bound of the mean pick up time is computed under the assumption that the probability of the rotary head’s next movement to any component position of the PCB feeder is equal. Similarly, the lower bound of the mean pick up time is computed under the assumption that the probability of the rotary head’s next movement to the component position of the PCB feeder is reciprocal of the rth power of movement distance. The result indicates that the value of r is between 0.8 and 1.0 and that the concentric PCB feeder configuration has the minimal mean pick up time.

(2)

˘ă݈! ֏

ܕѐϤٺξಞ࿸׶Ă઼̰γᚮۋ۰࠹ᚶയனĂঐ෱۰

๝рк̮චតඈЯ৵ĂౄјயݡϠ׻ฉഇഴൺăΐి஌՛Ă

ֹ଀̍ຽυืԣిϠய͌ณкᇹ̼۞யݡĂͽ႕֖ξಞ۞

ԣిតዏĄ҃дށܛߵ[1]ᄃ Wang ඈˠ[2]ࡁտ̚ૻአОה

࿪ྮڕ(printed circuit boardćPCB)̮І೧ăפІ̍ຽ̚Ă PCB щ྅ॡม͹ࢋߏפՙٺ፟ୠ͘ᓖொજॡมă࿪ྮڕொ

જॡม̈́࿬Іᇿொજॡม̚౵ਈॡ۞˘ีүຽॡมĂ൒҃

ࢋщ೧ٺPCB ˯۞࿬Іܧ૱ிкĂͷՏѨ۞௡྅Բณ̙Ξ ਕΪ௡྅˘ڕĂЯѩிк۞ޞщ྅̮ІĂໂѣΞਕౄј࿬

Іᇿវ᎕ᘀ̂Ăᄃொજ̮ІॡĂхдొ̶ྵܜ۞ொજ෼ᗓ ᄃॡมĂౄјፋ࣎PCB ௡྅үຽ۞ᙯᔣॡมΞਕ̙д፟ୠ

͘ᓖԹפ̈́щ྅̮І۞үຽॡมĂ҃ߏд࿬Іᇿொજͽ̈́

̮І྅๱үຽٙਈ෱۞ॡมĂ߇ଣ੅ྵዋ۞࿬Іᇿೀңԛ ې̮̈́І੨ཉયᗟĂͽഴ͌࿬Іᇿ۞ҫϡ۩มăொજ෼ᗓ ᄃॡมĂซ҃ഴ͌ፋវ೧פІүຽ۞ԆјॡมĂࢫҲಏҜ Ϡயјώߏܧ૱ࢦࢋ۞Ą

ᐌ඾ԫఙ۞ซՎ̈́யݡ۞͟ৈ౹າĂЧёЧᇹ۞PCB ೧ăפІሀ௡̝፟ୠన౯̈́үຽ͞ёĂ̙ᕝ۞జฟ൴̈́ࡁ տ΍ֽĂ፟ୠ͘ᓖ˯۞Թᐝ(rotary heads)̏Լซז˘ѨΞ Թפᇴ̮࣎ІĂ֭дԹפॡТॡซҖ׎΁̮І۞щ྅જ үĄ҃૞ϡ፟ሀ௡˵Լซј˞఼ϡ፟ሀ௡Ă˘έҋજ̼೧ă פІϠயր௚གྷϤ̙ТనؠĂӈΞ௡྅̙Тᙷݭ۞PCBĂ ࠤҌΞТॡϠய˘჌ͽ˯̙Тఢॾ۞PCBsĄ

ፂ࿅ΝࡁտពϯĂٸཉ̮І̝࿬Іᇿវ᎕࠹ྵٺPCB

҃֏ߏܧ૱ᘀ̂۞ĂໂٽౄјፋវүຽԆјॡม྆Ăхд

̂ొ̶ॡม঎෱ٺඈޞ࿬ІᇿொજಶҜĂ߇Լត࿬Іᇿೀ

ңԛېܮΞਕࢫҲ࿬Іᇿொજ෼ᗓ(ٕॡม)ĂϺΞࢫҲ PCB ፋវүຽԆјॡมĄЯѩĂώࡁտ೩΍ፖШଵЕ๪ሹ ݭ࿬Іᇿ̈́Т͕๪ଵЕᒖݭ࿬Іᇿ׌჌፟ݭĂᖣ඾࿬Іᇿ

ೀңԛې۞ԼචĂഴ͌࿬Іᇿ۞វ᎕ͽ̈́ொજ෼ᗓĂֹ࿬

Іᇿொજॡٙ঎෱۞ॡมഴҌ౵ҲĄ

Ϥٺ၁ᅫщଵ̮Ід࿬ІᇿҜཉॡĂӮົ૟ֹϡᐛத

ྵ੼۞̮Іٸཉٺ፟ୠ͘ᓖྵटٽԹפז۞ҜཉĂ҃Ϲആ

ֹϡᐛத෸੼۞̮ІĂд࿬Іᇿ̚۞ٸཉҜཉϺྵࠎ࠹

ܕĂٙͽநຐ۞࿬Іᇿ̮ІҜཉҶཉᑕ૟ߏ˭˘జԹפ̮

ІҜཉ۞፟தᄃ׎ொજ෼ᗓјͧͅᙯܼĂЯѩĂώࡁտӀ ϡ̮Ід࿬Іᇿ̚జԹפ۞፟தᄃ׎ொજ෼ᗓr Ѩ͞Ӕͅ

ͧ۞͞ёĂࢍზπӮொજ෼ᗓĂ༊r = 1 ॡܑϯ̮ІజԹפ

፟தᄃ׎ொજ෼ᗓјͧͅᙯܼć༊r = 2 ॡܑϯ̮ІజԹפ

፟தᄃ׎ொજ෼ᗓ۞π͞јͧͅᙯܼĂүࠎҿᕝ็௚ܜ୧ ݭăಏ˘๪ሹݭăፖШଵЕ๪ሹݭ̈́Т͕๪ଵЕᒖݭඈα

჌࿬ІᇿೀңԛېᐹК۞޽ᇾĂ֭ᄃ၁ᅫPCB ೧פІүຽ

ٙਈॡม࠹ͧྵĄᖣͽଣ੅ώࡁտٙనࢍ̝ҿᕝ޽ᇾ۞໤

ቁޘĄ

ဦ 1! ܜ୧ݭ࿬Іᇿ፟ݭሀ௡

PCB

ဦ 2! ᗕ፟ୠ͘ᓖ PCB ҋજ̼௡྅ր௚፟ݭ

˟ă͛ᚥаᜪ

˘ਠPCB ҋજ೧ăפІϠயր௚ֹٙ̚ϡ۞న౯͹ࢋ

ѣ̮ІԹᐝă፟ୠ͘ᓖă೧Іүຽέ̈́࿬ІᇿĂྍαี፟

௡న౯дԆј˘̮࣎І೧ăפІજүॡĂயϠ۞үຽॡม Ξ̶ࠎ̣࣎ొ̶҂ᇋ(1)̮ІԹᐝ೧Ї́פІॡมć(2)፟ୠ

͘ᓖொજॡมć(3)PCB ொજॡมć(4)࿬Іᇿொજॡม̈́(5)

૟̮І୊ᖼҌщ྅֎ޘٙ܅෱۞୊ᖼॡมĂ҃̂ొЊОה

࿪ྮڕҋજ೧ăפІϠயր௚дેҖॡĂࠎഴ̮͌І௡྅

ॡมĂ˯ࢗ۞(2)ă(3)ă(4)̈́(5)ีॡม૟Тॡ൴Ϡͷكѩ

፾ϲĂ֭ͽ׎̚౵ਈॡ۞˘ีүຽॡมࠎ໤Ą׎̚ௐ1 ี

̮І೧ăפІॡมĂдՏ˘̮࣎І۞щ྅ฉഇ̚࠰ࠎ׽ؠ

۞Ăͷ̙ᄃ׎΁ॡมТॡ൴Ϡćௐ(2)ีүຽࡶԼଳࣧሹё

፟ୠᓖॡĂٙਈॡมᄃ׎΁үຽ࠹ͧॡ࠹၆ໂൺĂ҃ௐ(5)

ีүຽਈॡϺ࠹၆ໂൺĂЯѩĂྵָ۞PCB ௡྅үຽӈߏ ௐ(3)ᄃ(4)˟ีүຽٙਈॡมດ࠹ܕͷດൺດָĂࡶ፟௡ଳ ϡ͘ᓖݭ፟ୠᓖĂ݋ߏௐ(2)ă(3)ᄃ(4)ඈˬีүຽٙਈॡม ດ࠹ܕͷດൺॡĂ௡྅үຽດָĄ

Leipala ̈́ Nevalainen [3]ֹٙϡ̝௡྅ሀ௡ଳϡ˘࣎

ܜ୧ݭ࿬Іᇿ̈́˘੸፟ୠ͘ᓖĂтဦ1Ă༊ PCB ˯ٙᅮࢋ

щ྅۞̮І჌ᙷᓄкĂ༊࿬Іᇿொજྵ׌ბ̮ІॡĂٽх дྵܜ۞ொજ෼ᗓĂͷ፟ୠ͘ᓖ۞ொજϺхд࿅кࢦኑĄ Johnsson ඈˠ[4]ଳϡ˘ܜ୧ݭ࿬Іᇿ੨Ъ׌੸፟ୠ͘ᓖĂ

૟˘ͯPCB ̶ࠎνΠ׌࣎ཏ௡ĂТॡેҖ೧ăפІүຽĂ тဦ 2Ă፟ୠ͘ᓖ۞ࢦኑொજ̂ณഴ͌Ăҭ࿬Іᇿொજྵ

׌ბ̮ІॡĂ̪хдྵܜ۞ொજ෼ᗓĂΩ Ahmadi ̈́ Kouevlis[5]૟࿬Іᇿ̶ࠎ׌̢࣎࠹πҖ۞ܜ୧ݭ࿬ІᇿĂ тဦ3Ă̶ҾҌٺ PCB ۞׌઎ĂТॡͷكѩ፾ϲүຽĂ൒

҃࢖྽ё፟ୠ͘ᓖд࿬Іᇿ̈́PCB มࢦኑொજॡĂ̪хд

(3)

ဦ 3! ᗕ࿬Іᇿ PCB ҋજ̼௡྅ր௚፟ݭ

PCB N1

N2

ဦ 4! ๪ݭ̮ІԹᐝ PCB ҋજ̼௡྅ր௚፟ݭ

ဦ 5! ๪ݭ࿬Іᇿ PCB ҋજ̼௡྅ր௚፟ݭሀ௡

ొ̶ॡม۞঎෱ĄЯѩĂBrad ඈˠ[6]ᄃ Ohno ඈˠ[7]ࡁտ

̚ଳϡܜ୧ݭ࿬ІᇿĂ̮҃ІԹᐝ̈́፟ୠ͘ᓖ݋ଳϡ๪ሹ ېĂԹᐝଵЕд๪׹఍Ăтဦ 4Ă༊ߙ˘̮ІԹᐝд࿬І ᇿ̚Թפ̮ІॡĂ෼ᗓྍԹᐝ౵ᅈ෼۞Ω˘ԹᐝϒдPCB

˯щ೧̮ІĂҭߏܜ୧ݭ࿬Іᇿؕ௣хд׌઎̮Іม۞ொ

જ෼ᗓ࿅ܜ۞৿ᕇĂ߇дWang ඈˠ[8]ࡁտ̚݋Ӏϡ๪ሹ ݭ࿬ІᇿĂ̮Іٸཉٺ๪׹˯Ăဦ 5Ăҭߏ༊̮І჌ᙷᇴ

෸кॡĂያܕ๪͕̚۩ొ̶ٙ঎෱۞న߉۩ม˵෸кĄ ЯѩĂдෲ˧Җඈˠ[9]۞ࡁտ̚Ă੫၆˯ࢗЧ჌ᙷ࿬

Іᇿ̝৿εΐͽԼච֭ܲ঻׎ᐹ๕ޢనࢍ˞ፖШଵЕё๪

ሹې࿬Іᇿ̈́Т͕๪ᒖݭ࿬ІᇿĂтဦ6 ᄃဦ 7 ٙϯĂд

ෲ˧Җඈˠ[10]ࡁտ̚ĂӀϡ࿬Іᇿٙ̚ѣ̮ІజԹפ̝

፟த࠰࠹ඈॡĂࢍზ࿬Іᇿ̚Ї׌̮ІมπӮொજ෼ᗓ˯

ࢨࣃĂҿᕝα჌࿬Іᇿม۞ொજड़தĂ֭дෲ˧Җඈˠ[11]

ࡁտ̚ซ˘ՎӀϡ̮ІజԹפ̝፟தᄃ׎ொજ෼ᗓπ͞ј

ͧͅ۞୧І˭Ăӈr = 1 ॡĂࢍზ࿬Іᇿ̚Ї׌̮ІมπӮ

A

S1 S4

S9

B

C

D

A B C D X-Y S S1

S4

S9

ဦ 6! ፖШଵЕё๪ሹݭ࿬Іᇿ௡྅፟ݭϯຍဦ

S1

S4

S9

B

C

D

A B C D X-Y S S1

S4

S9

ဦ 7! Т͕๪ᒖݭଵЕ࿬Іᇿ፟ݭϯຍဦ

ொજ෼ᗓ۞˭ࢨࣃĂ઱α჌࿬ІᇿπӮொજ෼ᗓ̝˭ࢨࣃ ඕڍ࠹म൑ೀĂͷᄃ˯ࢨࣃม̝ડม࿅̂Ă̙ٽડ̶α჌

࿬Іᇿม۞ᐹКĄЯѩĂώࡁտ૟Հซ˘Վనࢍ̮ІజԹ פ፟தᄃ׎ொજ෼ᗓr Ѩ͞јͧͅॡĂࢍზ࿬Іᇿ̚Ї׌

̮ІมπӮொજ෼ᗓ۞˭ࢨࣃĂ֭ଣ੅౵ዋ̝r ࣃĂͽഇ

ྍ˭ࢨࣃᄃ၁ᅫ೧פІүຽॡม۞म෼౵̼̈Ą

ˬă࿬Іᇿೀңԛې̶ژ 1. ፖШଵЕё๪ሹݭ࿬Іᇿ

ፖШଵЕё๪ሹݭ࿬ІᇿĂтဦ6 ٙϯĂϤᇴ࣎ፖШ ଵЕ̝๪ሹݭ̄࿬Іᇿ௡ЪјĂЧ࣎̄࿬Іᇿ࠰ΞТॡึ

ॡᛗٕਗ਼ॡᛗ͞Ш୊ᖼĂ҃ፋវ࿬ІᇿϺΞТॡШνٕШ ΠͪπொજĂҭ̙Ξ˯˭ொજĂͽܮ૟࿬Іᇿ̚ޞԹפ̝

̮ІொҌS1ळᇾ఍Ăֻ๪ሹݭ፟ጡ͘ᓖԹפĄ

೧פІүຽેҖॡĂፋវ࿬ІᇿăЧ࣎̄࿬Іᇿă๪

ሹݭ፟ୠ͘ᓖă͘ᓖ˯۞Թᐝ̈́X-Y үຽέߏТॡͷكѩ

፾ϲொજ۞Ă༊ޞԹפ̮Іăޞщ྅̮ІăԹᐝͽ̈́PCB

࠰̏ொજҌϒቁҜཉॡĂ˯̣ࢗี፟௡І૟Тॡઃͤொ

જĂޞ፟ୠ͘ᓖ˯̝ԹᐝેҖԆјԹפ̈́щ೧үຽॡẶ

(4)

ޢĂЧี፟௡І݋Чҋаᕩ੓ᕇĄ࠹၆ٺ፟ୠ͘ᓖҜཉĂ PCB ۞੓ᕇࠎν˯֎ᕇĂ׎੓ؕҜཉᄃ S9࠹ТĂ҃࿬Іᇿ

۞੓ᕇ݋ࠎፋវ࿬Іᇿ౵͕̄̚࿬Іᇿ۞ϒ˭̮͞ІॾĂ

ࡶ̄࿬Іᇿ࣎ᇴࠎઊᇴॡĂ݋פ͕̚ν̄͞࿬Іᇿ۞ϒ˭

̮͞Іॾࠎ੓ᕇĂ׎੓ؕҜཉᄃS1࠹ТĄΩγĂώࡁտ઄

నٙѣ࿬Іॾ྆ٙٸཉ۞̮І჌ᙷ࠰̙࠹ТĂ̮ІϤ̰҃

γ੨ཉٺЧ๪ሹ̝๪׹˯Ăӈ෼ᗓ੓ᕇດܕ۞̮ІॾດА ੨ཉֹϡᐛதດ੼۞̮ІĄ

2. Т͕๪ᒖېଵЕ࿬Іᇿ

ޙТ͕๪ᒖېଵЕ࿬ІᇿĂтဦ7 ٙϯĂϤٺТ͕๪

ݭې̝నࢍĂ༊౵̰ᒖ̄࿬Іᇿٙٸཉ۞̮Іᇴ̙ТॡĂ γᆸЧᒖ̄࿬Іᇿٙਕٸཉ۞౵̮̂Іᇴ࠰ົྫྷ඾ԼតĂ

׎೧פІүຽેҖ͞ёă̮І੨ཉ͞ёă๪ሹݭ፟ୠ͘ᓖă X-Y үຽέă̮ІԹᐝăPCB ඈ̝ொજ͞ёᄃЧ፟௡І੓

ᕇ࠰ᄃፖШଵЕё๪ሹݭ࿬Іᇿ࠹ТĂ઱ፋវ࿬Іᇿ̝ொ

જ͞ёߏڻ඾ݬۡٺ๪͕̝࢖྽˯ĂШ˯ٕШ˭ொજĂݒ

̙ਕνΠொજ۞Ąဦ7 ٙϯ̝፟ݭࠎ౵̰ᒖ̄࿬Іᇿٙٸ ཉ̝̮Іᇴࠎ3ă̄࿬ІᇿᓁВ̶ࠎ 5 ᆸͷᓁ̮І჌ᙷᇴ ࠎ78 ॡ̝፟௡ሀݭĄ

3. ࿬ІᇿೀңԛېᐹК޽ᇾ

࿅Νࡁտͧ̚ྵ࿬Іᇿೀңԛې̂ౌͽ၁ᅫ࣎९ֽ

ͧྵĂЯѩͧྵඕڍࠎ࣎९̝ඕڍĂώࡁտ݋ͽԹפЇ׌

̮Іม۞πӮொજ෼ᗓĂүࠎҿᕝ࿬Іᇿೀңԛې۞ᐹК

޽ᇾĂ൒҃д၁ᅫېڶ˭Ăֹϡᐛதྵ੼۞̮І఼૱ഇ୕

జٸཉٺ፟ୠ͘ᓖྵٽԹפז۞ҜཉĂͷϹആֹϡᐛத෸

੼۞̮Ід࿬Іᇿ̚ٸཉҜཉ˵ྵࠎ࠹ܕĂ߇ώࡁտనࢍ

༊̮ІజԹפ̝፟தᄃ׎ொજ෼ᗓr Ѩ͞јͧͅॡĂՐ଀

׌̮ІมπӮொજ෼ᗓĂүࠎෞҤ࿬ІᇿೀңݭېᐹК̝

޽ᇾࣃĄ

(˘) ̮Іมؾԛ෼ᗓม෼ᗓࢍზ͞ё

ፖШଵЕ๪ሹݭᄃТ͕๪ଵЕᒖݭ࿬Іᇿ̄࿬Іᇿ̚

࠹ዐ׌̮Іมொજ෼ᗓĂߏෛ̄࿬Іᇿ̮̚Іॾ࣎ᇴ m ᄃ̮Іॾᆵޘٙՙؠ۞Ăώࡁտ઄న̮Іॾᆵޘᄃ

̮Іۡश2cr ࠹ඈĄ༊̄࿬Іᇿ࣎ᇴ N ඈٺ 1 ॡĂܑ

ϯ่ѣ˘࣎̄࿬ІᇿĂ༊ N ඈٺ M ॡӈјࠎ˘ܜ୧ ݭ࿬ІᇿĄࠎ˞ᖎ̼ࢍზĂ΄M = N×mĂϤဦ 8 ̝ဦ ྋڱΞۢТ˘̄࿬Іᇿ̚Ă࠹ዐ׌̮Іม۞ӵ֎ θĂ ГӀϡˬ֎בᇴϒăዶؽؠநĂΞՐ଀̮Ідྍ̄࿬

Іᇿ̚๪ԛொજྮश۞Ηश RĂЯѩТ˘̄࿬Іᇿ

̚Ă࠹ዐ׌̮Іม۞ொજ෼ᗓӈΞϤё(1)~(3)Ր଀Ĉ

m

θ = ²π (1)

) csc(

* 2) csc(

* cr m cr

R θ π

=

= (2)

r r

R=r csc( )θ 2 θ2

θR

ဦ 8! ဦྋࢍზ̮Іொજ̝๪ԛྮश۞Ηश

m cr m m

a R

) csc(

*

* 2 2

π π π =

= (3)

(˟) ˭ࢨࣃࢍზ͞ё

α ჌ ࿬ І ᇿ ۞ π Ӯ ொ જ ෼ ᗓ ˯ ࢨ ࣃ Leipala ̈́ Nevalainen [3]ߏ޽࿬Іᇿ̚Տ˘̮ІజԹפ̝፟த࠰

࠹Т۞୧І˭ٙՐ଀̝πӮ෼ᗓĂϡͽෞҤፋវ೧פ Іүຽ۞౵मېڶĂ҃д၁ᅫېڶ˭Ăֹϡᐛதྵ੼

۞̮І఼૱ഇ୕జٸཉٺ፟ୠ͘ᓖྵٽԹפז۞Ҝ ཉĂͷϹആֹϡᐛத෸੼۞̮Ід࿬Іᇿ̚۞ٸཉҜ ཉ˵ྵࠎ࠹ܕĂЯѩ༊̮ІజԹפ̝፟தᄃ׎ொજ෼

ᗓ۞r Ѩ͞јͧͅॡĂٙՐ଀̝׌̮ІมπӮொજ෼

ᗓĂӈΞෛࠎ࿬Іᇿ̚Ї׌࣎ޞԹפ̮ІมĂπӮொ

જ෼ᗓ۞˭ࢨࣃĂ׎r ࣃࠎ˘ޮϒ̝ᇴࣃĂ҃ώࡁտ дᇶ̙҂ᇋPCB ၁ᅫொજ۞઄న˭Ăࢍზα჌࿬Іᇿ

̝˭ࢨࣃࢍზёྎࢗт˭Ĉ

(1) ็௚ܜ୧ݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃ

Яࠎ࿬Іᇿ̚Տ̮࣎Іᄃ׎ι̮Іม۞࠹၆Ҝཉ̙

Ⴝ࠹ТĂٙͽ̙Т੓ᕇٙՐ଀۞πӮொજ෼ᗓϺ̙Ⴝ

࠹ТĂдܜ୧ݭ࿬Іᇿ̚Ă݋υืӀϡ੨ཉٺ࿬Іᇿ

̚ௐ˘ॾӈn=1ĂҌ౵̚δॾӈ n=(M+1)/2 ۞̮І዇

߹༊ࣧᕇүࢍზĂࣧᕇӈࠎ፟ୠ͘ᓖ۞ԹІᕇҜཉĄ

ࢍზ͞ڱߏ૟Ч̮Іொજ෼ᗓ۞r Ѩ͞ࣆᇴෛࠎྍ̮

ІజԹפ۞፟தĂҭߏٙѣ̮ІజԹפ፟த۞ᓁЪݒ υืࠎ1ĂЯѩυืАࢍზ΍ܜ୧ݭ࿬Іᇿ̚Ăͽௐ x ࣎࿬Іॾࠎ੓ᕇॡĂ࿬Іᇿ̚׎΁̮ІொજҌޞԹ פҜཉ෼ᗓ̝r Ѩ͞ࣆᇴ۞ᓁ׶ SlnĂтё(4)ٙϯĂ ͽଯҤ΍׎΁̮ІజԹפ۞၁ᅫ፟தࣃĂ͞ਕՐ΍༊

ͽௐx ࣎࿬Іॾࠎ੓ᕇॡĂ࿬Іᇿ̚׎΁̮ІொજҌ ޞԹפҜཉ۞πӮொજ෼ᗓElnĂтё(5)ٙϯĂ౵ޢ ГπӮЧπӮொજ෼ᗓͽࢍზ༊ͽЇ˘࣎࿬Іॾࠎ

੓ᕇॡĂЧ̮ІொજҌޞԹפҜཉ۞πӮொજ෼ᗓ ElĂтё(6)ٙϯĄ

1] [ 2

1 ¹

ln= ∑⁻1 + ∑

=

−

= n

x

n M

n

y r

r r

l x y

S a (4)

(5)

1 ] [ 2

1 ¹

1 1 1

1 ln

ln= ∑⁻ + ∑

=

−

= −

−

− n x

n M

n

y r

r r

l S x y

E a (5)

2 ] [ 1

1 ln

+

= ∑

+

=

M E E

M

l n Ă ]

2 [M+1

ࠎ੼೻௑ཱི (6)

(2) ಏ˘๪ሹݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃ

ಏ˘̮ሹݭ࿬Іᇿྵ็௚ܜ୧ݭ࿬ІᇿՀ׍ѣ၆Ⴭ

۞পّĂϤٺᒖݭ࿬Іॾ۞ދّౕĂٙͽЇ˘̮Іॾ ࠎ੓ᕇٙՐ଀̝ඕڍ࠰࠹ТĂЯࠎдՏѨฟؕેҖ ೧ăפІүຽॡЧ፟௡࠰ืಶҜٺܐ̼ؕҜཉĂӈజ Ᏼࠎ੓ᕇ̝Ї˘̮Іॾ࠰ืொજҌ፟ୠ͘ᓖ۞ԹІ ᕇҜཉĂӈ̄࿬Іᇿϒ˭͞ҜཉĂჍࠎࣧᕇҜཉĂѩ ࠎᄃ˯ࢗࢍზ͞ё౵̂۞मளĂͷ༊̮มॾ࣎ᇴࠎ؈

ᇴٕઊᇴॡ۞ࢍზ͞ёϺரѣमளĂЯѩĂώࡁտ̶

Ҿጱ΍̮Іॾࠎ؈ᇴᄃઊᇴॡ۞ࢍზёĂтё(7)̈́ё (8)ٙϯĄҭߏૄٺ፟தᓁЪυืࠎ 1 ۞ࣧ݋Ă̪υื

Аࢍზ΍׎ι̮ІொજҌޞԹפҜཉ෼ᗓ̝r Ѩ͞ࣆ ᇴ۞ᓁ׶Ă̮Іॾࠎ؈ᇴॡͽS_c^evenܑϯĂтё(9)ٙ

ϯĂ̝ͅͽS_c^oddͽܑϯĂтё(10)ٙϯĂϡͽଯҤ΍

׎΁̮ІజԹפ۞၁ᅫ፟தࣃĂ͞ਕՐ΍ͽЇ˘࣎࿬

Іॾࠎ੓ᕇॡĂ࿬Іᇿ̚׎΁̮ІொજҌޞԹפҜཉ

۞πӮொજ෼ᗓE_c^evenٕE_c^oddĄ

1] [2

2 2 ¹ 1 1

+ ∑

=

−

=

− M

x r

r r cr ceven

x M

S a (7)

= ∑

−

= 2 1

1

1 2

M

x r

cr codd

x

S a (8)

1 ] [2

2 2 ¹

1 1

1 2

1 + ∑

=

−

= −

−

M

x r

r r ceven cr ceven

x M

S

E a (9)

= ∑

−

= −

− 2

1

1 1

1

1 2

M

x r

codd cr codd

S x

E a (10)

(3) ፖШଵЕ๪ሹݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃ ፖШଵЕ๪ሹݭ࿬Іᇿࣘ׍็௚ܜ୧ёІᇿᄃಏ˘

๪ሹݭ࿬Іᇿ၆ჍّĂЧ̄࿬ІᇿνăΠ࠹၆ჍĂ҃

Ч̄࿬Іᇿώ֗݋ᄃ˘ಏ˘๪ሹݭ࿬Іᇿ࠹ТĂፖШ ଵЕ๪ሹݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃࢍზՎូྎ

ࢗт˭Ą

c ࢍზፖШଵЕ๪ሹې࿬Іᇿ̚Ч̮ІజԹפ፟த ࢵА૟ፖШଵЕ̝౵ν̄͞࿬Іᇿෛࠎௐ 1 ࣎̄࿬

ІᇿĂ༊ᓁ̄࿬ІᇿЧᇴѣn ࣎ॡĂ౵Π̄͞࿬І ᇿӈࠎௐn ࣎̄࿬ІᇿĂЧ࣎̄࿬ІᇿТ๪ሹݭ࿬

ІᇿਠĂͽЇ˘̮Іઇࢍზٙ଀ඕڍ࠰࠹ТĂ઱ܧ ၆Ⴭ̝Ч̄࿬Іᇿ̝ࣧ̚ᕇٙࢍზ΍̝Ч̮Іజ Թפ፟த࠰̙࠹ТĂЯѩĂ૟ፖШଵЕ๪ሹې࿬І ᇿડ̶ࠎ၆Ⴭ̝νăΠ׌ొ̶Ă่ͽ׎̚˘ొЊ̚

ٙѣ۞ࣧᕇĂЧࢍზ˘Ѩፋវ࿬Іᇿ̚Ч̮ІజԹ פ۞፟தĂӈΞϡֽଯҤፋវ࿬Іᇿ̚Ї׌̮Іม

۞πӮொજ෼ᗓĂЧ̮ІజԹפ፟த۞ࢍზՎូ̈́

ࢍზёྎࢗт˭Ĉ

(i) ࢍზͽௐ 1 ࣎̄࿬Іᇿࠎ੓ᕇॡĂ፟ୠ͘ᓖԹפ

࿬Іᇿٙ̚ѣ̮Іொજ෼ᗓ r Ѩ͞ࣆᇴ۞ᓁ

׶ĂϤٺბᕇ̄࿬Іᇿ่ಏᙝ׍࠹၆Ⴭ۞̄࿬І ᇿĂ׎ࢍზёᄃܧბᕇ̄࿬Іᇿ̙ТĂ፾ϲࢍზ тё(11)̈́ё(12)Ă༊̄࿬Іᇿ̮̚Іॾ࣎ᇴࠎ

؈ᇴॡͽS_lc^odd_nܑϯĂ̝ͅͽS_lc^even_nܑϯĄ

∑ + ∑ +

= =

− +

= X

x

m

X

y r

r lc lc lceven

y a x

S a

1

2 ] [ 1

1 1( )

4 )

( 2

∑ +

+

− +

= 1 2

1 2 2 )

( 2 ) (

N

z r

lc r

r L

X ma

zL

m (11)

∑ + ∑ +

= =

− +

= X

x

m

X

y r

r lc lc lcodd

y a x

S a

1

2 ] [ 1

1 1( )

4 )

( 2

∑ +

− +

= 1 2

1 2 ) (

N

z r Lr

X zL

m (12)

(ii)ࢍზͽௐ n ࣎̄࿬Іᇿࠎ੓ᕇॡĂ፟ୠ͘ᓖԹפ

࿬Іᇿٙ̚ѣ̮Іொજ෼ᗓr Ѩ͞ࣆᇴ۞ᓁ׶Ă

׎̚ n≠1 тё(13)̈́ё(14)ٙϯĂ༊̄࿬Іᇿ̚

̮ І ॾ ࣎ ᇴ ࠎ ؈ ᇴ ॡ ͽS_lc^odd_nܑ ϯ Ă ͅ ̝ ͽ

n lceven

S ܑϯĄ

∑ + ∑ +

= =

− +

= X

x

m

X

y r

r lc n lc

lceven

y a x

S a

1

2 ] [ 1

1( ) 6 )

( 2

∑⁻ + ∑ +

=

−

= 1

2( ) ( )

2

n z

n N

n

k r

r kL

m zL

m

r r

lc L

X ma

2 4 2 )

(

3 +

+ (13)

∑ + ∑ +

= =

− +

= X

x

m

X

y r

r lc n lc

lcodd

y a x

S a

1

2 ] [ 1

1( ) 6 )

( 2

(6)

∑ + ∑

=2( ) = ( ) 2

z r k n kLr

m zL

m

Lr

X 2 4 +

+ (14)

(iii)Ϥٺ፟தᓁ׶υืࠎ 1Ăҭߏٙѣ̮І۞ொજ෼

ᗓr Ѩ͞ࣆᇴ۞ᓁЪ̙֭ࠎ 1ĂٙͽЧ̮ІజԹ פ̝፟தĂඈٺЧ̮І၁ᅫொજ෼ᗓ r Ѩ̝͞

ࣆᇴĂੵͽͽЧ̮Іٙд̝̄࿬Іᇿࠎ੓ᕇ ॡĂٙࢍზ̝ٙѣ̮ІజԹפॡ۞ொજ෼ᗓ r Ѩ͞ࣆᇴᓁ׶Ą

d ࢍზፖШଵЕ๪ሹې࿬Іᇿ̚Ї׌̮Іม۞πӮ

ொજ෼ᗓ

࿬Іᇿ̚Ї˘̮ІொજҌޞԹפҜཉ۞ഇ୕ொજ

෼ᗓĂඈٺྍ̮І۞၁ᅫொજ෼ᗓࢷͽྍ̮ІజԹ פ۞፟தĂ҃ٙѣഇ୕ொજ෼ᗓ۞ᓁ׶Ăӈࠎ࿬І ᇿ̚Ї˘̮ІொજҌޞԹפҜཉ(ӈ੓ᕇ)۞πӮொ

જ෼ᗓĄҭߏͽ̙Т۞̄࿬Іᇿࠎ੓ᕇॡĂٙՐ଀

Ч̮ІజԹפ۞፟தϺ̙࠹ТĂЯѩĂ࿬Іᇿ̚Ї

׌̮࣎Іม۞πӮொજ෼ᗓĂᑕྍߏ዇߹ͽ̙Т۞

̄࿬Іᇿࠎ੓ᕇॡٙࢍზ΍۞ഇ୕ொજ෼ᗓᓁ׶

۞ᓁπӮĂ׎ࢍზ͞ё̈́ՎូᄃࢍზЧ̮ІజԹפ

፟த۞͞ڱ࠹ТĂ̶Ҿྎࢗٺ˭Ĉ

(i) ࢍზͽௐ 1 ࣎̄࿬Іᇿࠎ੓ᕇॡĂፋវ࿬Іᇿ̚

Ї׌̮ІมπӮொજ෼ᗓтё(15)̈́ё(16)ٙ

ϯĂ༊̄࿬Іᇿ̮̚Іॾ࣎ᇴࠎ؈ᇴॡͽE_lc^odd_n

ܑϯĂ̝ͅͽE_lc^even_nܑϯĄ

∑

∑ +

=

− +

= −

2 ] [ 1

1 1

1

1 ( )

4 )

( ( 2 1

m

X

y r

lc X

x r

even lc lc lceven

y a x

a E S

1) 2 2 )

( 2 )

( 1 ¹

1

2 1 −

−

= −

+ +

∑ +

+ _r

lc r N

z r L

X ma

zL

m (15)

∑ + ∑

= =

− +

= −

− X x

m

X

y r

r lc odd lc

lc lcodd

y a x

a E S

1

2 ] [ 1

1 1

1 ( )

4 )

( ( 2 1

∑ +

+ + ⁻

= − −

1

2 1 2 11)

) (

N

z r Lr

X zL

m (16)

(ii)ͽௐ n ࣎̄࿬Іᇿࠎ੓ᕇॡĂፋវ࿬Іᇿ̚Ї׌

̮ІมπӮொજ෼ᗓĂтё(17)̈́ё(18)ٙϯĂ

༊̄࿬Іᇿ̮̚Іॾ࣎ᇴࠎ؈ᇴॡͽE_lc^odd_nܑ ϯĂ̝ͅͽE_lc^even_nܑϯĄ

∑ + ∑

= =

− +

= −

− X x

m

X

y r

r lc n lc

lceven n lceven

y a x

a E S

1

2 ] [ 1

1 1

1 ( )

6 )

( ( 2 1

∑ +

+z= r− k=n kLr−

m zL

m

2( ) 1 ( ) 1

2

2) 4 2 )

( 3

1 −1

−

+ _r+

lc r L

X

ma (17)

∑ + ∑

= =

− +

= −

− X x

m

X

y r

r lc n lc

lcodd n lcodd

y a x

a E S

1

2 ] [ 1

1 1

1 ( )

6 )

( ( 2 1

∑ +

+ ⁻

= −

−

= −

n N

n

k r

n

z r kL

m zL

m

1 1

2( ) 1 ( )

2

1

2 4

−

+ Lr

X ׎̚n=1,2,…,[

2 +1

m ] (18)

e ࢍზፋវ࿬Іᇿ̚Ї׌̮Іม۞πӮொજ෼ᗓ Яࠎͽ̙Т۞̄࿬Іᇿࠎ੓ᕇॡĂٙՐ଀Ч̮І

జԹפ۞፟த࠰̙࠹ТĂ҃ٙࢍზ΍۞ፋវ࿬І ᇿٙ̚ѣ̮ІĂொજҌ੓ᕇҜཉ۞ഇ୕ொજ෼ᗓ ᓁ׶Ϻ̙࠹ТĂЯѩĂ࿬Іᇿ̚Ї׌̮࣎Іม۞

πӮொજ෼ᗓĂᑕྍߏ዇߹ͽ̙Т۞̄࿬Іᇿࠎ

੓ᕇॡٙࢍზ΍۞ഇ୕ொજ෼ᗓᓁ׶۞ᓁπӮĂ тё(19)ٙϯĈ

2 ] [ 1

1+

∑

=

+

=

m E E

m

n oddeven lc

lc n (19)

(4) Т͕๪ଵЕᒖې࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃ Т͕๪ଵЕᒖݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨࣃࢍზ͞

ёĂᄃొ̶ፖШଵЕ๪ሹݭ࿬ІᇿπӮொજ෼ᗓ˭ࢨ ࣃࢍზ៍هĂ̈́Т͕๪ଵЕᒖݭ࿬ІᇿπӮொજ෼ᗓ

˯ࢨࣃࢍზ៍ه࿩ТLeipala ̈́ Nevalainen[3]Ăυื

АՐ଀࠹ዐ׌̮Іม۞၁ᅫொજ෼ᗓĂ͞ਕଯҤ΍ፋ វ࿬Іᇿ̚Ч̮࣎ІజԹפ̝፟தĂГϡֽࢍზЇ׌

࣎ޞԹפ̮Іม۞πӮொજ෼ᗓᄃԆј˘ͯОה࿪

ྮڕ۞ᓁொજ෼ᗓĂࢍზՎូྎࢗт˭Ą c ࢍზЧᒖ̄࿬Іᇿ̚࠹ዐ׌̮Іมொજ෼ᗓ Т͕๪ଵЕᒖې࿬Іᇿᄃ˯ࢗˬ჌࿬Іᇿมमள

̝˘ߏЧ࣎̄࿬Іᇿٙ̚੨ཉ۞̮Іॾᇴ࠰̙࠹

ТĂͽ࡭Ч࣎̄࿬Іᇿ̮̚Іม۞ொજ෼ᗓϺᐌ̝

ԼតĄΩϤٺТ͕๪ݭېనࢍࢨטĂ༊౵̰ᒖٙٸ ཉ۞̮Іᇴ̙ТॡĂγᆸЧᒖ̝̮Іॾ࣎ᇴ࠰ົྫྷ

඾ԼតĂЯѩĂ࿬Іᇿଂௐ2 ᆸ̄࿬ІᇿҌ౵γᆸ

̄࿬Іᇿ̝̮̚Іॾ࣎ᇴĂυืϤௐ1 ᆸ̄࿬Іᇿ

̝̮Іॾᇴ m₁ଯҤ΍ֽĄࢵАᅮՙؠ౵̰ᆸ̄࿬

(7)

Іᇿٙ̚੨ཉ۞̮Іॾ࣎ᇴm1ĂӀϡ̮Іॾ࣎ᇴࢍ

ზ΍׌࠹ዐ̮ІมĂ၆ᑕҌ୊ᖼค͕۞ӵ֎θ1Ăт ё(20)ٙϯĂГӀϡˬ֎בᇴϒăዶؽؠநĂՐ΍

౵̰ᆸ̄࿬Іᇿ̚Ă̮Іொજ̝ؾԛྮश۞Ηश R1Ăтё(21)ٙϯĂซ҃ଯҤ΍ௐ n ᆸ̄࿬Іᇿ̚Ă

̮Іொજ̝ؾԛྮश۞ΗशRnĂтё(22)ٙϯĂ౵

ޢГӀϡˬ֎בᇴϒăዶؽ̈́ͅבᇴؠநĂࢍზ΍

ௐn ᆸ̄࿬Іᇿ̚Ă׌࠹ዐ̮І၆ᑕҌ୊ᖼค͕۞

ӵ֎θnĂтё(23)ٙϯĂӈΞࢍზ΍ௐ n ᆸ̄࿬І ᇿٙ̚ਕٸཉ۞౵̮̂І࣎ᇴmnĂ̈́׌࣎࠹ዐ̮І ม۞ؾԛொજ෼ᗓan(1)Ăтё(24)̈́ё(25)ٙϯĄ

m1 1

2π

θ = (20)

rcsc2¹

1

= θ

R (21)

2) csc ) 1 n ( 2 (

r θ¹ +

−

n=

R (22)

csc 2 ) 1 ( 2 ( csc

2 ¹ θ¹

θ_n= ⁻ n− + ) (23)

2 ] [

n

mn

θ

= π (24)

n n

n m

a 2πR ) 1

( = (25)

d ࢍზТ͕๪ଵЕᒖݭ࿬Іᇿ̚Ч̮ІజԹפ̝፟

த

Чᒖ̄࿬Іᇿপّᄃಏ˘๪ሹݭ࿬Іᇿ࠹ТĂͽ̄

࿬ І ᇿ ̚ Ї ˘̮ І ॾ ౵ ࠎ ࣧᕇ ٙ Ր ଀ ̝ ඕڍ ࠹ ТĂҭߏ༊ͽ̙Т۞̄࿬Іᇿүࠎ੓ᕇॡĂЧ̮І ᄃ፟ୠ͘ᓖม۞۞ொજ෼ᗓ૟ྫྷ඾ԼតĂЯѩĂυ

ืͽЧᒖ̄࿬Іᇿ዇߹࠰үࠎ˘ѨࣧᕇĂЧࢍზ˘

Ѩፋវ࿬Іᇿ̚Ч̮ІజԹפ۞፟தĂ׎̚઄నТ

͕๪ଵЕᒖݭ࿬Іᇿ̚Ă౵̰ᆸ̄࿬Іᇿෛࠎௐ1 ᒖ̄࿬ІᇿĂ༊ᓁ̄࿬ІᇿЧᇴѣN ࣎ॡĂ౵γᆸ

̄࿬ІᇿӈࠎௐN ᒖ̄࿬ІᇿĂΩ઄న࠰ͽЧᒖ̄

࿬Іᇿϒ˭̝̮͞Іүࠎͽྍ̄࿬Іᇿࠎ੓ᕇॡ ฟؕેҖ̮І೧פІүຽॡ̝ࣧᕇĂ݋Ч̮ІజԹ פ፟த۞ࢍზՎូ̈́ࢍზё̶Ҿྎࢗт˭Ĉ (i) ࢍზௐᒖ̄࿬Іᇿ̚ĂЧ̮Іொજ෼ᗓ r Ѩ͞ࣆ

ᇴ۞ᓁ׶Ă׎ࢍზёෛྍᒖ̄࿬Іᇿٙ̚ਕٸཉ

۞౵̮̂Іᇴࠎઊᇴٕ؈ᇴ҃ѣ̙ٙТĂтё (26)̈́ё(27)ٙϯĂ༊̄࿬Іᇿ̮̚Іॾ࣎ᇴࠎ

؈ᇴॡͽS_ct^odd_nܑϯĂ̝ͅͽS_ct^even_nܑϯĄ

1] [2

2 2 ^] [ 1

1 1

+ ∑

=

−

=

− mⁿ

x r

nr r r ctn evenn

ct a m x

S (26)

= ∑

−

= 2 1

1 ( )

2

mn

x r

ctn n

ctodd

x

S a (27)

(ii)ࢍზௐ n ᒖ̄࿬Іᇿγᆸٙѣ̄࿬Іᇿ̚Ч̮

Іொજ෼ᗓr Ѩ͞ࣆᇴ۞ᓁ׶Ăࢍზ͞ڱᄃௐ n ᒖ࿬Іᇿ̚Ї˘̮ІொજҌγಛЧᒖ̄࿬Іᇿ

ٙ̚ѣ̮Іมொજ෼ᗓᓁ׶۞˯ࢨࣃࢍზ͞ڱ

៍ه࠹ҬLeipa ̈́ Nevalainen[3]Ă઱˯ࢨࣃ̚Я ࠎՏ̮࣎ІజԹפ۞፟த࠰࠹ТĂٙͽдࢍზё

̙̚҂ᇋ፟தĂͷᓁ෼ᗓߏͽЧ჌ொજ෼ᗓࢷͽ

ྍொજ෼ᗓ۞ᓁ̮Іᇴ҃଀Ă҃Ч̮Іொજ෼ᗓ r Ѩ͞ࣆᇴ۞ᓁ׶Ă݋ߏͽЧ̮Іொજ෼ᗓ r Ѩ

͞۞ࣆᇴࢷͽĂொજྍ෼ᗓ۞ᓁ̮ІᇴՐ଀Ăࢍ

ზётё(28)ٙϯĄ

∑ ∑ − −

+

= = +

−

= N

n x

n N

x r

x oddeven n ct

ctout

xR S X

S

1 1(2 )

1 2

∑ ∑

+

=

−

= N n x

n x

y r

ctx

ya

2 1

1 ( )

2 Ăn=1,2,…,N-1 (28)

(iii) ࢍზௐ n ᒖ̄࿬Іᇿ̰ᆸٙѣ̄࿬Іᇿ̚Ч̮

Іொજ෼ᗓ r Ѩ͞ࣆᇴ۞ᓁ׶Ăࢍზ͞ڱᄃ n ᒖ̄࿬Іᇿγᆸٙѣ̄࿬Іᇿ̚Ч̮Іொજ෼

ᗓ r Ѩ͞ࣆᇴᓁ׶͞ё࠹ТĂࢍზётё(29)

ٙϯĄ

∑ +∑ − ∑ ∑

= ⁻

=

−

=

−

= = 1

1

1 1

2 1 1

1 ²

) (

2 )

2 (

n x

Z

y r

ctx x r

oddeven n ct

ctin

ya xR

S Z

S ć

Z1 = min{ 2X+1 , mn-x }ć Z₂= min{ [m_x/2] , n-1-x }ć

n=2,3,…,N (29) (iv) Ϥٺፋវ࿬Іᇿ̚Ăٙѣ̮ІజԹפ፟த۞ᓁ

׶υืࠎ1ĂЯѩĂ༊ͽௐ n ᆸ̄࿬Іᇿࠎ੓ᕇ ॡĂЧ̮ІజԹפ۞፟தӈඈٺЧ̮Іொજ෼

ᗓr Ѩ̝͞ࣆᇴĂГੵͽ༊ͽௐ n ᆸ̄࿬Іᇿ ࠎ੓ᕇॡĂٙՐ଀̝ٙѣ̮Іொજ෼ᗓ r Ѩ͞

ࣆᇴ۞ᓁ׶S_ct_nĂтё(30)ٙϯĄ

inn n ct ctout n oddeven n ct

ct S S S

S = + + Ăn=1,2,…,N (30)

e ࢍზТ͕๪ଵЕᒖݭ࿬Іᇿ̚Ї׌̮Іม۞πӮ

ொજ෼ᗓ

ࢍზ͞ёᄃፖШଵЕ๪ሹݭ࿬ІᇿܕҬĂ઱ፖШଵ Е๪ሹݭ࿬Іᇿߏ̶νăΠ׌઎ࢍზĂ҃Т͕๪ଵ Еᒖݭ࿬Іᇿ݋̶ࠎ̰ᆸăγᆸᄃҋᆸඈˬᆸĂ҃

(8)

࿬Іᇿ჌ᙷ

m ܜ୧ݭ ಏ˘

๪ሹݭ

ፖШଵЕ

๪ሹݭ

Т͕๪

ଵЕᒖݭ

ፖШଵЕ๪ሹݭྵಏ˘

๪ሹݭ۞Լචड़த

Т͕๪ݭྵፖШଵЕ

๪ሹݭ۞Լචड़த

100 32 28 21 10 25% 52%

300 82 75 48 15 36% 69%

500 130 115 80 20 30% 75%

Տ˘ᆸ˫ѣ؈ᇴᄃઊᇴ۞मҾĂֹࢍზՀࠎኑᗔĂ ზёᄃࢍზՎូྎࢗт˭Ą

(i) ࢍზௐ n ᒖ̄࿬Іᇿ̚ĂЇ׌̮Іม۞πӮொજ

෼ᗓĂ׎ࢍზёෛྍᒖ̄࿬Іᇿٙ̚ਕٸཉ۞౵

̮̂Іᇴࠎઊᇴٕ؈ᇴ҃ѣ̙ٙТĂтё(31)̈́

ё(32)ٙϯĂ༊̄࿬Іᇿ̮̚Іॾ࣎ᇴࠎ؈ᇴॡ ͽE_ct^even_nܑϯĂ̝ͅͽE_ct^even_nܑϯĄ

1 ] [2

2 2 ^]

[ 1

1 1

1 2

1 + ∑

=

−

= −

−

mn

x r

nr r

ctn r ctn n cteven

x S m

E a (31)

= ∑

−

= −

2 ] [ 1

1 ( ) 1

2

mn

x r

ctn ctn n

ctodd

x a

E S (32)

(ii) ࢍზௐ n ᒖ̄࿬ІᇿγᆸЇ׌̮Іม۞πӮொ

જ෼ᗓĂࢍზ͞ڱᄃ˯ࢨࣃࢍზ͞ڱ࿩ТĂ઱˯

ࢨࣃ̚ЯࠎՏ̮࣎ІజԹפ۞፟த࠰࠹ТĂٙͽ дࢍზё̙̚҂ᇋ፟தĂ҃˭ࢨࣃࢍზё̚ĂЧ

̮І۞ഇ୕ொજ෼ᗓ݋υืГࢷͽྍ̮ІజԹ פ۞፟தĂࢍზётё(33)ٙϯĄ

∑ ∑ − −

+

= = +

−

= −

N n x

n N

x r

ctn x

oddeven n ct

crout

xR S E X

E

1 1 (2 ) 1

1 2

∑ ∑

+

=

−

= −

N n x

n x

y r

ctx ctn ya S

2 1

1 ( ) 1

2 Ă

n=1,2,…,N-1 (33) (iii) ࢍზௐ n ᒖ̄࿬Іᇿ̰ᆸЇ׌̮Іม۞πӮொ

જ෼ᗓĂࢍზ͞ڱᄃ˯ࢨࣃࢍზ͞ڱ࿩ТĂ઱

˯ࢨࣃ̚ЯࠎՏ̮࣎ІజԹפ۞፟த࠰࠹ТĂ

ٙͽдࢍზё̙̚҂ᇋ፟தĂ҃˭ࢨࣃࢍზё

̚ĂЧ̮І۞ഇ୕ொજ෼ᗓ݋υืГࢷͽྍ̮

ІజԹפ۞፟தĂࢍზётё(34)ٙϯĄ

∑ + ∑ −

= ⁻

=

−

= −

1 1

1

1 1

1

) 2 (

n x

n

x r

ctn x oddeven n ct

ctin

xR S E Z

E

∑ ∑⁻

= = −

2

1 1 1

2

) (

2

n x

Z

y r

ctx ctn ya

S , n=2,3,…,NĂ

Z1 = min{2X+1, mn-x }Ă

Z₂= min{[m_x/2], n-1-x} (34) (iv) ፋវ࿬Іᇿ̚ĂЇ׌̮Іม۞πӮொજ෼ᗓࢍ

ზётё(35)ٙϯĄ

N E E E E

N

n inn

n ct ctout n oddeven ct ct

∑ + +

= ⁼¹ (35)

αă̶ژඕڍ

ॲፂ˯ࢗࢍზ͞ёĂώࡁտ૟̶ژ༊̮ІజԹפ፟த ᄃ׎ொજ෼ᗓr Ѩ͞јͧͅॡĂᐌ۰ r ࣃăᓁ̮Іᇴ M ̈́

࿬Іᇿԛې۞ԼតĂ࿬Іᇿ̚Ї׌̮ІมπӮொજ෼ᗓ۞

ࢍზඕڍĂീྏ̝࿬Іᇿ჌ᙷࠎ็௚ܜ୧ݭ࿬Іᇿăಏ˘

̮ሹݭ࿬ІᇿăፖШଵЕ๪ሹݭ࿬ІᇿĂͽ̈́Т͕๪ଵЕ ݭ࿬ІᇿĄϤٺ̶ژᇴፂᘀ̂൑ڱ˘˘ЕᓝĂЯѩᎡᏴ׎

̚щ྅̮І჌ᙷᇴࠎ100ă300 ᄃ 500 ඈˬ௡ΐͽᄲځĂт ဦ9 Ҍဦ 11 ٙϯĂဦ 9 ܑϯ༊ᓁщ྅̮І჌ᙷᇴѣ 100

჌ॡĂα჌࿬Іᇿ̚Ї׌̮ІมπӮொજ෼ᗓ۞ᔌ๕ቢć ဦ10 ܑϯ༊ᓁщ྅̮І჌ᙷᇴѣ 300 ჌ॡć҃ဦ 11 ݋ܑ

ϯѣ500 ჌щ྅̮І჌ᙷᇴॡĄဦ̚˘ொજಏҜࠎ˘̮І ॾ۞ΗशĂŜܑϯܜ୧ݭ࿬Іᇿ̝πӮொજ෼ᗓᔌ๕ቢć ŕ޽ಏ˘๪ሹݭ࿬Іᇿć Ű޽ፖШଵЕ๪ሹݭ࿬ІᇿćŎ

޽Т͕๪ଵЕݭ࿬ІᇿĄ

Ϥဦ9 Ҍဦ 11 ̝ඕڍពϯĂ༊ r ࣃ̂ٺ 1.2 ॡĂ࿅ٺ

ૻአ෼ᗓᄃ፟தม̝ᙯܼĂಏ˘๪ሹݭăፖШଵЕ๪ሹݭ ᄃТ͕๪ଵЕᒖݭඈˬ࿬Іᇿมொજड़த࠹म൑ೀĂ̙ዋ آϡͽͧྵ׎มᐹКّć༊r ࣃд 0.8 Ҍ 1.0 ̝มॡĂٙՐ

଀̝πӮொજ෼ᗓྵΞૻአα჌࿬ІᇿมπӮொજ෼ᗓ̝

मளّĂܑ˘ٙϯࠎ༊r ࣃࠎ 0.9 ॡĂЧ࿬Іᇿ̚Ї׌̮

ІมπӮொજ෼ᗓᙯܼĂྍα჌࿬Іᇿมொજड़த۞ͧྵ

ࣃĂᄃ၁ᅫ೧ăפІүຽ̚Ăֹϡα჌࿬Іᇿщ೧˘̮І

ٙ܅෱̝ॡม(ொજ෼ᗓ)ͧྵࣃໂ࠹ܕĂӈ༊ r ࣃ̬ٺ 0.8~1 ̝มॡĂٙࢍზ΍̝࿬ІᇿπӮொજ෼ᗓĂՀዋЪ үࠎҿᕝ࿬ІᇿೀңԛېᐹК۞˭ࢨ޽ᇾࣃĂ֭ͽr = 0.9 ॡ౵ᐹĄ

ဦ12(a)~(d)ٙϯࠎ็௚ܜ୧ݭăಏ˘̮ሹݭăፖШଵ Е๪ሹݭĂͽ̈́Т͕๪ଵЕݭඈα჌࿬Іᇿ࿬Іᇿ༊r ࣃ

(9)

50 45 40 35 30 25 20 15 10

00.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

ဦ 9! 100 ჌щ྅̮І჌ᙷᇴॡ࿬ІᇿπӮொજ෼ᗓᔌ๕ቢ

50 45 40 35 30 25 20 15 10 5 0

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 ဦ 10! 300 ჌щ྅̮І჌ᙷᇴॡ࿬ІᇿπӮொજ෼ᗓᔌ๕

ቢ

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 250

200 150 100 50 0

ဦ 11! 500 ჌щ྅̮І჌ᙷᇴॡ࿬ІᇿπӮொજ෼ᗓᔌ๕ ቢ

ࠎ0.9 ॡĂᐌ۰ᓁ̮Іᇴ M ۞ԼតĂ׎˯ࢨࣃᄃ˭ࢨࣃ۞

ડมϯຍဦĂဦ̚Śܑϯٙѣ̮ІజԹפ̝፟த࠰࠹Т ॡĂٙࢍზ̝࿬ІᇿπӮொજ෼ᗓ˯ࢨࣃᔌ๕ቢćŜܑϯ

༊r = 0.9 ॡٙࢍზ̝˭ࢨࣃᔌ๕ቢćΩŕܑϯ r = 2 ॡĂ

࿬ІᇿπӮொજ෼ᗓ۞ᔌ๕ቢĄ

̣ăඕ! ኢ

ώࡁտӀϡ̮ІజԹפ̝፟தᄃ׎ொજ෼ᗓr Ѩ͞ј

ͧͅ۞͞ёĂࢍზ࿬Іᇿ̚Ї˘̮І۞πӮொજ෼ᗓĂͽ

ͧྵܜ୧ݭă๪ሹݭăፖШଵЕё๪ሹݭ̈́Т͕๪ଵЕё ᒖݭඈα჌࿬Іᇿม۞ᐹКĂࡁտඕڍពϯ༊ r ࣃд 0.8 Ҍ1.0 มĂٙՐ଀̝πӮொજ෼ᗓྵΞૻአα჌࿬Іᇿม πӮொજ෼ᗓ۞ͧྵᙯܼĂྵr = 2 ՀዋЪүࠎҿᕝ࿬Іᇿ πӮொજ෼ᗓ˭ࢨࣃ۞޽ᇾĄΩፖШଵЕё๪ሹݭ࿬Іᇿ

ྵಏ˘๪ሹݭ࿬ІᇿπӮொજ෼ᗓԼචड़த࠰྿ 45%ͽ

˯Ă҃Т͕๪ଵЕёᒖݭ࿬Іᇿ̝Լචड़த˫ͧፖШଵЕ ё๪ሹݭ࿬Іᇿ੼΍50%ͽ˯Ăͷώࡁտٙనࢍ̝ෞҤ͞

500 400 300 200 100 0

100 150 200 250 300 350 400 450 500 550 600

(a) 450400

350300 250200 150 10050

0 100 150 200 250 300 350 400 450 500 550 600

(b) 300

250 200 150 100 50

0 100 150 200 250 300 350 400 450 500 550 600

(c)

50 40 30 20 10

0 100 150 200 250 300 350 400 450 500 550 600

(d)

ဦ 12 (a)ܜ୧ݭ࿬Іᇿ˯ă˭ࢨࣃડมϯຍဦć(b)ಏ˘

๪ሹݭ࿬Іᇿ˯ă˭ࢨࣃડมϯຍဦć(c)ፖШଵЕ

๪ሹݭ࿬Іᇿ˯ă˭ࢨࣃડมϯຍဦć(d)Т͕๪ଵ Её࿬Іᇿ˯ă˭ࢨࣃડมϯຍဦ

ڱٙࢍზ΍α჌࿬Іᇿม۞ொજड़தͧྵࣃĂᄃ၁ᅫ೧ă פІүຽ̚Ăֹϡα჌࿬Іᇿщ೧˘̮Іٙ܅෱̝ॡม(ொ

જ෼ᗓ)ͧྵࣃໂ࠹ܕĂܑϯώࡁտٙనࢍ̝ҿᕝ࿬Іᇿೀ

ңԛېᐹК޽ᇾࣃ۞͞ёĂ׍ѣ၁ᅫᑕϡᆊࣃĂͷЯࠎ̙

ᅮТॡ҂ᇋPCB ᄃ፟ୠ͘ᓖ۞ொજॡมĂϺྵ็௚ҿᕝ͞

ё࠷ॡĄ

௑ཱི৶͔

M ᓁ̮І჌ᙷᇴ N ̄࿬ІᇿЧᇴ

m ̄࿬Іᇿ̮̚Іॾ࣎ᇴ