Calculo_eletrico_de_linhas_de_transmissa

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❈á❧❝✉❧♦ ❡❧étr✐❝♦ ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦ ✲ ◆♦t❛s ❞❡ ❛✉❧❛
CC© ❈❛r❧♦s ❑❧❡❜❡r ❞❛ ❈♦st❛ ❆rr✉❞❛∗
❈❊❋❊❚✲❘❏
✷✵ ❞❡ ♠❛✐♦ ❞❡ ✷✵✶✹
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❯♠❛ ✐❞❡✐❛ s♦❜r❡ ❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s ✷
✸ ❊st✉❞♦s ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦ ✸
✹ ❈á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❡❧étr✐❝♦s ✲ ♠♦❞❡❧❛❣❡♠ ❜ás✐❝❛ ✸
✺ ❉❡s❡♠♣❡♥❤♦ ❡❧étr✐❝♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ✶✶
✻ ▲✐♠✐t❡s ❞❡ tr❛♥s♠✐ssã♦ ✶✺
✼ ▼♦❞❡❧♦ ❞♦ q✉❛❞r✐♣♦❧♦ ✶✼
✽ ▼♦❞❡❧♦ ❞❡ ✢✉①♦ ❞❡ ♣♦tê♥❝✐❛ ✷✷
✾ ❈♦♠♣❡♥s❛çã♦ ❞❡ ❧✐♥❤❛s ✷✸
✶✵ ❈á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❡❧étr✐❝♦s ✲ ♠♦❞❡❧♦ ❞❡t❛❧❤❛❞♦ ✷✻
✶✶ ❊st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ tr❛♥s♠✐ssã♦ ❛tr❛✈és ❞❡ ♠❛tr✐③ Ybarra ✸✷
✶✷ ❘❡q✉✐s✐t♦s ❡❧étr✐❝♦s ❞❡ ♣r♦❥❡t♦ ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦ ✸✷
✶✸ ❈♦♠♣♦rt❛♠❡♥t♦ ♥ã♦✲❧✐♥❡❛r ❡♠ s✐st❡♠❛s ❞❡ tr❛♥s♠✐ssã♦ ✸✼
✶✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✸✽
❆ ❚❛❜❡❧❛ ❝♦♠♣❛r❛t✐✈❛ ❞❡ ♣❛râ♠❡tr♦s ✸✾
❇ ❈á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❡❧étr✐❝♦s ✲ ♠♦❞❡❧♦ s✐♠♣❧✐✜❝❛❞♦ ✸✾
❈ ❚ó♣✐❝♦s ❛✈❛♥ç❛❞♦s ✹✵
❉ ◗✉❡stõ❡s ❞❡ ❝♦♥❝✉rs♦s ✹✷
✶ ■♥tr♦❞✉çã♦
✶✳✶ ❙♦❜r❡ ❛ ❛♣♦st✐❧❛
❊st❡ ♠❛t❡r✐❛❧ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ s✉❜s✐❞✐❛r ❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❝á❧❝✉❧♦ ❡❧étr✐❝♦ ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✱
❧❡❝✐♦♥❛❞❛ ♥♦ ❈❊❋❊❚✲❘❏✳ P❛r❛ ♦ ❛ss✉♥t♦✱ ❡①✐st❡ ✉♠❛ ❧✐t❡r❛t✉r❛ ♠✉✐t♦ ✈❛st❛✱ ✐♥❝❧✉✐♥❞♦ ❛rt✐❣♦s✱
♥♦r♠❛s✱ t❡s❡s ❡ ❞✐ss❡rt❛çõ❡s✳ P❛rt✐✉✲s❡ ❞❛ ✐❞❡✐❛ ❞❡ r❡s✉♠✐r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s✱ ❝♦♥s✐❞❡r❛❞♦s ❜ás✐❝♦s✱
❞❡✐①❛♥❞♦ ♣❛rt❡s ❞❡ ♠❛✐♦r ♣r♦❢✉♥❞✐❞❛❞❡ ♣❛r❛ ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s✱ ❢♦r♠❛♥❞♦ ❛ss✐♠ ✉♠❛ ✏❡s♣✐r❛❧✑ q✉❡
r❡t♦r♥❛ ❛♦ â♥❣✉❧♦ ✐♥✐❝✐❛❧ ♠❛s ❝♦♠ ♣r♦❢✉♥❞✐❞❛❞❡✳
∗❝❛r❧♦s❦❧❡❜❡r❅❣♠❛✐❧✳❝♦♠ ✴ ❤tt♣✿✴✴s✐t❡s✳❣♦♦❣❧❡✳❝♦♠✴s✐t❡✴❝❛r❧♦s❦❧❡❜❡r✴ ✲ BY:© $\© P❡r♠✐t✐❞♦ ✉s♦ ♥ã♦ ❝♦♠❡r✲
❝✐❛❧✱ ❝✐t❛♥❞♦ ♦ ❛✉t♦r ❡ ❢♦♥t❡✳
✶
❉❡✈✐❞♦ ❛ ❞✐s❝✐♣❧✐♥❛ ♥ã♦ ❛❜r❛♥❣❡r ♦ ❝á❧❝✉❧♦ ♠❡❝â♥✐❝♦✱ ❝✉❥❛ ✐♥t❡r❛çã♦ ❝♦♠ ❛ ♣❛rt❡ ❡❧étr✐❝❛ é
♠✉✐t♦ í♥t✐♠❛✱ ❛❜♦r❞❛✲s❡ s♦♠❡♥t❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ♥❡st❛ ♣❛rt❡✱ ❝♦♠♦ ✢❡❝❤❛ ❡ ❛♠♣❛❝✐❞❛❞❡✱ ✜❝❛♥❞♦
❛♦ ❛❧✉♥♦ ❝♦♥s✉❧t❛r ❧✐✈r♦s ❝♦♠♦ ❬✶✻❪✱ ❡ ❛ ❛♣♦st✐❧❛ ❞❛ ♣❛rt❡ ♠❡❝â♥✐❝❛ ❬✹❪✳
Pr♦❝✉r♦✉✲s❡ ✐♥❝❧✉✐r r❡❢❡rê♥❝✐❛s ❛❞✐❝✐♦♥❛✐s✱ q✉❡ ❛♣❡s❛r ❞❡ ❡st❛r❡♠ ❢♦r❛ ❞♦ ❡s❝♦♣♦ ❞❛ ❣r❛❞✉❛çã♦✱
sã♦ ✐♥s♣✐r❛çã♦ ♣❛r❛ ♣♦♥t♦s ❞❡ ♣❛rt✐❞❛ ♣❛r❛ ❡st✉❞♦s s✉❜s❡q✉❡♥t❡s✳
✶✳✷ ◆♦t❛ s♦❜r❡ ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛ ❡ ❝♦♥✈❡♥çõ❡s
❚♦❞❛s ❛s ✉♥✐❞❛❞❡s sã♦ ♥♦ s✐st❡♠❛ ♠étr✐❝♦✱ ❡①❝❡t♦ q✉❛♥❞♦ ❛ ✉♥✐❞❛❞❡ é r❡❢❡rê♥❝✐❛ ✉s✉❛❧ ✭❝♦♠♦
♣♦r ❡①❡♠♣❧♦ ❛ ❡s♣❡❝✐✜❝❛çã♦ ❞❡ ❝❛❜♦s ✉s❛✲s❡ ▼❈▼ ♦✉ ❦❝♠✐❧✶✮✱ ♠❛s ♠❡s♠♦ ❡st❛s t❡♥❞❡♠ ❛ s❡r❡♠
s✉❜st✐t✉í❞❛s✳
❊♠ t♦❞❛s ❛s ❢ór♠✉❧❛s ❡ ❡q✉❛çõ❡s s✉♣õ❡✲s❡ q✉❡ ❛s ❣r❛♥❞❡③❛s ❡st❡❥❛♠ s❡♠ ♠ú❧t✐♣❧♦s ❡ s✉❜♠ú❧t✐✲
♣❧♦s✱ ♦✉ s❡❥❛✱ r❡❝♦♠❡♥❞❛✲s❡ ❛t❡♥çã♦ ❛♦ ♦♠✐t✐r ♠✐❧✐✱ ♠✐❝r♦✱ q✉✐❧♦✱ ♠❡❣❛❀ ❡♠ ✈ár✐❛s t❛❜❡❧❛s✱ ✉t✐❧✐③❛✲s❡
♠ú❧t✐♣❧♦s ❡ s✉❜♠ú❧t✐♣❧♦s ♣❛r❛ ❞❡✐①❛r ♦ t❡①t♦ ♠❛✐s ❧❡❣í✈❡❧✱ ❡✈✐t❛♥❞♦ ❛s ♣♦tê♥❝✐❛s ❞❡ ✶✵✳
❖ ❡st✉❞♦ ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦ ❡♥✈♦❧✈❡ ❛s ❡q✉❛çõ❡s ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦✱ ❛♦♥❞❡ ❛♣❧✐❝❛✲s❡✱ ♥♦
✈á❝✉♦✱ ❛s ❝♦♥st❛♥t❡s ❞❡ ♣❡r♠✐ss✐✈✐❞❛❞❡✱ ε0 = 8, 8541878·10−12 ❋✴♠✱ ❡ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛✱
µ0 = 4π 10
−7 ❍✴♠✳
✶✳✸ ❙♦❜r❡ ♦ ✉s♦ ❞❡ ❢❡rr❛♠❡♥t❛s ❞❡ ♣r♦❣r❛♠❛çã♦
❆♦ ❧♦♥❣♦ ❞♦ t❡①t♦ ❛❜♦r❞❛✲s❡ ❝á❧❝✉❧♦s ♣rát✐❝♦s✱ ❢❡✐t♦s ❝♦♠ ❛✉①í❧✐♦ ❞❡ ♣r♦❣r❛♠❛çã♦✳ ◆ã♦ s❡ tr❛t❛
❞❡ r♦t✐♥❛s ♣❛r❛ ✉s♦ ❝♦♠❡r❝✐❛❧✱ ❝♦♠♣✐❧❛❞❛s✱ ♠❛s s✐♠ ❝♦♥t❛s r❡❛❧✐③❛❞❛s ❞❡ ❢♦r♠❛ ♦r❞❡♥❛❞❛✳ ❆❧❣✉♥s
♣r♦❣r❛♠❛s q✉❡ ♣❡r♠✐t❡♠ ❡st❛ ♣r❛t✐❝✐❞❛❞❡ sã♦ ♦ ▼❛t❧❛❜✱ ❙❝✐❧❛❜✱ ❖❝t❛✈❡ ❡ ▼❛t❤❡♠❛t✐❝❛✳ ❈❛❞❛ ✉♠
t❡♠ s✉❛s ✈❛♥t❛❣❡♥s✱ ❡ ó❜✈✐♦ s❡✉ ❝✉st♦✱ s❡♥❞♦ ♦ ❙❝❧✐❛❜ ❡ ♦ ❖❝t❛✈❡ ❞❡ ❧✐✈r❡ ❞✐str✐❜✉✐çã♦ ❡ ♣❡r❢❡✐t❛♠❡♥t❡
❝❛♣❛③❡s ❞❡ s❡ r❡❛❧✐③❛r ♦s ❡st✉❞♦s✱ ✐♥❝❧✉s✐✈❡ ♠✉✐t♦ ♠❛✐s ❛✈❛♥ç❛❞♦s q✉❡ s❡ ♣r♦♣õ❡ ❛q✉✐✳
✷ ❯♠❛ ✐❞❡✐❛ s♦❜r❡ ❛s ❣r❛♥❞❡③❛s ❡♥✈♦❧✈✐❞❛s
❙♦♠❡♥t❡ ✈❡♥❞♦ ❡st❛ ❛♣♦st✐❧❛✱ ♦✉ ❛té ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ♥ã♦ t❡♠♦s ♥♦çã♦ ❞❛ ❣r❛♥❞❡③❛ q✉❡ é ✉♠❛
❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦✳ ◗✉❛❧ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ ✺✵✵ ❦❱❄ ◗✉❛❧ é ❛ ❝♦rr❡♥t❡ tí♣✐❝❛ ❞❡
❝✉rt♦✲❝✐r❝✉✐t♦❄ ◗✉❛♥t♦ ♣❡s❛ ✉♠ ❝❛❜♦❄ ❆ t❛❜❡❧❛ ✶ ❞á ✉♠ ✐❞❡✐❛ ❞❡st❡s ✈❛❧♦r❡s✱ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❡
❞✐✈❡rs❛s ❢♦♥t❡s✳ ❙ã♦ ✈❛❧♦r❡s ♠é❞✐♦s✱ s♦♠❡♥t❡ ♣❛r❛ ✉♠❛ ♦r❞❡♠ ❞❡ ❣r❛♥❞❡③❛✳
❚❛❜❡❧❛ ✶✿ ❖r❞❡♠ ❞❡ ❣r❛♥❞❡③❛ ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✳
P♦tê♥❝✐❛ tr❛♥s♠✐t✐❞❛ ✷✸✵ ❦❱✿ ✷✵✵ ▼❲
✸✹✺ ❦❱✿ ✺✵✵ ▼❲
✺✵✵ ❦❱✿ ✶ ●❲
✼✺✵ ❦❱✿ ✷ ●❲
❈♦♠♣r✐♠❡♥t♦s ❱ã♦ tí♣✐❝♦✿ ✸✵✵✲✺✵✵ ♠
❱ã♦ ❞❡ tr❛✈❡ss✐❛✿ ✶✵✵✵✲✷✵✵✵ ♠
▲❚ ✏❝✉rt❛✑✿ ❁ ✶✵✵ ❦♠
❈♦♠♣r✐♠❡♥t♦ ♠á①✐♠♦ s❡♠ s✉❜❡st❛çã♦ ✐♥t❡r♠❡❞✐ár✐❛✿ ✸✵✵ ❦♠
▲✐♥❤❛ ❞❡ ♠❡✐❛✲♦♥❞❛✿ ✷✷✺✵ ❦♠
❆❧t✉r❛ ❞❡ t♦rr❡ ▲✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦✿ ✸✵✲✺✵ ♠
❱ã♦ ❞❡ tr❛✈❡ss✐❛✿ ✶✵✵✲✸✵✵ ♠
❚❡♠♣❡r❛t✉r❛ ♥♦ ❝❛❜♦ ▲✐♠✐t❡ ♥♦♠✐♥❛❧✿ ✼✵✲✾✵❽
▲✐♠✐t❡ ❞❡ ❡♠❡r❣ê♥❝✐❛✿ ✶✵✵✲✶✸✵❽
▲✐♠✐t❡ ♣❛r❛ ❝❛❜♦s ❡s♣❡❝✐❛✐s✿ ✷✵✵❽
❉✐stâ♥❝✐❛s ❞❡
✐s♦❧❛♠❡♥t♦ ✭❡✜❝❛③✱
❢❛s❡✲♥❡✉tr♦✮
✺✵✵ ❦❱✿ ✷ ♠
✺✵✵ ❦❱ ✭❝♦♠ ❝♦♥s✐❞❡r❛çõ❡s ✉s✉❛✐s ❞❡ ♣r♦❥❡t♦✮✿ ✺✲✽ ♠
P❡s♦ ❧✐♥❡❛r ❞❡ ❝❛❜♦s ❆❈❙❘ ▲✐♥♥❡t ✭✸✸✻ ▼❈▼✮✿ ✻✽✽ ❦❣✴❦♠
❆❈❙❘ ❘❛✐❧ ✭✾✺✹ ▼❈▼✮✿ ✶✻✵✵ ❦❣✴❦♠
❆❈❙❘ ❚❤r❛s❤❡r ✭✷✸✶✷ ▼❈▼✮✿ ✸✼✻✵ ❦❣✴❦♠
❈❛♠♣♦ ❡❧étr✐❝♦ ▼á①✐♠♦ ♥♦ s♦❧♦ ✭❧✐♠✐t❡ ❞❛ ❢❛✐①❛✮✿ ✹✱✷ ❦❱✴♠
▼á①✐♠♦ ♥♦ s♦❧♦ ✭♦❝✉♣❛❝✐♦♥❛❧✮✿ ✽✱✸✸ ❦❱✴♠
❙✉♣❡r❢í❝✐❡ ❞♦ ❝❛❜♦✿ ✷✵ ❦❱✴❝♠
❉✐sr✉♣t✐✈❛✿ ✸✵ ❦❱✴❝♠
✶❝♠✐❧✿ ❝✐r❝✉❧❛r ♠✐❧✱ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦ ❝♦♠ ❞✐â♠❡tr♦ ❞❡ ✉♠ ♠✐❧és✐♠♦ ❞❡ ♣♦❧❡❣❛❞❛✱ s❡♥❞♦ ▼❈▼ ✐❣✉❛❧ ❛ ✶✵✵✵
❝✐r❝✉❧❛r ♠✐❧✳ ✶ ▼❈▼ ∼= 0, 5 mm2✳
✷
✸ ❊st✉❞♦s ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦
❯♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ é ✉♠ ❡❧❡♠❡♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ♣♦tê♥❝✐❛✱ ❧✐❣❛♥❞♦ ❢♦♥t❡s
❞❡ ❣❡r❛çã♦ ❝♦♠ ❝❛r❣❛s ❝♦♥s✉♠✐❞♦r❛s✳
❖ ♣r♦❥❡t♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ✐♥✐❝✐❛✲s❡ ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ tr❛♥s♣♦rt❛r ✉♠❛ q✉❛♥✲
t✐❞❛❞❡ ❞❡ ❡♥❡r❣✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳ ❆♣ós ❡st✉❞❛r ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❝❛r❣❛ ♥❛s ❧✐♥❤❛s ❡①✐st❡♥t❡s✱
♦❜s❡r✈❛✲s❡ ♦ ❡❢❡✐t♦ ❞❡ ✉♠❛ ♥♦✈❛ ❧✐♥❤❛ ♥♦ s✐st❡♠❛✱ ❝❤❡❣❛♥❞♦ ❛ ✉♠ ♥♦✈♦ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳
✸✳✶ ❚r❛♥s♠✐t✐r❄
P♦❞❡✲s❡ ❛❜r✐r ❡st❛ q✉❡stã♦ ❡♠ ❛❧❣✉♥s ♣r♦♥♦♠❡s✿ ♦ q✉ê✱ q✉❛♥❞♦✱ ❝♦♠♦✱ ♦♥❞❡ ❡ q✉❛♥t♦✳
❖ q✉ê tr❛♥s♠✐t✐r ❛ ✐♥t❡r❧✐❣❛çã♦ ❡♥tr❡ ❝❡♥tr♦s ❞❡ ❣❡r❛çã♦ ❡ ❝♦♥s✉♠♦✱ q✉❛♥❞♦ ✐♥✈❡✈✐t❛✈❡❧♠❡♥t❡ ❛
❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛ é ✐♥t❡r❡ss❛♥t❡✱ ♠❡s♠♦ ❝♦♠ ♦ ❝✉st♦ ❞❛ ❧✐♥❤❛✳
◗✉❛♥❞♦ tr❛♥s♠✐t✐r ❛ ♥❡❝❡ss✐❞❛❞❡ ❢✉t✉r❛ s✉r❣✐r✱ ♦✉ s❡❥❛✱ ♣r♦❥❡t❛♥❞♦ ♦ ❝r❡s❝✐♠❡♥t♦ ❞♦ ❝♦♥s✉♠♦
❡ ✐♥❝❧✉✐♥❞♦ ♦ t❡♠♣♦ ❞❡ ❝♦♥str✉çã♦✱ t❛♥t♦ ❞❛s ✉s✐♥❛s q✉❛♥t♦ ❞❛ ♣ró♣r✐❛ ❧✐♥❤❛✳
❈♦♠♦ tr❛♥s♠✐t✐r ❛ t❡❝♥♦❧♦❣✐❛ ❛ s❡r ✉s❛❞❛✱ ❞❡✜♥✐♥❞♦ s❡ ❛ ❧✐♥❤❛ s❡rá ❡♠ ❈❆ ♦✉ ❈❈✱ ❡ ♦s ♥í✈❡✐s
❞❡ t❡♥sã♦✳
❖♥❞❡ ♣❛ss❛ ❡✈❡♥t✉❛❧♠❡♥t❡ ❡①✐st❡ ❛ ♦♣çã♦ ❞❡ q✉❛✐s ❝❡♥tr♦s ❞❡ ❣❡r❛çã♦ ✐rã♦ ✐♥t❡r❧✐❣❛r q✉❛✐s ❝❡♥tr♦s
❞❡ ❝❛r❣❛ ✭❡①✳ ❇❡❧♦ ▼♦♥t❡ ❧✐❣❛ ❝♦♠ ❙✉❞❡st❡ ♦✉ ◆♦r❞❡st❡✮ ❡ ❞❡✜♥✐çã♦ ❞♦ tr❛ç❛❞♦ ❞❛ ❧✐♥❤❛✳
◗✉❛♥t♦ ❝✉st❛ tr❛♥s♠✐t✐r ♦ ❝✉st♦ ❡stá ❡♥✈♦❧✈✐❞♦ ❞❡s❞❡ ❛ ♣r✐♠❡✐r❛ q✉❡stã♦✱ ❞❡♣❡♥❞❡♥❞♦ ❛✐♥❞❛
❞❛ ❡❝♦♥♦♠✐❛ ❡ ❞❛ ♣♦❧ít✐❝❛ ❞❡ ❝♦♠❡r❝✐❛❧✐③❛çã♦ ✭❡①✳ ❣❛♥❤♦ ❡♠ ❡s❝❛❧❛ ♥❛ ❢❛❜r✐❝❛çã♦ ❞♦s ❝❛❜♦s✱
♦✉ r❡❣r❛s t❛r✐❢ár✐❛s✮✳
❊st✐♠❛✲s❡ q✉❡ ❡st❛ ❡♥❡r❣✐❛ ♦❜t✐❞❛ s❡❥❛ ❞✐str✐❜✉í❞❛✱ ❛♦ ❧♦♥❣♦ ❞❛ ✈✐❞❛ út✐❧ ❞❛ ❧✐♥❤❛✱ ❡♠ ✉♠ ♣❡r✜❧
❞❡ ❞❡♠❛♥❞❛✱ r❡s✉❧t❛♥❞♦ ♥❛ ❧✐♥❤❛ tr❛♥s♠✐t✐♥❞♦ ✉♠❛ ♣♦tê♥❝✐❛ ♠é❞✐❛✱ ❝♦♠ ❡✈❡♥t✉❛✐s ♥❡❝❡ss✐❞❛❞❡s
❞❡ s♦❜r❡❝❛r❣❛✳ P❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❞✐❞át✐❝♦✱ ♣♦❞❡♠♦s ❛ss✉♠✐r ✉♠❛ ♣♦tê♥❝✐❛ ❝♦♥st❛♥t❡✳
❆ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s ❞♦✐s ♣♦♥t♦s ❡stá s✉❥❡✐t❛ ❛♦ tr❛ç❛❞♦ ❞❛ ❧✐♥❤❛✱ ❛♦♥❞❡ ♦❜s❡r✈❛✲s❡ ❞❡s❞❡ ❛
t♦♣♦❣r❛✜❛ ❛té ❛ ✈✐❛❜✐❧✐❞❛❞❡ ❞❡ ❛q✉✐s✐çã♦ ❞♦s t❡rr❡♥♦s✳ ❆ ❞✐stâ♥❝✐❛ r❡❛❧ ♣♦❞❡ ✈❛r✐❛r ♥ã♦ ♠❛✐s ❞♦
q✉❡ ✶✵✪ ❞❡ ✉♠ tr❛ç❛❞♦ ❡♠ ❧✐♥❤❛ r❡t❛✳
❆ss✉♠✐♥❞♦ ❛ss✐♠ ❛ ♣♦tê♥❝✐❛ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❧✐♥❤❛✱ ❝❤❡❣❛✲s❡ ❛♦s ❝r✐tér✐♦s ❞❡ ❡s❝♦❧❤❛ ❞♦ t✐♣♦
✭❈❆ ♦✉ ❈❈✮ ❡ ♥í✈❡❧ ❞❡ t❡♥sã♦✳
✹ ❈á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❡❧étr✐❝♦s ✲ ♠♦❞❡❧❛❣❡♠ ❜ás✐❝❛
◆❡st❛ ♣❛rt❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ♦ ♠♦❞❡❧♦ ❜ás✐❝♦ ❞❡ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ♣❛r❛ ❡st✉❞♦ ❡♠ r❡❣✐♠❡
♣❡r♠❛♥❡♥t❡✳ ❆ss✉♠❡✲s❡ q✉❡ ❛ ❧✐♥❤❛ é tr✐❢ás✐❝❛✱ ❢❛③❡♥❞♦✲s❡ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♠♦♥♦❢ás✐❝❛✱ q✉❡ ❞❡
❛❝♦r❞♦ ❝♦♠ ♦ s✐st❡♠❛ ❞❡ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s é ❛♣❧✐❝á✈❡❧ ♣❛r❛ s✐st❡♠❛s ❡q✉✐❧✐❜r❛❞♦s ♦✉ ♥ã♦✳
■♥✐❝✐❛❧♠❡♥t❡ ❞❡♠♦♥str❛✲s❡ ❛ r❡❧❛çã♦ ❞❡ ♣❛râ♠❡tr♦s ❡♥tr❡ ❢❛s❡s✱ ❛♦♥❞❡ ❡①✐st❡♠ ❝♦♠♣♦♥❡♥t❡s
♣ró♣r✐❛s ✭q✉❡ ❛❢❡t❛♠ s♦♠❡♥t❡ ❛ ❢❛s❡ ❡♠ q✉❡stã♦✮ ❡ ❝♦♠♣♦♥❡♥t❡s ♠út✉❛s ✭q✉❡ ❛❢❡t❛♠ ❛s ❢❛s❡s
✈✐③✐♥❤❛s✮✳ P♦r r❡❝✐♣r♦❝✐❞❛❞❡✱ ❛s ❝♦♠♣♦♥❡♥t❡s ♠út✉❛s sã♦ s✐♠étr✐❝❛s✱ ♦✉ s❡❥❛✱ ♦ ❡❢❡✐t♦ q✉❡ ❛ ❢❛s❡ a
❝❛✉s❛ ♥❛ ❢❛s❡ b é ✐❣✉❛❧ ❛♦ ❡❢❡✐t♦ ❞❛ ❢❛s❡ b ♥❛ ❢❛s❡ a✳
❙❛❜❡✲s❡ ♣❡❧❛ t❡♦r✐❛ ❞❡ ❝✐r❝✉✐t♦s q✉❡ ✐♠♣❡❞â♥❝✐❛ ❡ ❛❞♠✐tâ♥❝✐❛ sã♦ ❣r❛♥❞❡③❛s r❡❝í♣r♦❝❛s✳ P♦r
❝♦♥✈❡♥çã♦ ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✱ ♥♦♠❡✐❛✲s❡ ❝♦♠♦ ✐♠♣❡❞â♥❝✐❛ ❛ ❝♦♠♣♦♥❡♥t❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♣♦r
✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ s❡♥❞♦ ❡♠ ❣❡r❛❧ ✉♠ ❡❧❡♠❡♥t♦ ❘▲ ❡♠ sér✐❡✷✳
◆♦♠❡✐❛✲s❡ ❝♦♠♦ ❛❞♠✐tâ♥❝✐❛ ❛ ❝♦♠♣♦♥❡♥t❡ tr❛♥s✈❡rs❛❧ ✭♣❛r❛❧❡❧❛ ♦✉ s❤✉♥t✮ ♣♦r ✉♥✐❞❛❞❡ ❞❡
❝♦♠♣r✐♠❡♥t♦✱ s❡♥❞♦ ❡♠ ❣❡r❛❧ ✉♠ ❡❧❡♠❡♥t♦ ❘❈ ❡♠ ♣❛r❛❧❡❧♦✱ s❡♥❞♦ ❛ r❡s✐stê♥❝✐❛ ❘✱ r❡♣r❡s❡♥t❛t✐✈❛
❞❛ ❝♦rr❡♥t❡ ❞❡ ❢✉❣❛✱ ❞❡s♣r❡③í✈❡❧✸✳ ❉❡st❛ ❢♦r♠❛ ♣♦❞❡✲s❡ ❡st✐♠❛r ❛ ✐♠♣❡❞â♥❝✐❛ ❡ ❛❞♠✐tâ♥❝✐❛ t♦t❛❧
❞❡ ✉♠❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❞✐r❡t❛♠❡♥t❡ s❡✉s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦✳
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ✐♠♣❡❞â♥❝✐❛✱ ❛ ❛❞♠✐tâ♥❝✐❛ é ❞❡✜♥✐❞❛ ♣❡❧♦ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦ Y = G+jB✱
s❡♥❞♦ G ❛ ❝♦♥❞✉tâ♥❝✐❛ ❡ B ❛ s✉s❝❡♣tâ♥❝✐❛✳
◆❛ s❡çã♦ ❈✳✶ ❛♣r❡s❡♥t❛✲s❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦❞❛s ❡q✉❛çõ❡s ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✱ t❛♠❜é♠
❝❤❛♠❛❞❛s ❞❡ ❡q✉❛çõ❡s ❞♦ t❡❧❡❣r❛✜st❛✳
◆❛ ♣rát✐❝❛ ❛♣r♦①✐♠❛✲s❡ ♦ ❝✐r❝✉✐t♦ ❧❛❞❞❡r ♣❛r❛ ❡❧❡♠❡♥t♦s ❞✐s❝r❡t♦s✱ s❡♥❞♦ ♦ ♠❛✐s s✐♠♣❧❡s ♦
❡q✉✐✈❛❧❡♥t❡ ✏♣✐✑ ✭✉♠❛ ✐♠♣❡❞â♥❝✐❛ ❡♠ sér✐❡ ❡ ❞✉❛s ❛❞♠✐tâ♥❝✐❛s ❡♠ ♣❛r❛❧❡❧♦ ♥❛s ❡①tr❡♠✐❞❛❞❡s✮✳
✷❊♠ ❧✐♥❤❛s ❈❈✱ ❛ ✐♥❞✉tâ♥❝✐❛ ♥ã♦ s❡ ❛♣❧✐❝❛ ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✱ ♠❛s ❡♠ ❡st✉❞♦s tr❛♥s✐tór✐♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦
♥❛ ♣r♦♣❛❣❛çã♦ ❞❡ s✉rt♦s✱ ❡❧❡ é ❞❡t❡r♠✐♥❛♥t❡✳
✸❊♠ ❧✐♥❤❛s ❈❈✱ ♣❡❧❛ ❢❛❧t❛ ❞❛ ❝♦rr❡♥t❡ ♣❡❧♦ ❡❢❡✐t♦ ❝❛♣❛❝✐t✐✈♦✱ ❛ r❡s✐stê♥❝✐❛ s❤✉♥t ❘ t♦r♥❛✲s❡ ♥♦✈❛♠❡♥t❡ r❡❧❡✈❛♥t❡✱
♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❝á❧❝✉❧♦ ❞❡ ❝♦♦r❞❡♥❛çã♦ ❞❡ ✐s♦❧❛♠❡♥t♦✳
✸
❋✐❣✉r❛ ✶✿ ❊①❡♠♣❧♦ ✐❧✉str❛t✐✈♦ ❞❡ s❡❧❡çã♦ ❞❡ ♥í✈❡❧ ❞❡ t❡♥sã♦✱ ❛ ♣❛rt✐r ❞❡ ♣r❡♠✐ss❛s ❞❡ ♣r♦❥❡t♦
❝♦♥s❡r✈❛❞♦r❛s ❬✶✹❪
✹✳✶ ❘❡s✐stê♥❝✐❛
❆ r❡s✐stê♥❝✐❛✱ ❝♦♠♦ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ é ❡♥s✐♥❛❞❛✱ é ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ r❡s✐st✐✈✐❞❛❞❡ ❞♦ ♠❛t❡r✐❛❧✱ ❛
s❡çã♦ tr❛♥s✈❡rs❛❧ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦✿
R = ρ
l
S
✭✹✳✶✮
s❡♥❞♦ ρ ❛ r❡s✐st✐✈✐❞❛❞❡ ❡ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ♦ s❡✉ ✐♥✈❡rs♦✿ σ = 1/ρ✱ l ♦ ❝♦♠♣r✐♠❡♥t♦ ❡ S ❛ s❡çã♦
tr❛♥s✈❡rs❛❧✳
❊♠ ❝♦rr❡♥t❡ ❛❧t❡r♥❛❞❛✱ ♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r ❞✐st♦r❝❡ ❛ r❡s✐stê♥❝✐❛ ❡❢❡t✐✈❛ ❞♦ ❝❛❜♦✿ ♦ ❡❢❡✐t♦ ❞❡
r❡♣✉❧sã♦ ❞❛s ❧✐♥❤❛s ❞❡ ❝♦rr❡♥t❡ ♣r♦✈♦❝❛ ✉♠ s✉❜❛♣r♦✈❡✐t❛♠❡♥t♦ ❞❛ s❡çã♦ tr❛♥s✈❡rs❛❧ ❞♦ ❝❛❜♦✳ ❊st❡
❡❢❡✐t♦ é ♠❛✐s ❡✈✐❞❡♥t❡ ❡♠ ❜✐t♦❧❛s ♠❛✐♦r❡s✱ ♣♦✐s ❡❧❡ ♥ã♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ ❞✐â♠❡tr♦✱ ❧♦❣♦ s❡♥❞♦
♣♦✉❝♦ ♣❡r❝❡❜✐❞♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ✐♥st❛❧❛çõ❡s r❡s✐❞❡♥❝✐❛✐s✳
❖s ❝❛❜♦s ✉s✉❛✐s ❡♠ ❈❆ sã♦ ❝♦♠♣♦st♦s ♣♦r ❞♦✐s ♠❛t❡r✐❛✐s✱ ❣❡r❛❧♠❡♥t❡ ✉♠ ♥ú❝❧❡♦ ❝♦♠ ✜♦s ♠❛✐s
r❡s✐st❡♥t❡ à tr❛çã♦ ❡ ✉♠❛ ❝♦r♦❛ ❝♦♠ ✜♦s ❞❡ ❜♦❛ ❝♦♥❞✉t✐✈✐❞❛❞❡✱ ❡ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❧❡✈❡ ❡ ❡❝♦♥ô♠✐❝♦✳
❊st❡ ❝♦♥❥✉♥t♦ ❛✉♠❡♥t❛ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ❡st✉❞♦✱ ♣♦r ❡①❡♠♣❧♦ ♥♦ ❝á❧❝✉❧♦ ♠❡❝â♥✐❝♦✱ ♠❛s ♥♦ ❝á❧❝✉❧♦
❞❛ r❡s✐stê♥❝✐❛ ♣♦ss✉✐rá ❜❛✐①❛ ✐♥✢✉ê♥❝✐❛✱ ♣♦✐s ♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r ✐rá ♣♦s✐❝✐♦♥❛r ❛ ❝♦rr❡♥t❡ ♥❛ r❡❣✐ã♦
❞❛ ❝♦r♦❛✱ ❡✈✐t❛♥❞♦ ♦ ♠❛t❡r✐❛❧ ❞♦ ♥ú❝❧❡♦✳
❖✉tr♦ ❡❢❡✐t♦ ✐♠♣♦rt❛♥t❡ é ❛ ✈❛r✐❛çã♦ ❞❛ r❡s✐stê♥❝✐❛ ♣❡❧❛ t❡♠♣❡r❛t✉r❛✳ ❊♠ ❣❡r❛❧ ❛ r❡s✐stê♥❝✐❛
❡♠ ❝❛tá❧♦❣♦s é t❛❜❡❧❛❞❛ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s tí♣✐❝♦s✱ ❝♦♠♦ ✼✺❽✱ ♠❛s ♦ ✈❛❧♦r ❡①❛t♦ ❞❡♣❡♥❞❡ ❞❛
♣ró♣r✐❛ ❝♦rr❡♥t❡✱ ❡♥tr❡ ♦✉tr♦s ❢❛t♦r❡s ❛♠❜✐❡♥t❛✐s✳
◆❛ s❡çã♦ ✶✵✳✶ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ♣ró♣r✐❛✱ ✐♥❝❧✉✐♥❞♦ ♦
❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r✳ ❖ ✈❛❧♦r ❝❛❧❝✉❧❛❞♦ s❡rá ♣ró①✐♠♦ ❛♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ❡♠ ❝❛tá❧♦❣♦s✹✳
❖❜s❡r✈❛✲s❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❝❛❜♦s é ❝♦♠♣♦st❛ ♣♦r ✜♦s ❡♥tr❡❧❛ç❛❞♦s✱ ❤❛✈❡♥❞♦ ❡♥tã♦ ❧❛❝✉♥❛s ♥♦
✐♥t❡r✐♦r ❞♦ ❝❛❜♦✳ ❖✉tr❛ ❝❛r❛❝t❡ríst✐❝❛ ❝♦♠✉♠ é ❛ ♣r❡s❡♥ç❛ ❞❡ ❞♦✐s ♠❛t❡r✐❛✐s ♥♦ ♠❡s♠♦ ❝❛❜♦✱ ❝♦♠♦
❛❧✉♠í♥✐♦ ❡ ❛ç♦✳ ❊st❛s ❡ ♦✉tr❛s ❝❛r❛❝t❡ríst✐❝❛s ❛❝r❡s❝❡♥t❛♠ ✉♠❛ ❝♦♠♣❧❡①✐❞❛❞❡ ♥♦ ❝á❧❝✉❧♦ ❡①❛t♦ ❞❛
r❡s✐stê♥❝✐❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❛♦ s❡ ❝♦♥s✐❞❡r❛r ♦s ❡❢❡✐t♦s ❞❛ t❡♠♣❡r❛t✉r❛✳
❊♠ ❣❡r❛❧ ❛s r❡s✐stê♥❝✐❛s sã♦ t❛❜❡❧❛❞❛s✱ ✐♥❝❧✉✐♥❞♦ ♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r ✭✏r❡s✐stê♥❝✐❛ ❈❆✑✮✳ ❚❛♠❜é♠
é ✉s✉❛❧ t❛❜❡❧❛r ❛ r❡s✐stê♥❝✐❛ ♣❛r❛ ❛❧❣✉♠❛s ❢❛✐①❛s ❞❡ t❡♠♣❡r❛t✉r❛✳
✹❊①✐st❡♠ ❛✐♥❞❛ ♦✉tr♦s ❢❛t♦r❡s q✉❡ ✐♥✢✉❡♥❝✐❛♠ ♥♦ ❝á❧❝✉❧♦ ❞❛ r❡s✐stê♥❝✐❛✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦ ♦ ❡❢❡✐t♦ ✏tr❛♥s❢♦r♠❛✲
❞♦r✑ ❞♦ ♥ú❝❧❡♦ ❞❡ ❛ç♦ ❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❛❞✐❝✐♦♥❛❧ ❞❡✈✐❞♦ à ❤❡❧✐❝♦✐❞❛❧ ❞♦s ✜♦s✳
✹
P❛r❛ ✉♠ ❝á❧❝✉❧♦ ✐t❡r❛t✐✈♦✱ é ♣r✉❞❡♥t❡ ✐♥✐❝✐❛r ♦ ❝á❧❝✉❧♦ ❞❛ r❡s✐stê♥❝✐❛ ❝♦♠ ✉♠ ✈❛❧♦r ❞❡ t❡♠♣❡r❛✲
t✉r❛ ♣ró①✐♠♦ ❞♦ ♥♦♠✐♥❛❧✱ ❡ ❛♣ós ❞❡t❡r♠✐♥❛r ❛ t❡♠♣❡r❛t✉r❛ r❡❛❧ ❞♦ ❝♦♥❞✉t♦r✱ r❡❛❧✐③❛r ❛ ❝♦rr❡çã♦✳
P❛r❛ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ ❢❡✐①❡ ❞❡ ❝♦♥❞✉t♦r❡s✱ ❛ r❡s✐stê♥❝✐❛ s❡rá ❞✐✈✐❞✐❞❛ ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❝❛❜♦s
❡♠ ❝❛❞❛ ❢❛s❡✳
❆ t❛❜❡❧❛ ✷ ❡①❡♠♣❧✐✜❝❛ ❛ r❡s✐st✐✈✐❞❛❞❡ ❞♦s ♠❛t❡r✐❛✐s ✉s❛❞♦s ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✱ ❜❡♠ ❝♦♠♦
♦✉tr♦s ♣❛râ♠❡tr♦s r❡❧❡✈❛♥t❡s ♣❛r❛ ♦ ♣r♦❥❡t♦✳
❚❛❜❡❧❛ ✷✿ ❈❛r❛❝t❡ríst✐❝❛s ❢ís✐❝❛s ❞❡ ❛❧❣✉♥s ♠❛t❡r✐❛✐s✳
❈♦♥❞✉t✐✈✐❞❛❞❡
■❆❈❙ ✭✪✮
❘❡s✐st✐✈✐❞❛❞❡
✭❲·mm2/m✮
❈♦❡✜❝✐❡♥t❡ ❞❡ ✈❛r✐❛çã♦ ❞❛
r❡s✐stê♥❝✐❛ ✭❽−1✮
▼❛ss❛
❡s♣❡❝í✜❝❛
✭g/cm3✮
❆❧✉♠í♥✐♦ ✶✸✺✵ ✻✶✱✵ ✵✱✵✷✽✷✻✹ ✵✱✵✵✹✵✸ ✷✱✼✵✺
❆❧✉♠í♥✐♦ ❧✐❣❛ ✻✷✵✶ ✺✷✱✺ ✵✱✵✸✷✽✹✵ ✵✱✵✵✸✹✼ ✷✱✻✾✵
❈♦❜r❡ ❞✉r♦ ❝♦♠❡r❝✐❛❧ ✾✼✱✵ ✵✱✵✶✼✼✼✺ ✵✱✵✵✸✽✶ ✽✱✽✾
❈♦❜r❡ ♣❛❞rã♦ ■❆❈❙ ✶✵✵✱✵ ✵✱✵✶✼✷✹✶ ✵✱✵✵✸✾✸ ✽✱✽✾
❆ç♦ ✲ ✵✱✶✼ ✲ ✼✱✾
❖❜s❡r✈❛✲s❡ q✉❡ ❛♣❡s❛r ❞♦ ❝♦❜r❡ ♣♦ss✉✐r ✉♠❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ♠❛✐s ❢❛✈♦rá✈❡❧✱ s✉❛ ♠❛ss❛ ❡ ♣r❡ç♦
✭❞❛ ♦r❞❡♠ ❞❡ 4× ♠❛✐s ❝❛r♦✮ ✐♥✈✐❛❜✐❧✐③❛♠ ❛ ❛♣❧✐❝❛çã♦ ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✳
✹✳✶✳✶ ❱❛r✐❛çã♦ ❝♦♠ ❛ t❡♠♣❡r❛t✉r❛
P❛r❛ ♦ ✉s♦ ♣r❡❝✐s♦ ❞❛ r❡s✐stê♥❝✐❛✱ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ♥♦ ❝á❧❝✉❧♦ ❞❛s ♣❡r❞❛s✱ ❞❡✈❡✲s❡ r❡❛❧✐③❛r ❛ ❝♦rr❡çã♦
♣❡❧❛ t❡♠♣❡r❛t✉r❛✳ ❊st❡ ❝á❧❝✉❧♦ ♣♦❞❡ s❡ t♦r♥❛r ❝♦♠♣❧✐❝❛❞♦✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛ r❡s✐stê♥❝✐❛ ✐rá
✐♥✢✉❡♥❝✐❛r ❛ ❝♦rr❡♥t❡✱ q✉❡ ♣♦r s✉❛ ✈❡③ ✐rá ❞✐t❛r ❛ t❡♠♣❡r❛t✉r❛ ❞♦ ❝♦♥❞✉t♦r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠
♦✉tr♦s ❢❛t♦r❡s✱ ❛❧❡♠ ❞♦s ❝❛❜♦s ❣❡r❛❧♠❡♥t❡ s❡r❡♠ ❝♦♠♣♦st♦s ♣♦r ❞♦✐s ♠❛t❡r✐❛✐s✳
❊♠ ❣❡r❛❧ ♦s ❢❛❜r✐❝❛♥t❡s ❢♦r♥❡❝❡♠ ♦s ✈❛❧♦r❡s ❞❡ r❡s✐stê♥❝✐❛ ✭❈❆ ♦✉ ❈❈✮ ♣❛r❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡
t❡♠♣❡r❛t✉r❛✳ ❆t❡♥t❡ ❡♠ ✉t✐❧✐③❛r ✉♠❛ r❡s✐stê♥❝✐❛ ♣❛r❛ ✉♠❛ t❡♠♣❡r❛t✉r❛ ♣ró①✐♠❛ às ❝♦♥❞✐çõ❡s ❞❡
♦♣❡r❛çã♦✳
❆ t❛❜❡❧❛ ✸ ✐❧✉str❛ ❛❧❣✉♥s ✈❛❧♦r❡s ❞❡ r❡s✐stê♥❝✐❛ ❈❆ ❡ ❈❈ ♣❛r❛ ❛❧❣✉♥s ❝❛❜♦s✳
❚❛❜❡❧❛ ✸✿ ❊①❡♠♣❧♦s ❞❡ ❛❧❣✉♥s ❝❛❜♦s ❝♦♠❡r❝✐❛✐s
❚✐♣♦ ❉❡♥♦♠✐♥❛çã♦ ❇✐t♦❧❛
✭▼❈▼✮
❙❡çã♦ tr❛♥s✈❡rs❛❧
t♦t❛❧ ✭♠♠➨✮
❉✐â♠❡tr♦
✭♠♠✮
❘❡s✐stê♥❝✐❛ ❈❈
✭❲/km20❽✮
❘❡s✐stê♥❝✐❛ ❈❆
✭❲/km75❽✮
❆❈❙❘ ❍❛✇❦ ✹✼✼ ✷✽✵✱✽✺ ✷✶✱✼✽ ✵✱✶✶✾✻ ✵✱✶✹✸✺
❆❈❙❘ ●r♦s❜❡❛❦ ✻✸✻ ✸✷✷✱✸ ✷✺✱✶✻ ✵✱✵✽✾✻ ✵✱✶✵✼✺
❆❈❙❘ ❘❛✐❧ ✾✺✹ ✺✷✻✱✽ ✷✾✱✺✾ ✵✱✵✺✾✼ ✵✱✵✼✸✸
❆❈❙❘ ❇✐tt❡r♥ ✶✷✼✷ ✼✷✻✱✹ ✸✹✱✶✻ ✵✱✵✹✹✽ ✵✱✵✺✺✽
❆❈❙❘ ❚❤r❛s❤❡r ✷✸✶✷ ✶✷✸✺✱✷ ✹✺✱✼✽ ✵✱✵✷✹✽ ✵✱✵✸✷✼
❆❆❈ ❙❛❣❡❜r✉s❝❤ ✷✷✺✵ ✶✶✸✾✱✺ ✹✸✱✾ ✵✱✵✷✺✺ ✵✱✵✸✹
❆❆❆❈ ✶✵✵✵ ✺✵✻✱✼ ✷✾✱✷ ✵✱✵✻✻✶ ✵✱✵✽✵✷
✹✳✷ ■♥❞✉tâ♥❝✐❛
❆ ✐♥❞✉tâ♥❝✐❛ é ♦ ❡❢❡✐t♦ ❞♦ ❝❛♠♣♦ ♠❛❣♥ét✐❝♦ s♦❜r❡ ✉♠ ❝✐r❝✉✐t♦✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r ❡①❡♠♣❧♦ ♣❡❧❛
❧❡✐ ❞❡ ❋❛r❛❞❛②✳ P♦❞❡✲s❡ t❡r ✐♥❞✉tâ♥❝✐❛ ♣ró♣r✐❛✱ q✉❛♥❞♦ ✉♠❛ ❧✐♥❤❛ ❞❡ ❝♦rr❡♥t❡ ♥♦ ❝♦♥❞✉t♦r ✐♥❞✉③
♣♦t❡♥❝✐❛❧ ❡♠ ♦✉tr❛ s❡çã♦ ❞♦ ♣ró♣r✐♦ ❝♦♥❞✉t♦r✱ ♦✉ ✐♥❞✉tâ♥❝✐❛ ♠út✉❛✱ q✉❛♥❞♦ ✉♠❛ ❝♦rr❡♥t❡ ❡♠ ✉♠
❝♦♥❞✉t♦r ❡①t❡r♥♦ ✐♥❞✉③ ❡st❡ ♣♦t❡♥❝✐❛❧✳
❆ss✐♠ ❝♦♠♦ ❛s ❝❛r❣❛s ❡❧étr✐❝❛s✱ t♦❞❛s ❛s ❝♦rr❡♥t❡s q✉❡ ♥ã♦ s❡❥❛♠ ❝♦♥st❛♥t❡s ✐♥❞✉③❡♠ ♣♦t❡♥❝✐❛❧
❡♠ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❝♦♥❞✉t♦r✱ ❡ s❡ ❡ss❡ ❡❧❡♠❡♥t♦ ❢❡❝❤❛r ✉♠ ❝✐r❝✉✐t♦✱ s✉r❣❡ ❛ ❝♦rr❡♥t❡ ✐♥❞✉③✐❞❛✳
▲♦❣♦✱ ✉♠❛ ❧✐♥❤❛ ♣♦❞❡ ✐♥❞✉③✐r ❡♠ ❝❡r❝❛s ♠❡tá❧✐❝❛s✱ ❝❛❜♦s ❛t❡rr❛❞♦s✱ ❡♥❝❛♥❛♠❡♥t♦s✱ ❡t❝✳
❆ ✐♥❞✉çã♦ t❛♠❜é♠ ❞❡♣❡♥❞❡rá s❡ ♦s ❡❧❡♠❡♥t♦s ❡st✐✈❡r❡♠ ♣❛r❛❧❡❧♦s✱ ❡♥tã♦ ❛ ✐♥❞✉çã♦ s❡rá ♠í♥✐♠❛
s❡ ♦s ❝♦♥❞✉t♦r❡s ❡st✐✈❡r❡♠ ♣❡r♣❡♥❞✐❝✉❧❛r❡s✳
✹✳✷✳✶ Pr❡♠✐ss❛s
❯♠❛ ❝♦♥s✐❞❡r❛çã♦✱ ❣❡r❛❧♠❡♥t❡ ♣♦✉❝♦ ❡✈✐❞❡♥❝✐❛❞❛✱ é s♦❜r❡ ❛ ❝♦rr❡♥t❡✿ ♣❛r❛ q✉❡ ❤❛❥❛ ✉♠❛ ❝♦rr❡♥t❡
❡❧étr✐❝❛ ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✱ s✉♣õ❡✲s❡ q✉❡ ❡❧❛ r❡t♦r♥❛ ♣❛r❛ ❛ s✉❛ ❢♦♥t❡ ❞❡ ❡♥❡r❣✐❛ ✭♦✉ ✏❢❡❝❤❛✑ ♦
s♦♠❛tór✐♦✱ ♥♦ ❝❛s♦ ❞❡ ✈ár✐❛s ❢♦♥t❡s✱ s❡❣✉✐♥❞♦ ❛s ▲❡✐s ❞❡ ❑✐r❝❤❤♦✛✮✳ ❊st❡ r❡t♦r♥♦ ♣♦❞❡ s❡r ♣♦r ✉♠
s❡❣✉♥❞♦ ❝♦♥❞✉t♦r ♦✉ ♣❡❧♦ s♦❧♦✱ ❢❡❝❤❛♥❞♦ ✉♠ ❧❛ç♦ ❞❡ ❝♦rr❡♥t❡✳
❖ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ ❧❛ç♦ ❞❡ ❝♦rr❡♥t❡ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❧❡✐ ❞❡ ❆♠♣ér❡✱ q✉❡
♥♦s ❢♦r♥❡❝❡rá ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♥❞✉tâ♥❝✐❛ ❞♦ ❝✐r❝✉✐t♦✳ ❊♥tã♦✱ ♥ã♦ ❢❛③ s❡♥t✐❞♦ ♣❡♥s❛r ❡♠ ✉♠
✺
❝♦♥❞✉t♦r s✐♥❣❡❧♦ ❝♦♠ ✉♠❛ ❝♦rr❡♥t❡✱ ♣♦✐s ❛ ❡q✉❛çã♦ só ❢❡❝❤❛ ❝♦♠ ✉♠❛ ❝♦rr❡♥t❡ r❡t♦r♥❛♥❞♦ ❡♠
s❡♥t✐❞♦ ❝♦♥trár✐♦✳
❖ ❝á❧❝✉❧♦ ❞❛ ✐♥❞✉tâ♥❝✐❛ ❡♠ ✉♠ ❝♦♥❞✉t♦r é ❞✐✈✐❞✐❞♦ ♥❛ s✉❛ ♣❛rt❡ ✐♥t❡r♥❛ ❡ ♥❛ ♣❛rt❡ ❡①t❡r♥❛✳
❊♠ ❛♠❜♦s✱ ♣❛rt❡✲s❡ ❞❛ ❧❡✐ ❞❡ ❆♠♣ér❡✳
P❛r❛ ❛ ✐♥❞✉tâ♥❝✐❛ ✐♥t❡r♥❛✱ ❝♦♠♦ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦ ✉♠ ❝♦♥❞✉t♦r ❝♦♠ ✉♠❛ s❡çã♦ ❝✐r❝✉❧❛r✱
❝♦♠ r❛✐♦ r✱ ❛♦♥❞❡ ❛tr❛✈❡ss❛ ✉♠❛ ❝♦rr❡♥t❡ I ❞✐str✐❜✉í❞❛ ✉♥✐❢♦r♠❡♠❡♥t❡✱ ♦❜té♠✲s❡ ✉♠ ✈❛❧♦r ❝♦♥s✲
t❛♥t❡ ❞❡ 0, 5 · 10−7 ❍✴♠ ❬✷✼❪✳ ❆ ♣❛r❝❡❧❛ ❞❛ ✐♥❞✉tâ♥❝✐❛ ❡①t❡r♥❛ é r❡❧❛❝✐♦♥❛❞❛ ❛♦ r❛✐♦ ❡ ❛ ❛❧t✉r❛✱
✉♥✐♥❞♦ ❛s ♣❛r❝❡❧❛s✿
Lii =
µ0
2π
(
1
4
+ ln
2h
r
)
✭✹✳✷✮
s❡♥❞♦ Lii ❛ ✐♥❞✉tâ♥❝✐❛ ♣ró♣r✐❛ ❞♦ ❝♦♥❞✉t♦r i✱ ❝♦♠ ❛ s♦♠❛ ❞♦ ✢✉①♦ ♠❛❣♥ét✐❝♦ ✐♥t❡r♥♦ ❡ ❡①t❡r♥♦✱ r
♦ r❛✐♦ ❞♦ ❝♦♥❞✉t♦r✱ h ❛ ❛❧t✉r❛ ❡ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛ ❞♦ ❛r µ0 = 4π · 10−7 H/m✳
❖❜s❡r✈❛♥❞♦ q✉❡ ♦ ❝❛❜♦ ♥ã♦ ♣♦ss✉✐ ❛❧t✉r❛ ❝♦♥st❛♥t❡✱ ❝♦♥t❡♥❞♦ ❛ ❢♦r♠❛ ❞❡ ✉♠❛ ❝❛t❡♥ár✐❛✳ P♦❞❡✲
s❡ ✉s❛r s❡♠ ♣r♦❜❧❡♠❛s ✉♠❛ ❛❧t✉r❛ ♠é❞✐❛ hm✱ ❝❛❧❝✉❧❛❞❛ ❞❡ ❞✉❛s ❢♦r♠❛s✿
hm = ht −
2
3
f = hv +
1
3
f ✭✹✳✸✮
s❡♥❞♦ ❛q✉✐ ht ❛ ❛❧t✉r❛ ❞♦ ❝❛❜♦ ♥❛ t♦rr❡✱ hv ❛ ❛❧t✉r❛ ❞♦ ❝❛❜♦ ♥♦ ♠❡✐♦ ❞♦ ✈ã♦ ❡ f ❛ ✢❡❝❤❛✱ s❡♥❞♦
❡ss❛ ❢ór♠✉❧❛ r❡❧❛t✐✈❛ ❛ ✉♠ ✈ã♦ ♥✐✈❡❧❛❞♦ ❬✹❪✳
❯s✉❛❧♠❡♥t❡ ❛ ❡q✉❛çã♦ ✭✹✳✷✮ é ♠❛♥✐♣✉❧❛❞❛ ❞❛ ❢♦r♠❛✿
Lii =
µ0
2π
(
ln e
1
4 + ln
2h
r
)
✭✹✳✹✮
Lii =
µ0
2π
ln
2h
r e
1
4
✭✹✳✺✮
Lii =
µ0
2π
ln
2h
r′
✭✹✳✻✮
❆ ✈❛r✐á✈❡❧ r′ ❝♦rr❡s♣♦♥❞❡ ❛♦ r❛✐♦ ❡q✉✐✈❛❧❡♥t❡ ❞♦ ❝♦♥❞✉t♦r ❛♦ s❡ ❝♦♥s✐❞❡r❛r ❛ ♣❛rt❡ ✐♥t❡r♥❛ ❞♦
✢✉①♦ ❬✷✼✱ ♣✳ ✺✷❪✱ ♣❛r❛ ✉♠ ❝❛❜♦ ❞❡ ❛❧✉♠í♥✐♦✱ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ é ✐❣✉❛❧ ❛♦ ❞♦ ❛r✱ ♥♦ q✉❛❧ µ = µ0✱
r′ = r e−1/4 ∼= 0, 7788r✳ P❛r❛ ❝❛❜♦s ❝♦♠ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ s✉♣❡r✐♦r ❛ µ0✱ ❝♦♠♦ ♦ ❛ç♦✺✱ r′ = r e−
µr
4 ✱
♥♦ q✉❛❧ µr ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ r❡❧❛t✐✈❛ ❞♦ ❝♦♥❞✉t♦r✱ µr = µ/µ0✳
❖ ✢✉①♦ ❡①t❡r♥♦ s❡rá ✐♥✢✉❡♥❝✐❛❞♦ ♣❡❧❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ❞♦ ❛r✱ ✐❣✉❛❧ ❛ µ0✳
✹✳✸ ■♠♣❡❞â♥❝✐❛ ♠út✉❛
❆ ✐♠♣❡❞â♥❝✐❛♠út✉❛ ❡♥tr❡ ❞♦✐s ❝♦♥❞✉t♦r❡s é ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛ ✐♥❞✉tâ♥❝✐❛✱ ❞❡✜♥✐❞❛ ♣❡❧❛s ❞✐stâ♥❝✐❛s
❡ ❛ ❝❛r❛❝t❡ríst✐❝❛ ♠❛❣♥ét✐❝❛ ❞♦ ❛r ✭❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❝♦♥❞✉t♦r ✐♥✢✉❡♥❝✐❛ s♦♠❡♥t❡ ♥❛ ✐♥❞✉tâ♥❝✐❛
✐♥t❡r♥❛✮✿
Lij =
µ0
2π
ln
Dij
dij
✭✹✳✼✮
s❡♥❞♦ Dij ❛ ❞✐stâ♥❝✐❛ ❞♦ ❝♦♥❞✉t♦r i ❛ ✐♠❛❣❡♠ ❞♦ ❝♦♥❞✉t♦r j✱ ❡ dij ❛ ❞✐stâ♥❝✐❛ ❞♦ ❝♦♥❞✉t♦r i ♣❛r❛
♦ ❝♦♥❞✉t♦r j✳
❖ ❡①❡♠♣❧♦ ❞❡ ✉♠ ❝❛❜♦ ❘❛✐❧ ✭∅29, 59 ♠♠✱ ❝♦♠♣♦st♦ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞❡ ❛❧✉♠í♥✐♦✱ µ = µ0✮✱
❛ ✉♠❛ ❛❧t✉r❛ ❞❡ ✷✵ ♠✱ s✉❛ ✐♥❞✉tâ♥❝✐❛ ♣ró♣r✐❛ s❡rá
Laa =
µ0
2π
ln
2h
r′
= 2 · 10−7 ln 2 · 20
0,02959/2 · 0, 7788 = 1, 6305 · 10
−6 H/m
❆ ✐♥❞✉tâ♥❝✐❛ ♠út✉❛ ❡♥tr❡ ❞♦✐s ❝❛❜♦s✱ ❞✐s♣♦st♦s ♥❛ ❤♦r✐③♦♥t❛❧ ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ❞❡ ✽ ♠✱ s❡rá
Lab =
µ0
2π
ln
Dab
dab
= 2 · 10−7 ln
√
402 + 82
8
= 1, 981 · 10−7 H/m
✺◆❛ r❡❢❡rê♥❝✐❛ ❬✶✾❪ ♦❜té♠✲s❡ ♣❛r❛ ❛ç♦ ✉s❛❞♦ ♥♦ ♥ú❝❧❡♦ ❞❡ ❝❛❜♦s ❆❈❙❘ ✈❛❧♦r❡s ❞❛ ♦r❞❡♠ ❞❡ µr ∼= 50✱ s❡♥❞♦
♣❧❛✉sí✈❡❧ ❝♦♥s✐❞❡r❛r ❡ss❛ ✈❛❧♦r ♣❛r❛ ❝❛❜♦s ♣❛r❛✲r❛✐♦s✳
✻
❉♦✐s ❝❛❜♦s ❞❡ ❛❧✉♠í♥✐♦✱ ❝♦♠ ✶ ❝♠ ❞❡ r❛✐♦✱ ✸✵ ♠ ❞❡ ❛❧t✉r❛ ❡ s❡♣❛r❛❞♦s ❛ ✶✵ ♠✱ ♣♦ss✉❡♠
✉♠❛ ✐♠♣❡❞â♥❝✐❛ ♠út✉❛ Żm✳ ❈❛❧❝✉❧❡ ❛ ✈❛r✐❛çã♦ ♣❡r❝❡♥t✉❛❧ ❞❡ Żm ❛♦ ✭❛✮ ❛♣r♦①✐♠❛r ♦s ❝❛❜♦s
♣❛r❛ ✺ ♠✱ ✭❜✮ ❛❜❛✐①❛r ♦s ❝❛❜♦s ♣❛r❛ ✶✵ ♠ ❞❡ ❛❧t✉r❛✳
❆ ✐♠♣❡❞â♥❝✐❛ ♠út✉❛ é ♣r♦♣♦r❝✐♦♥❛❧ às ❞✐stâ♥❝✐❛s✱ r❡❛❧ ❡ ✐♠❛❣❡♠✱ ❡ ♦ r❛✐♦ ♥ã♦ ✐♥✢✉❡♥❝✐❛
♥♦ r❡s✉❧t❛❞♦✿
Zm ∝ ln
Dij
dij
❋❛③❡♥❞♦ ❛ ❝♦♥t❛ s♦♠❡♥t❡ ❝♦♠ ♦ ❧♦❣❛r✐t♠♦✱ ♥❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧✿
Dij =
√
102 + 602 = 60, 8
dij = 10
Zm ∝ 1, 8055
◆❛ ❝♦♥❞✐çã♦ ✭❛✮✱ Zm(a) ∝ ln
√
52+602
5 = 2, 4884✱ ✉♠ ❛✉♠❡♥t♦ ❞❡ 1−
2,4884
1,8055 = 37, 8%✳
◆❛ ❝♦♥❞✐çã♦ ✭❜✮✱ Zm(b) ∝ ln
√
102+202
10 = 0, 8047✱ ✉♠❛ r❡❞✉çã♦ ❞❡ 1−
0,8047
1,8055 = 55, 4%✳
✹✳✹ ❉✐stâ♥❝✐❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛ ❡ r❛✐♦ ♠é❞✐♦ ❣❡♦♠étr✐❝♦
❈❤❛♠❛✲s❡ ❉▼● ❛ ❞✐st❛♥❝✐❛ ♠é❞✐❛ ❣❡♦♠étr✐❛✱ q✉❡ ♥❡st❡ ❝❛s♦ s❡rá ❛♣❧✐❝❛❞♦ às ❞✐stâ♥❝✐❛s ❡♥tr❡
❝♦♥❞✉t♦r❡s✳ ◗✉❛♥❞♦ tr❛t❛✲s❡ ❞❡ ❝♦♥❞✉t♦r❡s ❞❡ ✉♠❛ ♠❡s♠❛ ❢❛s❡✱ ♦✉ ❢❡✐①❡ ❞❡ ❝♦♥❞✉t♦r❡s✱ t❛♠❜é♠ é
❝❤❛♠❛❞♦ ❞❡ r❛✐♦ ♠é❞✐♦ ❣❡♦♠étr✐❝♦ ✭❘▼● ♦✉ ●▼❘✮✱ q✉❡ ♥❡st❡ ❝❛s♦ ✐rá r❡♣r❡s❡♥t❛r ✉♠ ❝♦♥❞✉t♦r
❡q✉✐✈❛❧❡♥t❡ ♣❛r❛ ❛s♣❡❝t♦s ❞❡ ✐♥❞✉tâ♥❝✐❛ ❡ ❝❛♣❛❝✐tâ♥❝✐❛✳
P❛r❛ n ❝♦♥❞✉t♦r❡s ❛rr✉♠❛❞♦s ❡♠ ♣♦s✐çõ❡s ❣❡♥ér✐❝❛s✱ ♦ ❘▼● s❡rá ✐❣✉❛❧ ❛
RMG = n2
√
√
√
√
n
∏
i=1
n
∏
j=1
dij =
n2
√
(d11 d12 · · · d1n)(d21 d22 . . . d2n) · · · (dn1 dn2 · · · dnn) ✭✹✳✽✮
s❡♥❞♦ dii ♦ r❛✐♦ ❞♦ ❝♦♥❞✉t♦r i✱ ❝♦♠ ❛ ❝♦rr❡çã♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ✐♥t❡r♥❛✱ r′i✱ ❡ dij ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ♦s
❝♦♥❞✉t♦r❡s i ❡ j✳
P❛r❛ ❢❡✐①❡s r❡❣✉❧❛r❡s✱ ♦✉ s❡❥❛✱ ❝♦♥❞✉t♦r❡s ❢♦r♠❛❞♦s ❡♠ ♣♦❧í❣♦♥♦s ❞❡ ❧❛❞♦ d✱ ♦ ❘▼● ❞♦ ❢❡✐①❡
s❡rá
RMG2 =
√
r′ d ✭✹✳✾❛✮
RMG3 =
3
√
r′ d2 ✭✹✳✾❜✮
RMG4 = 1, 09
4
√
r′ d3 ✭✹✳✾❝✮
♥♦ q✉❛❧ RMG2✱ RMG3 ❡ RMG4 sã♦ ♦s ❘▼●s ♣❛r❛ ❢❡✐①❡s ❞❡ ✷✱ ✸ ❡ ✹ ❝♦♥❞✉t♦r❡s ❡♠ ❢❡✐①❡s r❡❣✉❧❛r❡s✳
❆ ❡q✉❛çã♦ ♣❛r❛ ✉♠ ❢❡✐①❡ ❞❡ N ❝♦♥❞✉t♦r❡s✱ ❡s♣❛ç❛❞♦s ✐❣✉❛❧♠❡♥t❡ ❡♠ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ R
é ❞❡✜♥✐❞❛ ♣♦r
RMG =
N
√
r N RN−1 ✭✹✳✶✵✮
▲❡♠❜r❛♥❞♦ q✉❡ ♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r r′✱ só é ✐♥❝♦r♣♦r❛❞♦ ♥❛ ✐♠♣❡❞â♥❝✐❛✳ ▲♦❣♦
t❡r❡♠♦s ✉♠ ❘▼● ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ❡ ✉♠ ❘▼● ♣❛r❛ ❛ ❛❞♠✐tâ♥❝✐❛✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛
✉♠ ❢❡✐①❡ ❞❡ ✹ ❝♦♥❞✉t♦r❡s✱ t❡r❡♠♦s
RMGZ4 = 1, 09
4
√
r′ d3 ✭✹✳✶✶❛✮
RMGY4 = 1, 09
4
√
r d3 ✭✹✳✶✶❜✮
❉❡✜♥✐♥❞♦ ❝♦♠♦ M ❛ ♠❛tr✐③ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ ❣❡♦♠❡tr✐❛ ❞❛ ❧✐♥❤❛✱ t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③
❞❡ ❝♦❡✜❝✐❡♥t❡ ❞❡ ♣♦t❡♥❝✐❛✐s✿
L =
µ
2π
MZ ✭✹✳✶✷✮
s❡♥❞♦
MZii = ln
2hi
r′i
✭✹✳✶✸❛✮
MZij = ln
Dij
dij
✭✹✳✶✸❜✮
✼
❊ ❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ s❡rá
Z = R I+ j ωL = R I+ j ω
µ
2π
MZ ✭✹✳✶✹✮
❛❜r✐♥❞♦ ♦s t❡r♠♦s ❞❛s ♠❛tr✐③❡s✿
Z =


R 0 0
0 R 0
0 0 R

+ j ω
µ
2π



ln 2har′a
ln Dabdab ln
Dac
dac
ln Dbadba ln
2hb
r′
b
ln Dbcdbc
ln Dcadca ln
Dcb
dcb
ln 2hcr′c



✭✹✳✶✺✮
❙❡♥❞♦ R ❛ r❡s✐stê♥❝✐❛ ❞❡ ❝❛❞❛ ❝♦♥❞✉t♦r✱ ❝♦♥s✐❞❡r❛♥❞♦ ✐❣✉❛✐s✱ ❡ I ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ✭♥ã♦
❤❛✈❡rá r❡s✐stê♥❝✐❛ ♠út✉❛✮✳ ❖❜s❡r✈❛r q✉❡✱ ♣❛r❛ ❢❡✐①❡s ❞❡ ❝♦♥❞✉t♦r❡s✱ ❞✐✈✐❞✐r ❛ r❡s✐stê♥❝✐❛ ✐♥❞✐✈✐❞✉❛❧
♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❝♦♥❞✉t♦r❡s ❡ tr♦❝❛r ri ♣♦r RMGi✳
❙❡❣✉✐♥❞♦ ❝♦♠♦ ❡①❡♠♣❧♦ ❝♦♠♣❧❡t♦ ❛ ❧✐♥❤❛ ❞❡ ✺✵✵ ❦❱ ✏r❛q✉❡t❡✑✱ ❝✉❥♦ ♣❡r✜❧ é ✐❧✉str❛❞♦ ♥❛
✜❣✉r❛ ✷✱ ❡st❡ ❡①❡♠♣❧♦ ❢❛r❡♠♦s ♦ ❝á❧❝✉❧♦ ❝♦♠♣❧❡t♦ ❞♦s ♣❛râ♠❡tr♦s✱ ❝♦♠❡ç❛♥❞♦ ♣❡❧❛ ✐♠♣❡❞â♥❝✐❛
❝♦♥❢♦r♠❡ ❛❝❛❜♦✉ ❞❡ s❡ ♠♦str❛r ♥❡st❡ ❝❛♣ít✉❧♦✳
❆ ▲❚ ♣♦ss✉✐ ❢❡✐①❡s ❞❡ ✹ ❝❛❜♦s ❘❛✐❧✱ ❝✉❥♦s ♣❛râ♠❡tr♦s r❡❧❡✈❛♥t❡s ❥á ❢♦r❛♠ ❧❡✈❛♥t❛❞♦s ♥♦
❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✱ ❝♦♠ ✢❡❝❤❛ ❞❡ ✶✻ ♠✱ ❡ ♦ ❢❡✐①❡ é ✉♠ q✉❛❞r❛❞♦ ❞❡ ✹✺✱✼ ❝♠✱ ❝♦rr❡s♣♦♥❞❡♥t❡
❛♦ ♣❛❞rã♦ ❝♦♠❡r❝✐❛❧ ❞❡ ✶✽✑✳ ❖s ❝❛❜♦s ♣ár❛✲r❛✐♦s t❛♠❜é♠ ❡stã♦ ✐❧✉str❛❞♦s ♥❛ ✜❣✉r❛✱ ♠❛s ♣♦r
♦r❛ ♥ã♦ s❡rã♦ ❝♦♥s✐❞❡r❛❞♦s✳
❆ r❡s✐stê♥❝✐❛ ❞♦ ❢❡✐①❡ ✭❝♦♥s✐❞❡r❛♥❞♦ t❡♠♣❡r❛t✉r❛ ❞❡ ♦♣❡r❛çã♦ ❞❡ ✼✺❽✮ s❡rá 0,07334 =
0, 018325 ❲✴❦♠✳ P❛r❛ ❛ ✐♥❞✉tâ♥❝✐❛✱ ♣r✐♠❡✐r❛♠❡♥t❡ ❝❛❧❝✉❧❛✲s❡ ♦ ❘▼●✿
RMG = 1, 09 4
√
(
0, 02959
2
· 0, 7788
)
0, 4573 = 0, 1985 m
❯t✐❧✐③❛✲s❡ t❛♠❜é♠ ❛s ❛❧t✉r❛ ♠é❞✐❛s ❞♦s ❝❛❜♦s✿ ❛ ❢❛s❡ ❝❡♥tr❛❧ ❡stá ❛ 34− 2·163 = 23, 33 ♠✱ ❡ ❛s
❢❛s❡s ❧❛t❡r❛✐s ❡stã♦ ❛ 28− 2·163 = 17, 33 ♠✳
❈❛❧❝✉❧❛♥❞♦ ❛❣♦r❛ ❛s ♣❛r❝❡❧❛s ❣❡♦♠étr✐❝❛s r❡❢❡r❡♥t❡s às ✐♥❞✉tâ♥❝✐❛s ♣ró♣r✐❛s ♣❛r❛ ❝❛❞❛ ❢❛s❡✱
✉s❛♥❞♦ ❛ ❝♦♥✈❡♥çã♦ ❞❡ ✭❛✱❜✱❝✮ ♣❛r❛ ❡♥✉♠❡r❛r ❛s ❢❛s❡s✱ s❡♥❞♦ ✭❜✮ ❛ ❢❛s❡ ❝❡♥tr❛❧✿
Maa = ln
2 · 17, 33
0, 1985
= 5, 16277
Mbb = ln
2 · 23, 33
0, 1985
= 5, 46002
Mcc = Maa
❢❛③❡♥❞♦ ❛❣♦r❛ ❛s ♣❛r❝❡❧❛s r❡❢❡r❡♥t❡s às ✐♥❞✉tâ♥❝✐❛s ♠út✉❛s✱
Mab = ln
√
52 + (23, 33 + 17, 33)2
√
52 + (23, 33− 17, 33)2
= 1, 65731
Mbc = Mab
Mac = ln
√
(2 · 5)2 + (2 · 17, 33)2
2 · 5 = 1, 28298
P♦❞❡♥❞♦ s❡r ❞✐r❡t❛♠❡♥t❡ ✐♥s❡r✐❞♦s ❡♠ ✉♠ ♣r♦❣r❛♠❛✱ ♣r♦✈❡♥❞♦ ✉♠ ✈❡t♦r ❞❡ ❝♦✲
♦r❞❡♥❛❞❛s x ❡ h✱ ✐♠♣❧❡♠❡♥t❛✲s❡ ♥❛ ❢♦r♠❛ dij = sqrt[(xi − xj)2 + (hi + hj)2]✱
Dij = sqrt[(xi − xj)2 + (hi − hj)2] ❡ Mij = log[Dij/dij]✱ ❧❡♠❜r❛♥❞♦ ❞❛ ❝♦♥✈❡♥çã♦ ❞❛
❢✉♥çã♦ log[x] ❡♠ ❣❡r❛❧ s❡r ♦ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧✱ ln(x)✳
❆ ♠❛tr✐③ M s❡rá ❡♥tã♦
M =


5, 1627716 1, 6573122 1, 2829804
1, 6573122 5, 4600231 1, 6573122
1, 2829804 1, 6573122 5, 1627716


♦❜té♠✲s❡ ❛ ♠❛tr✐③ ✐♥❞✉tã♥❝✐❛ L ♠✉❧t✐♣❧✐❝❛♥❞♦ M ♣♦r µ02π ✱ ❡ ♥❛ s❡q✉ê♥❝✐❛ ❛ ♠❛tr✐③ Z ♠✉❧t✐♣❧✐✲
❝❛♥❞♦ L ♣♦r j ω ❡ s♦♠❛♥❞♦ ❛ ♠❛tr✐③ R✱ q✉❡ é ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❝♦♠ ❛s r❡s✐stê♥❝✐❛s ❞♦s
✽
❢❡✐①❡s✳ ❘❡s✉♠✐♥❞♦✱ t❡♠✲s❡✿
Z =R+ j ω
µ0
2π
M
=


0, 018325 + j0, 3892730 j0, 1249613 j0, 0967367
j0, 1249613 0, 018325 + j0, 4116857 j0, 1249613
j0, 0967367 j0, 1249613 0, 018325 + j0, 3892730


❲/km
♦❜s❡r✈❛♥❞♦ ❛t❡♥t❛♠❡♥t❡ ❛♦ ❡①♣r❡ss❛r ♦✉ ❝❛❧❝✉❧❛r ♦s ✈❛❧♦r❡s ❡♠ ❲✴♠ ♦✉ ❲✴❦♠✳
34
28
5
4 
 0,457
❋✐❣✉r❛ ✷✿ ❊①❡♠♣❧♦ ❞❡ ♣❡r✜❧ ❞❡ ▲❚✳
✹✳✺ ❈❛♣❛❝✐tâ♥❝✐❛ ❡ ❛❞♠✐tâ♥❝✐❛ tr❛♥s✈❡rs❛❧
❆ ❝❛♣❛❝✐tâ♥❝✐❛ ❞❛ ❧✐♥❤❛ t❛♠❜é♠ s❡rá ❞❡✜♥✐❞❛ ❛ ♣❛rt✐r ❞❡ s✉❛ ❣❡♦♠❡tr✐❛✻✳ P❛rt✐♥❞♦ ❞♦ ❡①❡♠♣❧♦
t❡ór✐❝♦ ❞❡ ✉♠ ❝❛❜♦ s✐♥❣❡❧♦ ♣♦❧❛r✐③❛❞♦ ❝♦♠ ✉♠ ♣♦t❡♥❝✐❛❧ V ❡♠ r❡❧❛çã♦ ❛♦ s♦❧♦✱ ❡st❡ ❝❛❜♦ t❡rá ✉♠❛
❝❛♣❛❝✐tâ♥❝✐❛ ❡♠ ❢✉♥çã♦ ❞♦ s❡✉ r❛✐♦ ❡ ❞❛ s✉❛ ❛❧t✉r❛✿
V =
q
2πε0
ln
2h
r
✭✹✳✶✻✮
C =
q
V
= 2π ε0
(
ln
2h
r
)−1
✭✹✳✶✼✮
❣❡♥❡r❛❧✐③❛♥❞♦ ♣❛r❛ ✉♠❛ ❧✐♥❤❛ ❝♦♠ n ❝♦♥❞✉t♦r❡s✱ ❞❡s❡♥✈♦❧✈❡✲s❡ ✉♠ r❡❧❛çã♦ ❣❡♦♠étr✐❝❛ ❞❡s❝r✐t❛
♣♦r ✉♠❛ ♠❛tr✐③ MY✱ s✐♠✐❧❛r ❛ MZ✿
C = 2π ε0 MY
−1 ✭✹✳✶✽✮
◆♦ q✉❛❧ ε0 ❛ ♣❡r♠✐ss✐✈✐❞❛❞❡ ❞♦ ❛r✱ ✐❣✉❛❧ ❛ 8, 85 · 10−12 ❋✴♠✳ ❆q✉✐ ♥ã♦ ❤á ✏❝❛♣❛❝✐tâ♥❝✐❛ ✐♥t❡r♥❛✑✱
❧♦❣♦ ♥ã♦ ❤á ❝♦rr❡çã♦ ❞♦ r❛✐♦ ❞♦s ❝♦♥❞✉t♦r❡s✱ ❝♦♠♦ ✈✐st♦ ♥❛ ❡q✉❛çã♦ ✹✳✻✱ ♠❛s ♦ t❡r♠♦ r❡❢❡r❡♥t❡ à
♠út✉❛ é r✐❣♦r♦s❛♠❡♥t❡ ✐❣✉❛❧✿
✻❊♠ ❬✷✼✱ ♣✳ ✼✷❪ ❞❡s❡♥✈♦❧✈❡✲s❡ ❛ t❡♦r✐❛ ❞❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❡♠ ▲❚s✱ ♠❛s ❝♦♠ ❛ ❛♣r♦①✐♠❛çã♦ ❡♠ ✏✉♥✐r✑ t♦❞❛s ❛s ❢❛s❡s
❡♠ ✉♠❛ ✏❞✐stâ♥❝✐❛ ♠é❞✐❛ ❣❡♦♠étr✐❝❛✳✑
✾
MY ii = ln
2hi
ri
✭✹✳✶✾❛✮
MY ij = ln
Dij
dij
✭✹✳✶✾❜✮
❆♦ ✐♥✈❡rt❡r✲s❡ ❛ ♠❛tr✐③MY✱ ♦❜s❡r✈❛✲s❡ ❛ ❢♦r♠❛çã♦ ❞❡ t❡r♠♦s ♥❡❣❛t✐✈♦s ❢♦r❛ ❞❛ ❞✐❛❣♦♥❛❧✱ ❞❡✈✐❞♦
❛♦ ♣r♦❝❡ss♦ ❞❡ ♣♦❧❛r✐③❛çã♦✿ ✉♠❛ ❝❛r❣❛ ❞❡ ♣♦❧❛r✐❞❛❞❡ ♣♦s✐t✐✈❛ ❡♠ ✉♠❛ ❢❛s❡ ✐rá ♣r♦✈♦❝❛r ❝❛r❣❛s ❞❡
♣♦❧❛r✐❞❛❞❡ ♥❡❣❛t✐✈❛ ♥❛s ♦✉tr❛s ❢❛s❡s✳
❆ ❛❞♠✐tâ♥❝✐❛ é ❞❡✜♥✐❞❛ ♣♦r✿
Y = G+ j ωC ✭✹✳✷✵✮
❉❡s❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣❛r❝❡❧❛ ❞❡ ❝♦♥❞✉tâ♥❝✐❛✱ ♦❜té♠✲s❡ ❛ ❢♦r♠❛ ✉s✉❛❧ ❞❛ ❛❞♠✐tâ♥❝✐❛ ♣❛r❛ ❧✐♥❤❛s ❈❆✿
Y = j ωC ✭✹✳✷✶✮
❙❡❣✉✐♥❞♦ ♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❛❞♠✐tâ♥❝✐❛✱ ♣♦❞❡✲s❡ ❛♣r♦✈❡✐t❛r ♣❛r❝✐❛❧♠❡♥t❡
❛ ♠❛tr✐③ M✱ r❡❝❛❧❝✉❧❛♥❞♦ ❛ ❞✐❛❣♦♥❛❧ ❝♦♥s✐❞❡r❛♥❞♦ ♦ r❛✐♦ r❡❛❧ ❞♦s ❝❛❜♦s✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ♦
❘▼●✿
RMG = 1, 09 4
√
(
0, 02959
2
)
0, 4573 = 0, 2113 m
❡ ♦s ❡❧❡♠❡♥t♦s ♣ró♣r✐♦s ❞❛ ♠❛tr✐③✿
Maa = ln
2 · 17, 33
0, 2113
= 5, 10027
Mbb = ln
2 · 23, 33
0, 2113
= 5, 39752
Mcc = Maa
t❡♠✲s❡ ❛ss✐♠ ❛ ♠❛tr✐③ M ❡ ❛ s✉❛ ✐♥✈❡rs❛✿
M =


5, 1002713 1, 6573122 1, 2829804
1, 6573122 5, 3975229 1, 6573122
1, 2829804 1, 6573122 5, 1002713


M
−1 =


0, 224171 −0, 0572269 −0, 0377949
−0, 0572269 0, 2204133 −0, 0572269
−0, 0377949 −0, 0572269 0, 224171


♦❜t❡♥❞♦✲s❡ ❛ ♠❛tr✐③ ❞❡ ❝❛♣❛❝✐tâ♥❝✐❛ C ♠✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r 2π ε0✱ ❡ ❛ ❛❞♠✐tâ♥❝✐❛ ♠✉❧t✐♣❧✐❝❛♥❞♦♣♦r j ω✱ ❝❛❧❝✉❧❛♥❞♦ ❞✐r❡t❛♠❡♥t❡✿
Y = j ω 2π ε0 M
−1
=


j4, 6994162 −j1, 1996776 −j0, 7923144
−j1, 1996776 j4, 6206408 −j1, 1996776
−j0, 7923144 −j1, 1996776 j4, 6994162

 ➭S/km
❆q✉✐ ♥♦✈❛♠❡♥t❡ ♣❛r❛ ❡✈✐t❛r ♦ ✉s♦ ❞❡ ✉♠ ❡①♣♦❡♥t❡✱ ♥♦ ❝❛s♦ 10−9 ❬❙✴♠❪✱ ♦♣t♦✉✲s❡ ❡♠
❡①♣r❡ss❛r ♦s ✈❛❧♦r❡s ✉t✐❧✐③❛♥❞♦ ♠ú❧t✐♣❧♦s ❡ s✉❜♠ú❧t✐♣❧♦s ❞❛s ✉♥✐❞❛❞❡s✳
✹✳✻ ❊❢❡✐t♦ ❞❛ tr❛♥s♣♦s✐çã♦
P❛r❛ ♦❜t❡r ✉♠ ❡q✉✐❧í❜r✐♦ ♥♦s ♣❛râ♠❡tr♦s ❞❛ ❧✐♥❤❛✱ ❛s ❢❛s❡s sã♦ tr♦❝❛❞❛s ❞❡ ♣♦s✐çã♦ ❡♠ ❛❧❣✉♥s
♣♦♥t♦s ❞❛ ❧✐♥❤❛✳ ▼❛t❡♠❛t✐❝❛♠❡♥t❡✱ s❡r❛ ❡q✉✐✈❛❧❡♥t❡ ❛ tr♦❝❛r ❧✐♥❤❛s ♥❛s ♠❛tr✐③❡s Z ❡ Y✳ ❙❡❥❛ ❛s
♠❛tr✐③❡s Z(1)✱ Z(2) ❡ Z(3) r❡❢❡r❡♥t❡s ❛ três tr❡❝❤♦s✿
Z
(1) =


Zaa Zab Zac
Zba Zbb Zbc
Zca Zcb Zcc

 ✭✹✳✷✷✮
✶✵
Z
(2) =


Zbb Zbc Zba
Zcb Zcc Zca
Zab Zac Zaa

 ✭✹✳✷✸✮
Z
(3) =


Zcc Zca Zcb
Zac Zaa Zab
Zbc Zba Zbb

 ✭✹✳✷✹✮
❙❡♥❞♦ ✉♠❛ tr❛♥s♣♦s✐çã♦ ✐❞❡❛❧ ✭♥♦ ❝❛s♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ ❝✐r❝✉✐t♦ s✐♠♣❧❡s✱ ❞✐✈✐❞✐❞❛ ❡♠ três
tr❡❝❤♦s ❞❡ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✮✱ ♣♦❞❡✲s❡ s✉♣♦r ✉♠ ❞❡s❡♠♣❡♥❤♦ ❡q✉✐✈❛❧❡♥t❡ ❞❛ ❧✐♥❤❛ ❡♠ ✉♠❛
♠❛tr✐③ ♠é❞✐❛✼✿
Z =
1
3
(
Z
(1) + Z(2) + Z(3)
)
=
1
3


Zaa + Zbb + Zcc Zab + Zbc + Zca Zac + Zba + Zcb
Zba + Zcb + Zac Zbb + Zcc + Zaa Zbc + Zca + Zab
Zca + Zab + Zbc Zcb + Zac + Zba Zcc + Zaa + Zbb

 ✭✹✳✷✺✮
P♦❞❡♠♦s ❞❡✜♥✐r ✉♠ t❡r♠♦ ❞❡ ✐♠♣❡❞â♥❝✐❛ ♣ró♣r✐❛✿
Zp =
1
3
(Zaa + Zbb + Zcc) ✭✹✳✷✻❛✮
❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ t❡♠♦s ✉♠❛ s✐♠❡tr✐❛ ❞♦ t✐♣♦ Zij = Zji✱ ✉♠ t❡r♠♦ ❞❡ ✐♠♣❡❞â♥❝✐❛ ♠út✉❛
Zm =
1
3
(Zab + Zbc + Zca) ✭✹✳✷✻❜✮
❛ ♠❛tr✐③ ❞❡ ✉♠❛ ❧✐♥❤❛ ✐❞❡❛❧♠❡♥t❡ tr❛♥s♣♦st❛ é ✐❣✉❛❧ ❛
Z =


Zp Zm Zm
Zm Zp Zm
Zm Zm Zp

 ✭✹✳✷✼✮
P❛r❛ ❛ ♠❛tr✐③ ❛❞♠✐tâ♥❝✐❛✱ s❡❣✉❡✲s❡ ❛ ♠❡s♠❛ ♠❡t♦❞♦❧♦❣✐❛✿
Y =


Yp Ym Ym
Ym Yp Ym
Ym Ym Yp

 ✭✹✳✷✽✮
s❡♥❞♦
Yp =
1
3
(Yaa + Ybb + Ycc) ✭✹✳✷✾❛✮
Ym =
1
3
(Yab + Ybc + Yca) ✭✹✳✷✾❜✮
❈♦♥t✐♥✉❛♥❞♦ ♥♦ss♦ ❡①❡♠♣❧♦✱ ♦❜té♠✲s❡✿
Zp = 0, 018325 + j0, 3967439 ❲/km
Zm = j0, 115553 ❲/km
Yp = j4, 6731578 ➭S/km
Ym = −j1, 0638899 ➭S/km
✺ ❉❡s❡♠♣❡♥❤♦ ❡❧étr✐❝♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦
✺✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❡♠ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s
❖ ♠ét♦❞♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s é ✉t✐❧✐③❛❞♦ ❡♠ s✐st❡♠❛s tr✐❢ás✐❝♦s ❡q✉✐❧✐❜r❛❞♦s ♦✉ ❞❡s❡q✉✐✲
❧✐❜r❛❞♦s✱ ❞❡ ❢♦r♠❛ ❛ ❞❡❝♦♠♣♦r ♦ ❡st✉❞♦ ❡♠ três ❝✐r❝✉✐t♦s ♠♦♥♦❢ás✐❝♦s✱ ♥♦ q✉❛❧ s❡✉s ❡q✉✐✈❛❧❡♥t❡s
✼❆q✉✐ ❝❛❜❡ ✉♠❛ ♦❜s❡r✈❛çã♦✱ ♥♦ q✉❛❧ ❛ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞♦s ❛❝❛❜❛ ❡q✉✐✈♦❝❛♥❞♦✲s❡✿ ✉♠❛ ❧✐♥❤❛ tr❛♥s♣♦st❛ ♣♦❞❡ s❡r
❝♦♥s✐❞❡r❛r ❝♦♠ ♣❛râ♠❡tr♦s ♠é❞✐♦s q✉❛♥❞♦ s❡♥❞♦ tr❛t❛❞❛ ✧♣♦r ✐♥t❡✐r❛✧✳ ❊st✉❞♦s ❝♦♠♦ ❞❡ ❢❛❧t❛s ♥♦ ♠❡✐♦ ❞❛ ❧✐♥❤❛
❛❝❛❜❛ ❞✐✈✐❞✐♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❡♠ ❞✉❛s ❧✐♥❤❛s ♣❛r❝✐❛❧♠❡♥t❡ tr❛♥s♣♦st❛s✦ ❖ ❡rr♦ ❛❞q✉✐r✐❞♦✱ ❞❡ ✉♠❛ ❧✐♥❤❛ s❡r ❛ss✉♠✐❞❛
❝♦♠♦ tr❛♥s♣♦st❛✱ é ♣❡q✉❡♥♦✱ ♠❛s ❛t❡♥t❛✲s❡ q✉❡ ✉♠ ❝á❧❝✉❧♦ ♠❛✐s ♣r❡❝✐s♦ ♠❡r❡❝❡ ✉♠ ♠♦❞❡❧♦ ♥ã♦ tr❛♥s♣♦st♦✳
✶✶
❚❤é✈❡♥✐♥ ♣♦❞❡♠ s❡r ❝♦♠❜✐♥❛❞♦s ♥♦ ❡st✉❞♦ ❞❡ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✱ ❢❛❧t❛s ❡ ❞❡❢❡✐t♦s ❡♠ ❣❡r❛❧✳ ◆❡st❛
s❡çã♦ ❛♣r❡s❡♥t❛✲s❡ ❝♦♠♦ r❡♣r❡s❡♥t❛r ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ♥❡st❡ s✐st❡♠❛✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s s♦✲
❜r❡ ❡st❛ ♠❡t♦❞♦❧♦❣✐❛ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✶✶✱ ✷✼❪✳
P❛r❛ ❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❛ ♠❛tr✐③ Z✱ ❞✐t❛ ❡♠ ❝♦♦r❞❡♥❛❞❛s ❞❡ ❢❛s❡✱ ♣❛r❛ ♦ s✐st❡♠❛ ❞❡
❝♦♦r❞❡♥❛❞❛s ❞❡ ♠♦❞♦✱ ♦✉ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s✱ ✉t✐❧✐③❛✲s❡ ❛ ♠❛tr✐③ A✱ ❞❡✜♥✐❞❛ ♣♦r
A =


1 1 1
1 a2 a
1 a a2

 ✭✺✳✶✮
♥♦ q✉❛❧ a = 1 120➦ ❡ a2 = 1 −120➦✱ ♦❜t❡♠✲s❡ ❛ ♠❛tr✐③ ❞❡ ✐♠♣❡❞â♥❝✐❛s ❡♠ ❝♦♦r❞❡♥❛❞❛s ❞❡ ♠♦❞♦✱
Z012✳ ❙❡ ❛s ♠❛tr✐③❡s Z ❡ Y ❝♦rr❡s♣♦♥❞❡r❡♠ ❛ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ✐❞❡❛❧♠❡♥t❡ tr❛♥s♣♦st❛✱
♦❜t❡♠✲s❡ ❛s ♠❛tr✐③❡s Z012 ❡ Y012 s♦♠❡♥t❡ ❝♦♠ t❡r♠♦s ♥❛ ❞✐❛❣♦♥❛❧✿
Z012 = A
−1
ZA =


Z0 0 0
0 Z1 0
0 0 Z2


=


Zp + 2Zm 0 0
0 Zp − Zm 0
0 0 Zp − Zm

 ✭✺✳✷✮
Y012 = A
−1
YA =


Y0 0 0
0 Y1 0
0 0 Y2


=


Ys + 2Ym 0 0
0 Ys − Ym 0
0 0 Ys − Ym

 ✭✺✳✸✮
P❛r❛ ❡st✉❞♦s ❞❡ ✢✉①♦ ❞❡ ♣♦tê♥❝✐❛ ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✱ ♦✉ ❡st✉❞♦ ❞❡ ❢❛❧t❛s s✐♠étr✐❝❛s✱ ✉t✐❧✐③❛✲
s❡ s♦♠❡♥t❡ ♦s ♣❛râ♠❡tr♦s ❞❡ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛✿
Z1 = Zp − Zm ✭✺✳✹❛✮
Y1 = Yp − Ym ✭✺✳✹❜✮
❝♦rr❡s♣♦♥❞❡♥t❡s ❛♦ ❡❧❡♠❡♥t♦ ♥❛ ♣♦s✐çã♦ ✭✷✱✷✮ ❞❛ ♠❛tr✐③✳ ❉❡st❡s ♣❛râ♠❡tr♦s q✉❡ s❡ ♦❜té♠ ❛ ✐♠♣❡✲
❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ Zc ❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣❛❣❛çã♦ γ✱ ✈✐st♦s ❛ s❡❣✉✐r✳
✺✳✷ ■♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛
❆ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛✽ é ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ ❜❛❧❛♥ç♦ ❡♥tr❡ ♦s ❝❛♠♣♦s ❡❧étr✐❝♦ ❡ ♠❛❣♥ét✐❝♦ ❞❛
❧✐♥❤❛✱ ♥♦ q✉❛❧ ✉♠❛ ❝❛r❣❛ r❡s✐st✐✈❛ ♥❡st❡ ✈❛❧♦r t❡rá ❛ ♠❛✐♦r ❡✜❝✐ê♥❝✐❛ ❞❡ ❛❜s♦rçã♦ ❞❡ ✉♠ ♣✉❧s♦✱
t❛♠❜é♠ ❞✐t♦ ❝♦♠♦ ✏❝❛s❛♠❡♥t♦ ❞❡ ✐♠♣❡❞â♥❝✐❛✑✳ ➱ ✉♠ ♣❛râ♠❡tr♦ ❡♠ ❝♦♠✉♠ ❝♦♠♦ ♦✉tr♦s t✐♣♦s ❞❡
❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦ ✭❡♠ ❘❋✱ ♠✐❝r♦♦♥❞❛s✱ ❝♦❛①✐❛❧ ♦✉ ♠✐❝r♦str✐♣✱ ❡t❝✮✳
➱ ❝❛❧❝✉❧❛❞❛ ♣❡❧♦s ♣❛râ♠❡tr♦s ❞❡ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛ Z1 ❡ Y1✱ s✐♠♣❧✐✜❝❛❞♦s ❛q✉✐ ❡♠ ❞✐❛♥t❡ ❝♦♠♦
Z ❡ Y ✿
Zc =
√
Z
Y
✭✺✳✺✮
s❡♥❞♦
Z = R+ j Xl = R+ j ω L ✭✺✳✻❛✮
Y = j Bc = j ω C ✭✺✳✻❜✮
♦s ❡q✉✐✈❛❧❡♥t❡s ♠♦♥♦❢ás✐❝♦s ♣❛r❛ ✉♠ ❡st✉❞♦ ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✾✱ ❝✉❥❛ ♣r❡♠✐ss❛ é ❞❡t❛❧❤❛❞❛ ♥❛
s❡çã♦ ✶✵✳✷✳
❯s✉❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛✲s❡ s♦♠❡♥t❡ ❛ ♣❛rt❡ r❡❛❧ ❞❡ Zc✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ ❡♥tã♦ ❛ ✉♠❛ ❧✐♥❤❛ s❡♠
♣❡r❞❛s✳ P♦ré♠✱ ❞❡✈❡✲s❡ ✉s❛r ♦ ❝á❧❝✉❧♦ ♣r❡❝✐s♦ ❞❡ Zc ❛♦ s❡ ❛♣❧✐❝❛r às ❢ór♠✉❧❛s ❞❡ ❧✐♥❤❛ ❧♦♥❣❛✱ ♥❛
s❡çã♦ ✺✳✻✳
✽❊♠ ✐♥❣❧ês r❡❢❡r❡♥❝✐❛❞♦ ❝♦♠♦ s✉r❣❡ ✐♠♣❡❞❛♥❝❡✱ ♦✉ ✐♠♣❡❞â♥❝✐❛ ❞❡ s✉rt♦✳
✾P❛r❛ ❡st✉❞♦s ❡♠ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s✱ ♣♦❞❡✲s❡ ❞❡❞✉③✐r ♦s ❡q✉✐✈❛❧❡♥t❡s ♣❛r❛ s❡q✉ê♥❝✐❛ ♥❡❣❛t✐✈❛ ❡ ③❡r♦✱ Zc2 ❡
Zc0 r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ sã♦ ❛♣❧✐❝á✈❡✐s ❡♠ ❡st✉❞♦s ❞❡ tr❛♥s✐tór✐♦s✳
✶✷
❆♦ ❝♦♥s✐❞❡r❛r ❛ ❧✐♥❤❛ ❝♦♠ ♣❡r❞❛ ❞❡s♣r❡③í✈❡❧ ✭r❡t✐r❛♥❞♦ R✮✱ ❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ s❡rá
❛♣r♦①✐♠❛❞❛♠❡♥t❡
Zc ∼=
√
Xl
Bc
∼=
√
L
C
✭✺✳✼✮
s❡♥❞♦ ❛ss✐♠ ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢r❡q✉ê♥❝✐❛✳
✺✳✸ P❛râ♠❡tr♦s ❞❡ ♣r♦♣❛❣❛çã♦
❆ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡♠♦♥str❛ ❛ ❞❡❢♦r♠❛çã♦ ❞❛ ♦♥❞❛ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛✳ ➱ ❞❡✜♥✐❞❛ ❝♦♠♦
γ =
√
Y Z =
√
(R+ j ω L)j ωC ✭✺✳✽✮
s❡♥❞♦ s✉❛ ✉♥✐❞❛❞❡ ❡♠ m−1✳ ❆ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣❛❣❛çã♦ ♣♦❞❡ s❡r ❞❡s♠❡♠❜r❛❞❛ ♥❛ ❢♦r♠❛ γ =
α + j β✱ s❡♥❞♦ α ❛ ❝♦♥st❛♥t❡ ❞❡ ❛t❡♥✉❛çã♦ ✭❡♠ ◆❡♣❡r✴♠✮ ❡ β ❛ ❝♦♥st❛♥t❡ ❞❡ ❢❛s❡ ✭❡♠ r❛❞✴♠✮✳
P♦❞❡✲s❡ ❡♥tã♦ ♦❜t❡r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ♦♥❞❛ ❞❛ ❧✐♥❤❛ λ✿
λ =
2π
β
✭✺✳✾✮
❈♦♥s✐❞❡r❛♥❞♦ ❛ ❧✐♥❤❛ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ s❡♠ ♣❡r❞❛s✱ γ ♣♦ss✉✐rá s♦♠❡♥t❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ❢❛s❡ β✿
γ ∼=
√
j ω L j ω C = j ω
√
LC ✭✺✳✶✵❛✮
β ∼= ω
√
LC ✭✺✳✶✵❜✮
❡ ❡st❡ ♣❛râ♠❡tr♦✱ ♣❛r❛ ❧✐♥❤❛s ❛ér❡❛s✱ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦ ♥í✈❡❧ ❞❡ t❡♥sã♦✱ s❡rá ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧
❛ ✵✱✵✵✶✸ r❛❞✴❦♠✳ P❛r❛ ❝❛❜♦s✱ ❡st❡ ✈❛❧♦r ♣♦❞❡ ✈❛r✐❛r ❡♥tr❡ ✵✱✵✵✹✻ ❛ ✵✱✵✵✾✶ r❛❞✴❦♠✳
❖✉tr♦ ♣❛râ♠❡tr♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞❛ ❧✐♥❤❛ é ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦ ❡❧étr✐❝♦✱ ♦✉ â♥❣✉❧♦ ❞❡ ❧✐♥❤❛✿
θ = β l ✭✺✳✶✶✮
q✉❡ ✐♥❞✐❝❛ ❛ ❞❡❢❛s❛❣❡♠ ♥❛t✉r❛❧ q✉❡ ♦❝♦rr❡rá ♥❛ tr❛♥s♠✐ssã♦✱ ♠❡s♠♦ q✉❡ s❡ ❝♦♥s✐❞❡r❡ ❛ ❧✐♥❤❛ ❝♦♠♦
s❡♠ ♣❡r❞❛s✳ ❊st❡ ❢❛t♦ é ❞❡✈✐❞♦ ❛♦ ♣r✐♥❝í♣✐♦ ❞❡ ❝✐r❝✉✐t♦ ❞✐str✐❜✉í❞♦✱ ♦✉ q✉❡ ❛ ❡♥❡r❣✐❛ tr❛♥s♠✐t✐❞❛
♣♦ss✉✐ ✈❡❧♦❝✐❞❛❞❡ ✜♥✐t❛ ❞❡ ♣r♦♣❛❣❛çã♦✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❧✐♥❤❛ ❛ér❡❛ ❞❡ ✸✵✵ ❦♠ t❡rá ✉♠ â♥❣✉❧♦
❞❡ ✵✱✸✾ r❛❞✱ ♦✉ ✷✷✱✸✹➦✳
❆ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ♣r♦♣❛❣❛çã♦ ♥❛ ❧✐♥❤❛ ♣❛r❛ ✉♠ ♦♥❞❛ ❞❡ ❢r❡q✉ê♥❝✐❛ f é ❝❛❧❝✉❧❛❞❛ ♣♦r v = λ f ✱ ❡
❝♦♥s✐❞❡r❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❧✐♥❤❛ s❡♠ ♣❡r❞❛s✱ t♦r♥❛✲s❡
v =
1√
LC
✭✺✳✶✷✮
s❡♥❞♦ ❛ss✐♠ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❢r❡q✉ê♥❝✐❛✱ ❡ é ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♥♦ ❡st✉❞♦ ❞❡ s✉rt♦s rá♣✐❞♦s ✭❡♥tr❡
✶✵✵ ❦❍③ ❡ ✶ ▼❍③✮✳ ❖❜s❡r✈❛✲s❡ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ♣r♦♣❛❣❛çã♦ é ❞❛ ♦r❞❡♠✱ ♠❛s ♥✉♥❝❛ ✐❣✉❛❧ ♦✉
s✉♣❡r✐♦r✱ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦✳
❖ ❡st✉❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s ✈✐❛❥❛♥t❡s é ❛❜♦r❞❛❞♦ ♣♦r ❡①❡♠♣❧♦ ❡♠ ❬✶✶✱ ♣✳ ✷✷✷❪ ❡ ❬✷✼✱ ♣✳
✶✷✵❪
✺✳✹ P♦tê♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛
❆ ♣♦tê♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ Pc é ❛ ♣♦tê♥❝✐❛ ❡♥tr❡❣✉❡ ♣❡❧❛ ❧✐♥❤❛ ♣❛r❛ ✉♠ ❝❛r❣❛ r❡s✐st✐✈❛✱ ❝♦♠ ✈❛❧♦r
✐❣✉❛❧ à ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛✳ P❛r❛ ❧✐♥❤❛s ❧♦♥❣❛s✱ é ✉♠ ❝r✐tér✐♦ ❛❞❡q✉❛❞♦ ♣❛r❛ ❡st✐♠❛r ❛ s✉❛
❝❛♣❛❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦✳ ➱ ❞❡✜♥✐❞❛ ♣♦r✿
Pc =
U20
Zc
✭✺✳✶✸✮
s❡♥❞♦ U0 ❛ t❡♥sã♦ ♠é❞✐❛ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛✱ ♦✉ s❡❥❛✱ ❝♦♥s❡❣✉❡✲s❡ ❡❧❡✈❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦✱
♠❛s s❛❝r✐✜❝❛♥❞♦ ❛ ❝♦♥✜❛❜✐❧✐❞❛❞❡ ✭✐♥❝❧✉✐♥❞♦ s♦❜r❡t❡♥sõ❡s✮ ❡ ❡❧❡✈❛♥❞♦ ♣❡r❞❛s ❝♦r♦♥❛✳
▼❛♥t❡♥❞♦ ❛ ❝♦♥s✐❞❡r❛çã♦ ❞❡ ❧✐♥❤❛ s❡♠ ♣❡r❞❛s✱ ❛ ♣♦tê♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ s❡rá ✉♠ ♥ú♠❡r♦ r❡❛❧✱
♦✉ s❡❥❛✱ ❡①♣r❡ss♦ ❡♠ ❲✳ ▼❡s♠♦ ♣❛r❛ ✉♠❛ ❧✐♥❤❛ ❝♦♠ ♣❡r❞❛s✱ é ✉s✉❛❧ ❡①♣r❡ss❛r s♦♠❡♥t❡ ❛ ♣❛rt❡
r❡❛❧✳
✶✸
P❛r❛ ♥♦ss♦ ❡①❡♠♣❧♦✱ ♣❛r❛ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛✱
Z1 = 0, 018325 + j0, 2811908 ❲/km
Y1 = j5, 7370477 ➭S/km
❡ ❡♠ s❡❣✉✐❞❛
Zc = 221, 506− j7, 2100622 ❲
γ = (0, 0413645 + j1, 2707934) · 10−6 Np/m
◗✉❛♥❞♦ ❛ ▲❚ é ❝❛❧❝✉❧❛❞❛ ✏s❡♠ ♣❡r❞❛s✑ ✭s❡♠ ❝♦♥s✐❞❡r❛r ❛ r❡s✐stê♥❝✐❛✮✱ Zc s❡rá ✉♠ ♥ú♠❡r♦
r❡❛❧ ❡ γ ✉♠ ♥ú♠❡r♦ ✐♠❛❣✐♥ár✐♦✳
❈♦♥s✐❞❡r❛♥❞♦ ❝♦♠♦ ✉♠❛ ▲❚ ❞❡ ✺✵✵ ❦❱✱ ❝♦♥s✐❞❡r❛♥❞♦ s♦♠❡♥t❡ ❛ ♣❛rt❡ r❡❛❧ ❞❡ Zc✱ ❛ ♣♦tê♥❝✐❛
❝❛r❛❝t❡ríst✐❝❛ s❡rá ✶✶✷✾ ▼❲✳ ❙❡ ✏❛♣❡rt❛r✑ ❛ t❡♥sã♦ ♠é❞✐❛ ♣❛r❛ ✺✷✺ ❦❱✱ ❛ ♣♦tê♥❝✐❛❡❧❡✈❛✲s❡
♣❛r❛ ✶✷✹✹ ▼❲✳
✺✳✺ ❘❡❛t✐✈♦ tr❛♥s✈❡rs❛❧ ❞❡ ❧✐♥❤❛
❯♠ ♣❛râ♠❡tr♦ r❡❧❡✈❛♥t❡ é ♦ r❡❛t✐✈♦ ❝❛♣❛❝✐t✐✈♦ q✉❡ ✉♠❛ ❧✐♥❤❛ ♣♦ss✉✐✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ❧✐♥❡
❝❤❛r❣✐♥❣✳ P♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ s✉s❝❡♣tâ♥❝✐❛ ♣❡❧♦ q✉❛❞r❛❞♦ ❞❛
t❡♥sã♦ ❞❡ ♦♣❡r❛çã♦✿
Qc = V
2 Bc ✭✺✳✶✹✮
❙❡♥❞♦ ✉s✉❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ❡♠ ▼✈❛r✴❦♠✳
P❛r❛ ♥♦ss♦ ❡①❡♠♣❧♦✱ s❡♥❞♦ Bc = 5, 7370477 · 10−9 ❙✴♠✱ ♦❜té♠✲s❡ ✶✱✹✸✹✸ ❦✈❛r✴♠✱ q✉❡
❡q✉✐✈❛❧❡ ❛ ✶✱✹✸✹✸ ▼✈❛r✴❦♠✳
❖❜s❡r✈❡ q✉❡ ❡st❛ ♣r❡♠✐ss❛ s✉♣õ❡ q✉❡ ♦ ♣❡r✜❧ ❞❡ t❡♥sã♦ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛ é ❝♦♥st❛♥t❡✱ ♦
q✉❡ ♥ã♦ é r❡❛❧✐st❛ ✲ ♦❜s❡r✈❡ ♣♦r ❡①❡♠♣❧♦ ♦ ❡❢❡✐t♦ ❋❡rr❛♥t✐✱ q✉❡ ❡❧❡✈❛ ❛ t❡♥sã♦ ♥❛ ❡①tr❡♠✐❞❛❞❡
❡♠ ❛❜❡rt♦✱ ❢♦r❛ ♦✉tr❛s ❝♦♥❞✐çõ❡s ♦♣❡r❛❝✐♦♥❛✐s ♥♦ q✉❛❧ ♦ ♣♦♥t♦ ❞❡ t❡♥sã♦ ♠❛✐s ❡❧❡✈❛❞❛ ♣♦❞❡
s❡r ♥♦ ♠❡✐♦ ❞❛ ❧✐♥❤❛✦
✺✳✻ ▼♦❞❡❧♦ ❞❡ ❝✐r❝✉✐t♦
❖ ❡q✉✐✈❛❧❡♥t❡ ♠♦♥♦❢ás✐❝♦ ✭♠♦❞❡❧♦ π✮ s❡rá ❝♦♠♣♦st♦ ♣❡❧❛ ✐♠♣❡❞â♥❝✐❛ Z1 ❡♠ sér✐❡ ❡ ❛ ❛❞♠✐tâ♥❝✐❛
Y1 ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s✱ ❡♠ ❝❛❞❛ ❡①tr❡♠✐❞❛❞❡✳ P❛r❛ ❧✐♥❤❛s ❝✉rt❛s ✭❛té ✷✵✵ ❦♠✮✱ ♠✉❧t✐♣❧✐❝❛✲s❡ ❛
✐♠♣❡❞â♥❝✐❛ ♣❡❧♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❧✐♥❤❛✿
Ze = Z l ✭✺✳✶✺✮
Ye2 =
Y l
2
✭✺✳✶✻✮
❆❝✐♠❛ ❞❡ ✷✵✵ ❦♠✱ ♦ ❡❢❡✐t♦ ❞❛ ♣r♦♣❛❣❛çã♦ t♦r♥❛✲s❡ ♠❛✐s ❡✈✐❞❡♥t❡✱ ♥❡❝❡ss✐t❛♥❞♦ r❡❛❧✐③❛r ✉♠❛
❝♦rr❡çã♦ ❤✐♣❡r❜ó❧✐❝❛✿
Ze = Zc sinh γ l ✭✺✳✶✼✮
Ye2 =
1
Zc
tanh
γ l
2
✭✺✳✶✽✮
♥♦ q✉❛❧ Ye2 ❥á é ❛ ♠❡t❛❞❡ ❞❛ ❛❞♠✐tâ♥❝✐❛ ❞❛ ❧✐♥❤❛✳ ◆❛t✉r❛❧♠❡♥t❡ ♣♦❞❡✲s❡ ✉s❛r ❛ ❢♦r♠✉❧❛çã♦ ❞❡
❧✐♥❤❛ ❧♦♥❣❛ ❞✐r❡t♦ ♣❛r❛ ❧✐♥❤❛s ❝✉rt❛s✳ ❖❜s❡r✈❛✲s❡ t❛♠❜é♠ q✉❡ Zc ❡ γ ❞❡✈❡♠ s❡r ♦s ✈❛❧♦r❡s ♣r❡❝✐s♦s✱
❝♦♥s✐❞❡r❛♥❞♦ ❛s ♣❡r❞❛s✱ ♣❛r❛ ♦❜t❡r✲s❡ ♦s ✈❛❧♦r❡s ❝♦rr❡t♦s ❞❡ Ze ❡ Ye2✳
◆ã♦ ❝♦♥❢✉♥❞❛ ♠♦❞❡❧♦ ❞❡ ❧✐♥❤❛ ❝♦♠ ❛ ♣ró♣r✐❛ ❧✐♥❤❛✳
❖ ♠♦❞❡❧♦ ❞❡ ❧✐♥❤❛ ❧♦♥❣♦ s❡r✈❡ ♣r❛ ❝❛❧❝✉❧❛r ❧✐♥❤❛s ❝✉rt❛s ❡ ❧✐♥❤❛s ❧♦♥❣❛s✱ ♦✉ s❡❥❛✱ ❡①✐st❡
✉♠❛ ♠❛❧ ✐♥t❡r♣r❡t❛çã♦ q✉❡ ❝❛❞❛ ❝♦♠♣r✐♠❡♥t♦ ♣♦ss✉✐ ✉♠ ♠♦❞❡❧♦✦ ❙♦♠❡♥t❡ ♦ ♠♦❞❡❧♦ ❞❡ ❧✐♥❤❛
❝✉rt❛ q✉❡ ♥ã♦ s❡ ❛❞❡q✉❛ ❛ ❧✐♥❤❛s ❧♦♥❣❛s✳
✶✹
Ze
Ye2 Ye2
I1
V1
I2
V2
❋✐❣✉r❛ ✸✿ ❘❡♣r❡s❡♥t❛çã♦ ♣♦r ❡q✉✐✈❛❧❡♥t❡ ♣✐✱ ❝♦♠ ❛s ❝♦♥✈❡♥çõ❡s ❞❡ t❡♥sõ❡s ❡ ❝♦rr❡♥t❡s✳
❖s ♣❛râ♠❡tr♦s Ze ❡ Ye2 sã♦ ♦s ✈❛❧♦r❡s ❛ s❡r❡♠ ✉s❛❞♦s ♣❛r❛ ✉♠ ❡st✉❞♦ ❞❡ r❡❞❡s ❡♠ ❡q✉✐✈❛❧❡♥t❡
♠♦♥♦❢ás✐❝♦✱ ✉t✐❧✐③❛♥❞♦ ♣♦r ❡①❡♠♣❧♦ ❡q✉✐✈❛❧❡♥t❡ ❚❤é✈❡♥✐♥ ❡ ♠❛tr✐③ Ybarra✳
❊✈❡♥t✉❛❧♠❡♥t❡✱ ♣❛r❛ ❞✐❢❡r❡♥❝✐❛r ❞❡♥tr♦ ❞❡ ✉♠ ♠❡s♠♦ ♣r♦❜❧❡♠❛✱ ♣♦❞❡✲s❡ ✉s❛r ❛ ❝♦♥✈❡♥çã♦
❞❡ ❧❡tr❛s ♠✐♥ús❝✉❧❛s ♣❛r❛ ♣❛râ♠❡tr♦s ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✭z ❡♠ ❲✴♠✱ y ❡♠ ❙✴♠✮
❡ ❧❡tr❛s ♠❛✐ús❝✉❧❛s ♣❛r❛ ♣❛râ♠❡tr♦s t♦t❛✐s ✭Z ❡♠ ❲ ❡ Y ❡♠ ❙✮✳ ◆♦✈❛♠❡♥t❡✱ ♠❡s♠♦ s❡♥❞♦
♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✱ s✉♣r✐♠✐✉✲s❡ ♦ ♣♦♥t♦ ✭s❡♥❞♦ ❝♦rr❡t♦ Ż✮✳
P❛r❛ ♦ ♥♦ss♦ ❡①❡♠♣❧♦✱ s✉♣♦♥❞♦ ✉♠❛ ❧✐♥❤❛ ❞❡ ✸✵✵ ❦♠✱ ♦❜té♠✲s❡
Ze = 5, 2343219 + j82, 339206 ❲
Ye2 = 0, 6986822 + j871, 12182 ➭S
❙❡ ✉s❛r♠♦s ❛ ❝♦♥s✐❞❡r❛çã♦ ❞❛ ▲❚ s❡♠ ♣❡r❞❛s✱ ❛s ❝♦rr❡çã♦ ❤✐♣❡r❜ó❧✐❝❛ ♣♦❞❡ s❡r ❢❡✐t❛ ❝♦♠
♠❛✐♦r ❢❛❝✐❧✐❞❛❞❡✿
Ze = 221, 5 sinh
(
j1, 2707936 · 10−6 · 300 · 103
)
= 221, 5j s❡♥ (0, 381238) = j82, 415911 ❲
Ye2 =
1
221, 5
tanh
(
j1, 2707936 · 10−6 300 · 10
3
2
)
=
1
221, 5
j t❣ (0, 190619) = j871, 13386 ➭S
❆s ✜❣✉r❛s ✹ ❡ ✺ ❞❡♠♦♥str❛♠ ❛ ❞✐❢❡r❡♥ç❛ ❞❛ ❝♦rr❡çã♦ ❤✐♣❡r❜ó❧✐❝❛ ♣❛r❛ ♦ ♠♦❞❡❧♦ ❧✐♥❡❛r ♣❛r❛
❡st❡ ❡①❡♠♣❧♦✱ ❛té ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✷✺✵✵ ❦♠✳ ❖❜s❡r✈❛✲s❡ q✉❡ ♣❛r❛ ✉♠ ❝❡r♦ ❝♦♠♣r✐♠❡♥t♦ ❛
r❡❛tâ♥❝✐❛ s❡ ❛♥✉❧❛ ❡ ❛ ❛❞♠✐tâ♥❝✐❛ t❡♥❞❡ ❛♦ ✐♥✜♥✐t♦✱ ♦✉ s❡❥❛ ❛ ❧✐♥❤❛ t♦r♥❛✲s❡ ❛✉t♦✲❝♦♠♣❡♥s❛❞❛✦
✻ ▲✐♠✐t❡s ❞❡ tr❛♥s♠✐ssã♦
❈♦♠♦ t♦❞♦ ❡q✉✐♣❛♠❡♥t♦✱ ✉♠❛ ❧✐♥❤❛ t❡♠ ❧✐♠✐t❡s ♦♣❡r❛t✐✈♦s✱ q✉❡ ♣♦❞❡♠ s❡r ❝♦♥s✐❞❡r❛❞♦s ♣❛r❛
r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡ ♦✉ tr❛♥s✐tór✐♦✳ P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠❛ s✐t✉❛çã♦ ❤✐♣♦tét✐❝❛ ❞❡ ❝✉rt♦✲❝✐r❝✉✐t♦✱
❛ ❧✐♥❤❛ ♣♦❞❡ s✉♣♦rt❛r ♦ ❞♦❜r♦ ❞❡ ❝♦rr❡♥t❡ ♥♦♠✐♥❛❧✱ ♦✉ ♥♦ ❝❛s♦ ❞❡ ✉♠ s✉rt♦ ♦r✐❣✐♥❛❞♦ ♣♦r ✉♠❛
❞❡s❝❛r❣❛ ❛t♠♦s❢ér✐❝❛✱ ♦ ✐s♦❧❛♠❡♥t♦ t♦❧❡r❛ ♠❛✐s q✉❡ ♦ ❞♦❜r♦ ❞❡ t❡♥sã♦ ♥♦♠✐♥❛❧✶✵✳
◆❡st❛ ❛♣♦st✐❧❛ ♣r✐♠❡✐r❛♠❡♥t❡ s❡rá tr❛t❛❞♦ ♦s ❧✐♠✐t❡s ♣❛r❛ ❝♦♥❞✐çã♦ ♥♦♠✐♥❛❧✳ ❯♠❛ r❡❧❛çã♦
❝♦♥❤❡❝✐❞❛ ♣♦r ❈✉r✈❛ ❞❡ ❙t✳ ❈❧❛✐r é ✐❧✉str❛❞❛ ♥❛ ✜❣✉r❛ ✻✱ ♦ q✉❡ ✐♥❞✐❝❛ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦
✶✵❖ ❡st✉❞♦ ❞❡ s♦❜r❡t❡♥sõ❡s tr❛t❛ ♣❡❧♦ ✈❛❧♦r ❞❡ ❝r✐st❛ ✭♦✉ ♣✐❝♦✮ ❡ ❢❛s❡✲♥❡✉tr♦✱ ❡♠ ✈❡③ ❞♦ ✈❛❧♦r ❡✜❝❛③ ✭❘▼❙✮
❢❛s❡✲❢❛s❡✱ ♦✉ s❡❥❛✱ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❞❡
√
2√
3
✶✺
500 1000 1500 2000
l @kmD
100
200
300
400
500
600
XL @WD
❋✐❣✉r❛ ✹✿ ▼♦❞❡❧♦ ❧✐♥❡❛r ❡ ❝♦rr❡çã♦ ❤✐♣❡r❜ó❧✐❝❛ ❞❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ Ze
500 1000 1500 2000
l @kmD
5
10
15
BC @mSD
❋✐❣✉r❛ ✺✿ ▼♦❞❡❧♦ ❧✐♥❡❛r ❡ ❝♦rr❡çã♦ ❤✐♣❡r❜ó❧✐❝❛ ❞❛ ♣❛rt❡ ✐♠❛❣✐♥ár✐❛ ❞❡ Ye2
❞❛ ❧✐♥❤❛ ✐❣✉❛❧ ❛ ♣♦tê♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ ✭❙■▲✮ ♣❛r❛ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✸✵✵ ♠✐❧❤❛s✳
❖s ❧✐♠✐t❡s ❞❛ ❧✐♥❤❛ q✉❡ ♥♦rt❡❛♠ ❡st❡ ❣rá✜❝♦✱ sã♦ ❞✐✈✐❞✐❞♦s ❡♠ três ❝r✐tér✐♦s✱ ❝❛❞❛ ✉♠ ✈á❧✐❞♦
♣❛r❛ ✉♠ ❝♦♠♣r✐♠❡♥t♦✳
✻✳✶ ▲✐♠✐t❡ tér♠✐❝♦
❖ ❧✐♠✐t❡ tér♠✐❝♦ é ❞❡t❡r♠✐♥❛♥t❡ ♣❛r❛ ❧✐♥❤❛s ❝✉rt❛s ✭❛té ✹✵ ❦♠✮✳ ❈♦♥s✐st❡ ❡♠ ❞♦✐s ❡❢❡✐t♦s✿ ♦
❛✉♠❡♥t♦ ❞❛ ✢❡❝❤❛ ♥♦s ❝❛❜♦s✱ r❡❞✉③✐♥❞♦ ❛s ❞✐stâ♥❝✐❛s ❞❡ s❡❣✉r❛♥ç❛ ❝♦♠ ♦ s♦❧♦ ♦✉ ♦✉tr♦s ♦❜❥❡t♦s❀ ❡
❛ ❞❡❣r❛❞❛çã♦ ❞♦ ♠❡t❛❧✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ♦s ❧✐♠✐t❡s ♣r❛t✐❝❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥❛ ♥♦r♠❛
❬✷❪✱ ❡ ♦s ❡st✉❞♦s sã♦ tr❛t❛❞♦s ♥❛ ❛♣♦st✐❧❛ ❞❡ ❝á❧❝✉❧♦ ♠❡❝â♥✐❝♦ ❬✹❪ ♦✉ ❡♠ ❧✐✈r♦s ❝♦♠♦ ❬✶✻❪✳
✶✻
❋✐❣✉r❛ ✻✿ ❈✉r✈❛s ❞❡ ❙t✳ ❈❧❛✐r ❬✷✺❪
Vs = 1 pu
Sr
l
Vr
❋✐❣✉r❛ ✼✿ ❊①❡♠♣❧♦ s♦❜r❡ ❧✐♠✐t❡ ❞❡ tr❛♥s♠✐ssã♦✳
✻✳✷ ▲✐♠✐t❡ ❞❡ r❡❣✉❧❛çã♦
✻✳✸ ▲✐♠✐t❡ ❞❡ ❡st❛❜✐❧✐❞❛❞❡
✼ ▼♦❞❡❧♦ ❞♦ q✉❛❞r✐♣♦❧♦
❯♠ q✉❛❞r✐♣♦❧♦ r❡❧❛❝✐♦♥❛ ❞♦✐s ♣❛r❡s ❞❡ ❣r❛♥❞❡③❛s ❡❧étr✐❝❛s✱ t❡♥sõ❡s ❡ ❝♦rr❡♥t❡s✱ ❛ss♦❝✐❛❞♦s ❛ ❞♦✐s
❜✐♣♦❧♦s✱ ✉♠ ❞❡ ❡♥tr❛❞❛ ❡ ♦✉tr♦ ❞❡ s❛í❞❛✳ ❖ q✉❛❞r✐♣♦❧♦ é ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ❛♦s ♠♦❞❡❧♦s ❝♦♥✈❡♥❝✐♦♥❛✐s
❞❡ ❝✐r❝✉✐t♦s✱ ❛♦♥❞❡ ♣❡❧❛ ❛♣r♦①✐♠❛çã♦ q✉❡ ❞✉❛s ❣r❛♥❞❡③❛s sã♦ ✈❛r✐❛♥t❡s✱ ❞❡t❡r♠✐♥❛✲s❡ ♦ ♦✉tr♦ ♣❛r
❞❡ ❣r❛♥❞❡③❛s✳
❖ ♠♦❞❡❧♦ ❞❡ q✉❛❞r✐♣♦❧♦ ❞❡ ♣❛râ♠❡tr♦s ❣❡♥❡r❛❧✐③❛❞♦s✱ ♦✉ ❆❇❈❉✱ r❡❧❛❝✐♦♥❛ t❡♥sã♦ ❡ ❝♦rr❡♥t❡
❞❡ ❡♥tr❛❞❛✱ V1 ❡ I1✱ ❝♦♠ t❡♥sã♦ ❡ ❝♦rr❡♥t❡ ❞❡ s❛í❞❛✱ V2 ❡ I2✱ ❡♠ ✉♠ ♠♦❞❡❧♦ ♠♦♥♦❢ás✐❝♦✱ ♥♦ q✉❛❧ ❛s
t❡♥sõ❡s ❛♣❧✐❝❛❞❛s sã♦ ❛s ❢❛s❡✲♥❡✉tr♦✳ ❆ ✜❣✉r❛ ✸ ♠♦str❛ ❛ ❝♦♥✈❡♥çã♦ ❞❡ t❡♥sõ❡s ❡ ❝♦rr❡♥t❡s✳ ❯s❛♥❞♦
❛ ❝♦♥✈❡♥çã♦ ❞❛ ❝♦rr❡♥t❡ I1 ❡♥tr❛♥❞♦ ♥♦ q✉❛❞r✐♣♦❧♦ ❡ ❛ ❝♦rr❡♥t❡ I2 s❛✐♥❞♦✿
V1 = AV2 +B I2 ✭✼✳✶❛✮
I1 = C V2 +D I2 ✭✼✳✶❜✮
[
V1
I1
]
= T
[
V2
I2
]
=
[
A B
C D
] [
V2
I2
]
✭✼✳✷✮
✶✼
✼✳✶ ▼♦❞❡❧♦ ❞❡ ❧✐♥❤❛ ❝✉rt❛
❉❡s❡♥✈♦❧✈❡♥❞♦ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡♥tr❛❞❛ ❡ s❛í❞❛✱ ♣❛r❛ ❧✐♥❤❛s ❝✉rt❛s✱ t❡r❡♠♦s
V1 =
(
V2
Y
2
+ I2
)
Z + V2 ✭✼✳✸❛✮
V1 =
(
Z Y
2
+ 1
)
V2 + Z I2 ✭✼✳✸❜✮
I1 = V1
Y
2
+ V2
Y
2
+ I2 ✭✼✳✹❛✮
I1 = V2 Y
(
1 +
Z Y
4
)
+
(
Z Y
2
+ 1
)
I2 ✭✼✳✹❜✮
❈♦♠♣❛r❛♥❞♦ ❝♦♠ ❛s ❡q✉❛çõ❡s ✭✼✳✶✮✱ t❡♠♦s ❝♦♠♦ ♣❛râ♠❡tr♦s
A =
Z Y
2
+ 1 ✭✼✳✺❛✮
B = Z ✭✼✳✺❜✮
C = Y
(
1 +
Z Y
4
)
✭✼✳✺❝✮
D = A ✭✼✳✺❞✮
s❡♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ AD −BC = 1✱ r❡♣r❡s❡♥t❛t✐✈❛ ❞❡ ✉♠ q✉❛❞r✐♣♦❧♦ s✐♠étr✐❝♦✳
✼✳✷ ▼♦❞❡❧♦ ❞❡ ❧✐♥❤❛ ❧♦♥❣❛
P❛r❛ ❧✐♥❤❛s ❧♦♥❣❛s✱ ❞❡s❡♥✈♦❧✈❡✲s❡ ❛s ❡q✉❛çõ❡s ❛ ♣❛rt✐r ❞❛ t❡♦r✐❛ ❞♦ ❡❧❡tr♦♠❛❣♥❡t✐s♠♦ ❬✶✶✱ ♣✳ ✷✶✶❪✱
❝❤❡❣❛♥❞♦ ♥❛ ❢♦r♠❛✿
V1 = V2 cosh(γ l) + I2 Zc sinh(γ l) ✭✼✳✻❛✮
I1 = I2 cosh(γ l) +
V2
Zc
sinh(γ l) ✭✼✳✻❜✮
s❡♥❞♦ ❡♥tã♦ ♦s ♣❛râ♠❡tr♦s ❞♦ q✉❛❞r✐♣♦❧♦✿
A = cosh(γ l) ✭✼✳✼❛✮
B = Zc sinh(γ l) ✭✼✳✼❜✮
C =
1
Zc
sinh(γ l) ✭✼✳✼❝✮
D = A ✭✼✳✼❞✮
s❡♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❧✐♥❤❛s ❧♦♥❣❛s t❛♠❜é♠ é ✈á❧✐❞♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❧✐♥❤❛s ❝✉rt❛s✳
❉♦ ♠♦❞❡❧♦ ❞♦ q✉❛❞r✐♣♦❧♦ é q✉❡ ♣♦❞❡✲s❡ ❝❛❧❝✉❧❛r ♦ ❝✐r❝✉✐t♦ π ❡q✉✐✈❛❧❡♥t❡ ❞❛ ❧✐♥❤❛ ❧♦♥❣❛✳ ❈♦♥✲
s✐❞❡r❛♥❞♦ ♦ ♠❡s♠♦ ❝✐r❝✉✐t♦ ❞❛ ✜❣✉r❛ ✸✱ ❛ ♣❛rt✐r ❞❛s ❡q✉❛çõ❡s ✭✼✳✸✮✱ tr♦❝❛♥❞♦ Z ♣♦r Ze ❡ Y ♣♦r
Ye✿
V1 =
(
Ze Ye
2
+ 1
)
V2 + Ze I2 ✭✼✳✽✮
❖❜t❡♠♦s ❛q✉✐
Ze = Zc sinh(γ l) ✭✼✳✾✮
♣❛r❛ ❛ ❛❞♠✐tâ♥❝✐❛
Ze Ye
2
+ 1 = cosh(γ l) ✭✼✳✶✵✮
Ye Zc sinh(γ l)
2
+ 1 = cosh(γ l) ✭✼✳✶✶✮
Ye
2
=
1
Zc
cosh(γ l)− 1
sinh(γ l)
✭✼✳✶✷✮
✶✽
❛♣r♦✈❡✐t❛♥❞♦✲s❡ ❞❡ ✉♠❛ r❡❧❛çã♦ ❤✐♣❡r❜ó❧✐❝❛✿
tanh
x
2
=
coshx− 1
sinhx
✭✼✳✶✸✮
❝❤❡❣❛♠♦s à r❡❧❛çã♦ ❛♣r❡s❡♥t❛❞❛ ♥❛ ❡q✉❛çã♦ ✭✺✳✶✼✮✿
Ye
2
=
1
Zc
tanh
γ l
2
✭✼✳✶✹✮
❖ ♠♦❞❡❧♦ ♣♦r q✉❛❞r✐♣♦❧♦ ❆❇❈❉ é ❛♣r♦♣r✐❛❞♦ q✉❛♥❞♦ s❡ ❢♦r♥❡❝❡ ❛ t❡♥sã♦ ❡ ❛ ❝♦rr❡♥t❡ ♥♦
r❡❝❡♣t♦r ✭V2 ❡ I2✮✳ P❛r❛ ✉♠❛ ♣♦tê♥❝✐❛ ❛♣❛r❡♥t❡ tr✐❢ás✐❝❛ Ṡ2 = S φ✱ ♣♦❞❡ s❡ ❛r❜✐tr❛r ✉♠❛ t❡♥sã♦
❞❡s❡❥❛❞❛ U0 ❡ ❝❛❧❝✉❧❛r ❛ ❝♦rr❡♥t❡✿
V̇2 =
U0√
3
✭✼✳✶✺❛✮
İ2 =
S2
U0
√
3
−φ ✭✼✳✶✺❜✮
♣♦❞❡♥❞♦ ♣♦r ❡①❡♠♣❧♦ ❡s❝♦❧❤❡r U0 ❛ t❡♥sã♦ ♥♦♠✐♥❛❧ ❞❛ ❧✐♥❤❛✱ s❡♥❞♦ q✉❡ ♥♦ q✉❛❞r✐♣♦❧♦ ❛ t❡♥sã♦
❞❡✈❡ s❡r ❢❛s❡✲t❡rr❛✱ ❡ Ṡ2 = Pc✱ ❛ ♣♦tê♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛✳ ❖✉tr❛s ♦♣çõ❡s sã♦ ❛r❜✐tr❛r ✉♠❛ ❝♦♥❞✐çã♦
❞❡ s♦❜r❡❝❛r❣❛✱ ❝✉rt♦✲❝✐r❝✉✐t♦ ✭V2 = 0✮ ♦✉ ❝✐r❝✉✐t♦ ❛❜❡rt♦ ✭I2 = 0✮✳
❊①❡♠♣❧♦✿ s❡❥❛ ♦ q✉❛❞r✐♣♦❧♦ r❡♣r❡s❡♥t❛t✐✈♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❡ tr❛♥s♠✐ssã♦✱ ❞❡✜♥✐❞♦ ♣♦r
Ȧ = Ḋ = 0, 9672 0, 23➦
Ḃ = 75, 15 83, 2➦ ❲
Ċ = j8, 633 · 10−4 S
❈❛❧❝✉❧❡ ❛s ♣❡r❞❛s ♥❛ ❧✐♥❤❛ ♣❛r❛ ✉♠❛ s❛í❞❛ ❝♦♠ ✹✵✵ ▼❲✱ ✸✹✺ ❦❱✳
❙♦❧✉çã♦✿
V2 =
345√
3
kV I2 = 669, 39 A
❋❛③❡♥❞♦ ❛ ♦♣❡r❛çã♦ ♠❛tr✐❝✐❛❧✱ ♦s ✈❛❧♦r❡s ❡♠ ♠ó❞✉❧♦ sã♦
V1 = 204, 98
√
3 = 355, 04 kV I1 = 670, 55 A
❆ ♣♦tê♥❝✐❛ ❛♣❛r❡♥t❡ s❡ráṠ1 = (412, 32 − j5, 48) ▼❱❆✱ s✉❜tr❛✐♥❞♦ ❛s ♣❛rt❡s r❡❛✐s✱ ∆P =
12, 32 ▼❲✳
❆❧❣✉♠❛s r❡❧❛çõ❡s tr✐❣♦♥♦♠étr✐❝❛s út❡✐s✿
sinh jβ = j s❡♥β
cosh jβ = cosβ
tanh jβ = j t❣β
sinhα = −j s❡♥ jα
coshα = cos jα
tanhα = −j t❣ jα
sinh(α+ jβ) = sinhα cosβ + j coshα s❡♥β
cosh(α+ jβ) = coshα cosβ + j sinhα s❡♥β
▲❡♠❜r❛♥❞♦ s❡♠♣r❡ ❞❡ ❝♦♥s✐❞❡r❛r ♦s ✈❛❧♦r❡s ❡♠ r❛❞✐❛♥♦s✳
✶✾
✼✳✸ ❆ss♦❝✐❛çã♦ ❡♠ ❝❛s❝❛t❛
❆tr❛✈és ❞❛ t❡♦r✐❛ ❞♦s q✉❛❞r✐♣♦❧♦s✱ ♣♦❞❡✲s❡ ❡st✉❞❛r ❛ ❛ss♦❝✐❛çã♦ ❞❡ ❧✐♥❤❛s ❡♠ ❝❛s❝❛t❛✳ ❙❡♥❞♦ ❞♦✐s
q✉❛❞r✐♣♦❧♦s Q1 ❡ Q2✱ ❛ ❛ss♦❝✐❛çã♦ ❡♠ sér✐❡ s❡rá ✐❣✉❛❧ ❛ Q = Q1 ·Q2✱ ♦✉✿
[
V1
I1
]
=
[
A1 B1
C1 D1
]
·
[
A2 B2
C2 D2
] [
V2
I2
]
= ✭✼✳✶✻✮
=
[
A1A2 +B1C2 A1B2 +B1D2
C1A2 +D1C2 C1B2 +D1D2
] [
V2
I2
]
s❡♥❞♦ q✉❡ ❛ ♦r❞❡♠ ❞♦s ❝✐r❝✉✐t♦s é r❡❧❡✈❛♥t❡✱ ❧♦❣♦ ❛ ❛ss♦❝✐❛çã♦ Q′ = Q2 · Q1✳ ❉❡ ♠❛♥❡✐r❛ ❣❡r❛❧✱
Q 6= Q′✳
❆ ❛ss♦❝✐❛çã♦ ❡♠ ❝❛s❝❛t❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ q✉❛❞r✐♣♦❧♦ ❡q✉✐✈❛❧❡♥t❡ ❞❡ ✉♠❛ ▲❚ ❝♦♠
❝♦♠♣❡♥s❛çã♦✳
❊①✳ s❡❥❛ ✉♠❛ ❧✐♥❤❛ ❝♦♠ ♣❛râ♠❡tr♦s ♣♦r ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ z = j0, 34 ❲/❦♠✱
y = j4, 8 ➭❙✴❦♠✱ ✭❛✮ ❝❛❧❝✉❧❡ ♦ q✉❛❞r✐♣♦❧♦ ♣❛r❛ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✻✵✵ ❦♠✱ ♦❜t❡♥❞♦ ♦s
♣❛râ♠❡tr♦s ❞❡ ❡♥tr❛❞❛ ♣❛r❛ ✉♠❛ s❛✐❞❛ ❞❡ ✼✺✵ ❦❱✱ ✷ ●❲✱ ✭❜✮ ❞✐✈✐❞❛ ❛ ❧✐♥❤❛ ❡♠ ❞♦✐s q✉❛❞r✐♣♦❧♦s
❞❡ ✸✵✵ ❦♠✱ ♦❜t❡♥❞♦ ♦ q✉❛❞r✐♣♦❧♦ ❡q✉✐✈❛❧❡♥t❡✱ ✈❡r✐✜❝❛♥❞♦ ❝♦♠ ❛ r❡s♣♦st❛ ❡♠ ✭❛✮✱ ✭❝✮ ❝❛❧❝✉❧❡
♦s ♣❛râ♠❡tr♦s ♥♦ ♠❡✐♦ ❞❛ ❧✐♥❤❛ ❛ ♣❛rt✐r ❞♦s ❝❛❧❝✉❧♦s ❡♠ ✭❜✮✳ ✭❞✮ ❞✐✈✐❞❛ ❛❣♦r❛ ❛ ❧✐♥❤❛ ❡♠ ✶✵
s❡❣♠❡♥t♦s ❡ ❧❡✈❛♥t❡ ♦ ♣❡r✜❧ ❞❡ t❡♥sã♦ ♣❛r❛ ❞✐✈❡rs❛s ❝♦♥❞✐çõ❡s ♦♣❡r❛❝✐♦♥❛✐s ✭❡♠ ✈❛③✐♦✱ ❝❛r❣❛
♥♦♠✐♥❛❧✱ ❡♠ s♦❜r❡❝❛r❣❛✮✳
❙♦❧✉çã♦✿ ♣❛r❛ t♦❞❛s ❛s ❡t❛♣❛s✱ s❡rá ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r ❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ ❡ ❛
❝♦♥st❛♥t❡ ❞❡ ♣r♦♣❛❣❛çã♦✿
Zc = 266, 1453❲
γ = j1, 2775 · 10−3 Np/km
✭❛✮ ♣❛r❛ ✻✵✵ ❦♠✱ ♦ q✉❛❞r✐♣♦❧♦ s❡rá
A = 0, 7203
B = j184, 6❲
C = j2, 606 · 10−3 S
✭❜✮ ♣❛r❛ ✸✵✵ ❦♠✱ ♦❜té♠✲s❡
A = 0, 9275
B = j99, 52❲
C = j1, 405 · 10−3 S
✭❝✮ r❡s♦❧✈❡♥❞♦ ♣❡❧♦ ▼❛t❧❛❜✱ t❡♥❞♦ ♣r❡✈✐❛♠❡♥t❡ ♦s ✈❛❧♦r❡s ❞❡ ✭❜✮ ♥❛ ♠❡♠ór✐❛✿
❃❃ q✸✵✵ ❂ ❬❛ ❜❀ ❝ ❛❪
q✸✵✵ ❂
✵✳✾✷✼✺ ✰ ✵✳✵✵✵✵✐ ✵✳✵✵✵✵ ✰✾✾✳✺✷✶✸✐
✵✳✵✵✵✵ ✰ ✵✳✵✵✶✹✐ ✵✳✾✷✼✺ ✰ ✵✳✵✵✵✵✐
❃❃ q✻✵✵ ❂ q✸✵✵ ✯ q✸✵✵
q✻✵✵ ❂
✶✳✵❡✰✵✷ ✯
✵✳✵✵✼✷ ✰ ✵✳✵✵✵✵✐ ✵✳✵✵✵✵ ✰ ✶✳✽✹✻✵✐
✵✳✵✵✵✵ ✰ ✵✳✵✵✵✵✐ ✵✳✵✵✼✷ ✰ ✵✳✵✵✵✵✐
P❡❧♦ ▼❛t❧❛❜ ♣♦❞❡✲s❡ ✏❡❧❡✈❛r ❛♦ q✉❛❞r❛❞♦✑✱ ♦❜t❡♥❞♦ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ ✭❙❡♠♣r❡ ❝♦♥s✉❧t❡ ♦
♠❛♥✉❛❧ ❞♦ ♣r♦❣r❛♠❛ ♣❛r❛ ❝♦♥st❛t❛r s❡ ✉♠❛ ❞❛❞❛ ❢✉♥çã♦ é ♣♦r ❡❧❡♠❡♥t♦ ♦✉ é ✉♠❛ ♦♣❡r❛çã♦
♠❛tr✐❝✐❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❡❧♦ ▼❛t❧❛❜ ❤á ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✏❫✑ ❡ ✏✳❫✑ ✭❝♦♠ ♣♦♥t♦✮✳✮✿
✷✵
❃❃ q✸✵✵ ❫ ✷
❛♥s ❂
✶✳✵❡✰✵✷ ✯
✵✳✵✵✼✷ ✰ ✵✳✵✵✵✵✐ ✵✳✵✵✵✵ ✰ ✶✳✽✹✻✵✐
✵✳✵✵✵✵ ✰ ✵✳✵✵✵✵✐ ✵✳✵✵✼✷ ✰ ✵✳✵✵✵✵✐
❊①tr❛✐♥❞♦ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❛ ♠❛tr✐③✱ ❆ ♥❛ ♣♦s✐çã♦ ✭✶✱✶✮✱ ❇ ♥❛ ♣♦s✐çã♦ ✭✶✱✷✮✱ ❈ ♥❛ ♣♦s✐çã♦ ✭✷✱✶✮✿
❃❃ q✻✵✵✭✶✱✶✮
❛♥s ❂
✵✳✼✷✵✸
❃❃ q✻✵✵✭✶✱✷✮
❛♥s ❂
✵✳✵✵✵✵❡✰✵✵ ✰ ✶✳✽✹✻✵❡✰✵✷✐
❃❃ q✻✵✵✭✷✱✶✮✯✶❡✸
❛♥s ❂
✵✳✵✵✵✵ ✰ ✷✳✻✵✻✷✐
❈❛❧❝✉❧❛♥❞♦ ❛ t❡♥sã♦ ♥♦ ♠❡✐♦ ❞❛ ❧✐♥❤❛✱ ❛ ♣❛rt✐r ❞❛ s❛í❞❛✿
V2 =
750 · 103√
3
= 433, 0 kV
I2 =
2 · 109
750 · 103
√
3
= 1539, 6 A
❆♣❧✐❝❛♥❞♦ ♦ q✉❛❞r✐♣♦❧♦ ❞❡ ✸✵✵ ❦♠✱ ❡♥❝♦♥tr❛✲s❡ ♥♦ ♠❡✐♦ ❞❛ ❧✐♥❤❛ Vm = 744, 5 20, 88➦ ❦❱✱
Im = 1552, 1 23, 07➦ ❆✳ ❆♣❧✐❝❛♥❞♦ ♠❛✐s ✉♠❛ ✈❡③ ♦ q✉❛❞r✐♣♦❧♦✱ ❡♥❝♦♥tr❛✲s❡ ♥♦ ✐♥í❝✐♦ ❞❛ ❧✐♥❤❛
V1 = 730, 9 42, 34➦ ❦❱✱ I1 = 1582, 2 45, 50➦ ❆✳
✭❞✮ ❈❛❧❝✉❧❛♥❞♦ ♦ q✉❛❞r✐♣♦❧♦ ❞❡ ✉♠❛ s❡çã♦ ❞❡ ✻✵ ❦♠✿
A = 0, 9971
B = j20, 38❲
C = j2, 8772 · 10−4 S
P♦❞❡✲s❡ ❛♣❧✐❝❛r ♦ s❡❣✉✐♥t❡ s❝r✐♣t ♥♦ ▼❛t❧❛❜✿
③ ❂ ✶✐✯✵✳✸✹❀
② ❂ ✶✐✯✹✳✽❡✲✻❀
❧ ❂ ✻✵❀ ✪ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ s❡❝❛♦
③❝ ❂ sqrt✭③ ✴ ②✮❀
❣❛♠❛ ❂ sqrt✭③ ✯ ②✮❀
❛ ❂ ❝♦s❤✭❣❛♠❛ ✯ ❧✮❀
❜ ❂ ③❝ ✯ s✐♥❤✭❣❛♠❛ ✯ ❧✮❀
❝ ❂ ✶✴③❝ ✯ s✐♥❤✭❣❛♠❛ ✯ ❧✮❀
q ❂ ❬❛ ❜❀ ❝ ❛❪❀
✈ ❂ ③❡r♦s✭✶✶✱✶✮❀
✈✭❡♥❞✮ ❂ ✼✺✵❡✸ ✴ sqrt✭✸✮❀
✐✷ ❂ ✷❡✾✴✼✺✵❡✸✴sqrt✭✸✮❀
t♠♣ ❂ ❬✈✭❡♥❞✮❀ ✐✷❪❀
❢♦r ✐✶ ❂ ✶✵✿✲✶✿✶✱
✷✶
t♠♣ ❂ q ✯ t♠♣❀ ✪ ❛♣r♦✈❡✐t❛ ❛ ✈❛r✐❛✈❡❧ ❞❡ ❡♥tr❛❞❛ ♣❛r❛ ❛ ♣r♦①✐♠❛ ✐t❡r❛❝❛♦
✈✭✐✶✮ ❂ t♠♣✭✶✮❀ ✪ ♣♦❞❡✲s❡ ❡①tr❛✐r t❛♠❜❡♠ ❛ ❝♦rr❡♥t❡✱ q✉❡ ❡st❛ ❡♠ t♠♣✭✷✮
❡♥❞
♣❧♦t✭❛❜s✭✈✮✳✯✶❡✲✸✳✯sqrt✭✸✮✮❀ ✪ ❞✐✈✐❞✐♥❞♦ ♣♦r ✶✵✵✵ ♣❛r❛ ❛❝❤❛r ❡♠ ❦❱
②❧❛❜❡❧✭✬❚❡♥s❛♦ ❬❦❱❪✬✮❀
❆ ✜❣✉r❛ ✽ ✐❧✉str❛ ❛❧❣✉♥s ❡①❡♠♣❧♦ ❞❡ ♣❡r✜s ❞❡ t❡♥sã♦✱ ❡♠ ♠ó❞✉❧♦✱ s✉♣♦♥❞♦ ❛ t❡♥sã♦ ❞❡
s❛í❞❛ ❡♠ ✼✺✵ ❦❱✳ ❖❜s❡r✈❛✲s❡ ♣❛r❛ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ s♦❜r❡❝❛r❣❛ ✭✹ ●❲✮ ✉♠❛ q✉❡❞❛ ❞❡ t❡♥sã♦
s✐❣♥✐✜❝❛♥t❡✱ ❡ ♦ ❡❢❡✐t♦ ❋❡rr❛♥t✐ ♣❛r❛ ✉♠❛ s❛í❞❛ ❡♠ ✈❛③✐♦✳
1 2 3 4 5 6 7 8 9 10 11
500
600
700
800
900
1000
1100
1200
T
en
sa
o 
[k
V
]
 
 
P = 0
P = 2 GW
P = 4 GW
❋✐❣✉r❛ ✽✿ ❊①❡♠♣❧♦ ❞❡ ♣❡r✜❧ ❞❡ t❡♥sã♦ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛✱ ❞✐✈✐❞✐❞❛ ❡♠ ✶✵ s❡çõ❡s✱ ♣❛r❛ ✉♠❛ s❛í❞❛
✜①❛ ❞❡ ✼✺✵ ❦❱ ❡ ❞✐✈❡rs❛s ❝♦♥❞✐çõ❡s ❞❡ ❝❛r❣❛✳
✼✳✹ ❆ss♦❝✐❛çã♦ ❡♠ ♣❛r❛❧❡❧♦
❖ q✉❛❞r✐♣♦❧♦ ❡q✉✐✈❛❧❡♥t❡ s❡rá ❞❛❞♦ ♣♦r
[
V1
I1
]
=
[
A1B2+A2B1
B1+B2
B1B2
B1+B2
C1 + C2 +
(A1−A2)(D2−D1)
B1+B2
B2D1+B1D2
B1+B2
]
[
V2
I2
]
✭✼✳✶✼✮
❙❡ tr❛t❛r ❞❡ ❞✉❛s ❧✐♥❤❛s ✐❞ê♥t✐❝❛s✱
[
V1
I1
]
=
[
A B2
2C+ D
] [
V2
I2
]
✭✼✳✶✽✮
✽ ▼♦❞❡❧♦ ❞❡ ✢✉①♦ ❞❡ ♣♦tê♥❝✐❛
P❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❛♣✉r❛❞♦✱ s❡r✐❛ ♥❡❝❡ssár✐♦ ✐♥s❡r✐r ♦ ♠♦❞❡❧♦ ❞❛ ▲❚ ♥♦ ❝♦♥t❡①t♦ ❞❡ ✉♠ s✐st❡♠❛
❞❡ tr❛♥s♠✐ssã♦✱ ❝♦♠ ❜❛rr❛s ❣❡r❛❞♦r❛s ❡ ❝❛r❣❛s✱ ✐♥t❡r❛❣✐♥❞♦ ❡♥tr❡ s✐✳ ❉❡ ❢♦r♠❛ s✐♠♣❧✐✜❝❛❞❛✱ ♣♦❞❡✲s❡
❛r❜✐tr❛r ❞✉❛s ❜❛rr❛s✱ ❛♦♥❞❡ ♥♦ ♠♦❞❡❧♦ ❞♦ q✉❛❞r✐♣♦❧♦ ❛ss✉♠❡✲s❡ ✉♠❛ ❜❛rr❛ ♣❛ss✐✈❛✱ ❝♦♠ t❡♥sã♦ ❡
❝♦rr❡♥t❡ ❝♦♥❤❡❝✐❞♦s✳ ❖✉tr❛ ❢♦r♠❛ ♣rát✐❝❛ ❞❡ ❡st✉❞❛r é ❛ss✉♠✐r ❞✉❛s ❜❛rr❛s ✑❢♦rt❡s➫➫✱ ❝♦♠ t❡♥sõ❡s
❞❡✜♥✐❞❛s✱ ❝❛❧❝✉❧❛♥❞♦✲s❡ ❛s ❝♦rr❡♥t❡s ❡ ♣♦tê♥❝✐❛s✳
✷✷
❙❡❥❛ ✉♠❛ ❧✐♥❤❛ ❝♦♥❡❝t❛♥❞♦ ❞✉❛s ❜❛rr❛s ❝♦♠ t❡♥sõ❡s ❞❡✜♥✐❞❛s✱ V1 ❡ V2✱ ❝✉❥♦ ♠ó❞✉❧♦s ❡ â♥❣✉❧♦s
♥ã♦ s❡❥❛♠ ❛❧t❡r❛❞♦s ♣❡❧❛ ✐♥s❡rçã♦ ❞❛ ❧✐♥❤❛✱ ❛ ❝♦rr❡♥t❡ ❡♥tr❡ ❛s ❜❛rr❛s s❡rá ❞❡t❡r♠✐♥❛❞❛ ❜❛s✐❝❛♠❡♥t❡
♣❡❧❛ ✐♠♣❡❞â♥❝✐❛ ❧♦♥❣✐t✉❞✐♥❛❧ ✭✉s❛♥❞♦ t❡♥sã♦ ❞❡ ❢❛s❡✮✱ ❛r❜✐tr❛♥❞♦ ♦ ✢✉①♦ ❞❛ ❜❛rr❛ ✶ ♣❛r❛ ✷✿
İ =
V1 − V2
Ż
√
3
✭✽✳✶✮
s❡♥❞♦ ❡st❛ ❝♦rr❡♥t❡ q✉❡ ❞❡t❡r♠✐♥❛rá ❛s ♣❡r❞❛s ❡ ♣❛rt❡ ❞♦ r❡❛t✐✈♦✳ ❖✉tr❛ ♣❛rt❡ s✐❣♥✐✜❝❛♥t❡ ❞♦ r❡❛t✐✈♦
❡st❛rá ♥❛ ❛❞♠✐tâ♥❝✐❛✱ s✉♣♦♥❞♦ ❡st❛ ❝♦♥❝❡♥tr❛❞❛ ❡♠ ❝❛❞❛ ❜❛rr❛✱ ♦❜té♠✲s❡ ❛ ❝♦rr❡♥t❡ ❡❢❡t✐✈❛ q✉❡
❡♥tr❛ ♦✉ s❛✐ ❞❡ ❝❛❞❛✱ I1 ❡ I2✿
I1 = I + IY 2 ✭✽✳✷❛✮
I2 = I − IY 2 ✭✽✳✷❜✮
❊①❡♠♣❧♦✿ ❈❛❧❝✉❧❡ ❛ ♣♦tê♥❝✐❛ tr❛♥s♠✐t✐❞❛ ❡ ♣❡r❞❛s ❡♠ ✉♠❛ ▲❚✱ ✸✹✺ ❦❱✱ ✐♠♣❡❞â♥❝✐❛ t♦t❛❧
❞❡ 6 + j50 ❲✱ ❛s ❜❛rr❛s ❝♦♠ t❡♥sõ❡s ✭❢❛s❡✲❢❛s❡✮ V1 = 345 0➦ ❦❱ ❡ V2 = 320 −10➦ ❦❱✳
❙♦❧✉çã♦✿ ▲❡♠❜r❛♥❞♦ ❡♠ ❝♦♥✈❡rt❡r V1 ❡ V2 ♣❛r❛ t❡♥sõ❡s ❢❛s❡✲♥❡✉tr♦✱ ♦✉ ❝♦♥✈❡rt❡♥❞♦ ❞✐r❡t♦
♥❛ ❡q✉❛çã♦✿
İ =
V̇1 − V̇2
Ż
√
3
= 723, 32 −21, 4➦ A
P♦❞❡✲s❡ ❝❛❧❝✉❧❛r ❛ ♣❡r❞❛ ❝♦♠♦ ∆P = 3RI2 = 9, 4 ▼❲✳
❆ ♣♦tê♥❝✐❛ tr❛♥s♠✐t✐❞❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ♣♦r Ṡ2 = 3 V̇2 İ∗ = (392, 93 + j79, 30) ▼❱❆✳
✭s❡♥❞♦ ❡ss❡ r❡❛t✐✈♦ s♦♠❡♥t❡ ♣❡❧❛ ♣❛rt❡ ❞♦ ▲ ❞❛ ❧✐♥❤❛✮✳ ❖✉ ❢❛③❡♥❞♦ ♣❡❧❛ ❢ór♠✉❧❛ ❛♣r♦①✐♠❛❞❛✱
P =
|V1| · |V2|
X
s❡♥ δ ∼= 383, 4 MW
✾ ❈♦♠♣❡♥s❛çã♦ ❞❡ ❧✐♥❤❛s
❆ ❝♦♠♣❡♥s❛çã♦ ❞❡ r❡❛t✐✈♦ ❡♠ ✉♠❛ ❧✐♥❤❛ ❝♦♥s✐st❡ ❡♠ ❜❛❧❛♥ç❛r ❛ ✐♠♣❡❞â♥❝✐❛ ♦✉ ❛ ❛❞♠✐tâ♥❝✐❛ ❝♦♠
❝❛♣❛❝✐t♦r❡s ❡♠ sér✐❡ ♦✉ r❡❛t♦r❡s ❡♠ ♣❛r❛❧❡❧♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡❧étr✐❝♦✱ ♦ ❡❢❡✐t♦
s❡rá ❞❡ ✏❡♥❝✉rt❛r✑ ❛ ❧✐♥❤❛✳
❈❛❞❛ t✐♣♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ é ❡s♣❡❝í✜❝❛ ♣❛r❛ ✉♠❛ ❝♦♥❞✐çã♦ ❞❛ ▲❚✿ ❛ ❝♦♠♣❡♥s❛çã♦ sér✐❡ é ❡s♣❡✲
❝í✜❝❛ ♣❛r❛ ❛ ❝♦♥❞✐çã♦ ❞❡ ♣❧❡♥❛ ❝❛r❣❛ ❡ ❛ ❝♦♠♣❡♥s❛çã♦ s❤✉♥t ♣❛r❛ ❛ ❧✐♥❤❛ ❡♠ ✈❛③✐♦✳ ❋♦r❛ ❞❡st❛s
❝♦♥❞✐çõ❡s✱ ❛ ❝♦♠♣❡♥s❛çã♦ t♦r♥❛✲s❡ ✉♠ ❡①❝❡ss♦ ❞❡ r❡❛t✐✈♦✱ ♠❛s ♦ s❡✉ ❝❤❛✈❡❛♠❡♥t♦ r❛r❛♠❡♥t❡ é
❛♣r♦♣r✐❛❞♦✳
❆ s♦❧✉çã♦ é ♦ ✉s♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝♦♠♣❡♥s❛çã♦ ❛t✐✈❛✱ s❡❥❛ r❡❛t♦r❡s ♦✉ ❝❛♣❛❝✐t♦r❡s ❝❤❛✈❡❛❞♦s
♣♦r ❡❧❡trô♥✐❝❛✱ ♦✉ ❛té ❡❧❡♠❡♥t♦s ❡❧❡trô♥✐❝♦s q✉❡ ❝♦♥tr♦❧❛♠ ❞✐r❡t❛♠❡♥t❡ ♦s r❡❛t✐✈♦s✳ ❉❡✈✐❞♦ ❛♦
❝✉st♦ ❡❧❡✈❛❞♦ ❞❡st❛s s♦❧✉çõ❡s✱ ♣♦❞❡✲s❡ t❛♠❜é♠ ✉t✐❧✐③❛r ❝♦♥✜❣✉r❛çõ❡s ♠✐st❛s ❞❡ ❡❧❡♠❡♥t♦s ♣❛ss✐✈♦s
❡ ❛t✐✈♦s✳ ▼❛✐♦r❡s ❞❡t❛❧❤❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶✺✱ ♣✳ ✻✷✼❪✳
P❛r❛ ❧✐♥❤❛s ♠✉✐t♦ ❧♦♥❣❛s✱ ❛ ❝♦♠♣❡♥s❛çã♦ é ❞✐str✐❜✉í❞❛ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛✱ ❝r✐❛♥❞♦✲s❡ s✉❜❡st❛çõ❡s
✐♥t❡r♠❡❞✐ár✐❛s✳
✾✳✶ ❈♦♠♣❡♥s❛çã♦ sér✐❡
❈♦♥s✐st❡ ❡♠ r❡❞✉③✐r ❛ r❡❛tâ♥❝✐❛ ❧♦♥❣✐t✉❞✐♥❛❧ ❞❛ ❧✐♥❤❛ ✉t✐❧✐③❛♥❞♦✲s❡ ❝❛♣❛❝✐t♦r❡s sér✐❡✱ r❡❞✉③✐♥❞♦ ❛
✐♠♣❡❞â♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡✳ ❖ ❡❢❡✐t♦ s❡rá ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠ ❡♥❝✉rt❛♠❡♥t♦ ❡❧étr✐❝♦✱ ❡❧❡✈❛♥❞♦ ❛ ❝❛♣❛❝✐✲
❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦✳
❙❡❥❛ ✉♠❛ ▲❚ ❝♦♠ ✉♠❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ Zc ♥♦ q✉❛❧
Zc ∼=
√
L
C
=
√
Xl
Bc
✭✾✳✶✮
❛ ❝♦♠♣❡♥s❛çã♦ sér✐❡ s❡rá ♣r♦♣♦r❝✐♦♥❛❧ à r❡❛tâ♥❝✐❛ ❧♦♥❣✐t✉❞✐♥❛❧✱ ♥❛ ❢♦r♠❛
Xc = ns Xl ✭✾✳✷✮
s❡♥❞♦ ns ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❝♦♠♣❡♥s❛çã♦ sér✐❡✳ ❉❡s❡♥✈♦❧✈❡♥❞♦✱ ♣♦❞❡✲s❡ ❞❡s❝r❡✈❡r ❛ ♥♦✈❛ ✐♠♣❡❞â♥❝✐❛
❝❛r❛❝t❡ríst✐❝❛ ♥❛ ❢♦r♠❛
Z ′c
∼=
√
Xl −Xc
Bc
= Zc
√
1− ns ✭✾✳✸✮
✷✸
❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣❛❣❛çã♦
β′ ∼= β
√
1− ns ✭✾✳✹✮
❖ ✉s♦ ❞❡ ❝❛♣❛❝✐t♦r❡s sér✐❡ ❞❡✈❡ s❡r ❢❡✐t♦ ❝✉✐❞❛❞♦s❛♠❡♥t❡ ♥❛ ♣r♦①✐♠✐❞❛❞❡ ❞❡ ✉s✐♥❛s✱ ❞❡✈✐❞♦ ❛♦
❡❢❡✐t♦ ❞❡ r❡ss♦♥â♥❝✐❛ s✉❜sí♥❝r♦♥❛ ✭♦✉ ❙❙❘ ✲ s✉❜s②♥❝❤r♦♥♦✉s r❡ss♦♥❛♥❝❡✮✳
❱❛♥t❛❣❡♥s ❡ ❞❡s✈❛♥t❛❣❡♥s✿
❼ ❆✉♠❡♥t❛ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ tr❛♥s♠✐ssã♦
❼ ❈♦♠♣❡♥s❛ ❛ ✐♥❞✉tâ♥❝✐❛❞❛ ❧✐♥❤❛ ✭XL −XC✮
❼ ❆♣r♦①✐♠❛ ❡❧❡tr✐❝❛♠❡♥t❡ ❛s ❜❛rr❛s✱ ❛✉♠❡♥t❛♥❞♦ ❛ ❡st❛❜✐❧✐❞❛❞❡
❼ ❊❧❡✈❛ ❛ t❡♥sã♦ ❞❡ ✉♠❛ ❧✐♥❤❛ ❝❛rr❡❣❛❞❛
❼ P♦❞❡ ♦r✐❣✐♥❛r ❡♠ r❡ss♦♥â♥❝✐❛s s✉❜✲sí♥❝r♦♥❛s ✭❙❙❘✮ ❝♦♠ ❛s ♠áq✉✐♥❛s ❣❡r❛❞♦r❛s✱ ❡♠ ❣❡r❛❧ ❡♠
♠áq✉✐♥❛s tér♠✐❝❛s✳
❼ ❖r✐❣✐♥❛ s♦❜r❡t❡♥sõ❡s ✈✐♦❧❡♥t❛s✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ✉♠❛ ♣r♦t❡çã♦ ❡s♣❡❝í✜❝❛ ✭❝❡♥t❡❧❤❛❞♦r❡s✱ ❞✐s✲
❥✉♥t♦r ❞❡ ❜②♣❛ss✱ ♣ár❛✲r❛✐♦s✮
❼ ❊q✉✐♣❛♠❡♥t♦ ♣❡s❛❞♦ q✉❡ ❡♥❝♦♥tr❛✲s❡ ♥♦ ♣♦t❡♥❝✐❛❧ ❞❛ ❧✐♥❤❛✱ s❡♥❞♦ ♥❡❝❡ssár✐♦ ✉♠❛ ❡str✉t✉r❛
❣r❛♥❞❡ ❞❡ s✉st❡♥t❛çã♦✳
❯♠ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡❝♥♦❧♦❣✐❛ é ♦ ❚❈❙❈ ✭❚②r✐st♦r ❝♦♥tr♦❧❧❡❞ ❙❡r✐❡s ❈❛♣❛❝✐t♦r✮ ♥♦ q✉❛❧ s✉❛
❝❛♣❛❝✐tâ♥❝✐❛ ✈❛r✐á✈❡❧ ♣♦❞❡ ♠✐♥✐♠✐③❛r ♦s ♣r♦❜❧❡♠❛s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞❡ ❙❙❘✳
❋✐❣✉r❛ ✾✿ ❈♦♥✜❣✉r❛çã♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ sér✐❡ ❡ ❚❈❙❈
✾✳✷ ❈♦♠♣❡♥s❛çã♦ ♣❛r❛❧❡❧❛ ✭s❤✉♥t✮
❆ ❝♦♠♣❡♥s❛çã♦ ❡♠ ❣❡r❛❧ é ❡s♣❡❝✐✜❝❛❞❛ ❡♠ ✉♠ ♣❡r❝❡♥t✉❛❧ r❡❧❛t✐✈♦ à ✐♠♣❡❞â♥❝✐❛ ♦✉ ❛❞♠✐tâ♥❝✐❛ ❞❛
❧✐♥❤❛✳ P♦❞❡✲s❡✱ ❛ ❣r♦ss♦ ♠♦❞♦✱ s✉❜tr❛✐r ❛s r❡❛tâ♥❝✐❛s ❞❛ ❧✐♥❤❛ ❝♦♠ ❛ ❞❛ ❝♦♠♣❡♥s❛çã♦ ♣❛r❛ ♦❜t❡r ♦
❡q✉✐✈❛❧❡♥t❡✳ ◆❛ ♣rát✐❝❛✱ ♦s ♠ó❞✉❧♦s ❞❡ ❝♦♠♣❡♥s❛çã♦ s❡rã♦ ✐♥st❛❧❛❞♦s ♥❛s ❡①tr❡♠✐❞❛❞❡s ❞❛ ❧✐♥❤❛✱
❞❡♥tr♦ ❞❛s s✉❜❡st❛çõ❡s✳
✾✳✸ ▼♦❞❡❧♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ ♣♦r q✉❛❞r✐♣♦❧♦s
❯♠ ♠ó❞✉❧♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ sér✐❡✴ ♣❛r❛❧❡❧♦ t❛♠❜é♠ ♣♦❞❡ s❡r ♠♦❞❡❧❛❞♦ ❝♦♠♦ ❝✐r❝✉✐t♦ ✉♠ ❝♦♠♦
✉♠ q✉❛❞r✐♣♦❧♦✳ ❯♠ ❝❛♣❛❝✐t♦r sér✐❡ Cs t❡r✐❛ ❝♦♠♦ ♣❛râ♠❡tr♦s ❆❇❈❉✿
A = 1 ✭✾✳✺❛✮
B =
1
j ω Cs
= −j ns Xl ✭✾✳✺❜✮
C = 0 ✭✾✳✺❝✮
D = 1 ✭✾✳✺❞✮
✷✹
❯♠ r❡❛t♦r s❤✉♥t Lp s❡r✐❛
A = 1 ✭✾✳✻❛✮
B = 0 ✭✾✳✻❜✮
C =
1
j ω Lp
= −j np Bc ✭✾✳✻❝✮
D = 1 ✭✾✳✻❞✮
s❡♥❞♦ np ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❝♦♠♣❡♥s❛çã♦ ♣❛r❛❧❡❧❛✳
Ze = R + j Xl
Ye = j Bc
I1
V1
I2
V2
- j Xc - j Xc
- j Bl- j Bl
❋✐❣✉r❛ ✶✵✿ ❘❡♣r❡s❡♥t❛çã♦ ♣♦r q✉❛❞r✐♣♦❧♦ ❞❡ ❝♦♠♣❡♥s❛çã♦ sér✐❡ ❡ ♣❛r❛❧❡❧♦ ❡♠ ❝❛❞❛ ❡①tr❡♠✐❞❛❞❡✳
❱❡♥❞♦ ❝♦♠♦ ❡①❡♠♣❧♦ ❛ ✜❣✉r❛ ✶✵✱ ✉s❛♥❞♦ ❛♠❜❛s ❛s ❝♦♠♣❡♥s❛çõ❡s✱ s❡♥❞♦ QLT ♦ q✉❛❞r✐♣♦❧♦
♦r✐❣✐♥❛❧ ❞❛ ❧✐♥❤❛✱ Qc ♦ q✉❛❞r✐♣♦❧♦ ❞♦ ❝❛♣❛❝✐t♦r sér✐❡ ❡ Ql ♦ q✉❛❞r✐♣♦❧♦ ❞♦ r❡❛t♦r s❤✉♥t✱ ♦
q✉❛❞r✐♣♦❧♦ ❡q✉✐✈❛❧❡♥t❡ s❡rá
Q = Qc ·Ql ·QLT ·Ql ·Qc
r❡s♣❡✐t❛♥❞♦✲s❡ ❛ ♦r❞❡♠ ❞♦s ❡❧❡♠❡♥t♦s ❞♦ ❝✐r❝✉✐t♦✳
❊①❡♠♣❧♦✿ ❊s♣❡❝✐✜q✉❡ ♦ ❜❛♥❝♦ ❞❡ r❡❛t♦r❡s ✭q✉❛♥t✐❞❛❞❡✱ t❡♥sã♦✱ ♣♦tê♥❝✐❛✱ ✐♥❞✉tâ♥❝✐❛ ❡
❧✐❣❛çã♦ ✕ ❞❡❧t❛ ♦✉ ❡str❡❧❛✮✱ ♣❛r❛ ✉♠❛ ❝♦♠♣❡♥s❛çã♦ s❤✉♥t ❞❡ ✸✵✪✱ ♣❛r❛ ✉♠❛ ❧✐♥❤❛ ❞❡ ✸✹✺ ❦❱
❝♦♠ ❡q✉✐✈❛❧❡♥t❡ ❞❡ Z = 10 + j250 ❲✱ Y = j10 ♠❙✳ ❆ss✉♠❛ q✉❡ ❛s ✉♥✐❞❛❞❡s sã♦ ♠♦♥♦❢ás✐❝❛s✳
✾✳✹ ❈♦♠♣❡♥s❛çã♦ ❞✐♥â♠✐❝❛
❖ ♣r♦❥❡t♦ ❞❛ ❝♦♠♣❡♥s❛çã♦ ♥✉♥❝❛ ❝♦♥t❡♠♣❧❛rá t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♦♣❡r❛❝✐♦♥❛✐s✱ ♦✉ s❡❥❛✱ ♦s
❡q✉✐♣❛♠❡♥t♦s ❡st❛rã♦ ❝❛❧✐❜r❛❞♦s s♦♠❡♥t❡ ♣❛r❛ ✉♠❛ ❝♦♥❞✐çã♦✱ ❡♠ ❣❡r❛❧ ♥❛ ♠é❞✐❛✳ ❖ ✉s♦ ❞❡ ❝♦♠✲
♣❡♥s❛çã♦ ✈❛r✐á✈❡❧ ♣❡r♠✐t❡ ❡❧❡✈❛r ❛ ❡✜❝✐ê♥❝✐❛✳
❖ ❝❤❛✈❡❛♠❡♥t♦ ♠❡❝â♥✐❝♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ❝♦♠♣❡♥s❛çã♦ s❡♠♣r❡ é ♣r♦❜❧❡♠át✐❝♦✱ ❞❡✈✐❞♦ ❛♦ s✉r❣✐✲
♠❡♥t♦ ❞❡ s♦❜r❡t❡♥sõ❡s✳ ❖ ✉s♦ ❞❡ ❡❧❡trô♥✐❝❛ ❞❡ ♣♦tê♥❝✐❛ ♣❡r♠✐t❡ ✉♠ ❝❤❛✈❡❛♠❡♥t♦ s✉❛✈❡✳ ❆❧❣✉♠❛s
t❡❝♥♦❧♦❣✐❛s sã♦✿
❼ ❙t❛t✐❝ ❱❛r ❈♦♠♣❡♥s❛t♦r ✭❙❱❈✮✿ ❈♦♠♣♦st♦ ♣♦r ✉♠ r❡❛t♦r ❡ ✉♠ ❜❛♥❝♦ ❞❡ ❝❛♣❛❝✐t♦r❡s✱ ❛♠❜♦s
❡♠ ♣❛r❛❧❡❧♦✱ ❝♦♥tr♦❧❛❞♦s ♣♦r t✐r✐st♦r❡s✳
❼ ❚❤②r✐st♦r ❈♦♥tr♦❧❧❡❞ ❙❡r✐❡s ❈❛♣❛❝✐t♦r ✭❚❈❙❈✮✿ ❇❛♥❝♦ ❞❡ ❝❛♣❛❝✐t♦r❡s sér✐❡ ❡♠ ♣❛r❛❧❡❧♦ ❝♦♠
✉♠ r❡❛t♦r✱ ❝❤❛✈❡❛❞♦ ♣♦r t✐r✐st♦r❡s✳
❼ ❙t❛t✐❝ ❈♦♠♣❡♥s❛t♦r ✭❙❚❆❚❈❖▼✮✿
❼ ❙t❛t✐❝ ❙②♥❝❤r♦♥♦✉s ❙❡r✐❡s ❈♦♠♣❡♥s❛t♦r ✭❙❙❙❈✮
❆ ❝♦♠♣❡♥s❛çã♦ ❞✐♥â♠✐❝❛ ♣♦❞❡ s❡r ♣❡r❢❡✐t❛♠❡♥t❡ ❝♦♠❜✐♥❛❞❛ ❝♦♠ ✉♠ ❜❛♥❝♦ ❞❡ ❝♦♠♣❡♥s❛çã♦
✜①❛✱ ♦t✐♠✐③❛♥❞♦ ♦s ❝✉st♦s✳
✷✺
✶✵ ❈á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❡❧étr✐❝♦s ✲ ♠♦❞❡❧♦ ❞❡t❛❧❤❛❞♦
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛✲s❡ ✉♠ ♠♦❞❡❧♦ q✉❡ ✐♥❝♦r♣♦r❛ ❡❧❡♠❡♥t♦s ❛❞✐❝✐♦♥❛✐s✱ ❝✉❥❛ ✐♥✢✉ê♥❝✐❛ ♣♦❞❡ s❡r
❞❡t❡r♠✐♥❛♥t❡ ❡♠ ❝❡rt❛s ❝♦♥❞✐çõ❡s ❡ ❡st✉❞♦s✳
✶✵✳✶ ▼♦❞❡❧♦ ❞❡ ✐♠♣❡❞â♥❝✐❛ ♣ró♣r✐❛✱ ❝♦♥s✐❞❡r❛♥❞♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r
❆ ♣r❡♠✐ss❛ ❞❡ ❝♦rr❡♥t❡ ✉♥✐❢♦r♠❡ ♥❛ ❡q✉❛çã♦ ✹✳✻ é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ✉s✉❛❧✱ ♣♦ré♠ ♣♦✉❝♦ ✉s❛❞❛ ♥❛
♣rát✐❝❛✳ P❛r❛ ✐♥❝♦r♣♦r❛r ♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r ♥♦ ❝á❧❝✉❧♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ✐♥t❡r♥❛✱ é ♥❡❝❡ssár✐♦ r❡s♦❧✈❡r
✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❬✶✼❪✱ ❝✉❥♦ r❡s✉❧t❛❞♦ é ✐❣✉❛❧ ❛
Zi =
j ω µ
2π ρ
I0(ρ)
I1(ρ)
✭✶✵✳✶✮
ρ = r
√
−j ω σ µ ✭✶✵✳✷✮
s❡♥❞♦ I0 ❡ I1 ❛s ❢✉♥çõ❡s ❞❡ ❇❡ss❡❧ ❞❡ ♣r✐♠❡✐r❛ ❡ s❡❣✉♥❞❛ ❡s♣é❝✐❡✶✶✱ σ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❞♦ ♠❛t❡r✐❛❧✱
❡ µ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛✳ ❊st❛ ❢ór♠✉❧❛ é ✈á❧✐❞❛ ♣❛r❛ ❝♦♥❞✉t♦r❡s ❞❡ s❡çã♦ ❝✐r❝✉❧❛r✱ ❡ ❥á
❢♦r♥❡❝❡ ❞✐r❡t❛♠❡♥t❡ ❛ r❡s✐stê♥❝✐❛ ❡ ❛ r❡❛tâ♥❝✐❛✳
P❛r❛ ❝❛❜♦s ❝♦♠♣♦st♦s✱ ♣♦❞❡✲s❡ ❞❡s♣r❡③❛r ♦ ❡❢❡✐t♦ ❞♦ ♠❛t❡r✐❛❧ ❞♦ ♥ú❝❧❡♦✱ ❝♦♥s✐❞❡r❛♥❞♦ s♦♠❡♥t❡
♦ ♠❛t❡r✐❛❧ ❞❛ ❝♦r♦❛✳ ❯♠ ❝á❧❝✉❧♦ ♠❛✐s ♣r❡❝✐s♦ ❝♦♥s✐❞❡r❛ ♦ ❝♦♥❞✉t♦r ❝♦♠♦ ✉♠ t✉❜♦✱ ❝♦♥❢♦r♠❡
❞❡s❝r✐t♦ ♥♦ ❛♥❡①♦ ❈✳✸✳ ❖ ✈❛❧♦r r❡❛❧ ❞❛ ✐♠♣❡❞â♥❝✐❛ ♣ró♣r✐❛ s❡rá ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ❛♦s ✈❛❧♦r❡s
❡♥❝♦♥tr❛❞♦s ❡♠ t❛❜❡❧❛✳
P❛r❛ ❛ ❝♦rr❡çã♦ ❞❛ r❡s✐stê♥❝✐❛ ♣❡❧❛ t❡♠♣❡r❛t✉r❛✱ ❛❥✉st❛✲s❡ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❞♦ ♠❛t❡r✐❛❧✱ s❡♥❞♦
♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ✈❛r✐❛çã♦ α ✭♥ã♦ ❝♦♥❢✉♥❞✐r ❝♦♠ ♦ ❝♦❡✜❝✐❡♥t❡ ❞❡ ❞✐❧❛t❛çã♦✮✿
σf = σ0[1 + α(θ0 − θ)] ✭✶✵✳✸✮
s❡♥❞♦ σ0 ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❞❡ r❡❢❡rê♥❝✐❛ ❡ θ0 ❛ t❡♠♣❡r❛t✉r❛ ♥♦ q✉❛❧ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ✐♥✐❝✐❛❧ s❡ r❡❢❡r❡✳
❖ ❝á❧❝✉❧♦ ❞❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ s❡rá
Z =


Zia 0 0
0 Zib 0
0 0 Zic

+ j ω
µ
2π
M ✭✶✵✳✹✮
M =



ln 2hara ln
Dab
dab
ln Dacdac
ln Dbadba ln
2hb
rb
ln Dbcdbc
ln Dcadca ln
Dcb
dcb
ln 2hcrc



✭✶✵✳✺✮
❡ ♥ã♦ é ♠❛✐s ♥❡❝❡ssár✐♦ ✉s❛r ♦ r❛✐♦ ❝♦rr✐❣✐❞♦ r′✱ ♣♦✐s s❡✉ ❡❢❡✐t♦ ❡stá ✐♥❝❧✉s♦ ♥♦s ❡❧❡♠❡♥t♦s Zi✱ ❡ ❛
♠❛tr✐③ M t♦r♥❛✲s❡ ú♥✐❝❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ❡ ❞❛ ❛❞♠✐tâ♥❝✐❛✳
P❛r❛ ✉♠❛ ❧✐♥❤❛ ❝♦♠ ❢❡✐①❡ ❞❡ ❝♦♥❞✉t♦r❡s✱ ❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ s❡rá ❢♦r♠❛❞❛ ♣♦r ❝❛❞❛ s✉❜❝♦♥✲
❞✉t♦r✳ P♦r ❡①❡♠♣❧♦✱ ✉♠❛ ❧✐♥❤❛ tr✐❢ás✐❝❛ ❝♦♠ ❢❛s❡s a✱ b ❡ c✱ ❝♦♠ ❝❛❞❛ ❢❡✐①❡ ❝♦♠ n s✉❜❝♦♥❞✉t♦r❡s✿
Z =













Za11 Za12 Za1n Za1b1 · · · Za1c1 Za1cn
Za21 Za22 Za2n · · ·
✳✳✳
Zan1 Zan2 Zann
Zb1a1
✳✳✳ Zb11 Zb1cn
✳✳✳
✳ ✳ ✳
Zc1a1 Zc11
Zcna1 · · · Zcnn













✭✶✵✳✻✮
♦❜s❡r✈❛✲s❡ q✉❡ é ❝♦♥s✐❞❡r❛❞♦ ♦ ❡❢❡✐t♦ ❡♥tr❡ ❝❛❞❛ s✉❜❝♦♥❞✉t♦r✱ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✳ P♦❞❡✲s❡ ♣❛rt✐❝✐♦♥❛r
❛ ♠❛tr✐③ ♣❡❧❛s ❢❛s❡s✱ s❡♥❞♦ ❝❛❞❛ s✉❜♠❛tr✐③ ❝♦♠ n× n ❡❧❡♠❡♥t♦s✶✷✿
Z =


Zaa Zab Zac
Zba Zbb Zbc
Zca Zcb Zcc

 ✭✶✵✳✼✮
♥♦ ✜♥❛❧ q✉❡r❡♠♦s r❡❞✉③✐r ❡st❛ ♠❛tr✐③ ♣❛r❛ ✉♠ ❡q✉✐✈❛❧❡♥t❡ ♣♦r ❢❛s❡✱ 3× 3✳
✶✶■♠♣❧❡♠❡♥t❛❞♦ ♥♦ ▼❛t❧❛❜ ❡ ❙❝✐❧❛❜ ❝♦♠♦ ❜❡ss❡❧✐✭✵✱①✮ ❡ ❜❡ss❡❧✐✭✶✱①✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
✶✷◆ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝❛❞❛ ❢❛s❡ t❡♠ q✉❡ t❡r ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ s✉❜❝♦♥❞✉t♦r❡s✱ ♣♦r ❡st❡ ♠ét♦❞♦ ♣♦❞❡✲s❡ t❡r
q✉❛❧q✉❡r ♣♦ss✐❜✐❧✐❞❛❞❡✱ só ♥ã♦ é ❡①♣♦st❛ ✉♠❛ ❢♦r♠❛ ✏t♦t❛❧♠❡♥t❡ ❣❡♥ér✐❝❛✑ ♣♦rq✉❡ s❡r✐❛ ✏✐♥♦✈❛çã♦ ❡♠ ❡①❝❡ss♦✑✳✳✳
✷✻
✶✵✳✷ ❘❡s✐stê♥❝✐❛✱ ✐♥❞✉tâ♥❝✐❛ ❡ ❝❛♣❛❝✐tâ♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡
❖ ❡q✉✐✈❛❧❡♥t❡ ♠♦♥♦❢ás✐❝♦ ❞❡ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛ ♣r❡ss✉♣õ❡ ✉♠ ❝✐r❝✉✐t♦ ❝♦♠ r❡s✐stê♥❝✐❛✱ ✐♥❞✉tâ♥❝✐❛
❡ ❝❛♣❛❝✐tâ♥❝✐❛✱ q✉❡ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ♣❡❧❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ❡ ❞❛ ❛❞♠✐tâ♥❝✐❛✿
Z1 = R1 + j ω L1 ✭✶✵✳✽✮
Y1 = j ω C1 ✭✶✵✳✾✮
s❡♥❞♦ R1✱ L1 ❡ C1 ♦s ❡q✉✐✈❛❧❡♥t❡s ♠♦♥♦❢ás✐❝♦s ✲ ❧❡♠❜r❡✲s❡ q✉❡ ❛ ❧✐♥❤❛ é tr✐❢ás✐❝❛✱ ❝♦♠ ❡❧❡♠❡♥t♦s
♣ró♣r✐♦s ❡ ♠út✉❛s✳
❆ ❡①tr❛çã♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ ❝✐r❝✉✐t♦ ♣r❡ss✉♣õ❡ t❛♠❜é♠ q✉❡ s✉❛ ❛♣❧✐❝❛çã♦ ♣❛r❛ ♦✉tr❛s ❢❛✐①❛s
❞❡ ❢r❡q✉ê♥❝✐❛ é ❧✐♥❡❛r ✲ ♦ q✉❡ ❞❡✈❡ s❡r ✉s❛❞♦ ❝♦♠ ♣r❡❝❛✉çã♦✳ P❛r❛ ✉♠❛ ❣❛♠❛ ❞❡ ❢r❡q✉ê♥❝✐❛s
❞❛s ♣r✐♠❡✐r❛s ❤❛r♠ô♥✐❝❛s✱ ♦ r❡s✉❧t❛❞♦ é ❜❡♠ ❛❝❡✐tá✈❡❧✱ ♣♦ré♠ ♣❛r❛ ❢r❡q✉ê♥❝✐❛s ❛❝✐♠❛ ❞❡ ✶✵ ❦❍③
❛ r❡s✐stê♥❝✐❛ t❡rá ✉♠ ❞❡s✈✐♦ ❝♦♥s✐❞❡rá✈❡❧ ❞❡✈✐❞♦ ❛♦ ❡❢❡✐t♦ ♣❡❧✐❝✉❧❛r✱ t♦r♥❛♥❞♦✲s❡ ❝♦♠♣❛rá✈❡❧ à
r❡❛tâ♥❝✐❛ ❞❛ ❧✐♥❤❛✶✸✳ ❆ ✜❣✉r❛ ✶✶ ✐❧✉str❛ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❧✐♥❤❛ ❝♦♠ ✈❛r✐❛çã♦ ❞❛ r❡s✐stê♥❝✐❛✱ ♣❛r❛
s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛ ❡ ③❡r♦✱ ❛té ✶ ▼❍③✳
❖ ❡❢❡✐t♦ é ♠❛✐s ✐♥t❡♥s♦ q✉❛♥❞♦ s❡ ♠♦❞❡❧❛ ♦ ❝✐r❝✉✐t♦ ❞❡ s❡q✉ê♥❝✐❛ ③❡r♦ ✲ ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❝♦♠
♦s ♣❛râ♠❡tr♦s ❞♦ s♦❧♦✳ ❊st❡ ❡❢❡✐t♦ ✐rá s❡ r❡✢❡t✐r ♥❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✶✸✳
❙♦♠❡♥t❡ ❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡✱ t❛♥t♦ ❡♠ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛ q✉❛♥t♦ ③❡r♦✱ ♣♦ss✉✐ ❝♦♠♣♦r✲
t❛♠❡♥t♦ ❧✐♥❡❛r ❡♠ ✉♠❛ ❛♠♣❧❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛ ✭❛té ✶ ▼❍③✮✳
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
−1
10
0
10
1
10
2
Frequencia (Hz)
R
es
is
te
nc
ia
 (
Ω
/k
m
)
 
 
R0
R1
❋✐❣✉r❛ ✶✶✿ ❊①❡♠♣❧♦ ❞❡ ✈❛r✐❛çã♦ ❞❛ r❡s✐stê♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡ ♣❡❧❛ ❢r❡q✉ê♥❝✐❛✳
✶✵✳✸ ❊❢❡✐t♦ ❞♦ s♦❧♦
❆s ❡q✉❛çõ❡s ✹✳✻ ❡ s❡❣✉✐♥t❡s ❛ss✉♠❡♠ q✉❡ ♦ s♦❧♦ é ✏✐❞❡❛❧✑✱ ♦✉ s❡❥❛ ♣♦ss✉✐ ❝♦♥❞✉t✐✈✐❞❛❞❡ ✐♥✜♥✐t❛ ♦✉
r❡s✐st✐✈✐❞❛❞❡ ③❡r♦✱ ♥♦ q✉❛❧ ❞❡st❛ ❢♦r♠❛ ❝♦♠♣♦rt❛rá ❝♦♠♦ ✉♠ ✏❡s♣❡❧❤♦✑ ♥♦ ♠ét♦❞♦ ❞❛s ✐♠❛❣❡♥s✳
❆♦ s❡ ❝♦♥s✐❞❡r❛r ♦ s♦❧♦ ❝♦♠ ✉♠❛ r❡s✐st✐✈✐❞❛❞❡ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ ❡ ❞❡ ❢❛t♦ ♣♦❞❡♠♦s t❡r ✈❛❧♦r❡s
❞❡ ✶✵ ❛ 10.000 ❲·♠✱ ♦ ❡❢❡✐t♦ ❞♦ ✏❡s♣❡❧❤♦✑ s❡rá ❞✐st♦r❝✐❞♦✳ ❆❧❣✉♠❛s t❡♦r✐❛s ✉s✉❛✐s sã♦ ❛ ❛♣r♦①✐♠❛çã♦
❞❡ P♦❧❧❛❝③❡❦ ❬✷✶❪✱ ❈❛rs♦♥ ❬✺❪ ❡ ❉❡r✐ ❬✻❪✱ ❡st❛ ú❧t✐♠❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ✏♣r♦❢✉♥❞✐❞❛❞❡ ❝♦♠♣❧❡①❛✑✿ ♦
❡❢❡✐t♦ ❞♦ s♦❧♦ é ❡♠❜✉t✐❞♦ ♥❛s ❡q✉❛çõ❡s ❡①✐st❡♥t❡s ❝♦♠♦ ✉♠ ♥ú♠❡r♦ ❝♦♠♣❧❡①♦✱ ♦✉ s❡❥❛✱ ❛ ♣❛r❝❡❧❛ h
s❡rá ✐❣✉❛❧❛✿
h′ = h+ d ✭✶✵✳✶✵✮
d =
1√
σ j ω µ
=
√
ρ
j ω µ
✭✶✵✳✶✶✮
✶✸❆❧❣✉♥s ♠♦❞❡❧♦s✱ ❝♦♠♦ ❞♦ ▼❛t❧❛❜ ❙✐♠P♦✇❡r❙②st❡♠s✱ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ ❧✐♥❤❛ é ❢✉♥❞❛♠❡♥t❛❞❛ ♥❛ r❡s✐stê♥❝✐❛ ❡
✐♥❞✉tâ♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡✱ ❝♦♠♦ ❞✐t♦ ❡♠ ❬✶✽❪✿ ✏❚❤✐s ♠♦❞❡❧ ❞♦❡s ♥♦t r❡♣r❡s❡♥t ❛❝❝✉r❛t❡❧② t❤❡ ❢r❡q✉❡♥❝② ❞❡♣❡♥❞❡♥❝❡ ♦❢
❘▲❈ ♣❛r❛♠❡t❡rs ♦❢ r❡❛❧ ♣♦✇❡r ❧✐♥❡s✳ ■♥❞❡❡❞✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ s❦✐♥ ❡✛❡❝ts ✐♥ t❤❡ ❝♦♥❞✉❝t♦rs ❛♥❞ ❣r♦✉♥❞✱ t❤❡ ❘ ❛♥❞
▲ ♠❛tr✐❝❡s ❡①❤✐❜✐t str♦♥❣ ❢r❡q✉❡♥❝② ❞❡♣❡♥❞❡♥❝❡✱ ❝❛✉s✐♥❣ ❛♥ ❛tt❡♥✉❛t✐♦♥ ♦❢ t❤❡ ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s✳✑ ❯♠ ❛rt✐❣♦ ❬✷✹❪
♣r♦♣õ❡ ✉♠ ♠♦❞❡❧♦ ♠❛✐s ❝♦♠♣❧❡t♦✳
✷✼
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
0
1
2
3
4
5
6
7
Frequencia (Hz)
In
du
ta
nc
ia
 (
m
H
/k
m
)
 
 
L0
L1
❋✐❣✉r❛ ✶✷✿ ❊①❡♠♣❧♦ ❞❡ ✈❛r✐❛çã♦ ❞❛ ✐♥❞✉tâ♥❝✐❛ ❡q✉✐✈❛❧❡♥t❡ ♣❡❧❛ ❢r❡q✉ê♥❝✐❛✳
s❡♥❞♦ σ ❛ ❝♦♥❞✉t✐✈✐❞❛❞❡ ❞♦ s♦❧♦✱ ω ❛ ❢r❡q✉❡♥❝✐❛ ❛♥❣✉❧❛r ❞♦ s✐st❡♠❛ ❡ µ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ♠❛❣♥ét✐❝❛
❞♦ s♦❧♦✱ ❡♠ ❣❡r❛❧ ♣ró①✐♠♦ ❛ µ0✳
P❡♥s❛♠❡♥t♦✿ ♥❛ ♣rát✐❝❛✱ ♦s ♣❛râ♠❡tr♦s ❞♦ s♦❧♦ ✈❛r✐❛♠ ❜❛st❛♥t❡✱ ❛♦ ❧♦♥❣♦ ❞❛ ❧✐♥❤❛✱ ❡ ❛té ❛♦
❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ♠❛s s❡♠♣r❡ r❡❛❧✐③❛♠✲s❡ ❡st✉❞♦s ❝♦♠ ♣❛râ♠❡tr♦s ✏❞❡t❡r♠✐♥íst✐❝♦s✑✳ ❈♦♥s✐❞❡r❡
♣♦r ❡①❡♠♣❧♦ ✉♠❛ tr❛♥s♣♦s✐çã♦✱ s✉♣♦st❛♠❡♥t❡ ✐❞❡❛❧✱ ❛♦♥❞❡ ✉♠ tr❡❝❤♦ ♣❛ss❛ ♣♦r ✉♠❛ r❡❣✐ã♦
❝♦♠ r❡s✐st✐✈✐❞❛❞❡ ρ1✱ ♦ s❡❣✉♥❞♦ tr❡❝❤♦ ♣❛ss❛ ♣♦r ✉♠❛ r❡s✐st✐✈✐❞❛❞❡ ρ2✳✳✳ q✉❛❧ s❡rá ♦ ❡❢❡✐t♦ ❞❡
s❡ ❛ss✉♠✐r ✉♠ ✈❛❧♦r ✏✜①♦✑❄
◗✉❛❧ s❡rá ♦ ❞❡s✈✐♦ ♥♦s ❝á❧❝✉❧♦s ❛♦ s❡ ❝♦♥s✐❞❡r❛r ✉♠ ✈❛❧♦r ❞❡ r❡s✐st✐✈✐❞❛❞❡ ❞✐❢❡r❡♥t❡❄ ◆ã♦
❤á ✉♠ ♠ét♦❞♦ ♣rát✐❝♦ ♣❛r❛ r❡s♦❧✈❡r ✐ss♦✱ s♦♠❡♥t❡ ✉♠ tr❛t❛♠❡♥t♦ ❡st❛tíst✐❝♦ ♣♦❞❡ ❛✈❛❧✐❛r ♦
❡rr♦✳
❖ ❡❢❡✐t♦ ❞♦ s♦❧♦ r❡❛❧ é ♠❛✐s r❡❧❡✈❛♥t❡ ♥♦ ❝á❧❝✉❧♦ ♥♦s ♣❛râ♠❡tr♦s ❞❡ s❡q✉ê♥❝✐❛ ③❡r♦✱ ❛❢❡t❛♥❞♦
♣❛rt✐❝✉❧❛r♠❡♥t❡ ♦s ❡st✉❞♦s ❞❡ ❢❛❧t❛s ♠♦♥♦❢ás✐❝❛s✱ ❡ s❡✉s ♠❡✐♦s ❞❡ ♠✐t✐❣❛çã♦ ✭❡①✳ r❡❧✐❣❛♠❡♥t♦
♠♦♥♦♣♦❧❛r✮✳
❊st❡ ♠♦❞❡❧♦ ♥ã♦ s❡ ❛♣❧✐❝❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ❛❞♠✐tâ♥❝✐❛✱ ♣♦✐s ♦ s♦❧♦ ♥ã♦ ❛❢❡t❛ s✐❣♥✐✜❝❛t✐✈❛♠❡♥t❡
❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❞❛ ❧✐♥❤❛✳
❊①✳✿ ♣❛r❛ ✉♠ s♦❧♦ ❞❡ 100 ❲·✱ ❛ ❞✐stâ♥❝✐❛ ❝♦♠♣❧❡①❛ ♣❛r❛ ✻✵ ❍③ s❡rá
d =
1
√
1/100j 2π 60 · 4π10−7
= 324, 87− j324, 87m
♣❛r❛ ✉♠ s♦❧♦ ❞❡ 10 ❲·✱ d = 102, 73− j102, 73 ♠✳ P❛r❛ ♦ s♦❧♦ ❞❡ 10 ❲·✱ ❝♦♠ ✉♠❛ ❢r❡q✉ê♥❝✐❛ ❞❡
✶ ❦❍③✱ d = 25, 16− 25, 16 ♠✳
❈❛❧❝✉❧❛♥❞♦ ❛ ✐♥❞✉tâ♥❝✐❛ ♣ró♣r✐❛ ❞❡ ✉♠ ❝❛❜♦✱ ❝♦♠ ✶ ❝♠ ❞❡ r❛✐♦ ❡ ✉♠❛ ❛❧t✉r❛ ♠é❞✐❛ ❞❡
✶✵ ♠✱ ♣r✐♠❡✐r♦ ❝♦♠ ♦ s♦❧♦ ✐❞❡❛❧✿
L =
µ0
2π
ln
2 · 10
0, 01
= 1, 5202 · 10−6 H/m
❝♦♠ ♦ s♦❧♦ ❞❡ 100 ❲·✿
L =
µ0
2π
ln
2 · (10 + 324, 87− j324, 87)
0, 01
= (2, 2887− j0, 1541) · 10−6 H/m
❡ss❛ ✏✐♥❞✉tâ♥❝✐❛ ❝♦♠♣❧❡①❛✑ ✐rá s❡ ❝♦♥✈❡rt❡r ❡♠ ✉♠❛ r❡s✐stê♥❝✐❛ ❛❞✐❝✐♦♥❛❧✳ ❉❡s♣r❡③❛♥❞♦ ❛
r❡s✐stê♥❝✐❛ ❞♦ ❝❛❜♦✱ ♦❜té♠✲s❡
ZL = j ω L = (0, 0581 + j0, 8629) · 10−3 ❲/m
✷✽
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
300
400
500
600
700
800
900
Frequencia (Hz)
Im
pe
da
nc
ia
 c
ar
ac
te
ris
tic
a 
(Ω
)
 
 
Zc0
Zc1
❋✐❣✉r❛ ✶✸✿ ❊①❡♠♣❧♦ ❞❡ ✈❛r✐❛çã♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛ ♣❡❧❛ ❢r❡q✉ê♥❝✐❛✳
❝♦♠ ♦ s♦❧♦ ❞❡ 10 ❲·
L =
µ0
2π
ln
2 · (10 + 102, 73− j102, 73)
0, 01
= (2, 0651− j0, 1478) · 10−6 H/m
❡ss❛ ❞✐❢❡r❡♥ç❛ t❡♥❞❡ ❛ s❡ ❛♥✉❧❛r q✉❛♥❞♦ ❝❛❧❝✉❧❛✲s❡ ❛ ✐♠♣❡❞â♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛ ✭Zp −
Zm✮✱ ♣♦ré♠ ♦ ❡❢❡✐t♦ s❡ ❛♠♣❧✐❛ ♥❛ s❡q✉ê♥❝✐❛ ③❡r♦ ✭Zp + 2Zm✮✳
✶✵✳✹ ❊❢❡✐t♦ ❞♦s ❝❛❜♦s ♣❛r❛✲r❛✐♦s
❖s ❝❛❜♦s ♣❛r❛✲r❛✐♦s ♣r♦t❡❣❡♠ ❛s ❢❛s❡s ♦✉ ♣♦❧♦s ❝♦♥tr❛ ❞❡s❝❛r❣❛s ❛t♠♦s❢ér✐❝❛s ❞✐r❡t❛s✳ ▼❛s s✉❛
♣r♦①✐♠✐❞❛❞❡ ♣r♦✈♦❝❛ ✉♠❛ ✐♥t❡r❛çã♦ ❡❧❡tr♦♠❛❣♥ét✐❝❛✳ ❊♠ ♥♦ss♦ ♠♦❞❡❧♦ ♦ ❝❛❜♦ s❡rá ✉♠❛ ✏❢❛s❡✑
❛❞✐❝✐♦♥❛❧✱ ❛❝r❡s❝❡♥t❛♥❞♦ ♠❛✐s ✉♠❛ ❧✐♥❤❛ ❡ ✉♠❛ ❝♦❧✉♥❛ ♥❛ ♠❛tr✐③✳
◆❡st❡ ♣♦♥t♦ é ❞❡t❡r♠✐♥❛♥t❡ ♦ t✐♣♦ ❞❡ ❧✐❣❛çã♦ ❞♦s ♣❛r❛✲r❛✐♦s✱ q✉❡ ♣♦❞❡♠ s❡r ❛t❡rr❛❞♦s ♦✉
✐s♦❧❛❞♦s✶✹✳ ❖ ♣❛r❛✲r❛✐♦ ❛t❡rr❛❞♦ t❡rá ♣♦t❡♥❝✐❛❧ ③❡r♦ ✭Vg = 0✮ ❡ t❡rá ❝♦rr❡♥t❡ ✐♥❞✉③✐❞❛✱ ❡♥q✉❛♥t♦
q✉❡ ✐s♦❧❛❞♦ ♥ã♦ ❤❛✈❡rá ❝♦rr❡♥t❡ ✭Ig = 0✮✱ ♠❛s t❡rá ♣♦t❡♥❝✐❛❧ ✐♥❞✉③✐❞♦✳ ❈❛❞❛ ❧✐❣❛çã♦ t❡♠ ✈❛♥t❛❣❡♥s
❡ ❞❡s✈❛♥t❛❣❡♥s✳ P❛r❛ q✉❛❧q✉❡r ♦♣çã♦✱ ❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ s❡rá ♥❛ ❢♦r♠❛✱ ♣♦r ❡①❡♠♣❧♦ ❝♦♠ ❞♦✐s
❝❛❜♦s ♣ár❛✲r❛✐♦s✿






va
vb
vc
vg1
vg2






=






zaa zab zac zag1 zag2
zba zbb zbc zbg1 zbg2
zca zcb zcc zcg1 zcg2
zg1a zg1b zg1c zg1g1 zg1g2
zg2a zg2b zg2c zg2g1 zg2g2












ia
ib
ic
ig1
ig2






✭✶✵✳✶✷✮
P❛r❛ ❝❛❜♦s ♣❛r❛✲r❛✐♦s ❝♦♥t✐♥✉❛♠❡♥t❡ ❛t❡rr❛❞♦s✱ ❞✐✈✐❞❡✲s❡ ❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ ✭♦✉ ♣❛rt✐❝✐♦♥❛✲
♠❡♥t♦✮ ❡♠ q✉❛tr♦ ♣❛rt❡s ❬✼✱ ♣✳ ✹✲✶✺❪✿
[
vu
vg
]
=
[
Zuu Zug
Zgu Zgg
] [
iu
ig
]
✭✶✵✳✶✸✮
♥♦ q✉❛❧ ❛ ♠❛tr✐③ r❡❞✉③✐❞❛ s❡rá
vu = Zred · iu ✭✶✵✳✶✹✮
Zred = Zuu − Zug · Zgg−1 · Zgu ✭✶✵✳✶✺✮
✶✹◆❛ ✈❡r❞❛❞❡ ❛ ✐s♦❧❛çã♦ ❞♦ ♣❛r❛✲r❛✐♦ é ♠í♥✐♠❛✱ s♦♠❡♥t❡ ♣❛r❛ ♥ã♦ ❝✐r❝✉❧❛r ❝♦rr❡♥t❡ ❡♠ ❝♦♥❞✐çõ❡s ♥♦r♠❛✐s✱ ♣♦✐s
♥❛ ✐♥❝✐❞ê♥❝✐❛ ❞❡ ✉♠❛ ❞❡s❝❛r❣❛ ❡❧❡ ❞❡✈❡ ❡s❝♦❛r ♣❛r❛ ♦ s♦❧♦
✷✾
♦ ♠❡s♠♦ ♠ét♦❞♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ♥❛ ♠❛tr✐③ M ❛♥t❡s ❞❡ ❞❡t❡r♠✐♥❛r ❛ ❛❞♠✐tâ♥❝✐❛✶✺✿
MY =
[
Muu Mug
Mgu Mgg
]
✭✶✵✳✶✻✮
MYred = Muu −Mug ·Mgg−1 ·Mgu ✭✶✵✳✶✼✮
C = 2π ε0MY
−1
red ✭✶✵✳✶✽✮
▲❡♠❜r❛♥❞♦ q✉❡ ♦s ❝❛❜♦s ♣❛r❛✲r❛✐♦s ❣❡r❛❧♠❡♥t❡ sã♦ ❞❡ ❛ç♦✱ ❝♦♠ ✈❛❧♦r❡s ❞❡ ♣❡r♠❡❛❜✐❧✐❞❛❞❡
r❡❧❛t✐✈❛ ❛❝✐♠❛ ❞❡ ✶✳ ◆ã♦ ❡①✐st❡ r❡❢❡rê♥❝✐❛s ❡①❛t❛s q✉❛♥t♦ ❛ ♣❡r♠❡❛❜✐❧✐❞❛❞❡ ❞❡st❡ t✐♣♦ ❞❡ ❛ç♦✱
s❡♥❞♦ ❛❝❡✐tá✈❡❧ ❝♦♥s✐❞❡r❛r ✈❛❧♦r❡s ❡♥tr❡ ✺✵ ❡ ✶✵✵ µ0✳
P♦r ❡①❡♠♣❧♦✱ ♣❛r❛ µr = 100✱ ♦ r❛✐♦ ❡q✉✐✈❛❧❡♥t❡ s❡rá
r′ = r e−
100
4 = 1, 3888 · 10−11 r
♦✉ s❡❥❛✱ ❜❡♠ ❞✐❢❡r❡♥t❡ ❞❡ ✵✱✼✼✽✽✦
❊①❡♠♣❧♦✿ ✉♠❛ ▲❚ ❝♦♠ ❝❛❜♦s ❋❛❧❝♦♥ ✭Rca = 0, 0448 ❲✴❦♠✱ � ✸✾✱✷✸ ♠♠✮✱ ✸ ❝❛❜♦s ♣♦r ❢❛s❡✱
❡s♣❛ç❛♠❡♥t♦ ✽✵ ❝♠✱ ❞✐s♣♦s✐çã♦ ❡♠ ♥❛❜❧❛✱ ❢❛s❡ ❝❡♥tr❛❧ ❛ ✶✺ ♠ ❞❡ ❛❧t✉r❛ ♠é❞✐❛✱ ❢❛s❡s ❧❛t❡r❛✐s ❛
✽ ♠ ❞❡ ❞✐stâ♥❝✐❛ ❤♦r✐③♦♥t❛❧ ❞♦ ❝❡♥tr♦✱ ✷✵ ♠ ❞❡ ❛❧t✉r❛ ♠é❞✐❛✱ ❝♦♠ ✷ ❝❛❜♦s ❣✉❛r❞❛ ✭❊❍❙ ✸✴✽✑✱
Rca = 4, 2324 ❲✴❦♠✱ � ✾✱✶✹ ♠♠✱ µ = 100µ0✮ ❛ ✻ ♠ ❞❡ ❞✐stâ♥❝✐❛ ❤♦r✐③♦♥t❛❧ ❞♦ ❝❡♥tr♦✱ ✸✺ ♠
❞❡ ❛❧t✉r❛ ♠é❞✐❛✳ ❖❜té♠✲s❡ ❝♦♠♦ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ ♣r✐♠✐t✐✈❛✿
Z =
















0.01493
❥✵✳✶✵✵✼✼ ❥✵✳✵✼✹✻✽ ❥✵✳✵✾✼✸✺ ❥✵✳✵✼✻✼✶
+j0.33657
❥✵✳✶✵✵✼✼
0.01493
❥✵✳✶✵✵✼✼ ❥✵✳✵✻✻✸✽ ❥✵✳✵✻✻✸✽
+j0.33657
❥✵✳✵✼✹✻✽ ❥✵✳✶✵✵✼✼
0.01493
❥✵✳✵✼✻✼✶ ❥✵✳✵✾✼✸✺
+j0.33657
❥✵✳✵✾✼✸✺ ❥✵✳✵✻✻✸✽ ❥✵✳✵✼✻✼✶
4.2324
❥✵✳✶✸✹✵✻
+j2.61155
❥✵✳✵✼✻✼✶ ❥✵✳✵✻✻✸✽ ❥✵✳✵✾✼✸✺ ❥✵✳✶✸✹✵✻
4.2324
+j2.61155
















❲/km
❛♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✶✵✳✶✼✱ ♦❜té♠✲s❡ ✭❡♠ ❲✴❦♠✮✿
Zred =


0.0123768 + j0.3382288 −0.0019213 + j0.1020155 −0.0024816 + j0.0762942
−0.0019213 + j0.1020155 0.0134680 + j0.3158326 −0.0019213 + j0.1020155
−0.0024816 + j0.0762942 −0.0019213 + j0.1020155 0.0123768 + j0.3382288
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
♦❜s❡r✈❛✲s❡ ❛❣♦r❛ ❛ ♣r❡s❡♥ç❛ ❞❡ ♣❛rt❡ r❡❛❧ ♥❛s ♠út✉❛s✱ ❞❡✈✐❞♦ ❛♦ r❡t♦r♥♦ ♣❡❧♦ ♣❛r❛✲r❛✐♦s✳ P❛r❛
♣r♦❣r❛♠❛r ❡♠ ▼❛❧t❛❜ ♦✉ ❙❝✐❧❛❜✱ ♦ ❝♦♠❛♥❞♦ s❡rá✿
③r❡❞ ❂ ③✭✶✿✸✱✶✿✸✮ ✰ ③✭✶✿✸✱✹✿✺✮✴③✭✹✿✺✱✹✿✺✮✯③✭✹✿✺✱✶✿✸✮
✶✵✳✺ ▼♦❞❡❧♦ ❣❡♥ér✐❝♦ ❞❡ r❡❞✉çã♦ ❞❡ ❢❡✐①❡s ❞❡ ❝♦♥❞✉t♦r❡s
❯♠ ♠ét♦❞♦ ❣❡r❛❧ é ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✼❪✱ ❡ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❡♠ q✉❛❧q✉❡r t✐♣♦ ❞❡ ❢❡✐①❡✳ P❛rt✐♥❞♦
❞❛ ♣r❡♠✐ss❛ q✉❡ ❛ s♦♠❛ ❞❛s ❝♦rr❡♥t❡s ♥♦ ❢❡✐①❡ é ✐❣✉❛❧ ❛ ❝♦rr❡♥t❡ ❞❛ ❢❛s❡✱ ❡ ❛ q✉❡❞❛ ❞❡ t❡♥sã♦ dV/dx
é ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✐❣✉❛❧ ♥♦ ❢❡✐①❡✱ ♦✉ s❡❥❛✱ ♣❛r❛ ✉♠ ❢❡✐①❡ ❞❡ n s✉❜❝♦♥❞✉t♦r❡s✿
I1 + I2 + · · ·+ In = If ✭✶✵✳✶✾✮
dv1
dx
=
dv2
dx
= . . . =
dvn
dx
✭✶✵✳✷✵✮
♣r♦❝❡❞❡✲s❡ ❝♦♠ ❛ s❡❣✉✐♥t❡ ♠❛♥✐♣✉❧❛çã♦ ♠❛tr✐❝✐❛❧✿
✶✳ ❉❡s❧♦❝❛✲s❡ ❛s ❧✐♥❤❛s ❡ ❛s ❝♦❧✉♥❛s ❞♦s s✉❜❝♦♥❞✉t♦r❡s ✷✱ ✸✱ ✳ ✳ ✳ ✱ n ♣❛r❛ ❛ ❡①tr❡♠✐❞❛❞❡ ❞❛ ♠❛tr✐③❀
✷✳ ❙✉❜tr❛✐r ❛ ❝♦❧✉♥❛ ❞♦ s✉❜❝♦♥❞✉t♦r ✶ ❞❛s ❝♦❧✉♥❛s ❞♦s s✉❜❝♦♥❞✉t♦r❡s ✷✱ ✸✱ ✳ ✳ ✳ ✱ n❀
✶✺P❛r❛ ❛ ✐♠♣❡❞â♥❝✐❛ ❞❡✈❡✲s❡ ❛♣❧✐❝❛r ❛ r❡❞✉çã♦ ❛♣ós s♦♠❛r ❛ r❡s✐stê♥❝✐❛✱ ✐♥❝❧✉✐♥❞♦ ❞♦s ❝❛❜♦s ♣❛r❛✲r❛✐♦s✳
✸✵
✸✳ ❙✉❜tr❛✐r ❛ ❧✐♥❤❛ ❞♦ s✉❜❝♦♥❞✉t♦r ✷ ❞❛s ❧✐♥❤❛s ❞♦s s✉❜❝♦♥❞✉t♦r❡s ✷✱ ✸✱ ✳ ✳ ✳ ✱ n❀
✹✳ P❡❧❛ ♦♣❡r❛çã♦ ♠❛tr✐❝✐❛❧ ❢❡✐t❛✱ ❡q✉✐✈❛❧❡✲s❡ ❛ ③❡r❛r ❛s ❝♦rr❡♥t❡s ♥♦s s✉❜❝♦♥❞✉t♦r❡s ✷✱ ✸✱ ✳ ✳ ✳ ✱
n✱ ♣r♦❝❡❞❡✲s❡ ❡♠ ❡❧✐♠✐♥❛r ❡st❡s s✉❜❝♦♥❞✉t♦r❡s✱ ✉s❛♥❞♦ ♦ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞♦s ❝❛❜♦s
♣ár❛✲r❛✐♦s ✭❡q✉❛çã♦ ✶✵✳✶✺✮❀
✺✳ ❖ s✉❜❝♦♥❞✉t♦r ✶ ❛❣♦r❛ r❡♣r❡s❡♥t❛ ♦ ❡q✉✐✈❛❧❡♥t❡ ❞♦ ❢❡✐①❡✳
❊①❡♠♣❧♦✿ ✉♠❛ ▲❚ s❡♠ ♣❡r❞❛s é ❝♦♠♣♦st❛ ♣♦r ❢❡✐①❡s ❞❡ três ❝❛❜♦s ♣♦r ❢❛s❡✱ ❝✉❥❛ ♠❛tr✐③
❝♦♠ ❝❛❞❛ s✉❜❝♦♥❞✉t♦r é ❛ss✐♠ r❡♣r❡s❡♥t❛❞❛✿
Z = j
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0, 9381 0, 6513 0, 6513 0, 4397 0, 4444 0, 4419 0, 3874 0, 3898 0, 3886
0, 6513 0, 9381 0, 6513 0, 4353 0, 4397 0, 4374 0, 3852 0, 3874 0, 3863
0, 6513 0, 6513 0, 9381 0, 4374 0, 4419 0, 4397 0, 3863 0, 3886 0, 3874
0, 4397 0, 4353 0, 4374 0, 9381 0, 6513 0, 6513 0, 4397 0, 4444 0, 4419
0, 4444 0, 4397 0, 4419 0, 6513 0, 9381 0, 6513 0, 4353 0, 4397 0, 4374
0, 4419 0, 4374 0, 4397 0, 6513 0, 6513 0, 9381 0, 4374 0, 4419 0, 4397
0, 3874 0, 3852 0, 3863 0, 4397 0, 4353 0, 4374 0, 9381 0, 6513 0, 6513
0, 38980, 3874 0, 3886 0, 4444 0, 4397 0, 4419 0, 6513 0, 9381 0, 6513
0, 3886 0, 3863 0, 3874 0, 4419 0, 4374 0, 4397 0, 6513 0, 6513 0, 9381
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❲/km
❙❡rá ❛♣❧✐❝❛❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♥❛ ú❧t✐♠❛ ❢❛s❡✱ r❡❢❡r❡♥t❡ ❛s ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ✼✱ ✽ ❡ ✾✱ ♣♦r ❥á ❡st❛r
♣♦s✐❝✐♦♥❛❞❛✳ ❙✉❜tr❛✐♥❞♦ ❛ ❝♦❧✉♥❛ ✼ ❞❡ ✽ ❡ ✾✱ ♦❜té♠✲s❡
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0, 9381 0, 6513 0, 6513 0, 4397 0, 4444 0, 4419 0, 3874 0, 0023 0, 0011
0, 6513 0, 9381 0, 6513 0, 4353 0, 4397 0, 4374 0, 3852 0, 0022 0, 0011
0, 6513 0, 6513 0, 9381 0, 4374 0, 4419 0, 4397 0, 3863 0, 0023 0, 0011
0, 4397 0, 4353 0, 4374 0, 9381 0, 6513 0, 6513 0, 4397 0, 0047 0, 0022
0, 4444 0, 4397 0, 4419 0, 6513 0, 9381 0, 6513 0, 4353 0, 0044 0, 0021
0, 4419 0, 4374 0, 4397 0, 6513 0, 6513 0, 9381 0, 4374 0, 0045 0, 0023
0, 3874 0, 3852 0, 3863 0, 4397 0, 4353 0, 4374 0, 9381 −0, 2868 −0, 2868
0, 3898 0, 3874 0, 3886 0, 4444 0, 4397 0, 4419 0, 6513 0, 2868 0, 0000
0, 3886 0, 3863 0, 3874 0, 4419 0, 4374 0, 4397 0, 6513 0, 0000 0, 2868
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❙✉❜tr❛✐♥❞♦ ❛❣♦r❛ ❛ ❧✐♥❤❛ ✼ ❞❛s ❧✐♥❤❛s ✽ ❡ ✾✱
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0, 9381 0, 6513 0, 6513 0, 4397 0, 4444 0, 4419 0, 3874 0, 0023 0, 0011
0, 6513 0, 9381 0, 6513 0, 4353 0, 4397 0, 4374 0, 3852 0, 0022 0, 0011
0, 6513 0, 6513 0, 9381 0, 4374 0, 4419 0, 4397 0, 3863 0, 0023 0, 0011
0, 4397 0, 4353 0, 4374 0, 9381 0, 6513 0, 6513 0, 4397 0, 0047 0, 0022
0, 4444 0, 4397 0, 4419 0, 6513 0, 9381 0, 6513 0, 4353 0, 0044 0, 0021
0, 4419 0, 4374 0, 4397 0, 6513 0, 6513 0, 9381 0, 4374 0, 0045 0, 0023
0, 3874 0, 3852 0, 3863 0, 4397 0, 4353 0, 4374 0, 9381 −0, 2868 −0, 2868
0, 0023 0, 0022 0, 0023 0, 0047 0, 0044 0, 0045 −0, 2868 0, 5736 0, 2868
0, 0011 0, 0011 0, 0011 0, 0022 0, 0021 0, 0023 −0, 2868 0, 2868 0, 5736
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❘❡❞✉③✐♥❞♦ ❛ ♠❛tr✐③ ✉s❛♥❞♦ ✭✶✵✳✶✼✮✱ t♦r♥❛♥❞♦✲s❡ ♣r♦✈✐s♦r✐❛♠❡♥t❡ ❝♦♠♦ 7× 7✿
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0, 9381 0, 6513 0, 6513 0, 4397 0, 4444 0, 4419 0, 3886
0, 6513 0, 9381 0, 6513 0, 4353 0, 4397 0, 4373 0, 3863
0, 6513 0, 6513 0, 9381 0, 4373 0, 4419 0, 4397 0, 3874
0, 4397 0, 4353 0, 4373 0, 9381 0, 6513 0, 6513 0, 4420
0, 4444 0, 4397 0, 4419 0, 6513 0, 9381 0, 6513 0, 4375
0, 4419 0, 4373 0, 4397 0, 6513 0, 6513 0, 9381 0, 4397
0, 3886 0, 3863 0, 3874 0, 4420 0, 4375 0, 4397 0, 7469
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❘❡♣❡t✐♥❞♦ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♣❛r❛ ❛s ♦✉tr❛s ❞✉❛s ❢❛s❡s✱ ❞❡✈❡✲s❡ ❝❤❡❣❛r ❛ s❡❣✉✐♥t❡ ♠❛tr✐③ ❡q✉✐✈❛❧❡♥t❡✿
Zred = j
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0, 7469 0, 4397 0, 3875
0, 4397 0, 7468 0, 4397
0, 3875 0, 4397 0, 7469
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 ❲/km
✶✵✳✻ ▼♦❞❡❧♦ ❞❡ ❝✐r❝✉✐t♦ ❞✉♣❧♦
P♦❞❡✲s❡ ♠♦❞❡❧❛r ✉♠❛ ❧✐♥❤❛ ❝♦♠ ❞♦✐s ✭♦✉ ♠❛✐s✮ ❝✐r❝✉✐t♦s ✉♥✐♥❞♦ ❛ ❢❛s❡ ❞❡ ❝❛❞❛ ❝✐r❝✉✐t♦✳ P♦ré♠✱ ♦
✉s♦ ❞♦ ❘▼● ❞❡✐①❛ ❞❡ t❡r ✈❛❧✐❞❛❞❡ ♣❛r❛ ❞✐stâ♥❝✐❛s ♠✉✐t♦ ❧♦♥❣❛s✳ ❙❡rá ♥❡❝❡ssár✐♦ ✉♠ tr❛t❛♠❡♥t♦
✸✶
♠❛tr✐❝✐❛❧✳
❙❡❥❛ ✉♠❛ ❧✐♥❤❛ ❞❡ s❡✐s ❝♦♥❞✉t♦r❡s ❣❡♥ér✐❝♦s✱ ❝♦♠ ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ t❡♥sã♦ ❡ ❝♦rr❡♥t❡ ♣♦r
✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦ r❡♣r❡s❡♥t❛❞❛ ❛❜❛✐①♦✿
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V1
V2
V3
V4
V5
V6
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=
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z11 z12 z13 z14 z15 z16
z21 z22 z23 z24 z25 z26
z31 z32 z33 z34 z35 z36
z41 z42 z43 z44 z45 z46
z51 z52 z53 z54 z55 z56
z61 z62 z63 z64 z65 z66
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I1
I2
I3
I4
I5
I6
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✭✶✵✳✷✶✮
❙❡♥❞♦ ❛❣♦r❛ ❡ss❡ s✐st❡♠❛ ❧✐❣❛❞♦ ❝♦♠♦ ✉♠ ❝✐r❝✉✐t♦ ❞✉♣❧♦✱ ♥♦ q✉❛❧ Va = V1 = V4✱ Vb = V2 = V5
❡ Vc = V3 = V6✳ P♦r s✉❛ ✈❡③✱ ❛s ❝♦rr❡♥t❡s s❡rã♦ s♦♠❛❞❛s✱ Ia = I1 + I4✱ Ib = I2 + I5 ❡ Ic = I3 + I6✳
❊ss❡ ♣r♦❝❡❞✐♠❡♥t♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞♦ ❡♠ ❬✸✱ ♣✳ ✶✵✽❪✱ ♣♦❞❡♥❞♦ ✐♥❝❧✉s✐✈❡ s❡r ✉s❛❞❛ ♣❛r❛ ♦
❝á❧❝✉❧♦ ♣r❡❝✐s♦ ❞❛ ✐♠♣❡❞â♥❝✐❛ ❞❡ ❢❡✐①❡ ❞❡ ❝♦♥❞✉t♦r❡s✳ ◆❛ ♠❡s♠❛ r❡❢❡rê♥❝✐❛ ❬✸✱ ♣✳ ✶✸✼❪ ❡st✉❞❛✲s❡ ♦
❞❡s❜❛❧❛♥ç♦ ❡♥tr❡ ♦s ❝✐r❝✉✐t♦s✱ q✉❡ ♣♦❞❡ ❝❛✉s❛r ♣♦r ❡①❡♠♣❧♦ ❝♦rr❡♥t❡s ❝✐r❝✉❧❛♥t❡s✳ ❯♠ ♣r♦❝❡❞✐♠❡♥t♦
♠❛✐s ❝♦♠♣❧❡t♦ é ❛❜♦r❞❛❞♦ ❡♠ ❬✼❪✳
P❡♥s❛♠❡♥t♦✿ ♣❛r❛ ♠♦❞❡❧❛r ✉♠❛ ❧✐♥❤❛ ❤❡①❛❢ás✐❝❛✱ ♣♦❞❡✲s❡ ♣❛rt✐r ❞❛ ❡q✉❛çã♦ ✶✵✳✷✶✱ ❛ss✉✲
♠✐♥❞♦ ♦s ✈❛❧♦r❡s V1 ❛ V6 ❢❛s♦r❡s s✐♠étr✐❝♦s ❞❡❢❛s❛❞♦s ❡♠ ✻✵➦✱ ❡ s❡❣✉✐♥❞♦ ❛ ♠❡s♠❛ ♠❡t♦❞♦❧♦❣✐❛
❞♦ ❝❛♣ít✉❧♦ ✹✳ ❈♦♠♦ ❝❤❡❣❛r ❛♦ ❡q✉✐✈❛❧❡♥t❡ ♠♦♥♦❢ás✐❝♦❄
✶✵✳✼ ❈á❧❝✉❧♦ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ s❡q✉ê♥❝✐❛ ③❡r♦
❈♦♠♦ ✈✐st♦ ♥❛s ❡q✉❛çõ❡s ✺✳✷ ❡ ✺✳✸✱ ❛ ✐♠♣❡❞â♥❝✐❛ ❡ ❛ ❛❞♠✐tâ♥❝✐❛ ❞❡ s❡q✉ê♥❝✐❛ ③❡r♦ é ♠✉✐t♦ ✐♥✲
✢✉❡♥❝✐❛❞❛ ♣❡❧❛ ❝♦♠♣♦♥❡♥t❡ ♠út✉❛ ✭Zm ❡ Ym✮✳ ◆❡st❡ ♣♦♥t♦ ❛ ♠♦❞❡❧❛❣❡♠ ❝♦rr❡t❛ ❞♦ s♦❧♦ s❡rá
❞❡t❡r♠✐♥❛♥t❡✱ ❡ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ s♦❧♦ ✐❞❡❛❧ ❞❡✐①❛ ❞❡ s❡r ❞❡s♣r❡③í✈❡❧✳
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥❛ s❡q✉ê♥❝✐❛ ♣♦s✐t✐✈❛✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r ✉♠❛ ✐♠♣❡❞â♥❝✐❛ ❝❛r❛❝t❡ríst✐❝❛
❞❡ s❡q✉ê♥❝✐❛ ③❡r♦✱ Zc0 =
√
Z0
Y0
✱ q✉❡ ❞❡t❡r♠✐♥❛rá ❛ ♣r♦♣❛❣❛çã♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❤♦♠♦♣♦❧❛r❡s✳
❈♦♥❢♦r♠❡ é ♠♦str❛❞♦ ❡♠ ❡st✉❞♦s ❞❡ ✢✉①♦ ❞❡ ♣♦tê♥❝✐❛ ❡ ❝♦♠♣♦♥❡♥t❡s s✐♠étr✐❝❛s✱ ❛ ❝♦♠♣♦♥❡♥t❡
❞❡ s❡q✉ê♥❝✐❛ ③❡r♦ é ✐♥✢✉❡♥t❡ ♥♦ ❝á❧❝✉❧♦ ❞❡ ❢❛❧❤❛s ❞❡ ❝✉rt♦✲❝✐r❝✉✐t♦✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❡♠ ❝✉rt♦
♠♦♥♦❢ás✐❝♦✱ s❡♥❞♦ ❡st❡ ♦ t✐♣♦ ♠❛✐s ❝♦♠✉♠ ❞❡ ♦❝♦rrê♥❝✐❛ ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✳
❊st✉❞♦s r❡❝❡♥t❡s ❜✉s❝❛♠ ♦t✐♠✐③❛r ❛ r❡❝✉♣❡r❛çã♦ ❞❛ ❧✐♥❤❛ ❢r❡♥t❡ ❛ ❞❡❢❡✐t♦s ♠♦♥♦❢ás✐❝♦s✱ r❡❛❧✐✲
③❛♥❞♦ ♦ r❡❧✐❣❛♠❡♥t♦ ♠♦♥♦♣♦❧❛r✳
✶✶ ❊st✉❞♦ ❞❡t❛❧❤❛❞♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ tr❛♥s♠✐ssã♦ ❛tr❛✈és
❞❡ ♠❛tr✐③ Ybarra
◆❡st❛ s❡çã♦ ❛♣r❡s❡♥t❛✲s❡ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦✱ ❝♦♠♣♦st♦ ♣♦r ✉♠ tr♦♥❝♦ ❝♦♠ ✷ ▲❚s✱ s✉❛s r❡s♣❡❝t✐✈❛s
❝♦♠♣❡♥s❛çõ❡s✱ ❡ ❞✉❛s ❜❛rr❛s ❞❡ ✉♠ s✐st❡♠❛ ✜❝tí❝✐♦✱ r❡♣r❡s❡♥t❛❞❛s ♣♦r s❡✉s ❡q✉✐✈❛❧❡♥t❡ ❚❤❡✈❡♥í♥✳
❙❡rá ❡st✉❞❛❞♦ ♦ ❡st❛❞♦ ❞♦ s✐st❡♠❛ ❡♠ três ❝♦♥❞✐çõ❡s✿ ✢✉①♦ ❝♦♠ ♣♦tê♥❝✐❛ ♥♦♠✐♥❛❧ ❞❛s ▲❚s✱ ♦
s✐st❡♠❛ ❡♠ ✈❛③✐♦✱ ❡♥❡r❣✐③❛❞♦ ♣♦r ✉♠❛ ❞❛s ❜❛rr❛s✱ ❡ ♦ ❡❢❡✐t♦ ❞❡ ❝✉rt♦✲❝✐r❝✉✐t♦ ❡♠ ✉♠❛ ❞❛s ▲❚s✳
◆♦✈❛♠❡♥t❡ s❡rá ✉s❛❞♦ ♦ ❡①❡♠♣❧♦ ❞❛ ▲❚ ✏r❛q✉❡t❡✑✱ ♣❛r❛ ✉♠ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✸✵✵ ❦♠ ❡ ❝♦♠♣❡♥✲
s❛❞❛ ❡♠ ✺✵✪✱ t❛♥t♦ sér✐❡ q✉❛♥t♦ ♣❛r❛❧❡❧♦✳
◆❛ ✜❣✉r❛ ✶✹ ❛♣r❡s❡♥t❛✲s❡ ✉♠❛ ❝♦♠♣❡♥s❛çã♦ ♣❛r❛❧❡❧❛ ❝♦♠ r❡❛t♦r ❞❡ ♥❡✉tr♦✱ ✉♠ ❡❧❡♠❡♥t♦ ✉s❛❞♦
♣❛r❛ ❝♦♥tr♦❧❡ ❞❡ ❛r❝♦ s❡❝✉♥❞ár✐♦✳ P♦r ♦r❛✱ s❡r❛ ♦❜s❡r✈❛❞♦ s❡✉ ❡❢❡✐t♦ ♥♦ ❝✉rt♦ ♠♦♥♦❢ás✐❝♦✱ s❡♠
q✉❡stõ❡s ❞❡ ❞❡s❧✐❣❛♠❡♥t♦ ❡ r❡❧✐❣❛♠❡♥t♦✳
✶✷ ❘❡q✉✐s✐t♦s ❡❧étr✐❝♦s ❞❡ ♣r♦❥❡t♦ ❞❡ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦
◆❡st❛ s❡çã♦ s❡rã♦ ❧✐st❛❞♦s ♦s r❡q✉✐s✐t♦s ❡❧étr✐❝♦s✱ ❛ ♣❛rt❡ ❞♦ ❝á❧❝✉❧♦ ❞♦s ♣❛râ♠❡tr♦s ❜ás✐❝♦s✱ ❢✉♥✲
❞❛♠❡♥t❛✐s ♣❛r❛ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ♦✉ s❡❣✉r❛♥ç❛ ❞♦ ♣r♦❥❡t♦✳ ❇❛s✐❝❛♠❡♥t❡ ♦s r❡q✉✐s✐t♦s sã♦
r❡❧❛❝✐♦♥❛❞♦s ❛♦ ❞❡s❡♠♣❡♥❤♦ ❡ ❛ s❡❣✉r❛♥ç❛✳
❊♥t❡♥❞❡✲s❡ ❝♦♠♦ ❞❡s❡♠♣❡♥❤♦ ♦s ❛s♣❡❝t♦s q✉❡ ❞❡s❝r❡✈❡r❛♠ ♦ ❡❢❡✐t♦ ❞❛ ❧✐♥❤❛ s♦❜ ❞✐✈❡rs❛s ❝♦♥✲
❞✐çõ❡s✱ ❝♦♠♦ ❡♠ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡ ❡ ❡♠ r❡❣✐♠❡ tr❛♥s✐tór✐♦✱ ❝♦♠♦ ❛♦ ♠❛♥♦❜r❛r ✉♠❛ ❝❤❛✈❡ ♦✉ ❛♦
✐♥❝✐❞✐r ✉♠❛ ❞❡s❝❛r❣❛ ❛t♠♦s❢ér✐❝❛✳
❖s r❡q✉✐s✐t♦s ❞❡ s❡❣✉r❛♥ç❛ tr❛❞✉③❡♠ ♦ ❡❢❡✐t♦ ❞❛ ❧✐♥❤❛ ♥♦ ❛♠❜✐❡♥t❡✱ ❡♠ ♣❡ss♦❛s ♦✉ ♦✉tr♦s s❡r❡s
✈✐✈♦s✱ ♥❛ ❢♦r♠❛ ❞❡ r❛❞✐❛çã♦ ♥ã♦✲✐♦♥✐③❛♥t❡✱ r✉í❞♦ ❡ ❛té r✐s❝♦s ❞❡ q✉❡❞❛ ❡ ♣♦❧✉✐çã♦ ✈✐s✉❛❧✳ P❛r❛ ❡st❡s
✸✷
Xeq1
ZLT
V1 = 1 pu
XT1
Xeq2
XT2
Xn
XRS
XCS
❋✐❣✉r❛ ✶✹✿ ❙✐st❡♠❛ ❞❡ tr❛♥s♠✐ssã♦ ❝♦♠ ❝♦♠♣❡♥s❛çã♦ sér✐❡ ❡ ♣❛r❛❧❡❧❛✱ ❝♦♠ ❤✐♣ót❡s❡ ❞❡ ❝✉rt♦
♠♦♥♦❢ás✐❝♦ ♥♦ ♠❡✐♦ ❞❡ ✉♠❛ ❞❛s ❧✐♥❤❛s✳
❡❢❡✐t♦s✱ ❛ ❞✐stâ♥❝✐❛ é ❡❧❡♠❡♥t♦ ❞❡t❡r♠✐♥❛♥t❡✱ ❡ ♦ q✉❡ ✈❛✐ ❡st✐♣✉❧❛r ❛ ❢❛✐①❛ ❞❡ ♣❛ss❛❣❡♠ ❞❛ ❧✐♥❤❛✱
s❡♥❞♦ ♣❛r❝❡❧❛ ✐♠♣♦rt❛♥t❡ ♥♦ ❝✉st♦ ✜♥❛❧✳
❙♦❜r❡ ♦ ❝r✐tér✐♦ ❡❧étr✐❝♦✱ ♣♦❞❡♠♦s t❛♠❜é♠ ❞✐✈✐❞✐r ♦s ❡❢❡✐t♦s ♥❛ ♦r✐❣❡♠✿ s❡❥❛ ♥❛ t❡♥sã♦✱ ❝♦♠♦
❡♠ ❧✐♥❤❛s ❊❍❱✱ ♦✉ ♥❛ ❝♦rr❡♥t❡✱ ♠❛✐s ❡✈✐❞❡♥t❡ ❡♠ ❧✐♥❤❛s ❞❡ ❞✐str✐❜✉✐çã♦✳
✶✷✳✶ ❊❢❡✐t♦s ♦r✐❣✐♥❛❞♦s ♣❡❧❛ t❡♥sã♦
✶✷✳✶✳✶ ❊❢❡✐t♦ ❝♦r♦♥❛
❖ ❡❢❡✐t♦ ❝♦r♦♥❛ é ❛ ❝❛✉s❛ ❞❡ ❞✐✈❡rs♦s ❢❡♥ô♠❡♥♦s ♣r❡s❡♥t❡s ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❡♠ ❧✐♥❤❛s ❞❡ ❡①tr❛✲❛❧t❛
t❡♥sã♦ ✭✸✹✺ ❦❱ ❡ s✉♣❡r✐♦r✮✱ ♠❛s ♣♦❞❡ ♦❝♦rr❡r ❡♠ ♥í✈❡✐s ❞❡ t❡♥sã♦ ♠❛✐s ❜❛✐①♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛
✐♥st❛❧❛çã♦✳
❖ ❡❢❡✐t♦ ❝♦r♦❛ é ✉♠❛ ❞❡s❝❛r❣❛ ♣❛r❝✐❛❧ q✉❡ ♦❝♦rr❡ ❡♠ ✉♠ ♠❡✐♦ ❣❛s♦s♦✱ ♥❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠
❣r❛❞✐❡♥t❡ ❞❡ ❝❛♠♣♦ ❡❧étr✐❝♦ ✐♥t❡♥s♦✱ ❣❡r❛❧♠❡♥t❡ ♣r❡s❡♥t❡ ❡♠ ❝♦♥❞✉t♦r❡s ❝♦♠ ♣❡q✉❡♥♦ r❛✐♦ ❞❡
❝✉r✈❛t✉r❛✱ ♠❛s ♥♦ q✉❛❧ ♥ã♦ ♣r♦✈♦❝❛ ❛ ❞✐sr✉♣çã♦ ❝♦♠♣❧❡t❛ ❞♦ ❣ás✳ ❆ ❣❡♦♠❡tr✐❛ ❞♦ ❝♦♥❞✉t♦r
♣r♦✈♦❝❛rá ✉♠❛ ❞❡❢♦r♠❛çã♦ ♥♦ ❝❛♠♣♦✱ t♦r♥❛♥❞♦ ❛ ❞❡s❝❛r❣❛ ❛✉t♦ss✉st❡♥t❛❞❛ ❡ ❝♦♠ ❛ ✐♦♥✐③❛çã♦
❝♦♥✜♥❛❞❛ ♣ró①✐♠❛ ❛♦ ❝♦♥❞✉t♦r✳
❉❡st❡ ❢❡♥ô♠❡♥♦ ♦r✐❣✐♥❛✲s❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡r❞❛s ❡❧étr✐❝❛s✱ ✐♥t❡r❢❡rê♥❝✐❛ ❡❧❡tr♦♠❛❣♥ét✐❝❛✱ ❡
r✉í❞♦ ❛✉❞í✈❡❧✳ ❖✉tr♦s ❛s♣❡❝t♦s sã♦ ❛ ❣❡r❛çã♦ ❞❡ ♦③ô♥✐♦✱ ❞❡❣r❛❞❛çã♦ ❞❡ ♠❛t❡r✐❛✐s ❡ s✉r❣✐♠❡♥t♦
❞❡ ✉♠ ❜r✐❧❤♦ ✈✐♦❧❡t❛✳
❈♦♠♦ ❡①❡♠♣❧♦ t❡ór✐❝♦✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ✉♠❛ ❣❡♦♠❡tr✐❛ ❝♦❛①✐❛❧ é ♦❜t✐❞♦ ♣❡❧❛ ❢ór♠✉❧❛✿
E =
1
4π ε0
λ
r2
✭✶✷✳✶✮
s❡♥❞♦ λ ❛ ❝❛r❣❛ ♣♦r ❝♦♠♣r✐♠❡♥t♦ ❞❛ ❧✐♥❤❛✱ ♦❜t✐❞❛ ♣❡❧❛ r❡❧❛çã♦ ❝♦♠ ❛ ❝❛♣❛❝✐tâ♥❝✐❛ ❧✐♥❡❛r ❡ ❛ t❡♥sã♦
♥❛ ❧✐♥❤❛✿ λ = C V ✳
❚♦♠❛♥❞♦ ❝♦♠♦ r ♦ r❛✐♦❡q✉✐✈❛❧❡♥t❡ ❞♦ ❝♦♥❞✉t♦r✶✻✱ E s❡rá ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ s✉♣❡r✜❝✐❛❧✱ ♥♦ q✉❛❧
♥ã♦ ♣♦❞❡rá✱ ❡♠ ❝♦♥❞✐çõ❡s ♥♦r♠❛✐s✱ ✉❧tr❛♣❛ss❛r ♦ ✈❛❧♦r ❝rít✐❝♦ ❞❡ ❝♦r♦♥❛✳ P♦r ❡①❡♠♣❧♦✱ ♣❡❧❛ ❧❡✐ ❞❡
P❡❡❦ ❬✷✵❪✱ ❡st❡ ❧✐♠✐t❡ s❡rá ✐❣✉❛❧ ❛
Ec = 3, 0 · 106 mδ
(
1 +
0, 0308√
δ r
)
✭✶✷✳✷✮
s❡♥❞♦ δ ❛ ❞❡♥s✐❞❛❞❡ r❡❧❛t✐✈❛ ❞♦ ❛r ❡ m ✉♠ ❢❛t♦r ❡♠♣ír✐❝♦ r❡❧❛t✐✈♦ à s✉♣❡r❢í❝✐❡ ❞♦ ❝❛❜♦✳
✶✷✳✶✳✷ ❘❛❞✐♦✲✐♥t❡r❢❡rê♥❝✐❛
❖ ❡❢❡✐t♦ ❝♦r♦♥❛ ♣r♦❞✉③ r✉í❞♦ ❡❧❡tr♦♠❛❣♥ét✐❝♦ ❡♠ ✉♠❛ ❛♠♣❧❛ ❢❛✐①❛ ❞❡ ❢r❡q✉ê♥❝✐❛✱ q✉❡ ❡st❡♥❞❡✲s❡
♣❡❧❛s ♦♥❞❛s ❞❡ rá❞✐♦ ❡ ❞❡ ❚❱✳ ❆t✉❛❧♠❡♥t❡ ♥ã♦ ❡①✐st❡ ❝♦♥s❡♥s♦ ✭♥♦r♠❛t✐③❛çã♦ ❛t✉❛❧✐③❛❞❛✮ q✉❛♥t♦
✶✻✑❡q✉✐✈❛❧❡♥t❡✑ ❞❡✈✐❞♦ ❛♦s ❝❛❜♦s ♣♦ss✉ír❡♠ ✉♠❛ ❣❡♦♠❡tr✐❛ ❞❡ ✜♦s q✉❡ ♥ã♦ ❡①❛t❛♠❡♥t❡ ♦ t♦r♥❛ ♣❡r❢❡✐t❛♠❡♥t❡
❝✐r❝✉❧❛r✳
✸✸
❛♦s ❧✐♠✐t❡s ❛ s❡r❡♠ ✐♠♣♦st♦s✱ ❡s♣❡❝✐✜❝❛♠❡♥t❡ q✉❛♥t♦ ❛ ♠❡❞✐çã♦ ❞❛ ✐♥t❡r❢❡rê♥❝✐❛✳ ■st♦ ❞❡✈❡✲s❡ ❛♦s
❡q✉✐♣❛♠❡♥t♦s✱ q✉❡ ✉s✉❛❧♠❡♥t❡ ♠❡❞❡♠ s♦♠❡♥t❡ ✉♠❛ ❢r❡q✉ê♥❝✐❛✱ ❡①✳ ✺✵✵ ❦❍③ ♦✉ ✶ ▼❍③✱ ♠❛s ❛
✐♥t❡r❢❡rê♥❝✐❛ ♥❡♠ s❡♠♣r❡ s❡ ❝♦♥❝❡♥tr❛ ❡♠ ✉♠ ✈❛❧♦r ✉s✉❛❧✳
✶✷✳✶✳✸ ❘✉í❞♦ ❛✉❞í✈❡❧
❖ ❡❢❡✐t♦ ♠❛✐s ♣❡r❝❡♣tí✈❡❧ ♥❛s ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦ ❡♠ ❝♦♥❞✐çõ❡s ♥♦r♠❛✐s é ♦ r✉í❞♦ ❛❝úst✐❝♦✳ ❖
r✉í❞♦ ❞❡ ❛❧t❛ ❢r❡q✉ê♥❝✐❛ ❛ss❡♠❡❧❤❛✲s❡ ❛ ✉♠ s♦♠ ❞❡ ✏❢r✐t❛❞❡✐r❛✑✱ ❝❛r❛❝t❡ríst✐❝♦ ❞♦ ❡❢❡✐t♦ ❝♦r♦♥❛ ❡♠
❝❛❜♦s ❡ ❢❡rr❛❣❡♥s ❞❡ ❧✐♥❤❛s✱ ❡♥q✉❛♥t♦ q✉❡ ♦ r✉í❞♦ ❞❡ ✶✷✵ ❍③✱ é ♠❛✐s ❣r❛✈❡✱ ♦r✐❣✐♥❛❞♦ ♥❛ ✈✐❜r❛çã♦ ❞♦s
♥ú❝❧❡♦s ❞❡ tr❛♥s❢♦r♠❛❞♦r❡s✱ ❡ ❡✈❡♥t✉❛❧♠❡♥t❡ t❛♠❜é♠ ♥❛s ❧✐♥❤❛s✳ ◆♦✈❛♠❡♥t❡✱ t❡♠♦s ❞♦✐s ❡❢❡✐t♦s
♦r✐❣✐♥❛❞♦s ❞❛ t❡♥sã♦ ✭❝♦r♦♥❛✮ ❡ ❞❛ ❝♦rr❡♥t❡ ✭✈✐❜r❛çã♦ ♠❛❣♥ét✐❝❛✮✳
✶✷✳✷ ❈❛♠♣♦ ❡❧étr✐❝♦
❆ ❧✐♥❤❛ ❡♠✐t✐rá ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ t♦❞❛ ❛ s✉❛ ✈✐③✐♥❤❛♥ç❛✱ s❡♥❞♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s✉❛ t❡♥sã♦✳ ❊st❡
❡❢❡✐t♦ é ❛t❡♥✉❛❞♦ s❡ ❛s três ❢❛s❡s ✭♦✉ ♦s ❞♦✐s ♣♦❧♦s✮ ❡st❛r❡♠ ♠❛✐s ♣ró①✐♠❛s ❡♥tr❡ s✐✱ ❢❛③❡♥❞♦ ❝♦♠
q✉❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❞✐st❛♥t❡ ❞❡ ❝❛❞❛ ❢❛s❡ ♦✉ ♣♦❧♦ s❡ ❛♥✉❧❡✳ P♦r r❛③õ❡s ó❜✈✐❛s ❤á ✉♠ ❧✐♠✐t❡ ♣rát✐❝♦
♥❛ ❛♣r♦①✐♠❛çã♦ ❞❛s ❢❛s❡s✳ ❖s ❝❛❜♦s ♣❛r❛✲r❛✐♦s t❛♠❜é♠ ✐♥t❡r❛❣❡♠ ❝♦♠ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ♣♦❞❡♥❞♦
❛t❡♥✉á✲❧♦ ❝♦♠♦ ✉♠❛ ❜❧✐♥❞❛❣❡♠✳ ■♥❝❧✉s✐✈❡ ❥á s❡ ✉t✐❧✐③❛ ❝❛❜♦s ❛t❡rr❛❞♦s ❛❜❛✐①♦ ❞❛s ❧✐♥❤❛s ♣❛r❛
❛t❡♥✉❛r ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❡♠ ár❡❛s ❝rít✐❝❛s✳
❖ ❡❢❡✐t♦ q✉❡ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ♣r♦✈♦❝❛ ❡♠ ♣❡ss♦❛s ❡ ♦❜❥❡t♦s é ❛ ✐♥❞✉çã♦ ❞❡ ❝♦rr❡♥t❡ ♣♦r ♣♦❧❛r✐✲
③❛çã♦✳ ❊st❡ ❡❢❡✐t♦ é ❛♠♣❧✐✜❝❛❞♦ ❞❡✈✐❞♦ à ❞✐st♦rçã♦ ❞♦ ❝❛♠♣♦ ♣r♦✈♦❝❛❞❛ ♣❡❧❛ ♣r❡s❡♥ç❛ ❞❛ ♣❡ss♦❛✱
♦✉ s❡❥❛✱ ♦ ❝❛♠♣♦ t❡♥❞❡ ❛ s❡ ❝♦♥❝❡♥tr❛r ❞❡ ✶✵ ❛ ✷✵ ✈❡③❡s ♥❛ ❝❛❜❡ç❛ ❬✽❪✱ ❝♦♠♣❛r❛❞♦ ❛♦ ❝❛♠♣♦ ♥❛
❛✉sê♥❝✐❛ ❞❡ ♦❜❥❡t♦s✳ P♦rt❛♥t♦✱ ♦ ❝❛♠♣♦ ❡❧étr✐❝♦ ❝❛❧❝✉❧❛❞♦ ♦✉ ♠❡❞✐❞♦ ✭❞❡ ✶ ❛ ✶✵ ❦❱✴♠✮ ❛♣❛r❡♥t❛
s❡r r❡❧❛t✐✈❛♠❡♥t❡ ❜❛✐①♦✱ ♠❛s ♥❛ ♣rát✐❝❛ ❡❧❡ ❡❧❡✈❛✲s❡ ♣❛r❛ ✷✵ ❛ ✷✵✵ ❦❱✴♠✳
❯♠ ❡①♣❡r✐♠❡♥t♦ ❛rtíst✐❝♦ ✭❤tt♣✿✴✴✇✇✇✳r✐❝❤❛r❞❜♦①✳❝♦♠✮ ❞❡♠♦♥str♦✉ ❛ ✐♥❞✉çã♦ ❡♠ ❧â♠♣❛❞❛s
✢✉♦r❡s❝❡♥t❡s ❞❡✈✐❞♦ ❛♦ ❝❛♠♣♦ ❡❧étr✐❝♦✳
❆❞♦t❛✲s❡ ♥♦ ❇r❛s✐❧ ❛ ♦r✐❡♥t❛çã♦ ❞♦ ■❈◆■❘P ❬✶✷❪✱ ♥♦ q✉❛❧ ❧✐♠✐t❛ ❛ ❡①♣♦s✐çã♦ ♦❝✉♣❛❝✐♦♥❛❧ ✭♦✉
s❡❥❛✱ ♣♦r ♣❡ss♦❛❧ q✉❛❧✐✜❝❛❞♦✮ ❡♠ ✶✵ ❦❱✴♠ ❛ ✺✵ ❍③✱ ♦✉ ✽✱✸✸ ❦❱✴♠ ❛ ✻✵ ❍③✱ ❡ ❡①♣♦s✐çã♦ ❞♦ ♣ú❜❧✐❝♦
❡♠ ❣❡r❛❧ ❡♠ ✺ ❦❱✴♠ ❛ ✺✵ ❍③✱ ♦✉ ✹✱✷ ❦❱✴♠ ❛ ✻✵ ❍③✳
✶✷✳✷✳✶ P♦❧❛r✐③❛çã♦ ❡ ✐♥❞✉çã♦ ❡♠ ❝❛❜♦s ♣ró①✐♠♦s
❖ ❝❛♠♣♦ ❡❧étr✐❝♦ t❛♠❜é♠ ♣r♦✈♦❝❛ ♣♦❧❛r✐③❛çã♦ ❡♠ ♦❜❥❡t♦s✱ ✐♥❝❧✉✐♥❞♦ ❝✐r❝✉✐t♦s✱ ❝❡r❝❛s ❡ ❝❛♥❛❧✐✲
③❛çõ❡s✳ ❙❡ ♦s ♦❜❥❡t♦s ❡st✐✈❡r❡♠ ✐s♦❧❛❞♦s✱ ❛ t❡♥sã♦ ✐♥❞✉③✐❞❛ t❡♥❞❡ ❛ s❡ ❞❡s❝❛rr❡❣❛r ❛♦ r❡❛❧✐③❛r ♦
❝♦♥t❛t♦ ❝♦♠ ♦ t❡rr❛✱ q✉❡ ♣♦❞❡ s❡r ♣♦r ❡①❡♠♣❧♦ ✉♠❛ ♣❡ss♦❛ ❛❜r✐♥❞♦ ✉♠❛ ❝❡r❝❛✱ ♦✉ ✉♠❛ ♠❛♥♦❜r❛
❞❡ ♠❛♥✉t❡♥çã♦ ❡♠ ✉♠ ❣❛s♦❞✉t♦✳✳✳
P♦❞❡✲s❡ s✐♠✉❧❛r ♦ ❡❢❡✐t♦ ❞❛ ♣♦❧❛r✐③❛çã♦ ❡♠ ♦✉tr♦s ❝♦♥❞✉t♦r❡s ✭❝❛❜♦s t❡❧❡❢ô♥✐❝♦s✱ ❧✐♥❤❛ ❞❡ ❞✐str✐✲
❜✉✐çã♦ ♦✉ r❡❞❡ ❞❡ ❞❛❞♦s✱ ❝❡r❝❛s ❡ ❡♥❝❛♥❛♠❡♥t♦s✮ ❛tr❛✈és ❞❡ ✉♠❛ ♠❛tr✐③✿ ❝❛❞❛ ❝❛❜♦ ♣❛r❛❧❡❧♦ ❡♥tr❛
❝♦♠♦ ✉♠❛ ❧✐♥❤❛ ❡ ✉♠❛ ❝♦❧✉♥❛ ❛❞✐❝✐♦♥❛❧ ♥❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ ❡ ❛❞♠✐tâ♥❝✐❛✳ P♦❞❡✲s❡ ✐♥❝❧✉s✐✈❡ ❛s✲
s✉♠✐r ✉♠❛ s✐♠✉❧❛çã♦ ❡♠ ❛❧t❛ ❢r❡q✉ê♥❝✐❛✱ s✉♣♦♥❞♦ ✉♠ s✐♥❛❧ ♦r✐❣✐♥❛❞♦ ❞♦ ❡❢❡✐t♦ ❝♦r♦♥❛✱ ✐♥❞✉③✐♥❞♦
✐♥t❡r❢❡rê♥❝✐❛ ❡♠ ✉♠❛ r❡❞❡ ❞❡ ❞❛❞♦s✱ ❡♥tr❡ ♦✉tr❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✳
✶✷✳✷✳✷ ❈♦rr❡♥t❡ ✐ô♥✐❝❛ ✭❈❈✮
◆❛ ♣r❡s❡♥ç❛ ❞❡ ❡❢❡✐t♦ ❝♦r♦♥❛✱ ♦❝♦rr❡ ❛ ❣❡r❛çã♦ ❞❡ í♦♥s ❞❛ ♠❡s♠❛ ♣♦❧❛r✐❞❛❞❡ ❞♦ ❡❧❡tr♦❞♦✱ q✉❡ s❡rã♦
r❡♣❡❧✐❞♦s✳ ◆♦ ❝❛s♦ ❞❛ ❝♦rr❡♥t❡ ❛❧t❡r♥❛❞❛✱ ❛ ✐♥✈❡rsã♦ ❞❡ ♣♦❧❛r✐❞❛❞❡ ♣r♦✈♦❝❛ ✉♠❛ ❛tr❛çã♦ ❞❡st❡s
í♦♥s ♥♦ ❝✐❝❧♦ s❡❣✉✐♥t❡✱ ♣♦ré♠ ❡♠ ❈❈ s❡♠♣r❡ ❤❛✈❡rá ♣r♦❞✉çã♦ ❡ r❡♣✉❧sã♦ ❞❡ í♦♥s✱ ♣r❡❡♥❝❤❡♥❞♦ ♦
❛♠❜✐❡♥t❡ ❡♠ t♦r♥♦ ❞♦ ❝♦♥❞✉t♦r✳
❆ ♣r♦♣❛❣❛çã♦ ❞♦s í♦♥s ♥♦ ❡s♣❛ç♦ é ❛ ❝♦rr❡♥t❡ ✐ô♥✐❝❛✱ q✉❡ ♣r♦✈♦❝❛ ✉♠ ❛✉♠❡♥t♦ ❞♦ ❝❛♠♣♦ ❡❧étr✐❝♦
♥♦ s♦❧♦✱ ❛✉♠❡♥t❛♥❞♦ ❛✐♥❞❛ ♠❛✐s ♦s ❡❢❡✐t♦s s♦❜r❡ s❡r❡s ✈✐✈♦s✳
❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ♦s í♦♥s t❡♥❞❡♠ ❛ ❛tr❛✐r ♣❛rtí❝✉❧❛s ♥♦ ❛r✱ ❝♦♠♦ ♣♦❧✉✐çã♦✱ ♣r♦✈♦❝❛♥❞♦ ♦ ❛❝ú♠✉❧♦
❛♥♦r♠❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❝❛❞❡✐❛s ❞❡ ✐s♦❧❛❞♦r❡s✱ ♠♦t✐✈♦ ♣❡❧♦ q✉❛❧ ♦ ✐s♦❧❛♠❡♥t♦ ❡♠ ❧✐♥❤❛s ❞❡ ❈❈ é
✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞❡ ♣r♦❥❡t♦✳
❖ ❧✐♠✐t❡ ❞❡ ♣r♦❥❡t♦ ✉s✉❛❧✱ ♥♦ ❧✐♠✐t❡ ❞❛ ❢❛✐①❛✱ é ❞❡ 5 nA/m2✳
✶✷✳✸ ❊❢❡✐t♦s ♦r✐❣✐♥❛❞♦s ♣❡❧❛ ❝♦rr❡♥t❡
✶✷✳✸✳✶ ❆♠♣❛❝✐❞❛❞❡
❆ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝♦rr❡♥t❡ ❞❡ ✉♠ ❝❛❜♦ ❞❡♣❡♥❞❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❞❡ três ❢❛t♦r❡s✿ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛✱
t❡♠♣❡r❛t✉r❛ ♠á①✐♠❛ ❡ ✢❡❝❤❛✳ ❖ ❡q✉✐❧í❜r✐♦ ❞❡st❡s três ❢❛t♦r❡s ✐♥❞✐❝❛ ❛ ♠❡❧❤♦r ❛♣❧✐❝❛çã♦ ❞♦ ❝❛❜♦✳
✸✹
❆ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛ tr❛❞✉③ ❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ♣❡r❞❛s✱ ❧♦❣♦ ❡♠ ❧✐♥❤❛s ❧♦♥❣❛s ❡st❡ ❢❛t♦r s❡rá
❞❡t❡r♠✐♥❛♥t❡✳ ❊✈❡♥t✉❛❧♠❡♥t❡ ✉♠ ❝❛❜♦ ❝♦♠ ♠❛✐♦r r❡s✐stê♥❝✐❛ ♣♦❞❡ s❡r ✉s❛❞♦ ❡♠ tr❡❝❤♦s ❡s♣❡❝í✜❝♦s✱
t❛✐s ❝♦♠♦ ✉♠❛ tr❛✈❡ss✐❛✱ ❛♦♥❞❡ ❛ ✢❡❝❤❛ s❡rá ❝rít✐❝❛✳
❆ ✢❡❝❤❛ ❞♦ ❝♦♥❞✉t♦r é ❞❡✜♥✐❞❛ ♣❡❧❛ t❡♠♣❡r❛t✉r❛ ❛t✉❛❧ ♥♦ ❝♦♥❞✉t♦r✱ ❡ ❛ tr❛çã♦ ♠❡❝â♥✐❝❛ ♥♦
q✉❛❧ ♦ ❝❛❜♦ ❡stá s♦❧✐❝✐t❛❞♦✳ ❆t✉❛❧♠❡♥t❡ ❡st✉❞❛✲s❡ ❛ ❡❧❡✈❛çã♦ ❞❛ tr❛çã♦ ❞❡ ♣r♦❥❡t♦✱ ❝♦♠ ♦ ❛❞✈❡♥t♦
❞❛ ♠♦♥✐t♦r❛çã♦ ♦♥✲❧✐♥❡ ❞❛ ❧✐♥❤❛ ♣♦❞❡✲s❡ ❛❝♦♠♣❛♥❤❛r ♦ ❞❡s❡♠♣❡♥❤♦✳
❆ t❡♠♣❡r❛t✉r❛ ❞♦ ❝❛❜♦ é ✐♥✢✉❡♥❝✐❛❞❛ ♣❡❧❛ ❝♦rr❡♥t❡ ❡ r❛❞✐❛çã♦ s♦❧❛r ❝♦♠♦ ❡❧❡♠❡♥t♦s ❞❡ ❡♥tr❛❞❛
❞❡ ❡♥❡r❣✐❛✱ ❡ ❛ ❞✐ss✐♣❛çã♦ ♣♦r ❝♦♥✈❡❝çã♦ ♥❛t✉r❛❧✱ ❝♦♥✈❡❝çã♦ ❢♦rç❛❞❛ ✭✈❡♥t♦✮ ❡ r❛❞✐❛çã♦✳ ❖ ❝♦♥❥✉♥t♦
❞❡st❡s ❡❧❡♠❡♥t♦s ♣r♦❞✉③ ✉♠ ❛❧❝❛♥❝❡ ❡st❛t✐st✐❝♦ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ❞♦ ❝❛❜♦✱ q✉❡ ♣♦r s✉❛ ✈❡③ ✐♥✢✉❡♥❝✐❛
♥♦s ❞♦✐s ❢❛t♦r❡s ❛♥t❡r✐♦r❡s✳
❯♠ ❛s♣❡❝t♦ ♠❛✐s ❝♦♠♣❧❡①♦ é ♦ ❝á❧❝✉❧♦ ❞❛ ❛♠♣❛❝✐❞❛❞❡ ❡♠ ❝♦♥❞✐çõ❡s tr❛♥s✐tór✐❛s✱ ❝♦♠♦ ❡♠ ❝✉rt♦✲
❝✐r❝✉✐t♦✳ ◆❡st❛ ♠♦❞❡❧❛❣❡♠ ♦ ❝❛❜♦ r❡❝❡❜❡ ✉♠ ♣✉❧s♦ ❞❡ ❡♥❡r❣✐❛ tér♠✐❝❛✱ ♥♦ q✉❛❧ s✉❛ ❞✐ss✐♣❛çã♦ é
r❡❧❛t✐✈❛♠❡♥t❡ ❧❡♥t❛✱ ❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡st❛ ❞✐♥â♠✐❝❛ é ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❝♦♥❞✐çõ❡s ❞❡ ❡♠❡r❣ê♥❝✐❛✳
✶✷✳✸✳✷ ❈❛♠♣♦ ♠❛❣♥ét✐❝♦
❆❞♦t❛✲s❡ ♥♦ ❇r❛s✐❧ ❛ ♦r✐❡♥t❛çã♦ ❞♦ ■❈◆■❘P✱ ♥♦ q✉❛❧ ❧✐♠✐t❛ ❛ ❡①♣♦s✐çã♦ ♦❝✉♣❛❝✐♦♥❛❧ ❡♠ ✺✵✵ ➭❚ ❛
✺✵ ❍③✱ ♦✉ ✹✷✵ ➭❚ ❛ ✻✵ ❍③✱ ❡ ❡①♣♦s✐çã♦ ❞♦ ♣ú❜❧✐❝♦ ❡♠ ❣❡r❛❧ ❡♠ ✶✵✵ ➭❚ ❛ ✺✵ ❍③✱ ♦✉ ✽✸ ➭❚ ❛ ✻✵ ❍③✳
✶✷✳✸✳✸ ■♥❞✉çã♦
❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ❛ ♣♦❧❛r✐③❛çã♦ ♣❡❧♦ ❝❛♠♣♦ ❡❧étr✐❝♦✱ ♥❛ s❡çã♦ ✶✷✳✷✱ ❛ ✐♥❞✉çã♦ ♠❛❣♥ét✐❝❛ s❡rá
♣r♦✈✐❞❛ ♣❡❧❛ ✐♥❞✉tâ♥❝✐❛ ♠út✉❛ ❡♥tr❡ ❝✐r❝✉✐t♦s✳ ◆❡st❡ ❝❛s♦✱ ❛ ♠❛tr✐③ ✐♠♣❡❞â♥❝✐❛ ✏❡①♣❛♥❞✐❞❛✑
✭✐♥❝♦r♣♦r❛♥❞♦ ♦s ❝♦♥❞✉t♦r❡s ❡①t❡r♥♦s✮ é q✉❡ ❞❡t❡r♠✐♥❛rá ♦ ❡❢❡✐t♦✱ ❛♦ ❝♦♥trár✐♦ ❞❛ ♣♦❧❛r✐③❛çã♦ q✉❡
é ✈✐st❛ ♣❡❧❛ ♠❛tr✐③ ❛❞♠✐tâ♥❝✐❛✳
✶✷✳✹ ▼❛♥✉t❡♥çã♦ ❡♠ ❧✐♥❤❛ ✈✐✈❛
❊♠ s✐st❡♠❛s ❝♦♠♦ ❞♦ ❇r❛s✐❧✱ ❝♦♠ ♣♦✉❝❛ t♦❧❡râ♥❝✐❛ à s❛í❞❛ ❞❡ ❧✐♥❤❛s✱ é ♥❡❝❡ssár✐❛ ❛ ♣rát✐❝❛ ❞❡
♠❛♥✉t❡♥çã♦ ❡♠ ❧✐♥❤❛ ✈✐✈❛✳ P❛r❛ ♦ ♣r♦❥❡t♦ ❞❡ ❧✐♥❤❛s✱ ♥ã♦ ❤á ✉♠❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡✜♥✐❞❛✱ s❡♥❞♦
♥❡❝❡ssár✐♦ ❛❞♦t❛r ❛ ♣rát✐❝❛ ❞❡ ❝❛❞❛ ❡♠♣r❡s❛✳
✶✷✳✺ ❉❡s❡♠♣❡♥❤♦ ❡♠ s♦❜r❡t❡♥sõ❡s
❖ ❡st✉❞♦ ❞❡ s♦❜r❡t❡♥sõ❡s ♣♦❞❡ s❡r r❡❛❧✐③❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♠ ❡st✉❞♦ ❞❡ ♣r♦♣❛❣❛çã♦ ❞❡ ♦♥❞❛s✳ ❆
♥♦çã♦ ❜ás✐❝❛ é ❞❡♠♦♥str❛❞❛ ♥❛ s❡çã♦ ❈✳✺✳
❯♠❛ s♦❜r❡t❡♥sã♦ é q✉❛❧q✉❡r t❡♥sã♦ tr❛♥s✐tór✐❛ ❡♥tr❡ ❢❛s❡ ❡ t❡rr❛✱ ♦✉ ❡♥tr❡ ❢❛s❡s✱ ❝✉❥♦ ✈❛❧♦r ❞❡
♣✐❝♦ s❡❥❛ s✉♣❡r✐♦r ❛♦ ✈❛❧♦r ❞❛ t❡♥sã♦ ♠á①✐♠❛ ❞♦ s✐st❡♠❛ ✭Vm
√
2√
3
♣❛r❛ ❢❛s❡✲t❡rr❛✱ Vm
√
2 ❡♥tr❡ ❢❛s❡s✮
◆♦ ❡st✉❞♦ ❞❡ s♦❜r❡t❡♥sã♦ ❡♥t❡♥❞❡✲s❡ ✉♠ r✐s❝♦ ❞❡ ❢❛❧❤❛✱ ♣♦r ♠❛♥♦❜r❛✱ ❞♦ ❞✐❡❧étr✐❝♦ r♦♠♣❡r✲
s❡✳ ❊♠ ❣❡r❛❧ ❛ss✉♠❡✲s❡ ✉♠ ✈❛❧♦r ❞❡ 10−3 ♣❛r❛ r✐s❝♦ ❞❡ ❢❛❧❤❛ ❡♥tr❡ ❢❛s❡✲t❡rr❛ ❡♠ ♠❛♥♦❜r❛ ❞❡
❡♥❡r❣✐③❛çã♦✱ ♦✉ s❡❥❛✱ ❝❤❛♥❝❡ ❞❡ ✶ ♠❛♥♦❜r❛ ❡♠ ✶✵✵✵ ❞❡ ❢❛❧❤❛r✳
❊♠ ❧✐♥❤❛s ❞❡ ❈❈✱ ♦ r✐s❝♦ tí♣✐❝♦ é ♥❛ ♦❝♦rrê♥❝✐❛ ❞❡ ❝✉rt♦✲❝✐r❝✉✐t♦ ❡♠ ✉♠ ❞♦s ♣♦❧♦s✱ ❤❛✈❡♥❞♦
s♦❜r❡t❡♥sã♦ ♥♦ ♣♦❧♦ r❡♠❛♥❡s❝❡♥t❡✳
✶✷✳✻ ❙♦❜r❡t❡♥sõ❡s tr❛♥s✐tór✐❛s ❞❡ ❢r❡♥t❡ rá♣✐❞❛ ✭s✉rt♦s ❛t♠♦s❢ér✐❝♦s✮
❙♦❜r❡t❡♥sõ❡s ♦r✐❣✐♥❛❞❛s ❡♠ ❧✐♥❤❛s ❞❡ tr❛♥s♠✐ssã♦✱ ♥♦ q✉❛❧ ♦♥❞❛s ✈✐❛❥❛♥t❡s ♣♦❞❡rã♦ ❝❤❡❣❛r ♥❛
s✉❜❡st❛çã♦ ❡ ❞❛♥✐✜❝❛r ♦s ❡q✉✐♣❛♠❡♥t♦s✳
❖r❞❡♠ ❞❡ ✶ ❛ ✶✵ ➭s ❞❡ t❡♠♣♦ ❞❡ ❢r❡♥t❡✱ ✺✵ ❛ ✶✵✵ ➭s ❞❡ t❡♠♣♦ ❞❡ ❝❛✉❞❛✳ ❖ t❡♠♣♦ ❞❡ ♥♦r♠❛ é
✶✱✷✴ ✺✵ ➭s✳
P❛râ♠❡tr♦ s✐❣♥✐✜❝❛♥t❡ ❡♠ s✐st❡♠❛s ❞❡ t❡♥sã♦ ❛té ✷✸✵ ❦❱✳
✶✷✳✼ ❙♦❜r❡t❡♥sõ❡s tr❛♥s✐tór✐❛s ❞❡ ❢r❡♥t❡ ❧❡♥t❛ ✭s✉rt♦s ❞❡ ♠❛♥♦❜r❛✮
P❛râ♠❡tr♦ s✐❣♥✐✜❝❛♥t❡ ❡♠ s✐st❡♠❛s ❞❡ t❡♥sã♦ ❛❝✐♠❛ ❞❡ ✷✸✵ ❦❱✳
❖r❞❡♠ ❞❡ ✶✵✵ ❛ ✺✵✵ ➭s ❞❡ t❡♠♣♦ ❞❡ ❢r❡♥t❡✱ ✶ ❛ ✺ ♠s ❞❡ t❡♠♣♦ ❞❡ ❝❛✉❞❛✳ ❖ t❡♠♣♦ ❞❡ ♥♦r♠❛ é
✷✺✵✴ ✷✺✵✵ ➭s✳
❖r✐❣❡♥s
Pr♦❝✉r❛✲s❡ ❡st✉❞❛r ❛s s♦❜r❡t❡♥sõ❡s ♥♦ t❡r♠✐♥❛❧ ❞❛ ♦r✐❣❡♠ ❞♦ s✉rt♦ ❡ ♥♦ t❡r♠✐♥❛❧ ♦♣♦st♦✱ ❡st❡
s❡❣✉♥❞♦ ❡♠ ❣❡r❛❧ ❛♣r❡s❡♥t❛rá ❛ ♠❛✐♦r s♦❜r❡t❡♥sã♦✳
✸✺
❋✐❣✉r❛ ✶✺✿ ❙♦❜r❡t❡♥sõ❡s ❞❡