❯❱ P❯

P PÓ❯

❱ ❮ P P❱ ❯❩

P❳Õ P ❱ ❳P P

❨❯ ❯❨

❯P PP♠♣s ❯♥rstár♦ ♦ ♠á

é♠Prárs

❯❱ P❯

P PÓ❯

❨❯ ❯❨

❱ ❮ P P❱ ❯❩

P❳Õ P ❱ ❳P P

❯P PP♠♣s ❯♥rstár♦ ♦ ♠á

é♠Prárs

❯❱ P❯

P PÓ❯

❨❯ ❯❨

❱ ❮ P P❱ ❯❩

P❳Õ P ❱ ❳P P

ssrtçã♦ s♠t à♥ ①♠♥♦r ♦ Pr♦r♠ Pósrçã♦♠ ♥♥r étr ❯P ♣r ♦t♥çã♦ ♦r str ♠ ♥♥r étr ♥ ár ♦♠♥çõs

❯P PP♠♣s ❯♥rstár♦ ♦ ♠á

é♠Prárs

Fujiyoshi, Daiyuki Maia, 1991- Modelagem através do método das diferenças finitasno domínio do tempo (fdtd) de solos dispersivosutilizando aproximações de padé validada comexperimentos em campo / Daiyuki Maia Fujiyoshi. - 2015.

Orientador: Rodrigo Melo e Silva deOliveira. Dissertação (Mestrado) - UniversidadeFederal do Pará, Instituto de Tecnologia,Programa de Pós-Graduação em EngenhariaElétrica, Belém, 2015.

1. Descargas elétricas - modelosmatemáticos. 2. Linhas elétricas subterrâneas.3. Condutividade elétrica. 4. Métodos desimulação. I. Título.

CDD 22. ed. 551.5632

Dados Internacionais de Catalogação-na-Publicação (CIP)Sistema de Bibliotecas da UFPA

❱

❯❱ P❯

P PÓ❯

❱ ❮ P P❱ ❯❩

P❳Õ P ❱ ❳P P

❯ ❨❯ ❯❨

❯ ❱ ❳P❱ P P PÓ❯ ❯❱ P ❯ ❯ P ❯ ❯Õ

P❱

❳

Pr♦ r ♦r♦ ♦ r PP❯P

Pr♦ r ❱t♦r ①♥r♦ ♠tr PP❯P

Pr♦ r ♦ã♦ P♦ r ❱r PP❯P

Pr♦ r r♦ ♥♥s ♠ ❯P

❱

Pr♦ r ♦ ♦♥çs Ps PP❯P

❱

rr♦ ts♦r♦

s♣♥ ♠ ♠♦ à ár t

♥ ♠ ♦♥çã♦

P♦r ♠♦ ♦ ♥♥sá s♦rç♦ s♦

s

❱

♦s ♠s ♣s ❨♦ ♠ r♠ã♦ ②

❱

r♠♥t♦s

rç♦ ♣r♠r♠♥t ♦s ♠s ♣s ❨♦ ♦ ♠ r♠ã♦ ②

t♦ ♠♥ ♠í ♣♦r s♠♣r ♠ ♣♦r♠ ♥♥tr♠ ♠ t♦♦s ♦s ♠♦♠♥t♦s

♠♥ ♥♦ ♣r♦♥ rtã♦ t♠é♠ ♦ ♠ ♠str r s

ós♦♦ srt♦r ♣st ♣♦r t♦♦s ♥♥t♦s r♦♥♠♥t♦s r♦s ❯♠

r♠♥t♦ s♣ à ♠♥ ♠ ♦♠♣♥r ♦r♥ Prs ♣♦r t♦♦ ♠♦r

r♥♦ ♣ê♥ ♥♥t♦ ♦ ♥♦ ♦rrr st ♠♣r♥♠♥t♦

rç♦ ♠♥s♠♥t ♦ ♠ ♦r♥t♦r Pr♦ r ♦r♦ r ♣♦r s

①♥t ♦r♥tçã♦ é♠ ss♦ ♣ s ♣ê♥ ♦♥s♦s q ã♦ é♠ ♦s s♣t♦s

♣r♠♥t ê♠♦s

rç♦ t♠é♠ ♦s ♠s ♠♦s ♦ ♦rtór♦ tr♦♠♥ts♠♦

t♦♥ ♠♦♥ ❲s♥t♦♥ ♦r♥ r♦ ♦♥ts ❲t♦♥ ss♦♥ Pr♦ ♠s

♥tr♥ts s♣♠♥t ①tr♥♦ ♠♥ rtã♦ ♦ ❲s♥t♦♥ ♣ s ss♥

♦♥trçã♦ ♥♦ s♥♦♠♥t♦ st tr♦ ♦ ♠♦♥ à ♦r♥ ♣ ♣rr ♥♦s

①♣r♠♥t♦s r③♦s ♠ ♠♣♦ ♦ t♦♥ ♣♦r s rçã♦ s♠♣r s s♣♦r

♠ r ♠é♠ ♦str rr ♦ ♥ ú♦ s♠♥t♦ ♥ ♥

♦q♠ ♣♦ t♦t ♣♦♦ ♣♦r ♦r♥r s ♦♥çõs r③r s ♠çõs ♥ ár

tr♦♥♦rt

♦str rr ♣r♦♥♠♥t ♦s ♠s ♠♦s rs ♦

♥tr♥♦♥ ♦ ②♦ ♥t ♦ Prá ♣♦r t♦♦ ♥♥t♦ ♦ à ♠♠

P♦r ♠ rç♦ ♦ P ❯P tr♦♥♦rt PP ♣ ♦s

♥çã♦ ♥tí ♣ ♥rstrtr ♦r♥ ♣r s♥♦r st tr♦

❱

st í♠♦♦s

x y z ❱t♦rs ❯♥tár♦s s rçõs x y z

~A ♠♣♦ ❱t♦r

Ax, Ay Az ♦♠♣♦♥♥ts ♦ ❱t♦r ~A ♣r ♦♦r♥s t♥rs

j ❯♥ ♠♥ár

A sr ♦♠♣①♦

A ❱t♦r ♦♠♣①♦

∇× ~A ♣r♦r ♦t♦♥ ~A

∇ · ~A ♣r♦r r♥t ~A

∂f∂α

r Pr f ♠ rçã♦ α

µ Pr♠ ♥ét

µ0 Pr♠ ♥ét ♦ ❱á♦

ε Pr♠ss étr

ε′ ε′′ Prts r ♠♥ár ε

εr Pr♠ss étr t

ε′r ε′′r Prts r ♠♥ár εr

ε0 Pr♠ss étr ♦ ❱á♦

σ ♦♥t étr

σDC ♦♥t étr ♣r ①s rqê♥s

σ(f) ♦♥t étr ♣♥♥t rqê♥ f

ρ sst étr

ρDC sst étr ♣r ①s rqê♥s

ρ(f) sst étr ♣♥♥t rqê♥ f

ω rqê♥ ♥r

❳

st s

P ♦♥♦t♦♥ Prt② t ②r

♥tr♥ ♠♦♠♥

st qrs t♦

♥ qr rr♦r

②♥tss ♥ ♥②ss ♦ r♦♥♥ ②st♠s

r♥s♥t r♦♥♥ sst♥

❳

♠ár♦

♥tr♦çã♦

♦♥tú♦ ♦ ♣ít♦

♥tr♦çã♦ r

t♦s st r♦

r♥③çã♦ st r♦

♥♠♥tçã♦ ór

♦♥tú♦ ♦ ♣ít♦

st♠s trr♠♥t♦

♥çã♦ ♥

♥ás r♥stór ♠ st♦♥ár♦ ♦s st♠s tr

r♠♥t♦

s♣rsã♦ ♦ ♦♦

trs s♣rs♦s

♦r♠çã♦ ❱sr♦♣♦ ♣r ♦♦s s♣rs♦s

♦♠ ♦s ♦♠ ♦♥t étr ♣♥♥t rqê♥

trés ♦ ét♦♦

♦♥tú♦ ♦ ♣ít♦

ét♦♦ s r♥çs ♥ts ♥♦ ♦♠í♥♦ ♦ ♠♣♦

s♣t♦s rs s♦r ♦ ét♦♦

❳

♦r♠çã♦ t♠át ♦ ♠ét♦♦

rtér♦s ♣rsã♦ st

♦♠ trs s♣rs♦s rtrár♦s ♣♦r ♦♥sr♥♦

s σ = σ(ω)

r sssã♦ s♦r ♣r♦①♠çã♦ ♥çõs trés ♦ P♦♥ô

♠♦ Pé

♦♠ trs s♣rs♦s rtrár♦s ♣ ♦ é

t♦♦

♦♥srçõs ♦r ♦s ①♣r♠♥t♦s

♦♥tú♦ ♦ ♣ít♦

t♣ ①♣r♠♥t

q♣♠♥t♦s ❯t③♦s ♦♥srçõs

♦r ♦ r♦r rt♦ ♥sã♦

♦r P♦♥t sst ♦♥t à í ♦ r♦r

♦r ♦ s♦só♣♦

♦r ♠♥tçã♦ ♦s q♣♠♥t♦s

❱çã♦ ♦♠ ♦ ♦♦ s♣rs♦ ♣ ♦ ét♦♦

♦♥tú♦ ♦ ♣ít♦

♦ ③çã♦ ♦s ①♣r♠♥t♦s ♠ ♠♣♦

♥ás r♥stór ♦ Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r

❯♠ st ❱rt

①♣r♠♥t♦ ♠ ♠♣♦

♠çã♦ ♥♠ér

st♦s

♥ás r♥stór ♦s Pr♦♠s

❳

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠ st

♥trr ❱rt♠♥t

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠ st

♥trr ♦r③♦♥t♠♥t

♥ás r♥stór ♦s Pr♦♠s

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠ st

♥trr ❱rt♠♥t

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s sts

❱rts tr♠♥t ♦♥ts

♥ás ♥ttt ♦s st♦s

MSE ♥ qr rr♦r

s ♦♠♣rts

♦♥srçõs ♥s

❳

st strçõs

♣rs♥tçã♦ ♦ á♦ t♥sã♦ ♥③ ♦rr♥t ♥t

①♠♣♦ ♠ r qr ♦ ♣r ♠ s♥ ♦♠ ♦r♠

♦♥ ♠ sr ssq♥t ♠ ♠ sst♠ trr♠♥t♦ ♦♠

♣♦st♦ ♣♦r ♠ st ♥trr rt♠♥t st r ♦ ♦t trés

♠ s♠çã♦ t③♥♦ ♦ ♠ét♦♦

♣rs♥tçã♦ ♦ st♣ ♠çã♦ ♣r tr♠♥r ♦♥t

♦r♦ ♦♠

rá♦ ♦♠♣r♥♦ ♦s s♥s t♥sã♦ ♥③ ♦t♦s ♣s s♠çõs

t③♥♦ ♦ ♠ét♦♦ ♦♠ r♥ts ♦rs ♣r♠ss r

t ♦ s♦♦ ♦♠♣r♦s ♦s s♦s ♦♠ εr =

♣t♦r ♣s ♣rs st♦ ♠ ♠♣♦ étr♦ stát♦ ~Ea

♣t♦ ❬❪

t♣ ①♣r♠♥t ♠♣♠♥t♦ ♣♦r ❱sr♦ t ♣r r ♣♥

ê♥ rsst ♦♠ rqê♥ ♣t♦ ❬❪

r♠ ♠♣â♥ ♦t♦ ♣rtr ♦ st♣ ①♣r♠♥t ♠♣♠♥

t♦ ♣♦r ❱sr♦ t ♠ ♠ s♦♦ ♦♠ ρDC = 495 Ω♠ ♣t♦ ❬❪

❱rçã♦ ρr ♦ ♦♥♦ ① ③ ③ ♣r ♦s s♦♦s st

♦s ♠ ❬❪ ♦♠♣rçã♦ ♥tr ♦ s♥ ①♣r♠♥t ♦s s♥s ♦t♦s

♣♦r s♠çõs ♦♠ ♣râ♠tr♦s ♦♥st♥ts ♣râ♠tr♦s ♣♥♥ts

rqê♥ ♣t♦ ❬❪

❳❱

♣rs♥tçã♦ é ❨

rá♦ ♦♠♣r♥♦ ♣r♦①♠çã♦ ♥çã♦ e−x ♣♦r ér ②♦r ♣♦r

♣♦♥ô♠♦ Pé ♦♠♣rçã♦ ♥tr s s ♣r♦①♠çõs ♥çã♦

♦r♥ rá♦ ♦ rr♦ s♦t♦

♦♥rçã♦ ①♣r♠♥t ♣r♦♣♦st ♣♦r ♥ ♣t♦ ❬❪

q♣♠♥t♦s t③♦s ♥♦s ①♣r♠♥t♦s ♠ ♠♣♦

P♥ r♦♥t ♦ r♦r srt♦ t③♦ ♥♦s ①♣r♠♥t♦s

♥ ♦ r♦r ♠♦ ♦ s♥ ♠♦♦ ♠t♠t♠♥t ♣r s

s♠çõs ♥♠érs

♥ ♦t♦ ♦ ♦♥tr ♦s q♣♠♥t♦s à r étr

❱st ér ♦ ♥tr♦ ♥♦♦ tr♦rástr♦♥♦rt stq

♥ ár ♦♥ ♦r♠ r③♦s ♦s ①♣r♠♥t♦s ♠ ♠♣♦

♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr

♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ ú♥ st rt♠♥t ♥trr

♠♥s ♦ ①♣r♠♥t♦ r③♦ ♠ ♠♣♦ stq ♥♦s ♣♦♥t♦s

♠çã♦ st ♥çã♦ ♦rr♥t stq ♥♦s rt♦s t♥sã♦

♦rr♥t

♣rs♥tçã♦ ♥♦ ♠♥t ♦ s♦tr ♦ st♣ ①♣r♠♥t ♦

sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ ú♥ st rt♠♥t ♥tr

r

r ♦t ♥♦ ①♣r♠♥t♦ ♣rtr r③ã♦ ♥tr vR(t) iR(t) ♣r

r ♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) =

2054 Ω

r ♦t ♥♦ ①♣r♠♥t♦ ♠ st ♥trr rt

♠♥t ♥ tr ♥ ♦ ♦r R = 45 Ω

rs ♥çã♦ σ(f) ♣r σDC = 0,02052 ♠ ♦ts ♣rtr ①

♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

❳❱

♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦s ♠ ♠♣♦ ♦t ♣♦r

s♠çã♦ ♥♠ér

♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦s ♠ ♠♣♦ ♦t♦ ♣♦r

s♠çã♦ ♥♠ér

♦♥rçã♦ ♦ ♥s♦ ♦ sst♠ trr♠♥t♦ ♠ st rt

♠♥t ♥trr t ♥ st ♥çã♦ ♦rr♥t ♦ rt♦

♦rr♥t

r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r

♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) =

2020 Ω

r ♦ ♦t ♥♦ ①♣r♠♥t♦ ♠ st rt ♥

tr ♥ ♦ ♦r 44,5 Ω rs ♥çã♦ σ(f) ♣r σDC =

0,02134 ♠ ♦ts ♣rtr ①♣rssã♦ rs♣t ♣r♦①♠çã♦

♣♦r ♣♦♥ô♠♦ Pé

♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t♦ ♣♦r

s♠çã♦ ♥♠ér

♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t ♣♦r

s♠çã♦ ♥♠ér

♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr

♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♦r③♦♥t♠♥t ♥trr

♠♥s ♦ sst♠ trr♠♥t♦ ♦r♠♦ ♣♦r ♠ st ♥trr ♦

r③♦♥t♠♥t

♣rs♥tçã♦ ♥♦ ♠♥t ♦ s♦tr ♦ st♣ ①♣r♠♥t ♦

sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♦r③♦♥t♠♥t ♥trr

r ♦t ①♣r♠♥t♠♥t ♣rtr r③ã♦ ♥tr vR(t) iR(t) ♣r

r ♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) =

2060 Ω

❳❱

st♦ ♣r♠♥r ♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦

♠ ♠♣♦ ♦t ♣♦r s♠çã♦ ♥♠ér t③♥♦ σDC = 0, 02134

♠ q é ♦ ♦r t③♦ ♥♦ s♦ ♠ st ♥trr rt♠♥t

sçã♦

rs ♥çã♦ σ(f) ♣r σDC = 0,01626 ♠ ♦ts ♣rtr ①

♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

♥s ♠♦s ♥♦s ①♣r♠♥t♦s ♠ ♠♣♦ ♦t♦s ♣♦r s♠çã♦ ♥♠é

r ♦rr♥t ♥t ♥sã♦ ♥③

r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r

♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) =

2070 Ω

r ♦ ♦t ♥♦ ①♣r♠♥t♦ ♠ st rt ♥

tr ♥ ♦ ♦r 42,5 Ω

rs ♥çã♦ σ(f) ♣r σDC = 0,02203 ♠ ♦ts ♣rtr ①

♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t♦ ♣♦r

s♠çã♦ ♥♠ér

♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t ♣♦r

s♠çã♦ ♥♠ér

♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr

♠♥t♦ ♦♠♣♦st♦ ♣♦r s sts rt♠♥t ♥trrs tr♠♥t

♦♥ts

t♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s sts

rt♠♥t ♥trrs t ♥♦ ♣♦♥t♦ ♥çã♦ ♦ s♥ ♦ srt♦

♥ ♦♥①ã♦ s s sts q ♦♠♣õ♠ ♦ sst♠ trr♠♥t♦ ♥s♦

❳❱

♣rs♥tçã♦ ♦ st♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦

♣♦r s sts rt♠♥t ♥trrs tr♠♥t ♦♥ts ♥♦

♠♥t ♦ s♦tr

r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r

♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) =

2160 Ω

♥s ♦t♦s ①♣r♠♥t♠♥t ♣♦r s♠çã♦ ♥♠ér ♦rr♥t

♥t ♥sã♦ ♥③

❳❱

st s

❱♦rs ♦s ♣râ♠tr♦s Aβ toβ τβ

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02052 ♠

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02134 ♠

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,01626 ♠

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02203 ♠

① rçã♦ σ(f) ♥ ♥ rqê♥ ♥s

♦♠♣rçã♦ ♦s ♦rs Vp Ip ♥tr ♦s ♦s ①♣r♠♥ts ♦s

♦s ♦t♦s ♥s s♠çõs ♥♠érs ♦♠ s♦♦s s♣rs♦s ♥ã♦

s♣rs♦s

♥ás ♦ MSE ♦ ♥tr ♦s ♦s ①♣r♠♥ts ♦s ♦s ♦

t♦s ♣rtr s s♠çõs ♥♠érs ♦♠ s♦♦s s♣rs♦s ♥ã♦

s♣rs♦s

♠♣♦ ①çã♦ s s♠çõs ♦♠ ♦ ♠♦♦ s♦♦ s♣rs♦

♦♠ ♦ ♠♦♦ s♦♦ ♥ã♦ s♣rs♦

Pçõs r♦♥s st tr♦

r ♦st♦ ❱ ②♦s ♥ ③r

♠♣s ♦♥t ♦r sqr r♦♥♥ rs ♥ ♦ rsstt② s♦s ♥♥ ♦ ♥

t♦♥ tr♦ sr ♦r♥ ♦ tr♦stts ♦ ♥♦ ♣♣

s♠♥t♦ r ②♦s P r♦

rú♦ ♥ r st♦ ♥♠ér♦①♣r♠♥t srt♦s ♣r♦♦♦s ♣♦r

srs t♠♦sérs ♠ ♠s trr♠♥t♦ étr♦ ♣r♦♦ ♣r ♣çã♦

♥♦ ❳❳ ♠♥ár♦ ♦♥ Pr♦çã♦ r♥s♠ssã♦ ♥r étr ❳❳

P ♦tr♦ ♦③ ♦ ç

②♦s ♥ r ❯♠ ét♦♦ ♣r st♠r Pr♠ss

étr t ♦ ♦♦ ♣r P♦ssí ♣çã♦ ♠ ♠♣♦ ♦ ♠♣ós♦

rsr♦ r♦♦♥s ♣t♦trô♥ ♦ ♦♥rss♦ rsr♦

tr♦♠♥ts♠♦ rt

trs ♣çõs

ts♥ rú♦ ②♦s ♥ r ❯♠

ét♦♦ s♣tr ♣r ♦③çã♦ ♥trs♦ ♠ ♠♥ts ♥♦♦r ♦ ♠

♣ós♦ rsr♦ r♦♦♥s ♣t♦trô♥ ♦ ♦♥rss♦ rsr♦

tr♦♠♥ts♠♦ rt

s♠♦

st tr♦ ♦ s♦♦ ♠③ô♥♦ rã♦ é♠ é ♠♦♦ ♦♠♦ ♠ ♠tr s

♣rs♦ ♥♦ q ♦♥t étr r ♦♠ rqê♥ Pr st♦ t③s ♠

♣r♦①♠♦r Pé ♥♦ ♦♠í♥♦ rqê♥ ♦ q é ♣♦ à qçã♦ ♠♣èr

qçõs s♣s ♣r t③r ♦ ♠♣♦ étr♦ ♥♦ ♦♠í♥♦ ♦ t♠♣♦ sã♦ ♦ts

trés ♠és t♠♣♦rs tr♥s♦r♠ ♥rs ♦rr s rst♦s ♦t♦s

trés s s♠çõs ♥♠érs r③s ♦♠ ♦ ♠ét♦♦ sã♦ ♦♠♣r♦s ♦♠

tr♥stór♦s ♦t♦s ①♣r♠♥t♠♥t Ps♦s ♣r♦♥③♦s ♣♦r ♥♦r♠s ♣r srs

t♠♦sérs sã♦ ♥t♦s ♠ sst♠s trr♠♥t♦ áss♦s ①♥t ♦♥♦râ♥

♦ ♦sr ♥♦ ♦ ♠ét♦♦ ♥♠ér♦ s♥♦♦

Prs ♦♦s ♠③ô♥♦s s♣rs♦s ♣s♦s srs t♠♦sérs ♠é

t♦♦ ♣r♦①♠♦r Pé ①♣r♠♥tçã♦ ♠ ♠♣♦

strt

♠③♦♥♥ s♦ ♦ é♠ s ♠♦ s s♣rs ♠tr ♦ t tr

♦♥tt② s ♣♥♥t ♦♥ t rq♥② ♦r ts ♠ s t Pés ♣♣r♦①♠

t♦♥ ♥ rq♥② ♦♠♥ s ♣♣ t♦ ♠♣èrs ♣ ♣t qt♦♥s

♦r tr ♥ t♠ ♦♠♥ r ♦t♥ ② ♣♣②♥ t♠ r♥ ♥ t ♥

rs ♦rr tr♥s♦r♠ rsts ♦t♥ ② ♥♠r s♠t♦♥s ♣r♦r♠ t

t ♠t♦ r ♦♠♣r t♦ tr♥s♥t rs♣♦♥ss ♠sr ♥ t♥♥

♣ttr♥ ♣ss r ♥t ♥ ss r♦♥♥ s②st♠s ①♥t r♠♥t s ♦

sr ts t ♦♣ ♥♠r ♠t♦

②♦rs s♣rs ♠③♦♥♥ s♦s t♥♥ ♣ss ♠t♦ Pés ♣

♣r♦①♠♥t ①♣r♠♥t ♠sr♠♥ts

♣ít♦

♥tr♦çã♦

♦♥tú♦ ♦ ♣ít♦

st ♣ít♦ ③s ♠ r ♥tr♦çã♦ s♦r ♦s sst♠s trr♠♥t♦ t♠

é♠ é ♣rs♥t ♠♦tçã♦ ♣çã♦ ♠♦♠ s♦♦s s♣rs♦s ♠

s♠çõs ♥♠érs ♣r ♥sr rs♣♦st tr♥stór r♠ ♣r♠♥♥t sss

sst♠s ♥ sã♦ ♣rs♥t♦s ♦s ♦t♦s ♦r♥③çã♦ ♦ ♣rs♥t tr♦

♥tr♦çã♦ r

❯♠ sst♠ trr♠♥t♦ t♠ ♦♠♦ ♣r♥♣ ♥çã♦ ♣r♦♣♦r♦♥r ♦♥çã♦ ♠s

♦rá ♣r q s ♦rr♥ts ♥ss ♦♠♦ s ♣r♦♥♥ts ♦♥t♥ê♥s ♠

sst♠s étr♦s ♦ srs t♠♦sérs ♠ ♣r ♦ s♦♦ ♠♥r ③ st

♦r♠ ♣rs♥ç ♠ trr♠♥t♦ étr♦ ♦♠♣♦♥♦ ♠ sst♠ étr♦ ♦

trô♥♦ é ♥s♣♥sá ♣r ♣r♦tçã♦ ♠♥t♥çã♦ ♦ ♥♦♥♠♥t♦ sts sst♠s

é♠ ss♦ ♦ trr♠♥t♦ étr♦ é ♠♣♦rt♥t ♣r sr♥ç s ♣ss♦s q ♦♣r♠

♦ ♥tr♠ ♦♠ ♦s sst♠s étr♦s ❬❪

Pr r ♦ s♠♣♥♦ ♦ sst♠ trr♠♥t♦ é ♥ssár♦ ♥sr s ♦♠♣♦r

t♠♥t♦ tr♦♠♥ét♦ t♥t♦ ♣r ♥t♦s ①s rqê♥s q♥t♦ ♣r rqê♥

s s ❬❪ st út♠♦ t♣♦ ♥t♦ ♣♦ss ♠♦r ♠♣♦rtâ♥ ♣♦s ♥♦r♠

♠♥t stã♦ ss♦♦s às ♦rr♥ts ♦♠ ♠♣t ♦♠♦ srs t♠♦sérs

♣♦r ss♦ é ♥ssár♦ ♠ s♦♠♥t♦ ♥t sss ♦rr♥ts tr♥stórs ♣r ♠♥

♠③r ♣♦ssís ♥♦s às ♣ss♦s ♦s q♣♠♥t♦s ♥tr ♦tr♦s t♦rs rs♣♦st

sss sst♠s ♣♥ ♦♥rçã♦ ♦♠étr ♦ trr♠♥t♦ ♦r♠ ♦♥

♦ ♣s♦ ♥t♦ ♦s ♣râ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦ ❬ ❪

♦♠ ♥tt♦ s♠r ♦ ♦♠♣♦rt♠♥t♦ tr♥stór♦ r♠ ♣r♠♥♥t ♠

♦ sst♠ trr♠♥t♦ st♦ s♥s t rqê♥ rs♦s ♠♦♦s q

s♦♦♥♠ ♥♠r♠♥t s qçõs ① sã♦ t③♦s ♥ trtr ❬ ❪

♦♥t♦ ①tã♦ ♦s rst♦s ♦t♦s ♣♦r ♠♦ s♠çõs ♥♠érs ♣♥

♥tr ♦tr♦s t♦rs ♠♦♠ t③ ♣r rtr③r ♦ s♦♦ ♠ ♣rtr

♣♥ê♥ ♦s ss ♣râ♠tr♦s tr♦♠♥ét♦s ♦♠ rqê♥ ♣♦ tr s♥

t♠♥t rs♣♦st tr♥stór ♦ trr♠♥t♦ étr♦ ❬❪

♦r♠♠♥t ♦t♠s ♦rs ♦♥st♥ts ♣r rsst ♥rs♦ ♦♥t

♣r♠ss rt ♦ s♦♦ rsst é st♠ trés ♠çõs

t③♥♦ ♥str♠♥t♦s ♦♥♥♦♥s q ♠♣r♠ s♥s ①s rqê♥s ♦ é

♦t ♦r♠ ♥rt trés ór♠ ♥ ❬❪ Pr ♣r♠ss rt

♦ts ♠ ♦r ♥tr ♣♥♥♦ ♦ t♦r ♠ ♦ s♦♦ ❬❪ ♠♦r

♣r♠ ♠♥ét ♣♦ss sr ♦♥sr à ♣r♠ ♦ á♦ s♠

♣rí③♦ ♣r ♠♦♠ ♦ s♦♦ ♥♦ts q ♦♥t ♣r♠ss é

tr ♣♦ss♠ ♠ ♥♦tór ♣♥ê♥ ♦♠ rqê♥ ❬ ❪ ♥s ♥ ♥

rqê♥s r♣rs♥tts s srs t♠♦sérs s ♣r♠r♦s tr♦s ①♣

r♠♥ts ❬ ❪ q rr♠ ♣♥ê♥ ♦s ♣râ♠tr♦s étr♦s ♦ s♦♦ ♦♠

rqê♥ ♦r♠ ♣♦s ♥♦ ♥í♦ ♦ sé♦ ❳❳ t♠♥t rs♦s ♣sqs♦rs

♥s♠ rçã♦ ♦s ♣râ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦ ♦♠ rqê♥ ss♠ ♦♠♦

♦ ♠♣t♦ st t♦ ♥ rs♣♦st tr♥stór ♦s sst♠s trr♠♥t♦ ❬❪

♠ ♠t♦s ❬ ❪ r③♦ st♦s ①♣r♠♥ts s♦r ♣♥ê♥

♦♥t ♣r♠ss étr ♦ s♦♦ ♦♠ rqê♥ ♥♦ q ♥sr♠s

rss ♠♦strs s♦♦s ♦♠ r♥ts t♦rs ♠ ♥ ♦ t♦r ♣rs♥t♦

♠ t ♦♠ ♦s ♦rs st♠♦s ♦s ♣râ♠tr♦s étr♦s ♦ s♦♦ ♣r rqê♥s

s♣ís ♥ ① ③ ③

♦s ♥♦s s♥♦ ♠s ♦rt♦rs ♦tt ❬ ❪ ♥s♦

rçã♦ ♦s ♣râ♠tr♦s tr♦♠♥ét♦s ♦♥t ♣r♠ss étr ♣r

♠ ♠♥ét ♦♠ rqê♥ ♣r r♦s ♠♦strs rs♦s t♣♦s

s♦♦s ♥ ♦r♠ ♣r♦♣♦sts ①♣rssõs ♣r srr ♣♥ê♥ ♦s ♣râ♠tr♦s

tr♦♠♥ét♦s ♦ s♦♦ ♦♠ rqê♥

♦♥♠r t ❬❪ ♠ ♣r♦♣sr♠ ①♣rssõs s♠♥íts ♣r r♣rs♥

tr rçã♦ ♦s ♣râ♠tr♦s étr♦s ♦♥t ♣r♠ss ♦ s♦♦ ♦♠

rqê♥ ♠♦♦ ♣r♦♣♦st♦ ♦ ♦t♦ ♣rtr r♣rs♥tçã♦ ♠ ♣♦rçã♦ ♦ s♦♦

♣♦r ♠ ♠♣â♥ ♥rs ♦♠♣♦st ♣♦r ♠ rt♦ s ①♣rssõs ♣r♦♣♦sts

♠ ❬❪ ♦r♠ ♦ts ♣rtr ♦s ♦s ①♣r♠♥ts ♣rs♥t♦s ♣♦r ♦tt ❬❪

P♦str♦r♠♥t ♠ P♦rt ❬❪ ♣♦ st♦s ①♣r♠♥ts s♦r ♦ ♦♠

♣♦rt♠♥t♦ s♣rs♦ ♦ s♦♦ ♣rtr ♥ás ♠♦strs t♣♦s s♦♦s s

♣râ♠tr♦s étr♦s ♦ s♦♦ ♦r♠ ♥s♦s ♥ ① rqê♥ ③ té ③

s r♥t♠♥t ❱sr♦ ♣♦ ♣r♠ tr♦s ❬ ❪ s♦r ♥ás

♣♥ê♥ ♦s ♣râ♠tr♦s étr♦s ♦ s♦♦ ♦♠ rqê♥ s rst♦s ①♣r

♠♥ts ♦r♠ ♦t♦s ♣rtr ♠çõs r③s ♠ ♠♣♦ ♠ t♣♦s s♦♦s

❯t③♥♦ ♦s ♦s q ♦tr♠ ♥sss tr♦s ♦s t♦rs ♣r♦♣sr♠ ♦r♠çõs

♠t♠áts q sr♠ rçã♦ rsst étr ♥rs♦ ♦♥t

étr ♣r♠ss étr ♦ s♦♦ ♦♠ rqê♥ ♦♥♦r♠ srá ♦r♦

sr ♦ ♣rs♥t tr♦ t③ ♦r♠çã♦ ❱sr♦♣♦ ♣r rsst

♣♥♥t rqê♥ ♣r ♠♦r ♦ ♦♠♣♦rt♠♥t♦ s♦♦s s♣rs♦s ♥♦s qs

♦♥t r ♠ ♥çã♦ rqê♥

trtr ❬ ❪ tr♦s ♥♠ér♦①♣r♠♥ts q ♥s♠ ♦ ♦♠♣♦rt

♠♥t♦ tr♥stór♦ trr♠♥t♦s étr♦s st♦s srs t♠♦sérs ♠♦♥str♠

q q♥♦ s ss♠ q ♦s ♣râ♠tr♦s ♦ s♦♦ sã♦ ♦♥st♥ts ♦s s♥s ♦t♦s ♥♠

r♠♥t ♣rs♥t♠ s♦s s♥t♦s ♠ rçã♦ ♦s s♥s ♠♦s ♠ ♠♣♦

st tr♦ é ♣r♦♣♦st ♠ ♠♦♠ ♣r ♠trs s♣rs♦s trés

♣♥ê♥ ♦♥t étr ♦♠ rqê♥ σ(f) ♣r ♦ ♠ét♦♦

♥tr♥ ♠♦♠♥ ❬❪ ❯t③s ór♠ ❱sr♦♣♦ ❬❪ ♣r

r♣rs♥tr ♦ ♦♠♣♦rt♠♥t♦ σ(f) ♦♥♦r♠ srá st♦ ♥♦ ♣ít♦ ♥sã♦

σ(f) ♥s qçõs ① é t trés ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦s Pé

❬❪ ♠ tr♠♦s jω ♥çã♦ ♠t♠át ♣r♦♣♦st ♣♦r ❱sr♦♣♦ st ♦r♠

♦ ♣♦ssí s♠♣r ♠♥♣çã♦ tr♥s♦r♠ ♥rs ♦rr s qçõs

① s♥♦ s♥ssár♦ ♣r ♥çã♦ ♦♥♦çã♦ q stá ♠♣ít

♥s ♠♥♣çõs ♠t♠áts r③s sts q ♥s s♠çõs ♥♠érs ♦

♦♠í♥♦ ♦♠♣t♦♥ é tr♥♦ trés té♥ P ❬❪ é♠ ss♦ t③s

té♥ ♦♥♦ ❬❪ ♣r ♠♦r q♠♥t s sts tr♦♦s ♠♦♦

tr ♠♦r ♥í srt③çã♦ ♦ ♦♠í♥♦ ♦♠♣t♦♥

♠♦♠ ♣r♦♣♦st ♥st tr♦ é trés tsts ①♣r♠♥ts ♥♦s

qs ♦r♠ ♣♦s ♣s♦s t♥sã♦ ♣r♦♥③♦s ♦♠ ♦r♠ ♦♥ srs

t♠♦sérs ♠ rs♦s sst♠s trr♠♥t♦s st♣ ①♣r♠♥t t③♦ ♥st

tr♦ é s♦ ♥ ♦♥rçã♦ ♣r♦♣♦st ♣♦r ♥ ♠ ❬❪

ssts q ♦s ①♣r♠♥t♦s ♠♣♠♥t♦s ♥st tr♦ ♦r♠ r③♦s ♥

é♠ ♥ trtr ♥ã♦ ♦r♠ ♥♦♥tr♦s st♦s ♣r♦♥♦s q ♥s♠

♦ ♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦ s♦♦s s♣rs♦s ♥st rã♦ ♠③ô♥

t♦s st r♦

♦♥sr♥♦ ♦ ♦♥t①t♦ ♣rs♥t♦ ♠ ♥♦ q ♣rs râ♥ ♠♦

♠ ♦ s♦♦ s♣rs♦ ♣ à ♥ás ♦ sst♠ trr♠♥t♦ st tr♦

♦t

• s♥♦r ♠ ♠♦♠ ♣r ♠♦s ♦♠ ♦♥t étr ♣♥♥t

rqê♥ s♦♦ ♠③ô♥♦ ♣r ♦ ♠ét♦♦ ♣♦ às qçõs ①

♠ ♠ rã♦ tr♠♥s♦♥

• ❱r ♠♦♠ ♣r♦♣♦st ♥st tr♦ ♣♦r ♠♦ tsts ①♣r♠♥ts

r③♦s ♠ ♠♣♦ ♣r sst♠s trr♠♥t♦ ♦♠ r♥ts ♦♥rçõs

♦♠étrs

r♥③çã♦ st r♦

st tr♦ stá ♦r♥③♦ ♠ ss ♣ít♦s ♥♥♦ ♥tr♦çã♦ ♦♥♦r♠

srt♦ sr

• ♣ít♦ ♥tr♦çã♦ r ♦ tr♦

• ♣ít♦ ♣rs♥t ♦s ♦♥t♦s ás♦s rt♦s ♦s sst♠s trr♠♥t♦

♦r ♦s s♣t♦s ís♦s q ♦♥r♥♠ ♦s ♠trs s♣rs♦s

• ♣ít♦ st r♠♥t ♦ ♠ét♦♦ ♣rs♥t ♠♦♠

♠♦s ♦♠ ♦♥t étr ♣♥♥t rqê♥ ♣r ♦ ♠ét♦♦

q é ♣r♥♣ ♦♥trçã♦ st tr♦

• ♣ít♦ ♣rs♥t ♦ ♠♦♦ st♣ t③♦ ♥♦s tsts ①♣r♠♥ts r

③♦s ♠ ♠♣♦ ♥ sã♦ ♣♦♥ts s ♣rçõs q ♠ sr ♦srs

♣r q ♦s ①♣r♠♥t♦s s♠ r③♦s ♦♠ ♠í♥♠ ♥trrê♥ ♣♦ssí

• ♣ít♦ ♣rs♥t♠s ♦s tsts ①♣r♠♥ts ♠♣♠♥t♦s ♣r r

♠♦♠ ♣r♦♣♦st ♥st tr♦ é♠ ss♦ é t ♦♠♣rçã♦ ♦s s♥s

t♥sã♦ ♦rr♥t ♠♦s ♦t♦s ♥♠r♠♥t ♣♦s ♠♦♦s s♦♦ s♣rs♦

♣r♦♣♦st♦ q ♥ã♦ s♣rs♦

• ♣ít♦ ♦♥té♠ s ♦♥srçõs ♥s ♦ ♣rs♥t tr♦

♣ít♦

♥♠♥tçã♦ ór

♦♥tú♦ ♦ ♣ít♦

st ♣ít♦ trts ♦s ♦♥t♦s r♦♥♦s ♦s sst♠s trr♠♥t♦

③♥♦ ♠ ♦r♠ ís ♥ás tr♦♠♥ét ♦s s♣t♦s q ♥♥♠

♥ rs♣♦st tr♥stór r♠ sss sst♠s ♥ é ♣rs♥t ♥çã♦

♠♦s s♣rs♦s ♣çã♦ st ♦♥t♦ ♦s s♦♦s s♣rs♦s

st♠s trr♠♥t♦

♥çã♦ ♥

s sst♠s trr♠♥t♦ étr♦ sã♦ strtrs ♦♥ts ♠ sst♠ étr♦

♦ q♣♠♥t♦ ♣♦tê♥ ♦ s♣♦st♦ trô♥♦ ♠ r sã♦ ♦♠♣♦sts ♣♦r

♠♥t♦s ♠tá♦s ♥trr♦s ♦♥t♦ ♠ ♠s stçõs ♣♦♠ sr ♦♥sttís

♣♦r ♦tr♦s ♠trs s strtrs q ♦♠♣õ♠ ♦ sst♠ trr♠♥t♦ ♣♦♠ sr

♣♥s ♠ st ♥trr rt♠♥t ♦ ♠ ♦♠♣① ♠ trr♠♥t♦ ♥

q ár♦s q♣♠♥t♦s sã♦ trr♦s s♠t♥♠♥t ❬ ❪

♣r♥♣ ♥ ♠ sst♠ trr♠♥t♦ é tr♥srr ♦r♠ ♠s

♥t ♣♦ssí ♦rr♥t tr♥stór ♣r trr r♥t ♦♦rrê♥ ♠ sr

t♠♦sér ♦ ♦tr♦ srt♦ r♠ tr♥stór♦ ♠ ♦♥çõs ♥♦r♠s ♦♣rçã♦ r

♠ ♦ ③ ♦ sst♠ trr♠♥t♦ q③r ♦ ♣♦t♥ ♥tr ♦s rs♦s

♥str♠♥t♦s ♦♥t♦s ♦r♠ rr ♠ rrê♥ ú♥ ♦t♠ P♦r

ss♦ ♦ sst♠ trr♠♥t♦ sr ♣r♦t♦ ♣r tr ♠ ♠♣â♥ ♠t♦ ♠♥♦r

♦ q ♠♣â♥ ♦ rt♦ ♥♦ q stá ♦♥t♦ st ♦r♠ s ♦rr♥ts ♣r♦

③s ♣♦ srt♦ ♦ ♣♦ ①♦ rs r♠③♥s tr♦stt♠♥t ♠ sr

s♦s ♣r ♦ s♦♦ ♠♥r ♠s ♥t ♣♦ssí ♦♥♠♥t ♣r♦♣♦r♦♥r

♠♦r sr♥ç às ♣ss♦s q str♠ s♦r ♠ trr trés q♣♦t♥

③çã♦ s♣rí ♦ s♦♦ ♣r♦tr ♦s q♣♠♥t♦s

♥ás r♥stór ♠ st♦♥ár♦ ♦s st♠s

trr♠♥t♦

♦♠♣♦rt♠♥t♦ tr♥stór♦ ♠ sst♠ trr♠♥t♦ ♣♥ s♠♥t

três t♦rs ♦♠tr ♦ sst♠ trr♠♥t♦ ♣r♦♣rs tr♦♠♥éts ♦ s♦♦

♦r♠ ♦♥ ♦s s♥s ♥t♦s ❬❪

tr♦s t♦rs q ♣♦♠ tr rs♣♦st tr♥stór ♦s sst♠s trr♠♥t♦

sã♦ rtrísts íss s♣s q sã♦ ♥trí♥ss ♦s s♦♦s ♥tr s st♠s

s♣rsã♦ ❬❪ ♦♥③çã♦ ♦ s♦♦ ❬❪

♥♦ ♠ trr♠♥t♦ étr♦ é st♦ ♦rr♥ts ♦♠ ♠♣ts s IP ≥

10 kA ♦s ♠♣♦s tr♦♠♥ét♦s ♥ rã♦ ♦ s♦♦ ♣ró①♠ ♦s ♠♥t♦s ♦ trr

♠♥t♦ ♣r♦♠♦♠ ♠ ♦♠♣♦rt♠♥t♦ ♥ã♦ ♥r ♣r trr trés ♣♥ê♥ ♦s

♣râ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦ s♠♥t ♦♥t étr ♦♠ ♠♣

t ♦ ♠♣♦ étr♦ st ♦♥çã♦ rtr③ ♦ t♦ ♦♥③çã♦ ♦ s♦♦ ♥♦ q

♦srs q q♥♦ ♥t♥s ♦ ♠♣♦ étr♦ ♥st rã♦ ① ♠ ♠t

rít♦ ♠♦ ♠♣♦ étr♦ rít♦ ECR ♥ã♦ ♥r é ♦sr ①

s ♣r ♦rs ECR é ❱♠ ♣♥♥♦ ♦ t♦r ♠ ♦

s♦♦ ❬❪ rs♦s tr♦s ♣r♦♣õ♠ ♠♦♦s ♣r srr ♦♥③çã♦ ♦ s♦♦ ❬❪

♦♥t♦ ♥st tr♦ ♦ t♦ ♦♥③çã♦ ♦ s♦♦ é s♣r③♦ ♣♦s s ♠♣ts

s ♦rr♥ts ♥ts ♥♦s tsts ①♣r♠♥ts ♦r♠ ①♦

s♦♦ é ♦♥sr♦ ♠ ♠♦ s♣rs♦ q♥♦ ♣♦ ♠♥♦s ♠ ♦s ♣râ♠tr♦s

tr♦♠♥ét♦s σ ε ♦ µ é ♥çã♦ rqê♥ ❬❪ trtr ♠t♦s tr♦s

sr♠ q ♣♥ê♥ sts ♣râ♠tr♦s ♦♠ rqê♥ é s♥t s♦r

t♦ ♣r rs♣♦st s♥s ts rqê♥s ♦♠♦ srs t♠♦sérs

sr ♦♥sr ♥ ♥ás ♦ ♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦ ♦s sst♠s trr

♠♥t♦ ❬ ❪ çã♦ st ♥ô♠♥♦ srá ♦r♦ ♦♠ ♠s

ts ♠ ♥♠♥tr ♠♦♠ ♠♦s ♦♠ ♦♥t étr

♣♥♥t rqê♥ ♠♦s s♣rs♦s ♣ ♦ ♠ét♦♦ q é ♣r♥♣

♦♥trçã♦ ♦ ♣rs♥t tr♦

♦♠♦ s ♥ás tr♦♠♥ét r③ ♥st tr♦ ♦ tó♣♦ sr

♦r r♠♥t s qçõs ① ♥ ♦r♠ r♥ ♠ s srã♦

st♦s tó♣♦s r♦♥♦s ♦ ♦♠♣♦rt♠♥t♦ tr♦♥â♠♦ ♣r ♦s sst♠s

trr♠♥t♦

qçõs ① ♥ ♦r♠ r♥ ♥♦ ♦♠í♥♦ ♦ t♠♣♦

s qçõs ① ♠♦♠ ♦ ♦♠♣♦rt♠♥t♦ ís♦ ♥t♥s rt♦s é

tr♦s ó♣t ♣çõs ♦♠étrs tr♦s tr♦♠♥ét♦s ♦tr♦s ♣r♦♠s q ♥

♦♠ ♣r♦♣çã♦ ♦♥s tr♦♠♥éts t♥t♦ ♣r ① ♦♠♦ ♣r ts rqê♥

s ❬❪ trtr rs♦s tr♦s ♦♠♣r♠ tsts ①♣r♠♥ts s♦çõs

♥♠érs s♠♥íts sss qçõs ♥♦s qs ♦srr♠s ①♥ts ♦♥♦r

â♥s ♥tr sss rst♦s ❬ ❪ ♥ ♣rtr sss qçõs ♣♦s

♥sr rs♣♦st ♦s sst♠s trr♠♥t♦ ①tçõs tr♦♠♥éts ♥♥♦

s♥s ♦♠ ts rqê♥s ts ♦♠♦ srs t♠♦sérs

♦r♠ r♥ r② é ♣♦r

∇× ~E = −∂ ~B

∂t

♠♣èr① é ♥ ♣ qçã♦

∇× ~H = ~J +∂ ~D

∂t.

s ①♣rssõs ~E é ♦ t♦r ♥t♥s ♠♣♦ étr♦ ❱♠ ~H é

♦ t♦r ♥t♥s ♠♣♦ ♠♥ét♦ ♠ ♦♥sr♥♦ ♠♦s s♦tró♣♦s ♥ã♦

s♣rs♦s ss q ♠ ♥s ①♦ ♠♥ét♦ é ♥ ♣♦r ~B = µ ~H

❲m2 ♦♥ µ é ♣r♠ ♠♥ét ♠ qçã♦ ♥s

♦rr♥t ♦♥çã♦ é ♣♦r ~J = σ ~E m2 ♦♥ σ é ♦♥t étr ♠

♥s ①♦ étr♦ é ♣♦r ~D = ε ~E m2 ♥ q ε é ♣r♠ss

étr ♠

r② ♥ q rçã♦ t♠♣♦r ~B ♣r♦♦ ♦ sr♠♥t♦ ♠♣♦

étr♦ r♥t ♠ t♦r♥♦ ♦ ♣ró♣r♦ t♦r ~B ♥♦♠♥t ♠♣èr ③ q

rçã♦ t♠♣♦r ~D ③ srr ♠ rçã♦ ♦ ♠♣♦ ♠♥ét♦ ♠ t♦r♥♦ ♦ t♦r

~D s s s ♥ts ♠♦str♠ q ♣rtr ①tçã♦ ♠ ♦♥t t♥sã♦ ♦

♠♣♦ étr♦ ①t♦ ♠ ♠ ♣♦♥t♦ ♣r♦♠♦ rçã♦ t♠♣♦r s ♦♠♣♦♥♥ts ♦

♠♣♦ ♠♥ét♦ ♥ts sts ♣r♦♠♦♠ rçã♦ t♠♣♦r ♦ ♠♣♦ étr♦ ♠

♣♦♥t♦s ♣ró①♠♦s ♦♠ ss♦ ♥s ♠ ♣r♦ss♦ ♣r♦♣çã♦ ♦♥ tr♦♠♥ét

rtr③♦ ♣♦r ♠ ♦♥

P♦ sr ♣r♦♦ q s ♦ sst♠ rts♥♦ ♦♦r♥s ♦r t③♦ ♦♠

♣çã♦ ♦ ♦♣r♦r r♥t ♥s qçõs é ♣♦ssí ♦tr s s♥ts

①♣rssõs ♣r ♦s ♠♣♦s ♥s ①♦ ♠♥ét♦ ♥s ①♦ étr♦

s ss

∇ · ~B = 0

∇ · ~D = ρv,

♦♥ ρv é ♥s ♦♠étr rs étrs ♠3

qçã♦ ③ q ♣rs♥ç ♠ r étr ρv ♣r♦♦ ♦ sr♠♥t♦

♥s rts ♦ ♠♣♦ étr♦ ~E ♣rtr st ♦srçã♦ ♥♦ts q s qçõs

① sts③♠ ♦ ♦♠♣♦rt♠♥t♦ srt♦ ♣ ♦♦♠ q stá ♥s

♠♣t♠♥t ♥♦ sst♠ qçõs ♥♦ ♣♦r ① P qçã♦ ♣♦s

♥rr q s ♥s ♦ ♠♣♦ t♦r ~H sã♦ s♠♣r s sr sê♥

♠♦♥♦♣ó♦s ♠♥ét♦s ♥ ♥tr③

♥ ① ♥ ♠ s tr♦ ❬❪ ♠ ♠♣♦ t♦r ~A ♦ ♥♦♠♥♦

♠♣s♦ ♥r q é ♦♥♦ t♠♥t ♣♦r t♦r ♣♦t♥ ♠♥ét♦ st ♠♣♦

t♦r é ♥♦ s♥t ♦r♠

∇× ~A = µ ~H.

♦r ssttí ♠ t♠s

∇× ~E = −∂

∂t

(

∇× ~A)

.

sr♥♦ é ♦t s♥t ①♣rssã♦

∇×

(

~E +∂ ~A

∂t

)

= 0.

♦♠♦ ♦ r♦t♦♥ ♠ r♥t ♠ ♠♣♦ sr qqr é ♥t♠♥t ♥♦

♣♦s ♥r ♠ sr φ ♥♦♠♥♦ ♣♦t♥ sr étr♦ t ♦r♠ q

~E +∂ ~A

∂t= −∇φ.

♥tã♦ ♣♦s ♥rr q

~E = −

(

∇φ +∂ ~A

∂t

)

.

♦♦ ♦ ♠♣♦ étr♦ ♣♦ sr ①♣rss♦ ♠ ♥çã♦ ♠ ♠♣♦ sr φ ♠

♠♣♦ t♦r ~A ①♣rssã♦ ♣♦ sr ♥tr♣rt ♦♠♦ ♠ ♦♠♣♦sçã♦

♠♦t③

♦♥sr♥♦ ♦ ♦♠♣♦rt♠♥t♦ tr♦♥â♠♦ ♦ ♠♣♦ étr♦ ~E ♦ ♣♦r

♥ã♦ é ♦♥srt♦ ♦♥♦r♠ ♣♦s ♥rr ♣ ①♣rssã♦ ♦♥t♦ ♣r ♦ s♦

tr♦stát♦ ∂ ~A∂t

= 0 ♦srs trés ♦ q ∇× ~E = 0 ①♣rssã♦ q

q ∮

c~E · d~l = 0 ♣♦rt♥t♦ ♥st s♦ ~E é ♠ ♠♣♦ t♦r ♦♥srt♦

tr♦♥â♠ ♠ sst♠s trr♠♥t♦

♦♠♣♦rt♠♥t♦ tr♦♥â♠♦ ♠ sst♠ trr♠♥t♦ ♣♦ sr ♦

♣rtr ♦s s♥ts ♣râ♠tr♦s tr♦♠♥ét♦s t♥sã♦ ♥③ v(t) ♦rr♥t ♥t

i(t) rsstê♥ trr♠♥t♦ R r♥s♥t r♦♥♥ sst♥ ♣♦ss

sts r♥③s ♣♦s ♥sr ê♥ ♦ sst♠ trr♠♥t♦ t♠é♠ é

♣♦ssí st♠r s rtrísts tr♦♠♥éts ♦ s♦♦

• ♥sã♦ ♥③

t♥sã♦ ♥tr ♦s ♣♦♥t♦s a b vab é ♥ ♣ ♥tr ♥ ♦ ♠♣♦ étr♦

~E ♣♦r ♠ ♠♥♦ rtrár♦ ♦ s

vab =

∫ b

a

~E · d~l (❱).

♦♥♦r♠ ♦ t♦ ♥tr♦r♠♥t ♠ ♠ ♥ô♠♥♦ tr♦stát♦ ♦ ♠♣♦ étr♦ é ♦♥

srt♦ ♣♦rt♥t♦ vab é ♦t♦ ♥♣♥♥t ♦ ♠♥♦ ♥trçã♦ s♦♦ ♣♦

sr ♦ ♣ r♥ç ♦ ♣♦t♥ étr♦ ♥tr ♦s ♦s ♣♦♥t♦s vab = φb − φa

♦♥t♦ ♦ às rtrísts tr♥stórs ♦ ♥ô♠♥♦ tr♦♠♥ét♦ ♣r♦♦♦

♣♦r ♠ s♥ rt rçã♦ ♦ ♦r vab(t) ♣♥ ♦ ♠♥♦ ♥trçã♦

♦ ♠ ♦trs ♣rs rs♣♦st tr♥stór ♠ sst♠ trr♠♥t♦ ♣♥

♦♠♦ t♥sã♦ étr é ♠ ♦ ♦♠ ss♦ ♠ ú♥♦ sst♠

trr♠♥t♦ ♣♦ tr ♥♥ts ♣♦sss ♦r♠s ♦♥ rs♣♦st tr♥stór

❯♠ ♦r♠ r ♣r♦r♠♥ ♦ trr♠♥t♦ é trés t♥sã♦ ♥③

st ♣râ♠tr♦ é ♦t♦ q♥♦ ♦ ♣♦♥t♦ b rrê♥ t♥sã♦ é ♦ ♠ ♠ ♣♦♥t♦

♥♥t♠♥t st♥t a ♥♦ q ~E ≈ 0 ss ♦r♠ t♥sã♦ ♥③

v(t) ♣♦ sr ♣♦r

v(t) = va(t) =

∫

∞

a

~E(t) · d~l (❱).

♥ ss♠ ♣♥♥♦ ♦ ♠♥♦ s♦♦ ♦té♠s r♥ts ♦r♠s ♦♥ ♦

s♥ v(t) P♦ré♠ q♥♦ ♦ ♣s♦ ♥t♦ ♥tr ♠ r♠ st♦♥ár♦ t♦♦s s ♦r♠s

v(t) ♠ ♦♥rr ♣r ♦ ♠s♠♦ ♦r ♥tr③ ♦♥srt ♦ ♠♣♦ ~E

♥♦s s♦s tr♦stát♦s

r ♣rs♥tçã♦ ♦ á♦ t♥sã♦ ♥③ ♦rr♥t ♥t

SOLO

Gerador de Surto

! = # $ ⋅ &')*

Haste de injeção

...) → ∞$ ≈ .

* + ! − )

SOLO

Gerador de Surto ! = # $ ⋅ &')

Haste de injeção

$(!) (!) -

• ♦rr♥t ♥t

s♥♦ ♣râ♠tr♦ ♥ssár♦ ♣r ♥sr ♦ ♦♠♣♦rt♠♥t♦ ♦ sst♠ tr

r♠♥t♦ é ♦rr♥t ♥t i(t) st ♣râ♠tr♦ é ♦ ♥♦ ♣♦♥t♦ ♥çã♦

♦rr♥t ♣ró①♠♦ à s♣rí ♦ s♦♦ ♣♦♥t♦ a ♥ ♣♦ sr tr♠♥♦

♣ ♥tr s♣rí ♥s ♦rr♥t ♦♥çã♦ ~J q trés

sçã♦ tr♥srs ♦ ♦ ♣rát s s♦♥s ♠çã♦ ♦té♠ ♦ ♦r ♦rr♥t

étr ♣rtr s ♦♠♣♦♥♥ts ♦ ♠♣♦ ♠♥ét♦ ~H q r ♠ t♦r♥♦ ♦ ♦

étr♦ Prt♥♦ ♠♣èr ♥ ♦r♠ ♥tr ♣♦s ①♣rssr

♦rr♥t q ♥♦ ♦ s♥t ♦r♠

i(t) =

∫∫

S

~J · d~S =

∮

c

~H · d~l ().

s q ♥♦s ♠trs ♦♥t♦rs ♦ tr♠♦ ∂∂t

∫∫

S~D · d~S ♠♣èr é

③r♦ ♣♦s ♥♦s ♠trs ♦♥t♦rs ♥s ♦rr♥t s♦♠♥t♦ é ♥

∂ ~D∂t

= 0

• sstê♥ trr♠♥t♦

rsstê♥ trr♠♥t♦ R é ♠ ♣râ♠tr♦ q q♥t ♠♣♦st

♣♦ sst♠ trr♠♥t♦ ♣r q ♦rr♥t étr s♦ ♣r ♦ s♦♦ ♦♥sr♥♦

s♥s ♦ ①íss♠s rqê♥s á♦ R é ♦ ♣ ①♣rssã♦

R =VDC

IDC

(Ω),

♦♥ VDC IDC sã♦ ♦s ♦rs ♦♥st♥ts ♦ qs ♦♥st♥ts t♥sã♦ ♥③

♦rr♥t ♥t rs♣t♠♥t ♦♥♦r♠ srá st♦ sr rsstê♥ trr

♠♥t♦ ♣♦ sr ♥r ♦sr♥♦ ♦ ♦r r♠ st♦♥ár♦

rsstê♥ trr♠♥t♦ R é ♥rs♠♥t ♣r♦♣♦r♦♥ à ♦♥t étr

♦♥♦r♠ srá st♦ ♣♦r ♣r ♦ s♦ ♠ st ♦ ss♦ ♦s sst♠s

trr♠♥t♦ ♠ s♦♦s ♦♠ ① ♦♥t ♣rs♥t♠ R ♦s ♣♦rt♥t♦

rqr♠ ♦♥rçõs ♠s ♦♠♣①s ♠♦r ♥ú♠r♦ tr♦♦s ♣r♦t♦ ♠s

trr♠♥t♦ t ♣r ♦tr R s♥t♠♥t ①♦

• r♥s♥t r♦♥♥ sst♥

r♥s♥t r♦♥♥ sst♥ ♦ rçã♦ t♥sã♦♦rr♥t é ♣

r③ã♦ ♣♦♥t♦ ♣♦♥t♦ ♥♦ t♠♣♦ ♥tr t♥sã♦ ♥③ v(t) ♦rr♥t ♥t i(t) ♦

s

TGR(t) =v(t)

i(t)(Ω).

♣rtr st ♣râ♠tr♦ é ♣♦ssí r③r ♥áss ♥♦ ♣rí♦♦ tr♥stór♦ ♥♦ r♠

st♦♥ár♦ ♦♠ ss♦ é ♣♦ssí ♥tr ♦s ♥st♥ts ♥♦s qs ♦ sst♠ trr

♠♥t♦ ♣r♦♦ ♠♦r ♣r q ♦rr♥t ♣♦ss r ♣r ♦ s♦♦

é ♣rs♥t♦ ♠ ①♠♣♦ r ♦t ♥♠r♠♥t trés ♦ ♠ét♦♦

♦ ♣r ♠ ♣s♦ t♠♦sér♦ ♠ ♠ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r

♠ st ♥trr rt♠♥t

r ①♠♣♦ ♠ r qr ♦ ♣r ♠ s♥ ♦♠ ♦r♠ ♦♥ ♠ sr ssq♥t ♠ ♠ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠st ♥trr rt♠♥t st r ♦ ♦t trés ♠ s♠çã♦ t③♥♦♦ ♠ét♦♦

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,60

10

20

30

40

50

60

TGR

( )

Tempo (s)

Simulado

trés ♥♦ts q ♥♦s ♥st♥ts ♥s á ♠ ♦rt rçã♦ ♥ r

♣r♦♦ ♣♦ ①♦ ♦r ♦rr♥t t♠é♠ ♣s ♦♠♣♦♥♥ts ts

rqê♥s ♣rs♥ts ♥♦ ♣rí♦♦ t♠♣♦ s ♦ ♣s♦ t♠♦sér♦ ♥t♦ ♣ós

♦ ♣rí♦♦ tr♥stór♦ ♦ s♥ r t♥ ♣r ♠ ♦r r♠ q

♦rrs♣♦♥ à rsstê♥ trr♠♥t♦ R

sr♥♦ q ♦s s♥s ♣♦ss♠ ♥r s♥t ♠ ♦♠♣♦♥♥ts rqê♥s

♠s ts é ♥♦tá q ♦ ♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦ ♠ sst♠ trr♠♥t♦

♥ã♦ ♣♦ sr r♣rs♥t♦ ♣♥s ♣♦r ♣râ♠tr♦s ① rqê♥ ♦♠♦ rsstê♥

trr♠♥t♦ s é ♠ ♣râ♠tr♦ ♠♣♦rt♥t ♣♦s q♥t ♦ ♦♥♦

♦ t♠♣♦ ♠♣♦st ♣♦ sst♠ trr♠♥t♦ ♦ ①♦ ♦rr♥t

Prâ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦

♦♠♣♦rt♠♥t♦ tr♦♥â♠♦ ♦ sst♠ trr♠♥t♦ stá rt♠♥t ♦

às ♣r♦♣rs tr♦♠♥éts ♦ s♦♦ sts ♣r♦♣rs ♣♦r s ③ ♣♥♠

ár♦s t♦rs ts ♦♠♦ ♦ t♣♦ s♦♦ ♦ t♦r ♠ ♦♠♣♦sçã♦ qí♠ ♣rs♥ç

♠t♠s t♠♣rtr ♦r♠ ♦♥ ♠♣t ♥ã♦♥rs ①

rqê♥ s♣rsã♦ ♦ s♥ ♥t♦

s ♣râ♠tr♦s s♠♥t t③♦s ♣r r♣rs♥tá♦ sã♦ ♦♥t étr

σ ♣r♠ss étr ε ♣r♠ ♠♥ét µ ♦r♠♠♥t ♦ s♦♦

♥ã♦ é ♠ ♠♦ ♠♥ét♦ ♣♦rt♥t♦ ♣r♠ é ♦♥sr à ♦ á♦

µ = µ0 = 4π × 10−7 ♠ ❬❪ ♦♥t♦ ♦♥t ♣r♠ss étr ♦

s♦♦ r♠ ♠ ♠ r♥ ① ♦rs ❬ ❪ ♦r♠ ♦♥ ♦s s♥s

♠♦s ♦ ♦s ♥♠r♠♥t é s♥s♠♥t t ♣♦r sts ♣râ♠tr♦s

• ♦♥t étr

♦♥t étr ♠ ♠tr é ♠ ♣râ♠tr♦ q rtr③

♦s étr♦♥s s ♠♦r♠ ♠ ♠ tr♠♥♦ ♠tr ♦♠ ♦ ♠♥t♦ t♠♣rtr

rçã♦ ♦s étr♦♥s ♠♥t ♣r♦♦ ♠♦r ♠♣♠♥t♦ ♦ s♦♠♥t♦ ♦s étr♦♥s

♦♠ ss♦ ♦♥t étr ♦ ♠tr ♠♥ ❬❪

♦♥t étr ♦ s♦♦ ♣r ①s rqê♥s σDC ♣♦ sr ♥ ♦♠♦

♦ ♥rs♦ rsstê♥ étr Rsolo ♠ ♥tr s s ♦♣♦sts ♠ ♣r♣í♣♦

♠♥sõs b × h × l ♠3 ♣r♥♦ ♦♠ ♠ ♠♦str ♦ s♦♦ ♥s♦

♥ é ♣♦r ♠ ①♣rssã♦ q ♥ ♦♥t é ♣♦r

σDC =l

RsoloA,

♦♥ A = b × h ♠2

r ♣rs♥tçã♦ ♦ st♣ ♠çã♦ ♣r tr♠♥r ♦♥t ♦r♦ ♦♠

Solo

Ω

s♦♦ ♠ s st♦ ♥tr s♦ é ♦♥sr♦ ♠ r ♠ ♠ ♦♥t♦r

tr trtr ❬ ❪ ♦s ♦rs σDC ♦t♦s ♣r ♦ s♦♦ r♠

♥tr 10−5 10−1 ♠

♠ ❬❪ ♥ ♣r♦♣õ ♠ qçã♦ ♣r ♥sr rsstê♥ trr♠♥t♦ R

trés s ♦♥rçõs ♦♠étrs ♦ trr♠♥t♦ rsst ♦ s♦♦ ♥

♦♠♦ ♦ ♥rs♦ ♦♥t étr st tr♦ ór♠ ♥ ❬❪ ♦

t③ ♣r st♠r ♦♥t étr ♦ s♦♦ ♣r ①s rqê♥s σDC

Pr ♥r ♦ trt♠♥t♦ ♠t♠át♦ ♥ ❬❪ ♦♥sr ♠ ♦♥t♦r í♥r♦

♣♦s♦♥♦ ♦ ♦♥♦ ♦ ①♦ x ♦♠ ♦♠♣r♠♥t♦ x = −l/2 x = l/2 ss♠s

q ♦ ♥tr ♠ ♦rr♥t I0 ♥♦ ♣♦♥t♦ ♠é♦ ♦ ♦♥t♦r é ♦t ♠ ♦rr♥t

s♦♠♥t♦ 2I(0)/l ♦ ♦♥♦ ♦ ♦♥t♦r st ♦r♠ ♦ ♣♦t♥ étr♦ é ♦

s♥t ♦r♠ ❬❪

V (x, y) =2I(0)ρ

2πlln α(x, y),

♦♥

α(x, y) =

√

(x + l/2)2 + y2 + (x + l/2)√

(x − l/2)2 + y2 + (x − l/2).

♥♦ ♦ ♦♠♣r♠♥t♦ ♦ ♦♥t♦r é ♠t♦ ♠♦r ♦ q ♦ s â♠tr♦ ♦ ♣♦t♥

♥♦ ♣♦♥t♦ ♠é♦ ♥ ①tr♠ ♦ ♦♥t♦r é ♦♥sr♥♦ y = a [(l/2)2 + a2]1/2 −

l/2 ≈ a2/l ❬❪

V (0, a) =2I(0)ρ

2πlln

(

l

a

)

V (l/2, a) =2I(0)ρ

4πlln

(

2l

a

)

.

♥tã♦ ♦♥sr♥♦ ♠ ♦rr♥t s♦♠♥t♦ ♥♦r♠ ♦ ♣♦t♥ ♥♦ ♣♦♥t♦ ♠é♦

♥♦ q I0 é ♥t♦ é ♣r♦①♠♠♥t s ③s ♠♦r ♦ q ♥ ①tr♠ ♦

♦♥t♦r

♥♦ ♦ ♣♦t♥ ♠é♦ ♦ ♣ ♥trçã♦ ♥tr x = 0 x = l/2 é

♦ ♣♦r 2I(0) ♦tê♠s s♥t ①♣rssã♦ ♣r rsstê♥ trr♠♥t♦ R ♦

♦♥t♦r ♠ ♠ ♠♦ ①t♥sã♦ ♥♥t ♠ t♦s rçõs

R =ρ

2πl

(

ln

[

l

a

(

1 +√

1 + (a/l)2)

]

+a

l−√

1 + (a/l)2

)

.

ss♠♥♦ q l ≫ a t♠♦s

R =ρ

2πl

[

ln2l

a− 1

]

.

s♦♥♦ rsst ♠ ♦tê♠s

ρ =2πlR

[

ln 2la− 1] .

♥♦s q ♦♥t étr é ♦ ♥rs♦ rsst s q

σDC =

[

ln 2la− 1]

2πlR,

♦♥ σDC é ♦♥t étr ♣r ①s rqê♥s ♠ l é ♦ ♦♠♣r♠♥t♦

♠ ♦ tr♦♦ st trr♠♥t♦ ♥trr♦ rt♠♥t a é ♦ â♠tr♦ ♠ ♦

tr♦♦ trr♠♥t♦ R é rsstê♥ trr♠♥t♦ Ω

srs q ①♣rssã♦ ♠ ♦♥sr ♣♥s ♣râ♠tr♦s stát♦s ♣r r

♦♥t ♣♦r ss♦ ♦ ♦r σDC é á♦ ♣♥s ♣r ①s rqê♥s

sçã♦ srá ♣rs♥t qçã♦ ❱sr♦♣♦ ❬❪ q sr rçã♦

rsst ♥rs♦ ♦♥t ♠ rçã♦ à rqê♥ é qçã♦ t③

♥ çã♦ ♠♦♠ ♦ s♦♦ s♣rs♦ ♣r♦♣♦st ♥st tr♦

• Pr♠ss étr

♣r♠ss étr é ♠ ♣râ♠tr♦ q stá ss♦♦ à ♣ r♠③

♥♠♥t♦ ♥r étr ♠ ♠ tr♠♥♦ ♠tr r♠③♥♠♥t♦ ♥r

stá ss♦♦ ♦ s♦♠♥t♦ rs ♣♦sts ♥ts ♦s át♦♠♦s q ♦♠♣õ♠

♦ ♠tr ♦r♠ q ♣çã♦ ♦ ♠♣♦ ♣r♦③ ♠ r♥ q♥t ♣♦♦s

étr♦s ♦♠ ss♦ ♥r é r♠③♥ ♥ ♦r♠ ♥r ♣♦t♥ étr

♦r♠ s♠r ♦ q ♦♥t ♦ s str ♠ ♦r ❬❪ ♣r♠ss ♠

♦ ♠tr é r♦♥ à ♣r♠ss ♦ s♣ç♦ r ♣♦r

ε = εrε0,

♦♥ ε0 é ♣r♠ss étr ♦ á♦ ♣♦ ♦r 136π

× 10−9 ♠ εr é

♣r♠ss étr rt ♦ ♠♦ ♦ ♠tr ♦♥sr♦

♥ê♥ st ♣râ♠tr♦ é ♦sr ♥♦ ♣rí♦♦ tr♥stór♦ ♦ s♥ st♦ ♣♦ sr

r♦ ♣rtr s qçõs ① ♦♥sr♥♦ ♦ s♦♦ s♣rs♦ ♠ tr♠♦s

♦♥t ♣♥♥t rqê♥ ~J = σ(t) ∗ ~E ♠♣èr ♥♦ ♦♠í♥♦

♦ t♠♣♦ é ♣♦r

∇× ~H = σ(t) ∗ ~E +∂(ε ~E)

∂t.

sr♥♦ ♥♦ ♦♠í♥♦ rqê♥ ♦♦♥♦ ♠ ê♥ ♥t♥s

♠♣♦ étr♦ ♦tê♠s

∇× H = (σ(ω) + jωε)E.

qçã♦ ♦srs q ♦♥t étr σ(ω) é ♣r♦♠♥♥t ♣r

s♥s ①s rqê♥s ωmaxε ≤ σ(ω)max/10 trtr ❬❪ ♦s ♦rs

♦t♦s ♣r ♣r♠ss rt ♦ s♦♦ stã♦ ♥ ① ♥tr 3 80 ♣♥♥♦ ♦

t♦r ♠ ♦ s♦♦

r rá♦ ♦♠♣r♥♦ ♦s s♥s t♥sã♦ ♥③ ♦t♦s ♣s s♠çõst③♥♦ ♦ ♠ét♦♦ ♦♠ r♥ts ♦rs ♣r♠ss rt ♦ s♦♦♦♠♣r♦s ♦s s♦s ♦♠ εr =

0,0 0,5 1,0 1,5 2,0 2,5 3,00

2

4

6

8

10

12

14

16

18

20Te

nsão

(V)

Tempo (s)

r = 5 r = 10 r = 30 r = 50

Pr rr ♥ê♥ ♣r♠ss étr ♦ s♦♦ ③s ♠ ♦♥♥t♦

s♠çõs ♦ sst♠ trr♠♥t♦ ♠ st ♥trr rt♠♥t tr♥♦

♣♥s ♦ ♦r εr ♦ s♦♦ ①♥♦ ♦ ♦r σDC = ♠ ssts q

st ♦r ♦♥t ♦ ♦t♦ ♣rtr ♦s tsts ①♣r♠♥ts r③♦s ♥st

tr♦ ♣rs♥ts ♦♠♣rçã♦ ♦s s♥s t♥sã♦ ♥③ ♦t♦s

♣♦r ♠♦ s♠çõs ♥♠érs t③♥♦ ♦ ♠ét♦♦ ♦♠ εr ♦ s♦♦

s s♦s s♠♦s ♦r♠ rt♦s ♥çã♦ ♠ ♣s♦ t♠♦sér♦ ♠ ♠

sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♥trr rt♠♥t

♣rtr rs ♣r♠ss étr rt ♥ã♦ ♣r♦♦ ♥ê♥

s♥ ♥s ♦r♠s ♦♥ t♥sã♦ ♥③ ♠s♠♦ ♠♦♥♦ s ♦r ♠

♠ r♥ ① sr♦s q ♦ s♦♦ t③♦ ♥♦s tsts ①♣r♠♥ts ♣♦ss t

♦♥t étr < σDC < ♠ ❬ ❪ ♦♠ ss♦ é s♣r♦

q ♣r♠ss ♥ã♦ t s♥t♠♥t ♦r♠ ♦♥ ♦s s♥s ♦t♦s t

♦♠♦ ♣♦ sr ♥r♥♦ trés

s♣rsã♦ ♦ ♦♦

r♠♥t ♠ ♥áss sst♠s trr♠♥t♦ st♦s srs t♠♦sérs

♦s ♣râ♠tr♦s tr♦♠♥ét♦s σ ε µ ♦ s♦♦ sã♦ ♦♥sr♦s ♦♥st♥ts ♦♥t♦

tr♦s ♣rs♥ts ♥ trtr ❬ ❪ ♠♦♥str♠ ♠ rçã♦ s

♥t ♦s ♣râ♠tr♦s étr♦s σ ε ♦ s♦♦ ♦♠ rqê♥ ♣r♥♣♠♥t ♥

♥ rqê♥ s srs t♠♦sérs ♥ é r♦ ♠ r♥ s♦ ♥

r♣♦st tr♥stór ♦s trr♠♥t♦s ♥tr ♦s s♥s ♦t♦s ①♣r♠♥t♠♥t ♦s s♥s

♦t♦s trés s♠çõs ♥♠érs q ♥ã♦ ♦♥sr♠ rçã♦ ♦s ♣râ♠tr♦s

♦ s♦♦ ♦♠ rqê♥ ❬❪

st ♦r♠ ③s ♥ssár♦ ♣r ♠♦♦s ♣rçã♦ ♣♥ê♥ ♦s ♣râ

♠tr♦s ♦ s♦♦ ♦♠ rqê♥ ♣r q s s♠çõs ♥♠érs ♣rs♥t♠ rs♣♦sts

♣ró①♠s ♦ ♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦ r ♦♠ ss♦ ♣r♠t♠ r ♦ s♠

♣♥♦ ♦s sst♠s trr♠♥t♦s ♦♠ ♠♦r ♣rsã♦

sçã♦ srá t ♠ r ♦r♠ s♦r ♦s ♠trs étr♦s

♦ ♥ srã♦ ♣rs♥t♦s ♦s s♣t♦s ís♦s rt♦s ♦s ♠trs s♣rs♦s ♠ s

♥ sçã♦ ♣rs♥ts ór♠ ❱sr♦♣♦ q sr rçã♦

rsst ♥ ♦♠♦ ♦ ♥rs♦ ♦♥t ♦ s♦♦ ♦♠ rqê♥

trs s♣rs♦s

♦r♠ ís ♦s ♠trs étr♦s

♥ts str ♠♦♠ ís ♦s ♠♦s s♣rs♦s é ♥ssár♦ rstr ♦s

♥♠♥t♦s ♦s ♠trs étr♦s

s ♠trs étr♦s s ♥ã♦ ♦♥té♠ rs étrs rs t ♦♠♦ é ♦sr♦

♥♦s ♦♥t♦rs ♣♦r ss♦ ss át♦♠♦s ♠♦és ♦♥stt♥ts sã♦ tr♠♥t ♥

tr♦s ♦♥sr♥♦ ♠ ♥ás ♠r♦só♣ ♥♦ sts ♠trs sã♦ s♠t♦s

♠♣♦s tr♦♠♥ét♦s ①tr♥♦s s rs ♣♦sts ♥ts ♦③s ♥ s♣rí

♦ ♠tr ♥ã♦ sã♦ r♠♦s ♦♠♦ ♦♦rr ♥♦s ♦♥t♦rs ♠s ♦s rs♣t♦s ♥tró

s ♦s át♦♠♦s ♦ ♠♦és ♣♦♠ sr s♦♦s s ♣♦sçã♦ ♦r♥ ss♠s ♠

s♦♠♥t♦ ♥♥ts♠ ❬❪ st ♥ô♠♥♦ ③ srr ♥ú♠r♦s ♣♦♦s étr♦s q

♣♦♠ sr ①♣rss♦s trés ♦ t♦r ♠♦♠♥t♦ ♣♦♦ d~p ♦ ♣♦r

d~p = Q~ℓ,

♦♥ Q é ♠♥t r ♣♦st ♥t át♦♠♦s ♦ ♠♦és ♦s

♥trós ♦r♠ s♦♦s t♦r♠♥t ♠ ~ℓ ♠ t♦r ♠♦♠♥t♦ ♣♦♦ t♦t

~pt ♠ ♠tr é ♦t♦ ♣ s♦♠tór t♦♦s ♠♦♠♥t♦s ♣♦♦s d~pi Pr ♠

♦♠ ∆v ♦♥ ①st♠ Ne ♣♦♦s étr♦s ♣♦r ♥ ♦♠ ~pt ♣♦ sr ①♣rss♦

s♥t ♠♥r

~pt =Ne∆v∑

i=1

d~pi.

♠ ♠ s ♠r♦só♣ ♦ ♦♠♣♦rt♠♥t♦ ♠rs ♣♦♦s étr♦s ♦r

♠♦s ♣♦♠ sr ♦♥t③♦s trés ♦ t♦r ♣♦r③çã♦ étr ~P st t♦r é

♥♦ ♣ s♥t ①♣rssã♦

~P = lim∆v→0

[

1

∆v~pt

]

= lim∆v→0

[

1

∆v

Ne∆v∑

i=1

d~pi

]

(C/m2).

♦♥sr♥♦ ♠ t♦r ♠♦♠♥t♦ ♣♦♦ ♠é♦ ♦ ♣♦r

d~p♠é♦ = Q~ℓ♠é♦

♣♦r ♠♦é ♦ t♦r ♣♦r③çã♦ étr ♣♦ sr rsrt♦ q♥♦ Ne ♣♦♦s s

tr♠ ♥♦s ♥ ♠s♠ rçã♦ ♦♠♦ ♦♦rr ♥♦ s♦ ♠ ♠tr st♦ ♠

♠♣♦ étr♦ ①tr♥♦ s♥t ♦r♠

~P = Ned~p♠é♦ = NeQ~ℓ♠é♦ .

♦r ♦♥srs ♠ ♦♥t t♥sã♦ ♦♥tí♥ ①t♥♦ ♠ ♣t♦r ♣s

♣rs st ♣t♦r t♠ ♠t s ♣♦rçã♦ ♥tr s ♣s ♣r♥ ♦♠ á♦

♦ ♣r♥♠♥t♦ ♦tr ♠t é t♦ ♣♦r ♠ ♠tr étr♦ rtrár♦

r ♣t♦r ♣s ♣rs st♦ ♠ ♠♣♦ étr♦ stát♦ ~Ea ♣t♦ ❬❪

st stçã♦ ♥q♥t♦ ♦ ♠♣♦ étr♦ ♦♥st♥t ♣♦ ~Ea ♥r s♦r ♦ ♣

t♦r ♥s ①♦ étr♦ ~D0 ♥ ♣♦rçã♦ ♣r♥ ♣♦ á♦ é ♣♦r

~D0 = ε0~Ea.

♣♦rçã♦ ♣r♥ ♣♦ ♠tr étr♦ ♦s ♣♦♦s étr♦s q sr♠ ♦ ♦

♠♣♦ étr♦ ①tr♥♦ ③♠ ♠♥tr ♥s ①♦ étr♦ ~D st♦ ♣♦ sr

①♣rss♦ ♣♦r

~D = ε0~Ea + ~P .

♥s ①♦ étr♦ ♥♦ étr♦ ♣♦ t♠é♠ sr r♦♥ ♦♠ ~Ea ♣

♣r♠ss étr stát εs ♠ ♥ q t♠s

~D = εs~Ea.

♦♠♣r♥♦ ♥rs q ~P ♣♦ss ♠ ♣♥ê♥ ♦♠ ♦ ♠♣♦

étr♦ ♣♦ ~Ea st ♣♥ê♥ é ①♣rss ♣♦r ❬❪

~P = ε0χe~Ea,

♦♥ χe é ♠ ♣râ♠tr♦ ♠♥s♦♥ ♠♦ ss♣t étr q ♣♥

rqê♥ f

stt♥♦ ♠ ♦♠♣r♥♦ ♦♠ ♦tê♠s q

~D = ε0~Ea + ε0χe

~Ea = ε0(1 + χe) ~Ea = εs~Ea.

♥s♥♦ s♥t rçã♦ é ♦sr

εs = ε0(1 + χe).

♥ é ♣♦ssí ♥r ♣r♠ss stát rt εsr

εsr =εs

ε0

= 1 + χe,

q ♥♦r♠♠♥t é rr s♠♣s♠♥t ♦♠♦ ♣r♠ss rt εr ♦ ♦♥st♥t

étr ♣r ♠ rqê♥ f ssts q s ①♣rssõs sã♦

ás ♣r ♠♣♦s stát♦s ♦ ♦♥çõs ①s rqê♥s

♦♠ ís ♣r ♦s ♠trs étr♦s s♣rs♦s ♣r ts

rqê♥s

♠ ♣r ♥r ♦♥srs ♦ ♠♣♦ étr♦ ♦♠ ♣♦r③çã♦ ① ♥♦

♦s ♠trs étr♦s sã♦ s♠t♦s ♠♣♦s tr♦♠♥ét♦s ♦♠ s♣tr♦ rtr

③♦ ♣♦r ♠ ♥ rqê♥s ♣r♠ss étr ♦ t♦r ♣♦r③çã♦ étr

~P (t) = Re[Pxejωtx] tê♠ ♦♥trçõs r♥ts ♣r rqê♥ ♥♦ ♣s♦ rtr

③♥♦ ♦ ♦♠♣♦rt♠♥t♦ ♠ ♠tr s♣rs♦ é♠ ss♦ ♥ê♥ ♠♣♦s

ts rqê♥s ♠♣õ ♠♥çs s♥ts ♥♦ ♦♠♣♦rt♠♥t♦ ♦♥t

étr ♦ ♠tr

♠ ❬❪ é t ♠ ♥♦ ♦♠ ♦ ♠♥s♠♦ ♦r♣♦♠♦ ♣r r③r ♠ s♥♦

♠♥t♦ ♠t♠át♦ ♣r ♠♦r ♦ ♦♠♣♦rt♠♥t♦ ♦ t♦r ♣♦r③çã♦ étr ♦

♦sçã♦ ♦s ♣♦♦s étr♦s ss ♣♦s ♠♣♦s tr♦♠♥ét♦s r♠ô♥♦s ♥♦

t♠♣♦ ♦♠ rqê♥ ♥r ω ♦♠♣♦♥♥t Px ♦t ♦ ♣r ♠ ♠♣♦ étr♦

~E(t) = Re[Exejωtx] = Re[Eejωtx] ♥ ♦r♠ r♠ô♥ t♠♣♦r é ♣♦r

Px(t) = P (t) = NeQℓ(t) =Ne

(

Q2

m

)

E(t)

(ω20 − ω2) + jω

(

dm

) ,

♦♥ m é ♠ss rt ♠ ♥ú♦ tô♠♦ st♦♥ár♦ d é ♦ ♦♥t ♠♦r

t♠♥t♦ ω0 =√

s/m ♦♥ s é ♦ ♦♥t t♥sã♦ ♥♦ ♠♦s ♦s ♦s

①♣rssã♦ ♣♦r E(t) ♦tê♠s

P (t)

E(t)=

Ne

(

Q2

m

)

(ω20 − ω2) + jω

(

dm

) ,

♣rtr ♣♦s ♥rr q

εE(t) = ε0E(t) + P (t),

♦♥ ε é ♣r♠ss ♦ ♠♦ ♥ú♠r♦ ♦♠♣①♦ ♥♦ ♠♦s ♦s ♦s

♣♦r E(t) ♦♠♣r♥♦ ♦♠ ♦tê♠s

ε = ε0 +P (t)

E(t)= ε0 +

Ne

(

Q2

m

)

(ω20 − ω2) + jω

(

dm

) = ε′ − jε′′,

♦♥ ε′ ε′′ sã♦ rs♣t♠♥t s ♣rts rs ♠♥árs ♣r♠ss étr

ε ♥ ♥♦ ♣♦r ε0 é ♣♦ssí ♥r ♣r♠ss rt εr ♥ q

t♠s q

εr =ε

ε0

= ε′r − jε′′r = 1 +

NeQ2

ε0m

(ω20 − ω2) + jω

(

dm

) .

P♦r ♦♠♣rçã♦ ε′r ε′′r ♣♦♠ sr rsrts s♥t ♠♥r

ε′r = 1 +

NeQ2

ε0m(ω2

0 − ω2)

(ω20 − ω2)

2+(

ω dm

)2

ε′′r =NeQ

2

ε0m

[

ω dm

(ω20 − ω2)

2+(

ω dm

)2

]

.

♣ós ♦srr q ♣r♠ss étr ♠ ♠♦ é ♥tr③ ♦♠♣① ♦

♣♦r ♠♣èr ♥♦ ♦♠í♥♦ rqê♥ ♣♦ sr ①♣rss ♣♦r

∇× H = σDCE + jωεE = (σDC + ωε′′)E + jωε′E.

♥♥♦ ♠ ♦♥t t♦t q é ♣♥♥t rqê♥ σ(ω) = σDC + ωε′′

①♣rssã♦ ♣♦ sr rsrt s♥t ♠♥r

∇× H = σ(ω)E + jωε′E.

♠ σ(ω) r♣rs♥t ♦♥t t♦t ♦♠♣♦st ♣ s♦♠ ♦♥t

stát ♦ r♠ st♦♥ár♦ σDC ♦♠ ♣r q r ♠ ♥çã♦

rqê♥ ωε′′ s ♣ ♦sçã♦ ♦s ♣♦♦s étr♦s q t♥♠ s ♥r ♦♠

♦ ♠♣♦ étr♦ ~E r♥t ♥♦ t♠♣♦

♥ ♥ ①♣rssã♦ ♥♦ts q ♦ rátr s♣rs♦ ♠ ♠♦ ♣♥ê♥

♠ ♣râ♠tr♦ tr♦♠♥ét♦ ♦♠ rqê♥ é ♦sr♦ t♥t♦ ♠ tr♠♦s ♦♥

t étr σ(ω) q♥t♦ ♠ tr♠♦s ♣r♠ss étr ε′ ♦♥t♦ st♦s

♣rs♥ts ♥ trtr ❬❪ sr♠ q ♥ê♥ ♣r♠ss ♥♦ ♦♠♣♦rt

♠♥t♦ tr♦♠♥ét♦ ♠ sst♠ trr♠♥t♦ é ♠♥♦r ♦ q ♦♥t

♣r♥♣♠♥t q♥♦ ♦ s♦♦ ♣♦ss t ♦♥t q é ♦ s♦ ♦ s♦♦ ♠③ô

♥♦ ♥s♦ ♥st tr♦ ♦ ss♦ st tr♦ ♦♥sr ♦ t♦ s♣rs♦

♣♥s ♠ tr♠♦s ♦♥t étr st ♦r♠ r③s s♥t♠♥t ♦

♥ú♠r♦ á♦s ♥♦♦s ♥ ♠♦♠ ♣r♦♣♦st ♥st tr♦ ♥ ♦ st♦

♦♠♣t♦♥ ♦ ♣r♦r♠ s♥♦♦ t♠é♠ é r③♦ ♦♥sr♠♥t

♠ ♥s tr♦s ❬❪ ♦s ♠♦s s♣rs♦s sã♦ ♠♦♦s t③♥♦ ss♣t

étr χe(ω) ♣♥ê♥ st ♣râ♠tr♦ ♦♠ rqê♥ ♥r ω ♣♦

sr ♥r ♣rtr ♥ q t♠s q

P (ω) = ε0χe(ω)E(ω).

♦r♠çã♦ ❱sr♦♣♦ ♣r ♦♦s s♣rs♦s

rs♦s tr♦s ❬ ❪ t③♠ ♦s ①♣r♠♥ts ♣r r

♣♥ê♥ ♦s ♣râ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦ ♦♠ rqê♥ ♣rtr sts

♥áss sã♦ s♦s rs♦s ♠ét♦♦s ♣r♦①♠çã♦ rs ♣♦r ♥çõs ♣r ♣r♦♣♦r

♠ ór♠ q sr ♦ ♦♠♣♦rt♠♥t♦ sss ♣râ♠tr♦s ♠ ♥çã♦ rqê♥

♠♦♠ ♠♦s ♦♠ ♦♥t étr ♣♥♥t rqê♥ ♣r ♦

♠ét♦♦ ♥♠ér♦ ♣r♦♣♦st♦ ♥st tr♦ t③ qçã♦ s♥♦ ♣♦r ❱sr♦ t

♠ ❬❪ ♣r ♠♦r rçã♦ rsst ρ(f) ♥rs♦ ♦♥t σ(f)

ssts q ♠ ❬❪ ♦s t♦rs ♣r♦♣õ♠ ♠ ①♣rssã♦ ♣r ρ(f) ♦tr ♣r

ε(f) ♦♥t♦ t♥♦ ♠ st q ♥ê♥ ♦♥t é ♦♠♥♥t ♠ rçã♦

à ♣r♠ss ♥♦s s♦♦s t③♦s ♥st tr♦ t ♦♠♦ str♦ ♣ st

tó♣♦ ♣rs♥t ♠t♦♦♦ ♠♣r ♦♥♦ ♦ s♥♦♠♥t♦ qçã♦ ♣r

ρ(f)

♠ ❬❪ é ♣ ♠ ♠t♦♦♦ ①♣r♠♥t ♣r tr♠♥r ♣♥ê♥

rsst ♦ s♦♦ ♦♠ rqê♥ ♥s ♦♥çõs ♥♦♥trs ♠ ♠♣♦ s t♦rs

♥sr♠ t♣♦s s♦♦s ♦♠ rsst ♣r ①s rqê♥s ρDC ♥tr

Ω♠ ♥♦s qs ♦ ♥s rçã♦ ss ♣râ♠tr♦ ♥tr♦ ♥ 102 4× 106

③ ♣rtr ♦s ♦s ♦t♦s ♥♦s ①♣r♠♥t♦s r③♦s ♦ ♣r♦♣♦st ♠ qçã♦

q sr ♣♥ê♥ rsst ♦♠ rqê♥

st♣ ①♣r♠♥t ♠♣♠♥t♦ ♠ ❬❪ é ♣rs♥t♦ ♥ s s♥s

t③♦s ♣r ♥tr ♥♦ sst♠ trr♠♥t♦ t③♦ ♦r♠

µs ♦♥ ♦ ♣r♠r♦ ♥ú♠r♦ s rr ♦ t♠♣♦ s ♦ r♥t ♦

s♥♦ ♥ú♠r♦ é ♦ t♠♣♦ ♠♦s ♠ ♠r♦ss♥♦s

r t♣ ①♣r♠♥t ♠♣♠♥t♦ ♣♦r ❱sr♦ t ♣r r ♣♥ê♥ rsst ♦♠ rqê♥ ♣t♦ ❬❪

Gerador de Pulso

Malha auxiliar

Cuba hemisférica

Solo

♣rtr tr♥s♦r♠ ♦rr ♦s s♥s t♥sã♦ ♦rr♥t ♦

♠♣â♥ trr♠♥t♦ Z(ω) = V (ω)/I(ω) ♦ sst♠ ♣rs♥ts ♦

r♠ ♠♣â♥ ♦t ♣rtr ♦ st♣ ①♣r♠♥t t③♦ ♠ ♠ s♦♦ ♦♠

ρDC = 495 Ω♠

r r♠ ♠♣â♥ ♦t♦ ♣rtr ♦ st♣ ①♣r♠♥t ♠♣♠♥t♦♣♦r ❱sr♦ t ♠ ♠ s♦♦ ♦♠ ρDC = 495 Ω♠ ♣t♦ ❬❪

Sinal da Fonte: Sinal da Fonte:

Frequência (Hz)

∠!"($%&'(

)Frequência (Hz)

!(") (+)

♠ ❬❪ ♦ sst♠ s♠sér♦ ♦ ♦♠♣r♦ ♠ rt♦ ♣r♦ ♦♠ ♠

tâ♥ G + jωC ♦t♥♦ s♥t ①♣rssã♦

G + jωC =1

Z=

2π(

1ρ

+ jωε)

(

1r1− 1

r2

) .

♣rtr é ♣♦ssí ♥sr ♦ ♦♠♣♦rt♠♥t♦ rsst ♠ ♥çã♦

rqê♥ ♦srs s rs ρr(f) = ρ(f)/ρDC ❱rs ♠

r♥ rçã♦ rsst ρr(f) ♥tr♦ ① rqê♥ ♥s ♥

♦srs q q♥t♦ ♠♦r ρDC ♠♥♦r σDC ♠♦r srá rçã♦ ρr ♦♠ rqê♥

r ❱rçã♦ ρr ♦ ♦♥♦ ① ③ ③ ♣r ♦s s♦♦s st♦s♠ ❬❪ ♦♠♣rçã♦ ♥tr ♦ s♥ ①♣r♠♥t ♦s s♥s ♦t♦s ♣♦r s♠çõs ♦♠♣râ♠tr♦s ♦♥st♥ts ♣râ♠tr♦s ♣♥♥ts rqê♥ ♣t♦ ❬❪

Tempo ( s)Te

nsão

(V)

Tensão Simulada: !(#) e %(#)Tensão MedidaTensão Simulada: ! e % constantes

trés ♥ás s rs ♣rs♥ts ♥ ♦s t♦rs ❬❪ ♣r♦♣s

r♠ s♥t ①♣rssã♦ q tr♠♥ rçã♦ ρ(f) ♦♠ rqê♥

ρ(f) = ρDC

1 + [1,2 × 10−6· ρ0,73

DC ] × [(f − 100)0,65]−1

,

♦♥ ρDC é rsst Ω♠ f é rqê♥ ♥s ③

♣rs♥ts ♦♠♣rçã♦ ♦ s♥ ♠♦ ♦♠ ♦s s♥s ♦t♦s ♥♠r♠♥t ♦♠ ρ

ε ♦♥st♥ts ♦♠ ♦s s♥s ♦t♦s ♦♠ ρ(f) ①♣rssã♦ ε(f) ❬❪ srs

①♥t ♦♥♦râ♥ ♥tr ♦s s♥s t♥sã♦ ♠♦ ♦t♦ ♣♦r s♠çã♦ t③♥♦

♣ít♦

♦♠ ♦s ♦♠

♦♥t étr ♣♥♥t

rqê♥ trés ♦ ét♦♦

♦♥tú♦ ♦ ♣ít♦

st ♣ít♦ ♣rs♥ts ♠ ♦r♠çã♦ ♥♠ér s♥♦ ♥st tr♦

♣r ♠♦♠ ♠trs s♣rs♦s trés ♦ ♠ét♦♦ ♦♥srs

♦♥t étr ♦♠♦ ♠ ♥çã♦ rqê♥ ♥ srã♦ ♦r♦s ♦ ♠ét♦♦

tr♦♥ ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé ♠ té♥ ♠t♠át ♣r

♥tr♣♦r ♥çõs rtrárs st té♥ ♣r♦①♠çã♦ é t③ ♥ ♠♦♠

σ(ω) é s ♦r♠çã♦ ♠t♠át q é ♣r♥♣ ♦♥trçã♦ st tr♦

ét♦♦ s r♥çs ♥ts ♥♦ ♦♠í♥♦ ♦ ♠♣♦

s♣t♦s rs s♦r ♦ ét♦♦

♠ét♦♦ ♥tr♥ ♠♦♠♥ é ♠ té♥ ♥♠ér t③

♣r s♦♦♥r ♦r♠ ♣r♦①♠ s qçõs ① ♥♦ ♦♠í♥♦ ♦ t♠♣♦ q

♦ s♥♦ ♣♦r ♥ ❨ ♠ ❬❪

st ♠ét♦♦ ♦♥sst ♥ srt③çã♦ t♠♣♦r s♣ ♦♠í♥♦ ♥ás ♦s

♠♣♦s étr♦ ♠♥ét♦ t③ ♣r♦①♠çã♦ r♥çs ♥ts ♥trs ♣r

r♣rs♥tr s rs ♣rs♥ts ♥s qçõs ① ♥ ♦r♠ r♥ Pr

r♣rs♥tr ♦ ♦♠í♥♦ ♥ás ♥s ♠ ♠ ♦♠♣t♦♥ s ♥ s

é ♠ é ❨ ♦♥sr♥♦ ♠ ♦♠í♥♦ ♥ás s

♠♥sõs s rsts s és sã♦ ∆x ∆y ∆z ♠ rss stçõs s és

♣♦ss♠ ♦r♠t♦ ú♦ ♦ s ∆x = ∆y = ∆z s ♣♦sçõs s♣s ♥♦ ♦♠í♥♦

♦♠♣t♦♥ sã♦ ♥s ♠ tr♠♦s ♦s í♥s srt♦s s és ♥♦t♦s ♣♦r i

j k ♦♥♦r♠ ♣♦ sr ♦sr♦ ♥ ♣♦sçã♦ ís ♦ ♥t♦ ♠

é s ♦♠♦ rrê♥ s♣ ♠s♠ é ♣♦r x = i∆x y = j∆y

z = k∆z

♣ss♦ t♠♣♦r é ♦♥t③♦ ♣♦ í♥ n ♦ ♥r♠♥t♦ t♠♣♦r é ♥♦ ♣♦r

∆t t ♦r♠ q ♦ ♥st♥t ís♦ ♥♦ q ♦s ♠♣♦s étr♦ ♠♥ét♦ sã♦ ♦s

é ♦ ♣♦r t = n∆t ♥r♠♥t♦ t♠♣♦r ∆t é ♥♦ ♣ ♦♥çã♦ ♦r♥t q

srá st ♥ sçã♦

ss ♠♥r ♠ ♥çã♦ f q ♣♥ ♦ s♣ç♦ x y z ♦ t♠♣♦ t é

r♣rs♥t♦ s♥t ♦r♠ srt

f(x, y, z, t) ≈ fnd (i, j, k).

r ♣rs♥tçã♦ é ❨

x

yz

( , , )i j k

Hx

Hz

Ez

Ex

Ey

Hy

yx

z

♦♥♦r♠ stá str♦ ♥ s ♦♠♣♦♥♥ts ♦ ♠♣♦ étr♦ sã♦ s♦s

♠ ♠ rst é ♠ rçã♦ às ♦♠♣♦♥♥ts ♦ ♠♣♦ ♠♥ét♦ st ♦♥

rçã♦ ♣r♠t ♣r s s r② ♠♣èr ♥s ♦r♠s r♥ ♥tr ♥♦

s♣ç♦ srt♦ é♠ ss♦ á ♠ s♦♠♥t♦ ♠♦ ♣ss♦ t♠♣♦r ♥tr ♦s ♠

♣♦s étr♦ ♠♥ét♦ ♣r♦♦♦ ♣ ♣r♦①♠çã♦ ♣♦r r♥çs ♥ts s rs

t♠♣♦rs

♦r♠çã♦ t♠át ♦ ♠ét♦♦

①♣♥♥♦ r② ♠ ♦♦r♥s rt♥rs ♦tê♠s s s♥ts

qçõs srs

∂Hx

∂t=

1

µ

(

∂Ey

∂z−

∂Ez

∂y

)

,

∂Hy

∂t=

1

µ

(

∂Ez

∂x−

∂Ex

∂z

)

,

∂Hz

∂t=

1

µ

(

∂Ex

∂y−

∂Ey

∂x

)

.

♠s♠ ♦r♠ ♣r ♠♣èr t♠♦s

∂Ex

∂t=

1

ε

(

∂Hz

∂y−

∂Hy

∂z− σEx

)

,

∂Ey

∂t=

1

ε

(

∂Hx

∂z−

∂Hz

∂x− σEy

)

,

∂Ez

∂t=

1

ε

(

∂Hy

∂x−

∂Hx

∂y− σEz

)

.

♦♥♦r♠ ♦ st♦ ♥tr♦r♠♥t ♣s ♦ ♦♥t♦ r♥çs ♥ts ♥

trs ♣r ♣r♦①♠r s rs ❯♠ ♣♦ssí ♣r♦①♠çã♦ ♥tr ♣r rs

♣♦r r♥çs ♥ts é ♥ ♣♦r

df(α)

dα≈

f (α + ∆α) − f (α − ∆α)

2∆α

.

①♣rssã♦ é ♣ ♥s qçõs ♦♠♥♦ ♦♠♦ ①♠♣♦

r t♠♣♦r ∂E(x,y,z,t)∂t

♣r♦①♠çã♦ ♣♦r r ♥tr é ♣♦r

∂E(x, y, z, t)

∂t≈

En+1(i,j,k) − En

(i,j,k)

∆t

.

♦♥sr♥♦ r s♣ ∂H(x,y,z,t)∂z

t♠s q

∂H(x, y, z, t)

∂z≈

Hn+ 1

2

(i,j,k+ 1

2)− H

n+ 1

2

(i,j,k− 1

2)

∆z

.

♣♥♦ ♥s ①♣rssõs ♦s ♦♥t♦s ♦sr♦s ♠ ♦té♠

s s s♥ts qçõs t③çã♦

Hn+ 1

2

x (i,j+ 1

2,k+ 1

2)= H

n− 1

2

x (i,j+ 1

2,k+ 1

2)+

+∆t

µ

[

Eny (i,j+ 1

2,k+1)

− Eny (i,j+ 1

2,k)

∆z

−

Enz (i,j+1,k+ 1

2)− En

z (i,j,k+ 1

2)

∆y

]

,

Hn+ 1

2

y (i+ 1

2,j,k+ 1

2)= H

n− 1

2

y (i+ 1

2,j,k+ 1

2)+

+∆t

µ

[

Enz (i+1,j,k+ 1

2)− En

z (i,j,k+ 1

2)

∆x

−

Enx (i+ 1

2,j,k+1)

− Enx (i+ 1

2,j,k)

∆z

]

,

Hn+ 1

2

z (i+ 1

2,j+ 1

2,k)

= Hn− 1

2

z (i+ 1

2,j+ 1

2,k)

+

+∆t

µ

[

Enx (i+ 1

2,j+1,k)

− Enx (i+ 1

2,j,k)

∆y

−

Eny (i+1,j+ 1

2,k)

− Eny (i,j+ 1

2,k)

∆x

]

,

En+1x (i+ 1

2,j,k)

= Enx (i+ 1

2,j,k)

(

1 − σ∆t

2ε

1 + σ∆t

2ε

)

+

+∆t

ε(

1 + σ∆t

2ε

)

Hn+ 1

2

z (i+ 1

2,j+ 1

2,k)

− Hn+ 1

2

z (i+ 1

2,j− 1

2,k)

∆y

−

Hn+ 1

2

y (i+ 1

2,j,k+ 1

2)− H

n+ 1

2

y (i+ 1

2,j,k− 1

2)

∆z

,

En+1y (i,j+ 1

2,k)

= Eny (i,j+ 1

2,k)

(

1 − σ∆t

2ε

1 + σ∆t

2ε

)

+

+∆t

ε(

1 + σ∆t

2ε

)

Hn+ 1

2

x (i,j+ 1

2,k+ 1

2)− H

n+ 1

2

x (i,j+ 1

2,k− 1

2)

∆z

−

Hn+ 1

2

z (i+ 1

2,j+ 1

2,k)

− Hn+ 1

2

z (i− 1

2,j+ 1

2,k)

∆x

,

En+1z (i,j,k+ 1

2)= En

z (i,j,k+ 1

2)

(

1 − σ∆t

2ε

1 + σ∆t

2ε

)

+

+∆t

ε(

1 + σ∆t

2ε

)

Hn+ 1

2

y (i+ 1

2,j,k+ 1

2)− H

n+ 1

2

y (i− 1

2,j,k+ 1

2)

∆x

−

Hn+ 1

2

x (i,j+ 1

2,k+ 1

2)− H

n+ 1

2

x (i,j− 1

2,k+ 1

2)

∆y

.

rtér♦s ♣rsã♦ st

s ♠ét♦♦s ♥♠ér♦s sã♦ ♥♦s ♣rtr ♣r♦①♠çõs ♣♦rt♥t♦ r♠ rr♦s

♥♠ér♦s q sã♦ ♠t♦s ♣rs♠ sr ♠♥♠③♦s ss ♦r♠ é ♥ssár♦

sts③r ♦♥çõs ♣r r♥tr q ♦ ♠ét♦♦ ♥♠ér♦ ♣rs♥t s♦çõs q ♥♠

♦♥rr ♦ rst♦ ís♦ sts rtér♦s stã♦ r♦♥♦s ♦s ♥r♠♥t♦s s♣s

∆x ∆y ∆z ♦ ♥r♠♥t♦ t♠♣♦r ∆t

♥tr ♦s rr♦s ♥♠ér♦s ♦ ♠ét♦♦ ♣r♦♦ st♦rçõs ♥ s ♦ s♥ ♣r♦

♣♥t t♦ st s♦ ♣ ♣r♦①♠çã♦ s rs q r t♠é♠ st♦rçã♦

♥ ♦ ♣r♦♣çã♦ ♦♥ st ♥ô♠♥♦ ♥ã♦ ís♦ é ♥♦♠♥♦ s♣rsã♦

♥♠ér ❬❪

Pr r③r ♦ t♦ st ♥ô♠♥♦ ♥♠ér♦ ♦ ♥r♠♥t♦ s♣ sr ♠t♦

♠♥♦r ♥♦ ♠á①♠♦ ③s ♠♥♦r ♦ q ♦ ♦♠♣r♠♥t♦ ♦♥ ♠í♥♠♦ ♦ s♥

♣r♦♣♥t ❬❪ ss ♦r♠ ♦ rtér♦ ♣r ♥r ♦ ♦r ♦s ♥r♠♥t♦s s♣s ∆x

∆y ∆z é ♦ ♣♦r

∆x,y,z ≤λmin

10

♦♥ λmin é ♦ ♠í♥♠♦ ♦♠♣r♠♥t♦ ♦♥ ss♦♦ à rqê♥ ♠á①♠ fmax ♦ s♥

①tçã♦ ♣s♦ rqê♥ ♠á①♠ é tr♠♥ ♣ ♠♦r rqê♥ ♦ s♥

q ♣♦ss ♥r s♥t sr q ♠♣ ♥♦ t♦ ♦ ♠ét♦♦

♥ã♦ sr ♣③ s♠r ♠♣s♦s st♦ rqrr q ♦ ♥r♠♥t♦ s♣ ♦ss ③r♦

á q rqê♥ ♠á①♠ ♠ ♠♣s♦ é ♥♥t λmin = 0

st ♥♠ér é r♥t ♥♥♦ ♦ ♥r♠♥t♦ t♠♣♦r ∆t ♣rtr

♦♥çã♦ ♦r♥t ❬❪ st rtér♦ ♦t♦ trés ♦ ♠ét♦♦ ❱♦♥♠♥♥ é

①♣rss♦ ♣♦r

∆t <1

vmax

√

(

1∆x

)2

+(

1∆y

)2

+(

1∆z

)2

♦♥ vmax é ♦ ♠á①♠ ♣r♦♣çã♦ ♦♥ ♦r♠♠♥t t③s

♦ ♠t s♣♦ ♠

♦ s♦tr s♥♦♦ ♥st tr♦ t③s ♦ ♠♦♦ ♦ ♦ ♥♦ ❬❪ ♣r

r♣rs♥tr q♠♥t ♦s ♠♥t♦s ♠tá♦s í♥r♦s ♦♠ r♦ ♠♥♦r q rst

é ❨

♦t♦ st tr♦ é ♥sr ♦s sst♠s trr♠♥t♦ ♠ ♠♥ts rt♦s

♣♦rt♥t♦ sr ♥ssár♦ ♠♣♠♥tr ♠ ♦♠í♥♦ ♦♠♣t♦♥ ♥♥t♦ ♥trt♥t♦

s ♠áq♥s ♣♦ss♠ ♠tçã♦ ♠♠ór ♣r♦ss♠♥t♦ ss ♦r♠ é ♥ssár♦

tr♥r ♦ ♦♠í♥♦ ♦♠♣t♦♥ t ♦r♠ q s ♣♦ss r♣rs♥tr ♦♥ tr♦

♠♥ét ♣r♦♣♥♦s ♣r ♦r ♦ s♣ç♦ ♥ás ♦♠ ♠í♥♠♦ r①õs st

tr♦ tr♥♠ ♦ ♦♠í♥♦ ♦♠♣t♦♥ é t t③♥♦ té♥ P ♦♥

♦t♦♥ Prt② t ②r ❬❪

♦♠ trs s♣rs♦s rtrár♦s

♣♦r ♦♥sr♥♦s σ = σ(ω)

r sssã♦ s♦r ♣r♦①♠çã♦ ♥çõs trés ♦

P♦♥ô♠♦ Pé

st tr♦ r③s ♣r♦①♠çã♦ ♥çã♦ ♠t♠át q ♠♦ s♣rsã♦

♦ s♦♦ srt ♥♦ ♣ít♦ ♣r♦①♠çã♦ é t t③♥♦ ♦ ♣♦♥ô♠♦ Pé ❬❪

q srá srt♦ ♥st ssçã♦

♣r♦①♠çã♦ Pé é ♥ ♣ r③ã♦ ♥tr ♦s ♣♦♥ô♠♦s ♣r ♠♦r ♠

♥çã♦ ♠t♠át rtrár st ♣r♦①♠çã♦ é s ♥ s♥t ①♣rssã♦

f(x) ≈

∑ni aix

i

∑mj bjxj

.

♠ s ts ♦t♦r♦ ❬❪ Pé ♠♦str q é ♥ssár♦ ♥r b0 = 1

♠ét♦♦ ♣r♦①♠çã♦ Pé ♣r♠t ♠ r♣rs♥tçã♦ tr♥t ♣r ♥çõs

♦♠♣s ♥♦r♠♠♥t ♦♠ ♠♥♦s tr♠♦s q sér ②♦r r③ã♦ ♦s

♣♦♥ô♠♦s ♠♥t ♦ r r ♣r♦①♠çã♦ Pé ♦♠ ss♦ ♦srs

t♠é♠ q ① ♦♥rê♥ ♦ ♣♦♥ô♠♦ Pé é ♠t♦ ♠s r ♦ q

① ♦♥rê♥ sér ②♦r ♦♠ ár♦s tr♠♦s ♣rs♥ts

♠ rá♦ strt♦ ♦♠♣r♥♦ ♣r♦①♠çã♦ ♠ ♥çã♦ ①♣♦♥♥ f(x) =

e−x t③♥♦ sér ②♦r ♦ ♣♦♥ô♠♦ Pé st r ♣rs♥ts ♠

♣r♦①♠çã♦ ♣ sér ②♦r t③♥♦ ss tr♠♦s s♥♦ ♥ ♣♦r

f(x) ≈5∑

k=0

f (k)(0)

k!xk = 1 − x +

x2

2!−

x3

3!+

x4

4!−

x5

5!.

♣♦♥ô♠♦ Pé t③♦ ♥st ♦♠♣rçã♦ é ♥♦ ♣♦r ♠ r③ã♦ ♥tr ♦s

♣♦♥ô♠♦s ♦ r s ♥çã♦ ♠t♠át é ♣♦r

f(x) ≈1 − 1

2x + 1

12x2

1 + 12x + 1

12x2

.

r rá♦ ♦♠♣r♥♦ ♣r♦①♠çã♦ ♥çã♦ e−x ♣♦r ér ②♦r ♣♦r♣♦♥ô♠♦ Pé ♦♠♣rçã♦ ♥tr s s ♣r♦①♠çõs ♥çã♦ ♦r♥ rá♦ ♦ rr♦ s♦t♦

0 1 2 3 4-1,0

-0,5

0,0

0,5

1,0

exp(

-x)

x

exp(-x) Série de Taylor Aprox. Padé

0 1 2 3 40

1

2

3

4

Erro

abs

olut

o

x

Série Taylor Aprox. Padé

sr♥♦ ♦s rá♦s ♥ rs q ♦ ♣♦♥ô♠♦ Pé ♦♥s

r♣rs♥tr ♠♦r ♥çã♦ f(x) = e−x t③♥♦ ♠ r③ã♦ ♣♦♥ô♠♦s ♦♠ ♠♥♦s

tr♠♦s ♥ ♣rtr ♥♦ts q ♦ rr♦ s♦t♦ ♣r♦①♠çã♦ ♣

sér ②♦r rs r♣♠♥t ♥q♥t♦ q ♦ rr♦ s♦t♦ ♣r♦①♠çã♦ t

trés ♦ ♣♦♥ô♠♦ Pé é ♣rt♠♥t ③r♦ ♠ qs t♦ ① ♦♥sr

st tr♦ t③s ♣r♦①♠çã♦ Pé ♣r r♣rs♥tr ♥çã♦ ♠t♠át

q ♠♦ ♦ ♦♠♣♦rt♠♥t♦ ♦♥t ♦ s♦♦ ♦♠♦ ♠ ♠♦ s♣rs♦

é♠ ♠♦r ① ♦♥rê♥ ♦ ♦r♠t♦ ♦ ♣♦♥ô♠♦ Pé é ♠s q♦

♣r ♥r ♦ ♠♦♦ ♣r ♠♦s s♣rs♦s ♥♦ ♦♥♥t♦ qçõs ♦ ♠ét♦♦

♦♥♦r♠ srá st♦ sr ssts q é ♣ té♥ ♠í♥♠♦s q

r♦s ♣r st♠r ♦ r♣♦ ♦♥ts ①♣rssã♦ Pé ♣r ♦ ♣r♦♠ q

♠♦♦

♦♠ trs s♣rs♦s rtrár♦s ♣

♦ ét♦♦

Pr ♦s ♠♦s s♣rs♦s tê♠s s s♥ts rçõs ♦♥sttts

~D(t) = ε ~E(t),

~J(t) = σ(t) ∗ ~E(t) =

∫ t

0

~E(t − τ)σ(τ) dτ

~B(t) = µ ~H(t).

st tr♦ ♦ ♦♠♣♦rt♠♥t♦ s♣rs♦ ♦ ♠♦ srá ♣r♦♦♦ ♣ rçã♦

♦♥t étr ♦♠ rqê♥ ♥r ♦ s σ = σ(ω) P♦rt♥t♦

♠♣èr ♠♦ ♦♥sr♥♦ ♦ ♠♦ s♣rs♦ á q ~J(t) = σ(t) ∗ ~E(t) =

∫ t

0~E(t − τ)σ(τ) dτ é ♥ ♣♦r

ε∂ ~E

∂t+

(∫ t

0

~E(t − τ)σ(τ) dτ

)

= ∇× ~H.

r② ♣r♠♥ ♥tr ♦♥♦r♠ ♣♦ sr st♦ sr

∂ ~B

∂t= −∇× ~E.

♦♥♦r♠ ♦ t♦ ♥tr♦r♠♥t ①♣rssã♦ sr rçã♦ ♦♥t

étr ♦♠ rqê♥ ❬❪ r♣rs♥t♥♦ ♦ ♦♠♣♦rt♠♥t♦ ♠ s♦♦ s♣rs♦

Pr ♠♦r st ♦♠♣♦rt♠♥t♦ ♥♦ ♠ét♦♦ ♥st tr♦ t③♦s ♠

♣r♦①♠çã♦ ♦♠♣① ♣♦r ♣♦♥ô♠♦ Pé ♣r σ(ω) ♣♦r

σ(ω) ≈a0 + a1(jω) + a2(jω)2 + ... + an(jω)n

1 + b1(jω) + b2(jω)2 + ... + am(jω)m.

♥♦ q f = ω2π ①♣rssã♦ ♣r♦♣♦st ♣♦r ❱sr♦ t ❬❪ é rsrt

s♥t ♦r♠

ρ(ω) = ρDC

1 + [1,2 × 10−6· ρ0,73

DC ] ×

[

( ω

2π− 100

)0,65]

−1

,

♦♥

σ(ω) = ρ(ω)−1,

♣rtr q r③s ♥tr♣♦çã♦ ♥çã♦ σ(ω) ♣ ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦

Pé ♦♠ n = m = 2 ♦t♥♦

σ(ω) ≈a0 + a1(jω) + a2(jω)2

1 + b1(jω) + b2(jω)2

s ♦♥st♥ts a0 a1 a2 b1 b2 ♣♦♠ sr ♦ts rs♦♥♦s ♠ sst♠ qçõs

♥rs ssts q st ♦♥♥t♦ ♦♥st♥ts é ♥çã♦ rqê♥ ♦ ♦r

σDC = 1/ρDC st tr♦ t③♦s té♥ ♥tr♣♦çã♦ ♣♦r ♠í♥♠♦s qr♦s

❬❪ ♣r s♦♦♥r ♦ sst♠ ♥r ♥♦♥trr ♦s ♦♥ts ♦ ♣♦♥ô♠♦ Pé

❯t③♥♦ ♠ ♥♦ ♦♠í♥♦ rqê♥ t♠s

Jy(ω) = σ(ω)Ey(ω) ≈a0 + a1(jω) + a2(jω)2

1 + b1(jω) + b2(jω)2Ey(ω),

♦ s

(

1 + b1(jω) + b2(jω)2)

Jy(ω) =(

a0 + a1(jω) + a2(jω)2)

Ey(ω)

ssts q ♦ ♣♦♥ô♠♦ Pé ♦ ♥♦ ♠ tr♠♦s jω ♣r♦♣♦st♠♥t ♣r

q ♠♥♣çã♦ tr♥s♦r♠ ♥rs ♦rr s ♠♥♦s ♦♠♣ ♣♥♦

tr♥s♦r♠ ♥rs ♦rr ♠ ♦tê♠s s♥t ①♣rssã♦ ♥♦ ♦♠í♥♦

♦ t♠♣♦

Jy(t) + b1J′

y(t) + b2J′′

y (t) = a0Ey(t) + a1E′

y(t) + a2E′′

y (t).

sr♥♦ ♥ ♦r♠ r♥çs ♥ts s t♦r♥

Jny

+ b1

Jn+1y − Jn−1

y

2∆t+ b2

Jn+1y − 2Jn

y + Jn−1y

∆t2=

= a0Eny

+ a1

En+1y − En−1

y

2∆t+ a2

En+1y − 2En

y + En−1y

∆t2.

♥s♥♦ s qçõs ① t♠s q

∂Dy

∂t+ Jy ≈ ε

(

En+1y − En

y

∆t

)

+(

Jn+ 1

2y

)

=(

∇× ~H)

n+ 1

2y .

srs q Jy sr ♦ ♥♦ ♥st♥t ís♦♦♠♣t♦♥ n + 1/2 ♥tã♦

Jn− 1

2y

+ b1J

n+ 1

2y − J

n− 3

2y

2∆t+ b2

Jn+ 1

2y − 2J

n− 1

2y + J

n− 3

2y

∆t2=

= a0En− 1

2y

+ a1E

n+ 1

2y − E

n− 3

2y

2∆t+ a2

En+ 1

2y − 2E

n− 1

2y + E

n− 3

2y

∆t2.

♦♦♥♦ ♦s tr♠♦s Jy ♦s ♥♦ ♠s♠♦ ♥st♥t ♠ ê♥ t♠s

(

b1

2∆t+

b2

∆t2

)

Jn+ 1

2y =

(

2b2

∆t2− 1

)

Jn− 1

2y

+

(

b1

2∆t−

b2

∆t2

)

Jn− 3

2y +

+a0En− 1

2y

+ a1E

n+ 1

2y − E

n− 3

2y

2∆t+ a2

En+ 1

2y − 2E

n− 1

2y + E

n− 3

2y

∆t2.

s

Jn+ 1

2y =

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y +

+a0

(

b12∆t

+ b2∆t2

)En− 1

2y

+a1

(

b12∆t

+ b2∆t2

)

En+ 1

2y − E

n− 3

2y

2∆t+

+a2

(

b12∆t

+ b2∆t2

)

En+ 1

2y − 2E

n− 1

2y + E

n− 3

2y

∆t2.

♣♥♦ s ♠és t♠♣♦rs ♣r ♦s tr♠♦s Ey ♦tê♠s

Jn+ 1

2y =

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y +

+a0

(

b12∆t

+ b2∆t2

)

(

Eny

+ En−1y

2

)

+a1

(

b12∆t

+ b2∆t2

)

(

En+1y

+Eny

2

)

−

(

En−1y

+En−2y

2

)

2∆t+

+a2

(

b12∆t

+ b2∆t2

)

(

En+1y

+Eny

2

)

− 2(

Eny+En−1

y

2

)

+(

En−1y

+En−2y

2

)

∆t2.

♦ ♥

Jn+ 1

2y =

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y +

+a0

2(

b12∆t

+ b2∆t2

)

(

Eny

+ En−1y

)

+a1

(

b12∆t

+ b2∆t2

)

(

En+1y

+ Eny

)

−

(

En−1y

+ En−2y

)

4∆t+

+a2

(

b12∆t

+ b2∆t2

)

(

En+1y

+ Eny

)

− 2(

Eny

+ En−1y

)

+(

En−1y

+ En−2y

)

2∆t2.

♦♦♥♦ ♦ tr♠♦ Ey ♠ ê♥ ♣r ♥st♥t t♠♣♦ t♠s

Jn+ 1

2y =

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y +

+

(

a1(

b12∆t

+ b2∆t2

)

4∆t+

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En+1y

+

+

(

a0

2(

b12∆t

+ b2∆t2

) +a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

Eny+

+

(

a0

2(

b12∆t

+ b2∆t2

) −a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En−1y

+

+

(

a2(

b12∆t

+ b2∆t2

)

2∆t2−

a1(

b12∆t

+ b2∆t2

)

4∆t

)

En−2y

.

stt♥♦ ♠ ♦tê♠s

ε

(

En+1y − En

y

∆t

)

+(

Jn+ 1

2y

)

=(

∇× ~H)

y=

(

ε

∆t+

a1(

b12∆t

+ b2∆t2

)

4∆t+

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En+1y +

(

−ε

∆t+

a0

2(

b12∆t

+ b2∆t2

) +a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

Eny +

(

a0

2(

b12∆t

+ b2∆t2

) −a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En−1y

+

+

(

a2(

b12∆t

+ b2∆t2

)

2∆t2−

a1(

b12∆t

+ b2∆t2

)

4∆t

)

En−2y

+

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y .

Pr t♦ s♠♣çã♦ ♠♥t♦ ê♥ ♦♠♣t♦♥ ♥s

s♥t qçã♦ ①r

Ψ0 =

(

ε

∆t+

a1(

b12∆t

+ b2∆t2

)

4∆t+

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

.

P♦r ♠ qçã♦ t③çã♦ ♣r En+1y é ♣♦r

En+1y =

(

∇× ~H)

y

Ψ0

−1

Ψ0

(

a0

2(

b12∆t

+ b2∆t2

) +a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2−

ε

∆t

)

Eny +

−1

Ψ0

(

a0

2(

b12∆t

+ b2∆t2

) −a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En−1y

+

−1

Ψ0

(

a2(

b12∆t

+ b2∆t2

)

2∆t2−

a1(

b12∆t

+ b2∆t2

)

4∆t

)

En−2y

−1

Ψ0

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 1

2y

+

−1

Ψ0

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 3

2y .

♦♥ Jn− 1

2y ♦ ♥ts En+1

y é ♦ ♣♦r

Jn− 1

2y =

(

2 b2∆t2

− 1)

(

b12∆t

+ b2∆t2

)Jn− 3

2y

+

(

b12∆t

− b2∆t2

)

(

b12∆t

+ b2∆t2

) Jn− 5

2y +

+

(

a1(

b12∆t

+ b2∆t2

)

4∆t+

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

Eny+

+

(

a0

2(

b12∆t

+ b2∆t2

) +a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En−1y

+

+

(

a0

2(

b12∆t

+ b2∆t2

) −a1

(

b12∆t

+ b2∆t2

)

4∆t−

a2(

b12∆t

+ b2∆t2

)

2∆t2

)

En−2y

+

+

(

a2(

b12∆t

+ b2∆t2

)

2∆t2−

a1(

b12∆t

+ b2∆t2

)

4∆t

)

En−3y

.

♠s♠ é ♣ ♣r ♥♦♥trr s qçõs t③çã♦ ♣r s ♠s

♦♠♣♦♥♥ts ♦ ♠♣♦ étr♦ s qçõs t③çã♦ ♣r ♦ ♠♣♦ ♠♥ét♦

♥s ♣♦r ♥ã♦ sã♦ ♠♦s

♠♣♦rt♥t ♥♦tr q ♠♣♠♥tçã♦ rqr ♦ s♦ rás

t ♣rsã♦ ♥♠ér s♦ ♦♥trár♦ rê♥ ♦ ♠ét♦♦ ♣r♦♣♦st♦ é ♦sr

Pr rs♦r t qstã♦ ♦ t③ ♥st tr♦ ♦t ❯ P t♣

Prs♦♥ ♦t♥♣♦♥t ♦♠♣tt♦♥s t ♦rrt ♦♥♥ ❬❪

♣ít♦

♦♥srçõs ♦r ♦s ①♣r♠♥t♦s

♦♥tú♦ ♦ ♣ít♦

♠t♦♦♦ ♠çã♦ ♠♣r ♦ st♣ ①♣r♠♥t ♦r♠ s♦s ♥♦ tr

♦ ♥ ❬❪ P♦rt♥t♦ ♥♠♥t srá t ♠ r ♣rs♥tçã♦ ♠t♦♦

♦ ♣r♦♣♦st ♣♦r ♠ s sã♦ ♣rs♥t♦s ♦s q♣♠♥t♦s t③♦s ♥♦s ①♣

r♠♥t♦s ♠ ♠♣♦ é♠ ss♦ sã♦ ts ♠s ♦♥srçõs s♦r ♦s q♣♠♥t♦s

♣r ♠♥♠③r ♥trrê♥s rí♦s ♦tr♦s t♦rs q ♣rq♠ ♦♥♦râ♥

♦s rst♦s ①♣r♠♥ts ♦♠ ♦s ♦t♦s ♥s s♠çõs ♥♠érs s ♣rçõs

sts q sã♦ ss ♠ ❬❪

t♣ ①♣r♠♥t

♠ s tr♦ ♥ ❬❪ ♣r♦♣õ ♠ ♠t♦♦♦ ♣r ♥sr ♦ ♦♠♣♦rt♠♥t♦

tr♥stór♦ sst♠s trr♠♥t♦ ♦♠♣r♥♦ rst♦s ①♣r♠♥ts s♠♦s

trés ♦ ♠ét♦♦ ♣r ♣s♦s srs t♠♦sérs ♣rs♥t

s ♦♥rçã♦ ①♣r♠♥t ♣r♦♣♦st ♠ ❬❪ ♦♠♣♦st ♣♦r ♠ r♦r srt♦

t♥sã♦ rsstê♥ ♥ sí ♦ r♦r ♣♦♥t rsst rt♦ ♦rr♥t rt♦

t♥sã♦

r ♦♥rçã♦ ①♣r♠♥t ♣r♦♣♦st ♣♦r ♥ ♣t♦ ❬❪

rt♦ ♦rr♥t é t③♦ ♦r♠ ♥r ♦ ♣♦♥t♦ rrê♥ ♦ r♦r

srt♦ t ♠♥r q ♦ rt♦ ♦rr♥t é ♦ ♣ trr ♥tr ♦ tr♦♦

♥çã♦ ♦ tr♦♦ rrê♥ ♦ rt♦ ♦rr♥t rt♦ t♥sã♦ t♠ ♠

ss ①tr♠♦s tr♠♥s ♣♦s♦♥♦ ♠ ♣q♥ stâ♥ st ♥çã♦

♦rr♥t ♦r♠♥♦ ♠ ♣ ♦♠ ♦ ♦t♦ r ♦ s♥ tr♥stór♦ t♥sã♦

♦ sst♠ trr♠♥t♦ ♦tr♦ ①tr♠♦ ♦ rt♦ t♥sã♦ é ♦ tr♦♦ r♠♦t♦

t♥sã♦ s tr♦♦s rrê♥ ♦s rt♦s ♦rr♥t t♥sã♦ ♠ str

s♥t♠♥t st♥ts ♦ ♣♦♥t♦ ♥çã♦ t ♦r♠ q ~E ≈ 0

♦rr♥t ♥t i(t) é ♠ ♦ ♥ st ♥çã♦ ♠ ♠ ♣♦♥t♦

♣ró①♠♦ à s♣rí ♦ s♦♦ t♥sã♦ ♥③ v(t) é ♠ ♦ ♥♦ ♣

♥tr st ♥çã♦ ♦ ①tr♠♦ ♠s ♣ró①♠♦ ♦ ♦ ♦ rt♦ t♥sã♦

♣rtr sts ♣râ♠tr♦s é ♣♦ssí r rsstê♥ trr♠♥t♦ R ♣r

♦♠♣♠♥tr ♥ás ♦ ♦♠♣♦rt♠♥t♦ tr♥stór♦ ♦s sst♠s trr♠♥t♦

q♣♠♥t♦s ❯t③♦s ♦♥srçõs

s q♣♠♥t♦s ♥ssár♦s ♣r ♠♣♠♥tr ♦ st♣ ①♣r♠♥t ♠♣r♦ ♥st

tr♦ sã♦ sts trr♠♥t♦ ♦s étr♦s rsst♦r ♦s♦só♣♦ ♦♠ s♦♥s

♠çã♦ ♦rr♥t t♥sã♦ ♥rs♦r ♥r tr r♦r srt♦ t♥sã♦

♣rs♥ts ♠ ♠♠ ♦s q♣♠♥t♦s t③♦s ♥st tr♦

r q♣♠♥t♦s t③♦s ♥♦s ①♣r♠♥t♦s ♠ ♠♣♦

Gerador de Surto de Tensão

Osciloscópio Digital

Amplificador de corrente

Bateria 12

Hastes de aterramento

Inversor de energia 12 !! − 127 #!

Cabos e terminais

$%('()*'+,) =2,2 /0

Sondas de medição

♦r ♦ r♦r rt♦ ♥sã♦

♦ s♥♦♠♥t♦ st tr♦ ♦ t③♦ ♠ r♦r srt♦ s

s♥♦♦ ♠ ❬❪ ♦ q t♥ às ♥♦r♠s ❬❪ t♥r ♥

qs ♦r ❱♦t st♥ ❬❪ ♣r tsts ♦♠ ♣s♦s t t♥sã♦ q r♣r♦③♠

s ♦r♠s ♦♥ s srs t♠♦sérs Pr r♣rs♥tr s srs ssq♥

ts s ♥♦r♠s ❬❪ ❬❪ tr♠♥♠ q ♦s t♠♣♦s s t♥sã♦ ♦♠

♦s tr♠♥s ♠ rt♦ rt♦ ♠ sr s 0,8 µs 50 µs rs♣t♠♥t st

stçã♦ é ♠s rít ♦ ♣♦♥t♦ st ♣r♦r♠♥ tr♥srê♥ ♦rr♥t ♣r

trr ♦ ♦ ♣q♥♦ t♠♣♦ s ♦ ♣s♦

é♠ ss♦ ♥st stçã♦ ♦ ♣s♦ t♠♦sér♦ ♣♦ss ♥r s♥t ♥s rqê♥

s ♠s ts ③ ♣r♦①♠♠♥t ③ st ♦srçã♦ st ♦♥

trçã♦ ♣r♥♣ st tr♦ ♣♦s ♥ss ♥ rqê♥ ♦ ♦♠♣♦rt♠♥t♦ s

♣rs♦ ♦ s♦♦ t♦r♥s ♥t ♦♥♦r♠ ♦ st♦ ♥ çã♦ sr ♦♥sr♦

♥ ♥ás ♦s sst♠s trr♠♥t♦

r P♥ r♦♥t ♦ r♦r srt♦ t③♦ ♥♦s ①♣r♠♥t♦s

Chave seletora Chave de disparo manual

Voltímetro

Chave seletora de tensão

Terminais de saída

♦r♠ ♦♥ ♦ r♦r srt♦ ♦ ♠♦ ♠t♠t♠♥t trés ①

♣rssã♦

p(t) = A1e−

“

t−to1τ1

”2

+ A2e−

“

t−to2τ2

”2

+ A3e−

“

t−to3τ3

”2

+ A4e−

“

t−to4τ4

”2

+ A5e−

“

t−to5τ5

”2

+

+B(

e−α1t− e−α2t

)

+ K,

♦♥ B = 6,338× 102 ❱ α1 = 1,5× 104 s−1 α2 = 1,2× 106 s−1 K = −1,16× 102 ❱ s

♣râ♠tr♦s Aβ toβ τβ sã♦ ♣rs♥t♦s ♥

❱♦rs ♦s ♣râ♠tr♦s Aβ toβ τβ

Prâ♠tr♦s❮♥ β

Aβ ❱ −2,112 × 102 2,535 × 102 2,535 × 101 −1,479 × 101 3,803 × 101

toβ s 0,25 × 10−6 1,14 × 10−6 1,78 × 10−6 2,5 × 10−6 3,45 × 10−6

τβ s 0,17 × 10−6 0,3 × 10−6 0,4 × 10−6 0,2 × 10−6 0,3 × 10−6

♣rs♥ts ♦♠♣rçã♦ ♥tr ♦ s♥ ♠♦ ♦ r♦r srt♦ t

③♦ ♥♦s tsts ①♣r♠♥ts ♦ s♥ ①tçã♦ ♠♦♦ ♥s s♠çõs ♥♠érs

srs ♠ ♦ ♦♥♦râ♥ ♥tr sts ♦s s♥s

r ♥ ♦ r♦r ♠♦ ♦ s♥ ♠♦♦ ♠t♠t♠♥t ♣r s s♠çõs ♥♠érs

0,0 0,5 1,0 1,5 2,0 2,5 3,00

100

200

300

400

500

600

700

800

900Sinal de excitação

Tens

ão (V

)

Tempo (s)

Medido Modelado para a simulação

♦r P♦♥t sst ♦♥t à í ♦ r♦r

♦♠♦ ♦ r♦r srt♦ ♣♦ss ① rsstê♥ ♥tr♥ é ♥ssár♦ t③r ♠

♣♦♥t rsst ♥tr ♦ r♦r ♦ sst♠ trr♠♥t♦ s♦ tst ♦r♠ q

♠♣t ♦r♠ ♦ ♣s♦ t♥sã♦ ♥t♦ s♠ ♣rsr♦s ♥♣♥♥t r

q é ♦♥t ♦ r♦r Pr st♦ ♦ ♦r rsstê♥ RS ♣♦♥t rsst sr

♠t♦ ♠♦r ♦ q ♦ ♠ó♦ ♠♣â♥ ♦ ♦♥♥t♦ ♦r♠♦ ♣♦r s♦♦ sst♠

trr♠♥t♦ st tr♦ t③♦s ♠ rsst♦r ♣♦♥t rsst ♦♠ ♦r ♥♦♠♥

RS(nominal) = 2,2 kΩ q é ♠t♦ ♠♦r ♦ q ♠♦r rsstê♥ ♥tr♥ ♦ r♦r

≈ 33Ω ♦ q rsstê♥ trr♠♥t♦ ♦sr ♥♦s ①♣r♠♥t♦s ♠ ♠♣♦

≈ 44 Ω ♦♠ st ♣r♦♠♥t♦ ♦r♠ ♦♥ ♦ s♥ ♥t♦ ①♣r♠♥t♠♥t

♥♦ s♦♦ é ♣rsr

♥ ♥st♥♦ r③ã♦ ♥tr t♥sã♦ ♥tr ♦s tr♠♥s ♦ rsst♦r vR(t)

♦rr♥t i(t) q trés ♦sr♥♦ ♦ ♦r r♠ ♦r qs

r ♥♦t♦s q ♥♦s ①♣r♠♥t♦s st rçã♦ ♠ r♥t ♦ t♠♣♦ ♣çã♦

♦ ♣s♦ ♦r♠ q s♦s s♥t♦s ♠ rçã♦ ♦ ♦r ♥♦♠♥ sã♦ ♦sr♦s

♦ st t♦ ♥s s♠çõs ♥♠érs t③♦s ♠ ♦r t♦ ♣r rsstê♥

♣♦♥t ♦ q ♥♦♠♥♦s rsstê♥ t RS(efetiva) ♠ ♠ ♦s tsts ①♣r

♠♥ts ♦ r ♠ rsstê♥ t RS(efetiva) ≈ 2,054 kΩ ♦ s ♦ s♦

♣r♥t ♦sr♦ ♦ ♣r♦①♠♠♥t 6,6 ♦♥♦r♠ srá st♦ ♥♦ ♣ít♦

t♦ RS(efetiva) ♣rs♥tr ♠ r♥ç ♦♥srá RS(nominal) é trí♦ ♦

♦♠♣♦rt♠♥t♦ ♥ã♦ ♥r ♦ ♦♠♣♦♥♥t q♥♦ s♠t♦ ♦ ♣s♦ tr♥stór♦ ♣

♦ çã♦ rsstê♥ t RS(efetiva) é ♠ t♦r tr♠♥♥t ♣r r

♦♥♦râ♥ ♥tr ①♣r♠♥t♦s s♠çõs

♦r ♦ s♦só♣♦

st tr♦ t③♦s ♦ ♦s♦só♣♦ t tr♦♥① P rr

♥ ③ t① ♠♦str♠ s ♦ q ♣♦ss qtr♦ ♥s tr

♠♥t s♦♦s ♥tr s ♦ s ♦ ♣♦♥t♦ rrê♥ ♥ é ♥♣♥♥t ♦s

♦tr♦s ♦♠ ss♦ ♦ ♦♣♠♥t♦ ♦♥t♦ ♥tr s rrê♥s ♦s ♥s ♦ ♠♥♦

♣r♠t ♦ s♦ ♠s ♠ ♥ s♠t♥♠♥t ♥s ♠çõs r③s ❱

rsstr q ♦ s♦ ♠s ♠ ♥ ♦♥♦♠t♥t♠♥t ♠♥ ♣♦ss

rr♦ ♥ ♠çã♦ s♠ ♥tr ♦s s♥s v(t) i(t) ♦t♦s ♥♦s ①♣r♠♥t♦s ♠

♠♣♦ ♥♦ ♦♥ s ♠çõs st s♠ ♥tr v(t) i(t) ♦♦rr

♦ ♥tr③ ♦♠♣① ♠♣â♥ ♦ sst♠ trr♠♥t♦ ♦ s♦♦

♦s tsts ①♣r♠♥ts s s♦♥s ♠çõs t③s ♦r♠ s♦♥ ♣r ts

t♥sõs ♠♦♦ tr♦♥① P VRMS Vp ♣r ♠r ♦ ♣s♦ t♥sã♦

♦ r♦r t♥sã♦ ♥♦s tr♠♥s ♣♦♥t rsst s♦♥ ♣r ①s t♥sõs ♠♦♦

tr♦♥① PP VRMS ♣r ♠r t♥sã♦ ♥③ s♦♥ ♦ t♣♦ t

♣r ♠çã♦ ♦rr♥t ♥t ♠♦♦ tr♦♥① P ADC Ap é♠

ss♦ ♦ t③♦ ♠ ♠♣♦r ♦rr♥t ♠♦♦ tr♦♥① P

♦r ♠♥tçã♦ ♦s q♣♠♥t♦s

Pr ♠♥r ♦s rí♦s ♣r♦♥♥ts r étr ♦ ♥ssár♦ t③r ♠ tr

t♦♠♦t ♠ ♦♥♥t♦ ♦♠ ♠ ♥rs♦r ♥r ❲ ❱DC ❱AC ♣r

♠♥tr ♦ r♦r srt♦ ♦s♦só♣♦ ♣♦r s ③ ♣♦ssr ♠ ♠♥tçã♦

♥♣♥♥t r étr t♠é♠ ♦ r♦r srt♦ ♦s♦só♣♦ t③♦

♥st tr♦ ♣♦ss ♠ tr ♥tr♥ ♠♥ ♣♦ss ♦♥t♠♥çã♦ s

♠çõs ♣r♦♦ ♣ r étr ♣♦ r♦r srt♦ trés ♠♥tçã♦

♣♦s ♦srr ♣r t♦ strçã♦ ♠ s♥ t♥sã♦ ♦♥t♠♥♦

♣♦ rí♦ ♣r♦♥♥t r étr q♥♦ ♦ ♦s♦só♣♦ é ♠♥t♦ rt♠♥t

♥ r

r ♥ ♦t♦ ♦ ♦♥tr ♦s q♣♠♥t♦s à r étr

Interferência

s ♣♦♥t♦s st♦s ♥st ♣ít♦ sã♦ ♠♣♦rtâ♥ ♥♠♥t ♣r q ♦s s♥s

♠♦s ♦t♦s ♣s s♠çõs ♥♠érs ♣rs♥tss♠ ♠ ♦ ♦♥♦râ♥ ❯♠

♦ ♦♥ sr t♦♠♦ ♦ r♦r srt♦ sr ♣♦s♦♥♦ ♣♦ ♠♥♦s

♠ ♠tr♦ ♠♦ ♠ ♦ ♦s♦só♣♦ s♦ ♦♥trár♦ s ♦rr♥ts q ♠ ♥♦s

rt♦s ♦ r♦r srt♦ ♥③rã♦ s♥s ♥♦ ♦s♦só♣♦ ♦♥t♠♥♥♦ ♦ rstr♦

♦s s♥s tr♥stór♦s s♣♠♥t t♥sã♦ ♥③

♣ít♦

❱çã♦ ♦♠ ♦ ♦♦

s♣rs♦ ♣ ♦ ét♦♦

♦♥tú♦ ♦ ♣ít♦

st ♣ít♦ sã♦ ♣rs♥ts s ♥áss ♥♦ ①♣r♠♥t♦s r③♦s ♠

♠♣♦ ♣r r ♠♦♠ ♦ s♦♦ s♣rs♦ ♣r♦♣♦st ♥♦ ♣ít♦ ♣r♠r♦

tst ①♣r♠♥t ♦ r③♦ ♥♦ rr♦ st ♦sã♦ ③s

♥ás rs♣♦st tr♥stór ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♥

trr rt♠♥t ♦ ♥♦ ♦r♠ t♦s ♦s ①♣r♠♥t♦s ♠

♠♣♦ ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st rt♠♥t ♥trr ♦

sst♠ ♦♠♣♦st♦ ♣♦r ♠ st ♦r③♦♥t♠♥t ♥trr ♦ ♦st♦

r③r♠s ♦s tsts ①♣r♠♥ts ♣r ♦ trr♠♥t♦ étr♦ ♦♠ ♠ st rt

♦ trr♠♥t♦ ♦♠ s sts rts tr♠♥t ♦♥ts ssts q

♠ t♦♦s tsts ①♣r♠♥ts ♦r♠ ♦ts três ♠♦strs ♦s s♥s t♥sã♦ ♦r

r♥t st ♦r♠ ♦ ♣♦ssí r ♦s rs♣t♦s s♥s ♠é♦s st ♣r♦♠♥t♦

♠♥♠③ sttst♠♥t ♦s rí♦s ♥♦s s♥s ♠♦s ♦♥♦r♠ t♦ ♥tr♦r♠♥t

♦s ①♣r♠♥t♦s ♦r♠ r③♦s ♦♠ s ♥ ♠t♦♦♦ s♥♦ ♣♦r ♥ ❬❪

♦♦s ①♣r♠♥t♦s ♦r♠ r♣r♦③♦s ♥♦ s♦tr ❬❪ q t③ ♦ ♠ét♦♦

❬❪ ♣r rs♦r ♥♠r♠♥t s qçõs ① ♥♦ ♦♠í♥♦ ♦♠♣

t♦♥ ♥ é t tr♥♠ ♦ ♦♠í♥♦ ♦♠♣t♦♥ trés té♥

P ❬❪ st ♦r♠ ♣♦s r♣rs♥tr s ♦♥s tr♦♠♥éts ♣r♦♣♥♦ ♣r

♦r ♦ s♣ç♦ ♥ás ♦ ♦ ♠ rs♦s tr♦s ♥♦s qs ♦ ♠♦

♥s♦ ♥ã♦ r ♦♥sr♦ s♣rs♦ σ ε µ ♦♥st♥ts ❬❪ st tr♦

st s♦tr ♦ ♠♦♦ ♣r ♥r ♠♦♠ s♦♦s s♣rs♦s ♣rs♥t ♥♦

♣ít♦ ♦♠♣r♦s ♦s s♥s ♠♦s ♦s s♥s ♦t♦s ♣s s♠çõs ♥♠érs

♦♠ s♦♦ s♠ s♣rsã♦ ♠♦♠ tr♦♥ ❬❪ ♦♠ s♦♦ s♣rs♦ ♠♦♠

♣r♦♣♦st ♥st tr♦ sr♦s q ♦s s♥s ♦t♦s ♥♠r♠♥t ♦♥sr♥♦

♠♦♠ ♦ s♦♦ s♣rs♦ ♣rs♥t♠ ♠♦r ♦♥♦râ♥ ♦♠ ♦s s♥s ♠♦s

♠ ♠♣♦

♦ ③çã♦ ♦s ①♣r♠♥t♦s ♠ ♠♣♦

st tr♦ ♦s ①♣r♠♥t♦s ♦r♠ r③♦s ♠ ♠ ár r♦r③ s♦

♣r♦①♠♠♥t ♠ × ♠ ♦③ ♥♦ ♥tr♦ ♥♦♦

tr♦rástr♦♥♦rt ♥♦ rr♦ r♠r ♠ é♠P é ♣rs♥t

st ér ♦ ♦ ♦♥ ♦r♠ r③♦s ♦s ①♣r♠♥t♦s ♠ ♠♣♦ ár ♦♥ ♦r♠

r③♦s ♦s ①♣r♠♥t♦s stá st ♦♠ ♠ rtâ♥♦ r♠♦

r ❱st ér ♦ ♥tr♦ ♥♦♦ tr♦rástr♦♥♦rt stq ♥ár ♦♥ ♦r♠ r③♦s ♦s ①♣r♠♥t♦s ♠ ♠♣♦

♦ ♠♥t ♠ q ♦r♠ ♠♣♠♥t♦s ♦s ①♣r♠♥t♦s ♦ ♥í ♠á①♠♦ ♦ rí♦

♠♦ ♦ ♣♥s 100 ♠❱ ♦♥tr♥♦ ♣r q ♦s s♥s ♠♦s r♥t ♦s ①♣

r♠♥t♦s ♣rs♥tss♠ ①♥t ♦♥♦râ♥ ♦♠ ♦s s♥s s♠♦s

♥ás r♥stór ♦ Pr♦♠ st♠ tr

r♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠ st ❱rt

♦ rr♦ ♥tr ♦ r③♦ ♦ ①♣r♠♥t♦ ♣r

♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♥trr rt♠♥t

①♣r♠♥t♦ ♠ ♠♣♦

st♣ ①♣r♠♥t ♠♣♠♥t♦ ♣r ♥sr ♦ ♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦

♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♥trr rt♠♥t é ♣rs♥

t♦ ♥ st ♥çã♦ ♦s tr♦♦s rrê♥ ♦s rt♦s t♥sã♦

♦rr♥t tê♠ 1,20 ♠ ♦♠♣r♠♥t♦ 7,9 ♠♠ r♦ ♦r♠ ♥trr♦s 0,90 ♠ rt

♠♥t ♦♠♣r♠♥t♦ ♦ ♦ étr♦ s♦ ♥♦ rt♦ t♥sã♦ é ♠ ♦ ♦

♠♣r♦ ♥♦ rt♦ ♦rr♥t ♣♦ss ♠ ♦♠♣r♠♥t♦ ♠♦s tê♠ 1,25 ♠♠

r♦ ♦r♠ ♣♦s♦♥♦s ♠ ♠ s♣rí ♦ s♦♦ s ♣♦♥t♦s ♠çã♦

t♥sã♦ ♥③ v(t) ♦rr♥t ♥t i(t) stã♦ st♦s ♥s s

sã♦ ♣rs♥ts s ♠♥s ♣r strr ♦ st♣ ♠♣r♦ ♥♦ ①♣

r♠♥t♦ ♠ ♠♣♦ sr s s♦♥s ♠çã♦ t♥sã♦ ♦rr♥t ♦s

rt♦s t♥sã♦ ♦rr♥t

r ♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦♦♠♣♦st♦ ♣♦r ♠ ú♥ st rt♠♥t ♥trr

Solo

Circuito de tensão

22m !(#$%&#'()= 2,2 +ΩGerador de Surto

0,9m

0,3m

Eletrodo de referência de corrente

0,9m

0,3m

Eletrodo de referência de

tensão

+ _

Medição da tensão induzida

Medição de corrente

0,9m

Haste de injeção

z

xy

Parte acima da superfície do soloParte enterrada no solo

r ♠♥s ♦ ①♣r♠♥t♦ r③♦ ♠ ♠♣♦ stq ♥♦s ♣♦♥t♦s ♠çã♦ st ♥çã♦ ♦rr♥t stq ♥♦s rt♦s t♥sã♦ ♦rr♥t

Sonda de medição de corrente

+

-

Haste de injeção de corrente

Sonda de medição de tensão

Circuito de corrente

Circuito de tensão

♠çã♦ ♥♠ér

é str r♣rs♥tçã♦ ♦ sst♠ trr♠♥t♦ ♠ st

rt ♥♦ ♠♥t ♦ s♦tr ♦♠í♥♦ ♦♠♣t♦♥ ♣r r♣rs♥tr st

①♣r♠♥t♦ ♦ ×× és ús ❨ ♦♠ rsts ∆x = ∆y = ∆z = 0,10

♠ s ♥r♠♥t♦s s♣s ∆x ∆y ∆z ♦ ♥r♠♥t♦ t♠♣♦r ∆t ♦r♠ ♦s

t ♦r♠ q t♥♠ ♦s rtér♦s st♦s ♥ çã♦

r ♣rs♥tçã♦ ♥♦ ♠♥t ♦ s♦tr ♦ st♣ ①♣r♠♥t ♦sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ ú♥ st rt♠♥t ♥trr

z yx

Circuito de corrente

!(#$#%&'()= 2054 +

Gerador de surto

Haste de injeção

Cálculo da corrente injetada Cálculo

da tensão induzida

-+

0,9m

Circuito de

tensão

♦♥♦r♠ ♦ st♦ ♥ sçã♦ ♣r ♥r rsstê♥ sí ♥s s♠

çõs s r rsstê♥ sí t RS(efetiva) ♦sr ♥ ♦♦ ♦ s

♥♦ ♠♦♠♥t♦ ♠çã♦ st ♣râ♠tr♦ é ♦t♦ ♣rtr ♦ ♦r r③ã♦ ♥tr

t♥sã♦ ♠ ♥♦s tr♠♥s ♦ rsst♦r vR(t) ♦rr♥t iR(t) q trés

♣rs♥ts ♦r♠ ♦♥ ♦t ♣ rçã♦ vR(t)/iR(t) srs q

r t♥ ♣r ♠ ♦r ♦♥st♥t RS(efetiva) ≈ 2054 Ω ♣rs♥t♥♦ ♠ r♥ç

♣r♦①♠♠♥t 6,6 ♦ ♦r ♥♦♠♥ RS(nominal) Ω

r r ♦t ♥♦ ①♣r♠♥t♦ ♣rtr r③ã♦ ♥tr vR(t) iR(t) ♣r r♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) = 2054 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,01000

1500

2000

2500

30001 Haste Verticalmente Enterrada

v R /

i R (

)

Tempo (s)

Medido

2054

♦♥♦r♠ ♦ st♦ ♥ sçã♦ ♣r♠ss étr ♥ã♦ t s♥

t♠♥t ♦ s♥ t♥sã♦ ♦t♦ st tr♦ ♦ ♦t♦ εr = 50 q stá ♥tr♦

① s ♣r♠ss étr rt ♣r st t♣♦ s♦♦ ❬❪

st♠t ♦♥t ♦ s♦♦ ♣r ①s rqê♥s σDC ♦ t ♣rtr ♦s

s♥s ♠♦s ♠ ♠♣♦ ♦♥♦r♠ ♦ st♦ ♥ çã♦ rsstê♥ étr

trr♠♥t♦ R ≈ 45 Ω é ♥r ♦sr♥♦ r ♦t♦ ♣rtr

♦s s♥s t♥sã♦ ♦rr♥t ♠♦s P♦r ♠ t③♥♦ ór♠ ♥

♦té♠s σDC ≈ 0,02009 ♠

r r ♦t ♥♦ ①♣r♠♥t♦ ♠ st ♥trr rt♠♥t ♥ tr ♥ ♦ ♦r R = 45 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,00

10

20

30

40

50

60

70

80

90

1001 Haste Verticalmente Enterrada

TGR

()

Tempo (s)

Medido

45

s s♥s t♥sã♦ ♦rr♥t ♦t♦s ♣ s♠çã♦ t③♥♦ st ♦r ♦♥

t ♣rs♥tr♠ ♠ ♣q♥♦ s♦ ♠ rçã♦ ♦s rs♣t♦s s♥s ♠♦s

Pr q ♦s s♥s ♦t♦s ♥♠r♠♥t ♠♦s ♣rs♥tss♠ ♠♦r ♦♥♦râ♥

③s ♠ st st ♦r ♣r σDC = 0,02052 ♠ ♣rs♥t♠s ♦s

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r st s♦ q ♦r♠ ♦t♦s ♦♥♦r♠

♦ st♦ ♥♦ ♣ít♦ ♣rs♥ts ♦ rs♣t♦ rá♦ σ(f)

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02052 ♠♦♥t ❱♦r

a0 2,064594047525060 × 10−2 + j0a1 0 − j1,423911859572707 × 10−8

a2 −4,698356060540265 × 10−16 + j0b1 0 − j6,321062023873333 × 10−7

b2 −1,067817572300194 × 10−14 + j0

r rs ♥çã♦ σ(f) ♣r σDC = 0,02052 ♠ ♦ts ♣rtr ①♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

0,0 0,5 1,0 1,5 2,00,020

0,021

0,022

0,023

0,024

0,025

0,026

f (S

/m)

Frequência (MHz)

Visacro-Alipio Polinômio de Padé

DC = 0,02052 S/m

st σDC é ♠♦t♦ ♣♦s rr♦s s♦s ♣s ♣r♦①♠çõs ts ♥♦

s♥♦♠♥t♦ ♠t♠át♦ ór♠ ♥ ♣ rtríst tr♦ê♥ ♦ s♦♦

t③♦ ♥♦s ①♣r♠♥t♦s st ♦r♠ ♦ ♦r ♦rr♦ ♣♦ sr ♥tr♣rt♦ ♦♠♦

♠ ♦♥t étr t q ♦ rt♦ ♥①r ♥♦ ♠♦♠♥t♦ ♣çã♦ ♦

srt♦

st♦s

s s ♣rs♥t♠s ♦s rá♦s ♦♠♣rt♦s ♦s s♥s ♦rr♥t ♥

t t♥sã♦ ♥③ rs♣t♠♥t ❱rs ♠♦r ♦♥♦râ♥ ♥tr s rs

♦ts ♥♦s ①♣r♠♥t♦s s♠çõs ♥♠érs ♦♠ s♦♦ s♣rs♦

♦♥♦r♠ st♦ ♥ sçã♦ ♦ rsst♦r sí ♣♦♥t rsst r♥t q

♦r♠ ♦♥ t♥sã♦ ♥t s ♥♣♥♥t r ♥①r ♣♦ sst♠

♥çã♦ ♣s♦ r♦r ♦♥t♦ à ♣♦♥t rsst t♠♦s s♥s

♦rr♥t ♠♦ s♠♦ ♦♥sr♥♦s σ ♦♥st♥t σ(f) ♦ts q ♦r♠

♦♥ ♦rr♥t ♥ã♦ ♣rs♥t♦ r♥ç s♥t ♥tr ♦ s♦ ♦♠ s♦♦ s♣rs♦

♦ s♦♦ s♠ s♣rsã♦ st♦ ♦♥t ♦ ♦ t♦ q ♣♦♥t rsst t♠ ♠♣â♥

♠t♦ ♠♦r ♦ q ♠♣â♥ trr ♦♥t ♠ sér s s♦s ♦sr♦s ♥s

♦r♠s ♦♥ ♦s s♥s s♠♦s sã♦ ♣r♦♦♦s ♣ rçã♦ t♠♣♦r rsstê♥

Rs ♣♦♥t rsst st tr♦ ♦♥srs Rs ♦♥st♥t

r ♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦s ♠ ♠♣♦ ♦t ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

Cor

rent

e (A

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

1 Haste Verticalmente Enterrada

♣♦ ♦ s♥ ♦rr♥t ♠♦ ♦ Ip,exp ≈ 0,393 ♦ ♣♦ ♦ s♥ ♦rr♥t

s♠♦ ♦ Ip,simu ≈ 0,401 ss♠ s♠çã♦ ①♣r♠♥t♦ ♣rs♥tr♠ s♦ ♥♦

♣♦ ♣r♦①♠♠♥t 2,04%

P♦ré♠ ♣rtr ♦srs q ♦ ♣♦ ♦ s♥ t♥sã♦ ♥③ ♠♦

♦ Vp,exp ≈ 17,169 ❱ s ♦rs t♥sã♦ ♣♦ ♣r ♦s s♦s s♠♦s ♦♠ ♦s

♠♦♦s s♦♦ s♣rs♦ s♦♦ ♥ã♦ s♣rs♦ ♦r♠ rs♣t♠♥t

Vp,simu() ≈ 17,152 ❱ Vp,simu() ≈ 17,613 ❱ P♦rt♥t♦ ♦ s♦ ♣rs♥t♦ ♣r ♦

s♦ s♠♦ ♦♠ s♣rsã♦ ♦ ♣r♦①♠♠♥t −0,1% ♥q♥t♦ q ♦ s♦ ♣r

♦ s♦ s♠♦ s♠ s♣rsã♦ ♦ 2,59% é♠ ss♦ ♣♥ rsstr q t♥t♦

♦ ♦r♠t♦ q♥t♦ s ♦ s♥ t♥sã♦ ♥③ s♠♦ ♦♥sr♥♦s s♣rsã♦

♦r♠ sst♥♠♥t ♠♦r♦s ♦♠ ♦ ♥♦♦ ♠♦♦ ♥♠ér♦ ♣r♦♣♦st♦

♥st tr♦

r ♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦s ♠ ♠♣♦ ♦t♦ ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00

2

4

6

8

10

12

14

16

18

201 Haste Verticalmente Enterrada

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

♥ás r♥stór ♦s Pr♦♠s

❯♠ s♥♦ ♦♥♥t♦ ①♣r♠♥t♦s ♦ r③♦ ♥♦ ♥♦ ♥tr

s sst♠s trr♠♥t♦ ♥s♦s ♦r♠ ♠ st rt♠♥t

♥trr ♠ st ♥trr ♦r③♦♥t♠♥t

♥s♦ rt♦ ♦ sst♠ trr♠♥t♦ ♠ st rt ♦ r♣t♦ ♣♦r ♦s

♠♦t♦s r ♦ ♠ét♦♦ ♣r s♦♦s ♦♠ r♥ts ♦♥çõs t♠♣rtr ♠

♦♠♣tçã♦ r③r st♠t ♦ σDC ♦ s♦♦ sr ♦t ♥s ♠s s♠çõs

♥♠érs

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠

st ♥trr ❱rt♠♥t

①♣r♠♥t♦ ♠ ♠♣♦

♣rs♥ts ♠ ♠♠ st ①♣r♠♥t♦ st♣ ①♣r♠♥t

♠♣r♦ ♣r ♥sr ♦ sst♠ ♦♠♣♦st♦ ♣♦r ♠ st rt é ê♥t♦ ♦ ♥s♦

r③♦ ♥♦ rr♦ çã♦

r ♦♥rçã♦ ♦ ♥s♦ ♦ sst♠ trr♠♥t♦ ♠ st rt♠♥t♥trr t ♥ st ♥çã♦ ♦rr♥t ♦ rt♦ ♦rr♥t

Haste de injeção de corrente

Circuito de corrente

♠çã♦ ♥♠ér

♥♦ ♠ st q ♦ st♣ ①♣r♠♥t é ♦ ♠s♠♦ çã♦ s ♠♥sõs

♦ ♠♥t ♦♠♣t♦♥ r♣rs♥tçã♦ ♦♠étr ♦ rt♦ ♥ã♦ ♦r♠ trs

♣r s♠çã♦ ♥♠ér st s♦ ♦r♠ ♥ssár♦s ♠♦r ♦s s♥ts ♣râ♠tr♦s

rsstê♥ sí t Rs(efetiva) ♦♥t ♣r ①s rqê♥s σDC

♥s♥♦ r ♦t ♣ r③ã♦ ♥tr vR(t) iR(t) ♠♦s ♥st ①♣r♠♥t♦

rs q rsstê♥ t é RS(efetiva) ≈ 2020 Ω ♦♠ ss♦ ♦ s♦

♠ rçã♦ ♦ ♦r ♥♦♠♥ RS(nominal) = 2200 Ω ♦ ♣r♦①♠♠♥t 8,18

r r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) = 2020 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,01000

1500

2000

2500

30001 Haste Verticalmente Enterrada

v R /

i R (

)

Tempo (s)

Medido

2020

Pr st♠r σDC ♦ r♦ ♦ ♦r ♠é♦ rsstê♥ trr♠♥t♦ R ♥ r

♦t ♣rtr ♦s s♥s ♠♦s ♠ ♠♣♦ sr♥♦

rs q R ≈ Ω ❯t③♥♦ qçã♦ ♦ts σDC = 0,02134

♠ ♣rs♥t♠s ♦s ♦♥ts ♣r st s♦

♣rs♥ts ♦ rs♣t♦ rá♦ σ(f)

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02134 ♠♦♥t ❱♦r

a0 2,136577121966814 × 10−2 + j0a1 0 − j1,925334004054197 × 10−8

a2 −6,678713048129872 × 10−16 + j0b1 0 − j8,324729493118694 × 10−7

b2 −1,510953471874635 × 10−14 + j0

r r ♦ ♦t ♥♦ ①♣r♠♥t♦ ♠ st rt ♥tr ♥ ♦ ♦r 44,5 Ω rs ♥çã♦ σ(f) ♣r σDC = 0,02134 ♠♦ts ♣rtr ①♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

0,0 0,5 1,0 1,5 2,0 2,5 3,00

10

20

30

40

50

60

70

80

90

1001 Haste Verticalmente Enterrada

TGR

()

Tempo (s)

Medido

44,5

0,0 0,5 1,0 1,5 2,00,020

0,021

0,022

0,023

0,024

0,025

0,026

0,027 f

(S/m

)

Frequência (MHz)

Visacro-Alipio Polinômio de Padé

DC = 0,02134 S/m

st♦s

s s sã♦ ♣rs♥t♦s ♦s rá♦s ♦♠♣rt♦s ♦s s♥s ♦rr♥t

t♥sã♦ ♥③ s♠♦ ♣ós qtr♦ ♠ss ♦ ♣r♠r♦ ①♣r♠♥t♦ çã♦ ♥

é r ♠♦r ♦♥♦râ♥ ♥tr s rs ♦ts ♥♦s ①♣r♠♥t♦s s♠çõs

♥♠érs ♦♠ ♠♦♠ ♦ s♦♦ s♣rs♦

♥s♥♦ ♥♦ts q Ip,exp ≈ 0,389 ♦ ♣♦ ♦ s♥ ♦rr♥t

s♠♦ ♦ Ip,simu ≈ 0,412 s♦ ♥♦ ♦r ♣♦ ♥tr ♦ s♥ ♠♦ ♦ s♥

♦t♦ ♥♠r♠♥t é ♣r♦①♠♠♥t 5,80%

r ♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t♦ ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

Cor

rent

e (A

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

1 Haste Verticalmente Enterrada

♥♦♠♥t ♣rtr rs q Vp,exp ≈ 16,540 ❱ Vp,simu() ≈

16,669 ❱ Vp,simu() ≈ 17,368 ❱ P♦rt♥t♦ ♦ s♦ ♣rs♥t♦ ♣r ♦ s♦ ♦♠

s♣rsã♦ ♦ ♣r♦①♠♠♥t 0,78% ♥q♥t♦ q ♦ s♦ ♣r ♦ s♦ s♠ s♣rsã♦

♦ 5,01% ♦♠♥t ♠ ♦r♠ ♦ ♥tr♦çã♦ ♦ ♠♦♦ s♣rs♦ ♥s

s♠çõs ♣r♦♦ ♠ ♠♦r s♥t ♥tr ♦s s♥s t♥sã♦ ♠♦ s♠♦

♠ tr♠♦s s♣ ♦ ♦♥♦ ♦ ♥tr♦ t♠♣♦ ♥s♦ ♣r ♦ ♣r♦♠

r ♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00

2

4

6

8

10

12

14

16

18

201 Haste Verticalmente Enterrada

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠

st ♥trr ♦r③♦♥t♠♥t

①♣r♠♥t♦ ♠ ♠♣♦

é str♦ ♦ st♣ ①♣r♠♥t ♠♣♠♥t♦ ♣r r ♦ ♦♠♣♦r

t♠♥t♦ tr♦♠♥ét♦ ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♥trr

♦r③♦♥t♠♥t

ss♠ ♦♠♦ ♥♦s ♠s s♦s st ♥çã♦ ♦s tr♦♦s rrê♥ ♦s

rt♦s t♥sã♦ ♦rr♥t ♠♠ 1,20 ♠ ♦♠♣r♠♥t♦ 7,9 ♠♠ r♦ st

♥çã♦ ♦rr♥t ♦ ♥trr ♦r③♦♥t♠♥t ♠ ♣r♦♥ 0,10 ♠

♦s tr♦♦s rrê♥ ♣r♠♥r♠ ♥trr♦s 0,90 ♠ rt♠♥t ♦ q

♦♥t st ♥çã♦ ♦ rsst♦r sí Rs é s♥♣♦ ♣♥s ♥ ♣♦rçã♦ q

stá ♥trr t ♦r♠ q q ♠ ♦♥tt♦ ♦♠ ♦ s♦♦ s ♠♥sõs ♦s rt♦s

t♥sã♦ ♦rr♥t sã♦ ê♥ts ♦s ♠s s♦s s ♣♦♥t♦s ♠çã♦ t♥sã♦

♥③ v(t) ♦rr♥t ♥t i(t) stã♦ st♦s ♥

r ♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦♦♠♣♦st♦ ♣♦r ♠ st ♦r③♦♥t♠♥t ♥trr

Solo

Circuito de tensão

22m !(#$%&#'()= 2,2 +ΩGerador de Surto

0,9m

0,3m

Eletrodo de referência de corrente

0,9m

0,3m

Eletrodo de referência de

tensão

_

Medição da tensão induzida

Medição de corrente

10cm

Haste de injeção

z

xy

Parte acima da superfície do soloParte enterrada no solo

+

r ♠♥s ♦ sst♠ trr♠♥t♦ ♦r♠♦ ♣♦r ♠ st ♥trr ♦r③♦♥t♠♥t

Valeta + haste de injeção de corrente

Haste de injeção de corrente enterrada

Valeta criada para enterrar a haste de injeção de corrente

ssts q ♣r ♥trrr st ♥ ♦r③♦♥t ♦ ♥ssár♦ rr ♠ t

♥♦ s♦♦ ♦♠ s ♠♥sõs ♠ × ♠ × ♠ st ♣r♦♠♥t♦

tr♦ s rtrísts ♦ s♦♦ ♥ ♣♦rçã♦ q ♥♦ st ♥çã♦ ♦rr♥t

♣r♥♣♠♥t ♦♠♣tçã♦ ss ♣♦rçã♦ ♦ s♦♦ ♠ r①♦ st trçã♦ r♦

s ♥ss ♠♦r ♦ ♦r σDC ♦ s♦♦ t③♦ ♥ s♠çã♦ t ♦r♠

q r♣rs♥t ♠ ♦♥t t q ♦ sst♠ trr♠♥t♦ ♥①r

♠çã♦ ♥♠ér

♣rs♥ts r♣rs♥tçã♦ st s♦ ♥♦ ♠♥t ♦ s♦tr

♦♠í♥♦ ♦♠♣t♦♥ ♣r r♣rs♥tr ♦ ♥s♦ ♦ sst♠ trr♠♥t♦ ♥

s♦ ♥st ssçã♦ ♦ ×× és ♠♥sã♦ s rsts ♣r♠♥r♠

♥trs ♦ s ∆x = ∆y = ∆z = 0,10 ♠

r ♣rs♥tçã♦ ♥♦ ♠♥t ♦ s♦tr ♦ st♣ ①♣r♠♥t ♦sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ♠ st ♦r③♦♥t♠♥t ♥trr

z

x

y

-+

Haste de injeção

Gerador de surto

!(#$#%&'()= 2060 +Cálculo da corrente injetada

Cálculo da tensão induzida

Circuito de

corrente

Circuito de

tensão

sr♥♦ r ♦t ♣ r③ã♦ ♥tr vR(t) iR(t) ♥rs q

rsstê♥ t é RS(efetiva) ≈ 2060 Ω ♦♠ ss♦ ♦ s♦ ♠ rçã♦ ♦ ♦r ♥♦♠♥

RS(nominal) = 2200 Ω ♦ ♣r♦①♠♠♥t 6,36

r r ♦t ①♣r♠♥t♠♥t ♣rtr r③ã♦ ♥tr vR(t) iR(t) ♣rr ♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) = 2060 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,01000

1500

2000

2500

30001 Haste Horizontalmente Enterrada

v R /

i R (

)

Tempo (s)

Medido

2060

♦♥♦r♠ ♦ st♦ ♥tr♦r♠♥t ♥st s♦ ♦ ♥ss trr ♦

♦r σDC ♦ s♦♦ ♦ trçã♦ s rtrísts ♦ s♦♦ q ♥♦ st

♥çã♦ ♥ss st ♦ σDC é ♦♥r♠ ♦ ♦♠♣rr ♦ s♥ t♥sã♦

♠♦ ♦♠ ♦s rs♣t♦s s♥s ♦t♦s ♣s s♠çõs q ♠♥té♠ σDC = 0, 02134

♠ ♦r t③♦ ♥♦ s♦ ♥tr♦r st ♦♠♣rçã♦ é ♣rs♥t ♥

r st♦ ♣r♠♥r ♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦♠ ♠♣♦ ♦t ♣♦r s♠çã♦ ♥♠ér t③♥♦ σDC = 0, 02134 ♠ q é ♦ ♦rt③♦ ♥♦ s♦ ♠ st ♥trr rt♠♥t sçã♦

0,0 0,5 1,0 1,5 2,0 2,502468

10121416182022

1 Haste Horizontalmente Enterrada

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

srs q ♦s s♥s ♦t♦s ♣s s♠çõs ♥ã♦ stã♦ r♣rs♥t♥♦ ♦rrt

♠♥t ♦ s♥ ♠♦ t♥t♦ ♠ tr♠♦s ♠♣t ♦♠♦ ♦r♠ ♦♥ ♣rtr

♦srs q Vp,exp ≈ 19,998 ❱ Vp,simu() ≈ 16,569 ❱ ♣r ♠♦

♠ ♦♠ s♦♦ s♣rs♦ Vp,simu() ≈ 18,855 ❱ ♣r ♠♦♠ ♦♠ s♦♦ ♥ã♦

s♣rs♦ st stçã♦ ♦ s♦ ♣rs♥t♦ ♥♦ s♦ ♦♠ s♣rsã♦ ♦ ♣r♦①♠

♠♥t 17,14% ♥q♥t♦ q ♦ s♦ ♣r ♦ s♦ s♠ s♣rsã♦ ♦ 5,71% ♠♦r

♦ s♥ ♦t♦ ♣♦ ♠♦♦ ♦ s♦♦ ♥ã♦ s♣rs♦ ♣rs♥t ♠ s♦ ♠♥♦r ♦r♠

♦♥ ♥ã♦ r♣rs♥t q♠♥t ♦ s♥ ♠♦ P♦r ♦tr♦ ♦ ♦ s♥ ♦t♦ ♣

s♠çã♦ ♦♠ ♦ ♠♦♦ s♦♦ s♣rs♦ ♣♦ss ♠ ♦r♠ ♦♥ q s ss♠

♦ ♦r♠t♦ ♦ s♥ ♠♦ ♣♦ré♠ ♦♠ ♠ ♠♣t ♠♥♦r

st ♦r♠ ③s ♦ st ♦ ♦r σDC ♣r 0,01626 ♠ ♣r q ♦s s♥s

t♥sã♦ ♣rs♥t♠ ♦ ♦♥♦râ♥ ♦♥♦r♠ srá str♦ ♠s ♥t

♣rs♥t♠s ♦s ♦♥ts ♣r st s♦ ♣rs♥ts ♦

rs♣t♦ rá♦ σ(f)

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,01626 ♠♦♥t ❱♦r

a0 1,628394721574459 × 10−2 + j0a1 0 − j1,491903561415482 × 10−8

a2 −5,666661742587516 × 10−16 + j0b1 0 − j8,324729493118694 × 10−7

b2 −1,510953471874635 × 10−14 + j0

♦♠♣r♥♦ ♦♠ r σ(f) ♦t ♥♦ s♦ ♦ sst♠ trr♠♥t♦ ♠

st rt ♦ Pr♦♠ sçã♦ ♦srs q ①

rçã♦ ♦♥t s♦r ♠ trçã♦ s♥t ♥tt♠♥t s♣rs

q ♠♣t ♦ s♥ t♥sã♦ t♠é♠ s♦r ♠ ♠♦çã♦ ①♣rss st

s♣♦sçã♦ é ♦♥r♠ ♥♦ s♥t tó♣♦ rst♦s

r rs ♥çã♦ σ(f) ♣r σDC = 0,01626 ♠ ♦ts ♣rtr ①♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

0,0 0,5 1,0 1,5 2,00,014

0,016

0,018

0,020

0,022

0,024

0,026

0,028

DC = 0,02134 S/m

f (S

/m)

Frequência (MHz)

Visacro-Alipio Polinômio de Padé

DC = 0,01626 S/m

st♦s

sã♦ ♣rs♥t♦s ♦s rá♦s ♦♠♣rt♦s ♦s s♥s ♦rr♥t ♥t

t♥sã♦ ♥③ ♣ós r③r ♦ st ♦ σDC srs ♠♦r ♦♥♦râ♥ ♥tr

♦s s♥s ♠♦s ♦s ♦t♦s trés s♠çã♦ q ♣ ♠♦♠ ♦ s♦♦

s♣rs♦

♥s♥♦ ♥♦ts q Ip,exp ≈ 0,384 ♦ ♣♦ ♦ s♥ ♦rr♥t

s♠♦ ♦ Ip,simu ≈ 0,405 s♦ ♥♦ ♦r ♣♦ ♥tr ♦ s♥ ♠♦ ♦ s♥

♦t♦ ♥♠r♠♥t é ♣r♦①♠♠♥t 5,36%

♦srs q Vp,exp ≈ 19,998 ❱ Vp,simu() ≈ 19,489 ❱ Vp,simu() ≈

21,946 ❱ P♦rt♥t♦ ♦ s♦ ♣rs♥t♦ ♣♦ ♠♦♦ ♦♠ s♦♦ s♣rs♦ ♦ ♣r♦①

♠♠♥t 2,55% ♣♦ ♠♦♦ ♦♠ s♦♦ ♥ã♦ s♣rs♦ ♦ 9,74%

st s♦ t♥t♦ ♦ ♦r ♣♦ ♦♠♦ ♦r♠ ♦♥ ♦ s♥ t♥sã♦ ♣r ♦ s♦

s♠ s♣rsã♦ ♣rs♥t♠ ♠ s♦ ♦ ♥trt♥t♦ ♦ s♥ t♥sã♦ ♦t♦ ♣

♠♦♠ ♦ s♦♦ s♣rs♦ ♣♦ss ①♦ s♦ ♥♦ ♣♦ ♦r♠ ♦♥ ♣rs♥t

①♥t ♦♥♦râ♥ ♦♠ ♦ s♥ ♠♦

r ♥s ♠♦s ♥♦s ①♣r♠♥t♦s ♠ ♠♣♦ ♦t♦s ♣♦r s♠çã♦ ♥♠ér ♦rr♥t ♥t ♥sã♦ ♥③

0,0 0,5 1,0 1,5 2,0 2,50,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

Cor

rent

e (A

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

1 Haste Horizontalmente Enterrada

0,0 0,5 1,0 1,5 2,0 2,502468

10121416182022

1 Haste Horizontalmente Enterrada

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

♥ás r♥stór ♦s Pr♦♠s

♦ ♦st♦ ♥sr♠s ♦s sst♠s trr♠♥t♦ ♦♠♣♦st♦s ♣♦r

♠ st rt♠♥t ♥trr s sts ♥trr rt♠♥t tr♠♥t

♦♥ts

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r ❯♠

st ♥trr ❱rt♠♥t

st♣ ①♣r♠♥t ♠♣r♦ ♣r ♥sr ♦ sst♠ ♦♠♣♦st♦ ♣♦r ♠ st

rt é ê♥t♦ ♦s ♥s♦s r③♦s ♥♦s s rr♦ ♥♦ ♦r♦s

♥ çã♦ ♥ sçã♦ rs♣t♠♥t

♠çã♦ ♥♠ér

♥♦ ♠ st q ♦ st♣ ①♣r♠♥t é ♦ ♠s♠♦ çã♦ sçã♦

s ♠♥sõs ♦ ♠♥t ♦♠♣t♦♥ r♣rs♥tçã♦ ♦♠étr ♦ rt♦

♥ã♦ ♦r♠ trs ♣r s♠çã♦ ♥♠ér st s♦ ♦r♠ ♥ssár♦s ♠♦r ♦s

s♥ts ♣râ♠tr♦s rsstê♥ sí t Rs(efetiva) ♦♥t ♣r ①s

rqê♥s σDC

♣rtr r ♦t ♣ r③ã♦ ♥tr vR(t) iR(t) ♠♦s ♥st ①♣r♠♥t♦

♣♦s ♥rr q rsstê♥ t é RS(efetiva) ≈ 2070 Ω st ♦r♠ ♦sr

s q ♦ s♦ ♠ rçã♦ ♦ ♦r ♥♦♠♥ RS(nominal) = 2200 Ω ♦ ♣r♦①♠♠♥t

5,91

r r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) = 2070 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,01000

1500

2000

2500

30001 Haste Verticalmente Enterrada

v R /

i R (

)

Tempo (s)

Medido

2070

♦sr♥♦ r ♦t ①♣r♠♥t♠♥t rs q

R ≈ Ω ❯t③♥♦ ór♠ ♥ ♦tê♠s q σDC = 0,02203 ♠

r r ♦ ♦t ♥♦ ①♣r♠♥t♦ ♠ st rt ♥tr ♥ ♦ ♦r 42,5 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,00

10

20

30

40

50

60

70

80

90

1001 Haste Verticalmente Enterrada

TGR

()

Tempo (s)

Medido

42,5

♣rs♥t♠s ♦s ♦♥ts ♦ ♣r♦①♠♦r Pé ♦ ♣ ①

♣rssã♦ ♣r st s♦ ♣rs♥ts ♦ rs♣t♦ rá♦ σ(f)

♦♥ts ♦ ♣r♦①♠♦r Pé ♣r σDC = 0,02203 ♠♦♥t ❱♦r

a0 2,216338608816205 × 10−2 + j0a1 0 − j1,522138834333961 × 10−8

a2 −4,908817129612966 × 10−16 + j0b1 0 − j6,321062023873333 × 10−7

b2 −1,067817572300194 × 10−14 + j0

r rs ♥çã♦ σ(f) ♣r σDC = 0,02203 ♠ ♦ts ♣rtr ①♣rssã♦ rs♣t ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé

0,0 0,5 1,0 1,5 2,00,020

0,022

0,024

0,026

0,028

f (S

/m)

Frequência (MHz)

Visacro-Alipio Polinômio de Padé

DC = 0,02203 S/m

st♦s

s s sã♦ ♣rs♥t♦s ♦s rá♦s ♦♠♣rt♦s ♦s s♥s ♦rr♥t

t♥sã♦ ♥③ sr q ♥♦♠♥t é r ♠♦r ♦♥♦râ♥ ♥tr s rs

♦ts ♥♦s ①♣r♠♥t♦s s♠çõs ♥♠érs ♦♠ ♠♦♠ ♦ s♦♦ s♣rs♦

♥s♥♦ ♥♦ts q Ip,exp ≈ 0,359 ♦ ♣♦ ♦ s♥ ♦rr♥t

s♠♦ ♦ Ip,simu ≈ 0,377 s♦ ♥♦ ♦r ♣♦ ♥tr ♦ s♥ ♠♦ ♦ s♥

♦t♦ s♠çã♦ ♥♠ér é ♣r♦①♠♠♥t 4,80%

r ♥ ♦rr♥t ♥t ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t♦ ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

Cor

rent

e (A

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

1 Haste Verticalmente Enterrada

rs q Vp,exp ≈ 15,025 ❱ Vp,simu() ≈ 15,023 ❱ Vp,simu() ≈

15,398 ❱ s♦ ♣rs♥t♦ ♣r ♦ s♦ ♦♠ s♦♦ s♣rs♦ ♦ ♣r♦①♠♠♥t

−0,01% ♥q♥t♦ q ♦ s♦ ♣r ♦ s♦ ♦♠ s♦♦ ♥ã♦ s♣rs♦ ♦ 2,48%

♥♦ ♦s s♦s ♦t♦s ♥st s♦ ♥♦♠♥t é r♦ q s♠çã♦ ♥♠ér

♦♠ ♦ ♠♦♦ s♦♦ s♣rs♦ ♣r♦♣♦st♦ ♥st tr♦ sr ♦r♠ q ♦

♦♠♣♦rt♠♥t♦ tr♦♠♥ét♦ ♦ s♦♦ ♥s♦

r ♥ t♥sã♦ ♥③ ♠♦ ♥♦ ①♣r♠♥t♦ ♠ ♠♣♦ ♦t ♣♦rs♠çã♦ ♥♠ér

0,0 0,5 1,0 1,5 2,0 2,5 3,00

2

4

6

8

10

12

14

16

181 Haste Verticalmente Enterrada

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

Pr♦♠ st♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s

sts ❱rts tr♠♥t ♦♥ts

①♣r♠♥t♦ ♠ ♠♣♦

sst♠ trr♠♥t♦ st♦ ♥st sçã♦ é ♦♠♣♦st♦ ♣♦r s sts ♣rs

rt♠♥t ♥trrs tr♠♥t ♦♥ts ♠s s sts ♣♦s

s♠ 1,20 ♠ ♦♠♣r♠♥t♦ ♦s qs 0,90 ♠ sã♦ ♥trr♦s st♣ ①♣r♠♥t é

ê♥t♦ ♦ sst♠ ♠ st rt♠♥t ♥trr ①t♦ ♣♦r ♠ s♥ st

q stá st 0,90 ♠ ♣r♠r s♣♦st ♣r♠♥t à st ♥çã♦ s

ts s sts stã♦ tr♠♥t ♦♥ts ♣♦r ♠ ♦ étr♦ ♦♥♦r♠ ♣♦ sr

♦sr♦ ♥ r♣rs♥tçã♦ sq♠át

♣rs♥ts ♠ ♠♠ ♦♥rçã♦ ♦ ①♣r♠♥t♦ ♦ sst♠

trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s sts rts srs ♦ t ♥ ♦♥①ã♦ ♥tr

s s sts ♦ ♣♦♥t♦ ♥çã♦ ♦rr♥t

r ♣rs♥tçã♦ sq♠át ♦ st♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦♦♠♣♦st♦ ♣♦r s sts rt♠♥t ♥trrs tr♠♥t ♦♥ts

Solo

Circuito de tensão

22m !(#$%&#'()= 2,2 +ΩGerador de Surto

0,9m

0,3m

Eletrodo de referência de corrente

0,9m

0,3m

Eletrodo de referência de

tensão

+ _

Medição da tensão induzida

Medição de corrente

0,9m

Haste de injeção

z

xy

Parte acima da superfície do soloParte enterrada no solo

0,9m

0,3m

r t♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s stsrt♠♥t ♥trrs t ♥♦ ♣♦♥t♦ ♥çã♦ ♦ s♥ ♦ srt♦ ♥ ♦♥①ã♦ ss sts q ♦♠♣õ♠ ♦ sst♠ trr♠♥t♦ ♥s♦

Duas hastes eletricamente

conectadas

Ponto de injeção do sinal do gerador de surto

♠çã♦ ♥♠ér

♣rs♥ts ♦ t ♦♥rçã♦ ♦♠étr ♦ ①♣r♠♥t♦ r

♣r♦③♦ ♥♦ ♠♥t ♦♠♣t♦♥ ♦ s♦tr ♦ ♦♠í♥♦ ♦♠♣t♦♥ ♦

×× és ♦♠ rsts ∆x = ∆y = ∆z = 0,10 ♠ Pr st s♦ ♦ ♦r

rsstê♥ t é RS(efetiva) ≈ 2160 Ω ♦♠ ss♦ ♦ s♦ ♠ rçã♦ ♦ ♦r ♥♦♠♥

♦ ♣r♦①♠♠♥t 1,82

r ♣rs♥tçã♦ ♦ st♣ ①♣r♠♥t ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦♣♦r s sts rt♠♥t ♥trrs tr♠♥t ♦♥ts ♥♦ ♠♥t ♦ s♦tr

z

xy

!(#$#%&'()= 2160 +

Gerador de surto

Cálculo da corrente injetada

Circuito de

corrente

Circuito de

tensão

0,9m

0,9m

-+

Cálculo da tensão induzida

Haste de injeção

r r ♦t ①♣r♠♥t♠♥t ♣ r③ã♦ ♥tr vR(t) iR(t) ♣r r♦ ♦r RS(efetiva) ♥ tr ♥ ♦ ♦r RS(efetiva) = 2160 Ω

0,0 0,5 1,0 1,5 2,0 2,5 3,01000

1500

2000

2500

30002 Hastes Verticalmente Enterradas

v R /

i R (

)

Tempo (s)

Medido

2160

s ♣râ♠tr♦s tr♦♠♥ét♦s ♦ s♦♦ ♣r st s♦ ♦r♠ ♦♥sr♦s s ♦

s♦ ♦ sst♠ trr♠♥t♦ ♠ st rt♠♥t ♥trr sçã♦

st♦ q ♦s ①♣r♠♥t♦s ♦r♠ r③♦s ♥♦ ♠s♠♦ ♥tr ♠ ①♣r♠♥t♦

♦tr♦ s rtrísts ♦ s♦♦ ♠ ♦♠♣tçã♦ t♠♣rtr t ♥ã♦ ♦r♠

trs ♦r♠ s♥t P♦rt♥t♦ ♥ s♠çã♦ rt ♦ sst♠ ♦♠♣♦st♦ ♣♦r

s sts ♦r♠ ♦t♦s ♦s s♥ts ♣râ♠tr♦s σDC = 0,02203 ♠ εr = 50

ssts q ♦s ♦♥ts ♦ ♣r♦①♠♦r Pé ♦r♠ ♠♥t♦s

st♦s

sã♦ ♣rs♥t♦s ♦s rá♦s ♦♠♣rt♦s ♦s s♥s ♦rr♥t ♥t

t♥sã♦ ♥③ st s♦ t♠é♠ ♥♦ts ♠♦r ♦♥♦râ♥ ♥tr s rs

t♥sã♦ ♦ts ♥♦s ①♣r♠♥t♦s s♠çõs ♥♠érs ♦♥sr♥♦ ♦ s♦♦ s♣rs♦

♣rtr ♦srs q ♦ ♣♦ ♦ s♥ ♦rr♥t ♠♦ ♦ Ip,exp ≈

0,351 ♦ ♣♦ ♦ s♥ ♦rr♥t s♠♦ ♦ Ip,simu ≈ 0,360 ss♠ ♦ s♦ ♥♦

♣♦ ♦rr♥t ♥tr s♠çã♦ ①♣r♠♥t♦ ♦ ♣r♦①♠♠♥t 2,62%

rs q ♦ ♣♦ ♦ s♥ t♥sã♦ ♠♦ ♦ Vp,exp ≈ 8,184

❱ s ♦rs t♥sã♦ ♣♦ ♣r ♦s s♦s s♠♦s ♦♠ s♦♦ s♣rs♦ ♥ã♦

s♣rs♦ ♦r♠ rs♣t♠♥t Vp,simu() ≈ 8,190 ❱ Vp,simu() ≈ 8,436 ❱

P♦rt♥t♦ ♦ s♦ Vp ♥ s♠çã♦ ♦♠ s♦♦ s♣rs♦ ♦ ♣r♦①♠♠♥t 0,07%

♥♦ s♦ ♦♠ s♦♦ ♥ã♦ s♣rs♦ ♦ 3,08% ♦♠♥t ♥♦ts q ♣r ♦ ♠♦♦

s♣rs♦ ♣r ♦ s♦♦ ♦ ♠♣♦rt♥t ♠♦r ♥ ♦♥♦r♠ ♣r ♦ s♣ ♣r

s ♥tr ♦s s♥s ♠♦ s♠♦ ♣r t♦♦ ♦ ♥tr♦ t♠♣♦ ♥s♦

r ♥s ♦t♦s ①♣r♠♥t♠♥t ♣♦r s♠çã♦ ♥♠ér ♦rr♥t♥t ♥sã♦ ♥③

0,0 0,5 1,0 1,5 2,0 2,5 3,00,00

0,05

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

Cor

rent

e (A

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

2 Hastes Verticalmente Enterradas

0,0 0,5 1,0 1,5 2,0 2,5 3,00

1

2

3

4

5

6

7

8

9

102 Hastes Verticalmente Enterradas

Tens

ão (V

)

Tempo (s)

Medido Simulado - sem dispersão Simulado - com dispersão

♥ás ♥ttt ♦s st♦s

MSE ♥ qr rr♦r

♦♠♣rçã♦ ♥tr ♦s ♦rs ♣♦ ♦s s♥s ①♣r♠♥ts s♠♦s ♣rs♥t

♠ ♥ás ♦ ♣♥s ♥♦ ♥st♥t ♦ ♣♦ ♦♥t♦ ♦srs q ♠♦♠

s♦♦ s♣rs♦ ♥♥ t♠é♠ ♥ ♦r♠ ♦♥ ♦ s♥ t♥sã♦ ♣r q é

r ♠♦r ♦♥♦râ♥ ♦♠ ♦ s♥ ♠♦

Pr r ♦♥♦râ♥ ♥tr s rs t♥sã♦ ♦ts ①♣r♠♥t♠♥t

s♠çã♦ ♥♠ér srá t③♦ ♦ ♥♦r ♥ qr rr♦r MSE ❬❪ st

♣râ♠tr♦ é ♦t♦ ♣ ♠é ♦ s♦ qrát♦ ♥tr ♠♦strs s ♥çõs

♦ts ♣r ♦s ♠s♠♦s ♥st♥ts ♣♦♥t♦ ♣♦♥t♦ ♥♦ t♠♣♦ á♦ ♦ MSE ♦♥t③

♦ s♦ ♣rs♥t♦ ♣♦s s♥s ♥♠ér♦s ♦t♦s ♣ ♠♦♠ ♦ s♦♦ s♣rs♦

♥ã♦ s♣rs♦ ♠ t♦♦s ♥st♥ts t♠♣♦ st ♦r♠ ♣r♠t r③r ♠ ♥ás

♦ q s ♥çõs ♦ts ♥♠r♠♥t st ♣râ♠tr♦ é ♦ ♣

s♥t ①♣rssã♦ ❬❪

MSE =N−1∑

i=0

[vexp(i) − vsimu(i)]2

N,

♦♥ i é ♦ í♥ ♠♦str ♦ s♥ N é ♦ ♥ú♠r♦ t♦t ♣♦♥t♦s ♦ s♥ vexp(i) é

iés♠ ♠♦str t♥sã♦ ♦t ①♣r♠♥t♠♥t vsimu(i) é iés♠ ♠♦str

t♥sã♦ ♦t ♥♠r♠♥t t③♥♦ ♦ ♠♦♦ s♦♦ s♣rs♦ ♦ ♥ã♦ s♣rs♦

s ♦♠♣rts

♣rs♥ts ① rçã♦ ♦♥t σ(f) ♥♦ s♣tr♦

rqê♥ ♥s♦ ♦ts q ♠♦r rçã♦ σ(f) é é

♦sr ♣r σ(f = 0 Hz) = 0,01626 ♠ q é ♠♥♦r ♦♥t r

♥st tr♦ ♦♥♦r♠ ♣♦s ♥rr st ♦♠♣♦rt♠♥t♦ é s♣r♦

♣♦s q♥t♦ ♠♥♦r ♦r σDC rçã♦ σ(f) é ♠s ♥t P♦rt♥t♦ q♥t♦ ♠♥♦r

♦♥t ♠♦r t♦r♥s ♠♣♦rtâ♥ ♠♦♠ ♦ s♦♦ s♣rs♦ ♣r

q ♦♥♦râ♥ ♥tr ♦s s♥s ♠♦ ♦ ♦t♦ ♥♠r♠♥t

① rçã♦ σ(f) ♥ ♥ rqê♥ ♥sσ(f = 0 Hz) σ(f = 2 MHz) ∆σ

Pr♦♠ st ❱rt Pr♦♠ st ❱rt Pr♦♠ st ♦r③ Pr♦♠ st ❱rt Pr♦♠ sts ❱rt

♣rs♥ts ♦♠♣rçã♦ ♥tr ♦s ♦rs ♣♦ t♥sã♦ Vp

♦rr♥t Ip ♦t♦s ♥♦ ♣rs♥t tr♦ ♣rtr ♦s ①♣r♠♥t♦s r③♦s ♠ ♠♣♦

s s♠çõs ♥♠érs ♦♠ ♦s ♠♦♦s s♦♦ s♣rs♦ s♦♦ ♥ã♦ s♣rs♦

♦♠♣rçã♦ ♦s ♦rs Vp Ip ♥tr ♦s ♦s ①♣r♠♥ts ♦s ♦s♦t♦s ♥s s♠çõs ♥♠érs ♦♠ s♦♦s s♣rs♦s ♥ã♦ s♣rs♦s

çã♦ ErSD ErSND

Pr♦♠ st Vp ❱ ❱rt Ip

Pr♦♠ st Vp ❱ ❱rt Ip

Pr♦♠ st Vp ❱ ♦r③ Ip

Pr♦♠ st Vp ❱ ❱rt Ip

Pr♦♠ sts Vp ❱ ❱rt Ip

♦♥♦r♠ ♣♦ sr st♦ ♥ ♠ t♦♦s ♦s s♦s ♥s♦s ♦s s♦s ♥♦s

♦rs Vp ♦t♦s ♣rtr ♦ ♠♦♦ ♦♠ s♦♦ s♣rs♦ sã♦ ♠♥♦rs ♦ q

♦s s♦s ♦s s♥s s♠♦s ♦♠ s♦♦ ♥ã♦ s♣rs♦ ♥ ♦srs q

♠♣t ♦s s♥s t♥sã♦ ♦t♦s ①♣r♠♥t♠♥t ♣s s♠çõs q t③♠

♠♦♠ s♦♦ s♣rs♦ é s♠♣r ♠♥♦r ♦ q ♠♣t ♦s rs♣t♦s

s♥s ♦t♦s trés s s♠çõs ♦♠ s♦♦ s♠ s♣rsã♦ st ♦♠♣♦rt♠♥t♦ é

r♦ t♠é♠ ♠ ♦tr♦s tr♦s ♣r ♦tr♦s t♣♦s s♦♦ ♣rs♥ts ♥ trtr

❬❪ r♦rç♥♦ ♦rê♥ ♦s rst♦s ♦t♦s ♣ ♠♦♠ ♣r♦♣♦st

♥♦ ♣rs♥t tr♦

♣rs♥t♠s ♦s ♦rs MSE ♦s s♥♦ sr

s q ♠ t♦♦s s♥s t♥sã♦ ♥s♦s ♦s ♦rs MSE ♦s ♣r ♦s s♦s

s♦♦s s♣rs♦s sã♦ ♠♥♦rs ♦ q ♦s ♦rs ♦t♦s ♣rtr ♦s s♦s

s♦♦s ♥ã♦ s♣rs♦s ♠♥♦r ♦r MSE ♦t♦ ♦ ♣r ♦ s♦ ♦♠ s♦♦

s♣rs♦ ♦ sst♠ trr♠♥t♦ ♦♠♣♦st♦ ♣♦r s sts rts tr♠♥t

♦♥ts Pr♦♠ ♦r MSE ♣r ♦ rs♣t♦ s♦ s♠ s♣rsã♦ ♦

q é ♣r♦①♠♠♥t ③s ♠♦r ♦ q ♦ s♦ ♦♠ s♣rsã♦ ♠♦r

r♥ç ♥♦ ♦r MSE ♥tr ♠♦♠ ♦ s♦♦ s♣rs♦ ♠♦♠ ♦ s♦♦

♥ã♦ s♣rs♦ é ♦sr ♥♦ s♦ ♠ st ♦r③♦♥t ♥♦ q ♦ts MSESD =

0,1809 MSESND = 1,5201 ♦ s ♦ MSE s♠çã♦ ♦♠ s♦♦ ♥ã♦ s♣rs♦ é

♣r♦①♠♠♥t ③s ♠♦r ♣♦rt♥t♦ ♣rs♥t ♠ s♦ ♠t♦ ♠♦r ♦ q

s♠çã♦ ♦♠ s♦♦ s♣rs♦

♥ás ♦ MSE ♦ ♥tr ♦s ♦s ①♣r♠♥ts ♦s ♦s ♦t♦s ♣rtr s s♠çõs ♥♠érs ♦♠ s♦♦s s♣rs♦s ♥ã♦ s♣rs♦s

MSE

Pr♦♠ st ❱rt

Pr♦♠ st ❱rt

Pr♦♠ st ♦r③

Pr♦♠ st ❱rt

Pr♦♠ sts ❱rt

♣rs♥t♠s ♦s t♠♣♦s ①çã♦ s s♠çõs ♥♠érs ♦♠

♦s ♠♦♦s s♦♦ s♣rs♦ s♦♦ ♥ã♦ s♣rs♦ ♠♣♠♥ts ♥st

tr♦ srs q s s♠çõs ♦♠ ♦ ♠♦♦ s♦♦ s♣rs♦ sã♦ ♠ ♠é

③s ♠s ♠♦rs ♦ q s rs♣ts s♠çõs ♦♠ ♠♦♦ s♦♦ ♥ã♦

s♣rs♦ st♦ é ♦ ♠♦r q♥t á♦s r③♦s ♥♦ ♠♦♦ s♦♦

s♣rs♦ t♠é♠ é s♦ ♣ t③çã♦ s rás t ♣rsã♦ ♥♠ér

s s♠çõs ♦r♠ r③s ♠ ♠ str ♦♠♣♦st♦ ♣♦r qtr♦ ♦♠♣t♦rs ♠

sr♦r três sr♦s ♥♦s qs ♦ ♣r♦ss♦r ♥t ♦r P❯ ③

♣♦ss qtr♦ ♥ú♦s ts ♠♠ór s s♠çõs ♥♠érs

♠♣♠♥ts ♥st tr♦ ♦r♠ ♥ssár♦s t③r ♦s ♥ú♦s s♣♦♥ís ♥♦

str ssts q ♦ sst♠ ♦♣r♦♥ t③♦ é ♦ r ♥① ♣r

ts

♠♣♦ ①çã♦ s s♠çõs ♦♠ ♦ ♠♦♦ s♦♦ s♣rs♦ ♦♠ ♦ ♠♦♦ s♦♦ ♥ã♦ s♣rs♦

TempoSD ♠♥ TempoSND ♠♥

Pr♦♠ st ❱rt Pr♦♠ st ❱rt Pr♦♠ st ♦r③ Pr♦♠ st ❱rt Pr♦♠ sts ❱rt

♦♥t♦ ♣rtr s s rs q ♠♦♠ ♣r♦♣♦st ♥st

tr♦ ♣r s♦♦s s♣rs♦s ♣♦ ♦ ♠ét♦♦ sr ♠s q♠♥t

♦s s♣t♦s ís♦s r♦♥♦s ♦s sst♠s trr♠♥t♦ étr♦ ♦ q ♠♦♠

q ♦♥sr ♦ s♦♦ ♥ã♦ s♣rs♦ ♦ ♥♦rr ♦ t♦ s♣rs♦ ♦ s♦♦ ♥s s♠çõs

♥♠érs ♦srs s♦s ♦s ♥ ♦r♠ q♥♦ ♦♠♣rs ♦ ♦r

♣♦ t♥sã♦ ♠ s♠ trtr ♦tr♦s tr♦s sr♠ q sts

s♦s ♣♦♠ sr ♥ ♠♦rs ❬ ❪ é♠ ss♦ rs ♠ r♥ç

s♥t ♥ ♦r♠ ♦♥ ♥tr ♦ s♥ ♦♠ ♠♦♦ s♦♦ ♥ã♦ s♣rs♦ ♦ s♥

①♣r♠♥t

♣ít♦

♦♥srçõs ♥s

st tr♦ ♦ s♥♦ ♠ ♠♦♠ ♣r ♠♦s s♣rs♦s trés

♣♥ê♥ ♦♥t étr ♦♠ rqê♥ ♥♥♦ ♦s t♦s ♥çã♦

σ(f) ♥♦ ♠ét♦♦ ❬❪ ♥sã♦ σ(f) ♥s qçõs ① ♦ t ♣

♥♦ ♣r♦①♠çã♦ ♣♦r ♣♦♥ô♠♦ Pé ♦ r ♠ tr♠♦s jω ♣r ♥çã♦

♠t♠át ♣r♦♣♦st ♣♦r ❱sr♦ t ❬❪ st ♦r♠ ♦ ♣♦ssí s♠♣r ♠

♥♣çã♦ tr♥s♦r♠ ♥rs ♦rr s qçõs ① s s♠çõs

♥♠érs ♦ ♦♠í♥♦ ♦♠♣t♦♥ é tr♥♦ trés té♥ P ❬❪ ♠♦

♠ s sts tr♦♦s ♦ r③ trés té♥ ♦♥♦ ❬❪ ♠♦♦

tr ♠♦r ♥í srt③çã♦ ♦ ♦♠í♥♦ ♦♠♣t♦♥

♦ ♣ít♦ ♦ ♣rs♥t♦ ♦ ♠♦♦ st♣ ①♣r♠♥t ♣r♦♣♦st♦ ♣♦r ♥

❬❪ q ♦ t♦♠♦ ♦♠♦ s ♣r ♠♣♠♥tçã♦ ♦s st♣s ♦s ♥s♦s r③♦s ♠

♠♣♦ ♥st tr♦ ♥ ♦r♠ ♣♦♥ts r♦♠♥çõs té♥s ♥♠♥t

♠♣♦rtâ♥ ♣r ♠♥♠③r t♦rs q ♣rq♠ ♦♥♦râ♥ ♥tr ♦s rst♦s

♥♠ér♦s ①♣r♠♥ts

♠♦♠ ♣r♦♣♦st q ♦ trés ♦♠♣rçõs ♥♠ér♦①♣r♠♥ts

rs♣♦st tr♥stór ♦s sst♠s trr♠♥t♦ ♦♠♣♦st♦s ♣♦r ♠ st ♥tr

r rt♠♥t r③♦ ♠ três ♦sõs ♠ st ♥trr ♦r③♦♥t♠♥t

s sts ♥trrs rt♠♥t tr♠♥t ♦♥ts s tsts ①♣r♠♥

ts ♦r♠ r③♦s ♥ ár r♦r③ ♦③ ♥♦ ♥tr♦ ♥♦♦ tr♦

rástr♦♥♦rt ♠ é♠ ♦ Prá ♥ q ♦ ♦sr♦ q ♦ s♦♦ ♣rs♥t t

♦♥t

♣rtr ♦s rst♦s ♣rs♥t♦s ♥♦ ♣ít♦ q♥♦ ♦♠♣r♦ ♦ ♠♦♦

s♦♦ ♥ã♦ s♣rs♦ ♦srs q ♦s s♥s ♦t♦s ♣♦ ♠♦♦ s♥♦♦ ♥st

tr♦ ♣rs♥tr♠ ♠♦r ♦♥♦râ♥ ♦♠ ♦s s♥s ♠♦s ♦ ♦♠♣rr ♦s ♦rs

♣♦ t♥sã♦ ♦ ♠♦r s♦ s♦t♦ ♦ ♠♦♦ s♦♦s s♣rs♦s ♦

P♦r ♦tr♦ ♦ ♦ ♠♦r s♦ s♦t♦ ♦sr♦ ♣r ♠♦♦ ♦♠ s♦♦s ♥ã♦

s♣rs♦s ♦ Pr ① ♦♥t ♦sr ♥st tr♦

st r♥ç ♥♦ ♦r ♣♦ t♥sã♦ é s♣r t ♦♠♦ srt♦ ♠ ❬❪ ♥♦

♥ss ♦ s rs ♦ts ♣♦s ♦s ♠♦♦s ♥♦ts ♠ s♣r ♥♦

s♦ ♠á①♠♦ ♥ ♠♦r MSESD,max = 0, 1809 MSESND,max = 1, 5201 s ♦

rr♦ ♠é♦ qrát♦ ♦ s♦♦ s♣rs♦ é ♠s ③s ♠♥♦r

P♦rt♥t♦ ♦♥s q trés s s♠çõs r③s ♦♠ ♦ ♠♦♦ s♦♦ s

♣rs♦ s♥♦♦ ♥st tr♦ str♠s ♥ã♦ só ♦s ♣♦s t♥sã♦ ♥③

♠s t♠é♠ ♦r♠ ♦♥ ♦s s♥s q ♦ ♣rt♠♥t ♦♠♣tí ♦♠ ♦s s

♥s ♠♦s ♠ ♠♣♦ ♥ rssts q ♦s ♦♥ts ♦ ♣♦♥ô♠♦ Pé

♦t♦s ♥st tr♦ rtr③♠ ♦ ♦♠♣♦rt♠♥t♦ s♣rs♦ ♦ s♦♦ é♠

❱s♥♦ ♦ ♣r♠♦r♠♥t♦ ♠♦♠ ♣r♦♣♦st ♥st tr♦ ♥♦s ♦♥tr

çõs st♠s ♠s ♣r♦♣♦sts tr♦s tr♦s

• ♥r ♣♥ê♥ ♣r♠ss étr ♦♠ rqê♥ ♣r q ♠♦

♠ ♣r♦♣♦st ♥st tr♦ s ♣á ♠ s♦♦s ♠♥♦s ♦♥t♦s

• ♦r ♦ t♦ ♦♥③çã♦ ♦ s♦♦ ♥t♠♥t ♦♠ ♦s t♦s s♣rs♦s

• ♣r ♠♦♠ ♠trs s♣rs♦s ♦s ♠ét♦♦s ♠sss ♦♠♦ ♦ P

P♦♥t ♥tr♣♦t♦♥ t♦ ♦r♠ q s♠ r♣rs♥ts q

♠♥t s ♦♠trs ♥ã♦rt♥rs ♦s sst♠s trr♠♥t♦

rê♥s ♦rás

❬❪ ♥ rt ♦♥t♦♥ ts ♥ r♥s♠ss♦♥ ②st♠s ♦r Pt♦♥s

♥

❬❪ ❱ ♦ trr♠♥t♦s étr♦s ♦♥t♦s ás♦s té♥s ♠çã♦ ♥str

♠♥tçã♦ ♦s♦s trr♠♥t♦ st rtr

❬❪ ♥r♠♥♥ ♥ ♠♣♥♦♦ trr♠♥t♦ étr♦ r

❯❩❩

❬❪ ♥ ♦ ♠t♦ ♦r ♥②③♥ t tr♥s♥t ♦r ♦ r♦♥♥ s②st♠s

s ♦♥ t ♥t r♥ t♠♦♠♥ ♠t♦ P ♣♦rt ♦♦

❬❪ ❱sr♦ ♦♠♣r♥s ♣♣r♦ t♦ t r♦♥♥ rs♣♦♥s t♦ t♥♥ r

r♥ts r♥s P♦r r② ♦ ♥♦ ♣♣ ♥r②

❬❪ r ♠♣s ♥② ♦ r♦♥ tr♦s r♥s P♦r r②

♦ ♥♦ ♣♣ ♥

❬❪ r t♥♥ sr ♥② ♦ r♦♥♥ rs P♦r r②

r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣ ②

❬❪ r ♥ ♥ tr♦♠♥t ♠♦ ♦r tr♥s♥ts ♥ r♦♥♥

s②st♠s r♥s P♦r ♦ ♥♦ ♣♣ t

❬❪ ❱sr♦ ♥ ♦rs ♠♦ ♦r s♠t♦♥ ♦ t♥♥rt

♥♥r♥ ♣r♦♠s r♥s P♦r r② ♦ ♥♦ ♣♣

♣r

❬❪ r ♦st♦ ❱ ②♦s ♥ ③r

♠♣s ♦♥t ♦r sqr r♦♥♥ rs ♥ ♦ rsstt② s♦s ♥♥ ♦

♥t♦♥ tr♦ ♦r♥ ♦ tr♦stts ♦ ♥♦ ♣♣

❬❪ ❱sr♦ ♣♦ ❱ ♥ Prr rs♣♦♥s ♦ r♦♥♥

tr♦s t♦ t♥♥ rr♥ts t t ♦ rq♥②♣♥♥t s♦ rsstt② ♥

♣r♠ttt② r♥st♦♥s ♦♥ tr♦♠♥t ♦♠♣tt② ♦ ♥♦

②

❬❪ ♠t♦s tr ♠sr♠♥ts ♦♥ s♦ t tr♥t♥ rr♥ts Pr♦

♦ P ♣♣

❬❪ P♦rt sr♠♥t ♥ ♠♦♥ ♦ s♦ tr♦♠♥t ♦r ♥ Pr♦

♥t ②♠ tr♦♠♥ ♦♠♣t t ❲

❬❪ ♦♥♠r ♥ ♠t ❯♥rs ♠♣♥ ♦r ♦s ♥s r

♥② ♥t rr r♦♣ ♣♦rt ♦r Pr♦ ② ♣t♠r

❬❪ ♠t♦s tr ♣r♦♣rts ♦ s♦s ♦r tr♥t♥ rr♥ts t r♦

rq♥s Pr♦ ♦② ♦ ♦♥♦♥ ♦ ♥♦ ♣♣

❬❪ ❱sr♦ ❱ ♠rãs rú♦ ❲ P♥t♦ ♥

í♣♦ rs♣♦♥s ♦ r♦♥♥ tr♦s t♦ t♥♥ rr♥ts t t

♦ rq♥②♣♥♥t rsstt② ♥ ♣r♠ttt② ♦ s♦ ♥ Pr♦ t ♥t ♦♥

t♥♥ Pr♦tt♦♥ r t② ♣t

❬❪ ❱sr♦ ♥ ♣♦ rq♥② ♣♥♥ ♦ s♦ ♣r♠trs ①♣r♠♥

t rsts ♣rt♥ ♦r♠ ♥ ♥♥ ♦♥ t t♥♥ rs♣♦♥s ♦ r♦♥♥

tr♦s r♥st♦♥s ♦♥ P♦r r② ♦ ♥♦ ♣r

❬❪ ♣♦ ♥ ❱sr♦ rq♥② ♣♥♥ ♦ s♦ ♣r♠trs t ♦♥ t

t♥♥ rs♣♦♥s ♦ r♦♥♥ tr♦s r♥st♦♥s ♦♥ tr♦♠♥t

♦♠♣tt② ♦ ♥♦ rr②

❬❪ ♣♦ r♦r ♦♥s♦ r ♥ sss

tr s ♦ r♦♥♥ tr♦s t rq♥② ♣♥♥t s♦ ♣r♠trs

tr P♦r ②st♠s sr ♦ ♣♣ rr②

❬❪ r ❱sr♦ ♣♦ ♥ ♦♥t t♥♥♥ ♦ts

♦r ♦ss② r♦♥ t t ♦ rq♥② ♣♥♥ ♦ tr ♣r♠trs ♦ s♦

tr♦♠♥t ♦♠♣tt② r♥st♦♥s ♦♥ ♦ ♥♦ t♦r

❬❪ s②♥ ♥ r t♦♥ ♦ t♥♥♥ ❱♦ts ♦♥

t♦♥t♦r r ♥s ♦t ♦ ♦ss② s♣rs r♦♥ P♦r

r② r♥st♦♥s ♦♥ ♦ ♥♦ ♣r

❬❪ ♦tt tr ♥ ♠♥t ♣r♦♣rts ♦ r♦ ♥ s♦ ❯ ♦ r

♣t ♥tr♦r ❲s♥t♦♥

❬❪ ♦tt rr♦ ♥ ♥♥♥♠ tr ♦♥st♥t ♥ tr

♦♥tt② ♠sr♠♥ts ♦ ♠♦st r♦ ♥ ♦rt♦r② ♠t♦ ♦r♥ ♦

♦♣②s sr ♦ ♥♦ ♣♣

❬❪ ❱sr♦ ♥ ♦s♦ s♣♦♥s ♦ r♦♥♥ tr♦s t♦ ♠♣s rr♥ts

♥ ①♣r♠♥t t♦♥ r♥s tr♦♠♥ ♦♠♣t ♦ ♥♦ ♣♣

❬❪ ❨ ♠r s♦t♦♥ ♦ ♥t ♦♥r② ♣r♦♠s ♥♦♥ ①s

qt♦♥s ♥ s♦tr♦♣ ♠ ♥t♥♥s ♥ Pr♦♣t♦♥ r♥st♦♥s ♦♥

♦ ♥♦ ♣♣ ②

❬❪ Pé r ré♣rs♥tt♦♥ ♣♣r♦é ♥ ♦♥t♦♥ ♣r s rt♦♥s rt♦♥

s P ss ♥♥ ♦ ♦r

❬❪ ♦♥ ♥ ♥② ♦♥♦t♦♥ P P ♥ ♥t

♠♣♠♥tt♦♥ ♦ t P ♦r rtrr② ♠ r♦ ♥ ♣t

♥♦♦② ttrs ♦ ss ♣♣

❬❪ ❨ ♥ ❨ ♦ ♥ ♠t♥ ♥ ♠♣r♦ ♥ ❲r

♣rs♥t♦♥ ♦r ♦♠♣tt♦♥s ♥t♥♥s ♥ Pr♦♣t♦♥ r♥

st♦♥s ♦♥ ♦ ♥♦ ♣♣ t

❬❪ ♦s ♦ ♦♥③t♦♥ r♥t ss♦t t sr ♦ r

r♥ts ♥t♦ ♦♥♥trt tr♦s P♦r r② r♥st♦♥s ♦♥ ♦

♥♦

❬❪ ♦♥♦s ♥ tt♦♣♦s ♦ ♦♥③t♦♥ ♥r t♥♥ ♠♣s ♦ts

Pr♦♥s ♦ ♥♦

❬❪ ❨ ♦ ❩♥ ❩♦ ❳ ♥ ❩♥ ♥ ♥

t ♥t ♦ ♦♥tr♣♦s ❲r ❯♥r t♥♥ rr♥t P♦r r②

r♥st♦♥s ♦♥ ♦ ♥♦

❬❪ ♥s ♥ ♥♥r♥ tr♦♠♥ts ♥ ♦♥ ❲② ♦♥s

♥

❬❪ ❱sr♦ ♥ P♦rt ♦ ♣r♠ttt② ♥ ♦♥tt② ♦r ♦♥ r

q♥② r♥ ♦ tr♥s♥t ♣♥♦♠♥ ♥ tr ♣♦r s②st♠s ♣rs♥t t t

♥t ②♠♣ ❱♦t ♥ r♥s r♠♥②

❬❪ r ♥ ♦r♥♦ ♦♠♣tt♦♥ ♥r♦♥♠♥t ♦r

♠t♥ t♥♥ tr♦s ♥ P♦r stt♦♥ ② ♥tr♥ ♠♦♠♥

t♦ tr♦♠♥t ♦♠♣tt② r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣

♦

❬❪ ♦ r ♦r♥♦ ♥

♠ Pr ♥ ①♣r♠♥t rsts ♦ ♦r t ♥ ♣♥t♦♠s

③ r♦ ♥ ♣t♦tr♦♥s ♦♥r♥

♥tr♥t♦♥ ♣♣ ♦

❬❪ ① ②♥♠ t♦r② ♦ t tr♦♠♥t P♦s♦♣ r♥

st♦♥s ♦ t ♦② ♦t② ♦ ♦♥♦♥ ♦ ♣♣

❬❪ rt rsst♥s ♦r s♥s t ♦♥♦♥

❬❪ rts ♥ ♦♥ rtér♦s ♠çõs trr♠♥t♦ s

r♥ç ♠ sstçõs ♥s ♦ ❱P ♠♥ár♦ ♦♥ Pr♦çã♦

r♥s♠ssã♦ ♥r étr ♦ ♥r♦

❬❪ s ♥ P ♥♥♥ r♦♥♣♥trt♥ rr ♦r rs♦t♦♥ ♠♣♣♥

♦ s♦ ♥ r♦ strtr♣② ♦♣②s Pr♦s♣ ♦ ♣♣

❬❪ ❲ ♥③ P♠♥t s♥ ①s ♣rt♠♥t ♦ r♥s♣♦rtt♦♥

♥ ♥r②

❬❪ tt♦s é♥s trr♠♥t♦ ♠ tr♦♠♥ts♠♦ ♣♦

t ♠♣♥s

❬❪ ♠ Pr♦♣♦st ♠ ♦♦ ♦♦ ♣r ♥ás ♦s ♦♠♣♦rt♠♥t♦s r♥

stór♦ st♦♥ár♦ st♠s trr♠♥t♦ s♥♦s ♦ ét♦♦ s

♦t♦r♦ ♠ ♥♥r étr ♥sttt♦ ♥♦♦ ❯♥rs

r ♦ Prá

❬❪ ♣t ♥ ♣r ♠♣s ♠♣♥ ♦ r♦♥♥ rs r♥s

P♦r ♣♣ ②st ♦ P ♥♦ ♣♣ ♦

❬❪ rs ♥srr ♥③ t♥r ♥ ♥r

rq♥②♣♥♥t ♥tr♥ ♠♦♠♥ ♦r♠t♦♥ ♦r s♣rs

trs tr♦♠♥t ♦♠♣tt② r♥st♦♥s ♦♥ ♦ ♥♦

♣♣

❬❪ ② ♥ rs Ps ♥r rrs ♦♥♦t♦♥ ♦r s♣rs

♠ s♥ ♥t♥♥s ♥ Pr♦♣t♦♥ r♥st♦♥s ♦♥ ♦

♥♦ ♣♣ ♥

❬❪ ssr ♣r♦♣t♦♥ ♦ ♥ tr♦♠♥t ♠♣s tr♦ s♦ ♥

♥ ♦ rq♥② ♣♥♥t ♣r♠trs ♣ ss♦♥ sr

♦r♣♦rt♦♥ ♥t rr

❬❪ ♦ ♥ ♥ss ♦♠♣tt♦♥ tr♦②♥♠s ♥tr♥

♠♦♠♥ r rt ♦s ♥

❬❪ ♥③♦s ♣♣ ♥②ss Pr♥t

❬❪ ❯ P st t♣Prs♦♥ ♦t♥♣♦♥t ♦♠♣tt♦♥s t ♦rrt

♦♥♥ tt♣♠♣r♦r ss♦ ♠ ♦st♦

❬❪ s♠♥t♦ r ②♦s P r♦

rú♦ ♥ r st♦ ♥♠ér♦①♣r♠♥t srt♦s ♣r♦♦♦s

♣♦r srs t♠♦sérs ♠ ♠s trr♠♥t♦ étr♦ ❳❳ ♠♥ár♦

♦♥ Pr♦çã♦ r♥s♠ssã♦ ♥r étr ❳❳P

♦tr♦ ♦③ ♦ ç

❬❪ rú♦ s♥♦♠♥t♦ ♠♣♠♥tçã♦ r♦r rt♦ s

♣çã♦ ♥ ♥ás ♠ér♦①♣r♠♥t ♦ ♦♠♣♦rt♠♥t♦ r♥stór♦ s

t♠s trr♠♥t♦ étr♦ ssrtçã♦ str♦ ♠ ♥♥r étr

❯♥rs r ♦ Prá PP

❬❪ ♦t tst t♥qs Prt ♥r ♥t♦♥s ♥ tst rqr♠♥ts ♥

tr♥t♦♥ t♥r

❬❪ t♥r ♥qs ♦r ❱♦t st♥ t ♣♣

❬❪ r ♦ t♦♦♦ ♣r ♥ás í♥ts st♠s trr

♠♥t♦ ♦♠♣①♦s ❯t③♥♦ ♦ ét♦♦ ♦♠♣tçã♦ Pr t♦♠át

s rs rts s ♦t♦r♦ ♠ ♥♥r étr Pr♦r♠

Pósrçã♦ ♠ ♥♥r étr ♥sttt♦ ♥♦♦ ❯♥rs

r ♦ Prá

❬❪ ②♥♠♥ ♥ ♦r ♥♦tr ♦♦ t ♠srs ♦ ♦rst r②

♥tr♥t♦♥ ♦r♥ ♦ ♦rst♥ ♦ ♥♦ ♣♣

❬❪ ♦r ♥ ♦♠♣rs♦♥ ♦ rq♥②♣♥♥t s♦

♠♦s ♣♣t♦♥ t♦ t ♥②ss ♦ r♦♥♥ s②st♠s tr♦♠♥t ♦♠♣

tt② r♥st♦♥s ♦♥ ♦ ♥♦ ♣♣