1

B GIAO THNG VN TITRNG I HC HNG HI

KHOA: IN IN T TU BINB MN: IN T - VIN THNG

BI GING

TRNG IN T V TRUYN SNG

TN HC PHN :TRNG IN T V TRUYN SNG

M HC PHN : 13205

TRNH O TO : I HC CHNH QUY

DNG CHO SV NGNH: IN T - VIN THNG

HI PHNG 2010

2

CHNG M U

0.1. Gii tch vct- Tch v hng ca hai vecto :

. . . . osA B B A A B c - Tch c hng ca hai vecto :

x sin .A B AB n( A , B v n lm thnh tam din thun)

- o hm khi (hay o hm khng gian) ca mt trng (c th l trng v hng P,

hoc trng vecto B ) l mt i lng (v hng hoc vecto) :gradP P l mt vecto

00 0

P P PP x y z

x y z

divB B l mt i lng v hng

. yx zBB B

Bx y z

xrotB B l mt vecto

00 0

x

x y z

x y z

Bx y z

B B B

Trong : ton t Nabla ( ) trong ta decac c dng

0 0 0x y zx y z

Ch hai cng thc chuyn i tch phn :

Cng thc Gauss . . .S V

B n ds divB dv ngha : nu 0divB th . . . 0

S V

B n ds divB dv tc l tch phn i lng B theo mt mt kn S c gi tr khc 0, ta gi trong mt kn S c ngun, cn khc i ta gi trong mt kn khng c g (ngun).

Cng thc Stok . .L S

B dl rotB ds ngha : nu 0rotB th . . 0

L S

B dl rotB ds tc l tch phn i lng B theo mt ng cong kn L c gi tr khc 0, ta gi B l trng xoy (ng sc khp kn), cn khc i ta gi l trng khng xoy.

0.2. in trng tnh- Trng tnh in l trng c to ra xung quanh cc in tch c nh ( E v D

khng thay i theo thi gian).- Trng tnh in l mt trng th.- L mt trng mang nng lng, c tng tc ln cc in tch. Cc in tch chuyn

ng ngang qua trng s c gia tc.

3

- nh l Gauss :

.S

D ds q Q Trng tnh in l mt trng c ngun :

( )div E - Trng tnh in khng phi l mt trng xoy, cc ng sc ca trng khng khp

kn trong khng gian :

0rotE - Vect cng in trng E v th (in th) ca trng U :

E gradU U 2 0U

- iu kin b :Thnh phn tip tuyn ca vect cng in trng lin tc ti b :

1 2t tE EThnh phn php tuyn ca vect in cm gin on ti b khi trn b mt phn cch

c in tch :

1 2n nD D

0.3. Dng in dn- Dng in dn l dng chuyn ng c hng ca cc ht mang in.- nh lut bo ton in tch :

ddivJ

dt

i vi dng in khng i :

0divJ - nh lut m :

J E- Cc nh lut Kirchoff (2 nh lut : tng cc dng in mt nt bng 0, v tng cc

st p trn mt mch vng bng tng cc sc in ng)

0.4. T trng tnh- T trng tnh l trng c to ra xung quanh cc dng in khng i ( B v H

khng thay i theo thi gian).- nh lut Ampe : Lu s ca vect cng t trng H theo mt ng cong kn L

bng tng i s ca cc dng in nm trong vng kn :

.L

H dl IT trng l mt trng xoy, cc ng sc t khp kn trong khng gian bao quanh cc dng in to ra t trng :

rotH J- Thng lng ca vect cm ng t qua mt mt kn bng 0 :

. 0S

B d s T trng l trng khng c ngun :

0divB - iu kin b :

4

Thnh phn php tuyn ca vect cm ng t lin tc ti b :

1 2n nB BThnh phn tip tuyn ca vect cng t trng gin on ti b khi trn mt

phn cch c dng in :

1 2t t sH H J

5

Chng 1TRNG IN T

1.1 H PHNG TRNH MAXELL

H phng trnh Maxell dng vi phn (ct 1 bng 4.1 tr. 46 gio trnh)

0

ErotH J

t

HrotE

t

div E

div H

Hoc

x

x

.

. 0

t

t

DH J

BE

D

B

H phng trnh Maxell dng tch phn (ct 3 bng 4.1 tr. 46 GT)

. .

.

.

. 0

l S

l s

S

S

dH dl I D ds

dt

E dl Bd st

E ds q

H ds

Trong :

E Vect cng in trng [V/m]D Vect in cm ( )D E [A/m2]H Vect cng t trng [A/m]B Vect t cm ( )B H [T=Wb/m2]J Mt dng in [A/m2] Mt in tch khi [C/m3]

ngha vt l ca cc phng trnh Maxell c tm tt nh trong ct 4 bng 4.1 tr.46 GT:

- phng trnh Maxell th nht c th bin i nh sau :

. ( ) dan dichl s s s

EH dl J d s Jd s Ed s i i

t t

6

trong dans

i Jd s l dng in dn-dng chuyn ng ca cc in tch, cn

J E (nh lut m tr.27)( E)

dich

s s

i Ed s d st t

l dng in dch-dng xut hin do c s

bin thin ca cng in trng theo thi gian. Khi nim v dng in dch c trnh by r rng v n gin mc 4.1 tr. 40 GT.

in trng bin thin theo thi gian sinh ra t trng xoy (bin thin trong khng gian vi ng sc khp kn)

- phng trnh Maxell th hai c th vit

.l s

dE dl Bd s

dt t

T trng bin thin theo thi gian sinh ra in trng xoy

- phng trnh Maxell th ba th hin in trng c ngun, ngun ca in trng l cc in tch.

- phng trnh Maxell th t th hin t trng khng c ngun. Trong t nhin khng c cc t tch t do.

- iu kin b tng qut ca trng in t trng bin thin :

1 2 1 2

1 2 1 2

t t n n

t t s n n

E E D D

H H J B B

Ch 1 : Nguyn l i ln ca cc phng trnh Maxell

mm

m

m

ErotH J

H E t

HJ J rotE Jt

div E

div H

trong mJ v m l i lng o

Ch 2 : in trng tnh v t trng tnh l cc trng hp ring ca trng in t bin thin, khi cc thnh phn o hm theo thi gian bng 0, H phng trnh Maxell trong cc trng hp ny s c cc bin dng nh cc kt qu kho st ca cc chng 1, 2, 3.

Ch 3 : Mt s cng thc ca gii tch vect cn n li mn Ton cao cp 2, v c ghi li ph lc 1 tr.270 GT.

1.2 NH L POYNTING

7

1.2.1 nh l Poynting :

s

dWP d s

dt

trong 2 2

( )2 2V

E HW dv

l nng lng in t tch t trong th tch V

. .V

P J E dv l cng sut tn hao nhit ca dng in trong V

xE H c gi l vect Poynting.

- Vect l vect mt thng lng nng lng chy qua mt S trong n v thi gian.Theo nh ngha, th nng lng ca trng in t mi im s lan truyn theo

phng ca vect , tc l phng php tuyn vi mt phng to bi hai vect E v H . Gi tr (tc thi) :

. .sin( , ) .E H

E H E H E H

(W/m2)Gi tr trung bnh :

* *

0

1 1 1(E ) Re(E )

2 2

T

m mtb dt H HT

- nh l Poynting ch ra rng : s bin i nng lng trng in t trong mt th tch V, mt phn do bin thnh nhit v mt phn do truyn lan thot ra mt bao bc th tch y.

- Cn gi l nh l Umv-Poynting.

1.2.2 Chng minh nh l Poynting tr.49Gi :

ErotH J

E tH H

rotEt

2 2

2 2

1( )

2

E H E HErotH HrotE J E E H J E

t t t t

2 2

E H ( )

1 ( ) 1 ( )

2 2

rotH rotE div E H

E E H HE H

t t t t

t suy ra cng thc 4.30 v 4.31

1.2.3 Cc v d minh ha nh l Poynting- Trong mi trng in mi l tng 0 , tc l 0J v 0P

nu 0dW

dt tc l 0

S

d s nng lng thot ra khi V, bao bi mt S

8

nu 0dW

dt tc l 0

S

d s nng lng thm nhp vo V, bao bi mt S- Kho st s truyn nng lng qua on dy dn (tr.52).

Cu hi n tp chng 1 :

1. H phng trnh Maxell v ngha vt l.2. Trnh by nguyn l i ln ca cc phng trnh Maxell.3. Pht biu nh l Poynting v nu ngha vt l.4. Chng minh nh l Poynting.

9

Chng 2SNG IN T PHNG

2.1 KHI NIM V SNG PHNG- mt ng bin : cng bin - mt ng pha : cng pha- sng in t phng : mt ng pha ng bin l mt phng (gn ng vng xa i

vi tt c cc ngun bc x)

2.2 SNG PHNG TRONG MI TRNG IN MI L TNG

2.2.1 Mt s gi thit - mi trng in mi l tng 0 - khng c ngun ngoi 0, 0J - chn h ta zi mt sng (x,y), 0z zE H

do yx x yE E i E i v yx x yH H i H i c th tch thnh 2 h thng A(gm Ex,Hy) v B(gm Ey,Hx)

2.2.2 Phng trnh sng (5.4 tr 59)- xut pht t h phng trnh maxell vi gi thit 2.2.1- trin khai hai v ca M1 v M2, ch sng truyn theo trc z nn

0x y

ch c 0

z

i vi , , ,x y x yE E H H

0x y z t

i vi ,z zE H

dn ra c h phng trnh 5.2 tr 58.- nhn c phng trnh sng i vi cc thnh phn sng, ly Ex lm in hnh,

xut pht t I.a

zH

y

y xH E

z t

vi phn 2 v :

( . )2 2

2 2

1( ) ( )

xEHy II bzt

y yx xH HE E

t t z z t z

- rt ra phng trnh sng i vi Ex :

2 2

2 2

1x xE E

t z

t 1

v

ta c pt 5.4 tr 59 :

2 2

2 2 2

10x x

E E

z v t

hay

2 2

2 20x x

E E

t

vi z

v

2.2.3 Nghim ca phng trnh sng Nghim ca 5.5 l hm s ty theo hai bin dng ( )F t v ( )F t v c hai

hm ny u c o hm bc 2 theo t v bng nhau.Nghim tng qut c dng :

10

1 2 1 2( ) ( ) ( ) ( )xz z

E F t F t F t F tv v

1( )F t biu din sng truyn theo hng z dng

2 ( )F t biu din sng truyn theo hng z m

2.2.4 Quan h gia cc thnh phn trng

T y xH E

z t

thay

1

1

( )

( )

x

y

zE F t vzH G t v

ta c ' '1 11

( ) ( )z zF t G tv vv

ly tch phn theo bin ( )zt v li c 1 11

( ) ( )z zF t G tv vv

ch : 1

v

v c 1 1( ) ( )z zF t G tv v

Nh vy i vi sng thun :

0x

y

EZ

H

gi l tr khng sng

2.2.5 Sng phng iu hai vi sng phng iu ha :

1( ) cos ( ) cos( )z zF t A t A t zv v v

K hiu : 2 2f

kv v

v gi l h s pha hay hng s sng, c th vit

0 0

( ) ( )0 0

cos( ) cos( )x

i t kz i t kzx

E E t kz E t kz

hay

E E e E e

Thnh phn Hy vung gc vi Ex v c gi tr bng 00

EZ .

Hnh 5.3a biu din s ph thuc ca cc thnh phn E v H theo thi gian ti mt im z c nh.Hnh 5.3b biu din s ph thuc ca E v H theo z mt thi im c nh.

2.3 SNG PHNG TRONG MI TRNG BN DN

2.3.1 Phn loi mi trng truyn sngThc t khng c mi trng in mi l tng, v do ch c mi trng truyn

sng bn dn.

Tnh bn dn ph thuc vo quan h tng i gia dng in dn Jdn ( )E v dng

in dch Jdch ( )E

t

.

Mi trng c coi l mi trng dn in, nu :