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 :