Formular fulltime TRƯỜNG ĐIỆN TỪ

MỘT SỐ CÔNG THỨC CƠ BẢN TRONG CÁC BÀI TẬP TRƯỜNG ĐIỆN TỪ

gradϕ =
G
divA =

G
a1 + h12

1 ∂ϕ
h1 ∂u1

1
h1h 2 h 3

G
a 2 + h13

∂ϕ
∂u2

G
a3

∂ϕ
∂u3

G
rotA =

 ∂ (h 2 h3A1 ) + ∂ (h1h3A2 ) + ∂ (h1h 2 A3 ) 

∂u 2
∂u 3
 ∂u1


∆ϕ = div(gradϕ ) =

1
h1h 2 h3




(

∂
∂u1

h 2 h3 ∂ϕ
h1 ∂u1

) + ...

L

α=

ω µε

( )

σ

1+

2

G
E =−gradϕ
G
G
B = rotA

∆ϕ = − ρεV
G
G
∆A = −µJ
2

ωε

−1

ω µε

β=

G
h 3a 3

∂

∂u1

∂
∂u 2

∂
∂u 3

h1A1

h 2A 2

h 3A 3

Hệ

h1

h2

h3

Đề các

1

1

1

Trụ
Cầu

1
1

r
r

1
rsinθ

G G
G G
G
G
D = εE B = µH J = σ E
dV = h1h 2 h 3 du1du2 du3

G
G
G
G
d A = h1du1a1 + h 2 du2a 2 + h 3du3a 3

C=

Q
U

µ0 =4π.10

L=

Φ
I

−7

( )
σ

2

ωε

(H/m)

η=

+1

G
G
P
=
(ε
−
ε
)E

0
V
GG
G G G
Wm = 12 ∫ H.BdV = 12 L.I2 ρpS =−an(P1 −P)
2
V∞
GG
G
γ = jωµ(σ+jωε) = α + jβ
PJ = ∫ EJdV ρpV = −divP
V
G
G
β =−gradϕ =β.as

GG
We = 12 ∫ E.DdV = 12 C.U2

ε0 = 361π 10−9(F/m)

1+

2

G
h 2a2

G
G

G
∆ A = grad(divA) − rot(rotA)

G
G
G
G
dS =±h2h3du2du3a1 ± h1h3dudu
1 3a2 ± h1h2dudu
1 2a3

G G
∗
D
v∫ s dS = q
G G ∗
H
v∫ d A = I

1
h 1h 2 h 3

G
h1a1

∞

jωµ

σ + jωε

Uniform Plane Wave and Antenna:

Hệ phương trình Maxwell

Waveguide:

(



E x = K12 −γ ∂∂Exz − jωµ ∂∂Hyz
c

∂ 2 E z
∂x 2

G•
 G• G 
E = η  H× a s 



G G
G
G
an × (H1 − H2 ) = Js
G G
G
a n × (E1 − E 2 ) = 0

G G G
a n .(D1 − D2 ) = ρS
G G G
a n .(B1 − B2 ) = 0
G G G
an .(J1 − J2 ) = − ∂∂ρtS

G G G
rotH = J + ∂∂Dt
G
G
rotE = − ∂∂Bt
G
divD = ρ V
G
divB = 0
G
divJ = − ∂∂ρtV

)

2


∂2H
z

+ ∂∂yE2z + K c2 E z = 0

∂x 2

 = C cos
H
z
 =
H
x

jβC mπ
Kc2 a

 =
H
y

jβC nπ
K c2 b

(

Pbx =

v∫

S

< Pr > dS

R bx = 2Pbx / I

x cos

2

)

-jβz

mπ
a

nπ
b

-jβz

mπ
a

nπ
b

-jβz

v
1 − (f th /f )

λ mn =

2

c

1 − (f th /f )

α

(

E x = − jβC
K2

mπ
a

E y = − jβC
K2

nπ
b

c

(

=

1
2

ω µ
2σ

)

− jAIβ3sinθ
4πωε

(

x sin

nπ
b

mπ
a

mπ
a

0

∫

b

0

(

−1
βr

2π

∫0 ∫0

)

+ β31r3 .e-jβr

j
β2 r 2

)

+ β2jr2 + β31r3 .e-jβr

u n sin θ d θ d φ

4πu m ax
Pbx

=

(

 = 12 −γ ∂H z − jωε ∂E z
H

y
∂y
∂x
K
c





)dS

α

d

=

)

ω µ σ
2 β

Các thông số

-jβz

( ) +( ) = =
K = ( ) +( ) = - β + ω µε
β = β = ( ) −( ) −( )

f th = v2

nπ
b

-jβz

nπ
b

-jβz

m
a

mπ
a

2
c

2

2

n
b

2

ωth
2π

2

nπ
b

ω

mn

 =0; E = η H



H
z
x
TM y ; E y = − ηTM H x
a

)

+ β 21r 2 .e -jβr

j
βr

4πu nm ax

π

2

>∫ C n g H t t d A
 ∫ ( E x Hˆ y − E y Hˆ x
 S

) ( y ) .e
cos ( x ) sin ( y ) .e
sin ( x ) cos ( y ) .e

mπ
a

P = 12 ηTE/TM ∫

2

w

(

Kiểu sóng tới TMmn

E z = Csin
c

λ

D=

0

 = 12 −γ ∂H z + jωε ∂E z
H
x
∂x
∂y
K



 ; E = −η H

E z =0; E x = ηTE H
y
y
TE x
v mn =

ε = ε0(εr − j ωσε )

R0 = Re(η)
; L0 = Im(η)
2πa
2πaω

(

E θ =

δ =1/α

2
m

AIβ 2 sinθ
4π

 3
E r = − jA2Iβπωεcosθ

λ = 2π/β

 =0
+ ∂∂yH2z + K c2 H
z

) ( y ) .e
sin ( x ) cos ( y ) .e
cos ( x ) sin ( y ) .e
nπ
b

∗

S

Kiểu sóng tới TEmn

mπ
a

 =
H
φ

G G
v = ω/β
Pz = 12 Re{Eì H}z dSz p
ã



E y = K12 Eyz + jà Hxz
c

TẹT cuỷa nguyeõn toỏ anten thaỳng

Gã
G Gã
H = 1 a s ì E



2

2

m

a

v

v
th

2

2

n
b

2

TE = ωµβ ; ηTM = ωεβ ; γ = jβ

 |2 +|H
 |2 dxdy
|H
x
y


Lossless Transmission Line:

 =U
 e− jβz +U
 e jβz = U

 e jβd +U
 e− jβd = U
 cos(βz) − jZ I sin(βz) = U
 cos(βd) + jZ I sin(βd)
U
t1
fx1
t2
fx2
1
0 1
2
0 2

I = U t1 e− jβz − U fx1 e jβz =
Z
Z
0

 =
U
t2

 +Z I
U
2
02
2

dmax = 2βθ + k λ2

0

 =
U
fx2

 −Z I
U
2 02
2


U
t2
Z0




e jβd − UZfx2 e− jβd = − j UZ 1 sin(βz) + I1cos(βz) = j UZ 2 sin(βd) + I2cos(βd)
0

0

1+ n
0 1 − n

Z(d ) = Z

dmin = 2βθ + λ4 + k λ2

0

Z2 + jZ0 tgβd
0 Z0 + jZ2 tgβd

=Z

SWR = 11+− nn2
2

Umax = Ut2(1+n2)

SWR − 1
n 2 = SWR
+1

Umin = Ut2 (1− n2 )

Imax =

 *}= 1 (U )2 − 1 (U )2
P = 12 Re{U.I
t
fx
2Z
2Z
0

0

n = n2∠θ − 2βd

n 2 = ZZ2 +−ZZ0 = n2∠θ
2

Umax
Z0
Z0 =

0

Imin =
L0
vp =
C0

;

Umin
Z0
c
εr µ r