Lect2 电磁场课件EM

TEM, TE and EM Waves
structures that have more than one conductor
may support TEM waves let us consider the a transmission line or a waveguide with its cross section being uniform along the z-direction

b a
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EE 41139
Microwave Technique
TEM, TE and EM Waves
the electric and magnetic fields can
be written as
E( x, y, z) [e t e t ( x, y ) ze z ( x, y )]e jz (1) H( x, y, z) [e t h t ( x, y ) zh z ( x, y )]e jz ( 2)
TEM, TE and EM Waves
we can define the wave impedance for the
TEM mode:
Z TEM Ex (16 ) Hy
i.e., the ratio of the electric field to the
EE 41139 Microwave Technique 19
why is TEM mode desirable?
a closed conductor cannot support
TEM wave as the static potential is either a constant or zero leading to et 0
k
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Microwave Technique
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TEM, TE and EM Waves
now let us consider the Helmholtz’s
2 2 2 2 2 2 ( k ) E x 0, k Ex 0 2 2 2 y z x
H z jH x jE y x
EE 41139 Microwave Technique 8
TEM, TE and EM Waves
H z E z jH x j ( jH x ) / ( j ) x y E z H z 2 2 ( k )H x j j y x E z H z Hx ( ) (9) y x k2 j
no dispersion, signals of different
frequencies travel at the same speed, no distortion of signals solution to Laplace’s equation is relatively easy
magnetic field, note that the components must be chosen such that E x H is pointing to the direction of propagation
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TEM, TE and EM Waves
2 V12 1 2 1 E dl (14)
while the current flowing on a
conductor is given by
EE 41139
I H dl (15 )
C
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Microwave Technique
H z jH y jEx , ( 6 ) y H z jH x jE y , ( 7) x H y H x jE z , ( 8)owave Technique
TEM, TE and EM Waves
each of the four transverse
2e t 0 (13) t
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Microwave Technique
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TEM, TE and EM Waves
Knowing that 2 et 0 and D t e t 0 t , we have 2( x, y) 0 t the voltage between two conductors is
given by
2 V12 1 2 1 E dl (14)
while the current flowing on a conductor is
given by
EE 41139
I H dl (15 )
C
Microwave Technique 16
Lecture 2
TEM, TE and TM Waves
Coaxial Cable
Grounded Dielectric Slab Waveguides Striplines and Microstrip Line Design Formulas of Microstrip Line
each of these structures is characterized
by a propagation constant and a characteristic impedance; if the line is lossy, attenuation is also needed
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2e t 0 (13) t
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Microwave Technique
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TEM, TE and EM Waves
Knowing that
2 t et 0
and
D t e t 0
, and we have
2( x, y) 0 t
equations can be written as E jH, H jE Therefore,
E z jE y jH x , ( 3) y E z jEx jH y , ( 4 ) x E y Ex jH z , ( 5 ) x y
equation
2 2 note that z 2 and therefore, for TEM
wave, we have
2 2 Ex 0 2 y 2 x
Microwave Technique 12
EE 41139
TEM, TE and EM Waves
components can be written in terms of and , e.g., consider Eqs. (3) and (7): E
y
z jE jH y x
H z jH x jE y x
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TEM, TE and EM Waves
each of the four transverse
components can be written in terms of and , e.g., consider Eqs. (3) and (7): E
y
z jE jH y x
TEM, TE and EM Waves
this is also true for
E y, therefore, the transverse components of the electric field (so as the magnetic field) satisfy the two-dimensional Laplace’s equation
Where et and ht
are the transverse components and et and ht are the longitudinal components
Microwave Technique 5
EE 41139
TEM, TE and EM Waves
in a source free region, Maxwell’s
for TEM field, the E and H are related
by
h ( x, y ) 1 Z TEM z e( x, y ) (17)
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Microwave Technique
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why is TEM mode desirable?
cutoff frequency is zero
EE 41139
Microwave Technique
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Lecture 2
An Approximate Electrostatic
Solution for Microstrip Line The Transverse Resonance Techniques Wave Velocities and Dispersion
if a waveguide has more than 1
dielectric, TEM mode cannot exists as cannot be zero in all regions
c
j j
E z H z Ey ( ) ( 12) 2 y x k
c

k c is called the cutoff wavenumber
Microwave Technique 10