TEM, TE and TM Waves

1
Lecture 2

• TEM, TE and TM Waves
• Coaxial Cable
• Grounded Dielectric Slab Waveguides
• Striplines and Microstrip Line
• Design Formulas of Microstrip Line

2
Lecture 2

• An Approximate Electrostatic Solution for
  Microstrip Line
• The Transverse Resonance Techniques
• Wave Velocities and Dispersion

3
TEM, TE and EM Waves

• transmission lines and waveguides are primarily
  used to distribute microwave wave power from one
  point to another
• each of these structures is characterized by a
  propagation constant and a characteristic
  impedance if the line is lossy, attenuation is
  also needed

4
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

5
TEM, TE and EM Waves

• the electric and magnetic fields can be written
  as
• Where and are the transverse
  components and and are the
  longitudinal components

6
TEM, TE and EM Waves

• in a source free region, Maxwells equations can
  be written as
• Therefore,

7
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)

8
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)

9
TEM, TE and EM Waves
10
TEM, TE and EM Waves

• Similarly, we have
• is called the cutoff wavenumber

11
TEM, TE and EM Waves

• Transverse electromagnetic (TEM) wave implies
  that both and are zero (TM,
  transverse magnetic,
• 0, 0 TE, transverse electric,
• 0, 0)
• the transverse components are also zero unless
  is also zero, i.e.,

12
TEM, TE and EM Waves

• now let us consider the Helmholtzs equation
• note that and therefore, for TEM
  wave, we have

13
TEM, TE and EM Waves

• this is also true for , therefore, the
  transverse components of the electric field (so
  as the magnetic field) satisfy the
  two-dimensional Laplaces equation

14
TEM, TE and EM Waves

• Knowing that and
• , and we have
• while the current flowing on a conductor is given
  by

15
TEM, TE and EM Waves

• this is also true for , therefore, the
  transverse components of the electric field (so
  as the magnetic field) satisfy the
  two-dimensional Laplaces equation

16
TEM, TE and EM Waves

• Knowing that and
• , we have
• the voltage between two conductors is given by
• while the current flowing on a conductor is given
  by

17
TEM, TE and EM Waves

• we can define the wave impedance for the TEM
  mode
• i.e., the ratio of the electric field to the
  magnetic field, note that the components must be
  chosen such that E x H is pointing to the
  direction of propagation

18
TEM, TE and EM Waves

• for TEM field, the E and H are related by

19
why is TEM mode desirable?

• cutoff frequency is zero
• no dispersion, signals of different frequencies
  travel at the same speed, no distortion of
  signals
• solution to Laplaces equation is relatively easy

20
why is TEM mode desirable?

• a closed conductor cannot support TEM wave as the
  static potential is either a constant or zero
  leading to
• if a waveguide has more than 1 dielectric, TEM
  mode cannot exists as cannot be zero in all
  regions

21
why is TEM mode desirable?

• sometime we deliberately want to have a cutoff
  frequency so that a microwave filter can be
  designed

22
TEM Mode in Coaxial Line

• a coaxial line is shown here
• the inner conductor is at a potential of Vo volts
  and the outer conductor is at zero volts

23
TEM Mode in Coaxial Line

• the electric field can be derived from the scalar
  potential F in cylindrical coordinates, the
  Laplaces equation reads
• the boundary conditions are

24
TEM Mode in Coaxial Line

• use the method of separation of variables, we let
  
• substitute Eq. (21) to (18), we have
• note that the first term on the left only depends
  on r while the second term only depends on f

25
TEM Mode in Coaxial Line

• if we change either r or f, the RHS should remain
  zero therefore, each term should be equal to a
  constant

26
TEM Mode in Coaxial Line

• now we can solve Eqs. (23) and (24) in which only
  1 variable is involved, the final solution to Eq.
  (18) will be the product of the solutions to Eqs.
  (23) and (24)
• the general solution to Eq. (24) is

27
TEM Mode in Coaxial Line

• boundary conditions (19) and (20) dictates that
  the potential is independent of f, therefore
  must be equal to zero and so as
• Eq. (23) is reduced to solving

28
TEM Mode in Coaxial Line

• the solution for R(r) now reads

29
TEM Mode in Coaxial Line

• the electric field now reads
• adding the propagation constant back, we have

30
TEM Mode in Coaxial Line

• the magnetic field for the TEM mode
• the potential between the two conductors are

31
TEM Mode in Coaxial Line

• the total current on the inner conductor is
• the surface current density on the outer
  conductor is

32
TEM Mode in Coaxial Line

• the total current on the outer conductor is
• the characteristic impedance can be calculated as

33
TEM Mode in Coaxial Line

• higher-order modes exist in coaxial line but is
  usually suppressed
• the dimension of the coaxial line is controlled
  so that these higher-order modes are cutoff
• the dominate higher-order mode is mode,
  the cutoff wavenumber can only be obtained by
  solving a transcendental equation, the
  approximation is often
  used in practice

34
Surface Waves on a Grounded Dielectric Slab

• a grounded dielectric slab will generate surface
  waves when excited
• this surface wave can propagate a long distance
  along the air-dielectric interface
• it decays exponentially in the air region when
  move away from the air-dielectric interface

35
Surface Waves on a Grounded Dielectric Slab

• while it does not support a TEM mode, it excites
  at least 1 TM mode
• assume no variation in the y-direction which
  implies that
• write equation for the field in each of the two
  regions
• match tangential fields across the interface

36
Surface Waves on a Grounded Dielectric Slab

• for TM modes, from Helmholtzs equation we have
• which reduces to

37
Surface Waves on a Grounded Dielectric Slab

• Define

38
Surface Waves on a Grounded Dielectric Slab

• the general solutions to Eqs. (32) and (33) are
• the boundary conditions are
• tangential E are zero at x 0 and x
• tangential E and H are continuous at x d

39
Surface Waves on a Grounded Dielectric Slab

• tangential E at x0 implies B 0
• tangential E 0 when x implies C 0
• continuity of tangential E implies
• tangential H can be obtained from Eq. (10) with

40
Surface Waves on a Grounded Dielectric Slab

• tangential E at x0 implies B 0
• tangential E 0 when x implies C 0
• continuity of tangential E implies

41
Surface Waves on a Grounded Dielectric Slab

• continuity of tangential H implies
• taking the ratio of Eq. (34) to Eq. (35) we have

42
Surface Waves on a Grounded Dielectric Slab

• note that
• lead to
• Eqs. (36) and (37) must be satisfied
  simultaneously, they can be solved for by
  numerical method or by graphical method

43
Surface Waves on a Grounded Dielectric Slab

• to use the graphical method, it is more
  convenient to rewrite Eqs. (36) and (37) into the
  following forms

44
Surface Waves on a Grounded Dielectric Slab

• Eq. (39) is an equation of a circle with a radius
  of , each interception point
  between these two curves yields a solution

45
Surface Waves on a Grounded Dielectric Slab

• note that there is always one intersection point,
  i.e., at least one TM mode
• the number of modes depends on the radius r which
  in turn depends on the d and
• h has been chosen a positive real number, we can
  also assume that is positive
• the next TM will not be excited unless

46
Surface Waves on a Grounded Dielectric Slab

• In general, mode is excited if
• the cutoff frequency is defined as

47
Surface Waves on a Grounded Dielectric Slab

• once and h are found, the TM field components
  can be written as for

48
Surface Waves on a Grounded Dielectric Slab

• For
• similar equations can be derived for TE fields

49
Striplines and Microstrip Lines

• various planar transmission line structures are
  shown here

50
Striplines and Microstrip Lines

• the strip line was developed from the square
  coaxial

51
Striplines and Microstrip Lines

• since the stripline has only 1 dielectric, it
  supports TEM wave, however, it is difficult to
  integrate with other discrete elements and
  excitations
• microstrip line is one of the most popular types
  of planar transmission line, it can be fabricated
  by photolithographic techniques and is easily
  integrated with other circuit elements

52
Striplines and Microstrip Lines

• the following diagrams depicts the evolution of
  microstrip transmission line

53
Striplines and Microstrip Lines

• a microstrip line suspended in air can support
  TEM wave
• a microstrip line printed on a grounded slab does
  not support TEM wave
• the exact fields constitute a hybrid TM-TE wave
• when the dielectric slab become very thin
  (electrically), most of the electric fields are
  trapped under the microstrip line and the fields
  are essentially the same as those of the static
  case, the fields are quasi-static

54
Striplines and Microstrip Lines

• one can define an effective dielectric constant
  so that the phase velocity and the propagation
  constant can be defined as
• the effective dielectric constant is bounded by
  
• , it also depends on the
  slab thickness d and conductor width, W

55
Design Formulas of Microstrip Lines

• design formulas have been derived for microstrip
  lines
• these formulas yield approximate values which are
  accurate enough for most applications
• they are obtained from analytical expressions for
  similar structures that are solvable exactly and
  are modified accordingly

56
Design Formulas of Microstrip Lines

• or they are obtained by curve fitting numerical
  data
• the effective dielectric constant of a microstrip
  line is given by

57
Design Formulas of Microstrip Lines

• the characteristic impedance is given by
• for W/d 1
• For W/d 1

58
Design Formulas of Microstrip Lines

• for a given characteristic impedance and
  dielectric constant , the W/d ratio can be
  found as
• for
  W/dlt2

59
Design Formulas of Microstrip Lines

• 
  for W/d gt 2
• Where
• And

60
Design Formulas of Microstrip Lines

• for a homogeneous medium with a complex
  dielectric constant, the propagation constant is
  written as
• note that the loss tangent is usually very small

61
Design Formulas of Microstrip Lines

• Note that where x is
  small
• therefore, we have

62
Design Formulas of Microstrip Lines

• Note that
• for small loss, the phase constant is unchanged
  when compared to the lossless case
• the attenuation constant due to dielectric loss
  is therefore given by
• Np/m (TE or TM) (55)

63
Design Formulas of Microstrip Lines

• For TEM wave , therefore
• Np/m (TEM) (56)
• for a microstrip line that has inhomogeneous
  medium, we multiply Eq. (56) with a filling
  factor

64
Design Formulas of Microstrip Lines

• 
  (57)
• the attenuation due to conductor loss is given by
  
• (58) Np/m where
• is called the surface resistance of the
  conductor

65
Design Formulas of Microstrip Lines

• note that for most microstrip substrate, the
  dielectric loss is much more significant than the
  conductor loss
• at very high frequency, conductor loss becomes
  significant

66
An Approximate Electrostatic Solution for
Microstrip Lines

• two side walls are sufficiently far away that the
  quasi-static field around the microstrip would
  not be disturbed (a gtgt d)

67
An Approximate Electrostatic Solution for
Microstrip Lines

• we need to solve the Laplaces equation with
  boundary conditions
• two expressions are needed, one for each region

68
An Approximate Electrostatic Solution for
Microstrip Lines

• using the separation of variables and appropriate
  boundary conditions, we write

69
An Approximate Electrostatic Solution for
Microstrip Lines

• the potential must be continuous at yd so that
• note that this expression must be true for any
  value of n

70
An Approximate Electrostatic Solution for
Microstrip Lines

• due to fact that
• if m is not equal to n

71
An Approximate Electrostatic Solution for
Microstrip Lines

• the normal component of the electric field is
  discontinuous due to the presence of surface
  charge on the microstrip,

72
An Approximate Electrostatic Solution for
Microstrip Lines

• the surface charge at yd is given by
• assuming that the charge distribution is given by
  on the conductor and zero elsewhere

73

• multiply Eq. (63) by cos mpx/a and integrate from
  -a/2 to a/2, we have

74
An Approximate Electrostatic Solution for
Microstrip Lines

• the voltage of the microstrip wrt the ground
  plane is
• the total charge on the strip is

75
An Approximate Electrostatic Solution for
Microstrip Lines

• the static capacitance per unit length is
• this is the expression for

76
An Approximate Electrostatic Solution for
Microstrip Lines

• the effective dielectric is defined as
• , where is obtained from
  Eq. (64) with
• the characteristic impedance is given by

77
The Transverse Resonance Techniques

• the transverse resonance technique employs a
  transmission line model of the transverse cross
  section of the guide
• right at cutoff, the propagation constant is
  equal to zero, therefore, wave cannot propagate
  in the z direction

78
The Transverse Resonance Techniques

• it forms standing waves in the transverse plane
  of the guide
• the sum of the input impedance at any point
  looking to either side of the transmission line
  model in the transverse plane must be equal to
  zero at resonance

79
The Transverse Resonance Techniques

• consider a grounded slab and its equivalent
  transmission line model

80
The Transverse Resonance Techniques

• the characteristic impedance in each of the air
  and dielectric regions is given by
• and
• since the transmission line above the dielectric
  is of infinite extent, the input impedance
  looking upward at xd is simply given by

81
The Transverse Resonance Techniques

• the impedance looking downward is the impedance
  of a short circuit at x0 transfers to xd
• Subtituting
  , we have
• Therefore,

82
The Transverse Resonance Techniques

• Note that , therefore, we have
• From phase matching,
• which leads to
• Eqs. (65) and (66) are identical to that of Eq.
  (38) and (39)

83
Wave Velocities and Dispersion

• a plane wave propagates in a medium at the speed
  of light
• Phase velocity, , is the speed
  at which a constant phase point travels
• for a TEM wave, the phase velocity equals to the
  speed of light
• if the phase velocity and the attenuation of a
  transmission line are independent of frequency, a
  signal propagates down the line will not be
  distorted

84
Wave Velocities and Dispersion

• if the signal contains a band of frequencies,
  each frequency will travel at a different phase
  velocity in a non-TEM line, the signal will be
  distorted
• this effect is called the dispersion effect

85
Wave Velocities and Dispersion

• if the dispersion is not too severe, a group
  velocity describing the speed of the signal can
  be defined
• let us consider a transmission with a transfer
  function of

86
Wave Velocities and Dispersion

• if we denote the Fourier transform of a
  time-domain signal f(t) by F(w), the output
  signal at the other end of the line is given by
• if A is a constant and y aw, the output will be

87
Wave Velocities and Dispersion

• this expression states that the output signal is
  A times the input signal with a delay of a
• now consider an amplitude modulated carrier wave
  of frequency

88
Wave Velocities and Dispersion

• the Fourier transform of
  is given by
• note that the Fourier transform of s(t) is equal
  to

89
Wave Velocities and Dispersion

• The output signal , is given by
• for a dispersive transmission line, the
  propagation constant b depends on frequency, here
  A is assume to be constant (weakly depend on w)

90
Wave Velocities and Dispersion

• if the maximum frequency component of the signal
  is much less than the carrier frequencies, b can
  be linearized using a Taylor series expansion
• note that the higher terms are ignored as the
  higher order derivatives goes to zero faster than
  the growth of the higher power of

91
Wave Velocities and Dispersion

• with the approximation of

92
Wave Velocities and Dispersion

• Eq. (67) states that the output signal is the
  time-shift of the input signal envelope
• the group velocity is therefore defined as

93
Wave Velocities and Dispersion

• consider a grounded slab and its equivalent
  transmission line model