8
Transmission line
Dr. Maha Raouf
2022/2023
Introduction to waveguides
Introduction to waveguides
Coaxial Cable Waveguides
Rectangular Elliptical
Circular
Planar Transmission Lines
Microstrip lines Striplines
Propagation Modes
Coaxial Cable Waveguides
Transverse Electric and Magnetic mode Transverse Electric mode Transverse Magnetic mode
(TEM mode) (TE mode) (TM mode)
Hz≠0, Ez=0 Hz=0, Ez≠0
Hz=Ez=0
Propagation Modes
Microstrip lines Striplines
Transverse Electric and Magnetic mode Transverse Electric and Magnetic mode
(Quasi TEM mode) (TEM mode)
Hz=Ez=0
Comparison
Characteristics Coax Waveguide Stripline Microstrip
Propagation modes TEM TE, TM TEM Quasi TEM
Bandwidth High Low High High
Loss Medium Low High High
Power Capacity Medium High Low Low
Physical Size Large Very Large Medium Small
Fabrication Medium Medium Easy Very Easy
Component Integration Hard Hard Easy Easy
General Solutions For
different modes in waveguides
General Solutions For TEM, TE, And TM Waves
We will find general solutions to Maxwell’s equations for the specific cases of TEM, TE, and TM
wave propagation in cylindrical transmission lines or waveguides
Assume time-harmonic fields with an 𝒆𝒋𝝎𝒕 dependence and wave propagation along the z-axis
The electric and magnetic fields
Where
• ത 𝑦) represent the transverse (𝑥,
𝑒ҧ 𝑥, 𝑦 and ℎ(𝑥, ො 𝑦)
ො electric and magnetic field components
• 𝑒𝑧 and ℎ𝑧 are the longitudinal electric and magnetic field components
• The wave propagation (+z or –z )direction can be adjusted by replaced –β by +β
• If conductor or dielectric loss is present, the propagation constant will be complex; jβ replaced with
γ = α + jβ
General Solutions For TEM, TE, And TM Waves
Assuming that the transmission line or waveguide region is source free, we can write Maxwell’s
equations as
Solving Plan:
- Represent transverse component
(Ex , Ey ,Hx ,Hy) in term of
longitudinal component (Ez ,Hz)
General Solutions For TEM, TE, And TM Waves
Where
cutoff wave number
wave number of the material filling the transmission line or
waveguide region
General Solutions For TEM, TE, And TM Waves
TEM Waves
Transverse electromagnetic (TEM) waves are characterized by Ez = Hz = 0.
0
0
The cutoff wave number
𝐾𝑐2 = 0 (𝐾 2 = 𝛽 2 )
Transverse fields are also all zero, in which case
we have an indeterminate result
General Solutions For TEM, TE, And TM Waves
The Helmholtz wave equation for Ex
similar result also applies to Ey
𝐸ത 𝑥, 𝑦, 𝑧 = 𝑒ҧ 𝑥, 𝑦 𝑒 −𝑗𝛽𝑧
General Solutions For TEM, TE, And TM Waves
ℎ𝑧 = 0 𝑓𝑜𝑟 𝑇𝐸𝑀 𝑚𝑜𝑑𝑒
So 𝒆ത is conservative field and the transverse fields of a TEM wave are thus the same as the static fields
In the electrostatic case, we know that the electric field can be expressed as the gradient of a scalar
potential, ∅(𝒙, 𝒚)
The voltage between two conductors
The current flow on a given conductor can be found from Ampere’s law
General Solutions For TEM, TE, And TM Waves
The wave impedance of a TEM mode
Combining the results of ZTEM gives a general expression for the transverse fields as
Important Notes:
TEM waves can exist when two or more conductors are present.
Plane waves are also examples of TEM waves
Characteristic impedance, Z0, relates traveling voltage and current and is a function of the line
geometry as well as the material filling the line
Wave impedance relates transverse field components and is dependent only on the material
constants
General Solutions For TEM, TE, And TM Waves
The procedure for analyzing a TEM line can be summarized as follows:
1. Solve Laplace’s equation, for ∅(𝑥, 𝑦). The solution will contain several unknown constants.
2. Find these constants by applying the boundary conditions for the known voltages on the
conductors.
3. Compute 𝑒ҧ , 𝐸ത , ℎത , and 𝐻
ഥ
4. Compute V and I .
5. Find the propagation constant, and the characteristic impedance is given by Z0 = V/I .
General Solutions For TEM, TE, And TM Waves
TE Waves
Transverse electric (TE) waves (H waves) are characterized by 𝑬𝒛 = 𝟎 and 𝑯𝒛 ≠ 𝟎.
General Solutions For TEM, TE, And TM Waves
From the Helmholtz wave equation
This equation must be solved subject to the boundary conditions of the specific guide geometry.
The TE wave impedance can be found as
General Solutions For TEM, TE, And TM Waves
TM Waves
Transverse magnetic (TM) waves (E waves) are characterized by 𝑯𝒛 = 𝟎 and 𝑬𝒛 ≠ 𝟎.
General Solutions For TEM, TE, And TM Waves
From the Helmholtz wave equation
This equation must be solved subject to the boundary conditions of the specific guide geometry.
The TM wave impedance can be found as
General Solutions For TEM, TE, And TM Waves
The procedure for analyzing TE and TM waveguides can be summarized as follows:
1. Solve the reduced Helmholtz equation, for hz or ez . The solution will contain several unknown
constants and the unknown cutoff wave number, kc.
2. Find the transverse fields from hz or ez .
3. Apply the boundary conditions to the appropriate field components to find the unknown
constants and kc.
4. The propagation constant is given by β and the wave impedance by ZTE or ZTM.
Microstrip line
Microstrip line
In most practical applications, however, the dielectric
substrate is electrically very thin (d ≪ λ), and so the fields are
quasi-TEM.
Then the phase velocity and propagation constant can be
expressed as:
Where
𝝐𝒆 the effective dielectric constant of the microstrip line
Microstrip line
Formulas for Effective Dielectric Constant, Characteristic Impedance, and Attenuation
The effective dielectric constant of a microstrip line is given approximately by
The characteristic impedance:
Microstrip line
Design equations for Microstrip line:
Microstrip line
Attenuation due to dielectric loss:
𝛼𝑑(𝑚𝑖𝑐𝑟𝑜𝑠𝑡𝑟𝑖𝑝) = 𝛼𝑑 ∗ 𝑓𝑖𝑙𝑙𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟
𝜖𝑟 (𝜖𝑒 − 1)
𝐹𝑖𝑙𝑙𝑖𝑛𝑔 𝑓𝑎𝑐𝑡𝑜𝑟 =
𝜖𝑒 (𝜖𝑟 − 1)
The attenuation due to conductor loss
surface resistivity of the conductor
Microstrip line
Example:
Solution
𝑊 8𝑒 𝐴
= 2𝐴
𝑑 𝑒 −2
𝑍0 𝜖𝑟 + 1 𝜖𝑟 − 1 0.11 𝑊ൗ = 0.9654
𝐴= + 0.23 + = 2.142 𝑑
60 2 𝜖𝑟 + 1 𝜖𝑟
So, the condition W/d> 2 is satisfied
Microstrip line
Microstrip line
The total loss on Microstrip line:
𝜖𝑒 = 6.665 𝜖𝑟 = 9.9 𝐾0 = 209.4 𝑚−1 tan 𝛿 = 0.001
αd = 0.255 Np/m
αC = 0.0108 Np/cm
𝑙 = 8.72 𝑚𝑚 The total loss on line is 0.0116412 Np
