Transmission Lines and Waveguides Mode of guided waves: Modes of wave Propagation along the Lines Transverse Electro-Magnetic (TEM) Wave Transverse Electric (TE) Wave , h-Wave Transverse Magnetic (TM) Wave, e-Wave Transmission Line or Waveguide region is source free: => Maxwell Curl Equations are

Solution for Wave Propagation Modes For time harmonic waves propagating along the lines( z-axis), Electric and magnetic field can be written as, in cartesian coordinate system (x,y,z),

TEM Waves

Wave Impedance of a TEM mode

Voltage: Potential Difference between two Conductors Current: from Ampere’s Circuital law, Characteristic Impedance

TE Wave (h-wave)

TE Wave (h-wave) Wave Impedance of a TE mode

TM Wave (e-wave)

TM Wave (e-wave) Wave Impedance of a TM mode

Complex Propagation Constant (dielectric loss)

Parallel Plate Waveguide Boundary Condition TEM mode: TE mode: TM mode:

TEM mode: 1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

TEM mode: 2. Find Fields from Potential 3. Compute V and I

TEM mode: 4. Characteristic Impedance and Propagation

Time average Poynting Vector TEM mode: 5. Transmitted Power Time average Poynting Vector Time average Power transmitted to (+z) direction along the line,

TM mode: 1. Solve the scalar Helmholtz Eq. for axial electric field 2. Find Constants by Applying B.C.

3. Find transverse Field Components TM_n mode

Time average Poynting Vector Time average Power transmitted to (+z) direction along the line,

TE mode: 1. Solve the scalar Helmholtz Eq. for axial magnetic field 2. Find transverse Field Components TEn mode

3. Find Constants by Applying B.C. on

Time average Poynting Vector Time average Power transmitted to (+z) direction along the line,

Cut-off frequency for TM and TE mode Minimum Cut-off Frequency

Rectangular Waveguide Rectangular Waveguide can’t propagate TEM waves Propagate only TE & TM wave For TEM, With Boundary Condition, Equipotential Surface (a Conductor Surface) Rectangular Waveguide can’t propagate TEM waves

Rectangular Waveguide (TM modes) Scalar Wave Equation for electric field axial component Separation of variables

Boundary Condition

The TM mode with lowest cutoff frequency: TM11 lowest cutoff frequency of TM11: Wave Impedance of TM mode

Rectangular Waveguide (TE modes) Scalar Wave Equation for magnetic field axial component B.C on tangential electric fields:

Wave Impedance of TE mode

The TE mode with lowest cutoff frequency: TE10 lowest cutoff frequency of TE10: TE10 +TE11 +TM11 Only TE10 No propagation The Dominant Mode of Rectangular Waveguide is TE10 Only TE10 mode can propagate when

Dominant Mode TE10 Field Components

Dominant Mode TE10 Time average Poynting Vector Time average Power transmitted to (+z) direction along the line,

Coaxial Line Boundary Condition TEM mode can propagate TEM mode: TE mode: TM mode:

TEM mode: 1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

TEM mode: 2. Find Fields from Potential

Wave Impedance 3. Compute V and I Characteristic Impedance

Scalar Wave Equation for magnetic field axial component Higher Order Mode (TE mode): Scalar Wave Equation for magnetic field axial component Separation of variables

Bessel’s Differential Equation 1st kind 2nd kind

( ) , = ¢ - Þ ú û ù ê ë é + F µ a k Y b J solution nontrivial for D C , = ¢ - Þ ú û ù ê ë é + F µ a k Y b J solution nontrivial for D C e c n f r Can be determined Approximate Solution for n=1

Circular Waveguide can’t propagate TEM waves Propagate only TE & TM wave For TEM, With Boundary Condition, Equipotential Surface (a Conductor Surface) Circular Waveguide can’t propagate TEM waves

Scalar Wave Equation for axial component --same with Higher order mode of Coaxial Line

B.C on tangential electric fields: 1. TE mode

B.C on tangential electric fields: 2. TM mode

Dominant Mode of the Circular Waveguide:TE11 Wave Impedance

Stripline Boundary Condition TEM mode can propagate

TEM mode: 1. Solve the Laplace Equation for Electrostatic Potential with Boundary Condition

General solution for B.C at y=b/2 can be written as linear combination of general solutions each region.

On Strip, Equipotential as V0 Not exact Erroneous Approx.

On Strip, Total Charge On Strip of unit length,

On Strip, given surface charge density Not exact, Reasonable Approx.

Characteristic Impedance

Phase unbalance for TEM mode Microstripline Phase unbalance for TEM mode 1. TEM mode can’t propagate 2. Hybrid TE-TM mode -> Quasi TEM mode Approx. when d<<wavelength

Fourier Series Solution Quasi TEM mode Approx. Fourier Series Solution

On Strip,

On Strip, given surface charge density

Characteristic impedance for microstrip transmission lines (assumes nonmagnetic dielectric)

Wave Velocity and Dispersion - Distortionless transmission - Transfer Function for Distortionless transmission Distortion Amplitude distortion Phase distortion - Dispersion

Wave Velocity and Dispersion f(t): Band limited signal with highest freq. fm

Wave-front for TEM Energy Transmission

Power transmitted along the Line Standing Wave Across the Line For TEm0 mode

Definition of Voltage and Current for Network Analysis and Synthesis 1)TEM mode

TE10 of Rectangular Waveguide 2)non-TEM mode TE10 of Rectangular Waveguide Transverse Electric Field for TE10 can’t define unique Voltage & Current waves

For a non-TEM mode with wave impedance Zw, Transverse Fields are Define Voltage wave proportional to Transverse Electric field, Current wave proportional to Transverse Magnetic field. Not Unique Definition

Forward power transmission Arbitrary

The Concept of Impedance 1. Intrinsic Impedance of the Medium the ratio of Electric Field to Magnetic Field 2. Wave Impedance of the particular Mode the ratio of transverse Electric Field to transverse Magnetic Field 3. Characteristic Impedance of Transmission Line the ratio of Voltage Wave to Current Wave Impedance at some point of Transmission Line the Ratio of Voltage to Current