Lecture 2

Review lecture 1 Wavelength: Phase velocity: Characteristic impedance: Kerchhoff’s law Wave equations or Telegraphic equations L, R, C, G ?

Derive the transmission line parameters (L, R, C, G) in terms of the electromagnetic fields Retrieve the telegrapher equations using these parameters  1.4 Field analysis of transmission lines Example: Voltage : V 0 e  j  z Current: I 0 e  j  z

Work (W) and power (P) H* multiplies the two sides of the first Maxell’s equation: E multiplies the two sides of the conjugated second Maxell’s equation: Add the above two equations and utilize We obtain (J=Js+σE): Integrate the above formula in volume V and utilize divergence theory, we have the following after reorganize the equation

Poynting law: Source power: Output power: Loss power: Stored magnetic energy: Stored electric energy: R, G L C

Calculate the time-average stored magnetic energy in an isotropic medium ( the results are valid for any media ) Calculate magnetic energy

Transmission line parameter: L The time-average stored magnetic energy for 1 m long transmission line is And circuit line gives. Hence the self inductance could be identified as

Transmission line parameter: C Similarly, the time-average stored electric energy per unit length can be found as Circuit theory gives, resulting in the following expression for the capacitance per unit length:

Transmission line parameter: R The power loss per unit length due to the finite conductivity of the metallic conductors is The circuit theory gives, so the series resistance R per unit length of line is (R s = 1/  is the surface resistance and  is the skin depth, H is the tangential field)

Transmission line parameter, G The time-average power dissipated per unit length in a lossy dielectric is Circuit theory gives, so the shunt conductance per unit length can be written as

Transmission line parameters for different line types

From fields to circuit theory

Derive Telegrapher Equations from Field Analysis x y aρ  b μ,μ, The fields inside the coaxial line will satisfy Maxwell's curl equations Expanding the above equations in cylindrical coordinates and then gives the following vector equations （ TEM waves ， E z = H z = 0, no ϕ -dependence ）

Using the above equations, we obtain Eliminate h(z) and g(z) Voltage between the two conductors: Current on the inner conductor: Substitute the coefficients by L, G, C The same telegrapher equations as derived from distributed theory. x y aρ  b μ,μ,

Waves in lossless coaxial waveguides Helmholtz equation Electric and magnetic fields (TEM): For one-way propagation, eg, along +z axis

Voltage between the two conductors at z = 0 Current on the inner conductor (Ampere’s circuit law) at z = 0: Waves in lossless coaxial waveguides

Propagation Constant, impedance and Power Flow for the Lossless coaxial Line Propagation constant: (for a lossless medium) TEM transmission lines have the same form of propagation constant as that for plane waves in a lossless medium. Characteristic impedance: Power flow (computed from the Poynting vector): (Match the circuit theory) Wave impedance:

Homework 1. The fields of a traveling TEM wave inside the coaxial line shown left can be expressed as where  is the propagation constant of the line. The conductors are assumed to have a surface resistivity R s, and the material filling the space between the conductors is assumed to have a complex permittivity  =  ’ - j  " and a permeability μ = μ 0 μ r. Determine the transmission line parameters (L,C,R,G). x y aρ φ b μ,μ, 2. For the parallel plate line shown left, derive the R, L, G, and C parameters. Assume w >> d. d rr  y z x w

Surface resistance and surface current of metal Energy entering a conductor: The contribution to the integral from the surface S can be made zero by proper selection of this surface (Snell law --> refraction angle ≅ 0). Therefore, From vector identity, we have The energy absorbed by a conductor: metal dielectric Evanescent

Grad, Div and Curl in Cylindrical Coordinates