# Chapter 2 Summary

# Concepts

- A transmission line is a two-port network connecting a generator to a load. EM waves traveling on the line may experience ohmic power losses, dispersive effects, and reflections at the generator and load ends of the line. These transmission-line effects may be ignored if the line length is much shorter than λ.

- TEM transmission lines consist of two conductors that can support the propagation of transverse electromagnetic waves characterized by electric and magnetic fields that are transverse to the direction of propagation. TEM lines may be represented by a lumped-element model consisting of four line parameters (R′, L′, G′, and C′) whose values are specified by the specific line geometry, the constitutive parameters of the conductors and of the insulating material between them, and the angular frequency ω.

- Wave propagation on a transmission line, which is represented by the phasor voltage V˜(z) and associated current I˜(z), is governed by the propagation constant of the line, γ=α+j β, and its characteristic impedance Z 0. Both γ and Z 0 are specified by ω and the four line parameters.

- If R′=G′=0, the line becomes lossless (α=0). A lossless line is generally nondispersive, meaning that the phase velocity of a wave is independent of the frequency.

- In general, a line supports two waves, an incident wave supplied by the generator and another wave reflected by the load. The sum of the two waves generates a standing-wave pattern with a period of λ/2. The voltage standing-wave ratio S, which is equal to the ratio of the maximum to minimum voltage magnitude on the line, varies between 1 for a matched load (Z L=Z 0) to ∞ for a line terminated in an open circuit, a short circuit, or a purely reactive load.

- The input impedance of a line terminated in a short circuit or open circuit is purely reactive. This property can be used to design equivalent inductors and capacitors.

- The fraction of the incident power delivered to the load by a lossless line is equal to (1−|Γ|2).

- The Smith chart is a useful graphical tool for analyzing transmission-line problems and for designing impedance-matching networks.

- Matching networks are placed between the load and the feed transmission line for the purpose of eliminating reflections toward the generator. A matching network may consist of lumped elements in the form of capacitors and/or inductors, or it may consist of sections of transmission lines with appropriate lengths and terminations.

- Transient analysis of pulses on transmission lines can be performed using a bounce-diagram graphical technique that tracks reflections at both the load and generator ends of the transmission line.

# Mathematical and Physical Models

TEM Transmission Lines

L′C′=μ ε
G′C′=σ ε
α=ℜ e(γ)=ℜ e((R′+j ω L′)(G′+j ω C′)) (Np/m)
β=ℑ m(γ)=ℑ m((R′+j ω L′)(G′+j ω C′)) (rad/m)
Z 0=R′+j ω L′γ=R′+j ω L′G′+j ω C′ (Ω)
Γ=z L−1 z L+1

Step Function Transient Response

V 1+=V g Z 0 R g+Z 0
V∞=V g R L R g+R L
Γ g=R g−Z 0 R g+Z 0
Γ L=R L−Z 0 R L+Z 0

Lossless Line

α=0
β=ω L′C′
Z 0=L′C′
u p=1 μ ε(m/s)
λ=u p f=c f 1 ε r=λ 0 ε r
d max=θ r λ 4 π+n λ 2
d min=θ r λ 4 π+(2 n+1)λ 4
S=1+|Γ|1−|Γ|
P av=|V 0+|2 2 Z 0[1−|Γ|2]

# Important Terms

Provide definitions or explain the meaning of the following terms:

- admittance Y

- air line

- attenuation constant α

- bounce diagram

- characteristic impedance Z 0

- coaxial line

- complex propagation constant γ

- conductance G

- current maxima and minima

- dispersive transmission line

- distortionless line

- effective relative permittivity ε eff

- guide wavelength λ

- higher-order transmission lines

- impedance matching

- in-phase

- input impedance Z in

- load impedance Z L

- lossless line

- lumped-element model

- matched transmission line

- matching network

- microstrip line

- normalized impedance

- normalized load reactance x L

- normalized load resistance r L

- open-circuited line

- optical fiber

- parallel-plate line

- perfect conductor

- perfect dielectric

- phase constant β

- phase opposition

- phase-shifted reflection coefficient Γ d

- quarter-wave transformer

- short-circuited line

- single-stub matching

- slotted line

- Smith chart

- standing wave

- standing-wave pattern

- surface resistance R s

- susceptance B

- SWR circle

- telegrapher’s equations

- TEM transmission lines

- time-average power P av

- transient response

- transmission-line parameters

- two-wire line

- unit circle

- voltage maxima and minima

- voltage reflection coefficient Γ

- voltage standing-wave ratio (VSWR or SWR) S

- wave equations

- wave impedance Z(d)

- waveguide

- WTG and WTL