# Chapter 2: Transmission lines and waveguides - PowerPoint PPT Presentation

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Chapter 2: Transmission lines and waveguides. 2.1 Generation solution for TEM, TE and TM waves 2.2 Parallel plate waveguide 2.3 Rectangular waveguide 2.4 Circular waveguide 2.5 Coaxial line 2.6 Surface waves on a grounded dielectric slab 2.7 Stripline 2.8 Microstrip

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Chapter 2: Transmission lines and waveguides

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Chapter 2: Transmission lines and waveguides

2.1 Generation solution for TEM, TE and TM waves

2.2 Parallel plate waveguide

2.3 Rectangular waveguide

2.4 Circular waveguide

2.5 Coaxial line

2.6 Surface waves on a grounded dielectric slab

2.7 Stripline

2.8 Microstrip

2.9 Wave velocities and dispersion

2.10 Summary of transmission lines and waveguids

### 2.1 Generation solution for TEM, TE and TM waves

Closed waveguide

Two-conductor TL

Electromagnetic fields (time harmonic eiωt and propagating along z axis):

where e(x,y) and h(x,y) represent the transverse (x,y) E and H components, while ezand hzare the longitudinal E and H components.

In the case of source free, Maxwell’s equations can be written as:

With the e-iβz dependence, the above vector equations can be divided into six component equations and then solve the transverse fields in terms of the longitudinal components Ez amd Hz:

The cutoff frequency:

(1) TEM waves (Ez = Hz = 0)

0

0

• Propagation constant:

(kc = 0, no cutoff)

• The Helmholtz equation for Ex:

For a e-jz dependence, and the above equation can be simplified

Laplace equations, equal to static fields

Similarly,

• Transverse magnetic field:

0

• Wave impedance:

0

• Note:

• Wave impedance, Z

• relates transverse field components and is dependent only on the material constant.

• For TEM wave

• Characteristic impedance of a transmission line, Z0:

• relates an incident voltage and current and is a function of the line geometry as well as the material filling the line.

• For TEM wave: Z0= V/I

(V incident wave voltage

I incident wave current)

(2) TE waves (Ez = 0 and Hz0)

• The field components can be simplified as:

•  is a function of frequency and TL/WG structure

• Solve Hz from the Helmholtz equation

Because , then

where . Boundaries conditions will be used to solve the above equation.

• TE wave impedance:

(3) TM waves (Hz = 0 and Ez0)

• The field components can simplified:

•  is a function of frequency and TL/WG structure

• Solve Ez from Helmholtz equation:

Because , then

where . Boundaries conditions will be used to solve the above equation.

• TE wave impedance:

(4) Attenuation due to dielectric loss

Total attenuation constant in TL or WG = c + d.

c: due to conductive loss; calculated using the perturbation method; must be evaluated separately for each type.

d: due to the dielectric loss; calculated from the propagation constant.

Taylor expansion (tan << 1)

2.2 Parallel plate waveguide

w >> d (fringing fields and any x variation could be ignored)

• Formed from two flat plates or strips

• Probably the simplest type of guide

• Support TEM, TE and TM modes

• Important for practical reasons.

(a) TEM modes (Ez = Hz = 0)

• Laplace equation for the electric potential  (x,y)

for

Boundary conditions:

• The transverse field , so that we have

• Characteristic impedance:

• Phase velocity:

(b) TM modes (Hz = 0)

• The transverse ez(x,y) satisfies

Bn = 0

Boundary conditions: ez(x,y) = 0 at y = 0, d.

k > kc traveling wave

k = kc ?

k < kcevanescent wave

Cutoff frequency

• Propagation constant:

• The field components:

Wave impedance:

Power flow:

(n > 0)

(c) TE modes (Ez = 0)

An = 0

The transverse hz(x,y) satisfies

And boundary Ex(x,y) = 0 at y = 0, d.

where the propagation constant

• Wave impedance:

• Cutoff frequency :

Parallel plate waveguide

TEM

TM

TE