Sketch the k - βdiagram for the infinite periodic structure shown below. Assume Z0=100 Ω, d= 1.0

step 1 It has been asked to sketch two directions. One is 0 1 1 bar 0 and the other one is 2 bar 2 bar

Sketch the k - βdiagram for the infinite periodic structure...

Key Concepts

Periodic Structures

Periodic structures consist of repeating units or cells that impart unique wave propagation characteristics. Their repetitive nature leads to effects such as band formation and bandgaps, which are essential in controlling and understanding how waves propagate through the medium.

Dispersion Diagram

A dispersion diagram graphically represents the relationship between wave frequency (or wavenumber) and the phase shift per unit cell (or propagation constant). It provides insight into the allowed and forbidden modes of energy propagation and is essential for analyzing the dynamic behavior of periodic systems.

Bloch-Floquet Theorem

The Bloch-Floquet theorem states that in a periodic structure, the wave solutions can be expressed as a product of a periodic function and a plane wave. This principle is fundamental in deriving the dispersion relation and understanding the propagation characteristics of infinite periodic systems.

Passbands and Stopbands

Passbands and stopbands are frequency ranges within which waves are allowed or forbidden to propagate, respectively, due to the periodicity of the structure. The dispersion diagram clearly indicates these regions, with passbands representing frequencies with real propagation constants and stopbands indicating evanescent or decaying modes.

Phase Shift Per Unit Cell

The phase shift per unit cell, often represented by the product ?d in periodic structures, quantifies the change in phase of the wave as it traverses a single unit. This parameter is crucial in developing the dispersion relation and directly influences the wave's overall propagation behavior in a periodic system.

Transcript

00:01 It has been asked to sketch two directions.

00:06 One is 0 1 1 bar 0 and the other one is 2 bar 2 bar 4 3 in a hexagonal unit cell.

00:20 So the above two directions can be plotted within a hexagonal unit cell.

00:26 So the above directions are plotted within a hexagonal unit cell having a reduced scale coordinate scheme.

00:34 For the direction 0 1 1 bar 0, so we will choose this reduced scale coordinate system or coordinate scheme to represent that it in a hexagonal unit cell.

00:55 Starting with the first direction that is 0 1 1 bar 0, we will first look into the projections on all the coordinate axes in terms of unit cell parameters and the axes are a1, a2, a3 and c.

01:11 So these are the four axes of a hexagonal unit cell.

01:16 The intercepts are or the projections are 0 1 1 minus 1 and 0.

01:24 The following are the steps involved plot the direction 0 1 1 bar 0.

01:34 First let us begin at the origin of the coordinate system that is at the point o.

01:44 So let this be the hexagonal unit cell with a reduced scale coordinate scheme and this is our origin proceed one unit along the a2 axis to point p.

02:05 So we move towards a2 axis one point that is point p.

02:10 Then from point p proceed one unit parallel in the direction parallel to minus a3 axis that means we have to move one unit to the direction along the direction minus a3 to a point q.

02:38 The 0 1 1 bar 0 direction is represented by a vector that extends from point o that is the origin to point q.

02:54 So if you draw a vector like this from o to q this would represent the direction 0 1 1 bar 0.

03:03 So coming to the second direction that is 2 bar 2 bar 4 3 the projections on the coordinate axis in terms of unit cell parameters are the projections are minus 2 along a1 minus 2 units along a2 4 units along in terms of unit cell parameters.

03:42 To draw the vector representing the direction 2 bar 2 bar 4 3 following are the steps involved...

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