12. Draw the field configuration of dominant mode in rectangular waveguide.

12. Draw the field configuration of dominant mode in rectangular waveguide.

   March 10, 2014       legend

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The dominant mode in rectangular waveguide is TE₁₀
The field pattern is as shown below

The radiation pattern of TE₁₀ mode is as follows

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11. what is waveguide cutoff frequency , guide wavelength , phase velocity , group velocity , propagation constant

11. what is waveguide cutoff frequency , guide wavelength , phase velocity , group velocity , propagation constant

   March 05, 2014       legend

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Lets explain cutoff frequency , guide wavelength , group velocity , phase velocity and propagation constant of a waveguide.

Cutoff frequency :

Cutoff frequency is the frequency below which attenuation occurs and above which propagation takes place.Each mode have a specific cutoff frequency.

For TE_mn modes the cutoff frequency is given by
\[f_{c}=\frac{1}{2\sqrt{\mu \varepsilon}}\sqrt{{(\frac{m}{a})}^{2}+{(\frac{n}{b})}^{2}}\]
Guide Wavelength :

It is the distance traveled by the wave in order to undergo a phase shift of 2π radians.
It is related to propagation constant β as

\[\lambda_{b}=\frac{2\pi}{\beta}\]
Wavelength in waveguide is different from wavelength in free space.

Relation between cutoff frequency and guide wavelength :

The relationship between two is as follows
\[\frac{1}{\lambda_{0}^{2}}=\frac{1}{\lambda_{g}^{2}}+\frac{1}{\lambda_{c}^{2}}\]
Also it can be written as
\[\lambda_{g}=\frac{\lambda_{0}}{\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}}\Big)^{2}}\]
☞ When ⋋₀ ≪ ⋋_c , then ⋋_g= ⋋₀
☞ When ⋋₀ = ⋋_c , then ⋋_g becomes ∞  
☞ When ⋋₀ > ⋋_c , then ⋋_g becomes imaginary ,that means no propagation in the waveguide

Where ⋋₀ is the free space wavelength.

Phase velocity :

The phase velocity of a wave is the rate at which the phase of the wave propagates in space.
The phase velocity is given by
\[v_{p}=\frac{\omega}{k}\]
Where k = wave number

Also it is given as
\[v_{p}=\frac{c}{\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}}\]

Group velocity :

If there is modulation in carrier, the modulation envelope travels at a velocity slower than the carrier. This velocity of the modulation envelope is called as group velocity.

Or in other words

The group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space.

It is given as
\[v_{g}=\frac{\partial \omega}{\partial k}\]
Also it is given as
\[v_{g}=c\sqrt{1-\Big(\frac{\lambda_{0}}{\lambda_{c}}\Big)^{2}}\]

Relationship between group velocity and phase velocity :
\[v_{p}v_{g}=c^{2}\]

Propagation constant :

The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction

For a TE_mn mode it is given by


\[\beta=\sqrt{\mu\epsilon}\sqrt{\omega^{2}-\omega_c^2}\]

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10. What is rectangular waveguide

10. What is rectangular waveguide

   February 27, 2014       legend

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Rectangular waveguide is one of the types of waveguides.The shape of it is a hollow metallic tube with rectangular cross section as shown in the figure with dimension 'a' along x axis and dimension 'b' along y axis.

 

♦ Electric and magnetic fields of the signal is confined within the waveguide and no power is lost.

♦ Since normally waveguide are air filled a dielectric loss exists but negligible.Also some power lost as heat at walls of the waveguide but small.

♦ Possible to send several electromagnetic waves simultaneously through waveguide

♦ A rectangular waveguide supports TE and TM modes but not TEM waves

♦  A given waveguide have a cut off frequency for each mode, below which attenuation takes place.

♦ The dominant mode in rectangular waveguide is TE₁₀

♦ The mode supported by the waveguide is determined by the dimensions of the waveguide and the dielectric inside the waveguide.

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35. Explain about Ideal Low Pass Filter (ILPF) in frequency domain.

35. Explain about Ideal Low Pass Filter (ILPF) in frequency domain.

   February 27, 2014       legend

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Lowpass Filter:

The edges and other sharp transitions (such as noise) in the gray levels of an image contribute significantly to the high-frequency content of its Fourier transform. Hence blurring (smoothing) is achieved in the frequency domain by attenuating us the transform of a given image.

G (u, v) = H (u, v) F(u, v)

where F (u, v) is the Fourier transform of an image to be smoothed. The problem is to select a filter transfer function H (u, v) that yields G (u, v) by attenuating the high-frequency components of F (u, v). The inverse transform then will yield the desired smoothed image g (x, y).

Ideal Filter:

A 2-D ideal lowpass filter (ILPF) is one whose transfer function satisfies the relation

\[H(u,v)=\begin{cases}1 & D(u,v) \leq D_{0}\\0 & D(u,v) > D_{0}\end{cases}\]

where D is a specified non negative quantity, and D(u, v) is the distance from point (u, v) to the
origin of the frequency plane; that is,
\[D(u,v)=\sqrt{(u^{2}+v^{2})}\]
Figure 3 (a) shows a 3-D perspective plot of H (u, v) u a function of u and v. The name ideal filter indicates that oil frequencies inside a circle of radius D₀ are passed with no attenuation, whereas all frequencies outside this circle are completely attenuated.

Fig.3 (a) Perspective plot of an ideal lowpass filter transfer function; (b) filter cross
section.

The lowpass filters are radially symmetric about the origin. For this type of filter, specifying a cross section extending as a function of distance from the origin along a radial line is sufficient, as Fig. 3 (b) shows. The complete filter transfer function can then be generated by rotating the cross section 360 about the origin. Specification of radially symmetric filters centered on the N x N frequency square is based on the assumption that the origin of the Fourier transform has been centered on the square.

For an ideal lowpass filter cross section, the point of transition between H(u, v) = 1 and H(u, v) = 0 is often called the cutoff frequency. In the case of Fig.3 (b), for example, the cutoff frequency is Do. As the cross section is rotated about the origin, the point Do traces a circle giving a locus of cutoff frequencies, all of which are a distance Do from the origin. The cutoff frequency concept is quite useful in specifying filter characteristics. It also serves as a common base for comparing the behavior of different types of filters.

The sharp cutoff frequencies of an ideal lowpass filter cannot be realized with electronic components, although they can certainly be simulated in a computer.

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9. What is degenerate mode in waveguide.

9. What is degenerate mode in waveguide.

   February 25, 2014       legend

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In a waveguide when two or more modes have the same cut off frequency then they are said to be degenerate modes.

In a rectangular waveguide the TE_mn and TM_mn with m ≠ 0 and n ≠ 0 are degenerate modes.

For a square waveguide for which a = b, all the TE_pq, TE_qp, TM_pq, TM_qp modes are degenerate.

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8. what is dominant mode in waveguides

8. what is dominant mode in waveguides

   February 23, 2014       legend

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For a waveguide the dominant mode is the mode with the lowest cut off frequency.

Single mode propagation reduce losses, so we use dominant mode propagation.The waveguide can be operated only in the dominant mode for a certain range of frequency, this range of frequency extend from cut off frequency of dominant mode to the cut off frequency of next higher mode.

In rectangular waveguide the dominant mode is TE₁₀, and for circular waveguide the dominant mode is TE₁₁.

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7. what are modes in waveguides

7. what are modes in waveguides

   February 23, 2014       legend

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Lets explain what are modes in waveguide mean.For every type of waveguide the electromagnetic waves inside the waveguide can have an infinite number of distinct electromagnetic field patterns or configurations, these distribution of electric and magnetic fields in a waveguide is called modes.

The characteristics of these modes depend upon the cross-sectional dimensions of the conducting waveguide, the type of dielectric material inside the waveguide, and the frequency of operation. Waveguide modes are typically classed according to the nature of the electric and magnetic field components E_z and H_z.These components are called the longitudinal components of the fields.So the types of modes in a waveguide are

✎ TE modes. Transverse-electric modes, sometimes called H modes. These modes have E_z = 0 and H_z ≠ 0 at all points within the waveguide, which means that the electric field vector is always perpendicular (i.e., transverse) to the waveguide axis. These modes are always possible in waveguides with uniform dielectrics.
✎ TM modes. Transverse-magnetic modes, sometimes called E modes. These modes have H_z = 0 and E_z ≠ 0 at all points within the waveguide, which means that the magnetic field vector is perpendicular (i.e., transverse) to the waveguide axis. Like TE modes, they are always possible in waveguides with uniform dielectrics.
✎ EH modes. EH modes are hybrid modes in which neither E_z nor H_z are zero, but the characteristics of the transverse fields are controlled more by E_z than H_z . These modes are often possible in waveguides with inhomogeneous dielectrics.
✎ HE modes. HE modes are hybrid modes in which neither E_z nor H_z are zero, but the characteristics of the transverse fields are controlled more by H_z than E_z . Like EH modes, these modes are often possible in waveguide with inhomogeneous dielectrics.
✎ TEM modes.Transverse-electromagnetic modes, often called transmission line modes. These modes can exist only when a second conductor exists within the waveguide, such as a center conductor on a coaxial cable. Because these modes cannot exist in single, closed conductor structures, they are not waveguide modes.

A waveguide will have a definite cut off frequency for each mode.Also it is possible to propagate several modes within a waveguide.

The different modes are named with subscript m and n, for example TE_mn where m is the number of half wave variations across x axis and n is the number of half wave variations across y axis.

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