lecture5.2

Chapter V

Guided Waves and Wave Guides

Transverse Electromagnetic Waves (TEM):

As discuss earlier, for TEM waves E_z and H_z are zero and therefore from equations (11) to (14), we observe that all the components of fields will become zero inside waveguide unless h2 = 0 i.e.

---(16)

---(17)

Above eq. gives the same expression for the propagation constant as it was for plane wave in unbounded medium.

The velocity of propagation of TEM wave is,

The ratio of the fields E0

x and H0y is known as wave impedance and this ratio can be written from equation (9b)

and (10a)

by putting E0

z and H0z equal to zero.

Thus we have the wave impedance for TEM as,

Also we know

Using above, Eq. (19) can be written as,

---(20)

Which is same as intrinsic impedance of dielectric medium.

If we use E0

z and H0z equal to zero in Eq. (9a)

and (10b),

then we have, Using above two equations

---(21)

Combining (19) and (21), we get following formula for TEM wave propagating in +z direction,

It may be noted that the single conductor waveguides cannot support

TEM waves. The reason as follows: We know magnetic lines of force

always form the closed loops. If TEM wave exist inside the waveguide then B and H will form the closed loops in the transverse plane.

However, modified form of Ampere’s law says that the line integral of M.F. Over closed loop is equal to the sum of longitudinal conduction current and longitudinal displacement current through loop.

However without an inner conductor there is know longitudinal

conduction current inside the waveguide. Also by definition, the for TEM Waves E0Z is zero and therefore the value of longitudinal displacement current will also be zero for the TEM waves inside the waveguide.

Transverse Magnetic Waves: TM waves have zero H_z component. To study the behaviour of TM waves, we solve following Eq. for E_z component (subject to boundary conditions of guide),

Specially for E_z, we write

---(23)

---(24)

we get ---(25) ---(26) ---(27) ---(28) Combing (27) and (28), we get

---(29)

Where,

---(30) Eq (30) gives gradient of E0

The wave impedance for TM wave can be written as (using (25) to (28),

---(31)

Knowing the values of E0

x and E0y as discussed earlier, we can use above Eq. to find H0

y and H0x. For TM wave we can write

It may be noted that the solutions of Helmholtz equation for

A given waveguide with some specific boundary conditions exist

Only for discrete values of h. There may be infinity of discrete values Of h however the solutions exist only for some specific values of h. Those values of h for which solution exist are known as Eigen values Or characteristic values of the boundary value problem.

Each of the Eigen value gives us the characteristic properties of TM Mode of given waveguide.

The propagation constant can be written as

For we get

---(34).

The frequency f_c is known as cut-off frequency.

The value of f_c for a particular mode in a waveguide depends upon Eigen value of mode.

Using (34) in (33), we get

So we have,

---(36)

---(38)

---(39) Where f is the frequency of the plane wave in the unbounded

medium with characteristics and is velocity of light in medium.

We can write following relation

---(40)

Where, is the cutt off wavelength.

---(41) From above eq we observe that the phase velocity of TM wave in

a waveguide is greater then the velocity of the wave in the unbounded Dielectric medium and also it is frequency dependent.

It means single conductor waveguides are dispersive medium.

The group velocity is written as

---(42)

Where,

The wave impedance of TM can be written, using (37) in (31), as

---(44)

The wave impedance of propagating TM mode in a loss less dielectric waveguide is purely resistive and is always less than the intrinsic

Impedance of the dielectric medium.

(b) i.e. Operating frequency is less than the Cutt-off frequency. In this situation γ will be real

---(45)

Since all field components contain the factor the wave diminishes rapidly with z and is called evanescent wave. A waveguide is called high pass filter.

Using (46) in (31), we get

Transverse Electric Waves: For TE wave E_z = 0.

For understanding TE waves, we solve following Eq. for H_z

---(47)

Combining (48) and (49), we get

---(52)

Transverse components of E.F. Can be written in terms of transverse component of M.F through the definition of wave impedance,

---(53)

Also we can write

For TE waves also, we have two situation, depending upon whether operating frequency is greater or lower as compared to cut-off

frequency. (a)

In this situation,

---(55)

Wave impedance, using (55)

---(56)

The wave impedance of propagating TE modes in a waveguide with a lossless dielectric is purely resistive and always larger than the

intrinsic impedance of the dielectric medium.

---(57) Wave impedance

---(58)