Rectangular waveguides are th one of the earliest type of the transmission lines. They are used in many applications. A lot of components such as isolators, detectors, attenuators, couplers and slotted lines are available for various standard waveguide bands between 1 GHz to above 220 GHz.

A rectangular waveguide supports TM and TE modes but not TEM waves because we cannot define a unique voltage since there is only one conductor in a rectangular waveguide. The shape of a rectangular waveguide is as shown below. A material with permittivity e and permeability m fills the inside of the conductor.

A rectangular waveguide cannot propagate below some certain frequency. This frequency is called the cut-off frequency.

Here, we will discuss TM mode rectangular waveguides and TE mode rectangular waveguides separately. Let’s start with the TM mode.

TM Modes

Consider the shape of the rectangular waveguide above with dimensions a and b (assume a>b) and the parameters e and m. For TM waves H_z = 0 and E_z should be solved from equation for TM mode;

Ѳ_xy E_z⁰ + h² E_z⁰ = 0

Since E_z(x,y,z) = E_z⁰(x,y)e^-gz, we get the following equation,

If we use the method of separation of variables, that is E_z⁰(x,y)=X(x).Y(y) we get,

Since the right side contains x terms only and the left side contains y terms only, they are both equal to a constant. Calling that constant as k_x², we get;

where k_y²=h²-k_x²

Now, we should solve for X and Y from the preceding equations. Also we have the boundary conditions of;

E_z⁰(0,y)=0

E_z⁰(a,y)=0

E_z⁰(x,0)=0

E_z⁰(x,b)=0

From all these, we conclude that

X(x) is in the form of sin k_xx, where k_x=mp/a, m=1,2,3,…

Y(y) is in the form of sin k_yy, where k_y=np/b, n=1,2,3,…

So the solution for E_z⁰(x,y) is

(V/m)

From k_y²=h²-k_x², we have;

For TM waves, we have

From these equations, we get

where

Here, m and n represent possible modes and it is designated as the TM_mn mode. m denotes the number of half cycle variations of the fields in the x-direction and n denotes the number of half cycle variations of the fields in the y-direction.

When we observe the above equations we see that for TM modes in rectangular waveguides, neither m nor n can be zero. This is because of the fact that the field expressions are identically zero if either m or n is zero. Therefore, the lowest mode for rectangular waveguide TM mode is TM₁₁ .

Here, the cut-off wave number is

and therefore,

The cut-off frequency is at the point where g vanishes. Therefore,

Since l=u/f, we have the cut-off wavelength,

At a given operating frequency f, only those frequencies, which have f_c<f will propagate. The modes with f<f_c will lead to an imaginary b which means that the field components will decay exponentially and will not propagate. Such modes are called cut-off or evanescent modes.

The mode with the lowest cut-off frequency is called the dominant mode. Since TM modes for rectangular waveguides start from TM₁₁ mode, the dominant frequency is

The wave impedance is defined as the ratio of the transverse electric and magnetic fields. Therefore, we get from the expressions for E_x and H_y (see the equations above);

The guide wavelength is defined as the distance between two equal phase planes along the waveguide and it is equal to

which is thus greater than l, the wavelength of a plane wave in the filling medium.

The phase velocity is

which is greater than the speed of light (plane wave) in the filling material.

Attenuation for propagating modes results when there are losses in the dielectric and in the imperfectly conducting guide walls. The attenuation constant due to the losses in the dielectric can be found as follows:

TE Modes

Consider again the rectangular waveguide below with dimensions a and b (assume a>b) and the parameters e and m.

For TE waves E_z = 0 and H_z should be solved from equation for TE mode;

Ѳ_xy H_z + h² H_z = 0

Since H_z(x,y,z) = H_z⁰(x,y)e^-gz, we get the following equation,

If we use the method of separation of variables, that is H_z⁰(x,y)=X(x).Y(y) we get,

Since the right side contains x terms only and the left side contains y terms only, they are both equal to a constant. Calling that constant as k_x², we get;

where k_y²=h²-k_x²

Here, we must solve for X and Y from the preceding equations. Also we have the following boundary conditions:

at x=0

at x=a

at y=0

at y=b

From all these, we get

(A/m)

From k_y²=h²-k_x², we have;

For TE waves, we have

From these equations, we obtain

where

As explained before, m and n represent possible modes and it is shown as the TE_mn mode. m denotes the number of half cycle variations of the fields in the x-direction and n denotes the number of half cycle variations of the fields in the y-direction.

Here, the cut-off wave number is

and therefore,

The cut-off frequency is at the point where g vanishes. Therefore,

Since l=u/f, we have the cut-off wavelength,

At a given operating frequency f, only those frequencies, which have f>f_c will propagate. The modes with f<f_c will not propagate.

The mode with the lowest cut-off frequency is called the dominant mode. Since TE₁₀ mode is the minimum possible mode that gives nonzero field expressions for rectangular waveguides, it is the dominant mode of a rectangular waveguide with a>b and so the dominant frequency is

The wave impedance is defined as the ratio of the transverse electric and magnetic fields. Therefore, we get from the expressions for E_x and H_y (see the equations above);

The guide wavelength is defined as the distance between two equal phase planes along the waveguide and it is equal to

which is thus greater than l, the wavelength of a plane wave in the filling medium.

The phase velocity is

which is greater than the speed of the plane wave in the filling material.

The attenuation constant due to the losses in the dielectric is obtained as follows:

After some manipulation, we get

Example:

Consier a length of air-filled copper X-band waveguide, with dimensions a=2.286cm, b=1.016cm. Find the cut-off frequencies of the first four propagating modes.

Solution:

From the formula for the cut-off frequency