Rectangular Waveguides - Basic Theory

Electric and Magnetic Fields in a Rectangular waveguide:

• For any wave, γ2x + γ2

y + γ2z = γ2

0 = −ω2µε=⇒β2x + β2

y + β2z = β2

0 = +ω2µε (From Helmholtz wave equa-tion)

• For a wave in rectangular waveguide, we need to apply boundary conditions Etangential = 0 andHnormal = 0. Then we get,

Ey = A1 sin(mπax)cos(nπby)e±jβzz

Ex = A2 cos(mπax)sin(nπby)e±jβzz

In the above equation ± indicates the fact that the wave can propagate in either −z or +z direction.From Poynting theorem we get,

Hx = − A1

Zmn0

sin(mπax)cos(nπby)e±jβzz

Ex =A2

Zmn0

cos(mπax)sin(nπby)e±jβzz,

where Zmn0 indicates the characteristic impedance corresponding to the mnth mode (either TE or TMmode, which we will study after CAT II exam).

• Now, Let's closely observe the below equation:

Ey = A1 sin(mπax)cos(nπby)e±jβzz

= A1

1

2j

ejmπa x−e−jmπa x︸ ︷︷ ︸waves along ±x

12

ej nπb y + e−jnπb y︸ ︷︷ ︸

waves along ±y

e±jβzz︸ ︷︷ ︸waves along ±z

As we can see, Ey can be decomposed into waves propagating along +x,−x,+y,−y,+z, and − zdirections. We can decompose other �elds also like this. Important point to observe here is, propagationconstants along various directions are given as

γx = 0 + jβx = 0 + jmπ

a

γy = 0 + jβy = 0 + jnπ

b(1)

1

• OK ... Then what is γz? ... If you know answer to this question, that means you understood theconcept of wave propagation in waveguides ...

Propagation Constant and Phase Velocity along z direction:

We know from the previous section that γ2x + γ2

y + γ2z = γ2

0 = −ω2µε. Substituting (1) in this equation gives

γ2x + γ2

y + γ2z = −ω2µε

⇒ γ2z = −γ2

x − γ2y − ω2µε

⇒ γ2z = −

(jmπ

a

)2

−(jnπ

b

)2

− ω2µε

⇒ γ2z =

(mπa

)2

+(nπb

)2

− ω2µε

⇒ γz = 0 + jβz =

√(mπa

)2

+(nπb

)2

− ω2µε. (2)

Once propagation constant is known, we can obtain velocity from the equation

v =ω

β=⇒ vphase =

ω

βz=

ω√ω2µε−

[(mπa

)2+(nπb

)2]Cuto� Frequency, Wavelength, and Propagation Constants:

If a wave propagates along ±z directions, then the corresponding propagation constant γz should be a pureimaginary term like 0 + jβz. So, wave propagation occurs only when the term inside the square root of theabove equation becomes negative number, i.e.,(mπ

a

)2

+(nπb

)2

− ω2µε < 0

⇒(mπa

)2

+(nπb

)2

< ω2µε

⇒ ω2µε >(mπa

)2

+(nπb

)2

⇒ ω>1√µε

√(mπa

)2

+(nπb

)2

= ωcutoff . (3)

So, signals with frequency greater than ωcutoff only can propagate. Of course, we can similarly de�ned cuto�wavelength too. Cuto� wavelength is given as

λcutoff =c

fcutoff=

cωcutoff

2π

=1√µε

2π

1√µε

√(mπa

)2+(nπb

)2⇒ λcutoff=

2π√(mπa

)2+(nπb

)2 . (4)

And, �nally we can also de�ne one more parameter, cuto� propagation constant as shown below:

βcutoff =2π

λcutoff=

√(mπa

)2

+(nπb

)2

(5)

2

Rectangular Waveguide �lled with Lossy Dielectric Medium:

Until now, we assumed that the rectangular waveguide is �lled with loss-less medium (e.g., vacuum). However,some times we may need to �ll it with lossy dielectric materials. In such cases, waves get attenuated as theypropagate along z direction. so, we need to calculate that attenuation constant (i.e., αz). So, let's calculatethat parameter in this section. From (2),

γlossyz =

√(mπa

)2

+(nπb

)2

− ω2µε

=

√(mπa

)2

+(nπb

)2

− ω2µ0µrε0εr (1− j tan δ)

=

√[(mπa

)2

+(nπb

)2

− ω2µ0ε0εr

]+ jω2µ0ε0εr tan δ, assuming µr = 1

=√γ2z,loss−less + jω2µ0ε0εr tan δ

=√(jβz,loss−less)

2+ jβ2

0,loss−less tan δ

=√−β2

z,loss−less + jβ20,loss−less tan δ

= jβz,loss−less ×

√1−

jβ20,loss−less tan δ

β2z,loss−less

≈ jβz,loss−less ×

(1− 1

2

jβ20,loss−less tan δ

β2z,loss−less

)

=

(jβz,loss−less +

1

2

β20,loss−less tan δ

βz,loss−less

)

=1

2

β20,loss−less tan δ

βz,loss−less︸ ︷︷ ︸αz,lossy

+j βz,loss−less︸ ︷︷ ︸βz,lossy

.

So, for a rectangular waveguide �lled with lossy dielectric material, real and imaginary parts of the propaga-tion constant are given as

αz,lossy =1

2

β20,loss−less tan δ

βz,loss−less

βz,lossy = βz,loss−less = ω2µ0ε0εr (6)

Guided Wavelength:

Since the wave is propagating (in other words guided) along z direction, guided wavelength is nothing but wavelength along z direction. We already know propagation constant along z direction. So, guided wavelength isgiven from βz as

λguided =2π

βz.

3

We also know from (2) that

jβz =

√(mπa

)2

+(nπb

)2

− ω2µε

⇒ −β2z =

(mπa

)2

+(nπb

)2

− ω2µε

⇒ β2z =ω2µε−

[(mπa

)2

+(nπb

)2]

⇒ βz =

√ω2µε−

[(mπa

)2

+(nπb

)2].

So, guided wavelength is given as

λguided =2π

βz=

2π√ω2µε−

[(mπa

)2+(nπb

)2] . (7)

Summary

• γ2x + γ2

y + γ2z = γ2

0 = −ω2µε =⇒β2x + β2

y + β2z = β2

0 = +ω2µε

• γx = 0 + jβx = 0 + jmπa, & γy = 0 + jβy = 0 + j nπ

b

• γz = 0 + jβz =√(

mπa

)2+(nπb

)2 − ω2µε

• ωcutoff = 1√µε

√(mπa

)2+(nπb

)2• For lossy Rectangular waveguide

αz,lossy =1

2

β20,loss−less tan δ

βz,loss−less

βz,lossy = βz,loss−less = ω2µ0ε0εr

• λguided = 2πβz

= 2π√ω2µε−

[(mπa )

2+(nπb )

2]

4