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LESSON 2
CIRCULAR WAVEGUIDES
2.1 Introduction
The mathematical treatment of circular waveguides is similar to those for a rectangular
waveguides only that they are represented in cylindrical coordinate system.
2.2.1 Circular waveguides
Consider a circular waveguide of radius as shown in Fig. 1.8.
z
r

Fig. 2.1 Circular waveguide
We assume complex exponential variation of fields 𝐸 = 𝐸𝑜 𝑒 [𝑖(𝜔𝑡−𝑘𝑧)] and 𝐵 =
𝐵𝑜 𝑒 [𝑖(𝜔𝑡−𝑘𝑧)]
In order to get the general equation governing propagation of electromagnetic wave we
use Maxwell’s modified equations since the waveguide is hollow ( = J = 0) . These are
∇ ∙ 𝐸⃗ = 0
(1.4)
⃗ = ∇ ∙ 𝜇𝑜 𝐻
⃗ =∇∙𝐻
⃗ =0
∇∙𝐵
(1.5)
⃗
𝜕𝐵
⃗
∇x𝐸⃗ = − 𝜕𝑡 = −𝑖𝜔𝐻
⃗
⃗ = 𝜕𝐷 = 𝑖𝜔𝜀𝑜 𝐸⃗
∇x𝐻
𝜕𝑡
(1.6)
(1.7)
Taking the curl of eq. 1.6 gives
⃗)
∇x∇x𝐸⃗ = −𝑖𝜔𝜇𝑜 (∇𝑥𝐻
(1.8)
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Substituting eq. 1.7 into eq. 1.8 gives
∇x∇x𝐸⃗ = 𝜔2 𝜇𝑜 𝜀𝑜 𝐸⃗
(1.9)
⃗⃗ = ∇. (∇ ∙ 𝑀
⃗⃗ ) − ∇2 𝑀
⃗⃗ to the left hand side of eq. 1.9 gives
Applying the identity ∇x∇x𝑀
∇. (∇ ∙ 𝐸⃗ ) − ∇2 𝐸⃗ = 𝜔2 𝜇𝑜 𝜀𝑜 𝐸⃗
(1.10)
Substituting eq. 1.4 into eq. 1.10 gives
∇2 𝐸⃗ + 𝜔2 𝜇𝑜 𝜀𝑜 𝐸⃗ = 0
(1.11)
To express eq. 1.11 in terms of H, we take the curl of eq. 1.7
⃗ = 𝑖𝜔𝜀𝑜 (∇𝑥𝐸⃗ )
∇x∇x𝐻
(1.12k)
⃗⃗ = ∇. (∇ ∙ 𝑀
⃗⃗ ) − ∇2 𝑀
⃗⃗ gives
Substituting eq. 1.6 into eq. 1.12k and apply ∇x∇x𝑀
⃗ ) − ∇2 𝐻
⃗ = 𝑖𝜔𝜀𝑜 (−𝑖𝜔𝜇𝑜 𝐻
⃗ ) = 𝜔2 𝜀𝑜 𝜇𝑜 𝐻
⃗
∇. (∇ ∙ 𝐻
(1.13k)
Substituting eq. 1.4 into eq. 1.13k gives
⃗ + 𝜔2 𝜀𝑜 𝜇𝑜 𝐻
⃗ =0
∇2 𝐻
(1.14k)
Longitudinal (z-directed) component of the vector field may be separated giving
∇2 𝐸𝑧 = −𝜔2 𝜇𝜀𝐸𝑧
(2.1)
∇2 𝐻𝑧 = −𝜔2 𝜇𝜀𝐻𝑧
(2.2)
And
𝜕2
1 𝜕
1 𝜕2
𝜕2
Expanding the Laplacian in cylindrical coordinates ∇2 = 𝜕𝑟 2 + 𝑟 𝜕𝑟 + 𝑟 2 𝜕𝜃2 + 𝜕𝑧 2 eqns.
2.2 and 2.2 become
𝜕2 𝐸𝑧
𝜕𝑟 2
1 𝜕𝐸
1 𝜕2 𝐸
𝜕2 𝐸
1 𝜕𝐻
1 𝜕2 𝐻
𝜕2 𝐻
+ 𝑟 𝜕𝑟𝑧 + 𝑟 2 𝜕𝜃2𝑧 + 𝜕𝑧 2𝑧 = −𝜔2 𝜇𝜀𝐸𝑧
(2.3)
And
𝜕2 𝐻𝑧
𝜕𝑟 2
+ 𝑟 𝜕𝑟𝑧 + 𝑟 2 𝜕𝜃2𝑧 + 𝜕𝑧 2𝑧 = −𝜔2 𝜇𝜀𝐻𝑧
(2.4)
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Since propagation is in the z direction, the fourth term gives
𝜕2 𝐻𝑧
1 𝜕𝐻
1 𝜕2 𝐻
𝜕2 𝐻𝑧
1 𝜕𝐻
1 𝜕2 𝐻
𝜕2 𝐻𝑧
=
𝜕𝑧 2
𝜕2 (𝐻𝑧𝑜 𝑒 −𝑖𝑘𝑧 )
𝜕𝑧 2
+ 𝑟 𝜕𝑟𝑧 + 𝑟 2 𝜕𝜃2𝑧 − 𝑘 2 𝐻𝑧 + 𝜔2 𝜇𝜀𝐻𝑧 = 0
𝜕𝑟 2
= −𝑘 2
(2.5a)
+ 𝑟 𝜕𝑟𝑧 + 𝑟 2 𝜕𝜃2𝑧 + (𝜔2 𝜇𝜀 − 𝑘 2 )𝐻𝑧 = 0
𝜕𝑟 2
(2.5b)
If we let 𝑘𝑐2 = 𝜔2 𝜇𝜀 − 𝑘 2 and eq. 2.5b becomes
𝜕2 (𝐻𝑧 )
𝜕𝑟 2
1 𝜕(𝐻𝑧 )
+𝑟
𝜕𝑟
1 𝜕2 (𝐻 )
+ 𝑟 2 𝜕𝜃2𝑧 + 𝑘𝑐2 𝐻𝑧 = 0
(2.6)
Hence we may seek a solution to eq. 2.6 of the form
𝐻𝑧 = 𝑅(𝑟)𝑃()
(2.7)
where R is a function of r and P is a function of  which gives
𝑃(𝜃) = 𝐴𝑠𝑖𝑛(𝑛𝜃) + 𝐵𝑐𝑜𝑠(𝑛𝜃)
(2.12)
𝑅(𝑟) = [𝐶𝐽𝑛 (𝑘𝑐 𝑟) + 𝐷𝑌𝑛 (𝑘𝑐 𝑟)]
(2.15)
and
and Jn(kcr)and Yn(kcr) are the Bessel functions of order n. Ignoring 𝐷𝑌𝑛 (𝑘𝑐 𝑟) since it
decreases as kcr  0, and A and B are independent of one another and one of them can be
zero. Assuming B=0 and A 0, full solution to eq. 2.6 is
𝐻𝑧 = 𝑅(𝑟)𝑃() = 𝐶𝐽𝑛 (k 𝑐 r)[𝐴𝑠𝑖𝑛(𝑛𝜃)]
(2.16)
Including the time dependence, the full solution is
𝐻𝑧 = 𝐶𝐽𝑛 (k 𝑐 r)[𝐴𝑠𝑖𝑛(𝑛𝜃)]𝑒 𝑖(𝜔𝑡−𝑘𝑧)
(2.17)
The variation of Jn(kcr) with kcr is given in Fig. 2.2
Jn(kca)
Yn(kca)
(kca)
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2.2.2 TM modes in a circular waveguide
For TM modes, Hz = 0. For a hollow cylinder Yn(kcr) is invalid since it goes to infinity
when r = 0 and the field must always be finite. Also constants A and B control the
amplitude of the sin n and cos n terms, which are independent.
That is, because of the azimuthal symmetry of the circular waveguide, both the sin n
and cos n terms represent valid solutions, and both may be present in a specific problem.
The actual amplitudes of these terms will depend on the excitation of the waveguide.
From a different viewpoint, the coordinate system can be rotated about the z-axis to
obtain an Hz with either A = 0 or B = 0. If we let B=0, eq. 2.17 reduces to
𝐸𝑧 = 𝐶𝐽𝑛 (𝑘𝑐 𝑟)[𝐴𝑠𝑖𝑛(𝑛𝜃)]𝑒 𝑖(𝜔𝑡−𝑘𝑧)
(2.18)
The boundary condition is that E should vanish at the surface (r = a)
𝐽𝑛 (𝑘𝑐 𝑎) = 0
(2.19)
Which provide values of k and hence propagation constant  which in turn gives k. For
the wave to be sustained k must be real such that (𝑘𝑐2 = 𝜔2 𝜇𝜀 − 𝑘 2 )
𝜔2 𝜇𝜀 > 𝑘 2
(2.20)
The characteristic values of knm of k, where knma is the mth root of 𝐽𝑛 (𝑘𝑐 𝑎), define the
TEnm modes (n is the number of half sinusoid in the radial direction and m is the number
of full sinusoid in the circumferential direction).
J0(kca)
J1(kca)
0
1
2
J2(kca)
3
4
5
6
7
8
Fig. 2.2 Roots to TMnm modes in Circular waveguide
9
10
11
(kca)
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The cut-off wave vector is
𝑘𝑐 =
𝑝𝑛𝑚
𝑎
The cut-off frequencies of these modes are obtained from 𝜔𝑐 =
𝑘
𝑘𝑐
√𝜇𝜀
or
𝑝
𝑓𝑐 = 2 𝑐𝜇𝜀 = 2𝑎𝑛𝑚𝜇𝜀
√
(2.21)
√
For frequencies above fc propagation constant has real and imaginary parts given by
𝜔2
𝜔2
𝑝
2
= +i where  = 0 and 𝛽 = 𝑘 = √ 𝑐 2 − 𝑘𝑐2 = √ 𝑐 2 − ( 𝑛𝑚
)
𝑎
(2.22)
And the guide wavelength is kc=2/c
2𝜋
2𝜋
𝑔 =  = 𝑘 =
1
2
√𝜔2 −𝑘𝑐2
𝑐
=
1
2
2
√𝜔2 −(𝑝𝑛𝑚 )
𝑎
𝑐
where 𝑜 is the free-space wavelength.
(2.23)
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2.2.2 TM modes field patterns in a circular waveguide
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TE modes in a circular waveguide
For TE modes, Ez = 0. For a hollow cylinder Yn(kcr) is invalid since it goes to infinity
when r = 0 and the field must always be finite and letting B =0 as before, eq. 2.17 reduces
to
𝐻𝑧 = 𝐶𝐽𝑛′ (𝑘𝑐 𝑟)[𝐴 sin(𝑛)] 𝑒 𝑖(𝜔𝑡−𝑘𝑧)
(2.24)
The boundary condition is that E should vanish at the surface (r = a)
𝐽𝑛′ (𝑘𝑐 𝑎) = 0
J’0(kca
)
(2.25)
J’1(kca)
J’2(kca
)
0
1
2
3
4
5
6
7
8
9
10
11
(kca)
Fig. 2.2 Roots to TEnm modes in Circular waveguide
The expressions for fc, c and g are similar to those for TM. Table 1.1 gives the roots of
eqs. 2.8 and 2.9.
Table 1.1 TE modes
𝑇𝐸𝑛𝑚 modes
n
m=1
m=2
𝑘𝑐1 𝑎
𝑘𝑐2 𝑎
0
3.832
7.015
1
1.841
5.332
2
3.054
6.705
m=3
𝑘𝑐3 𝑎
10.174
8.536
9.963
The field patterns for TE modes are shown in Fig. 2.3
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