PH 531, FALL 2009

Authored by: Jessica McCartney

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DERIVATION OF FIELD COMPONENTS

The Ez component equation is used to derive the radial and azimuthal components, Er

and Eϕ, of the electric field for Transverse Magnetic (TM) modes. TM modes are those

for which the magnetic field is perpendicular (transverse) to the direction of propagation,

and Hz=0. The Hz component equation is used to derive the radial and azimuthal

components, Hr and Hϕ, of the electric field for Transverse Electric (TE) modes.

Transverse Electric modes are those for which the electric field is perpendicular

(transverse) to the direction of propagation, and Ez=0.

From Cronin’s Microwave and Optical Waveguides (1995), “the transverse components

Er, Eϕ, Hr, Hϕ can …be derived from the longitudinal components Ez and Hz. From this

we may therefor conclude that in solving the wave equation … it is only necessary to

determine the longitudinal components”. The TM mode is apparently the easiest to

solve for, so this mode is generally solved for first, with the TE mode being derived

directly from the TM mode. This method will be followed here. To find the the

longitudinal components of the electric and magnetic fields in a cylindrical waveguide,

Ez and Hz, it is necessary to start from two of Maxwell’s equations.

𝛁 × 𝑬 = − 𝜕𝑩

𝜕𝑡 (𝟏)

(Faraday’s Law of Induction)

And

𝛁 × 𝑯 = 𝐽𝑓 + 𝜕𝑫

𝜕𝑡 (𝟐)

(Ampere’s Circuit Law with Maxwell’s correction)

To get equation (1) in terms of H, H must be substituted in for B in (1) using the

relationship B=μ0(H+M). As there is assumed to be no magnetization B=μ0(H+M)=

μ0(H+0)= μ0(H+0)= μ0H, equation (1) becomes:

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𝛁 × 𝑬 = − 𝜇𝜕𝑯

𝜕𝑡 (𝟑)

Likewise, to get (2) in terms of E, E is substituted in for D using the relationship D=ϵE,

and as there is assumed to be no free current (Jf), equation (2) becomes:

𝛁 × 𝑯 = 𝜖 𝜕𝑬

𝜕𝑡 (𝟒)

The equation for the Electrical field in a cylindrical waveduide, in cylindrical coordinates,

starts with the wave equation.

∇2𝐄 = 𝜇𝜖𝜕2𝑬

𝜕𝑡2 (𝟓)

According to Holt’s Introduction to Electromagnetic Fields and Waves, the wave

equation for E can be derived from Maxwell’s Equations in the following manner:

1. Taking the curl of both sides of equation (3):

𝛁 × 𝛁 × 𝑬 = −𝜇 𝜕

𝜕𝑡 𝛁 × 𝑯 (𝟔)

2. Substituting the right-hand-side (RHS) of equation (4) in for the curl of H,

this becomes:

𝛁 × 𝛁 × 𝑬 = −𝜇 𝜕

𝜕𝑡 𝜖

𝜕𝑬

𝜕𝑡 = −𝜇𝜖

𝜕2𝑬

𝜕𝑡2 (𝟕)

3. According to Wikipedia, “Electromagnetic wave equation”, the left-hand-side (LHS) of equation (7) can be rewritten using a vector identity as:

𝛁 × 𝛁 × 𝑬 = 𝛁 𝛁 ∙ 𝑬 − 𝛁𝟐𝑬 = −𝜇𝜖𝜕2𝑬

𝜕𝑡2 (𝟖)

4. Using another of Maxwell’s Equations (Gauss’s Law), with charge density

ρ=0 as the propagating medium is assumed to be uncharged:

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𝛁 ∙ 𝑬 =𝜌

𝜖0=

0

𝜖0= 0 (𝟗)

5. Substituting for the divergence of E in (8) using (9), the RHS of (8) is equal

to the negative Laplacian of the electric field. The negative signs cancel, and all that remains is the wave equation, (5), reproduced here for completeness.

0 − 𝛁𝟐𝑬 = −𝜇𝜖𝜕2𝑬

𝜕𝑡2→ ∇2𝐄 = 𝜇𝜖

𝜕2𝑬

𝜕𝑡2 (𝟓)

If the waveguide mode is propagating in the z-direction, then in cylindrical coordinates

E0=(E0r, E0ϕ, 0), where the radial and azimuthal components of the electric field will be

functions of r and ϕ. The z-dependence of the field is assumed to be given by the

equation for the electric field, Ez=E0z(ρ, ϕ)ei(ωt-kzz).

In order to solve for the electrical components for a circular waveguide, the first step is

to rewrite the Laplacian of E (LHS of (5)) in cylindrical coordinates,

𝛁𝟐𝑬 = 𝝆 1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬𝜌

𝜕𝜌 +

1

𝜌2 𝜕2𝑬𝜌

𝜕𝜑2− 𝐸𝜌 + 2

𝜕𝑬𝜑

𝜕𝜑 +

𝜕2𝑬𝜌

𝜕𝑧2

+ 𝝋 1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬𝜑

𝜕𝜌 +

1

𝜌2 𝜕2𝑬𝜑

𝜕𝜑2− 𝐸𝜑 + 2

𝜕𝑬𝜌

𝜕𝜑 +

𝜕2𝑬𝜑

𝜕𝑧2

+ 𝒛 1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬𝑧

𝜕𝜌 +

1

𝜌2 𝜕2𝑬𝑧

𝜕𝜑2 +

𝜕2𝑬𝑧

𝜕𝑧2 (𝟏𝟎)

Since only the longitudinal component Ez needs to be determined, the ρ and ϕ terms

will go away, and eq. (5) becomes:

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1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬𝑧

𝜕𝜌 +

1

𝜌2 𝜕2𝑬𝑧

𝜕𝜑2 +

𝜕2𝑬𝑧

𝜕𝑧2 = 𝜇𝜖

𝜕2𝑬𝒛

𝜕𝑡2 (𝟏𝟏)

Substituting in Ez=E0z(ρ, Φ)ei(ωt-kzz).

1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬0𝑧 𝜌, 𝜑 𝑒𝑖(𝜔𝑡−𝑘𝑧𝑧)

𝜕𝜌 +

1

𝜌2 𝜕2𝑬0𝑧 𝜌, 𝜑 𝑒𝑖(𝜔𝑡−𝑘𝑧𝑧)

𝜕𝜑2 +

𝜕2𝑬0𝑧 𝜌, 𝜑 𝑒𝑖(𝜔𝑡−𝑘𝑧𝑧)

𝜕𝑧2

= 𝜇𝜖𝜕2𝑬0𝑧 𝜌, 𝜑 𝑒𝑖(𝜔𝑡−𝑘𝑧𝑧)

𝜕𝑡2 (𝟏𝟐)

Taking derivatives of exponential terms where possible, and then separating out

exponential terms, the equation becomes:

1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬0𝑧 𝜌, 𝜑

𝜕𝜌 𝑒𝑖 𝜔𝑡−𝑘𝑧𝑧 +

1

𝜌2 𝜕2𝑬0𝑧 𝜌, 𝜑

𝜕𝜑2 𝑒𝑖 𝜔𝑡−𝑘𝑧𝑧

+ 𝑬0𝑧 𝜌, 𝜑 𝑘2 −𝑒𝑖 𝜔𝑡−𝑘𝑧𝑧 = 𝜇𝜖 𝑬0𝑧 𝜌, 𝜑 𝜔2 −𝑒𝑖 𝜔𝑡−𝑘𝑧𝑧 (𝟏𝟑)

Cancelling out exponential terms, the equation becomes:

1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬0𝑧 𝜌, 𝜑

𝜕𝜌 +

1

𝜌2 𝜕2𝑬0𝑧 𝜌, 𝜑

𝜕𝜑2 + −𝑘2 𝑬0𝑧 𝜌, 𝜑 = 𝜇𝜖 −𝜔2 𝑬0𝑧 𝜌, 𝜑 (𝟏𝟒)

Gathering non-differential terms to the RHS:

1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕𝑬0𝑧 𝜌, 𝜑

𝜕𝜌 +

1

𝜌2 𝜕2𝑬0𝑧 𝜌, 𝜑

𝜕𝜑2 = 𝜇𝜖 −𝜔2 + 𝑘2 𝑬0𝑧 𝜌, 𝜑 (𝟏𝟓)

This equation needs to be solved using separation of variables. Rewriting E0z as a

product function E0z(ρ, ϕ)= Ρ(ρ)Φ(ϕ) and substituting this in, the equation becomes:

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1

𝜌

𝜕

𝜕𝜌 𝜌

𝜕Ρ(ρ)Φ(ϕ)

𝜕𝜌 +

1

𝜌2 𝜕2Ρ(ρ)Φ(ϕ)

𝜕𝜑2 = 𝜇𝜖 −𝜔2 + 𝑘2 Ρ ρ Φ ϕ (𝟏𝟔)

In order to complete variable separation, both sides of the equation are multiplied by ρ2:

𝜌𝜕

𝜕𝜌 𝜌

𝜕Ρ(ρ)Φ(ϕ)

𝜕𝜌 +

1

1 𝜕2Ρ(ρ)Φ(ϕ)

𝜕𝜑2 = 𝜌2 𝑘2 − 𝜇𝜖𝜔2 Ρ ρ Φ ϕ (𝟏𝟕)

Then both sides are divided by Ρ(ρ)Φ(ϕ) (after constants are factored out of differential

terms), to yield:

𝜌

P(ρ)

𝜕

𝜕𝜌 𝜌

𝜕Ρ(ρ)

𝜕𝜌 +

1

Φ(ϕ) 𝜕2Φ(ϕ)

𝜕𝜑2 = 𝜌2 𝑘2 − 𝜇𝜖𝜔2 (𝟏𝟖)

Rearranging to get ρ and ϕ terms on different sides,

𝜌

P(ρ)

𝜕

𝜕𝜌 𝜌

𝜕Ρ(ρ)

𝜕𝜌 − 𝜌2 𝑘2 − 𝜇𝜖𝜔2 = −

1

Φ ϕ 𝜕2Φ ϕ

𝜕𝜑2 (𝟏𝟗)

From Cronin’s Microwave and Optical Waveguides (1995), “The left hand side is

function of R [ρ] only and the right-hand side is a function of ϕ only. Each side must

therefore be independently equal to a common constant.” Setting this common

constant equal to m2, the equations become:

𝜌

P ρ

𝜕

𝜕𝜌 𝜌

𝜕Ρ ρ

𝜕𝜌 − 𝜌2 𝑘2 − 𝜇𝜖𝜔2 = 𝑚2 (𝟐𝟎)

−1

Φ(ϕ) 𝜕2Φ(ϕ)

𝜕𝜑2 = 𝑚2 (𝟐𝟏)

The equation for ϕ, (21), can be rewritten as:

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𝜕2Φ(ϕ)

𝜕𝜑2 + Φ(ϕ)𝑚2 = 0 (𝟐𝟐)

According to Magnusson et al. Transmission Lines and Wave Propagation (2001), “ [the

equation above] may be recognized as a homogenous linear differential equation of the

second order with constant coefficients, the solution to which is” :

Φ ϕ = C1sin𝑚𝜑 + C2cos𝑚𝜑 (𝟐𝟑)

In order for the solution to have a single value, Φ must repeat at intervals of 2π, and m

must be an integer.

After multiplying both sides by Ρ ρ)/ρ2 the equation for ρ (20) can be rewritten as:

1

ρ

𝜕

𝜕𝜌 𝜌

𝜕Ρ ρ

𝜕𝜌 + 𝜇𝜖𝜔2 − 𝑘2 −

𝑚2

𝜌2 P ρ = (𝟐𝟒)

Defining a new variable 𝑥=ρ 𝜇𝜖𝜔2 − 𝑘2 and using the product rule to separate the

differential term, eq. (24) becomes

𝜕2Ρ 𝑥

𝜕𝑥2+

1

Ρ 𝑥

𝜕Ρ 𝑥

𝜕Ρ 𝑥 + Ρ 𝑥 1 −

𝑚2

𝑥2 = (𝟐𝟓)

According to Cronin’s Microwave and Optical Waveguides (1995), this corresponds to a

Bessel equation of order n (See Appendix for Bessel function plot). According to

Magnusson et al. Transmission Lines and Wave Propagation (2001), this has a general

solution of the form:

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𝑃 𝑥 = 𝐴𝑚 𝐽𝑚 𝑥 + 𝐵𝑚𝑌𝑚 𝑥 (𝟐𝟔)

Where Jm and Ym represent Bessel functions of the first and second kind, respectively.

Defining a new variable h= 𝜇𝜖𝜔2 − 𝑘2, the equation becomes

𝑃 𝜌 = 𝐴𝑚 𝐽𝑚 𝑕𝜌 + 𝐵𝑚𝑌𝑚 𝑕𝜌 (𝟐𝟕)

The Ym term would approach infinity at ρ=0, implying an infinite field on the axis of the

waveguide. As this is an impossibility, the Ym term can be dropped from the solution.

Then the final solution of Ρ ρ) is:

𝑃 𝜌 = 𝐴𝑚 𝐽𝑚 𝑕𝜌 (𝟐𝟖)

(Note: Different books will use different arbitrary lettering for the constant h)

Substituting the solutions for Ρ ρ) (28) and Φ(ϕ) (23) into the assumed solution Ez=E0z(ρ,

Φ)ei(ωt-kzz), the overall solution for H is:

E0z ρ,Φ = 𝐴𝑚 𝐽𝑚 𝑕𝜌 C1sin𝑚𝜑 + C2cos𝑚𝜑 𝑒𝑖 𝜔𝑡−𝑘𝑧 (𝟐𝟗)

According to Cronin’s Microwave and Optical Waveguides (1995), “Since the form of

this equation [wave equation for H] is identical to the wave equation for E the solution

that we arrive at for Hz is just [the solution to the wave equation for E] with Hz replacing

Ez”, so one can simply substitute Hz in for Ez in the solution to the wave equation for E

and declare that this is the answer:

H0z ρ, Φ = 𝐴𝑚 𝐽𝑚 𝑕𝜌 C1sin𝑚𝜑 + C2cos𝑚𝜑 𝑒𝑖 𝜔𝑡−𝑘𝑧 (𝟑𝟎)

In order to actually derive the solution to the wave equation for H, one can start with the

electromagnetic wave equation for H and repeat the same steps as those followed for E

to arrive at the same conclusion. (This will not be repeated here, to save space.)

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However, one must remember that the boundary conditions are different at the

waveguide walls when the equation is written in terms of H. According to Magnusson et

al. Transmission Lines and Wave Propagation (2001), the values of the constants h and

k in equation( 30), “are fixed by the boundary conditions that the components of E which

are parallel to the conducting guide wall and the component of B, and hence H, which is

normal to the wall (the radial component of H), shall vanish along that conducting

surface.”

The radial and azimuthal components of E and H can be determined using the curl

equations for E and H and assuming the fields are sinusoidal travelling-wave functions.

According to J.F. Kiang, “TE Models of Cylindrical Waveguide”, the ρ and ϕ

components can then be written in terms of Ez and Hz as:

𝐸𝜌 =1

𝑘𝑧2 − 𝑘2

±𝑗𝑘𝑧

𝜕𝐸𝑧

𝜕𝜌+

𝑗𝜔𝜇

𝜌

𝜕𝐻𝑧

𝜕𝜑 (𝟑𝟏)

𝐸𝜑 = −1

𝑘𝑧2 − 𝑘2

∓𝑗𝛽𝑧

𝜌

𝜕𝐸𝑧

𝜕𝜑+ 𝑗𝜔𝜇

𝜕𝐻𝑧

𝜕𝜌 (𝟑𝟐)

𝐻𝜌 = −1

𝑘𝑧2 − 𝑘2

𝑗𝜔𝜖

𝜌

𝜕𝐸𝑧

𝜕𝜑∓ 𝑗𝑘𝑧

𝜕𝐻𝑧

𝜕𝜌 (𝟑𝟑)

𝐻𝜑 = −1

𝑘𝑧2 − 𝑘2

𝑗𝜔𝜖𝜕𝐸𝑧

𝜕𝜑± 𝑗

𝑘𝑧

𝜌

𝜕𝐻𝑧

𝜕𝜌 (𝟑𝟒)

The component equations can be rewritten as explicit functions of time, according to

Magnusson et al. Transmission Lines and Wave Propagation (2001), and J.F. Kiang,

“TE Models of Cylindrical Waveguide as:

𝐸𝜌 = 𝑗𝜔𝜇𝑛

𝑕2𝜌𝐽𝑛 𝑕𝜌 𝐴𝑛 sin 𝑛𝜑 − 𝐵𝑛cos 𝑛𝜑 𝑒∓𝑗𝑘𝑧𝑧 (𝟑𝟓)

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𝐸𝜑 = 𝑗𝜔𝜇

𝑕𝐽𝑛 𝑕𝜌 𝐴𝑛 cos 𝑛𝜑 − 𝐵𝑛sin 𝑛𝜑 𝑒∓𝑗𝑘𝑧𝑧 (𝟑𝟔)

𝐻𝜌 = ∓𝑘𝑧

𝜔𝜇𝐸𝜑 (𝟑𝟕)

𝐻𝜑 = ±𝑘𝑧

𝜔𝜇𝐸𝜌 (𝟑𝟖)

CUTOFF FREQUENCY

Applying the boundary condition Eϕ=0 at the guide wall (ρ=R) results in the expression

below, where h’mn is the nth root of the derivative of the mth-order Bessel function.

𝑕 =𝑕′𝑚𝑛

𝑅 (𝟑𝟗)

Substituting this relationship into the definition for h leads to the propagation constant.

Setting the propagation constant equal to zero yields the cutoff frequency:

𝑓 =1

2𝜋 𝜇𝜖

𝑕′𝑚𝑛

𝑅 (𝟒𝟎)

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DISPERSION EQUATION

From Magnusson et al. Transmission Lines and Wave Propagation (2001), the

longitudinal component of Poynting’s vector for a cylindrical waveguide can be written:

𝑃1𝑧 = 𝐸𝜑𝐻𝜌 (𝟒𝟎)

Which, when time-averaged over a number of cycles, reduces to:

𝑃1𝑎𝑣𝑔 =𝛽′

01𝜔𝜇𝐴′

012

2𝑕′012 𝐽1

2(𝑕′01𝑟)𝑎𝑧(𝟒𝟏)

Integrating equation (41) over the transverse cross section of the waveguide produces

the formula for transmitted power,

𝑃𝑡𝑟 =𝛽′

01𝜔𝜇𝐴′

012𝜋𝑟𝑎

2

2𝑕′012 𝐽0

2(𝑕′01𝑟𝑎)(𝟒𝟐)

The power dissipated over a short section of the waveguide wall is found by squaring

the Hz component function (written out as an explicit function of time), with ρ equal to

the radius of the waveguide, ra, and multiplying by surface resistance, denoted as Rs.

The resulting equation is then integrated over a cylindrical strip and simplified to

produce the attenuation function, α:

𝛼 =𝑅𝑠𝑕′01

2

𝜔𝜇𝛽′01𝑟𝑎(𝟒𝟑)

Substituting the cutoff frequency equation in for ω and simplifying yields the following

version of the attenuation function, which was important in early microwave

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development in terms of sending signals over long distances, as it shows that the

attenuation drops of as the carrier frequency is raised above the cutoff frequency:

𝛼 = 𝜋𝜖

𝜍

𝑓′𝑐012

𝑓32

(𝟒𝟒)

TE MODES

In a TEmn mode, the subscript m represents the order of the Bessel function, and n

represents the rank of the root. For circular waveguides, the hmn roots (eigenvalues) are

not regularly spaced, in contrast to the rectangular guide. According to Kraus,

Electromagnetics, 4th Edition, the TE01 mode should really be designated the TE02 mode

since it represents the second root of the Bessel function.

TE01 Mode

In this mode, the electrical field strength depends only on the azimuthal angle. The

TE01 mode, according to Magnusson et al. Transmission Lines and Wave Propagation

(2001), was the subject of intense interest due to the unique properties of its dispersion

function. The TE01 mode is the simplest possible circularly symmetrical TE mode. Early

on, it was glommed on to by the radio industry as a means of transmitting information

over long distances using microwaves. Its dispersion function drops of continuously as

frequency is raised. This results in very practical applications, as very low dispersion

can be achieved using a carrier frequency much greater than the cutoff frequency. The

TE01 is hence sometimes referred to as the low-loss mode. The properties of the

attenuation function can be attributed to the way that the mode fields fail to cohere to

the guide walls at high frequencies. This can result in problems with mode conversion if

the guide bends to go around corners. The

It has interesting properties beyond the waveguide as well. From Ramo et. Al. Fields

and Waves in Communication Electronics (1984), “In the TE01 mode, the electric field

lines do not end on the guide walls, but form closed circles surrounding the axial time-

varying magnetic field.” TE01 mode also has applications in measuring microwave

frequencies using wavemeters.

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TE11 Mode

The TE11 mode is the lowest order possible for a circular waveguide, and is the

fundamental mode of a circular waveguide, due to the fact that it has the lowest cutoff

frequency. It is sometimes referred to as the dominant mode. In this mode, the

Electrical field strength depends on both the radius and the azimuthal angle. Hence,

this mode is quite similar to the TE01 mode in a rectangular guide.

According to J. F. Kiang’s website on TE modes in cylindrical waveguides, the

magnitude of the field is the largest in the center of the waveguide and decreases

radially outwards. However, when the power in the waveguide increases, the field

strength increases, but the profile remains constant.

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REFERENCES AND WORKS CITED

Cronin, Nigel J. Microwave and optical waveguides. Bristol: Institute of Physics Pub.,

1995. Print.

"Electromagnetic wave equation -." Wikipedia, the free encyclopedia. Web. 01 Dec.

2009. <http://en.wikipedia.org/wiki/Electromagnetic_wave_equation>.

Holt, Charles A. Introduction to Electromagnetic Fields and Waves. New York: John

Wiley & Sons, 1963. Print.

Magnusson, Philip C., Gerald C. Alexander, Vijai K. Tripathi, and Andreas Weisshaar.

Transmission Lines and Wave Propagation, Fourth Edition. Null: CRC, 2000.

Print.

Ramo, Simon. Fields and waves in communication electronics. New York: Wiley, 1984.

Print.

"TE Modes of Cylindrical Waveguide." 台大電機系計算機中心. Web. 02 Dec. 2009.

<http://cc.ee.ntu.edu.tw/~jfkiang/electromagnetic%20wave/demonstrations/

demo_35/im2005_demo_35.htm>.