Electromagnetic (EM) waveguides are hollow structures with metallic walls and central core that is either evacuated or filled with pressurized dielectric gas. It is mainly used for transmission of microwave power and signals; however, with some structural modifications, it can also be used for

Question

Electromagnetic (EM) waveguides are hollow structures with metallic walls and central core that is either evacuated or filled with pressurized dielectric gas. It is mainly used for transmission of microwave power and signals; however, with some structural modifications, it can also be used for acceleration of charged particles. The cross section of EM waveguides is either rectangular with core sides a and b where a>b or circular with core radius a.

(a) For a uniform rectangular evacuated EM waveguide:

(1) State the dispersion relationship (ω,[latex]k_z[/latex]) for a given mode mn.
(2) State the general cutoff frequency [latex]\left(ω_c ight)_{mn}[/latex] for a given mode mn.
(3) For the [latex]TM_{mn}[/latex] mode where [latex]B_z = 0[/latex] everywhere and [latex]\left. {E}_z ight|_{x=0, a}= \left. {E}_z ight|_{y=0, b}=0 ext { determine the lowest cutoff frequency }\left(\omega_c ight)_{m n}[/latex] as well as the z component [latex]E_z[/latex] of the electric field E.
(4) For the [latex]TE_{mn}[/latex] mode where [latex]E_z = 0[/latex] everywhere and [latex]\partial \mathcal{B}_z /\left.\partial z ight|_{x=0, a}= \partial \mathcal{B}_z /\left.\partial z ight|_{y=0, b}=0 ext { determine the lowest cutoff frequency }\left(\omega_{\mathrm{c}} ight)_{m n}[/latex] as well as the z component [latex]B_z[/latex] of the magnetic field B.
(5) Determine the ratio between the lowest TM cutoff frequency and the lowest TE cutoff frequency.
(6) Determine the cutoff frequency [latex]ω_c[/latex] of the waveguide.

(b) A uniform rectangular evacuated EM waveguide is used for transmission of microwave power. The longer side a of the rectangular cross section of the waveguide core is 8.05 cm, the shorter side b is 3.26 cm. For the waveguide determine: (1) the 5 lowest cutoff frequencies of the [latex]TE_{mn}[/latex] modes, (2) the 5 lowest cutoff frequencies of [latex]TM_{mn}[/latex] modes, and (3) the 5 lowest modes (either TE or TM) that will be allowed to propagate in the waveguide.

(c) The waveguide of (b) is used for transmission of microwave power in the S microwave band at ν = 2856 MHz. State the modes of the 2856 MHz microwave input that will be allowed to propagate in the EM waveguide.

(d) Determine the phase velocity [latex]υ_{ph}[/latex] and group velocity [latex]υ_{gr}[/latex] of the 2856 MHz microwaves propagating in the waveguide of (b).

Accepted Answer

(a) Parameters of uniform rectangular evacuated EM waveguide (see Prob. 279):

(1) Dispersion relationship (ω,[latex]k_z[/latex]) for a given mode mn is expressed as follows

[latex]\omega^2=\left(\omega_{\mathrm{c}} ight)_{m n}^2+c^2 k_z^2,[/latex] (13.165)

where [latex]ω_{mn}[/latex] is the microwave angular frequency, [latex]\left(ω_c ight)_{mn}[/latex] is the cutoff frequency, and [latex]k_z[/latex] is the propagation coefficient.

(2) In general as well as for special transverse magnetic (TM) and transverse electric (TE) modes the cutoff frequency [latex]\left(ω_c ight)_{mn}[/latex] for a given mode mn is expressed as

[latex]\left(\omega_{\mathrm{c}} ight)_{m n}=c \sqrt{k_x^2+k_y^2}=\pi c \sqrt{\left(\frac{m}{a} ight)^2+\left(\frac{n}{b} ight)^2} .[/latex] (13.166)

(3) Lowest cutoff frequency [latex]\left(\omega_c^{\mathrm{TM}} ight)_{m n} ext { for } \mathrm{TM}_{m n}[/latex] modes. The [latex]\mathrm{TM}_{m n}[/latex] modes are: (i) characterized by [latex]B_z=0[/latex] everywhere in the core of the waveguide and (ii) governed by the Dirichlet-type boundary condition on the boundary between the core and wall of the waveguide, i.e., [latex]\left. {E}_z ight|_{x=0, a}=\left. {E}_z ight|_{y=0, b}=0 ext {. The } \mathrm{TM}_{m n}[/latex] solution of the wave equation for [latex]E_z[/latex] accounting for the Dirichlet-type boundary condition is

[latex] {E}_z(x, y, z, t)=\sum_m \sum_n\left( {E}_{z 0} ight)_{m n} \sin \frac{m \pi}{a} x \sin \frac{n \pi}{b} y e^{i\left(k_z z-\omega t ight)},[/latex] (13.167)

with [latex]\left(E_{z0} ight)_{mn}[/latex] the amplitude of the TM wave. Based on (13.166) and (13.167) we note that the lowest non-trivial TM mode will be characterized by m = 1 and n = 1 to give the following cutoff frequency [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TM}} ight)_{11}[/latex].

[latex]\left(\omega_{\mathrm{c}}^{\mathrm{TM}} ight)_{11}=\pi c \sqrt{\frac{1}{a^2}+\frac{1}{b^2}}=\frac{\pi c}{a} \sqrt{1+\left(\frac{a}{b} ight)^2}[/latex] (13.168)

and z component [latex]E_z[/latex] of the electric field E

[latex] {E}_z(x, y, z, t)=\left( {E}_{z 0} ight)_{11} \sin \frac{\pi}{a} x \cos \frac{\pi}{b} y e^{i\left(k_z z-\omega t ight)} .[/latex] (13.169)

(4) Lowest cutoff frequency [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{m n} ext { for } \mathrm{TE}_{m n} ext { modes. The } \mathrm{TE}_{m n}[/latex] modes are (i) characterized by [latex]E_z[/latex] = 0 everywhere in the core of the waveguide and (ii) governed by the Neumann-type boundary condition on the boundary between the core and wall of the waveguide, i.e., [latex]\mathrm{d} \mathcal{B}_z /\left.\mathrm{d} z ight|_{x=0, a}=\mathrm{d} \mathcal{B}_z /\left.\mathrm{d} z ight|_{\mathrm{y}=0, b}=0 ext {. The TE }_{m n}[/latex] solution of the wave equation for [latex]B_z[/latex] accounting for the Neumann-type boundary condition is

[latex]\mathcal{B}_z(x, y, z, t)=\sum_m \sum_n\left(\mathcal{B}_{z 0} ight)_{m n} \cos \frac{m \pi}{a} x \cos \frac{n \pi}{b} y e^{i\left(k_z z-\omega t ight)},[/latex] (13.170)

with [latex]\left(B_{z0} ight)_{mn}[/latex] the amplitude of the TE wave.
Based on (13.166) and (13.167) we note that the lowest non-trivial TE mode for a>b will be characterized by m = 1 and n = 0 to give the following cutoff frequency [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}[/latex].

[latex]\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}=\pi c \sqrt{\frac{1}{a^2}}=\frac{\pi c}{a}[/latex] (13.171)

and z component [latex]B_z[/latex] of the magnetic field B

[latex]\mathcal{B}_z(x, y, z, t)=\left(\mathcal{B}_{z 0} ight)_{10} \cos \frac{\pi}{a} x e^{i\left(k_z z-\omega t ight)}[/latex] (13.172)

(5) Using (13.168) and (13.169), the ratio between the lowest cutoff frequency [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TM}} ight)_{11}[/latex] for the TM modes and the lowest cutoff frequency [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}[/latex] for the TE modes is given as

[latex]\frac{\left(\omega_{\mathrm{c}}^{\mathrm{TM}} ight)_{11}}{\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}}=\sqrt{1+\left(\frac{a}{b} ight)^2} .[/latex] (13.173)

Since a>b, we note that [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TM}} ight)_{11}>\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}[/latex] and this tells us that the lowest cutoff frequency for all modes in the rectangular waveguide is the [latex]TE_{10}[/latex] mode.

(6) Cutoff frequency [latex]ω_c[/latex] for a given waveguide is defined as the cutoff frequency of the lowest mode mn that can propagate through a waveguide. Thus, the [latex]TE_{10}[/latex] is the mode with the lowest cutoff frequency of all modes in a uniform rectangular EM waveguide and therefore the cutoff frequency for rectangular waveguides is from (13.166) given as [latex]\left(\omega_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}=\pi c / a[/latex]. Note that the cutoff frequency of a rectangular waveguide is inversely proportional to a, the long side of the rectangular waveguide cross section but does not depend on b, the short side of the rectangular waveguide cross section.
Waveguide for transmission of a given radiofrequency (RF) are usually designed such that, at the given RF, the only mode they transmit is the [latex]TE_{10}[/latex] mode. This means that the cutoff frequencies of all TM modes as well as the cutoff frequencies of all TE modes above the [latex]TE_{10}[/latex] mode exceed the given RF.

(b) The general dispersion equation used for the determination of the cutoff frequency [latex]\left(ν_c ight)_{mn}[/latex] of the mn mode in a rectangular EM waveguide is given as

[latex]\left(v_{\mathrm{c}} ight)_{m n}=\frac{1}{2 \pi}\left(\omega_{\mathrm{c}} ight)_{m n}=\frac{c}{2} \sqrt{\left(\frac{m}{a} ight)^2+\left(\frac{n}{b} ight)^2} .[/latex] (13.174)

The same equation (13.174) is used for the special [latex]TM_{mn}[/latex] and [latex]TE_{mn}[/latex] modes; however, we must recognize the lower limits on m and n for each of the special modes as a result of the boundary conditions. As shown in (a), the lower limits on m and n are as follows: for [latex]TM_{mn}[/latex] modes m ≥ 1 and n ≥ 1, while for [latex]TE_{mn}[/latex] m ≥ 1 and n ≥ 0 or m ≥ 0 and n ≥ 1.
We now use (13.175) to determine the cutoff frequencies for a set of [latex]TE_{mn}[/latex] and [latex]TM_{mn}[/latex] modes for various values of m and n starting with the lowest allowed values. The results are shown in Table 13.3. We also rank the [latex]TE_{mn}\ and\ TE_{mn}[/latex] modes for the first five cutoff frequencies starting with the lowest value.
According to Table 13.3:

(1) Five lowest cutoff frequencies

[latex]\left(v_{\mathrm{c}}^{\mathrm{TE}} ight)_{m n} ext { for } \mathrm{TE}_{m n} ext { modes are: } \mathrm{TE}_{10}[/latex] (1863 MHz), [latex]\mathrm{TE}_{20}(3726 \mathrm{MHz}), \mathrm{TE}_{01}(4600 \mathrm{MHz}), \mathrm{TE}_{11}(4964 \mathrm{MHz}), \mathrm{TE}_{30}[/latex]

(5590 MHz).
(2) Five lowest cutoff frequencies for the [latex]TM_{mn}[/latex] modes are: [latex]TM_{11}[/latex] (4964 MHz), [latex]TM_{21}[/latex] (5921 MHz), [latex]TM_{31}[/latex] (7240 MHz), [latex]TM_{41}[/latex] (8759 MHz), [latex]TM_{12}[/latex] (9389 MHz).
(3) Combined order of 5 lowest special modes in the waveguide is as follows:

[latex]\mathrm{TE}_{10}(1863 \mathrm{MHz}), \mathrm{TE}_{20}(3726 \mathrm{MHz}), \mathrm{TE}_{01}(4600 \mathrm{MHz}), \mathrm{TM}_{11}[/latex] (4964 MHz), [latex]TE_{11}[/latex]

(4964 MHz).

(c) The input microwaves to be transmitted through the waveguide have a frequency ν = 2856 MHz that is below the lowest [latex]TM_{11}[/latex] mode cutoff, but is above the waveguide cutoff of 1863 MHz (lowest cutoff for [latex]TE_{mn}[/latex] modes). Since the second lowest cutoff for [latex]TE_{mn}[/latex] modes is at 3736 MHz [latex]\left(TM_{20} ight)[/latex] exceeding 2856 MHz, the input microwaves can propagate only with the [latex]TE_{10}[/latex] mode. This “single mode operation” reflects the standard approach to design of transmission waveguides where the cross sectional dimensions are chosen such that they allow only one mode and all the other modes are excluded.

(d) As derived in Prob. 279 [(13.161) and (13.162)], the phase velocity [latex]υ_{ph}[/latex] and group velocity [latex]υ_{gr}[/latex] of microwaves propagating in a uniform rectangular EM waveguide with angular frequency ω or frequency ν are, respectively, expressed as

[latex]v_{\mathrm{ph}}=\frac{c}{\sqrt{1-\frac{\left(\omega_{\mathrm{c}} ight)_{mn}^2}{\omega^2}}}=\frac{c}{\sqrt{1-\frac{\left( u_{\mathrm{c}} ight)_{m n}^2}{ u^2}}}[/latex] (13.175)

and

[latex]v_{\mathrm{ph}}=c \sqrt{1-\frac{\left(\omega_{\mathrm{c}} ight)_{m n}^2}{\omega^2}}=c \sqrt{1-\frac{\left(v_{\mathrm{c}} ight)_{m n}^2}{v^2}} .[/latex] (13.176)

As shown in (b), microwaves of frequency ν = 2856 MHz propagate through rectangular EM waveguide (a = 8.05 cm and b = 3.26 cm) in only one mode (transverse electric [latex]TE_{10}[/latex] mode) for which the cutoff frequency is [latex]\left(v_{\mathrm{c}}^{\mathrm{TE}} ight)_{10}=1863 \mathrm{MHz}[/latex], as shown in Table 13.6.

(1) Phase velocity [latex]υ_{ph}[/latex] is calculated using (13.175) as follows

[latex]\begin{aligned} v_{ ext {ph }} & =\frac{c}{\sqrt{1-\frac{\left(\omega_{\mathrm{c}} ight)_{m n}^2}{\omega^2}}}=\frac{c}{\sqrt{1-\frac{\left(v_{\mathrm{c}} ight)_{m n }^2}{v^2}}}=\frac{c}{\sqrt{1-\left(\frac{1863}{2856} ight)^2}} \ & =1.32 c=3.96 imes 10^8 \mathrm{~m} / \mathrm{s}>c .\quad (13.177) \end{aligned}[/latex]

(2) Group velocity [latex]υ_{gr}[/latex] is calculated using (13.176) as

[latex]\begin{aligned} v_{\mathrm{ph}} & =c \sqrt{1-\frac{\left(\omega_{\mathrm{c}} ight)_{m n}^2}{\omega^2}}=c \sqrt{1-\frac{\left(v_{\mathrm{c}} ight)_{m n}^2}{v^2}}=c \sqrt{1-\left(\frac{1863}{2856} ight)^2} \ & =0.76 c=2.27 imes 10^8 \mathrm{~m} / \mathrm{s}

Question 13.6.Q2: Electromagnetic (EM) waveguides are hollow structures with m......

Related Answered Questions

Verified Answer:

Verified Answer:

Two types of waveguide are used for transmission of microwave power and signals: rectangular and circular. Waveguides are usually oriented with their central axes parallel to the z axis of the Cartesian coordinate system for rectangular waveguides and cylindrical coordinate system for circular ...

Verified Answer:

In a clinical linear accelerator electrons are generated by an electron gun and injected into the acceleration waveguide with a relatively low kinetic energy (EK)0 between 20 keV and 150 keV (depending on the gun design). At the end of the acceleration process the electrons exit the acceleration ...

Verified Answer:

Verified Answer:

Verified Answer:

Verified Answer:

The theory of waveguides used for charged particle acceleration in linear accelerators (linacs) is very complex and relies on many approximations and empirical relationships. However, the acceleration waveguide theory is based solidly on the theory of uniform circular EM waveguides that is well ...

Verified Answer:

Electromagnetic (EM) waveguides are used for transmission of microwave power and signals as well as for charged particle acceleration in linear accelerators (linacs). (a) Briefly describe at least five notable differences between EM waveguides used for: (1) Transmission of microwave power and ...

Verified Answer:

Verified Answer: