Dirichlet and Neumann boundary conditions in cylindrical waveguides

[tworitdash]

TL;DR Summary

The amplitude of the field patterns for a circular cross-section waveguide is not presented well in the book of Balanis (Advanced Engineering Electromagnetics). However, when I was doing the mathematics for Mode matching techniques, I stumbled upon a set of boundary conditions to solve the amplitude. Although the fact that I know it and the amplitude is not that important for my analysis, I want to get a better idea of what those mean.

The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is:

$$F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z}$$

There is no mention of how to solve for the constant $$A_{mn}$$. Then, from a paper related to the mode matching technique (https://www.researchgate.net/publication/221908717_Computer_Aided_Design_of_Waveguide_Devices_by_Mode-Matching_Methods Appendix A.2 and A.3), only for the tangential components of the fields (Both TE and TM), they have some expressions with a scalar potential function which looks like this:
$$\Phi_{TE} = (N_{mn}^{TE})^{1/2} J_m(\beta_{\rho}\rho) \cos(m\phi) = \Bigg(\frac{1}{|\frac{\pi}{2} ((\chi_{mn}^{'})^2 - m^2) J_m^2(\chi_{mn}^{'})|} \Bigg )^{\frac{1}{2}}J_m(\beta_{\rho}\rho) \cos(m\phi)$$ ,
$$\Phi_{TM} = (N_{mn}^{TM})^{1/2} J_m(\beta_{\rho}\rho) \cos(m\phi) = \Bigg(\frac{1}{|\frac{\pi}{2} \chi_{mn}^2 J_m^{'2}(\chi_{mn})|} \Bigg)^{\frac{1}{2}} J_m(\beta_{\rho}\rho) \cos(m\phi)$$

$$\chi_{mn}^{'}$$ are the nth zeros of the derivative of the Bessel function of the first kind of order m and $$\chi_{mn}$$ are the nth zeros of the mth order Bessel function of the first kind.

And, this is a solution to the differential equation:

$$\Delta_t \Phi_n + \beta_c^2 \Phi_n = 0$$ with boundary conditions like:

$$\int\int_{S} |\nabla \Phi_n|^2 dS = \beta_c^2 \int\int_S \Phi_n^2 dS = 1$$ The constants in the expressions of equations for the potentials given above depend only on the modes. And, somehow the first boundary condition integral exactly looks like the power carried by the mode of interest (without the frequency dependent term). And these integrals give us those constants.

What is the importance of this boundary condition? I can only imagine an amplitude term as the square root of power term, the amplitude here looks like that. However, it is presented without the frequency-dependent term of the square root of the power. Or, is it an arbitrary constant that was found to be unique for each mode? I want to understand the boundary conditions. Do I need to consider them to find the amplitude of the field patterns for a circular cross-section waveguide?

If I use these constants in the field equations and want to find the power again, the constants will cancel out with the integral involved in the power calculation which is (for TE):

$$P = Z_{mn}(f, m, n) \int\int_S (h_t . h_t^*) dS = N_{mn} \frac{1}{N_{mn}} dS = 1$$ which contradicts the power flow. Now, the power flow doesn't depend on the integral involving the field pattern over either radial or azimuthal direction.

Where $$h = (Y_{mn})^{\frac{1}{2}} \nabla \Phi$$

Or, do I miss something?

One more point:

The paper suggests to write the fields like the following:

$$h = (P_{mn})^{\frac{1}{2}} (Y_{mn})^{\frac{1}{2}} \nabla \Phi$$ which does not create any contradiction while finding out the power. However, if this is the case, then the constants N are of no importance again. They aren't the amplitude terms $$A_{mn}$$ I was looking for.

 

[marcusl]

The amplitudes are determined by the driving conditions just as in simple systems like a guitar string. There the modes describe all possible vibrational modes of the string, while their amplitudes depend on how the string is plucked or driven. Plucking it at different points along its length will change the combination of harmonics excited (changing the relative amplitudes) and plucking hard or softly will scale all of the amplitudes up or down. The same holds for the wave guide. Driving a guide with a stub will excite a different set of modes than driving it with a loop, and the set changes, as well, when the positions of the drivers change. These conditions determine the relative ratios of the A_mn. The excitation power scales them all up or down.

 

[tworitdash]

The amplitudes are determined by the driving conditions just as in simple systems like a guitar string. There the modes describe all possible vibrational modes of the string, while their amplitudes depend on how the string is plucked or driven. Plucking it at different points along its length will change the combination of harmonics excited (changing the relative amplitudes) and plucking hard or softly will scale all of the amplitudes up or down. The same holds for the wave guide. Driving a guide with a stub will excite a different set of modes than driving it with a loop, and the set changes, as well, when the positions of the drivers change. These conditions determine the relative ratios of the A_mn. The excitation power scales them all up or down.

Hi marcusl,

Thank you so much for the answer. I just realized that the boundary conditions are given for a normalized field distribution where they normalize the power with normalized field quantities. The amplitude of this potential function (Not the field patterns) doesn't change anything overall for the tangential fields.