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Contents
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| Week 1: |
Introduction. Time-domain formulation of the cavity problem. Extraction of a self-adjoint operator, which act on the coordinates only, from the system of the Maxwell’s equations. |
| Week 2: |
Definition of the space of solutions as a Hilbert functional space. Operator eigenvalue equation in the space of solutions. Derivation of an operator equation for the self-adjoint operator. Equivalent boundary-value problems for the Laplacian |
| Week 3: |
Definition of the solenoidal and irrotational eigenvectors of the self-adjoint operator in the space of solutions. Modal basis. |
| Week 4: |
Normalization conditions. Orthogonality of the basis elements. Weyl theorem for the space of solutions. Weyl theorem. Completeness of the set of the solenoidal and irrotational eigenvectors in the space of solutions. |
| Week 5: |
Projecting of the cavity fields, sought for, onto the elements of the basis in the space of solutions. Modal decompositions of the fields where the basis elements are already known as the vectors functions of coordinates but the modal amplitudes are unknown yet scalar functions of time t. |
| Week 6: |
Projecting of the Maxwell’s equations onto the basis elements.
Derivation of the evolutionary equations for the modal amplitudes. Cauchy problem for the modal amplitudes.
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| Week 7: |
Modal basis for the rectangular cavity. Two sets of the solenoidal modes. Two sets of the irrotational modes modes. |
| Week 8: |
The modal decompositions for cavity fields in the general case. Cauchy problems for the modal amplitudes. Evolution of the modal amplitudes in the general case. |
| Week 9: |
Free and force oscillations in the cavities. Application the matrix exponential for solving the systems of evolutionary equations. Transient processes. |
| Week 10: |
Midterm examination. |
| Week 11: |
Excitation of a cavity by a sinusoidal signal which has a beginning in time. Exact solution. Resonances. Transient processes. Regular oscillations as an asymptotic. |
| Week 12: |
Excitation of a cavity by an instant signal describable by theHeaviside step function. Exact solution. Resonances. Transient processes. Regular oscillations as an asymptotic. |
| Week 13: |
Excitation of a cavity by a transient signal describable by the double eexponential function. Exact solution Reaction of the solenoidal and irrotational modes on the force funtion. |
| Week 14: |
Coupled oscillations in a cavity. Coupling of the solenoidal modes. Coupling of the solenoidal an irrotational modes. |
| Week 15*: |
Possible ways of further extensions of the evolutionary approach to electromagnetics. Dynamical constitutive relations for the Lorentz and Debye media. Convolution integrals. |
| Week 16*: |
Final examination. |
| Textbooks and materials: |
1. O.A. Tretyakov, “Evolutionary approach to electromagnetics in time domain,” Lecture notes, Part 1, Cavity problem.
2. O.A. Tretyakov,.’’Essentials of nonstationary and nonlinear electromagnetic field theory,’’.Chapter 3 in book Analytical and numerical methods in electromagnetic wave theory, M. Hashimoto, M. Idemen, O. A. Tretyakov, Eds., Tokyo, Science House Co. Ltd., 1993.
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| Recommended readings: |
1. F. Erden and O. A. Tretyakov, .” Excitation by a transient signal of the real-valued electromagnetic fields in a cavity,”.JEMWA, vol. 17, no. 12, 1665-1682, 2003.
2. Aksoy, S. and O. A. Tretyakov, “Study of a time variant cavity system,” J. Electromagn. Waves Appl., Vol. 16, No. 11, 1535-1553, Nov. 2002.
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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