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Notes 13

ECE 5317-6351 Microwave Engineering. Fall 2011. Fall 2011. Prof. David R. Jackson Dept. of ECE. Notes 13. Transverse Resonance Method. Transverse Resonance Method. This is a general method that can be used to help us calculate various important quantities:.

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Notes 13

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  1. ECE 5317-6351 Microwave Engineering Fall 2011 Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 13 Transverse Resonance Method

  2. Transverse Resonance Method This is a general method that can be used to help us calculate various important quantities: • Wavenumbers for complicated waveguiding structures (dielectric-loaded waveguides, surface waves, etc.) • Resonance frequencies of resonant cavities We do this by deriving a “Transverse Resonance Equation (TRE).”

  3. Transverse Resonance Equation (TRE) R To illustrate the method, consider a lossless resonator formed by a transmission line with reactive loads at the ends. x x = x0 x = L R = reference plane at arbitrary x = x0 We wish to find the resonance frequency of this transmission-line resonator.

  4. TRE (cont.) R x x = L x = x0 Examine the voltages and currents at the reference plane: I r I l R + V r- + V l- x = x0

  5. R I r I l + V r- + V l- x x = x0 TRE (cont.) Define impedances: Boundary conditions: Hence:

  6. TRE (cont.) R TRE Note about the reference plane: Although the location of the reference plane is arbitrary, a “good” choice will keep the algebra to a minimum. or

  7. Example Derive a transcendental equation for the resonance frequency of this transmission-line resonator. L x We choose a reference plane at x = 0+.

  8. Example (cont.) L R x Apply TRE:

  9. Example (cont.)

  10. Example (cont.) After simplifying, we have Special cases:

  11. Rectangular Resonator Derive a transcendental equation for the resonance frequency of a rectangular resonator. z PEC boundary Orient so that b < a < h h y a x b The structure is thought of as supporting RWG modes bouncing back and forth in the z direction. The index p describes the variation in the z direction. We have TMmnp and TEmnp modes.

  12. Rectangular Resonator (cont.) h We use a Transverse Equivalent Network (TEN): z We choose a reference plane at z = 0+. Hence

  13. Rectangular Resonator (cont.) Hence h z

  14. Rectangular Resonator (cont.) Solving for the wavenumber we have Hence Note: The TMz and TEz modes have the same resonance frequency. TEmnp mode: or The lowest mode is the TE101 mode.

  15. Rectangular Resonator (cont.) TE101 mode: Note: The sin is used to ensure the boundary condition on the PEC top and bottom plates: The other field components, Ey and Hx, can be found from Hz.

  16. Rectangular Resonator (cont.) Practical excitation by a coaxial probe z Lp (Probe inductance) PEC boundary h R L C y Tank (RLC) circuit a x b Circuit model

  17. Rectangular Resonator (cont.) Lp (Probe inductance) Q = quality factor of resonator R L C Tank (RLC) circuit Circuit model

  18. Rectangular Resonator (cont.)

  19. x h z Grounded Dielectric Slab Derive a transcendental equation for wavenumber of the TMx surface waves by using the TRE. Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction.

  20. x H TMx E z Grounded Dielectric Slab

  21. TMx Surface-Wave Solution R h The reference plane is chosen at the interface. TEN: x

  22. TMx Surface-Wave Solution (cont.) TRE:

  23. TMx Surface-Wave Solution (cont.) Letting We have or Note: This method was a lot simpler than doing the EM analysis and applying the boundary conditions!

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