The presence of material bodies complicates Maxwell’s equations. The fields in material media are related to each other through constitutive relations

The presence of material bodies complicates Maxwell’s equations.
The fields in material media are related to each other through constitutive relations

1. Overview Rectangular waveguides TE solution TM solution Cutoff frequency and cutoff wavelength

2. Rectangular waveguides Used for the frequency range 320 MHz to 333 GHz At 320MHz: 148.39cm x 73.9cm At 333 GHz: 0.086cm x 0.043cm The standard WR-90 X-band waveguide is used for 8.2-12.5GHz and has dimensions 2.29 cm x 1 cm. Applications: mainly to couple transmitters and receivers to antenna Rectangular metal waveguide

3. Unique circuits and currents do not exist and the waveguide is analysed as a field problem rather than as a distributed circuit parameter problem Rectangular waveguide supports TE nm and TM nm modes The integers n and m determine the number of standing wave interference maxima along the two transverse directions Each mode has its characteristic frequency below which it cannot exist (propagate) The cut-off frequency is defined by the geometry (size) of the waveguide

4. Rectangular waveguide Boundary conditions for the E field: Tangential E field components are zero on the metal boundary

5. TM modes – rectangular waveguide For the TM modes Hz=0 Wave equation is solved for E z k c is the cut-off wavenumber To solve (1) the technique known as separation of variables is used. The Ez component of the field is given as: The boundary conditions are: E z =0 at x=0,a E z =0 at y=0,b (1) where a b

6. TM modes - solution TM mode field solution ( see Appendix B for more detail ): k c is the cut-off wavenumber The cutoff frequency is: and

7. TM modes - solution The cutoff wavelength is: The propagation constant is given by: It is a function of dimensions and integers n and m only!

8. The transverse field components for the TMnm mode are:

9. TM modes - solution TM mode field solution: Observe the field expressions on the previous slide for n=0 or m=0. There is no field solution when n=0 or m=0. The lowest propagating mode is the n=m=1 mode! The cutoff wavelenght of the lowest TM propagating mode is:

10. x [m] y [m] TM 11 mode field profile Waveguide: a=22.9mm, b=10mm

11. TM 11 mode field profile

12. TM 21

13. TM 12

14. TM 22 TM 32

15. TE modes for rectangular waveguides - solution Ez=0 k c is the cut-off wavenumber The cutoff frequency is:

16. TE modes - solution The cutoff wavelenght is: The propagation factor is given by:

17. TE modes - solution TE mode field solution: The lowest propagating mode is the n=1, m=0 mode The cutoff wavelenght of the lowest TE propagating mode is: A typical guide has dimensions a=2b. For the wavelengths from l =a to l =2a only TE 10 - mode propagates – this is a dominant mode of the metal waveguides

18. The dominant TE 10 mode The lowest propagating mode is the n=1, m=0 mode TE 10 mode field solution: The cutoff wavelength of the lowest TE propagating mode is:

19. The dominant TE 10 mode A typical guide has dimensions a=2b. Lets calculate cutoff frequencies for the first few modes: For the wavelengths from l =a to l =2a only TE 10 - mode propagates – this is a dominant mode of the metal waveguides

20. Mode exists is b >0 a =22.3mm b =10mm

21. Waveguide Dimensions [mm] Useful TE10 range [GHz] WR340 86x43 2.2-3.3 WR90 23x10 8.2-12.4 WR28 7x3.6 26.5-40 WR10 2.54x1.27 75-110 WR4 1.02x0.508 170-260

22. General solution: At y=0, E=0: At y=b, E=0: Final solution: The wave equation to be solved: Constants A and B are found by satisfying boundary conditions: k y =transversal prop.constant Appendix A: TE modes of the planar waveguides

23. Appendix B: TM mode derivation for the rectangular waveguide Maxwell’s equations are: If we assume that the fields propagate along z coordinate as e -j b z then above equations (1,2) can be simplified by replacing derivatives along z as d/dz=-j b. We can also eliminate Ez and Hz from above set of 6 equation and re-express transversal field components (Ex, Ey, Hx, Hy) in terms of Ez and Hz as: (1) (2)

24. (3) If we know either Ez or Hz we can find TE or TM field components.

25. Appendix B: TM mode derivation for the rectangular waveguide At the metal boundary x=0, for all y E z =0 so: For the TM mode we know that Hz=0 so we need to solve wave equation for Ez. The solution can be found by assuming that the field is given in the form where

26. At x=a, for all y values we get: Two solutions are possible for the above equation: A 1 =0 (trivial solution) or sin( k x a )=0. If A 1 =0 then we will completely remove function f(x) from the field solution, so it must be

27. Similarly for y=0, and all x, E=0: Two solutions are possible for the above eq : B 1 =0 or sin( k y b )=0. If B 1 =0 then we will completely remove function f(y) from the field solution, so it must be For y=b, and all x E=0:

28. The solution for Ez is: Other field components are found from Maxwell’s equation eq(3) by replacing the above solution and assuming Hz=0, i.,e: For more detail see: (1) David M. Pozar , Microwave Engineering ; (2) Simon Ramo , Fields and wave in communication electronics ; (3) Robert Collin, Foundations for Microwave Engineering