1. Hanyang University 1/29 Antennas & RF Devices Lab. Partially filled wave guide Jeong Gu Ho

2. Hanyang University 2/29 Contents A. Derive the one governing equation and two auxiliary wavenumber equations for the propagation constant of mode B. plot the propagation constant of mode versus the normalized frequency for the following specific waveguide : a=2.286[cm], b=1.016[cm], h=b/3[cm], in other words, reproduce the Fig. 8-16. C. Derive the on governing equation and two auxiliary wavenumber equations for the propagation constant of mode by using the transverse resonance method (TRM) D. derive the propagation constant of mode

3. Hanyang University 3/29 In differential form, Maxwell’s equation can be written as Gauss law Gauss law for magnetism Faraday’s law Ampere law

4. Hanyang University 4/29 Time varying electromagnetic fields Assuming homogeneous equation. Vector identity Faraday’s law Ampere law

5. Hanyang University 5/29 For source free region (, ) For source free region and lossless media (, ) Maxwell equation Uncoupled second-order differential equation for E,HVector wave equation Simplest form of the vector wave equations

6. Hanyang University 6/29 Time-harmonic electromagnetic fields (time variations of the form( ) Time harmonic Time varying Vector wave equation

7. Hanyang University 7/29 Solution to the wave equation (time - harmonic form) –Source free and lossless media(,, ) –In rectangular coordinates, a general solution for E can be written as Scalar wave equations Time harmonic wave equation

8. Hanyang University 8/29 Separation of variable method Propagation constant Using boundary conditions

9. Hanyang University 9/29 Traveling wave Standing wave Propagation for z axis Standing wave Traveling wave Separation of variable method

10. Hanyang University 10/29 Assuming all constants real and inserting time domain +Z direction

11. Hanyang University 11/29 Rectangular coordinate system (the electric field in terms of the vector potentials A and F in source free region) In source free (J=0) TEM electric field in terms of the vector potentials A and F Magnetic flux density B electric flux density D

12. Hanyang University 12/29 Result

13. Hanyang University 13/29 The need for TM and TE modes are modes of propagation in waveguides.

14. Hanyang University 14/29 TM Separation-of- variables method Scalar wave equation Vector wave equation Rectangular waveguide equation

15. Hanyang University 15/29 Once A z is found, next step is to find the E and H field components

16. Hanyang University 16/29 Rectangular waveguide Separation-of- variables method Standing wave traveling wave Satisfy the following set of equation. : Scalar potential function TE

17. Hanyang University 17/29 For +z traveling wave On the bottom and top wall On the left and right wall Boundary condition 1 2 3 4

18. Hanyang University 18/29 x component of the electric field 1

19. Hanyang University 19/29 3 Boundary conditions For +z traveling waves (result)

20. Hanyang University 20/29 TE +z mn

21. Hanyang University 21/29 Dielectric region 1 Dielectric region 2 1 2 1 2 TE y

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23. Hanyang University 23/29 Boundary conditions

24. Hanyang University 24/29 Boundary conditions

25. Hanyang University 25/29 Result 1

26. Hanyang University 26/29 Result 2 Result 2 / Result 1

27. Hanyang University 27/29 Transverse resonance method (TRM) –The cross section of the waveguide or traveling wave antenna structure is represented as a transmission line system. –The formulations of this method are much simpler when applied to finding the propagation constants. –The objective here is to analyze the waveguide geometry.

28. Hanyang University 28/29 Wave impedance of TE z mn modes

29. Hanyang University 29/29 Antennas & RF Devices Lab. Thank you for your attention

30. Hanyang University 30/29 Vector potential A magnetic – flux density B is always solenoidal. Vector identity

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32. Hanyang University 32/29 Vector potential A and F

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37. Hanyang University 37/29 Cutoff frequency of given mn mode Traveling in the +z direction

38. Hanyang University 38/29 Cutoff occurs when =0

39. Hanyang University 39/29 Scalar wave equation Transverse wave equation Transverse direction wave number