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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 41 ECE

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Microstrip Line Dominant quasi-TEM mode: I0I0 We assume a purely x -directed current and a real wavenumber k x0. 2

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Fourier transform of current: Microstrip Line (cont.) 3

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Hence we have: Microstrip Line (cont.) 4 z = 0, z = 0

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Integrating over the -function, we have: Microstrip Line (cont.) where we now have 5

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Enforce EFIE using Galerkin’s method: Microstrip Line (cont.) x y The EFIE is enforced on the red line. Substituting into the EFIE integral, we have where Recall that (testing function = basis function) 6

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Microstrip Line (cont.) Since the Bessel function is an even function, Since the testing function is the same as the basis function, 7

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Using symmetry, we have This is a transcendental equation of the following form: Note: Microstrip Line (cont.) 8

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Branch points: Microstrip Line (cont.) Hence Note: The wavenumber k z 0 causes branch points to arise. 9

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Poles ( k y = k yp ): Microstrip Line (cont.) or 10

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Branch points: Poles: Microstrip Line (cont.) 11

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Note on wavenumber k x0 For a real wavenumber k x0, we must have that Otherwise, there would be poles on the real axis, and this would correspond to leakage into the TM 0 surface-wave mode of the grounded substrate. The mode would then be a leaky mode with a complex wavenumber k x0, which contradicts the assumption that the pole is on the real axis. Microstrip Line (cont.) Hence 12

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If we wanted to use multiple basis functions, we could consider the following choices: Microstrip Line (cont.) 13 Fourier-Maxwell Basis Function Expansion: Chebyshev-Maxwell Basis Function Expansion:

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Microstrip Line (cont.) We next proceed to calculate the Michalski voltage functions explicitly. 14

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Microstrip Line (cont.) 15

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Microstrip Line (cont.) 16

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At Hence Microstrip Line (cont.) 17

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Microstrip Line (cont.) E. J. Denlinger, “A frequency- dependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp , Jan Low-frequency results 18 h w Effective dielectric constant Parameters: w/h = 0.40 Dielectric constant of substrate ( r )

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Microstrip Line (cont.) Frequency variation 19 h w E. J. Denlinger, “A frequency- dependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp , Jan Effective dielectric constant Parameters: r = 11.7, w/h = 0.96, h = cm rr Frequency (GHz)

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Characteristic Impedance Microstrip Line (cont.) h w Original problem Equivalent problem (TEM) We calculate the characteristic impedance of the equivalent homogeneous medium problem. 20 h w 1) Quasi-TEM Method

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Using the equivalent TEM problem: Microstrip Line (cont.) (The zero subscript denotes the value when using an air substrate.) Simple CAD formulas may be used for the Z 0 of an air line. 21 h w

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2) Voltage-Current Method Microstrip Line (cont.) h w y z V

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Microstrip Line (cont.) 23

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Microstrip Line (cont.) 24

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Microstrip Line (cont.) 25 where The final result is

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Example (cont.) 26 We next calculate the function

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Example (cont.) Because of the short circuit, Therefore Hence At z = 0: 27

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Example (cont.) Hence 28 or

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Example (cont.) Hence where 29

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Example (cont.) 30 We then have Hence

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3) Power-Current Method Microstrip Line (cont.) Note: It is possible to perform the spatial integrations for the power flow in closed form (details are omitted). 31 h w y z

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4) Power-Voltage Method Microstrip Line (cont.) Note: It is possible to perform the spatial integrations for the power flow in closed form (details are omitted). 32 h w y z V + -

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Comparison of methods: Microstrip Line (cont.) At low frequency all three methods agree well. As frequency increases, the VI, PI, and PV methods give a Z 0 that increases with frequency. The Quasi-TEM method gives a Z 0 that decreases with frequency. The PI method is usually regarded as being the best one for high frequency (agrees better with measurements). 33

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Microstrip Line (cont.) r = 15.87, h = mm, w/h = Quasi-TEM Method 34 E. J. Denlinger, “A frequency- dependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp , Jan

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Microstrip Line (cont.) F. Mesa and D. R. Jackson, “A novel approach for calculating the characteristic impedance of printed-circuit lines,” IEEE Microwave and Wireless Components Letters, vol. 4, pp , April

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Microstrip Line (cont.) V e = effective voltage (average taken over different paths). VI PI PV V eIV eI PV e r = 10, h = mm, w/h = 1 (Circles represent results from another paper.) B Bianco, L. Panini, M. Parodi, and S. Ridella, “Some considerations about the frequency dependence of the characteristic impedance of uniform microstrips,” IEEE Trans. Microwave Theory and Techniques, vol. 26, pp , March GHz

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