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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 41 ECE 6341 1 Slide 2 Microstrip Line Dominant quasi-TEM mode: I0I0 We assume a purely x -directed current and a real wavenumber k x0. 2 Slide 3 Fourier transform of current: Microstrip Line (cont.) 3 Slide 4 Hence we have: Microstrip Line (cont.) 4 z = 0, z = 0 Slide 5 Integrating over the -function, we have: Microstrip Line (cont.) where we now have 5 Slide 6 Enforce EFIE using Galerkins 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 Slide 7 Microstrip Line (cont.) Since the Bessel function is an even function, Since the testing function is the same as the basis function, 7 Slide 8 Using symmetry, we have This is a transcendental equation of the following form: Note: Microstrip Line (cont.) 8 Slide 9 Branch points: Microstrip Line (cont.) Hence Note: The wavenumber k z 0 causes branch points to arise. 9 Slide 10 Poles ( k y = k yp ): Microstrip Line (cont.) or 10 Slide 11 Branch points: Poles: Microstrip Line (cont.) 11 Slide 12 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 Slide 13 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: Slide 14 Microstrip Line (cont.) We next proceed to calculate the Michalski voltage functions explicitly. 14 Slide 15 Microstrip Line (cont.) 15 Slide 16 Microstrip Line (cont.) 16 Slide 17 At Hence Microstrip Line (cont.) 17 Slide 18 Microstrip Line (cont.) E. J. Denlinger, A frequency- dependent solution for microstrip transmission lines, IEEE Trans. Microwave Theory and Techniques, vol. 19, pp. 30-39, Jan. 1971. Low-frequency results 18 h w Effective dielectric constant 5.0 6.0 7.0 8.0 9.0 10.0 Parameters: w/h = 0.40 Dielectric constant of substrate ( r ) Slide 19 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. 30-39, Jan. 1971. Effective dielectric constant 7.0 8.0 9.0 10.0 11.0 12.0 Parameters: r = 11.7, w/h = 0.96, h = 0.317 cm rr Frequency (GHz) Slide 20 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 Slide 21 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 Slide 22 2) Voltage-Current Method Microstrip Line (cont.) h w y z V + - 22 Slide 23 Microstrip Line (cont.) 23 Slide 24 Microstrip Line (cont.) 24 Slide 25 Microstrip Line (cont.) 25 where The final result is Slide 26 Example (cont.) 26 We next calculate the function Slide 27 Example (cont.) Because of the short circuit, Therefore Hence At z = 0: 27 Slide 28 Example (cont.) Hence 28 or Slide 29 Example (cont.) Hence where 29 Slide 30 Example (cont.) 30 We then have Hence Slide 31 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 Slide 32 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 + - Slide 33 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 Slide 34 Microstrip Line (cont.) r = 15.87, h = 1.016 mm, w/h = 0.543 Quasi-TEM Method 34 E. J. Denlinger, A frequency- dependent solution for microstrip transmission lines, IEEE Trans. Microwave Theory and Techniques, vol. 19, pp. 30-39, Jan. 1971. Slide 35 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. 283-285, April 2005. 35 Slide 36 Microstrip Line (cont.) V e = effective voltage (average taken over different paths). VI PI PV V eIV eI PV e r = 10, h = 0.635 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. 182-185, March 1978. 8.0 12.0 16.0 4.0 1.0 GHz 58 52 46 40 36