ECE 6341. Spring 2014. Prof. David R. Jackson ECE Dept. Notes 41. Microstrip Line. I 0. Microstrip Line. Dominant quasiTEM mode :. We assume a purely x directed current and a real wavenumber k x 0. Microstrip Line (cont.). Fourier transform of current:. Microstrip Line (cont.).
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I0
Microstrip Line
Dominant quasiTEM mode:
We assume a purely xdirected current and a real wavenumber kx0.
Fourier transform of current:
y
x
Enforce EFIE using Galerkin’s method:
The EFIE is enforced on the red line.
where
(testing function = basis function)
Recall that
Substituting into the EFIE integral, we have
Since the testing function is the same as the basis function,
Since the Bessel function is an even function,
Using symmetry, we have
This is a transcendental equation of the following form:
Note:
Note on wavenumber kx0
For a real wavenumber kx0, we must have that
Otherwise, there would be poles on the real axis, and this would correspond to leakage into the TM0 surfacewave mode of the grounded substrate.
The mode would then be a leaky mode with a complex wavenumber kx0, which contradicts the assumption that the pole is on the real axis.
Hence
If we wanted to use multiple basis functions, we could consider the following choices:
FourierMaxwell Basis Function Expansion:
ChebyshevMaxwell Basis Function Expansion:
We next proceed to calculate the Michalski voltage functions explicitly.
10.0
Lowfrequency results
9.0
8.0
Effective dielectric constant
w
7.0
h
6.0
5.0
Dielectric constant of substrate (r)
E. J. Denlinger,“A frequencydependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp. 3039, Jan. 1971.
Parameters: w/h = 0.40
12.0
r
11.0
10.0
w
Effective dielectric constant
h
9.0
Frequency variation
8.0
7.0
Frequency (GHz)
Parameters: r = 11.7, w/h = 0.96, h = 0.317 cm
E. J. Denlinger,“A frequencydependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp. 3039, Jan. 1971.
w
h
Characteristic Impedance
1) QuasiTEM Method
w
h
Original problem
Equivalent problem (TEM)
We calculate the characteristic impedance of the equivalent homogeneous medium problem.
Using the equivalent TEM problem:
(The zero subscript denotes the value when using an air substrate.)
w
h
Simple CAD formulas may be used for the Z0 of an air line.
We next calculate the function
z
w
y
h
3) PowerCurrent Method
Note: It is possible to perform the spatial integrations for the power flow in closed form (details are omitted).
z
w
y
+
h
V

4) PowerVoltage Method
Note: It is possible to perform the spatial integrations for the power flow in closed form (details are omitted).
Comparison of methods:
QuasiTEM Method
r = 15.87, h = 1.016 mm, w/h= 0.543
E. J. Denlinger,“A frequencydependent solution for microstrip transmission lines,” IEEE Trans. Microwave Theory and Techniques, vol. 19, pp. 3039, Jan. 1971.
F. Mesa and D. R. Jackson, “A novel approach for calculating the characteristic impedance of printedcircuit lines,” IEEE Microwave and Wireless Components Letters, vol. 4, pp. 283285, April 2005.
(Circles represent results from another paper.)
58
VI
PV
52
PI
46
VeI
PVe
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. 182185, March 1978.
40
1.0
12.0
4.0
8.0
16.0
GHz
r = 10, h = 0.635 mm, w/h= 1
Ve= effective voltage (average taken over different paths).