A. Determining the Characteristic Impedance of a Microstrip Geometry
This example simulates a microstrip line on a printed circuit board. The geometry of the structure is shown in Figure 1. The board is made of a dielectric with e _{r}=4.0. The trace is excited by a 1V s ource at one end, and is terminated by a resistor at the other end. To determine the characteristic impedance Z_{0} of the transmission line, we need to determine the input impedance when the load side is shorted or open, respectively. The input impedance Z_{in }of a transmission line is given by,
where (x_{f}, y_{f}) specifies its position, I
denotes the electric current magnitude, and d
(x) is the Dirac delta function. J^{int} is the impressed current
source. After the E fields along the source edges are obtained, the voltage
drop along the current filament can be calculated. Thus, the input impedance
Z_{in }can be obtained.
The input file for SIFT5 is as follows(Source frequency is 300 MHZ,
the load side is shorted):
# example 5: Use EMAP5 to determine the input impedance of
# a microstrip antenna when it is shorted at the load side.
# the unit is set to be one millimeter
unit 1 mm
# the dimension of the board is 60 mm * 54 mm * 2 mm
boundary 0 0 0 60 54 2
# use uniform mesh along the X axis
celldim 0 60 5 x
# use uniform mesh along the Y axis, the fields near the traces
# changes dramatically, thus use fine mesh near the trace region.
celldim 0 14 7 y
celldim 14 18 4 y
celldim 18 36 2 y
celldim 36 40 4 y
celldim 40 54 7 y
# use uniform mesh along the X axis
celldim 0 2 2 z
# the permittivity of the substrate is 4.0.
dielectric 0 0 0 60 54 2 4.0 0.0
# the active trace
conductor 20 20 2 40 24 2 5 10 10
# the load side is shorted
conductor 40 20 0 40 24 2 5 5 5
# the ground plane
conductor 0 0 0 60 54 0 10 10 10
# the source
einter 20 22 0 20 22 2 300 z 1.0 0
# print the E field along the source edge
output 20 22 0 20 22 2 z E5.out
When the load side is open, we need to delete the following lines:
# the load side is shorted
conductor 40 20 0 40 24 2 5 5 5
Table 1 shows inductance and capacitance obtained by EMAP5 when the load side is shorted or open, respectively. The characteristic impedance Z_{0} of the trace then can be determined. Three frequencies have been investigated. At each fre quency, the calculated value of Z_{0 }is 56.4 W. We can put a 56.4 W resistor at the load side to terminate the transmission line. Theoretically, there should be no reflection if Z_{0} is 56.4 W . Figure 2 shows the numerical results obtained by EMSIM when the trace is terminated with a 56.4 W resistor. It is evident that the transmission line is almost perfectly matched.
















B. Determining the Input Impedance of a Microstrip
Line with a Resistive Load.
In this example, the configuration is the same as shown in Figure 1.
Now however, the load is a 50 W resistor. A
load Z_{L }can be modeled as an element with finite conductivity
given by s =l/(Z_{L}S), where
l is its length, and S is the cross section. If the load is treated
as a lumped element, its contribution to the finite element matrix is as
follows [8]:
where (x_{L}, y_{L}) is the position of the load impedance.
Only edges coinciding with the load are affected by the load.
The following line should be added to the input file for SIFT5:
resistor 40 22 0 40 22 2 50
The above line defines an edge coincidingd with a 50 ohm resistor. If one wants to model the resistor as two edges, each edge should have a value of 25 ohms.
Figure 3 shows the impedance obtained by EMAP5 and compares them with results obtained by EMSIM. Since the characteristic impedance of the microstrip line is about 56.4 W, the 50 W load does not match the microstrip line perfectly. As shown in Figure 3, the input impedance is not exactly 50 W due to the mismatch. The EMAP5 results agree very well with the EMSIM results for this example.
References:
[1] Jianming Jin, The Finite Element Method in Electromagnetics, pp. 324325, New York: John Wiley & Sons Inc, 1993.