Obtaining the S-parameters of microstrip line discontinuities

This document describes the process of extracting S-parameters from a simple structure: a microstrip line discontinuity, changing from a high impedance line to a low impedance line, and vice-versa.

The transition between lines of differing impedances will naturally produce a reflected (S11) and transmitted (S12) wave. This example is simple enough for us to find the reflected and transmitted signals using a transmission line model:

where the impedance Z of a microstrip line is estimated to be [1]

(3)

Where

is approximately

These values will be used to verify the results obtained with LC. For more complicated structures, we won't have this convenient check, at least not with the same accuracy as with this simple discontinuity.

Terminology

Setup to extract S-parameters using LC

Geometry Layout

The design starts with a microstrip over a ground plane (figure 1) and substrate extending to infinity, simulated by the boundary conditions. The substrate used is a material with a permittivity $\epsilon = 5$ , approximating that of duroid, a material commonly used in CU-Boulder microstrip fabrication.

**Figure 1:** A view of the microstrip line cross section. The microstrip width varies as it progresses along the line, from 2mm to 6mm or more.
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The microstrip is built so that the signal propagates from the source region to the incident measurement and hits the discontinuity. Then the signal splits into reflected and transmitted waves, to be measured by the two probe regions. Initial simulations were done with a 4mm wide microstrip at the far end (figure 2).

**Figure 2:** Top view of the microstrip line.
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Source and probe setup

For microstrip analysis the quantities of interest are voltage, current, charge, and magnetic flux (figure 3). The source block produces a voltage pulse, and two dimensional visualizations of the propagating signal usually measure the voltage at various points on the microstrip.

Placement

The source block is placed near one end of the grid, positioned to inject a signal into the microstrip. The injected signal has a frequency content in the desired range, from 0Hz up to at least 10Ghz, and injects the signal as a voltage source.

The probes are placed in sets of four - the traditional quartet used for the LC pulse measurements - as shown in figure 3. The S-parameter analysis only uses the voltage and current measurements, but we might want to use the other probes (charge and magnetic flux) when examining line impedance or other parameters. Note that the probes are not specified as ideal; for instance, although we only need a to specify single infinitely thin line of integration to find the voltage, in this analysis a thick pillar is used. The probe is set to find the average voltage in the entire pillar. This is done partly because it gives a a smoother measurement value, and partly because it is convenient to design and place all the probes if they have the same width and depth.

Defining Pulses

Now that all the probes are defined, they are grouped together into pulse sets. Three pulses are defined using define||pulses, as shown in table 1. The terms ``narrow'' and ``wide'' refer to the pulses placed on the narrow or wide part of the microstrip (see figure 1. An anomaly seen by looking at this table is that the same set of probes is used to look at both the incident and reflected waves! This is because the ``narrow'' probe sees both the incident and the reflected wave, at different times. So to get the two signals we have to extract them by cutting out certain time ranges from the probe readings. This cutting occurs later, after the simulation is run and the probe readings are made.

**Figure 3:** The four probes required for a full ``pulse'' analysis, laid out for a microstrip line
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**Table 1:** Grouping of probe blocks to form pulse sets
Pulse	Narrow	Narrow	Wide
	(incident)	(reflected)	(transmitted)
Voltage Probe	volts.narrow	volts.narrow	volts.wide
Current Probe	current.narrow	current.narrow	current.wide
Charge Probe	charge.narrow	charge.narrow	charge.wide
Magnetic Flux Probe	magflux.narrow	magflux.narrow	magflux.wide

Results

Finally the simulation is started. The LC probes produce some time domain data, which can then be processed to make S-parameter graphs.

Time Domain

Using the analysis||plot pulses window in LC, we can take a look at all the data collected by the probes. A glance at the raw voltages (figure 4) clearly shows the overlap of the incident and reflected pulse: both the ``narrow'' and ``narrow.reflected'' probes see both the incident and reflected signals. By going back to define||pulses, we can place certain time ranges into each pulse, so that each pulse set contains only the data for a single pulse.

**Figure 4:** ``raw'' plot of LC simulation voltage probes
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**Figure 5:** ``time'' plot of LC simulation voltage probes, after time ranges are specified
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The timesteps containing each pulse are specified in the pulse start and pulse end fields, which can be set manually or by experimenting with the Guess From Data and

Set
From Graph

commands. The time ranges should be chosen so that each pulse definition has a clearly specified signal. After this, using analysis||plot pulses again and looking at ``time'' plots instead of ``raw'' plots (figure 5) we can easily see which signal corresponds to what pulse.

Frequency Domain

Now that the pulses are well defined, all that remains unfinished is to use analysis||S parameters. Specifying the proper pulses as incident, reflected, and transmitted, we can produce either S11 or S21, or both-see figure 6.

**Figure 6:** Voltage based S-parameters for the example circuit
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The analytical results are found using equations 1, 2, and 3. For a microstrip 2mm away from the groundplane, on a substrate with $\epsilon_r = 5$ ,with a width varying from 2mm to 4mm, we get:

Which produces results within 16% of the numerical simulation. For a more accurate result we might want to clean up the signals in figure 3 as well as the parts selected in Define||Pulses.

Design issues

Because this is numerical analysis and not analytical analysis, the model is not ideal; certain simulation artifacts must be recogized and somehow dealt with.

Examining the Padding Size

The width of the groundplane, substrate, and surrounding air must be chosen to keep most of the signal energy away from the boundaries; there is some distortion caused by waves propagating along the boundaries which we want to avoid. A reasonable amount of spacing around a microstrip line is to make the ``padding'' as thick as the microstrip width or the substrate height, whichever is larger. Figure 7 shows that the effect of the boundary conditions does not produce a noticable effect until the total grid has shrunk down to 10mm in size, corresponding to a padding of about the width of the wider part of the microstrip line. If the 4mm microstrip was the only line we wanted to look at, we could use a smaller simulation grid than is shown in figure 1 and reduce the simulation run time; however, later on we will look at other, much wider, lines and will need this extra padding. Therefore all simulations will use the 20mm wide grid.

**Figure 7:** Distortion of a signal due to boundary conditions.
$\begin{figure} \epsfig {file=paddingplots.eps, width=0.7\linewidth}\end{figure}$

Placement of Source and Probe Blocks

The source and probe blocks are placed on the microstrip line; there are restrictions on the locations we can use, because we need to produce a clean signal for analysis.

Source Placement

The ``raw'' output from a source will not produce a perfect waveguide mode, unless someone labors to derive the exact field values for a given waveguide (figure 8 and configures the sources appropriately.

**Figure 8:** Typical shape of the electric fields for a microstrip waveguide mode
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An alternative is to inject a signal using a simple block source. The fields produced by this source (figure 9) contain a bunch of modes: one mode is the propagating mode that we want (figure 8) and the other modes will either propagate at a different velocity or not propagate at all. So, if we let this block shaped signal travel down the microstrip, many of the unwanted modes will simply decay and die out. Other modes will propagate at different velocity; typically the strongest mode is the one we wanted in the first place.

**Figure 9:** Fields generated by a block source distribution
$\begin{figure} \epsfig {file=zaxissourcing.eps,width=\linewidth}\end{figure}$

Using this process to generate the desired modes requires that the signal propagate a distance away from the source. This requirement means that the source cannot be placed directly next to the incident wave probe or the structure we are studying (in this case, the discontinuity). So there must be a short propagation distance between the source and the probes/structure.

The effect of placing a source too close to the probe is shown in figure 10. The distance from source to probe has a drastic effect on the high frequency, less so at low frequency. This indicates that the source block is injecting a wide frequency range into the microstrip, but much of the higher frequency content quickly decays. Since S-parameters are only relevant for propagating signals and not decaying modes, we must place the source at least 14mm away from the first probe - and possibly further if the high frequency S21 parameter is wanted to any accuracy.

**Figure 10:** S11 (reflection) and S21 (transmission) parameters for the microstrip discontinuity setup, placing the source block at various distances from the incident probe block
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Probe Placement

The probes are placed reasonably close to the discontinuity; what does ``reasonable'' mean in this context? The microstrip discontinuity causes field distortion with the same net effect as with the block source: parts of the field constitute a guided wave mode, and propagate away, while other parts of the field quickly decay. Because of this effect, we will get better results if the probe is placed further away from the discontinuity - if placed too near the discontinuity, then the transient fields will effect the measurement. However, by placing the probe a long distance away you will naturally require more simulation grid space and time. The goal in this case is to place the probe as close to the point of interest (in this case, a microstrip discontinuity) while minimizing distortion from transient fields.

We can find a good spot to measure by taking multiple measurements at progressive points along the waveguide. Looking at figure 11, we see that the transmitted wave produces a large number of high frequency modes that quickly decay out, leaving only the smooth gaussian profile of the original signal, 15mm out from the discontinuity. The forward traveling wave evidently carried all the transients with it, since the reflected wave (figure 12) shows no real distortion with distance.

Once the ideal probe locations are found, the other testing probes can be eliminated and the grid shrunk down to enclose only these probes. Make sure at this point that boundary reflections will not effect the results too dramatically.

**Figure 11:** Transmitted signal seen at different probe distances
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**Figure 12:** Reflected signal seen at different probe distances
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LC snapshots

This section contains a graphical view of various parts of LC showing, visually, how to perform some of the feats described in previous sections.

Define Pulse dialog boxes

These pictures show the dialog boxes as they appear once all the probes have been properly assigned to the three pulse definitions, as specified in table 1.

Defining the incident pulse

Defining the reflected pulse

Defining the transmitted pulse