Suppose one wants to measure or simulate the characteristic impedance of something similar to a transmission line. This can be for example a 75 Ω coax cable or a via structure on HFSS. The characteristic impedance can be calculated form its measured or simulated S parameters references to 50 Ω.
According to https://en.wikipedia.org/wiki/Image_impedance, and assuming the symmetry hypothesis which allows to simply discard the second result, the characteristic impedance can be calculated as:
These formulas can be conveniently entered into an Excel spreadsheet.
Alternative non recommended method using transfer S parameters
Transfer S parameters can also be used for this calculation. However, this method is NOT recommended because the calculations are cumbersome.
Expressing the characteristic impedance as a reflection coefficient from 50 Ω, and recalling that by definition its invariant through the system, the following can be written:
Forsk Atoll antenna planning software is not so hard to use when a suitable map and terrain height data is available for import into Atoll. However such ready to use data is rather hard to find and finding such data is the top question asked in the comments of most Atoll tutorials.
Here is a tutorial to make a pretty convenient map with QGIS, usefull for all needs besides just Atoll, and to export relevant data towards Atoll.
QGIS
Installation
Install QGIS uing the OSGeo4W bundle. Ensure the following is checked for install:
QGIS
matplotlib
numpy
If you see the following or similar error message, check the installed packages in OSGeo4W.
Map and height map preparation
The steps of this part are long, but thanksfully must be done only once and can be reused for several projects. A starting file is given here: QGIS-base.qgz. When using this file, skip straight until Bookmark needed area.
Coordinate reference system
Project -> New (Ctrl + N)
Project -> Properties
Select the WGS 84 / UTM zone corresponding to the area of the study. For France, its UTM 31N (EPSG 32631). Antenna coverage studies are usually on a scale sufficiently small to make UTM practical, and the distances coordinates makes life much easier when estimating distances, for instance for antenna simulation radius settings.
OpenStreetMaps and transparency
By default, the OpenStreetMaps vector layers are opaque. Configuration is needed in order to make it transparent to overlay it on satellite imagery.
In case for whatever reason you want to do the previous step manually, here are the instructions. In most case, you ought better import the settings as shown in previous section.
Add OSM map using Vector Tiles -> OpenStreetMap vector:
Using QuickMapServices plugin, add satellite picture using QuickMap Services -> ESRI -> ESRI Satellite. The satellite view is not immediately visible because the layer is added behind the previouly added one:
Go to OpenStreetMaps vector properties and :
Uncheck background:
Uncheck fills:
Uncheck patterns:
Uncheck landcovers with exception of outlines:
Uncheck water areas with exception of outlines:
Uncheck transportation areas:
Uncheck oceans:
Uncheck leaf types:
Uncheck waterways :
Recheck all outlines in case some outline were accidentally unchecked in previous steps:
Contour maps
Using the browser pane on left, add Maptiler Topo.
From Maptiler topo, keep only Contours, and place it between satellite layer and overlayed map.
Check again in project properties that project coordinate reference system is the wanted one.
Bookmark needed area
View the extent of the area you need, next go View -> New Spatial Bookmark… (Ctrl B):
Height map preparation
Double-click on bookmark to be sure the display canvas matches it. Download height maps using the SRTM downloader icon:
Click set canvas extent, and put « ./ » in output path:
Usually, STRM downloader downloads only a single SRTM maps since their map cutting is quite big:
Here, KeePass and Ctrl+Alt+A can be quite useful.
After the layer being downloaded, you should see it. Reorder the layers to check it matches other layers:
Next, uncheck the height layer but keep it:
Map and height map export
Map export
Go to Project -> Export -> Export Map to Image:
Select first the suitable scale and resolution (good values are 1:40000, 300 dpi) and next select your bookmark:
Save as JPG format, convenient because satellite images hard to compress in PNG.
Height map export
Settings :
Select « Raw Data ».
Select « Erdas Imagine Images(.img).
Select CRS WRS 84 / UTM correct zone.
Extent : select bookmarks.
Resolution : round the best (lowest) value and use it for both horizontal and vertical because Atoll can’t handle different resolution on the axes.
Atoll
Coordinate reference system
In your project, go do Document -> Properties…:
Next, select WGS 84 / UTM correct zone for projection and display:
Map import
In Geo tab, create an Offline Maps folder for the created map:
Import the map into Offline Maps:
Height map import
Next, import the height file:
Note both maps covers the same area of the QGIS bookmark.
Once the correctness of the height map import is checked, this layer can be hidden, still it will be correctly taken into accout for calculations.
Atoll example
With map and height data generated by the previously described methods, the following quick draft of DVB-T digital television coverage simulation was performed on an area where reception is difficult. Transmitters positions come from ANFR Cartoradio1 and powers come from a forum 2,3. Of course, this quick draft must be refined, particularly concerning the radiation diagrams of some antennas. Nevertheless, the reception hole in the Ennuyé valley around Bésignan can be seen immediately due to the blocade of the Ventoux transmitter by the south mountain. The others transmitters are not significant on a wide scale because their are low power district transmitters.
This blog page is an English translation and adaptation of a part of my PhD thesis. Numbers in brackets refers to the original bibliography, they will be replaced in a future revision.
Impedance matching is performed by LC ladder networks. This method allows to synthesize low impedances (around 5 Ω) on the same PCB than the standard 50 Ω output (no need for a second PCB with high permittivity). Moreover, this method is more compact than quarter-wave transformer.
Exact value calculation was performed by numerical optimization. Manual calculation would be too difficule because the output impedance of the transistor is not a pure resistance1. However, numerical optimization needs to know the number of components of the ladder, because the ADS optimizer is not able to add components when needed, but is only able to determine their value. Moreover, an initial estimate of the values of the components of the LC ladder is useful for the optimizer to converge quicker towards the solution. Calculation method is the one of [84], adapted for the needs of the PhD thesis.
Inductors and capacitors are assumed ideal and lossless, as well as the microstrip junctions. The effets of the polarization networks of the transistors are also ignored. Such effects are absolutely not negligeable, but will be easily corrected by the numerical optimizer in the final phase of the design.
A simple empirical method is commonly used [32], but it doesn’t allows a priori calculation of the order and of the mismatch of the matching network.
In [116] and [29], tables of [84] are used to calculate a low-pass matching network of Chebychev type. Unfortunately these tables does not provide values for very broadband impedance matching network (1:6 ratio for the amplifier module of the PhD thesis).
For these reasons this page describes in detail the calculation of such impedance matching networks. The calculation method is the one of [84], adapted for the needs of the PhD thesis.
As usual, f is the usual frequency in s^-1 and \omega the angular pulsation in rad \cdot s^-1. Calculations will use mainly \omega.
In a first time, the matching network is calculated for the center frequency \omega_m=1 and source impedance R_S=1. This normalization is not mandatory, but allows to compare intermediate results with those of [84] to test the good operation of the Python program which was written during the PhD thesis.
The reflexion coefficient of a LC ladder (output) matching network of type Chebychev, seen from the source, is2:
with \omega_0=sqrt(\omega_a+\omega_b), \omega_a the beginning of the passband, and \omega_b the end of the passband.
Next figure shows the reflection coefficient seen from the source of an example of an (output) LC matching network going from 5 Ω towards 50 Ω from 1 to 2,5 GHz. These values are approximately those of the first wideband amplifier of the PhD thesis.
Fig. 1. Example of reflection coefficient seen from the source of an LC matching network. See text for parameters.
In previous expression, \epsilon is chosen such as:
|\Gamma(f=0)|^2=((Z_2-Z_1)/(Z_2+Z_1))^2
This last condition is needed because LC ladders have no effect in DC. So, the transfert function is entirely determined by the order and the &&Z_2/Z_1&& ratio.
In the passband, maximum reflection coefficient and maximum insertion losses are respectively:
The first step of the calculation is to determine the first n such as |\Gamma_max|^2 is less than the requirements. This calcul is done numerically, by testing all the integers n from 1 until this requirement is met.
This n is half the number of elements of the final network [84].
Next, variable change p = j · ω is performed. This variable change enables to simplify greatly the calculations to come. Then, the square of the magnitude of the reflection coefficient is factored as such:
with a and b two polynomials whose roots have negative real parts3.
With this factorization, reflection coefficient (and not only his squared norm) can be calculted as such:
\Gamma(p) = (a(p)) / (b(p))
At the beginning of our work on the subject, factorization was performed numerically. This method was thereafter discarded due to numerical instability problems for high orders. This is why a semi-analytic method was taken, inspired by [47, 84]. Roots of the numerator and of the denominator are calculated analytically. Next, factorized polynomianls are calculated by taken only roots with negative real parts.
The calculation, more long than complex, won’t be detailed. The roots of the numerator and of the denominator are given by the following formulas:
{: ( +-j sqrt(Delta omega^2 \cdot cos((pi)/(2 \cdot n) \cdot (1 + 2 \cdot k))) , k in [1, n] ),
( +-j sqrt(Delta omega^2 \cdot cos(1/n \cdot arccos(j/epsilon))) , k in [1, 2 \cdot n] ) :}
The first equation give directly the set of needed roots, since the numerator has double imaginary roots. However, negative real part roots need to be selected from all the roots given by second equation. This point is easily done numerically.
Fig. 2. Roots of the numerator in the example. The roots of the numerator are double and purely imaginary.Fig. 3. Roots of the denominator in the example. The roots of interest are marked in blue, while the ones in red are ignored.
A polynomial is defined by the set of its roots, but up to a multiplicative factor. The next step is to determine this multiplicative factor. Details of the calculation won’t be given here, but only the result:
with &&a_1&& and &&b_1&& the polynomials initially determined.
Next, the input impedance, normalized4 with respect to Z1, is calculated as follows:
Z(p) = (b(p) + a(p))/(b(p) - a(p))
This impedance is then expanded into a continued fraction through successive divisions:
Z(p) = g_1 \cdot p + 1 / (g_2 \cdot p + 1/ (g_3 \cdot p + ... + 1 / (g_m \cdot p + g_(m+1))))
This expression immediately leads to an LC network. The odd gm values are the normalized values of the inductances, while the even gm values are the normalized values of the capacitances. This denormalization is performed according to the following equations5:
{: ( L = g / (2 pi f_0) \cdot Z_1 ),
( C = g / (2 pi f_0) \cdot 1 / Z_1 ) :}
The last &&g_m&& is the load resistance, which is also normalized. Its value has been known for a long time, but it can be interesting to recalculate it to verify that there is no significant error due to numerical inaccuracies.
It is not even a pure impedance. See the blog pages to come! ↩
Such polynomials are called Hurwitz polynomials. The reasons why a and b must satisfy this condition go beyond the scope of this thesis. The reader is encouraged to refer to a book on network synthesis [12, 56, 73]. ↩
This point has been forgotten to be mentioned in the PhD thesis pdf. Sorry. ↩
There is a typo in these formulas in the PhD thesis pdf. Sorry. ↩
Many thanks, in the order of appearance in the LinkedIn discussion, to Dr. Pierre-Antoine Garcia, Theunis Beukman, Benoit Derat, Hüseyin Yiğit, Andreas Barchanski for the insightful LinkedIn discussion which led to this post.
Recently on LinkedIn, a fellow colleague asked whether the group delay of an antenna could be calculated by the simulated complex gain. Sure it can, but with the right precautions.
What is tried to be measured ?
A reasonable asumption when dealing with an antenna is that there will be an other antenna facing it. The transmission coefficient S21 between the two antennas can be written as follows, using the Friis transmission equation https://en.wikipedia.org/wiki/Friis_transmission_equation :
Note this form is slightly different from the usual one, since the amplitude is taken instead of the power. Rearranging in function of the frequency gives:
S_{21} = G_a \cdot G_b \cdot {c} / {4 \cdot \pi \cdot f} \cdot 1 / d \cdot e^{-j \cdot 2 \cdot \pi \cdot d / c \cdot f}
For the group delay, only phase matters, hence:
"phase"(S_{21}) = "phase"(G_a) + "phase"(G_b) - 2 \cdot \pi \cdot d / c \cdot f
Group delay is calculated using the derivative of the phase by \tau_g = - 1 / (2 \cdot \pi) \cdot (d)/(df) "phase"(S_{21}) , which leads to ask which terms are constant with frequency and which terms changes.
Phase center of antennas
A non-trivial varying term is the distance between the antennas. Antennas are often big enough compared to the wavelength to forbid the use of a random point to calculate the phase shift from the distance. The right point to take is the phase center, which is the center of the spherical wavefronts at infinity. The phase center depends on the direction, but most importantly for group delay calculations, it depends on frequency1.
The group delay formula can be rewritten taking this into account:
"phase"(S_{21}) = "phase"(G_a) + "phase"(G_b) - 2 \cdot \pi \cdot (d_"geom" - \Delta_1 - \Delta_2) / c \cdot f
were d_"geom" is the geometrical distance between the antennas and \Delta_1 and \Delta_2 are the positions of the phase centers relative to the geometrical reference point. This reference can be any point. The sign of \Delta_1 and \Delta_2 is taken positive when the phase center gets closer to the other antenna.
Since the interest in only in the first antenna and not in the second, this formula can be normalized by taken all parameters of the second antenna to reference values: unit gain, phase center coinciding with geometrical center, and zero distance. The meaning of the last hypothesis is important to emphasize: the physical distance cannot be zero, it’s just a way to separate the effects depending of the antenna from the rest. So:
"phase"(S_{21}) = "phase"(G_a) + 2 \cdot \pi \cdot \Delta_1 / c \cdot f
Knowing which terms are constant and which are not, the derivative can be calculated:
This last equation shows that not only the phase of the gain must be taken into account, but also the move of the phase center.
Conclusion
The group delay of an antenna can be calculated using simulation results, but the total phase shift change must be taken into account, including the move of the phase center.
Even antennas with a symmetry plane like the ideal dipole can have a varying phase center for directions outside of their symmetry plane. ↩
F4INX corner
