Update from 2024-11-05: Steve from Microwaves 101 also tackled this topic (https://www.microwaves101.com/encyclopedias/branchline-coupler-port-definition), proposing a different port-naming convention for the branchline coupler. Steve names the direct and coupled outputs in reverse compared to the approach I use. His convention offers some interesting arguments: the output he names "forward" exhibits a higher bandwidth and flatter phase response.
Upon further analysis, it appears that the operating principles of branchline couplers differ significantly from those of coupled line couplers. Consequently, the terms "direct" and "coupled," commonly used for coupled line couplers, no longer carry the same meaning.
For now, I’ll retain the "geometric" convention I initially used, as it seems more common, although I’m not entirely convinced it’s more "correct." Examples that follow this convention can be found in this research, this diagram, this article, and this paper).
Many thanks to Steve for raising this interesting topic. Thanks also ChatGPT for improving my English.
Introduction
IQ modulators and quadrature couplers are often used together to perform a frequency translation while rejecting the unwanted sideband. However, the documentation often does not clarify whether the quadrature signal has a +90° or -90° phase shift, and whether the IQ modulator expects a quadrature input with a +90° or -90° phase shift.
The purpose of this page is to clarify this sign issue.
Upconverter side
Here, we assume an upconverter is used. The principle of an IQ upconverter is to perform a complex frequency translation as follows:
By application of the trigonometric identity &&sin(x) = cos(x - pi / 2)&&:
I + j \cdot Q = cos(\omega_b \cdot t) + j \cdot cos(\omega_b \cdot t - \pi / 2)
This clearly shows that the Q input has a phase lag compared to the I input, which is a -90° phase.
Coupler side case 1: coupled line coupler
Let’s assume that the coupled line coupler QCH-451+ from Mini-Circuits is used. The datasheet indicates which port is the quadrature output but not whether its phase is +90° or -90°.
Using ChatGPT and Plotly, the following plots were easily generated:
It’s clear from the plots than the quadrature output has a +90° phase (lead) compared to the in-phase output.
Besides clarifying the +/-90° sign issue, it is useful to check to what physical element correspond each port number. The hypothesis is that this coupler is made with an equivalent to coupled lines performed using an LC network, as pure coupled lines are obviously too long to fit in the component.
Analysis of the transmission curves enables to deduce the connexions:
At low frequencies, total transmission from port 1 to port 3 and no transmission to the other ports. Thus both ports are connected to the same first transmission line and ports 2 and 3 are connected to the second line.
Port 4 is not connected at low frequencies but is coupled in the band. So port 4 is the coupled port of the other line.
Port 2 is not connecter at low frequencies and remains approximately isolated in the band. So port 2 is the coupled output of the second line.
This leads to the following schematic, compared to the datasheet information:
The symmetry of the schematic is coherent with the symmetry of the table given in the Mini-Circuits datasheet:
The +90 phase shift stays even at low frequencies. This is coherent with the similar behaviour of an high-pass CR network, which is part of the equivalent schematic at low frequencies.
This part was performed in two steps. First, using scikit-rf with a some help of ChatGPT, a Python script was created to produce preliminary plots and an s4p file. The second step was to reuse the plotting code from the previous section, with some help from ChatGPT to refactor the duplicate parts, to make the following plots:
Contrary to coupled lines, which have a very broadband quadrature effect, the branch-line coupler has a narrowband quadrature effect. This is still useful for narrowband applications.
It’s clear from the plots than the coupled output has a -90° phase (lag) compared to the direct output.
Conclusion
Quadrature couplers are important for making image reject mixers, but the actual phase sign is sometimes not clearly stated. According to the theory, IQ mixers needs a -90° phase (lag) for the Q input. For coupled lines couplers, the coupled output has a +90° (lead) phase compared to the uncoupled output. Conversely, for branchline couplers, the coupled output has a -90° (lag) phase compared to the uncoupled output. These points must be considered when selecting the wanted mixer side.
The designer has several tools and formulas available to calculate the characteristic impedance of microstrip lines. Some are highly precise but rather complex like the Hammerstad and Jensen formulas1,2,3, while others are rather simple but with questionable accuracy like the IPC-2141 formulas4,5,6. While approximations can be useful for the first steps of a design, their accuracy must be evaluated before use. The authors of Qucs3 made some comparison, but this comparison don’t include the common IPC formulas5,6. A comparison of the most common microstrip calculation formulas is shown here.
While Hammerstad and Jensen formulas1 stay the gold standard, other formulas can be safely used but that IPC-2141 formulas4,5,6 must be used with extreme caution.
The geometrical parameters of the microstrip line studied are defined in Fig. 14: w is the width of the microstrip line, h the height of the substrate, t the thickness of the strip. Non-geometrical parameters are the relative permittivity of the substrate. We’ll note for the characteristic impedance of free space.
To simplify the analysis, the strip conductivity is assumed to be infinite, the substrate losses to be null, the thickness t to be null and the frequency to be null. Note that the infinite conductivity hypothesis implies the absence of low-frequency dispersion7,8. These assumptions are often used in RF and microwave design, and are of good accuracy for the impedance calculation.
Reference formula for Microstrip
Evaluating the accuracy of models needs a reference to compare to. Hammerstad and Jensen formulas are the most accurate closed-form formulas, and their accuracy is higher than manufacturing processes. They are commonly used in CAD software3,9 in which the accuracy of the models of single microstrip lines without discontinuities is recognized. However, these formulas are so complex than their practical use for a comparison can lead to type errors during implementation, which can lead to inaccuracies. 3D EM simulations can be as accurate as needed, but their setup and calculation time is time consuming when accuracy is needed. Measurements are highly expensive due to the need of accuracy in both manufacturing and characterization of board materials. So, for this setup, a well-known software is used, TXLine from AWR. This software implements probably Hammerstad and Jensen formulas and its wide use almost guarantees that the implementation is bug free.
Formulas to be compared
The formulas to be compared are the following:
TXLine
TXLine is a small software, lightweight and free to use, from AWR, which allows to calculate the characteristic impedance of transmission lines like microstrip and striplines. A screenshot is shown Fig. 2.
Fig. 2 – Screenshot of TXLine software.
Since TXLine has no direct option to handle the infinite conductivity, the zero thickness or the zero frequency, they were approximated as follows: conductivity 10^99\ "S"\cdot"m"^-1, thickness 0.1 µm or 0.001 µm (depending on cases), and frequency 1 Hz. The high conductivity can be considered as infinite for all practical purposes, and is high enough to avoid the impact of low-frequency dispersion on the calculation results 7,8. The thickness also is low enough to approximate the zero-thickness condition.
Hammerstad and Jensen
According to Hammerstad and Jensen formulas1,2,3 the calculation is made in two steps. First, the effective dielectric constant is calculated:
Thereafter, the following formulas are used to calculate the characteristic impedance of the microstrip line, assuming the line is in an homogeneous medium of dielectric constant :
Note that in reference1, the formulas are not very clear about whether or should be used, but this point was double-checked in the Excel spreadsheet formulas.
These formulas are used in these calculators10,11.
Wheeler 1965
Wheeler’s formulas are sometimes encountered in technical literature3,12. They are rather accurate for most uses, as will be seen in following sections. Their main problem is the undesirable impedance step. Note that, contrary to Hammerstad and Jensen formulas, the effective dielectric constant is not used in the characteristic impedance formulas.
We have not found any calculator using these formulas.
Wheeler 1977
Wheeler 1977 formulas are seen more often than Wheeler 1965 formulas in literature4,13,14,15. They are attributed sometimes to Wadell because he made a good summary in his book13. Note that these formulas are incorrectly typed a reference4, where a pi^2 was incorrectly replaced by a sqrt(pi), and an extra square root is present, as pointed by an other reference16.
They are used in several microwave calculators17,18,19,20,21. Calculator21 has a mistake in the handling of epsilon_(r,"eff"), which can be diagnosed by calculating impedances with epsilon_r=1.
No explicit formula is given for epsilon_(r,"eff"). The reason is that it can be calculated with: epsilon_(r,"eff")=[(Z_0(h,w,epsilon_r=1))/(Z_0(h,w,epsilon_r=epsilon_r))]^2. This makes these formulas both simple and complex at the same time. In a programming language, it is easy to define a function and to use it two times to calculate . In an old school Excel sheet this would lead to use twice plus one many columns, which more than Hammerstad & Jansen formulas.
Hammerstad 1975 formulas
Often seen in websites22 or in lectures on microwave techniques23,24, Hammerstad’s 1975 formulas are the following22,25,26,27,28. They are often attributed to Bahl, who made an improvement to their strip thickness correction26,27, not investigated in this article, but the zero-thickness formulas originates from Hammerstad25.
We have not found any calculator using this formula.
IPC-2141 formulas
The IPC-2141 formulas4,5,6,28 are very popular. However, despite their popularity, they should be used with extreme caution, as will be demonstrated during our comparison, because their range of validity is extremely narrow, and they have a bad asymptotic behavior.
Although they are widely quoted, their narrow range of validity is much less often quoted28: 0.1<w/h<2.0 and 1<epsilon_r<15. This range becomes even narrower because it enables to synthesize 50 Ω lines only for 3.9<epsilon_r<15.
Contrary to other formulas (Hammerstad and Jensen, Wheeler 1965, Wheeler 1977, Schneider and Hammerstad 1975), IPC-2141 formulas have a totally nonsense asymptotic behavior. All other formulas were found to have an accuracy better than 1.5 % for epsilon_r=4.5 and w/h ratios ranging from 0.001 to 1000. On the contrary, IPC-2141 formulas give a nonsense negative impedance result for w/h>7.5. This is a large line for most applications, but it still happens in some designs, and give an idea of the problem.
Fig. 1 compares the asymptotic behavior of IPC-2141 formulas with the good formulas. While IPC-2141 have a very bad asymptotic behavior for large lines, all good formulas have a good asymptotic behavior on the very wide range 0.001<w/h<1000, well outside of their guaranteed validity range.
Fig. 3 – Asymptotic behavior of formulas.
Despite the problems of IPC-2141 formulas, they are used in several online calculators33,34,35,36,37,38,39,40. Some calculators33,34,35 give warnings when using IPC-2141 formulas outside of their validity range like shown in Fig. 3. On the contrary, some other calculators36,37,38,39,40 give neither a warning nor a validity range, including a calculator on a renowned website38. Worse, some calculators38,39,40 even give nonsense negative impedance when fed with proper values without any warning.
Fig. 4 – Screenshot of a microstrip line impedance calculator33 raising a warning when trying to calculate impedances outside IPC-2141 validity range.
It should be mentioned that a calculator33 not only gives the validity range of the IPC-2141 formula and warns when trying to enter parameters outside of this range, but it also gives accuracy data.
Comparison results
Fig. 5 – Error of all formulas.Fig. 6 – Error of formulas without IPC-2141.
From the Fig. 5 graph, a clear outsider in inaccuracy is the IPC-2141 formula, reaching up to 44 % inaccuracy! This formula will be commented later. Fig. 6 graph, without the IPC-2141 formula on a reduced scale, shows the relative error of the remaining formulas.
The H&J formulas are probably the formulas used in TXLine software since the calculated difference between them is lesser than 0.06 % ! Note that this calculated error is always positive, because TXLine took a line thickness which is non-zero, although very thin, contrary to our calculation. Since we are comparing the H&J formulas to themselves, this benchmark does not prove that they are the most accurate. However, since they are believed to be the most accurate in all the recent literature, we’ll stick to that conclusion. They still have the inconvenient of their complexity, which can lead to potential typing mistakes.
Hammerstad 1975 formulas are at the second place for accuracy: error less than 0,38 % on tested values. However, they still have the inconvenience of their complexity, potentially leading to typing mistake, and of the different expressions for different subdomains.
Wheeler 1965 formulas are at the third place for accuracy: error less than 0.59 % on tested values. However, the calculation step makes them troublesome to use in some cases. And, for use in Excel spreadsheets, it makes mandatory to duplicate the formulas. The 0,06 % accuracy gained from Wheeler 1977 is not worth it.
Wheeler 1977 formulas, although not very known, are rather simple and rather accurate: the error is lesser than 0.66 % on tested values. The absence of impedance steps in these formulas make them interesting.
Schneider formulas are less complex than Hammerstad 1975 formulas but less accurate. Their error is less than 1.6 %. Their interest is mainly historical.
The real strange point is the accuracy of the IPC-2141 formulas. Their relative error reach 44 % on the tested values! A closer look reveals that most of the error happens when the normalized width u is higher than 2. When plotted on a narrower range, like in Fig. 7, the relative error is much lower: less than 2.1 %. This is precisely the range of values of the 50 Ω lines.
Fig. 7 – Error of all formulas on a narrow w/h range.
Review of calculators
The following table sums up some microstrip calculators and the formulas which they use. Only microstrip calculators for which the formula was told or could be inferred from JavaScript source code were included.
While H&J formulas are the gold standard for calculations, several other formulas give a good trade-off between accuracy and simplicity which are sufficient for most applications. Wheeler 1977 is the clear winner of this tradeoff with 0.66 % error when epsilon_(r,"eff") is not needed. However, on old school Excel sheets, when epsilon_(r,"eff") is needed, the need to duplicate the calculation makes it not anymore convenient than H&J formulas. In this case, Hammerstad 1975 is the winner of this tradeoff with 0.38 % error.
IPC-2141 formulas have severe issues and must be used with extreme caution.
E. Hammerstad and O. Jensen, “Accurate Models for Microstrip Computer-Aided Design,” in Microwave symposium Digest, 1980 IEEE MTT-S International, 1980. ↩↩2↩3↩4
Analog Devices, “Microstrip and stripline design,” 2009 ↩↩2↩3↩4
Institute for Interconnection and Packaging Electronic Circuits, Standard IPC-2141A, “Controlled Impedance Circuit Boards and High Speed Logic Design”, 2004. ↩↩2↩3↩4
D. R. McMahill, “Microstrip analysis/synthesis calculator,” 16 February 2020. [Online]. Available: http://mcalc.sourceforge.net/. [Accessed 13 April 2020]. ↩↩2
H. A. Wheeler, “Transmission-line properties of parallel strips separated by a dielectric sheet,” IEEE transactions on microwave theory and techniques, vol. 13, no. 2, pp. 172-185, 1965. ↩
B. C. Wadell, Transmission line design handbook, 1991. ↩↩2
H. A. Wheeler, “Transmission-line properties of a strip on a dielectric sheet on a plane,” IEEE transactions on microwave theory and techniques, vol. 25, no. 8, pp. 631-647, August 1977. ↩
E. O. Hammerstad, “Equations for microstrip circuit design,” in Proc. European Microwave Conf., 1975. ↩↩2
I. J. Bahl and D. K. Trivadi, “A designer’s guide to microstrip line,” Microwaves, pp. 174-182, May 1977. ↩↩2
I. J. Bahl and R. Garg, “Simple and accurate formulas for a microstrip with finite strip thickness,” Proceedings of the IEEE, vol. 65, no. 11, pp. 1611-1612, November 1977. ↩↩2
S. H. Hall, G. W. Hall and J. A. McCall, High-speed digital system - a handbook of interconnect theory and design practices, Wiley, 2000. ↩↩2↩3
This page was suggested by Hadrien, who has had recent experience in MLC capacitor variations with voltage. Did you know your capacitor nominal value can drop 80% when you apply a DC voltage to it? Worse, there does not seem to be any standards for voltage variations like there are for temperature variations.
This page is primarily discussing MLCCs, or multi-layer cerami capacitors, in Class 2. Class 2 uses exotic dielectrics such as barium titanate (BaTiO) with some other strange additives in order to get high dielectric constant in the thousande (to increase capaitance density). Barium titanate is a ferro-electric material, which is the source of voltage/capacitance misery.
From here down, thanks to Hadrien!
Case study
Since this page was originally written for Microwaves101, the case study to illustrate this problem was taken from it. Suppose we want to redesign the good old breadboard RF pulsed (https://www.microwaves101.com/encyclopedias/breadboard-rf-modulator) source from 2006 with more recent components (first version of this page written in 2019). Start by the capacitors. Capacitors values which were at this time most often made with electrolytic capacitors are nowadays more often made with MLCC (multiple layers caramic) capacitors.
To replace the 4.7 µF capacitor, a simple search gives this component:
Not good! If we need the 4.7 µF capacitance value, we’ll be soon in trouble.
Let’s search a bigger capacitor. We search the biggest value among MLCC capacitors having 10 V rating.
https://fr.rs-online.com/web/c/passifs/condensateurs/condensateurs-ceramique-multicouche/?applied-dimensions=4294244793,4291386526&pn=1
We find some nice models:
Let’s plot together their bias characteristics1,2,3,4,5,6:
The values of the capacitors at 5 V range from 28.7 to 45.9 µF. Still low for 100 µF capacitors but all would be more than sufficient to replace the 4.7 µF capacitor of the original design.
Personnally, to keep some safety margin from the maximum voltage, I don’t like using a capacitor at a voltage where its capacitance drops more than 50 % of its nominal value. So, if I had to redesign this board, I would continue searching capacitors, even when the Murata and Kemet capacitors are close matches.
Conclusion of case study
As a conclusion, the “100 µF” capacitors of our study, when a DC voltage is applied to them, have an actual capacitance value much lower than their nominal value. This capacitance value would be still sufficient for our application because we oversized the capacitor. However, to keep a sufficient margin from voltage breakdown, I find careful to not use a capacitor at a voltage where its capacitance drop is more than 50 %. In designs where the capacitors are undersized, the capacitance drop effect on applied voltage can be a bad surprise and lead to severe troubles. Such problems are pretty hard to debug if not aware because the cap meters instruments most often measure at 0 V.
An other interesting point is what happens when we combine both biasing and temperature effects. Let’s see an other curve provided by TDK:
The temperature dependence is better when biased. This leads to the conclusion that temperature coefficient is most often less troublesome than voltage coefficient for MLCC capacitors.
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.
F4INX corner
