From my point of view operating with complex impedance in rectangular form is more simple. But probably this is just a matter of habit and depends more on what calculations you have to deal with more often. Isn't it?

Take two any complex circuits and parallel connect them. If ever doing such impedance math do you know why polar math is faster and simpler in these situations.

Rectangular math can sometimes be preferred and sometimes not but as RF technician must you be able to know what math to perform and when.

Feeding a PC for doing calculation, then doesnt it matter but at lab work must often much of the math be solved on the fly and basic knowledge is needed.

What kind of antenna impedance matching, frequency or wideband and if it should be done with or without a Smith chart and pen or just using brain does often make math selection easy.

Frequency is affecting type of math because for certain frequencies do not some parameters and other are added.

In general:

-30 MHz and nominal value printed on big capacitor and inductors, Farad and Henry, is used as reliable parameters.

Below 100-3000 MHz and nominal value is of limited interest. It is not ideal environment and components are not ideal.

For impedance matching must VAN be used and matching network reactive components must be calculated using its measured Touchstone files.

Above 10 GHz and all is about angles and wavelengths at a PCB with no typical reactive components. Stub filters rules, bandpaass lowpass or as matching network between to reactive loads. Pure resistive loads do not exist above 30 MHz why it often is complex impedanses over wide frequencies at both ends of a transmission line.

More ideal math used below 30 MHz have limited use at higher frequencies as well as the opposite.

What math to select have never been a complicated choice. You chose same math language as you colleagues and customers are using if you want to be understood.

Especially if for a frequency range is not always any angels given, just assumed. Signal transformers, ac relays, speakers, have you ever seen its impedance given as a rectangular value? It is often just magnitude given.

Speaker and microphone impedance, as it vary a lot over frequency is often an average value given called nominal impedance and no angle included.

https://en.wikipedia.org/wiki/Electrical_characteristics_of_dynamic_loudspeakersIf dealing with signal processing do I see it more natural to describe it as a dynamic varying signal, an amplitude with an rotating angle.

If describing a modulated RF signal do we often have similar situation, an amplitude with a angle that varies over time and over covered bandwidth. Is often described as a carrier wave and with an additional math describing modulating signal effect on carrier signal.

A unmodulated signal can be written: V(t)=A cos(2 x pi x f ) but it is getting more complicated to fully describe a signal behavior for for modulation type FM

https://en.wikipedia.org/wiki/Frequency_modulationPSK is a modulation that even include phase in its name. Simplest PSK can be described as two alternating carriers, both same frequency but with opposite phase but many more phases can be added.

What I don't understand is why you think that the Polar representation is better than rectangular one?

To understand that must be understood what impedance describes.

Impedance is a vector with an angle which is repeated regularly with frequency, as result of phase difference between ac voltage and current.

That is pretty much whole idea with the impedance unit, a vector and a angle.

Resistance is not directly measurable as you probably know, it is a fictive unit and indirect result of measuring applied DC voltage and resulting current values and relative phase.

Impedance is not needed to be translated to rectangular resistance and reactance, it is just one of several ways to do sub-definitions.

If you tries to measure impedance using directly measurable properties, can it be done in a number of ways.

Power factor and voltage are two measurable units which can be directly measured and can be used to calculate impedance value.

Most common is to measure ac voltage and current magnitudes and its phase difference from which impedance can be calculated as magnitude and angle.

You're the first people that I see who like to talk about antenna complex impedance in Polar form instead of rectangular one.

From a bit more educating perspective: "We will often find it convenient to express this value in polar form."

https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Book%3A_AC_Electrical_Circuit_Analysis%3A_A_Practical_Approach_(Fiore)/02%3A_Series_RLC_Circuits/2.3%3A_Series_ImpedanceI am not alone. We are a lot of people that are skilled enough to understand why and how it it is less brain-consuming to calculate vectors in polar format.

You have already got several links where you can learn about this.

When you know a stub length and stub properties you can calculate a stub input impedance, after that you can just easy combine two impedance in rectangular form in order to get resulting impedance. Isn't it?

"When you know" is the part you must know how to calculate. "After that combine" - there is not much to combine in a simple stub-matching case. If you assume you is skilled enough to teach me what math to use, you must have at least beginners skills in this field.

In school do they usually show how to do a stub matching using a Smith chart printed at a paper for drawing of arcs using compass.

It is not for no reason a Smith chart is called an "RF engineers analogue computer ".

Just to show how simple it can be for an impedance matching using stubs will I describe an example and include detailed description how I solve this. It should make it more understandable how simple this actually is.

To make this example somewhat realistic, a bit above a school example will we assume a real unknown FR4 PCB, 1mm single side, and a VNA to do measurements with.

No complex filters just a single frequency matching will be done.

We can take as an example our ideal dipole and say that we want to stub match it to a 50 Ohm transmission line and antenna resonant frequency is 5 GHz.

To solve this impedance matching is it so simple that just a virtual Smith chart is needed, the one you have in your memory.

If using a virtual Smith chart and a virtual clock do I not need any real Smith chart at paper nor a pen as this is very simple math.

Call it a rough estimations but result is often just as good as PC-calculated result.

I am not clever or special skilled in math why I always tries to simplify needed calculations and I will try to explain that as well.

It saves me a lot of time in the RF lab to be able to calculate simpler things in my head.

Can you shut your eyes and see a virtual clock? I use a clock mainly because I finds it simple to estimate angles using hour hands.

Can you shut your eyes and do the same for a Smith chart? See a virtual diagram and even assume your written arcs?

If using real paper charts long enough, you will learn this too. No skill, just practicing.

I do not use a calculator for calculation at this level. I will try to show how I instead solves needed math.

We can follow a standard formula often learnt in school starting with transforming real impedance part using a quarter wave transformer.

Normalizing is of limited value this time. Finally, her starts the calculation to impedance match an ideal dipole to 50 Ohm characteristic impedance:

5 GHz wavelength is around 60 mm in length in free space but we must take in account the FR4 PCB dielectric so length is shorter then 60 mm.

A 15 mm long thin strip cu tape is placed on PCB and VNA is connected in one end of this strip using bottom side CU layer as ground. Now is the tape cut with an exacto until 0.25 wavelength is found. Lets for simple numbers assume length was found to be 10 mm.

Transformer impedance is calculated: (50=0.5x100) (72x0.5) =36 which happens to be square of 6 and needed transformer impedance is then 60 Ohm.

Soldering a smd 60 Ohm resistor in opposite end of the thin cu line and connect to ground and adjust width for 60 Ohm using VNA. After that width is found can resistor be removed.

We now know where we are at Smith chart as we have described a half circle around 60 Ohm at resistive center line and the arc is rotating is counter-vise from starting antenna impedance and have exactly reached the admittance line that is crossing 50 Ohm impedance line (and also crosses 0 Ohm). There are two such important admittance circles in a Smith chart and each have two alternative ways to find 50 Ohm.

It is easy to calculate impedance for this point as it is a special case due to the quarter wave transforming but we have no use for it for the moment.

It is now only remaining to add a pure reactive serial load. I select a serial shortcut stub and select a tape-width corresponding to 50 Ohm characteristic impedance. We know what width it is for actual PCB as we have previously decided trace width for impedance corresponding to 60 Ohm.

50 Ohm is 40% wider then measured 60 Ohm width given by this quote: (60x60)/(50x50) => 36/25 => 14/10 => 40%.

Remember the inverse square relation, it is useful to remember.

Actual angle to reach impedance for 50 Ohm do I estimate to be two hours, from 5 to 3 o'clock which is 1/6 wavelength. 40/6 = 20/3 or 6 mm.

I cut the cu tape length slightly longer and add a 50 Ohm resistor as load at end of resulting matching net.

Now can I check using my still connected VNA if the stub needs to be adjusted or else just save actual result.

No calculator was harmed during this process.

No rectangular math was needed, it is as most about angels and vectors lengths in wavelength and if using real compass and Smith chart is calculation not much more then two connected arcs written at the paper.

If you prefer rectangular math, practice by calculate same impedance match circuit using head calculation and show how it went to calculate this stub matching. By trying will you learn that rectangular math do sometimes sucks big.

I can do that kind of math as well but it is a it would harm my brain.

Many other stub types can be selected to achieve same final matching result and there is at least 4 admittance lines that we can use to find 50 Ohm, but I think this example is the simplest alternative to understand.

Maybe was above description a bit hard to follow . It is as a practical job much faster then what I can describe it.If assumed that I oversimplified, it is not so. It is even simpler if using a real Smith chart but it takes longer time using a real pen instead of a virtual one.

A very similar impedance matching lesson is shown in linked relative short two-part video, a pretty basic 5 minutes lesson anyone that need basic mastering of impedance matching will have use of.

Here is also a less virtual Smith chart used and a lot of clutter is done around Smith char, which I recommend. Much recommended video as well as several others i same series. Notice that math is still very simple, no calculator needed.

https://youtu.be/GdSV2-6sYqk84.56 Ω and 84.56 ∠+30.17° Ω are different impedance's. Because first one 84.56 Ω is a scalar value.

What I originally wrote was that an ideal dipole have a feeding impedance around 83 Ohm.

That is fully correct as a technical definition. It is a magnitude and not a coordinate. A value that can have a specific direction but it is not required.

It is no need to express angle for an impedance and is neither always used.

Compare sqrt(3

^{2}x4

^{2})=5. If 3 and 4 are ac voltage and current, what is 5? It is impedance magnitude expressed in Ohm.

It is not a angle-less value even if its actual angle not is expressed. If doing more complex impedance matching have you maybe heard about effective impedance or modulus impedance which is similar to absolute impedance.

About absolute relative modulus:

http://www.mathwords.com/a/absolute_value_of_a_complex_number.htmNo angle exist for these expressions, can not be expressed in rectangular format, neither polar format as it only is a magnitude. Impedance expressed for wideband antennas, speakers and transformer can not be given including angle but can still be very useful values.

Some single frequency transformers do even have its impedance given in percent.

For another example, some time ago I was playing with my custom software and firmware for a vector network analyzers. All math the same was performed in rectangular form.

I have also designed a number of software for the RF industry. Mainly for measurement chambers active measurements, especially EMI related, as well a for lab-bench simpler antenna design and impedance matching software matching using lossy discrete components.This later software must calculate optimal impedance topology using real world component impedances and do needed math as fast as possible to make result appear "live" at PC screen. I have now sold that software since 20 years. A bit beautified now as first versions was written in assembler in DOS and had a commandoline as only GUI, but it was back then good enough to develop first embedded cellphones antennas and its matching network.

For this software, I do mostly do as when doing head-calculations, most math is done using simplified integers and if possible is shift operations used instead of conventional math to keep calculation speed up as much as possible. As similar calculations are repeated for each sweep, maybe 10 times/second if VNA is sweeping fast enough, is not precision calculations needed for for each sweep. It is enough to compensate if there is something changed in data stream from VNA and each sweep have often just minor changes compared to previous sweep.That is why calculation for this software not seems to take any time at all, optimized wideband or multiband component selection and optimized topology takes a fraction of time relative VNA sweep time.

Numeric presentation is in whatever format actual user prefer and most IEEE-488 compatible VNAs are supported.

Here is a presentation, showing me doing a retuning of an existing antenna in real time. Develop and verify a discrete matching network in less then 10 minutes.

Most people that is using this software do not need any instructions but not all knows about Wheelers cap, which is shown in this video.

https://youtu.be/RyMFun_KhAcThis software, AnTune, is else mainly focused as an aid for optimized embedded antenna design.