Dear John:

At the first, let's get remember these facts:

1- Coaxial cabling is sometimes used in DC and low-frequency AC circuits as well as in high-frequency circuits, for the excellent immunity to induced “noise” that it provides for signals.

2- When the period of a transmitted voltage or current signal greatly exceeds the propagation time for a transmission line, the line is considered electrically short. Conversely, when the propagation time is a large fraction or multiple of the signal’s period, the line is considered electrically long.

3- A signal’s wavelength is the physical distance it will propagate in the timespan of one period. Wavelength is calculated by the formula λ=v/f, where “λ” is the wavelength, “v” is the propagation velocity, and “f” is the signal frequency.

3- A rule-of-thumb for transmission line “shortness” is that the line must be at least 1/4 wavelength before it is considered “long.”

3- In a circuit with a “short” line, the terminating (load) impedance dominates circuit behavior. The source effectively sees nothing but the load’s impedance, barring any resistive losses in the transmission line.

4- In a circuit with a “long” line, the line’s own characteristic impedance dominates circuit behavior. The ultimate example of this is a transmission line of infinite length: since the signal will never reach the load impedance, the source only “sees” the cable’s characteristic impedance.

5- When a transmission line is terminated by a load precisely matching its impedance, there are no reflected waves and thus no problems with line length.

For more information about your topic, you could benefit from this valuable article at:

https://www.allaboutcircuits.com/textbook/alternating-current/chpt-14/standing-waves-and-resonance/

I hope it will be helpful...

With my best regards...

Thank you for your comment. But my question is: why in impedance matching circuit with use L section stub, the shorter transmission lines has lower BW.

So The term “impedance matching” is rather straightforward. It’s simply defined as the process of making one impedance look like another. Frequently, it becomes necessary to match a load impedance to the source or internal impedance of a driving source.

Sir i think you could benefit from this valuable articles about your topic at:

1-- https://www.electronicdesign.com /technologies/communications/article/21796367/back-to-basics-impedance-matching-part-1.

2-- https://studylib.net/doc/25266752 /chapter-3-impedance-matching.

Information about (Stub tuners) - as you know - are basic laboratory tools used for matching load impedances to provide for maximum power transfer between a generator and a load, and introducing a mismatch into an otherwise matched system. Typical applications include power and attenuation measurements, tuned reflectometer systems and providing a DC return for single-ended mixers and detectors.

Stub tuners are impedance transformers that are designed to introduce a variable shunt susceptance into a coaxial transmission line. They consist of one or more short-circuited, variable length lines (stubs) connected at right angles to the primary transmission line. Each stub must be movable over one-half wavelength at the lowest frequency of operation in order to provide all possible shunt susceptances; therefore, the low frequency limit of a tuner is determined by the frequency at which the maximum stub travel equals one-half wavelength. Tuners, particularly multiple-stub models, are still usable below this limit. Other than the limitation of the connectors, there is no higher frequency limit; however, various models are offered for size convenience.

The spacing between the stubs of multiple-stub tuners determines the range of impedances that can be matched and the ease of tuning . The stub spacing of Maury double- and triple-stub tuners has been selected for general broadband applications. Triple-stub tuners are more convenient to use since tuning sensitivity is relatively independent of stub spacing.

Maury produces a comprehensive line of broadband stub tuners designed to satisfy the majority of applications. These tuners are available in double- and triple-stub configurations with frequency ranges extending from 0.2 to 18.0 GHz.

I hope it will be helpful...

When the transmission line has a length same as that of the wavelength of the wave propagating through it, the input impedance will be same as the load impedance.

Line length doesn't matter when the line is properly terminated, the input impedance will remain unchanged with varying lengths of line. High SWR on a line with high matched-line loss will produce significant additional loss on the line

The key concept to your answer is the transformation Q. Seeing this graphically on the Smith Chart is helpful by constructing constant Q contours and maintaining an impedance transfer with a given number of lines while adjusting their Zo and electrical length. Those are the two key parameters permitting a line to provide Z translation. At the same time, bound your Q in each step from Z1 to Z2... etc until hitting your target Final Z. Clearly, the bandwidth (BW) will be extended as the number of lines increases. However, since each line has a finite unloaded Q or loss, more lines extends BW at the price of loss. You really seek wide band at minimum loss. Slightly different problem. Take a look at Tchebychev transmission line matching forms. See also Pozar for more details.

Here is a chart plot that demonstrates the significant difference in Q transform from a low Z to 50 ohms at 1 GHz. The set of three lines and stubs are compared to just one line stub plus a single line. The Q transfer from each step in Z clearly constrains the operating bandwidth for the case with just one line and stub.

Dear Alan victor, Would you provide me a reference to cover transformation Q in Smith chart?

Sure John Jin,

I will put a reply together on this topic. Standby.

Alan

Here is a treatment on these topics. Also included in a pdf attached.

https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_III_-_Networks_(Steer)/07%3A_Chapter_7/7.4%3A_Stepped-Impedance_Transmission_Line_Transformer

I really appreciate your kindness and the time you took to prepare the pdf file.