Transmission line resonator

[Esquer]

TL;DR

Resonance open circuited quarter wavelength

Hello everyone,
I am studying transmission line resonators and in particular I was considering a particular case of resonator where I have an open circuited quarter wavelength transformer of high impedance terminating a regular 50 $\Omega$ line. If I look at the input impedance from the open end of the circuit I get that its real value reaches a maximum when the resonant frequency condition is matched (the one of the quarter wavelength) and at the same time the imaginary part is 0 and I think to understand this quite well. What surprised me is that there are other points in frequency where the imaginary part of the impedance becomes 0 while the real part reaches exactly 50 $\Omega$ and this happens exactly between two consecutive resonances. I was wondering what is the physical meaning of these points. Are they anti-resonance point? Or it means that at these specific frequencies my whole circuit just behaves like a regular 50 $\Omega$ line?
I hope I was clear enough and that some of you might help me to understand.
Thanks and happy holidays to everyone!

 

[Paul Colby]

I think one is concerned about the fraction of power that is available to that delivered to the load. Looking at the impedance of the open end of the resonator isn't really of interest is it? Power is reflected at the open end only and it isn't a load of interest. I find impedance matching obscure at times but I think it all boils down to net real power delivered to a load and not reflected from the load.

 

[tech99]

Suppose your line has Z0 = 500 Ohms. When you connect it to a 50 Ohm load, it has a standing wave ratio of 500/50 =10. As we alter the electrical length of the line, the impedance we see now varies between 50 Ohms and 50 x 10^2 = 5000 Ohms. At intermediate points, the resistance is also intermediate but is accompanied by reactance.
The two conditions, 50 and 5000 Ohms, are resonances, because reactance is zero, and correspond approximately to series and parallel resonance of an LCR circuit.
The 50 Ohm condition will repeat when the line is every multiple of half a wavelength. The 5000 Ohm condition will repeat every odd number of quarter waves.
You might notice a similarity with the vibration of a tuning fork (quarter wave) and a guitar string (half wave).

 

[trurle]

Summary:: Resonance open circuited quarter wavelength

Hello everyone,
I am studying transmission line resonators and in particular I was considering a particular case of resonator where I have an open circuited quarter wavelength transformer of high impedance terminating a regular 50 $\Omega$ line. If I look at the input impedance from the open end of the circuit I get that its real value reaches a maximum when the resonant frequency condition is matched (the one of the quarter wavelength) and at the same time the imaginary part is 0 and I think to understand this quite well. What surprised me is that there are other points in frequency where the imaginary part of the impedance becomes 0 while the real part reaches exactly 50 $\Omega$ and this happens exactly between two consecutive resonances. I was wondering what is the physical meaning of these points. Are they anti-resonance point? Or it means that at these specific frequencies my whole circuit just behaves like a regular 50 $\Omega$ line?
I hope I was clear enough and that some of you might help me to understand.
Thanks and happy holidays to everyone!

These two are current and voltage resonances. Current resonance happening at quarter wavelength TL is normal transformer mode. Although it is theoretically possible to utilize voltage resonance too, it is usually not practical because impedance matching at voltage resonance require longer TL length, resulting in greater losses for no gain.

 

[zoki85]

I think one is concerned about the fraction of power that is available to that delivered to the load. Looking at the impedance of the open end of the resonator isn't really of interest is it?

Well, one can also be concerned about magnitude of resonant overvoltage developing due to presence of high impedance/open end. Especially in a high power systems where too high voltage can lead to isolation breakdown and destructions

 

[Esquer]

The two conditions, 50 and 5000 Ohms, are resonances, because reactance is zero, and correspond approximately to series and parallel resonance of an LCR circuit.

But then why when I calculate the scattering matrix of my system I see resonances only for the quarter wavelength condition? In other words why can I only probe the parallel resonance of the RLC? Is it related to the definition of the scattering matrix which is the ratio between the input voltage and the reflected one and has nothing to do with currents?

 

[Baluncore]

I hope I was clear enough and that some of you might help me to understand.

I do not understand the question.

Do you have an open-ended quarter wave stub, or do you have a quarter wave transformer matching the line to a terminal load.

Please post a diagram of the lines, lengths and impedance values so we can see how it is actually configured.

 

[elguimour91]

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