From Microwave point of view: prove that less than lambda/4 length required for impedance transformation(inductive to capacitive and vice versa) in left handed/metatransmission lines? Can it be proved from the impedance graphs for short and open circuit RH and LH transmission lines? Images given
Some clarification: when you say "lambda/4", I assume you mean one-quarter wavelength (Bd = 90 degrees) of an unloaded right-handed transmission-line mode.
Yes, it can be demonstrated using impedance graphs, in a specific frequency range. However, it is useful to first consider the one-dimensional dispersion of a realistic left-handed transmission-line (TL) mode in the first Brillouin zone.
In isolation, this mode's lower frequency will occur at a phase of 180 degrees, while its higher frequency will occur at a phase of zero degrees. This means that over some frequency range, the phase constant Beta of the left-handed TL mode will have a larger value that that of an equivalent unloaded right-handed TL mode. See the frequency range indicated in the attached image.
Inserting a larger value of Beta and corresponding Bloch impedance (analogous to characteristic impedance) of the left-handed TL mode into the equations you've shown will indicate that the desired impedance could possibly (depending on the Bloch impedance) attain the same value over a smaller distance d (less than the physical length required for your "lambda/4").
Thank you Stuart ! I didnt think of other concepts being needed to be applied to prove this.. So I will read more about the Bloch impedance and Brillouin Zone and try to analyze your answer with my understanding !
Hi Roy,
The wavelength in the concept of "one-quarter wavelength" is operating wavelength(that is Guided Wavelength) of EM waves in transmission lines, rather than the wavelength in free space. Different transmission lines usually have different operating wavelength, whether they are right-handed TLs or left-haneded lines. Certainly, we can design a LH-TL or a RH-LH to have the same guided wavelength with a specific RH-TL or another LC-TL through carefull design, but it is not a neccesary work in many cases. Consequently, for answering your question, it will be the first step to make sure that the so-called "one-quarter wavelength" is for the LH-TL case or for the RH-TL case.
The crytical difference between LH-TL and RH-TL is in the sign of the phase shift constant. As we know, the LH-TL has a negative phase shift constant, but the RH-TL has a positive phase shift constant. This difference is usually the breakthrough to analyze properties of LH-TLs.
As for the bloch impedance of the LH-TL, it can be considered as the characteristics impedance in the case of RH-TL, and the block impedance has a positive value like the characteristic impedance. The block impedanceis usually considered when people design the impedance matching of a LH-TL with a conventional RH-TL, I don't think this concept can help you to understand the difference between the RH-TL and the LH-TL.