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1/4 wavelength theories for HF


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From:  W Y Liu [SMTP:eenwyl-at-sun.leeds.ac.uk]
Sent:  Friday, June 05, 1998 12:41 PM
To:  tesla-at-pupman-dot-com
Subject:  1/4 wavelength theories for HF

Hi List,

I would like to contribute the following info, which I hacked from somewhere
else. I hope that it is useful for you as it seems to be. If not, please 
accept my apology and forget about it.  At this stage I am afraid that I do 
not have enough knowledge to answer any technical question.


What follows is linear approximated line models for standing wave:-


                (transmission medium)
                         |
                         V
                        
                ----------------------
                                     |
                                    ---
                                    | |
        Zin(L) ==>       Zo         | | ZL
                                    | |
                                    ---
                                     |
                ----------------------
                
                |<------- L -------->|
                
         
         

           ZL  + j * Zo * tan (2*Pi*L/lamda)
Zin(L) =   ---------------------------------
           Zo  + j * ZL * tan (2*Pi*L/lamda)
           
, where lamda  (normally equal to c/f) is the wavelength in meters;
        ZL     is the load impedance in ohms attached; and
        Zo     is the characteristic impedance of the transmission line in
               ohms (depending on the medium of transmission) ;
        Zin(L) is the equivalent input impedance looking at distance L 
               from ZL.
        
Note
----

1) For an electrical network working at a frequency beyond meter waves or 
millimeter waves, where the impedance of both ends of the transmission 
medium are not properly matched, it is expected that the voltage or current
variation occurs in the form of standing wave occurs along the medium of 
transmission. That is, Zin(L1) is not likely equal to Zin(L2).

2) From this formula, we can see that when L/lamda = 1/4 and ZL = infinity 
(i.e. electrically open circuit),  we reach the peak of the standing wave where 
there should normally be a maxima.

3) For low frequency signal components, where the lamda is normally far 
greater than L, this formula is not needed. In most cases, where the wavelength
of the signal exceeds considerably the physical size of the transmission
network, the formula f = 1/(2*Pi*sqrt(LC)) or the like will be more 
appropriate in working out the natural frequency.

4) The above model is NOT sufficently applicable for the nonlinear circuit,
including transmission media with elastic impedance, voltage dependent
impedance, solitary waves, etc.

Bye-bye