Electrical Communication is a free textbook on the basics of communication technology. See the editorial for more information....

Home Transmission Lines Equivalent T Section of a Uniform Line
 
del.icio.us Reddit Facebook StumbleUpon MySpace Twitter Search the VIAS Library  |  Index

Equivalent T Section of a Uniform Line

It was shown on page 146 that any complex network could be represented by a T section (or a π section). Thus, using equations 19, 20, and 21 of Chapter 5, the equivalent T section for a transmission line can be determined. The equivalent T section (Fig. 21) for a transmission line having uniformly distributed constants can be calculated directly if the characteristic impedance Z0 and the propagation constant per section y are known.

Figure 21. Equivalent T network for a uniform transmission line or non-loaded cable.

On page 206 the impedance measured at the sending terminals of an open-circuited line was shown to be Zoc = Z0 coth γl, and on page 207 the impedance of a short-circuited line was Zsc = Z0 tanh γl. From these relations,

because, from hyperbolic trigonometry, coth x = 1/tanh x. In terms of the symmetrical T section of Fig. 21, equation 82 becomes, from equations 47 and 48, page 156,

when simplified by substituting the value* of Z0 from equation 49? page 156 (page 340 of reference 7). From equation 83,

and this equation contains Zo, which is given in terms of Z1 and Z2 by equation 49, page 156. Substituting this value for Z1 in ZQ = 0.5*sqrt(4Z1Z2 + Z12) obtained from equation 49.

Solving, and substituting coth x = 1/tanh x, csch x = sqrt(coth2 x-1), and csch x - 1/sinh x,

Substituting equation 86 in equation 84,

since tanh

Equations 86 and 87 give the values of Z1 and Z2 for the T network of Fig. 21 equivalent at one frequency to a uniform transmission line having distributed constants. Similar equations can also be derived for a π equivalent network.


Home Transmission Lines Equivalent T Section of a Uniform Line

Last Update: 2011-05-18