 on line (ohms) 2 = electrical wavelength at the operating frequency l = transmission line length Multiplying all bracketed terms by (erl/2) results in 2.2.13.3 Reactances using transmission lines A transmission line can be made to behave like a re- actance by making the terminating load a short circuit (ZL Bear in mind that by definition = 0). In this case, Equation 2.2.54 becomes sinh and cosh = Substituting for sinh rl and cosh rl in the above equation = jZo tan 2 (2.2.55) results in When 2/4 (2.2.52) Zin = jZo tan 271 = (2.2.55a) Equation 2.2.52 is a very important equation because it enables us to investigate the properties of a transmission and is inductive. line. If the total length of the line is l then Z1 becomes the When 2/4 < l > 2/2 input impedance of the line. Hence, Equation 2.2.52 becomes Zin = (2.2.55b) (2.2.53) and is capacitive. Equation 2.2.55 follows a tangent curve and like any tangent curve it will yield positive and negative values. 2.2.13.2 Input impedance of low loss Therefore Equation 2.2.55 can be used to calculate in- ductive and capacitive reactances. Adjustment of Z and transmission lines line length, l, will no doubt set the required values. From Equation 2.2.42, we know that r = a + jB. When Example 2.2.13 x<B, r = jB. From Equation 2.2.22, we know that B = 2n/A. From A 377 O transmission line is terminated by a short circuit mathematical tables¹7 we know that sin (jB) = j sin B and at one end. Its electrical length is 2/7. Calculate its input cos (jB) = COS B. If we now substitute the above facts into impedance at the other end. 17 You can also check this for yourself if you take the series for sin x and COS x and substitute jB in place of X. CHAPTER 2.2 Transmission lines Solution Another interesting case arises when l = 2/4. In this Using Equation 2.2.55a case, Equation 2.2.54 becomes = j377 x 1.254 =j472.80 Therefore the transmission line behaves like a trans- Similar reactive effects can also be produced by using an former where open-circuited load18 and applying it to Equatio