ere sl is still the short length of line referred to in Figure 2.2-8. It is easier to find r using the current phasors rather than the voltage phasors; so, using I = V1/Z and I2 = V2/Z (2.2.40) Equating the two expressions for I1/I2 gives or alternatively (2.2.40a) (Z/4) 81 (Z/4) 81 R & I (Z/4) (Z/4) Z2 = 1/y8l Fig. 2.2-13 Equivalent circuit of a very short length of line. CHAPTER 2.2 Transmission lines comparison with the propagation results calculated here Subtracting 1 from each side and dividing by ol gives for the discussion that follows after this answer. Equation 2.2.41 will be used to calculate the propagation constant r, and Equation 2.2.42 will be used to derive the atten- uation constant a and the phase constant B in all the calculations that follow. (a) At 100 Hz and as ol approaches zero y=V(R+jwL)(G+juL) (2.2.41) R + jwL = (23 j(2 X 100 125uH)) Since r is complex consisting of a real term a for am- = (23 + j0.08) km-1 plitude and B for phase, we can also write r =a+jB=/(R+jwL)(G+jwL) = (2.2.42) G + jwC = mS X 100 X 48 nF)) If the expression for r (Equation 2.2.42) is examined = +j0.030) mS km-superscript(1) more closely, it can be seen that there are two regions where r tends to be resistive and constant. The first Hence region occurs at very low frequencies when R >> jwL and G jwC. This results in 1(23+j0.08)(4+j0.030) x10-3 = 0.30 /0.01 r=V(R)(G) (2.2.43) = (0.30 nepers + 1.66 X km-1 In this region r is a real number which does not depend on W. Since the real part of r is a, the attenuation index, there is no amplitude distortion in the very low frequency range. The second region occurs at very high frequencies when jwL >> R and jwC >> G. This results in = 75.830 / 2.06 rad r=jwV(L)(C) (2.2.44) In this region r is purely imaginary and is proportional to W. (b) At 500 Hz Since the imaginary part of r is B, the phase index, it means that there is no dispersion (because B is proportional to w) in R + jwL = (23 j(2 X 500 the high frequency range. The region is very useful for pulse = (23 +j0.39) Q km 