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Chapter 2.2 Transmission lines > 2.2.12 Propagation constant (g)of transmission... - Pg. 59

Transmission lines CHAPTER 2.2 or reflection loss specifies the fraction of incident power not reaching the load and is equal to À10 log (1 À G 2 ). The current I 1 splits into two parts: I 2 and a part going through Z 2 . By the current divider rule, the split is I 2 ¼ giving Z 2 I 1 Z 2 þ Z 1 =2 þ Z 0 2.2.12 Propagation constant (g) of transmission lines 2.2.12.1 Introduction In Section 2.2.8, we saw that signals on transmission lines suffer attenuation, phase or time delay, and often fre- quency distortion. In this section, we will show the re- lationships between these properties and the primary constants (R, G, L and C ) of a transmission line. I 1 Z 1 Z ¼ 1 þ þ 0 I 2 2Z 2 Z 2 Substituting the definitions for Z 1 and Z 2 and the formula for Z 0 derived above gives I 1 1 ¼ 1 þ ðR þ juLÞðG þ juLÞðdlÞ 2 I 2 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðR þ juLÞðG þ juLÞdl q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ ðR þ juLÞðG þ juLÞdl 1 þ ðR þ juLÞðG þ juLÞðdlÞ 2 2 Also I 1 /I 2 ¼ e gdl . To use these two expressions for I 1 /I 2 to find g we must first expand e gdl into a Taylor series. 2.2.12.2 The propagation constant (g) in terms of the primary constants To find the propagation constant (g) we start with the same equivalent circuit (Figure 2.2-8) used for the der- ivation of Z 0 . It is re-drawn in Figure 2.2-13 with the voltage and current phasors indicated. The propagation constant, as defined, relates V 2 and V by