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Unformatted text preview: EE 311 - Lecture 7 Impedance on lossless lines Reflection coefficient Impedance equation Shorted line example Assigned reading: Sec 2.5.1 of Ulaby 1 Impedance on lossless lines For lossless lines, = j L C = j ; we will focus on lossless lines almost always from here on out Thus V ( z ) = V + e- jz + V- e jz (1) I ( z ) = 1 Z ( V + e- jz- V- e jz ) (2) where V + and V- are two phasor constants and Z = p L/C for a lossless line The impedance on a lossless line is then Z ( z ) = V ( z ) I ( z ) = Z V + e- jz + V- e jz V + e- jz- V- e jz which is a function of z ! Thus, the impedance on a transmission line changes depending where you measure it on the line 2 Alternate form Note that we can factor out V + e- jz on the top and bottom to get: Z ( z ) = Z V + e- jz V + e- jz 1 + V- V + e 2 jz 1- V- V + e 2 jz = Z 1 + e 2 jz 1- e 2 jz where we have defined = V- V + as a voltage reflection coefficient Note that if = 0 (we have no V- wave), Z ( z ) = Z meaning that the impedance is no longer a function of z This is usually the case when we talk about infinitely long lines, where + propagating waves can propagate forever without generating a- traveling wave Finite length lines terminated in an impedance, however, will typically have 6 = 0 as we will see next... 3 Finding gamma Consider a Xmission line of length l terminated in an impedance Z L . A picture is below The two parallel lines in this picture represent a transmission line. It could be any type of line (parallel plate, coax, etc.) but we are told the Z and the , inside the line which is all we need to know to talk about voltages, currents, and impedances Since weve drawn a circuit impedance Z L at z = 0, the impedance on the transmission line at z = 0 must be Z L V g I i Z g Z in Z Z L ~ V i ~ ~ + +-- V L ~ I L ~ +- Transmission line Generator Load z = - l z = 0 4 Using our previous formula for Z ( z ) and plugging in z = 0 we get Z (0) = Z 1 + 1- = Z L which can be solved to obtain = Z L- Z Z L + Z Note in general is complex; Ulaby writes = | | e j r Knowing we can find the impedance anywhere else on the line, for example at z =- l , by using Z ( z ) = Z 1 + e 2 jz 1- e 2 jz since all quantities in this equation are now known (note = LC = ) Another way of writing this equation is Z ( z ) = Z 1 + ( z ) 1- ( z ) where ( z ) = e 2 jz 5 Impedance equation without We can also plug in = Z L- Z Z L + Z and simplify some to get Z ( z ) = Z Z L- jZ tan( z ) Z- jZ L tan( z ) so that Z (- l ) = Z Z L + jZ tan( l ) Z + jZ L tan( l ) This last formula is probably the most useful of all the equations since it directly gives the impedance at z =- l if we know Z , Z L , , and l Notice the dependence on l occurs only in the tan(...
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