Reflections on a mismatched transmission line Reflections.doc (4/1/00) Introduction The transmission line equations are given by

Size: px

Start display at page:

Download "Reflections on a mismatched transmission line Reflections.doc (4/1/00) Introduction The transmission line equations are given by"

Kerry Horn
5 years ago
Views:

1 Reflections on a mismatched transmission line Reflections.doc (4/1/00) Introdction The transmission line eqations are given by, I z, t V z t l z t I z, t V z, t c z t (1) (2) Where, c is the per-nit-length capacitance obtained by taking the capacitance C of some section of length z and dividing the capacitance C by that length, that is, c C / z. All other sections of the line will be characterized by this per-nit-length capacitance since the line is niform. imilarly, for l, the per-nitlength indctance, where l / z. It is of importance to note that both c and l are static field parameters that operate at dc (zero Hz). These parameters can be calclated sing the theory of electrostatic and magnetostatic field theory. In addition, these parameters (l and c) represent the transverse electromagnetic (TEM) field strctre for non-static excitation, that is, ac. V(z,t) is the voltage between the two condctors comprising the transmission line, and I(z,t) is the crrent that flows throgh these two condctors. Time domain analysis of lossless transmission lines This section will consider signal sorces that are general in natre, that is, they have general waveshapes. The transmission line eqations given by eqations 1 and 2 can be combined to yield two additional eqations where each new eqation involves only the line voltage or the line crrent only. Differentiating eqation 1 with respect to distance z, and eqation 2 with respect to time t, the following is obtained, V ( z, t) I( z, t) l 2 z zt I( z, t) V ( z, t) c 2 zt t (3) (4) bstitting eqation 4 into eqation 3 gives, similarly, V ( z, t) V ( z, t) lc z t I( z, t) I( z, t) lc z t (5) (6) Phase velocity 2 The first thing to note abot eqations 5 and 6 is the prodct lc. This has the dimensions of 1 velocity. Ths, the velocity, of the signal travelling down a transmission line, is taken as,

2 1 m/s (7) lc For transmission lines contained within a homogeneos medim characterized by the permiability, and the permittivity, lc (8) oltion to eqations 5 and 6 are, z z V ( z, t) V t V t (9) z z I( z, t) I t I t (10) z where, V t and z I t represent forward travelling waves of voltage and crrent as sitting on a point on each waveform, the point mst move in the positive z direction for increasing time in order to keep the argments of the fnctions constant, that is, in order to stay sitting at the same position on the waveform. z imilarly, V t and z I t represent backward travelling waves of voltage and crrent, moving in the negative z direction. Characteristic impedance of the transmission line The forward travelling crrent and voltage waves are related, as are the backward travelling waves of crrent and voltage. This relationship is given by z z V t I t (11) z z V t I t (12) where, l c (13) and Z 0 is the characteristic impedance of the line. Impedance is generally sed, bt more strictly speaking the term shold be characteristic resistance of the line. By sbstitting eqations 11 and 12 back into the transmission line eqations (eqations 1 and 2), the general form of the soltions to eqations 5 and 6 become, z z V ( z, t) V t V t V z V z I( z, t) t t Z Z 0 0 (14) (15)

3 oad reflection coefficient Consider Figre 1 that shows a parallel wire transmission line of length, that is terminated in a load resistance R, and driven by a voltage sorce having an open-circit waveform of V () t and an internal resistance R. It is reqired to determine the voltage at the beginning of the transmission line V0, voltage at the load termination V, t, the crrent at the beginning of the transmission line I0, crrent into the load termination I, t as fnctions of time. t, the t and the Figre 1 The terminated transmission line The voltage reflection coefficient at the load Referring to Figre 2, at z =, the total voltage in terms of the total crrent is V, t R I(, t) (16) Expressing this condition in terms of forward and reflected waves, R R Z Z V t V t V t V t 0 0 (17) The left and right hand sides of eqation 17 are not satisfied nless R, in which case the line is matched, and also, there is no backward wave (the V - component is zero). In order to resolve this discrepancy broght abot by eqation 17, we mst modify this eqation. In order to do so, the following statements are made; We have forward and backward travelling waves existing at the load ome resolving factor in eqation 17 that absorbs the V - component is reqired. For the transmission line a voltage reflection coefficient at the load can be defined. This is the ratio of the backward and forward travelling waves at z =, given by V t / V t / (18)

4 Figre 2 Voltage reflection coefficient at the load bstitting eqation 18 into 17, eqation 17 becomes R V t 1 V t 1 Where, (19) R Z or the reflection coefficient at the load expressed in terms of the characteristic impedance and the load resistance is R (21) R The introdction of the reflection coefficient into eqation 17 resolves the apparent discrepancy. It is of interest to note that a crrent coefficient at the load can also be defined from eqation 15 and eqation 18. That is, I t / (22) I t / Eqation 22 shows that the crrent reflection coefficient is the negative of the voltage reflection coefficient at the load. Forward and backward voltage waves compared with forward and backward crrent waves Forward and backward voltage waves given by eqation 14 have the same positive sign, that is z Vf ( z, t) V t and, z Vb z t V t. This is becase both are in the direction assmed for z z the total voltage V ( z, t) V t V t. (20)

5 However, forward and backward crrent waves have opposite signs. This is shown by eqation 15, where V z V z I( z, t) t t. The forward crrent wave V z I f ( z, t) t is directed in the positive z direction as is the total line crrent, bt the backward wave (, ) V z Ib z t t is directed in the opposite direction (negative z). The applet shows the reflected wave at the load for both a plse and more instrctively, for a ramp. The reflection of waves by the load discontinity can be viewed as a mirror that prodces, as a reflected wave V -, a replica of V+, that is flipped arond and all points on the V - waveform are the corresponding points on the V + waveform mltiplied by the load reflection coefficient. The total voltage at the load is the sm of all of the individal voltages existing at the load at each instant in time. The applet shows this total voltage for all instants over the time that the plse or ramp traverses back and forth down the line for three reflections. The voltage reflection coefficient at the sorce Consider Figre 3, at z = 0, the portion of the transmission line at the sorce. Figre 3 Voltage reflection coefficient at the sorce When the sorce is connected to the line a forward travelling wave is propagated down the line. There will be no backward travelling wave appearing at the sorce ntil the initial forward travelling wave has reached the load and prodced a backward travelling wave that has retrned back to the sorce. The time for the forward wave to reach the load is T = /. The portion of the reflected wave reqires an additional time T = / to move back from the load to the sorce. Therefore, for 0t 2, no backward travelling waves will appear at the sorce z = 0 and for any time less than 2/ the total voltage and crrent at z = 0 will consist only of the forward travelling wave or if the wave has flly emerged, nothing at all. The applet allows for the plse or ramp width and line length to be varied so that an emerging plse can still be emerging by the time the reflected voltage wave has retrned (back and forth several times if so desired). Ths, the total voltage and crrent at the sorce can be written as

6 0 V 0, t V t 0t 2 (23) 0 V 0 t I 0, t I t 0t 2 (24) ince the ratio of the total voltage and crrent on the line is Z 0 for 0t 2, the line appears to have an inpt resistance Z 0 over this time-interval giving rise to a redction in the vale of the sorce voltage. That is, the forward travelling wave is related to the sorce voltage V (t) by R V 0, t V 0, t RI t V t V 0, t Zo so that, 2 V 0, t V t 0 t< (25) R The applet has V 0, t 20 pixels and is fixed at this vale regardless of the ratio. R At the sorce, similar to the case for the load, a reflection coefficient can be obtained which is given by R (26) R This sorce reflection coefficient is the ratio of the incoming incident wave (which is the reflected wave from the load) and the reflected portion of this wave (that portion that is sent back towards the load). A forward travelling wave is therefore initiated as the sorce in the same manner as at the load. As only resistive dividers are sed at the sorce and at the load, the waveshapes are retained and only redced in amplitde by a factor. The applet shows the total voltage at the sorce as the sm of the vales of the individal voltage waves existing at the sorce at any time dring the corse of three reflections. The process of repeated reflections contines as re-reflections at the sorce and load. At any time the total voltage (or crrent) at any point on the line is the sm of the vales of the individal voltage waves (crrent waves) existing on the line at that point and time. The applet shows this as the distribted voltage history waveform where one is sitting at some point on the front of the voltage waveform and riding the wave. AT 04/01/2000 References: Introdction to Electromagnetic Fields, Clayton R.Pal, Keith W. Whites, yed A. Nasar, WCB McGraw- Hill, third edition, 1998, IBN: International Edition IBN:

Second-Order Wave Equation

Second-Order Wave Equation Second-Order Wave Eqation A. Salih Department of Aerospace Engineering Indian Institte of Space Science and Technology, Thirvananthapram 3 December 016 1 Introdction The classical wave eqation is a second-order