PSpice Application Notes

Derivation of the Reflection Coefficient Expression Using the Principle of Energy Conservation As mentioned before, for a step input injected into the transmission line, the nature of termination of the transmission line is immaterial until the propagating wave reaches at the end of the transmission line. Until that point, during its traversal, the propagating waveform is blind to the kind of termination on the transmission line and can only see Z0 at each point it reaches. As far as the starting point x= 0 of the transmission line is concerned, until a round trip delay time = 2T d =2n√(LC), the impedance seen by the voltage source at x= 0 is Z0 itself, irrespective of the type of termination. Hence, a constant current=V/(Rs+Z0) is continuously injected into the transmission line entry point all during this time (T d =transmission line delay, n is the number of LC segments, and L and C are the unit inductance and capacitance values per segment). The voltage at this point x= 0, i.e., right at the entry point of the transmission line, it remains V*Z0/(Rs+Z0) during this time, irrespective of the kind of termination of the transmission line. Starting right from the step input application instant, for a time duration= the round trip transmission line delay time = 2T d =2n√(LC), the energy injected by the voltage source into the transmission line is V*I*t=V 1 I 1 2n√(LC), where V 1 ,I 1 are the voltage and current, respectively, at point x= 0. That is, the entry point of the transmission line, n, is the number of LC segments, and L and C are the unit inductance and capacitance values per segment. By the principle of energy conservation, this is the total available energy to be stored in the inductors and capacitors within the transmission line and to be dissipated in the final load resistance R L , with which the transmission line is terminated during this time duration. This is illustrated in Figure 8. Voltage, current at this point=V1,I1 Rs Source L1 L2 C1 C2 Ln Cn RL Transmission line consisting of "n" LC segments Total energy injected into the transmission line during time 2Td from step application = Total energy stored in the LC network durin ghtis time + Total energy dissipated in the load resistance during this time. Energy injected during 2Td=V*I*t=V 1 I 1 2n√(LC) Figure 8: Transmission line illustrating use of the principle of energy conservation. When the step input finally reaches the end of the transmission line, depending on the value of the termination resistance RL, the voltage at the end point becomes V2. Now the current drawn by the load resistance settles at I2=V2/RL. Starting from the nearest LC segment to the output load, the current in the inductors and the voltage across the capacitors start settling to I2 and V2, respectively, and this keeps on happening on all the inductors and capacitors gradually moving towards the source. Actually, this is, in fact, the mechanism of reflection. I will discuss reflections more later on. After a round dip delay time of around 2Td=2n√(LC), the currents in all the inductors and voltage across all the capacitors should have settled to I2 and V2, respectively. Now, the total energy stored at the end of 2Td in the LC network is given + 2 1 2 1 LI 2 CV 2 ( )*n = + 2 1 2 1 LI 2 CV 2 2 ( )*n , where n is the number of LC segments and L and C are the unit inductance and capacitance per segment, analogous to the inductance/unit length and capacitance/unit length. But I2=V2/RL, thus the total energy stored at the end of 2T d in the LC network is given by 2 1 2 L ( + 2 1 CV 2 2 )*n ( ) 2 . Regarding the energy dissipated in the load resistance RL during 2Td, note that current starts flowing across RL only after the first Td has been completed, i.e. it is= 0 for the first Td, and it only flows during a time period of Td after that. Thus, the energy dissipated in RL during this time period is given by: ܴ ܮ ܸ 2 *Td = *n√(LC) 2 ܴ ܮ ܸ 2 2 www.cadence.com 5 Accurately Modeling Transmission Line Behavior with an LC Network-based Approach

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