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Now, using the principle of energy conservation as mentioned in the previous slide, the energy injected by the voltage source into the transmission line during 2T d is V*I*t=V1*I1*2n√(LC) = The total available energy stored in the inductors and capacitors within the transmission line during 2T d + total energy dissipated in the final load resistance RL during 2T d , i.e.: 1. V 1 I 1 2n√(LC)= { 1 L( 2 ) 2 + 1 CV 2 2 }*n + 2 2 *n√(LC) 2 2 2. We know I 1 =V 1 /Z0. Thus, 1 2 2n√(LC)= { 1 L( 2 ) 2 + 1 CV 2 2 }*n + 2 2 *n√(LC) 2 2 0 3. Dividing both sides by n√(LC), 1 2 0 2 2 0 2 2 *2 = *Z0 + + 2 1 2 1 ( 2 ) 2 4. i.e., ܼ0 ܸ 1 2 2ܼ0 1 *2 = V 2 2 * { } + ܴ ܮ 1 + ܼ0 2ܴ ܮ 2 2 5. ܼ0 ܸ 1 2 *2 = V 2 2 * { } + + 2ܴܮܼ 0 2 3 3 2ܴܮ 2 4ܴܮ ܼ0 4ܴ ܮ ܼ0 6. 4ܴܮ V 1 2 V 2 2 * { } + + ܼ 0 2 2 2 ܴܮ 2 2ܴ ܮ ܼ0 = 7. 4ܴܮ V 1 2 V 2 2 * { } + ܼ0 2 ܴܮ 2 = ( ) 8. 2ܴ ܮ V 1 V 2 * { } + ܼ0 ܴܮ = ( ) -I 9. Or, 2ܴܮ V 2 V 1 * + ܼ0 ܴܮ = ( ) -II This is the expression for the voltage at the output of the transmission line V 2 after the first delay T d , i.e., after the transmitted step input first reaches the output load. Reflection coefficient is given by the ratio of the amplitude of the reflected wave (V 2 -V 1 ) to the incident wave (V 1 ), i.e. reflection coefficient ρ = (V 2 -V 1 ) / V 1 . Setting 2ܴܮ V 2 V 1 * + ܼ0 ܴܮ = ( ) ' , 2ܴܮ V 1 V 1 V 1 * - + ܼ0 ܴܮ ρ = ( ) { }/ , i.e., 2ܴܮ - ܼ0 - ܴܮ + ܼ0 ܴܮ ρ = { } . Thus, ܴ ܮ - ܼ0 + ܼ0 ρ = ( ) ܴ ܮ -III When the Reflected Wave from the Load Side Reaches at the Source: Source Reflections We saw previously that when a step input is first injected into the transmission line, once it reaches the output end, the voltage at the output end reaches a value V2 depending on the value of the load termination resistance, whose expression was derived using the principle of energy conservation. Now, 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=V2/RL 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, and in this manner the new voltage V2 propagates towards the source end gradually, LC segment by LC segment. Thus, the voltage step which started out as V1 at the transmission line input gets modified to V2 upon reaching the end of the transmission line. The reflected step injected back into the transmission line from the load end has an amplitude of V2 - V1, and this step propagates towards the source end. The step on reaching the source end modifies the voltage at the source end, i.e. at the input of the transmission line from the source side, to a new voltage V3 due to reflection at the source end depending on the source resistance. To find the expression for this new voltage V3 at the input of the transmission line after the first reflection at the source end, we can again make use of the principle of energy conservation and also the principle of superposition. There is already a voltage V1 right from the instant of application of the initial step input from the source at the input of the transmission line. This has to be superimposed with the voltage step at the source side due to source reflection, to find the net new voltage V3 at the input of the transmission line, i.e. at x= 0. To find the step voltage change alone at the source end due to the effect of reflection at the source, let us first short out the main source. The effective circuit is illustrated in Figure 9. The amplitude of the step injected back into the transmission line due to reflection at the load is V2-V1=V2': www.cadence.com 6 Accurately Modeling Transmission Line Behavior with an LC Network-based Approach