PSpice Application Notes

Figure 14: Simulation waveform of a short-circuited transmission line When the Step Reaches the End of the Transmission Line (Transmission Line Terminated in a Capacitor) The source resistance is 200ohms. Here, the output of the transmission line is terminated in a capacitor of value C. This is a special scenario. When the transmission line is terminated in a resistance= R, the injected step input on reaching the end of the transmission line is met by a constant impedance=resistance R at that instant. But in the case of a capacitance termination, the capacitor provides a time-varying impedance to the injected step input arriving at the transmission line end. This is explained below. Right at the instant the step reaches the capacitor, the impedance provided by the capacitor is zero because the capacitor acts as a short at that instant, preventing any instantaneous change of voltage across it. So, right at the instant the step reaches the capacitor, the system is analogous to a transmission line whose output is short- circuited. Now as the capacitor starts charging, the impedance provided by the capacitor, i.e., the ratio of the current into the capacitor to the voltage, keeps on changing, as per the variation in the current into the capacitor and the voltage across it. It is possible to derive the equation governing the voltage across the load capacitor, which gives the nature of variation of this voltage with respect to time. This is a particularly important result. This derivation is shown later on. Please note that this equation holds true from t=T0+T d , (where T0 is the instant of application of the original step input) to t=T0+3T d . This is because the capacitor starts charging from the instant where the original step reaches the end of the transmission line, i.e., t=T0+T d , and it can charge exactly as per this equation for a duration=round-trip delay time 2T d , after which the waveform gets disturbed by the reflected wave from the source end in response to the wave initially reflected from the load end. Derivation of the Equation for the Voltage Across the Capacitor Load at the Output of a Transmission Line Let's revisit equation num "II," which was presented earlier: V 2 V 1 * 2ܴܮ = (ܼ0 + ܴܮ) . This is the expression for the voltage at the output of the transmission line V2 after the first delay T d , i.e., after the transmitted step input first reaches the output load. V1 is the amplitude of the step initially injected into the transmission line, given by V*Z0/(Z0+RS), where V is the amplitude of the step from the main source. From the instant the originally injected step reaches the load end, to the instant when the reflected step from the source end in response to the reflection from the load end reaches back at the load end, the duration is 2T d . At any time during this time interval, if the value of the load resistance RL changes, V2 will change accordingly as per V 2 V 1 * 2ܴܮ = (ܼ0 + ܴܮ) . Let's replace RL in the equation by a generic load impedance ZL, where ZL is given by V2/I2, i.e. the ratio of the voltage across the load to the current going into the load. In this derivation, we are going to make use of the idea that the capacitor presents itself as a time-variable impedance, which is dependent on the voltage across it. So, we have to consider the instantaneous impedance ZL(t). 1. Let V2(t) be the voltage across the capacitor at time t. Now, V 2 (t) V 1 * 2ܼܮ(ݐ) = (ܼ0 + ܼܮ(ݐ)) 2. For a capacitor, the current through it I = CdV/dt, where V is the voltage across it. Thus, the impedance of the capacitor=V/I =V/(CdV/dt). Applying this principle for ZL(t), ZL(t) =V2(t)/(CdV2(t)/dt), where C is the value of the load capacitor, and V2(t) the instantaneous voltage across the capacitor at time t. www.cadence.com 10 Accurately Modeling Transmission Line Behavior with an LC Network-based Approach

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