PSpice Application Notes

3. Substituting for ZL, 2V 2 (ݐ)/CdV 2 (ݐ)/dt V 2 (t) V 1 * = (ܼ0 + V 2 (ݐ)/( CdV 2 (ݐ)/dt )) 4. Z0+ V 2 (t)/(CdV 2 (t)/dt) =2V 1 /(CdV 2 (t)/dt) 5. Z0CdV 2 (t)/dt+V 2 (t) =2V 1 6. Taking Laplace transform on both sides, we get V 2 (s) + sZ0CV 2 (s) =2V 1 (s)/s. Please note it is assumed initial voltage at time t= 0 across the capacitor C is 0V here. 7. i.e., V 2 (s)(1+sZ0C) =2V 1 (s)/s 8. So V 2 (s) = 2V 1 (s)/(s(1+sZ0C)) = 2V 1 (s)/(sZ0C(s+1/Z0C)) 9. Making use of partial fractions, 2V 1 (s)/(sZ0C(s+1/Z0C)) = A/sZ0C + B/(s+1/Z0C) 10. Thus, A(s+1/Z0C) + BsZ0C =2V 1 (s) 11. Letting s= 0 and s= -1/Z0C, we get A=2V 1 (s)Z0C and B = -2V 1 (s) 12. Substituting for A and B in the equation for V 2 (s) , V 2 (s) =2V 1 (s)Z0C/sZ0C - 2V 1 (s)/(s+1/Z0C) 13. i.e., V 2 (s) =2V 1 (s)/s - 2V 1 (s)/(s+1/Z0C) 14. Taking inverse Laplace transform, V 2 (t) = 2V 1 (t) - 2V 1 (t)(e -t/ Z0C ) = 2V 1 - 2V 1 (e -t/ Z0C ) , because V1 is a constant. 15. Thus, finally we get: V 2 (t) = 2V 1 (1-e -t/ Z0C ) This is the equation governing the voltage across the load capacitor, which gives the nature of variation of this voltage with respect to time. As noted before, 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+Td, 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. Thus, it is clear that capacitor charging follows a typical simple RC-type exponential charging, with the time constant being Z0C. We know that for such a charging profile, the voltage across a capacitor varies as per V final (1-e -t/RC ), and the current into it varies as per I initial (e -t/RC ). Now, taking the ratio, the impedance provided by the capacitor follows a profile given by k(e t/ RC -1), where k is a constant=V final /I initial . The simulated waveform for this impedance variation profile is given for a load cap of 2pF below in Figure 15: Figure 15: Simulated waveform for impedance variation profile for a load cap of 2pF Even this impedance sees an exponential time-variation profile, with a time constant=Z0C itself. Thus, at the output of the transmission line, the voltage exponentially charges from 0V (what the output voltage of the transmission line would've been for a shortened transmission line) to whatever voltage the output end would have gone to were the transmission line output open-circuited, with a time constant=Z0C. www.cadence.com 11 Accurately Modeling Transmission Line Behavior with an LC Network-based Approach

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