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Figure from:Transient Sensitivity Computation for MOSFET Circuits

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which is discontinuous; hence, dq/dp is discontinuous, by recalling that g also equals C(v, — v). Now, since the charge sensitivity is the integral of the current sensitivity, there must be an impulse in the current sensitivity at the time when u(t) = 3. What this example demonstrates, is that when there are loops of capacitors and voltage sources, and one of the capacitor charge models is of the type assumed here, then the current can possibly have dis- continuities and the current sensitivity can possibly have impulses. which is continuous. Computing the time derivatives it can be seen that 0(4) and i(t) are discontinuous. The voltage sensitivity response can be computed as

Figure 17 which is discontinuous; hence, dq/dp is discontinuous, by recalling that g also equals C(v, — v). Now, since the charge sensitivity is the integral of the current sensitivity, there must be an impulse in the current sensitivity at the time when u(t) = 3. What this example demonstrates, is that when there are loops of capacitors and voltage sources, and one of the capacitor charge models is of the type assumed here, then the current can possibly have dis- continuities and the current sensitivity can possibly have impulses. which is continuous. Computing the time derivatives it can be seen that 0(4) and i(t) are discontinuous. The voltage sensitivity response can be computed as

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for the sensitivity circuit branch equations. The two terms on the right in (30) will be treated separately in order to facilitate the actual implementation latter on. Further def- initions of variables and terms that will be needed are listed below:
for the sensitivity circuit branch equations. The two terms on the right in (30) will be treated separately in order to facilitate the actual implementation latter on. Further def- initions of variables and terms that will be needed are listed below:
Fig. 1. RC circuit for sensitivity test example.
Fig. 1. RC circuit for sensitivity test example.
Fig. 4. Sensitivity response with respect to capacitance (partially normal- ized), simulation result (thin line), and analytic result (thick line). Fig. 3. Sensitivity response with respect to conductance, simulation result (thin line), and analytic result (thick line).
Fig. 4. Sensitivity response with respect to capacitance (partially normal- ized), simulation result (thin line), and analytic result (thick line). Fig. 3. Sensitivity response with respect to conductance, simulation result (thin line), and analytic result (thick line).
sitivity with respect to W,, the width of depletion transis- tor M,, are shown as the thin line in Fig. 7. For compar- ison and accuracy verification this sensitivity response was also calculated by perturbation; that is, doing an addi- tional simulation with the width perturbed and computing AV,/AW, from the results. For the simulations that were used in this perturbation computation the maximum time step was restricted to 4ps, which requires there to be at least 1000 time points in the 4-ns simulation. This was done to insure that the perturbation sensitivity would be very accurate. The thick line in Fig. 7 depicts this pertur- bation sensitivity. Likewise, in Fig. 8 the baseline sensi- tivity with respect to Wg and the perturbation sensitivity are shown, here the perturbation sensitivity shows a small amount of numerical noise. In Figs. 9 and 10 is shown, for W, and Wz, respectively, the previous baseline sensi-
sitivity with respect to W,, the width of depletion transis- tor M,, are shown as the thin line in Fig. 7. For compar- ison and accuracy verification this sensitivity response was also calculated by perturbation; that is, doing an addi- tional simulation with the width perturbed and computing AV,/AW, from the results. For the simulations that were used in this perturbation computation the maximum time step was restricted to 4ps, which requires there to be at least 1000 time points in the 4-ns simulation. This was done to insure that the perturbation sensitivity would be very accurate. The thick line in Fig. 7 depicts this pertur- bation sensitivity. Likewise, in Fig. 8 the baseline sensi- tivity with respect to Wg and the perturbation sensitivity are shown, here the perturbation sensitivity shows a small amount of numerical noise. In Figs. 9 and 10 is shown, for W, and Wz, respectively, the previous baseline sensi-
Fig. 8. Baseline sensitivity response (thick line) and perturbation sensitiv- ity response (thin line) for Wz.
Fig. 8. Baseline sensitivity response (thick line) and perturbation sensitiv- ity response (thin line) for Wz.
Fig. 10. Baseline sensitivity response (thick line) and restricted time step sensitivity response (thin line) for W,. Fig. 9. Baseline sensitivity response (thick line) and restricted time step sensitivity response (thin line) for W,.
Fig. 10. Baseline sensitivity response (thick line) and restricted time step sensitivity response (thin line) for W,. Fig. 9. Baseline sensitivity response (thick line) and restricted time step sensitivity response (thin line) for W,.
Fig. 12. Responses of output nodes of inverter chain. Fig. 13. Sensitivity responses of inverter chain for W,. are shown in Fig. 13 for W, and in Fig. 15 for W,. Pertur- bation sensitivities computed with the baseline tolerances are shown in Fig. 14 for W, and in Fig. 16 for W,. These sensitivity waveforms have discontinuities and are more complex than those shown previously. The underlying rea- sons for this will be discussed in a later section. Observe however that the comparison between the sensitivity sim- ulation and the perturbation results are essentially the same. These sensitivity waveforms were also compared to sensitivity simulations done with very small maximum time step control and those results were also essentially the same.
Fig. 12. Responses of output nodes of inverter chain. Fig. 13. Sensitivity responses of inverter chain for W,. are shown in Fig. 13 for W, and in Fig. 15 for W,. Pertur- bation sensitivities computed with the baseline tolerances are shown in Fig. 14 for W, and in Fig. 16 for W,. These sensitivity waveforms have discontinuities and are more complex than those shown previously. The underlying rea- sons for this will be discussed in a later section. Observe however that the comparison between the sensitivity sim- ulation and the perturbation results are essentially the same. These sensitivity waveforms were also compared to sensitivity simulations done with very small maximum time step control and those results were also essentially the same.
Fig. 14. Perturbation sensitivity responses of inverter chain for W,.
Fig. 14. Perturbation sensitivity responses of inverter chain for W,.
Fig. 16. Perturbation sensitivity responses of inverter chain for Wg.
Fig. 16. Perturbation sensitivity responses of inverter chain for Wg.
One can also write the circuit equations in terms of U as shown
One can also write the circuit equations in terms of U as shown
Fig. 19. Circuit responses; charge, first time derivative of charge (current). and second time derivative of charge.
Fig. 19. Circuit responses; charge, first time derivative of charge (current). and second time derivative of charge.
Fig. 20. Circuit responses; voltage and first time derivative of voltage.
Fig. 20. Circuit responses; voltage and first time derivative of voltage.
Fig. 21. Sensitivity responses; voltage, charge, and current. Specifically, in the MOSFET model the charge has dis- continuous derivatives with respect to voltage and is a lin- ear function of the width parameter. The response sensi- tivities are easily derived and provided below. The voltage sensitivity is
Fig. 21. Sensitivity responses; voltage, charge, and current. Specifically, in the MOSFET model the charge has dis- continuous derivatives with respect to voltage and is a lin- ear function of the width parameter. The response sensi- tivities are easily derived and provided below. The voltage sensitivity is
In order to demonstrate impulses in the sensitivity cir- cuit, examine the circuit in Fig. 22 where C is a linear capacitor and C,, is a capacitor with charge model as given in (51). Assuming there is not any charge stored in the circuit at time zero, the solution for time greater than zero is given by
In order to demonstrate impulses in the sensitivity cir- cuit, examine the circuit in Fig. 22 where C is a linear capacitor and C,, is a capacitor with charge model as given in (51). Assuming there is not any charge stored in the circuit at time zero, the solution for time greater than zero is given by
Fig. 22. Circuit for second example. trolling the time steps. For the sensitivity responses the discontinuous behavior starts appearing in the voltage and current sensitivities, but the charge remains continuous. The discontinuous voltage sensitivity behavior was ob- served previously in the inverter chain example, note Fig. 13-Fig. 16.
Fig. 22. Circuit for second example. trolling the time steps. For the sensitivity responses the discontinuous behavior starts appearing in the voltage and current sensitivities, but the charge remains continuous. The discontinuous voltage sensitivity behavior was ob- served previously in the inverter chain example, note Fig. 13-Fig. 16.