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where
is the
frequency component of the voltage
of the capacitor in
Substituting (11.57) into (11.56) gives
Note (11.58) is identical to the result given in Table 11.2. The above
analysis reveals that the sensitivities of periodically switched linear cir-
cuits developed using the adjoint network approach can also be derived
using the sensitivity network technique. The differences between the
sensitivity networks of linear time-invariant and periodically switched
linear circuits also become apparent. For each element of a linear time-
invariant circuit, there is only one corresponding sensitivity network.
However, for each element of a periodically switched linear circuit, there
are an infinite number of sensitivity networks.
2.4 Numerical Examples
In this section, two periodically switched linear circuits are used to
demonstrate the effectiveness of adjoint network based sensitivity analy-
sis method. The first example is the stray-insensitive switched capacitor
integrator shown in Fig.11.9 with circuit parameters given in Table 11.4
The operational amplifier is modeled as a single-pole device with unity
gain frequency 700 kHz. Its equivalent circuit and the adjoint network
are shown in Fig.10.2. The sensitivity of the output with respect to at
1 kHz was computed using the adjoint network approach and the results
are plotted in Fig.11.10 for the real part and Fig.11.11 for the imaginary
part. The normalized sensitivity of the magnitude of the response to
is shown in Fig.11.12.
For the purpose of comparison, the same sensitivity was computed
using Watsnap, a commercial CAD tool for periodically switched linear
circuits [8] that calculates sensitivity using the direct sensitivity analysis
approach presented earlier in this chapter. The result is :
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