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input is only The accuracy of numerical algo-
rithms must be sufficiently higher than this limit in order to analyze
these circuits.
2. Modeling of Clocked Quantizers
Quantizers are the key element of over-sampled sigma-delta modula-
tors. For a reliable operation, switched capacitor and switched current
networks are designed in such a way that the network variables of these
circuits reach their steady value of the clock phase before reaching the
end of the clock phase. The quantizer of over-sampled sigma-delta mod-
ulators implemented using switched capacitor or switched current tech-
niques thereby also operates in a clocked manner, i.e. the output of the
quantizer changes only at the clocking instants, and remains unchanged
during the clock phase, as shown in Fig.9.3. This observation suggests
that a clocked quantizer can be modeled effectively using the following
behavioral model:
where and are the input and output of the quantizer at
respectively, and are two constant reference voltages.
One of the main advantages of modeling quantizers using behavioral
models is the elimination of the difficulties encountered in circuit-level
modeling of quantizers.
3. Modeling of Unclocked Quantizers
The analysis of preceding sigma-delta modulators with clocked quan-
tizers is significantly simplified as we only need to worry the output of
the quantizers at clocking instants. For sigma-delta modulators with un-
clocked quantizers, the exact time instants at which the quantizers flip
must be determined with high precision. As pointed out in Chapter 1
that sigma-delta modulators with unclocked quantizers fall into the cat-
egory of circuits with both externally clocked and internally controlled
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