Environmental Engineering Reference
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must be computed until a steady state is reached. The results of the
computation in the transient portion, however, must be discarded. Only
those in the steady-state is used in FFT analysis to yields distortion.
This process could be very time consuming, especially for periodically
switched nonlinear circuits because the time-domain analysis of these
circuits is costly, as detailed in Chapter 7. To speed up time-domain
analysis, trigonometric polynomial [122] and envelope-following integra-
tion algorithms [123] have been proposed. Finite difference method [124]
and shooting methods [125, 126, 127, 128], which compute the steady-
state response directly, have also been investigated extensively.
As compared with the time domain approaches, frequency domain
methods compute the distortion directly. These methods are more ef-
ficient computationally for circuits containing mild nonlinearities, i.e.
the characteristics of the nonlinear elements can be depicted adequately
using a finite number of the terms of their Taylor series expansion at
the operating point. Analytical approaches [129, 130, 131] f o r distortion
analysis are pen-and-paper approaches. They are effective for circuits of
small size only. Harmonic balance approach [132, 133, 134] obtains the
distortion components by solving the determining equations derived from
balancing the harmonics of the response of circuits to a sinusoidal input
numerically. This approach is computationally expensive because the
determining equations are nonlinear algebraic equations and the num-
ber of unknowns is directly proportional to the characteristics of the
nonlinearities in the circuits, i.e. the order of harmonics
K
considered
in approximation of the response of the circuit
where is the frequency of the sinusoidal input. is then plugged
into the nonlinear equations depicting the circuits and a power series
in is obtained. Making use of the identity of power series that
a power series is zero if and only if all the coefficients of the power
series are identically zero, a set of nonlinear algebraic equations with
the unknowns are derived. These nonlinear alge-
braic equations are called determining equations. They are solved using
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