Environmental Engineering Reference
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where
is the variation of the phasor of
at
Writing (11.31) using (11.33) gives
Eq.(11.34) characterizes the relationship between the variation of the
branch voltages and currents of the original circuit
N
and those of its
adjoint network It is termed the
incremental weak form
of Tellegen's
theorem for periodically switched linear circuits in the phasor domain. It
is particularly useful in deriving the sensitivity of periodically switched
linear circuits, as will be seen shortly. Also note that the perturbation of
circuit element value only occurs in
N.
In what follows we apply (11.34)
to the elements of the periodically switched linear circuit of Fig.11.5 and
derive the sensitivity of the response of the circuit.
Resistors
: A linear resistor in
N
and its counterpart in
are char-
acterized by
Let there be a perturbation in the resistor value. Representing
the voltage and current of the resistor using Taylor series expansion
to the first-order and substitute the results into (11.31)
Capacitors
: The behavior of a linear capacitor of periodically switched
linear circuits in the phasor domain is characterized by
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