The equivalent circuit for the current sensor with additional filter ( R _f and C _f) is depicted in Figure 1.30. For the steady-state analysis, the driving current source is placed in ( a + bt) form by rewriting equation (1.50). It is understood that

(A.1)

(A.2)

Given an unknown initial voltage, V _?, the output voltage in Laplace transformation form is

(A.3)

where

During the on-time D T _s, the output voltage in time domain is given as

(A.4)

while during the off-time it is given as

(A.5)

where f ₁( t) and f ₂( t) are the inverse transforms of their corresponding transfer functions in (A.3).

The continuity of states at the time-domain transition boundaries requires

(A.6)

(A.7)

Both unknowns, V _?1 and V _?2, can be solved from (A.6) and (A.7) and expressed in terms of V _in, V _o, D, and other components. The steady-state duty cycle is then determined by

(A.8)

We define an implicit function

(A.9)