A dc-dc converter is a device that takes as an input an unregulated dc voltage and provides a regulated dc voltage as its output. The output voltage may be lower (buck converter), higher (boost converter), or the same as the input voltage. Switching dc-dc converters have a semiconductor active switch (BJT or FET) that is closed periodically with a duty cycle d in a pulse width modulated (PWM) manner. For a boost converter, averaging techniques can be used to arrive at the following state equations (Van Dijk, 1995):

where L and C are, respectively, the values of internal inductance and capacitance; i_L is the current through the internal inductor; R is the resistive load connected to the converter; E_s is the dc input voltage; and the capacitor voltage, u_C, is the converter’s output.

a. Write the converter’s equations in the form

ẋ = Ax + Bu

y = Cx

assuming d is a constant.

b. Using the A, B, and C matrices of Part a, obtain the converter’s transfer function U_C(s)/E_s(s).

dir L = -(1 - d)u. + E, dt duc C 3 (1- d)i dt R