INTRODUCTION

The prior chapters on DC-DC converters presented circuit models without dynamic characteristics. Those models present no indication of how the converter behaves under transient conditions. The neglect of the dynamic behavior, however, provided an elegant and highly useful body of theory that provides the circuit designer with a valuable commodity: a starting place.

Once the inductance and capacitance values are determined from the CCM or DCM equations, those circuit parameters are used in simulation to observe the transient behavior of the circuit and to understand the effects of an increase or decrease in inductance and/or capacitance.

The mathematical transient features of a circuit must be expressed bydifferential equations. To facilitate the simulation process, the differential equations are formulated into the state-space model of the circuit.

STATE-SPACE MODELING OF LINEAR SYSTEMS

The state-space model of a linear system is a set of simultaneous, or coupled, first-order differential equations. An example of two coupled, first-order differential equations is the equation set

in which x ₁ and x ₂ are the state variables, a ₁₁ a ₂₂ are constant coefficients, and f(t) is a forcing function. The state-space model is a reformulation of the coupled equations into the matrix-vector form

In the case of Equation 6.1, vectors x and u in the state-space model are

and matrices A and B are

In general, for n first-order linear differential equations with m forcing functions, the state-space model is expressed...