Constraints on the Optimal Design of Switching Power Supplies

When optimizing the design of switching power supplies and their subcircuits, equations or inequality constraints are often used to represent the various technical performance indicators and other structural or process requirements that the design results should meet. For example, the output voltage ripple of the main circuit must be less than a certain value, the phase margin and gain margin of the closed-loop system must be greater than a certain specified value to ensure the relative stability of the system; the magnetic density in the core of the transformer and inductor components should be less than the allowable value, and the total area of ​​the winding should be less than the window area, etc. These constraints are nonlinear (or linear) functions of all (or part) of the design variables. Assume that there are a total of constraints, which can be expressed as inequalities:

gi(X)≥0,i=1,2,3,……,m (14-16)

For example, if the switching power supply output ripple voltage △UO≤1%V is specified, it is related to the following parameters: switching frequency, inductance, capacitance, duty cycle of the output filter, equivalent series resistance of the filter capacitor, etc. The analytical relationship between △UO and design variables can be derived, expressed as △UO(X). Then the ripple voltage requirement is expressed by the following inequality:

g(X)=△UO(X)/UO-0.01≤0 (14-17)

This constraint is an inequality not greater than 0. If a negative sign is added before the constraint function g(X), the inequality not greater than 0 can be converted to not less than 0. There is no substantial difference between the two. Therefore, in general, in a mathematical model, various inequality constraints can be uniformly expressed using equations (14-16). In some cases, the model may also contain p equality constraints.

gi(X)=0,i=1,2,……,p

The geometric meaning of the five inequality constraints represented by equations (14-16) can be explained in two dimensions by a planar feasible domain, as shown by the shaded lines in the figure (here it is assumed that there are five inequality constraints). The points in the feasible domain represent possible design solutions that meet the technical performance (or other) requirements, that is, feasible solutions. The boundaries of the feasible domain satisfy equations (14-18)

gi(X)≥0,i=1,2,……,m (14-18)

As can be seen from the figure, under constraints, the optimal point of the quadratic objective function falls on the boundary of the feasible region (the example in the figure is the point X* that falls on g2(X) = 0), and the solution is unique. The solutions to other unconstrained optimization problems are different.

Figure 2. Quadratic objective function and feasible region (constrained optimization problem)

Usually, there are upper and lower limits or non-negative constraints on the design variables, as shown in Equation (14-19) and Equation (14-20).

Where xi, ai, and bi represent the i-th variable and its lower and upper limits, respectively.

For problems where the design variables are discrete values, additional discrete variable constraints should be added. For example, if q of the n variables should take discrete values, then the discrete variable constraints can be expressed as

In the formula, Di represents the i-th discrete value, that is, a discrete point on the i-dimensional coordinate.