Averaged Non-Linear Buck PWM Switch Model
This a non-linear (also called large-signal) model because you have the product of two variables, \$d\$ (the duty ratio) and \$I_c\$ the current in terminal \$c\$ but also \$d\$ and the voltage between terminals \$a\$ and \$p\$. You can linearize these sources following two methods:


perturbation: you add a small perturbation to each of the variables. This is a low-amplitude modulation which maintains the system in its linear state. It is usually denoted as a letter with a ^ caret (or a hat) on top of it. Let's see, for the first source, you would write:


\$I_a+\hat{i_a}=(D+\hat{d})(I_c+\hat{i_c})\$

You expand and collect the ac and dc terms. A dc term multiplied by an ac term (with an hat) becomes an ac term. Cross-products of ac terms are deleted because a) they are non-linear and b) small by small is smaller. When you apply this to the above equation, you have

\$I_a+\hat{i_a}=DI_c+D\hat{i_c}+\hat{d}I_c+\hat{d}\hat{i_c}\$

the dc equation is \$I_c=DI_c\$ and the ac small-signal equation is \$\hat{i_a}=D\hat{i_c}+\hat{d}I_c\$. The large-signal or non-linear source \$I_a\$ is thus linearized by the paralleling of two current sources \$D\hat{i_c}\$ and \$\hat{d}I_c\$

The second option is to use partial differentiation: you simply calculate the sensitivity of the non-linear expression to each of its variables. For \$I_a\$, you would write

\$\hat{i_a}=\frac{\partial{I_a(d,I_c)}}{\partial{d}}\hat{d}+\frac{\partial{I_a(d,I_c)}}{\partial{I_c}}\hat{i_c}\$

You obtain the same result as in the above line. Please note that the dc equation is gone. The nice thing is that it can be automated in a math solver and this very practical when you have many variables in complex expressions.

If you want to discover small-signal modeling, check out this PPT:

http://cbasso.pagesperso-orange.fr/Downloads/PPTs/Chris%20Basso%20APEC%20seminar%202013.pdf