\( \DeclareMathOperator{\abs}{abs} \)

peak limiting current mode control

determine d

(%i1) e1: jc = j0 + m1 * d;
\[\mathrm{\tt (\%o1) }\quad \mathit{jc}=d\cdot \mathit{m1}+\mathit{j0}\]

determine j1

(%i2) e2: j1 = jc + m2 * (1 - d);
\[\mathrm{\tt (\%o2) }\quad \mathit{j1}=\left( 1-d\right) \cdot \mathit{m2}+\mathit{jc}\]

solve for j1 and d

(%i3) s1: linsolve([e1, e2], [j1, d]);
\[\mathrm{\tt (\%o3) }\quad [\mathit{j1}=\frac{\mathit{j0}\cdot \mathit{m2}-\mathit{jc}\cdot \mathit{m2}+\mathit{m1}\cdot \left( \mathit{m2}+\mathit{jc}\right) }{\mathit{m1}},d=\frac{\mathit{jc}-\mathit{j0}}{\mathit{m1}}]\]
(%i4) ej1: s1[1];
\[\mathrm{\tt (\%o4) }\quad \mathit{j1}=\frac{\mathit{j0}\cdot \mathit{m2}-\mathit{jc}\cdot \mathit{m2}+\mathit{m1}\cdot \left( \mathit{m2}+\mathit{jc}\right) }{\mathit{m1}}\]

find the steady state j0, labeled J0

(%i5) e3: ev(ej1, j1 = j0);
\[\mathrm{\tt (\%o5) }\quad \mathit{j0}=\frac{\mathit{j0}\cdot \mathit{m2}-\mathit{jc}\cdot \mathit{m2}+\mathit{m1}\cdot \left( \mathit{m2}+\mathit{jc}\right) }{\mathit{m1}}\]
(%i6) s2: linsolve(e3, j0);
\[\mathrm{\tt (\%o6) }\quad [\mathit{j0}=-\frac{\mathit{jc}\cdot \mathit{m1}+\left( \mathit{m1}-\mathit{jc}\right) \cdot \mathit{m2}}{\mathit{m2}-\mathit{m1}}]\]
(%i7) J0: rhs(s2[1]);
\[\mathrm{\tt (\%o7) }\quad -\frac{\mathit{jc}\cdot \mathit{m1}+\left( \mathit{m1}-\mathit{jc}\right) \cdot \mathit{m2}}{\mathit{m2}-\mathit{m1}}\]

introduce perturbations

(%i8) e4: ev(ej1, j1 = J0 + dj1, j0 = J0 + dj0);
\[\mathrm{\tt (\%o8) }\quad \mathit{dj1}-\frac{\mathit{jc}\cdot \mathit{m1}+\left( \mathit{m1}-\mathit{jc}\right) \cdot \mathit{m2}}{\mathit{m2}-\mathit{m1}}=\frac{-\mathit{jc}\cdot \mathit{m2}+\mathit{m1}\cdot \left( \mathit{m2}+\mathit{jc}\right) +\mathit{m2}\cdot \left( \mathit{dj0}-\frac{\mathit{jc}\cdot \mathit{m1}+\left( \mathit{m1}-\mathit{jc}\right) \cdot \mathit{m2}}{\mathit{m2}-\mathit{m1}}\right) }{\mathit{m1}}\]

solve for dj1

(%i9) e4: ratsimp(e4);
\[\mathrm{\tt (\%o9) }\quad -\frac{\left( \mathit{jc}+\mathit{dj1}\right) \cdot \mathit{m1}+\left( \mathit{m1}-\mathit{jc}-\mathit{dj1}\right) \cdot \mathit{m2}}{\mathit{m2}-\mathit{m1}}=\frac{-\mathit{jc}\cdot {{\mathit{m1}}^{2}}+\left( \left( \mathit{jc}-\mathit{dj0}\right) \cdot \mathit{m1}-{{\mathit{m1}}^{2}}\right) \cdot \mathit{m2}+\mathit{dj0}\cdot {{\mathit{m2}}^{2}}}{\mathit{m1}\cdot \mathit{m2}-{{\mathit{m1}}^{2}}}\]

final result

(%i10) solve(e4, dj1);
\[\mathrm{\tt (\%o10) }\quad [\mathit{dj1}=\frac{\mathit{dj0}\cdot \mathit{m2}}{\mathit{m1}}]\]
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