## 4 Application Scenario: Step-Down Converter
### Modeling  
According to Figure 2, a step-down converter is combined with a fuse for excessive current protection, a variable load \(R_{\text{S}}=\tilde{R}_{\text{S}}+\Delta R_{\text{S}}\), and the variably connectable additional resistor \(\tilde{R}_{\text{C}}\). For simplicity of the following state equations, the diode forward bias is assumed to be zero. In future work, it can be included in a straightforward manner by means of an additional system parameter in the corresponding voltage loop equation.  
Here, the equations (denoted for brevity without explicitly mentioning time arguments)  
\[u_{\rm E}=u_{R_{\rm L}}+u_{\rm L}+u_{R_{\rm C}}+u_{\rm C}\quad\text{and}\quad u_{ \rm C}+u_{R_{\rm C}}=u_{R_{\rm S}}\enspace, \tag{38}\]  
with  
\[u_{R_{\rm S}}=u_{R_{\rm S}}+\Delta u_{R_{\rm S}}\enspace, \tag{39}\]  
describe the two voltage loops, while  
\[i_{R_{\rm S}}+i_{\rm C}=i_{\rm L} \tag{40}\]  
results from [PERSON]'s node equation. The component equations of all Ohmic resistances are given by  
\[u_{R_{i}}=R_{i}\cdot i_{R_{i}} \tag{41}\]  
with \(i\in\{{\rm L},{\rm S},{\rm C}\}\). Furthermore, the inductivity and capacity are represented by  
\[u_{\rm L}=L\cdot\frac{\rm d}{{\rm d}t}i_{\rm L}\quad\text{and}\quad i_{\rm C}= C\cdot\frac{\rm d}{{\rm d}t}u_{\rm C}\enspace. \tag{42}\]  
Here, the physical energy storage is expressed by the current \(i_{\rm L}\) and the voltage \(u_{\rm C}\), which results in the ODE  
\[\frac{\rm d}{{\rm d}t}i_{\rm L}=\frac{1}{L}\biggl{(}-\biggl{(}\dot{R}_{\rm L} +\frac{R_{\rm S}R_{\rm C}}{R_{\rm S}+R_{\rm C}}\biggr{)}\cdot i_{\rm L}- \biggl{(}1-\frac{R_{\rm C}}{R_{\rm S}+R_{\rm C}}\biggr{)}\cdot u_{\rm C}+u_{ \rm E}\biggr{)} \tag{43}\]  
to describe variations of the magnetic field energy and in  
\[\frac{\rm d}{{\rm d}t}u_{\rm C}=\frac{R_{\rm S}}{C(R_{\rm S}+R_{\rm C})}\cdot i _{\rm L}-\frac{1}{C(R_{\rm S}+R_{\rm C})}\cdot u_{\rm C} \tag{44}\]  
for changes of the electric field energy. The corresponding state-space representation with  
\[{\bf x}=\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}i_{\rm L}\\ u_{\rm C}\end{bmatrix} \tag{45}\]  
results in  
\[\dot{\bf x} =\begin{bmatrix}-\frac{1}{L}\biggl{(}\ddot{R}_{\rm L}+\frac{R_{ \rm S}R_{\rm C}}{R_{\rm S}+R_{\rm C}}\biggr{)}&\frac{1}{L}\biggl{(}\frac{R_{ \rm C}}{R_{\rm S}+R_{\rm C}}-1\biggr{)}\\ \frac{R_{\rm S}}{C(R_{\rm S}+R_{\rm C})}&\frac{1}{C(R_{\rm S}+R_{\rm C})} \end{bmatrix}\cdot{\bf x}+\begin{bmatrix}\frac{1}{L}\\ 0\end{bmatrix}\cdot u_{\rm E} \tag{46}\] \[={\bf A}\cdot{\bf x}+{\bf b}\cdot u_{\rm E}\enspace.\]  
For the simulations in the following subsection with \({\bf x}(0)={\bf 0}\), the parameters in Equation (46) are set to \(L=1\,{\rm H}\), \(\ddot{R}_{\rm L}=100\,\Omega\) (together mimicking a real inductor with non-zero internal resistance; which can be implemented for a test rig with the help of an operational amplifier-based gyrator circuit), \(C=2\,{\rm mF}\), \(R_{\rm C}\in[0.1\;;\,0.6]\,\Omega\), and \(R_{\rm S}\in[0.1\;;\,3]\,\Omega\).  
Figure 2: Step-down converter.  
In addition, the input voltage of the system is defined as a piecewise constant, periodic signal of duration \(T=5\,\mathrm{ms}\) according to  
\[u_{\mathrm{E}}=\begin{cases}5\,\mathrm{V}&\mathrm{for}\quad(i-1)\cdot T\leq t<(i- 1)\cdot T+\eta\cdot T\\ 0\,\mathrm{V}&\mathrm{for}\quad(i-1)\cdot T+\eta\cdot T\leq t\leq i\cdot T\;, \;\;i\in\mathbb{N}\end{cases}\;, \tag{47}\]  
where the constant parameter \(\eta=0.6\) specifies the length of the duty cycle.