# Set-Point Control of a Spatially Distributed Buck Converter
## 2 Physical Modeling  
### Classical Buck Converter  
The schematics of a conventional buck converter is sketched in Figure 1 (left). The converter consists of two semiconductor elements (transistor, diode) and two reactive elements (inductor, capacitor). In this paper, the load is ohmic, i.e., a resistor with the resistance \(Z\). The associated network model of the converter is shown in Figure 1 (right). The transistor with the freewheeling diode is modeled by an ideal switch with two positions \(d\in\{0,1\}\). Therefore, the buck converter is a switching converter. The ohmic losses of the inductor and the capacitor are modeled by a resistance \(R\) and a conductance \(G\), respectively. In Figure 1 (right), the resulting passive network to model the non-ideal reactive elements is drawn in blue color.  
Based on the network model we can derive the circuits equations  
\[L\frac{\mathrm{d}\,i(t)}{\mathrm{d}\,t}+R\,i(t)+v(t) = E\,d(t), \tag{1a}\] \[C\frac{\mathrm{d}\,v(t)}{\mathrm{d}\,t}+G\,v(t)+\tfrac{1}{2}v(t) = i(t). \tag{1b}\]  
These equations form a system of first order ordinary differential equations (ODEs). In the model-theoretic context, system (1) is a lumped parameter model [23]. With the state  
Figure 1: Classical buck converter: circuit diagram (**left**), network model (**right**), where the passive part is drawn in blue color.  
vector \(x=(x_{1},x_{2})^{T}=(i,u)^{T}\), we can rewrite the implicit system of ordinary differential equations as a two-dimensional state-space model  
\[\dot{\mathbf{x}}(t)=\mathbf{A}\,\mathbf{x}(t)+\mathbf{b}\,d(t),\] (2a) where the system matrix \[\mathbf{A}\] and the input vector \[\mathbf{b}\] are given by \[\mathbf{A}=\begin{pmatrix}-\frac{R}{L}&-\frac{1}{L}\\ \frac{1}{C}&-\frac{G+1/Z}{C}\end{pmatrix},\quad\mathbf{b}=\begin{pmatrix}\frac{E}{ L}\\ 0\end{pmatrix}. \tag{2b}\]  
The circuit Equation (1) can be transformed from time to frequency domain using Laplace transform  
\[\underbrace{\begin{pmatrix}sL+R&1\\ -1&sC+G+\frac{1}{Z}\end{pmatrix}}_{\mathbf{M}(s)}\begin{pmatrix}I(s)\\ V(s)\end{pmatrix}=\begin{pmatrix}E\\ 0\end{pmatrix}D(s),\]  
where we use capital letters for the transformed signals. The characteristic polynomial of this system  
\[\det\mathbf{M}(s)=LC\,s^{2}+\left(LG+RC+\frac{L}{Z}\right)s+1+R\Big{(}G+\frac{1}{Z }\Big{)}\]  
is a second order polynomial. For meaningful physical parameter values (i.e., \(L,C,Z>0\) and \(R,G\geq 0\)), it is a Hurwitz polynomial corresponding to a damped harmonic oscillator. In the lossless case, i.e., \(R=0\) and \(G=0\), the characteristic polynomial simplifies to  
\[\det\mathbf{M}(s)=LC\,s^{2}+\frac{L}{Z}\,s+1. \tag{3}\]  
This polynomial has real roots if, and only if, \(4 CZ^{2}\leq L\).  
For a converter described by (2), one may consider three types of models [1; 18; 24; 25]. In the switched model, the input signal \(d\in\{0,1\}\) takes discrete values corresponding to the positions of the switch. In practice, the system is excited via pulse-width modulation (PWM) with a fixed switching period \(T\). If the average of all signals is taken over one switching period, one obtains the averaged model. The averaged model has the same structure as the original system (2). The averaged input signal \(d\) is called duty ratio or duty cycle with values from the interval \(d\in[0,1]\). For \(T\to 0\) the averaged model is transformed into the continuous model. In our simulations we will use the switched model with PWM excitation.  
To compute the equilibrium \(\mathbf{x}^{0}=(i^{0},u^{0})^{T}\) of the averaged model, we set \(\dot{\mathbf{x}}\equiv\mathbf{0}\). From (2) we obtain  
\[\mathbf{x}^{0}=-\mathbf{A}^{-1}\mathbf{b}\,d=\frac{E}{GRZ+Z+R}\begin{pmatrix}GZ+1\\ Z\end{pmatrix}d\rightarrow\begin{pmatrix}\frac{E}{Z}\\ E\end{pmatrix}d\]  
for the lossless case with \(R=0\) and \(G=0\). With \(d\in[0,1]\) and \(u^{0}=E\,d\), the operating point of the output voltage lies in the range between \(0\) and \(E\), i.e., the buck converter can reduce the input voltage \(E\) to an average output voltage \(0\leq u^{0}\leq E\).