# Set-Point Control of a Spatially Distributed Buck Converter
## 2 Physical Modeling
### Distributed Buck Converter  
The passive elements of the classical buck converter network model drawn in Figure 1 (right) with blue color can be interpreted as a lumped parameter model of a transmission line. Replacing these elements by a transmission line results in the distributed buck converter shown in Figure 2, see [8; 11]. On the left side of the transmission line, we have the electronic switch (single pole changeover). The ohmic load is on the right side of the transmission line. The transmission line can be modeled by the telegrapher's equations [26; 27]:  
\[\frac{\partial}{\partial z}v(z,t)+L^{\prime}\frac{\partial}{ \partial t}i(z,t)+R^{\prime}i(z,t) = 0, \tag{4a}\] \[\frac{\partial}{\partial z}i(z,t)+C^{\prime}\frac{\partial}{ \partial t}v(z,t)+G^{\prime}v(z,t) = 0. \tag{4b}\]  
The current \(i\) and the voltage \(u\) do not only depend on the time, but also on the spatial position \(z\in(0,l)\), where \(l\) denotes the length of the transmission line (see Figure 2). In this model, the lumped parameters \(L\), \(C\), \(R\), and \(G\) are replaced by distributed parameters \(L^{\prime}\), \(C^{\prime}\), \(R^{\prime}\), \(G^{\prime}\) that describe inductance, capacitance, resistance, and conductance per length, respectively. These equations form a system of first order partial differential equations (PDEs). In a control-theoretic context, this system is also called a distributed parameter model [23; 28]. The dynamics of the transmission line is influenced by the boundary conditions. On the left boundary, i.e., at \(z=0\), we have the switched input voltage \(E\,d(t)\). On the right boundary, i.e., at \(z=l\), we have the ohmic load \(Z\). This leads to the Dirichlet boundary conditions  
\[v(0,t) = E\,d(t), \tag{4c}\] \[v(l,t)-Z\,i(l,t) = 0. \tag{4d}\]  
Overall, Equation (4) constitute a boundary value problem. Next, we want to transform the PDE model of the transmission line into frequency domain. Applying the Laplace transform w.r.t. time \(t\) to Equation (4a,b) yields  
\[\frac{\partial}{\partial z}V(z,s)+(sL^{\prime}+R^{\prime})I(z,s) = 0, \tag{5a}\] \[\frac{\partial}{\partial z}I(z,s)+(sC^{\prime}+G^{\prime})V(z,s) = 0. \tag{5b}\]  
where we assume vanishing initial values, i.e., \(i(\cdot,0)=0\) and \(v(\cdot,0)=0\). With this transformation, the PDEs are transformed into a system of first order linear homogeneous ODEs. Using an exponential function \(\mathrm{e}^{\lambda z}\) (Euler ansatz), system (5) is associated with the system matrix  
\[\mathbf{M}= \begin{pmatrix}sL^{\prime}+R^{\prime}&\lambda\\ \lambda&sC^{\prime}+G^{\prime}\end{pmatrix} \]  
having the characteristic equation  
\[\det(\mathbf{M})=(sL^{\prime}+R^{\prime})(sC^{\prime}+G^{\prime})-\lambda^{2}=0.\]  
In the lossless case, i.e., \(R^{\prime}=0\) and \(G^{\prime}=0\), the characteristic equation simplifies to  
\[\det(\mathbf{M})=L^{\prime}C^{\prime}s^{2}-\lambda^{2}=0\quad\text{with}\quad \lambda_{1,2}=\pm s\sqrt{L^{\prime}C^{\prime}}. \tag{6}\]  
The transmission line can then also be described by the _characteristic impedance_\(Z_{0}=\sqrt{L^{\prime}/C^{\prime}}\) and the propagation delay \(\tau=\sqrt{L^{\prime}C^{\prime}}\). Figure 2: Distributed buck converter with a transmission line. **Remark 1**. : _The characteristic polynomials (3) and (6) are both second order polynomials, but have a different structure. Although the reactive elements of the classical converter are assumed to be lossless, we have a damping term in (3) resulting from the load resistance. The characteristic polynomial (6) resulting from the lossless transmission line has no damping term at all, because the damping due to the load is not part of the PDE itself but of the boundary condition (4d)._  
**Remark 2**. : _In our modeling, we considered ohmic losses of the reactive elements and the transmission line, respectively. However, we assumed that the switching is ideal. In practice, the switch is usually implemented by a transistor together with a freewheeling diode, but could also be realized with two transistors.