Maximilian Schafer

A multidimensional transfer function model for frequency dependent transmission lines

Measurements of the input impedance of electrical transmission lines indicate a frequency dependency of the line parameters. This behavior can be simulated by adjustments to a suitable transfer function model for frequency independent parameters. To this end, functional transformations in time and space set up a transfer function model as a decomposition into individual modes. Adjusting the parameters for each single mode to measured input impedances faithfully describes the frequency dependent behavior. The presented approach does not only model the input impedance, it is also suitable for time domain simulations.

Transfer function models for distributed-parameter systems with impedance boundary conditions

ABSTRACT A transfer function description is derived for a general class of linear distributed parameter systems dependent on time and one spatial variable. Suitable functional transformations are the Laplace transformation for the time variable and the Sturm– Liouville transformation for the space variable. A practical problem is the determination of the eigenfunctions of the Sturm– Liouville transformation since these depend on the type and the parameters of the boundary conditions. This contribution shows that the design of a transfer function model can be separated from the correct treatment of the boundary conditions. The presented approach exhibits strong parallels to state feedback techniques from control theory. Examples for an electrical transmission line demonstrate how terminations with arbitrary complex impedances can be considered without redesigning the transmission line model.

A transfer function model for 1D diffusion processes

The dynamics of linear diffusion processes can be modelled by multi-dimensional transfer functions in a spatiotemporal frequency domain. Finite spatial domains with simple and impedance type boundary conditions are possible. The resulting discrete time synthesis algorithm captures the essential behaviour with a restricted number of modes. Dynamic diffusion processes are of interest for molecular communication as shown by several simulation examples.

An analytical fault model for direct current lines

DC grids of system voltages above 20V demand different safety concepts compared to conventional AC grids. More elaborate protection devices have to be developed to detect not only high-power, but also low-power faults. The discrimination between faults and load variations can be supported by model-based machine learning methods. For this purpose, this contribution developes time-invariant and linearized system-models with chains of two-ports including possible faults. To consider faults in transmission lines occuring after steady-state system conditions, the initial distribution of voltages and currents is modeled by spatially concentrated equivalent sources. This approach leads to an analytic frequency domain solution without spatial discretization. Measurements on a two-core cable compare favourably with this closed form model.

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

This contribution presents an approximate solution of the enzyme kinetics problem for the case of excess of an enzyme over the substrate. A first order perturbation approach is adopted where the perturbation parameter is the relation of the substrate concentration to the total amount of enzyme. As a generalization over existing solutions for the same problem, the presented approximation allows for nonzero initial conditions for the substrate and the enzyme concentrations as well as for nonzero initial complex concentration. Nevertheless, the approximate solution is obtained in analytical form involving only elementary functions like exponentials and logarithms. The presentation discusses all steps of the procedure, starting from amplitude and time scaling for a non-dimensional representation and for the identification of the perturbation parameter. Suitable time constants lead to the short term and long term behaviour, also known as the inner and outer solution. Special attention is paid to the matching process by the definition of a suitable intermediate layer. The results are presented in concise form as a summary of the required calculations. An extended example compares the zero order and first order perturbation approximations for the short term and long term solution as well as the uniform solution. A comparison to the numerical solution of the initial set of nonlinear ordinary differential equations demonstrates the achievable accuracy.

Resonant electric arcs in DC microgrids with low system impedance in the VLF-band

Various fault scenarios have been analyzed by running a number of differently combined 380 VDC microgrid tests. These tests represented a common grid topology with low system impedance at grid resonance points within the single- or lower double-digit kHz range. At serial arc faults, self-excited resonant modes of the arc plasma column have been observed. They lead to an increased arc column stability compared to non-resonant arcs with colored noise behavior. These characteristics require a special focus on pattern recognition methods for arc fault sensors along with extended suitability tests for mechanical and hybrid switchgear concerning the altered stability of switching arcs. The use of small-signal models for system components such as source and load converters as well as for arcs with regard to large-signal DC operating points and converter control modes is helpful in order to describe the reaction of the system in the event of a malfunction. This is essential for the development of suitable protective components and algorithms.

Calculation of the transformation kernels for the functional transformation method

The main challenge in the application of the Functional Transformation Method is the determination of the transformation kernels for the forward and inverse Sturm-Liouville transformation. This paper presents a straightforward way for their calculation from the eigenfunctions and eigenfrequencies of the underlying physical system. The proposed form of the eigenfunctions is based on a beneficial representation of the matrix exponential. A suitable approach for the calculation of eigenfrequencies is also presented. Two extended examples demonstrate the proposed algorithms.