Non-linear transmission line circuits


# of variables # of modes # of jumps
n 3 2
Type Continuous dynamics Guards & Invariants Resets
hybrid non-polynomial linear polynomial identity

Note: The class of the benchmarks is determined by the input i(t).


Flow* n=2 line_circuit_2.model
Flow* n=2 line_circuit_4.model
Flow* n=2 line_circuit_6.model
Flow* n=2 line_circuit_8.model
Flow* n=2 line_circuit_10.model
Flow* n=2 line_circuit_12.model

Model description

We study a non-linear resistor circuit which is shown in the figure below.


The circuit is composed of n+1 non-linear resistors and the same number of capacitors. Each non-linear resistor consists of a diode and a unit resistor (r = 1). For simplicity, we assume that all capacitors have unit capacitance C = 1. For each diode, the I-V characteristic is given by I = \exp(K\cdot V) - 1. The value of K is 40 in the original model, but we set it as 5 to relieve the stiffness of the behavior. The current source i(t) in the figure is the input, and v_1 is the single output of the circuit. Therefore, the whole circuit system can be described by the following ODE.

 \left\{ \begin{array}{lcl} \dot{v}_1 & = & -2\cdot v_1 + v_2 + 2 - \exp(5\cdot v_1) - \exp(5\cdot (v_1 - v_2)) + i(t) \\ \dot{v}_2 & = & -2\cdot v_2 + v_1 + v_3 + \exp(5\cdot (v_1 - v_2)) - \exp(5\cdot (v_2 - v_3)) \\ & \cdots & \\ \dot{v}_{n-1} & = & -2\cdot v_{n-1} + v_{n-2} + v_n + \exp(5\cdot (v_{n-2} - v_{n-1})) - \exp(5\cdot (v_{n-1} - v_n)) \\ \dot{v}_n & = & -v_n + v_{n-1} - 1 + \exp(5\cdot (v_{n-1} - v_n)) \end{array} \right.

Scalable continuous and hybrid benchmarks can be built based on the circuit according to the size n and various types of the input. Here, we consider a discontinuous input i(t) which is defined by

     \[ i(t) = \left\{ \begin{array}{ll} 2, & t\leq 1 \\ 3 - t, & 1 < t \leq 2 \\ 1, & t > 2 \end{array} \right. \]

Reachability settings

The initial set under consideration is defined by  v_i \in [0,0.02] for  i=1,\dots,n .


The following figure shows an overapproximation of the reachable set for the case  n=6 over the time horizon  [0,3] computed by Flow*:



[1] M. Rewienski, J. White. A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. In IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Volume 22(2), pages 155–170, IEEE, 2003.