Subfield: Electromagnetism

Nonlinear Dynamics in Electric Circuits

Please read the general instructions in the separate envelope before you start this problem.

Introduction

Bistable non-linear semiconducting elements (e.g. thyristors) are widely used in electronics as switches and generators of electromagnetic oscillations. The primary field of applications of thyristors is controlling alternating currents in power electronics, for instance rectification of AC current to DC at the megawatt scale. Bistable elements may also serve as model systems for self-organization phenomena in physics (this topic is covered in part B of the problem), biology (see part C) and other fields of modern nonlinear science.

Goals

To study instabilities and nontrivial dynamics of circuits including elements with non-linear $I-V$ characteristics. To discover possible applications of such circuits in engineering and in modeling of biological systems.

Part A.Stationary states and instabilities

Fig. 1 shows the so-called S-shaped $I-V$ characteristics of a non-linear element $X$. In the voltage range between $U_{\mathrm{h}}=4.00 \mathrm{~V}$ (the holding voltage) and $U_{\text {th }}=10.0 \mathrm{~V}$ (the threshold voltage) this $I-V$ characteristics is multivalued. For simplicity, the graph on Fig. 1 is chosen to be piece-wise linear (each branch is a segment of a straight line). In particular, the line in the upper branch touches the origin if it is extended. This approximation gives a good description of real thyristors.

<img_4550>

Figure 1: $I-V$ characteristics of the non-linear element $X$.
Context question:
A.1 Using the graph, determine the resistance $R_{\mathrm{on}}$ of the element $X$ on the upper branch of the $I-V$ characteristics, and $R_{\text {off }}$ on the lower branch, respectively. The middle branch is described by the equation

$$
I=I_{0}-\frac{U}{R_{\text {int }}}
\tag{1}
$$

Find the values of the parameters $I_{0}$ and $R_{\text {int }}$.
Context answer:
\boxed{$R_{int}=2.00 $ , $I_{0}=6.00$}


Extra Supplementary Reading Materials:

The element $X$ is connected in series (see Fig.2) with a resistor $R$, an inductor $L$ and an ideal voltage source $\mathcal{E}$. One says that the circuit is in a stationary state if the current is constant in time, $I(t)=$ const.

<img_4387>

Figure 2: Circuit with element $X$, resistor $R$, inductor $L$ and voltage source $\mathcal{E}$.
Context question:
A.2 What are the possible numbers of stationary states that the circuit of Fig. 2 may have for a fixed value of $\mathcal{E}$ and for $R=3.00 \Omega$ ? How does the answer change for $R=1.00 \Omega$ ?
Context answer:
For $R=3.00 \Omega$, one always gets exactly one intersection.

For $R=1.00 \Omega$, one gets 1,2 or 3 intersections depending on the value of $\mathcal{E}$.

Question: A.3 Let $R=3.00 \Omega, L=1.00 \mu \mathrm{H}$ and $\mathcal{E}=15.0 \mathrm{~V}$ in the circuit shown in Fig.2 Determine the values of the current $I_{\text {stationary }}$ and the voltage $V_{\text {stationary }}$ on the non-linear element $X$ in the stationary state.

Solution:
The stationary state is on the intermediate branch, one can thus use the corresponding equation:



$$
\begin{aligned}
I_{\text {stationary }} & =\frac{\mathcal{E}-R_{\text {int }} I_{0}}{R-R_{\text {int }}} \\
& =3.00 \mathrm{~A} \\
U_{\text {stationary }} & =R_{\text {int }}\left(I_{0}-I\right) \\
& =6.00 \mathrm{~V}
\end{aligned}
$$



Final answer: $I_{\text {stationary }}=3.00$ , $U_{\text {stationary }}=6.00$ A,V