Edu

One of the rather familiar electronic devices that exhibit properties similar to those of a memristor (dependence of resistance on current at previous moments of time) is a gas-discharge lamp. Resistance $R$ of such lamp is determined by the concentration of free electrons $n_e$, which changes with current flow: $\begin{cases}R=\dfrac{F}{n_\mathrm e}\\ \dfrac{\mathrm dn_\mathrm e}{\mathrm dt}=-\beta n_\mathrm e+\alpha VI,\end{cases}$Here $F$ is determined by the geometry of the tube (in the simplest model, conductivity is proportional to concentration), the $-\beta n_\mathrm e$ term is responsible for the neutralization of free electrons as the result of collisions, and the last tern $\alpha VI$ is responsible for the thermal emission of electrons.

A1 What is the minimum voltage $V_{\min}$ that must be applied to a discharge lamp for it to light up?

The lamp is connected to a source with EMF $\mathcal E$ and internal resistance $r$.

A2 Find the expressions for the equilibrium current through the lamp and the voltage across the lamp.

Suppose now the lamp is supplied with an alternating voltage $V(t)=V_0\cos\omega t$. The internal resistance of the voltage source is still equal to $r$.

B1 Express the current and voltage on the lamp at some time through the concentration of electrons $n_\mathrm e$.

B2 Considering $n_\mathrm e$ to be nearly constant, average the equation for the concentration over time and find the equilibrium value of the electron concentration $n_{0}$.

In fact, the electron concentration depends on time, and in a first approximation this dependence can be represented as:$n_\mathrm e(t)=n_{ 0}+n_{1}\cos(2\omega t+\phi),$where $n_1\ll n_{0}$.

B3 Find $n_1$ and $\phi$.

A change in concentration results in a change in lamp resistance. We define its maximum relative change as:$\delta\equiv\frac{R_\max-R_\min}{R_\mathrm{avg}}.$

B4 Determine $\delta$ for the lamp assuming $\delta\ll1$. What does this smallness condition correspond to in terms of the initial variables of the problem?