= Tunnel-diode load-line stability
{title2=$1/R+I'(V)>0$}
A <current-voltage characteristic> determines possible steady <operating points>; it does not alone determine their <linear stability>. Consider an ideal <voltage source> $V_s$ feeding a <tunnel diode> through a <resistor> $R>0$, with a positive constant <capacitance> $C$ across the junction. Assume a quasistatic device relation $I(V)$ and neglect <inductance>, transit delays, <noise> and heating. <Kirchhoff's current law> gives the original one-state model
$$
C\dot V=\frac{V_s-V}{R}-I(V).
$$
A steady <operating point> $V_*$ satisfies $I(V_*)=(V_s-V_*)/R$: the device curve intersects the descending supply line. This is the source-side counterpart of a <resistive load line>. A small perturbation $V=V_*+\delta V$ obeys
$$
C\,\delta\dot V=-\left(\frac1R+I'(V_*)\right)\delta V,\qquad
\lambda=-\frac{1/R+I'(V_*)}{C}.
$$
Thus the <equilibrium point> is locally attracting if $1/R+I'(V_*)>0$ and unstable if the inequality reverses. Equality requires higher-order analysis. Even on a <negative differential resistance> segment, a sufficiently small supply <electrical resistance> can stabilize this model. The assumption about the <electrical circuit> matters: an inductive or delayed circuit has additional state variables and may behave differently.
Choose an original toy forward characteristic, with no fitted device parameters:
$$
x=\frac{V}{V_0},\qquad
I(V)=I_0\,i(x),\qquad
i(x)=x^3-6x^2+10x=x\bigl((x-3)^2+1\bigr),
$$
where $V_0=0.10\,\mathrm V$, $I_0=1.0\,\mathrm{mA}$ and $0\le x\le3.6$. For every $x>0$, $i(x)>0$, so this entire forward curve absorbs positive total <power>. Nevertheless
$$
I'(V)=\frac{I_0}{V_0}(3x^2-12x+10)<0
\quad\Longleftrightarrow\quad
2-\frac{\sqrt6}{3}<x<2+\frac{\sqrt6}{3}.
$$
The model therefore separates <negative differential resistance> from negative total $V/I$. The corresponding peak and valley are at approximately $0.1184\,\mathrm V$ and $0.2816\,\mathrm V$; these voltages deliberately differ from the historical peak.
With $R=100\,\Omega$ and $V_s=0.60\,\mathrm V$, the supply line is $I/I_0=6-x$. The intersection equation factors exactly:
$$
i(x)+x-6=(x-1)(x-2)(x-3)=0.
$$
The three <operating points> are $(0.10\,\mathrm V,5\,\mathrm{mA})$, $(0.20\,\mathrm V,4\,\mathrm{mA})$ and $(0.30\,\mathrm V,3\,\mathrm{mA})$. Their values of $1/R+I'(V_*)$ are respectively $20$, $-10$ and $20\,\mathrm{mS}$. The outer two attract nearby trajectories; the middle one repels them. This is <bistability> in the specified one-state <electrical circuit>.
\Image[/tunnel-diode-toy-load-line-stability.png]
{title=Chosen <tunnel diode> curve and supply-line <linear stability>}
{description=Original calculated <current-voltage characteristics> and trajectories of the stated quasistatic model. Filled green points attract nearby trajectories; the open red point repels them. The same negative-slope point becomes attracting when the supply <electrical resistance> is reduced and the source voltage adjusted. All values are chosen, not measured data or a fit to a historical device. Both panels use $C=10\,\mathrm{nF}$ and the same toy device curve.}
{source=tunnel-diode-toy-load-line-stability.py}
For the second panel, choose $R=25\,\Omega$ and $V_s=0.30\,\mathrm V$. Now $I/I_0=12-4x$ and
$$
i(x)+4x-12=(x-2)\bigl((x-2)^2+2\bigr)=0.
$$
There is exactly one real <operating point>, again $(0.20\,\mathrm V,4\,\mathrm{mA})$. Here $1/R+I'(V_*)=20\,\mathrm{mS}>0$, despite the device's own slope $I'(V_*)=-20\,\mathrm{mS}$. With $C=10\,\mathrm{nF}$, the perturbation decays with time constant $C/(20\,\mathrm{mS})=0.50\,\mathrm{\mu s}$; the unstable middle point of the first panel has growth time $1.0\,\mathrm{\mu s}$.
The one-state autonomous model supports switching between attracting <operating points> after a perturbation or parameter change; it cannot produce a nonconstant periodic orbit. A proposed <oscillator> needs additional dynamics. This limitation is useful: a load-line intersection establishes a possible steady reading, the slope test establishes local attraction in a specified model, and neither calculation alone predicts an oscillation frequency for a real <tunnel diode>.
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