There can be multiple configurations/circuits of a relaxation oscillator based on state-changing device. On a simplistic level, two basic configurations exist – (a) two state-changing devices in series, called D-D, where D stands for device (Figure 1A), and (b) a state-changing device in series with a resistance, called D-R (Figure 1B). In the former, the charging and discharging rates are equal, but they are different in the latter. The functioning of the two circuits is as follows.

Figure 1: (A) D-D oscillator circuit configuration and its equivalent circuit. (B) D-R oscillator circuit configuration and its equivalent circuit. (C) Load line graph for the D-R circuit and the region of operation of D-R oscillator. When the stable points (green filled points) are outside the region of operation, the system shows oscillations.

In case of two devices in series D-D, the two devices must be in opposite conduction states (one metallic and the other insulating) all the time for oscillations to occur. If the threshold voltages *v*_{l} and *v*_{h} are equal for the devices and the following condition holds:

$${V}_{l}+{V}_{h}={V}_{DD}$$(1)

and at *t*=0, the devices are in different conduction states, then any time one device switches, the other will make the opposite transition as well. As *g*_{dm} ≫*g*_{di}, the devices can be considered as switches, which are open in insulating state and closed in metallic with conductance *g*_{dm}. The mechanism of oscillations is essentially charging and discharging of the internal capacitances of the devices. The device in metallic state connects the circuit and charges (discharges) the lumped internal capacitance. The voltage at the output node increases (decreases) and eventually reaches the threshold voltage. Because of (1), both devices will switch at the same time, causing their behavior to switch. The charging (discharging) becomes discharging (charging), and the cycle continues. The modeling of a D-D oscillator is as follows. All the lowercase voltages referred in the paper are normalized voltages with respect to *V*_{DD}, which means *v*_{h} =*V*_{h}/*V*_{DD} and *v*_{l} =*V*_{l}/*V*_{DD}. Also *v*_{dd} is used as normalized and hence *v*_{dd} =1.

The single D-D oscillator can be described by the following set of piecewise linear differential equations:

$c{v}^{\prime}=\{\begin{array}{ll}\mathrm{(}{v}_{dd}-v\mathrm{)}{g}_{1dm}\hfill & \text{charging}\hfill \\ -{g}_{2dm}\hfill & \text{discharging}\hfill \end{array},$

where *g*_{1dm} and *g*_{2dm} are metallic conductances of the two devices, respectively. As *g*_{di} ≫*g*_{dm}, there is no term involving *g*_{di} in the equations. The equation can be re-written as follows:

$c{v}^{\prime}=-g\mathrm{(}s\mathrm{)}v+p\mathrm{(}s\mathrm{)}$

where *s* denotes the conduction state of the device (0 for metallic and 1 for insulating) and *g*(*s*) and *p*(*s*) depend on the device conduction state *s*, as follows:

$\begin{array}{l}g\mathrm{(}s\mathrm{)}=\{\begin{array}{ll}{g}_{1dm}\mathrm{,}\hfill & s=0\hfill \\ {g}_{2dm}\mathrm{,}\hfill & s=1\hfill \end{array}\\ p\mathrm{(}s\mathrm{)}=\{\begin{array}{ll}{g}_{1dm}\mathrm{,}\hfill & s=0\hfill \\ \mathrm{0,}\hfill & s=1\hfill \end{array}.\end{array}$

For D-R oscillators, the oscillations occur due to a lack of stable point as seen in the load line graph of Figure 1C. Solid sloped lines are the regions of operation of the device in insulating and metallic states, respectively. The system does not enter the dashed sloped lines as a transition occurs to the other conduction state at the red points. The stable points, denoted by green points, are the points where the load line intersects the I-V curve of the device. These stable points in each state lie outside the region of operation of the circuit, and hence, the circuit shows self-sustained oscillations. This is a much more practical configuration from an electrical implementation point of view, as the conditions required for oscillations are not very strict. Following a similar analysis as in the D-D oscillator case, the dynamics of the single D-R oscillator can be described as follows:

$c{v}^{\prime}=\{\begin{array}{ll}\mathrm{(}{v}_{dd}-v\mathrm{)}{g}_{dm}-v{g}_{s}\hfill & \text{charging}\hfill \\ -v{g}_{s}\hfill & \text{discharging}\hfill \end{array},$

which can be re-written as follows:

$c{v}^{\prime}=-g\mathrm{(}s\mathrm{)}v+p\mathrm{(}s\mathrm{)},$

where

$\begin{array}{l}g\mathrm{(}s\mathrm{)}=\{\begin{array}{ll}{g}_{dm}+{g}_{s}\mathrm{,}\hfill & s=0\hfill \\ {g}_{s}\mathrm{,}\hfill & s=1\hfill \end{array}\\ p\mathrm{(}s\mathrm{)}=\{\begin{array}{ll}{g}_{dm}\mathrm{,}\hfill & s=0\hfill \\ \mathrm{0,}\hfill & s=1\hfill \end{array}\end{array}$

and *s* denotes the conduction state of the system as before. Detailed analysis of configurations and modeling of such oscillators can be found in Ref. [25].

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