# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 3 Linear and Nonlinear Filters
### Metal-Oxide Variator  
Semiconductors are very sensitive to overvoltage transients and noise in the electronic circuits, and the majority of sources of these transients are the lines attached to the electrical feeding. These lines, through the general and several times unpredictable routes they follow until the power input of the devices, are very susceptible to electromagnetic interference, despite all the actions taken to protect them. Filters to suppress these transients and noise, even low-energy ones, must be extremely quick in responding to the rise of voltage and current, absorbing high rates of energy and maintaining the continuity of the regular electronic process during, and after, the occurrence. Components that apply to these exigencies are the variable resistors, called varistors (variable-resistor), which are composed of metal oxides. Variators are transient voltage surge suppressors, or TVSSs, whose resistance decreases with increasing voltage [24], i.e., varistors are components that depend on the voltage on their terminals, with a symmetrical V/I (voltage/current) characteristic curve as exemplified at Figure 10. Considering a circuit or device to protect against transients, the varistor is connected in parallel to the circuit, building a shunt that will divert the voltage excess. The value of the voltage to be diverted should be defined accordingly to the device/circuit normal operating voltage. In other words, the suppressor will deviate part of the energy (voltage and current), allowing just the necessary voltage to feed the device. As the curve in Figure 10 illustrates, the resistance is very high for the range of defined maximum operating voltages (behaving like an open circuit), but after a threshold, the resistance tends to decrease with a nonlinear behaviour, which is called \"protection level\". At region A, the varistor behaves almost like an open circuit (\(R>1\) Mohm), and at region C as a resistor with a low resistance value (\(R\)\(\sim\)1 ohm) [24; 25]. At region B, its behaviour is characterized by the equation  
\[I=\beta.V^{\alpha} \tag{12}\]  
where \(\alpha\) denotes the nonlinear varistor behaviour, and \(\beta\) is the ceramic constant that depends on the varistor type and manufacturing. The Equation (12) is a power-law where the nonlinear factor \(\alpha\) typically varies from \(\sim\)10 to as high as \(\sim\)30 for varistors, which have response times of \(<\)25 ns [24; 25]. Figure 10: Typical voltage/current (V/I) symmetrical curve of metal-oxide varistors (not dependent on the polarity of the voltage applied, linear scale). A typical varistor curve of a commercial product is illustrated in Figure 11, where, for simplicity, only the positive quadrant is shown in a log-log plot. Both \(\alpha\) and \(\beta\) can be calculated from the technical specifications of each commercial varistor, considering the values of voltage and current for the points \(a\) and \(b\) (Figure 11) and using the following equations. \[\alpha=\frac{\log\left(\frac{I_{b}}{I_{a}}\right)}{\log\left(\frac{V_{b}}{V_{a} }\right)} \tag{13}\]  
\[\beta=\frac{I_{b}}{{V_{b}}^{\alpha}} \tag{14}\]  
It is important to consider that in computational simulations the values of \(\alpha\) and \(\beta\) could slightly vary, depending on the points \(a\) and \(b\) considered in the varistor curve, and the manufacturer specifications. The value of \(\beta\), calculated in Equation (14), should use \(V\) and \(I\) closer to \(b\) at Figure 11, as shown by experimental results (see Section 4). One could argue that \(\beta\) is not really a \"constant\". It is true. It is approximately constant in region B (Figure 10). Anyway, the value obtained will be enough for simulations, as we will show. Variators are polycrystalline ceramics made by mixing metal oxides and additives. The most common varistor is done with zinc oxide, ZnO, which offers a resistance highly dependent on voltage, besides great energy capacity. The grains of zinc oxide are conductive, but the other additives are highly resistant.