Dimensional Dependence of Voltage Coefficient of Resistance (α_v) for PMNPT Based Varistors

Abstract

This paper describes design of a structured composite varistor (voltage variable resistor) exhibiting a positive voltage coefficient of resistance. This structured composite consists of a thick film resistor printed on a lead magnesium niobate-lead titanate substrate. In this device the variation of resistance due to an applied voltage is produced through the coupling of piezo electric strain to a thick film resistor. It has been found that this resistor can give rise to a positive voltage coefficient of resistance when used as a three terminal or four terminal device. A mathematical model is developed to establish the variation of resistance with an applied voltage. Dimensional dependence of voltage coefficient of resistance (α _v ) was investigated by varying the width of the thick film resistor while keeping its length constant. It was found that α _v is inversely proportional to the width of the thick film resistor.

Appendix

For PMNPT, the Piezoelectric stain coefficient d ₃₁ is negative and d ₃₃ is positive. As mentioned earlier, for the converse piezoelectric effect in which an applied field E produces proportional strain s, the value of d coefficient is numerically the same as that, in the case of direct effect [6].

$$ {\text{Strain}}\;s\;\text{along the direction of the electric field}\;E,s = d_{33} E $$

(1)

$$ {\text{But}},E = V_{1} /d $$

(2)

The strain produced in the PMNPT disc is coupled to the rubber-graphite thick film through Poisson’s ratio p.

$$ {\text{Therefore}},{\text{ strain in the thick film}},\;s^{{\prime }} = d_{33} \left( {V_{1} /d} \right)p $$

(3)

$$ {\text{Total change in length of the thick film, }}\varDelta L = d_{33} p\left( {V_{1} /d} \right)L $$

(4)

$$ {\text{Number of contacts between the graphite particles in the thick film along the length }}L, = L/\left( {D + t_{0} } \right) $$

(5)

$$ {\text{Therefore}},{\text{ the thickness of the rubber film between two conducting particles}},{\text{ after the expansion due to piezoelectric strain}}, = t_{0} + d_{33} p\left( {V_{1} /d} \right)\left( {D + t_{0} } \right) $$

(6)

Due to change in the thickness of the film, there is change in current flow, between the conducting granules, Assuming the current flow to be due to field emission, as given by Fowler–Nordheim equation, the current I that flows when a voltage V ₂ is applied between the two terminals of the thick film, may be written as,

$$ \text{I} =\text{a}[\text{AE}^{{\prime}^{2}}\exp^{{{-}(\text{b}/{E}^{\prime})}}]$$

(7)

$$ {\text{Number of graphite particles in a length of the thick film}} = L/\left( {D + t_{0} } \right) $$

(8)

$$ {\text{Therefore, voltage between each gap of conducting particles}} = \left( {V_{2} /L} \right)\left( {D + t_{0} } \right) $$

(9)

E′ is the electric field present between two graphite particles, due to voltage V ₂

Electric field across the thin rubber film between two conducting particles, E′ is given by

$$ \text{E}^{\prime} = \left[ {\left( {\text{V}_{2} /\text{L}} \right)\left( {\text{D} + \text{t}_{0} } \right)} \right]/\left[ {\text{t}_{0} + \left( {\text{d}_{33} \text{p}\left( {\text{V}_{1} /\text{d}} \right)\left( {\text{D} + \text{t}_{0} } \right)} \right)} \right] $$

(10)

Resistance of one gap between the conducting particles, R _g , is given by

$$ \text{R}_{\text{g}} = \left[ {\left( {\text{V}_{2} /\text{L}}\right)\left( {\text{D} + \text{t}_{0} } \right)}\right]/\text{a}\left[ {\text{AE}^{{\prime}^{2}}\exp^{{{-}(\text{b}/\text{E}^{\prime})}}} \right] $$

(11)

Number of contacts in series along the length of the film = L/(D + t ₀)

$$ \text{Re} {\text{sistance due to such serial contacts}} = R_{g} L/\left( {D + t_{0} } \right) $$

(12)

$$ {\text{No}} . {\text{ of contacts in parallel along the width of the film}} = w/\left( {D + t_{0} } \right) $$

(13)

Resistance due to such parallel contacts = (R _g L)/w

$$ {\text{Number of parallel contacts along the thickness}} = h/\left( {D + t_{0} } \right) $$

(14)

$$ {\text{Therefore}},{\text{ total resistance of the film}} = \left[ {\left( {R_{g} L} \right)/\left( w \right)} \right]\left[ {\left( {D + t_{0} } \right)/\left( h \right)} \right] $$

(15)

Substituting Eqs. (10, 11) in Eq. (15), Eq. (16) can be obtained, which describes the relation between the variation of thick film resistance with respect to the voltage applied to the PMNPT disc, as given below:

$$ R = \tfrac{{\left( {\left( {\frac{{V_{2} }}{wh}} \right)\left( {D + t_{0} } \right)^{2} } \right)}}{{\left[ {a\left[ {A\left[ {\frac{{\left( {\frac{{V_{2} }}{L}} \right)\left( {D + t_{0} } \right)}}{{t_{0} + \left( {d{}_{33}p\left( {\frac{{V_{1} }}{d}} \right)\left( {D + t_{0} } \right)} \right)}}} \right]^{2} } \right]\exp - \left[ {\frac{{b\left[ {t_{0} + \left( {d{}_{33}p\left( {\frac{{V_{1} }}{d}} \right)\left( {D + t_{0} } \right)} \right)} \right]}}{{\left( {\frac{{V_{2} }}{L}} \right)\left( {D + t_{0} } \right)}}} \right]} \right]}} $$

(16)