Analysis of dc resistive network in presence of one non-linear element

 

Objectives

 

• To understand the volt (V) - ampere (A) characteristics of linear and nonlinear elements.

 

• Concept of load-line and analysis of dc resistive network having a single non-linear element using load-line analysis.

 

L.9.1 Introduction

The volt-ampere characteristic of a linear resistance is a straight line through the origin as in fig. 3.2 (see Lesson-3) but the characteristic for non-linear element for example, diodes or lamps is not linear as in fig. 3.3 (lesson-3). Temperature effects cause much non-linearity in lamps that are made of metals. Most materials resistance increases (or decreases) with rise (or fall) in temperature. On the other hand, most nonmetals resistance decreases or increases with the rise or fall in temperature. The typical tungsten lamp resistance rises with temperature. Note, as the voltage across the lamp increases, more power is dissipated and in turn rising the filament temperature. Further note, that the increments of voltage produce smaller increments of current that causes increase resistance in the filament element. Opposite effects can be observed in case of carbon filament lamp or silicon carbide or thermistor. Additional increments of voltage produce large increments of current that causes decrease resistance in the element. Fig.9.1 shows the characteristics of tungsten and carbon filaments.

Let us consider a simple circuit shown in fig. L.9.2(a) that consists of independent sources, combination of linear resistances, and a nonlinear element. It is assumed that the nonlinear element characteristics either defined in terms of current () (flowing through it) and voltage () (across the nonlinear element) relationship or and relationships of nonlinear element can be expressed as mathematical expression or formula. For example, consider the actual (non-ideal) VI()it()nlvt(())nlVoltagevt()currenti− relationship of the typical diode can be expressed as 01⎛⎞⎜⎟⎝⎠=−VaIIe, where is constant ( for germanium diode 'a'=0.026 and silicon diode 'a'=0.052). Assume that the network (fig.9.2(a)) at the terminals '''aAandB' is replaced by an equivalent Thevenin network as shown in fig. 9.2(b). From an examination of the figure one can write the following expression:

ThABThABVIRV=×+ Thevenin terminal voltage = load voltage. ,ThABThABorVIRV−×=⇒

If the nonlinear element characteristic is given (note, no any analytical expression is available) then one can adopt graphical method called load-line analysis to determine the branch variables (ABnlABnIiandVv=) of nonlinear element as shown in fig. L.9.2(a). This resulting solution is frequently referred to as the operating point () for the nonlinear element characteristic (in the present discussion, we consider a nonlinear element is a resistor). This method is quite simple and useful to analysis the circuit while the load has a nonlinear VQ2RI− characteristic. It is very easy to draw the source characteristic using the intercepts at points and (),0()ABThABvtVViiopencircuitcondition====()0,ABvtV==ABii==

 

ThNThVIR=(short− in two axes. It is obvious that the values of voltage () and current (&mincircuitedatABteralsABVABI) at the terminals of the source are exactly same as the voltage across and current in the load as indicated in fig. 9.2(a). The point of intersection of the load and the source characteristic represents the only condition where voltage and current are same for both source and load elements. More-specifically, the intersection of source characteristic and load characteristic represents the solution of voltage across the nonlinear element and current flowing through it or operating point ()of the circuit as shown in fig.9.2(c). Application of load-line analysis is explained with the following examples. Q

L.9.3 Application of load-line method

Example-L.9.1: The volt-ampere characteristic of a non-linear resistive element connected in the circuit (as shown Fig.9.3(a)) is given in tabular form.

Table: volt-ampere characteristic of non-linear element

nlV 0 2 4 6 8 10 12 14 15
nlI	0	0.05	0.1	0.2	0.6	1.0	1.8	2.0	4.0