Voltage-Current Characteristics and Nonlinearities

Introduction

Recall that Ohm's law provides a relationship between the voltage and current for a resistor. Ohm's law states that, for the resistor shown in Fig. 1: $v = i \cdot R$ Figure 1. Resistor

In this section, we will look at Ohm's law from the standpoint of a voltage-current relationship. This will provide some continuity with our presentation of voltage current relationships for other components, such as capacitors, inductors, diodes, and transistors later on. We will also look at Ohm's law from a graphical standpoint (an i-v curve, or plot of voltage as a function of current). This will also allow us to introduce the concept of non-ideal resistors.

Figure 2 shows a graph of v vs. i according to equation (1); the resulting plot is a straight line with slope R. Equation (1) thus describes the voltage-current relationship for a linear resistor. Linear circuit elements are wonderful since they allow us to design what are called linear circuits or linear systems1). Linear circuits are much easier to deal with mathematically than nonlinear circuits, but there is one problem with them: they don't actually exist—all real circuits are nonlinear to some extent.

Figure 2. Linear resistor voltage vs. current characteristic.

As with any other component, all resistors are nonlinear, to some extent. That is, the voltage-current relationship is not exactly a straight line for all values of current (for example, all electrical devices will fail if enough current is passed through them). Figure 3 shows a possible nonlinear voltage-current relationship. However, many nonlinear resistors exhibit an approximately linear voltage-current characteristic over some range of voltages and currents; this is also illustrated in Fig. 3. We will assume for now that any resistor we use is operating within a range of voltages and currents over which its voltage-current characteristic is linear and can be approximated by equation (1).

Figure 3. Nonlinear voltage vs. current characteristic.

Important Points:

  • Ohm's law provides a voltage-current characteristic for linear resistors.
  • The voltage and current must be assigned according to the passive sign convention.
  • Ohm's law is an approximation to real resistor behavior. The i-v characteristic of any real resistor will be nonlinear to some extent. For example, very high currents will tend to burn out a resistor, changing its i-v characteristics.
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We will define linear systems linearity more rigorously later on. For now, we will just say that, for a linear system, the parameters change in proportion to one another. For example, in Ohm's law, if we double the current, the voltage doubles. So, for linear systems, plots of voltage-current relations will tend to be straight lines. Nonlinear systems don't do this, which makes them much harder to design. For that reason, even if the system is nonlinear, engineers really, really, try to pretend that it is. More on this later, but be warned—the mathematics is very cool, but challenging.