# Voltage-Current Characteristics

## Introduction

In any electric circuit, our typical goal is to move charges around to perform some useful task. This involves both voltage differences and currents. We create voltage differences in the circuit, which provides energy differences (or electromotive forces) which move charges around, creating currents. The trick in any design problem is to provide the voltages needed to get the right amount of current to go where we want it.

But how do we control the relationship between voltage and current in order to get the desired voltage and current? The answer is to choose or create an electrical component with the desired relationship between voltage and current.

Engineers describe the behavior of most electrical components in terms of the mathematical relationship between voltage and current for the component. The general idea is as shown in Fig. 1—the circuit component shown there has two terminals; there is some voltage difference between the terminals, and some current through the component. For any voltage difference we apply across the terminals, we will get some resulting current through the device. This relationship is called the voltage-current characteristic, or the i-v characteristic (since we usually represent current by the letter i and voltage by the letter v) of the component. Ohm's law is an example of an i-v characteristic; it provides a functional relationship between the voltage across a resistor and the current through a resistor.

#### Analogy:

If we push on a mass, it will accelerate in the direction that we pushed. The applied force and the resulting acceleration characterize the dynamics of the mass—according to Newton's laws, $F{\rm{ }} = {\rm{ }}m \times a$. We can use this relationship to decide how hard to push the mass to give us the acceleration that we want or need.

Often, the voltage-current characteristic is presented in graphical form as a plot of current as a function of voltage, or vice-versa. When presented as a plot, the characteristic is often called the i-v curve for the component.

Notice that the definition of voltage and current in Fig. 1 is consistent with the passive sign convention. Almost invariably, the voltage-current characteristics for a device depend upon the voltage and current of the device being assigned relative to the passive sign convention. If you do not use the passive sign convention, the voltage-current relations will not be correct, and your design will not work! This is why it is so important that you follow the passive sign convention when defining voltages and currents in your circuit.

#### Important Points:

• A circuit element's behavior is defined by its voltage-current characteristic. This is a functional or graphical representation of the relationship between the voltage across the element and the current through the element.

• The voltage-current characteristics for components are used to design circuits containing these components. Selection of components with the desired characteristics allows us to set the voltages and currents in a circuit so that the circuit performs the desired function.

• In order to correctly apply the voltage current characteristic in your design, you must follow the passive sign convention when defining voltages and currents in your circuit. If you do not follow the passive sign convention, you will be applying incorrect relations between the voltages and currents in your circuit and it is very unlikely that your design will work!

1. A diode has a voltage-current relation given by:

• $i = {I_S}{e^{v/{V_T}}}$
• Where IS is called the saturation current and VT is the thermal voltage. If a diode's saturation current is $7 \times {10^{ - 16}}$ amps, and the thermal voltage is 0.025 volts, determine the current through the diode if the voltage across he diode is:
• ${\rm{v = 0}}{\rm{.1 volts}}$
• ${\rm{v = 0}}{\rm{.7 volts}}$
• ${\rm{v = 2 volts}}$
2. Plot the i-v curve over a range of voltages $0 < v < 0.8V$ for the diode of problem 1. Plot voltage on the horizontal axis and current on the vertical axis.

3. A mysterious circuit has two terminals, as shown to the left below. The circuit is tested to determine its i-v characteristics by applying voltages of 2, 3, and 4 volts to the terminals and measuring the resulting current. The resulting data are plotted, and the i-v curve for the circuit is estimated to be as shown to the right below.
• What current would you expect if you applied 1V across the terminals?
• What voltage would you need to apply across the terminals to get about 0.8A of current?
• Estimate a straight line functional relationship between voltage and current for the circuit. (Hint: the form of the equation will be ${\rm{V = mi + b}}$, where m is the slope of the line, and b is the y-intercept.)

• $3.8 \times {10^{ - 14}}$A
• $1.0 \times {10^{ - 3}}$A
• $164.77$A
• ${\rm{V }} = {\rm{ }}2.5i{\rm{ }} + {\rm{ }}1.5$