Resistance

Resistor Color Code

The most commonly used type of resistor consists of four bands, so let's explain the meaning of each of these bands. To determine the properties of a resistor, we must read it from left to right. The first two bands correspond to the first two digits. The third band is the multiplier, which simply indicates how many zeros we must add to the first two digits. Finally, the fourth band is the tolerance, or the error of the resistor. Each color is paired with a number, which is compiled in Table 1 below.

Table 1 - Resistor color code.

To understand the application of these values better, let's analyze the resistor on the outermost right side in Figure 1.

• Three-band resistors consist of two digits and a multiplier, as the tolerance band is always "no color."
• Five-band resistors are the same as four-band resistors, only with three digits rather than two.
• Six-band resistors are the same as five-band resistors, only they have an additional band denoting the temperature coefficient.

Resistance Symbol

In equations, resistance is represented by the letter "R", which is quite straightforward. It becomes a little more tricky when we have to include resistors in an electric circuit, where just like every other component, they have their own symbol. Depending on the source and the country of origin, a fixed resistor can be represented either as a zigzag line (American style) or a rectangular box (International style), as pictured in Figure 2 below. An easy way to associate the zigzag line with resistance is imagining current passing through a straight line versus such a rapidly changing path - it will obviously be much harder for the electric charge to get through it.

Fig. 2 - Resistor symbols, rectangular box is the international version, zigzag is the American version.

A fixed resistor simply implies that its resistance remains constant and cannot be adjusted. Sometimes, these symbols have additional lines and arrows drawn around them. That indicates a very specific type of resistor with special functionality, which we won't be dealing with in this course.

Resistance and Resistivity

Resistance and resistivity are often thought to be the same thing, which is definitely not the case. The two concepts are intertwined, but resistance describes the opposition of current due to matter, while resistivity quantifies the structural properties of matter.

Resistivity is usually represented by the symbol \(\rho\). The factors impacting a material's resistivity are the atomic structure and the temperature. Resistivity has nothing to do with the shape of an object. Values of resistivity for various materials have been determined in labs and tabulated, usually, at \(0\, ^\circ \mathrm{C}\) or room temperature (\(20\, ^\circ \mathrm{C}\)). The resistivity of some of the more commonly used materials can be seen in Table 2 below.

Table 2 - Resistivity of various materials at room temperature.

Resistance Formula

The main factors influencing an object's resistance are its shape and the material of which it is made. We can easily obtain the expression for resistance by considering a cylinder, like the one in Figure 3.

Fig. 3 - The resistance of a cylinder is directly proportional to its length and resistivity, and inversely proportional to its cross-sectional area.

Mathematically, the expression for resistance is

\[ R=\frac{\rho \ell}{A}\]

where \(R\) is the resistance measured in ohms \(\left(\Omega\right)\), \(\ell\) is the length of the cylinder measured in meters (\(\mathrm{m}\)), \(A\) is the cross-sectional area of the cylinder in meters squared \(\left(\mathrm{m^2}\right)\), and \(\rho\) is the resistivity of the material measured in ohm-meters \(\left(\Omega\,\mathrm{m}\right)\). This equation can be applied to other, more complex shapes.

Essentially, resistance explains how easily electrical charges move through this cylinder. Increasing its length will increase the overall number of collisions between the atoms in the material and the electric charges, as they will travel a longer distance. Similarly, if we increase the diameter of the cylinder, it's easier for the current to move through it, therefore experiencing less resistance. Finally, if a material's resistivity is higher, it means that the material can more strongly resist electric current, so the resistance of the whole cylinder will be higher.

Let's apply the resistance equation to an example problem.

Electrical Resistance and Conductance

When dealing with electric circuits, electrical resistance and conductance are important characteristics that describe their behavior. Both of these properties are closely related to the electric current and ability of a component to move through it. Let's look at each of them separately.

Resistance

Resistance can be measured using an ohmmeter or calculated using Ohm's law:

\[R=\frac{\Delta V}{I},\]

where \(R\) is the resistance in ohms (\(\Omega\)), \(V\) is the voltage in volts (\(\mathrm{V}\)), and \(I\) is the current in amperes (\(\mathrm{A}\)).

If we know the type of circuit and individual resistance of each resistor, we can find the total resistance of an electrical system using the following relations:

• If the resistors are in series (i.e. next to each other), you add the value of each resistor together: \[R_\mathrm{series}=\sum_{n}R_n=R_1+R_2+ \cdots,\]

• If the resistors are in parallel, the rule for finding the total resistance is as follows: \[\frac{1}{R_\mathrm{parallel}}=\sum_{n}\frac{1}{R_n} =\frac{1}{R_1}+\frac{1}{R_2}+\cdots.\]

Conductance

Conductance is simply the opposite property of resistance. If resistance tells us the extent a component opposes current, conductance explains how easily current can flow through said component.

Considering it's the inverse of resistance, mathematically conductance \(G\) can be expressed as

\[G=\frac{1}{R}=\frac{I}{\Delta V}.\]

It's measured in the units of siemens (\(\mathrm{S}\)), which are the inverse of an ohm: \(1\,\mathrm{S}=1\,\frac{1}{\Omega}.\)