# Circuit Theory/Resistors

## Resistors

Mechanical engineers seem to model everything with a spring. Electrical engineers compare everything to a Resistor. Resistors are circuit elements that resist the flow of current. When this is done a voltage appears across the resistor's two wires.

A pure resistor turns electrical energy into heat. Devices similar to resistors turn this energy into light, motion, heat, and other forms of energy.

Current in the drawing above is shown entering the + side of the resistor. Resistors don't care which leg is connected to positive or negative. The + means where the positive or red probe of the volt meter is to be placed in order to get a positive reading. This is called the "positive charge" flow sign convention. Some circuit theory classes (often within a physics oriented curriculum) are taught with an "electron flow" sign convention.

In this case, current entering the + side of the resistor means that the resistor is removing energy from the circuit. This is good. The goal of most circuits is to send energy out into the world in the form of motion, light, sound, etc.

## Resistance

Resistance is measured in terms of units called "Ohms" (volts per ampere), which is commonly abbreviated with the Greek letter Ω ("Omega"). Ohms are also used to measure the quantities of impedance and reactance, as described in a later chapter. The variable most commonly used to represent resistance is "r" or "R".

Resistance is defined as:

${\displaystyle r={\rho L \over A}}$

where ρ is the resistivity of the material, L is the length of the resistor, and A is the cross-sectional area of the resistor.

## Conductance

Conductance is the inverse of resistance. Conductance has units of "Siemens" (S), sometimes referred to as mhos (ohms backwards, abbreviated as an upside-down Ω). The associated variable is "G":

${\displaystyle G={\frac {1}{r}}}$

Before calculators and computers, conductance helped reduce the number of hand calculations that had to be done. Now conductance and it's related concepts of admittance and susceptance can be skipped with matlab, octave, wolfram alpha and other computing tools. Learning one or more these computing tools is now absolutely necessary in order to get through this text.

## Resistor terminal relation

A simple circuit diagram relating current, voltage, and resistance

The drawing on the right is of a battery and a resistor. Current is leaving the + terminal of the battery. This means this battery is turning chemical potential energy into electromagnetic potential energy and dumping this energy into the circuit. The flow of this energy or power is negative.

Current is entering the positive side of the resistor even though a + has not been put on the resistor. This means electromagnetic potential energy is being converted into heat, motion, light, or sound depending upon the nature of the resistor. Power flowing out of the circuit is given a positive sign.

The relationship of the voltage across the resistor V, the current through the resistor I and the value of the resistor R is related by ohm's law:

[Resistor Terminal Relation]

${\displaystyle V=R*I}$

A resistor, capacitor and inductor all have only two wires attached to them. Sometimes it is hard to tell them apart. In the real world, all three have a bit of resistance, capacitance and inductance in them. In this unknown context, they are called two terminal devices. In more complicated devices, the wires are grouped into ports. A two terminal device that expresses Ohm's law when current and voltage are applied to it, is called a resistor.

## Resistor Safety

Resistors come in all forms. Most have a maximum power rating in watts. If you put too much through them, they can melt, catch fire, etc. Resistance is an electrical passive element which oppose the flow of electricity.

## Example

Suppose the voltage across a resistor's two terminals is 10 volts and the measured current through it is 2 amps. What is the resistance?

If ${\displaystyle v=iR}$ then ${\displaystyle R=v/i=10V/2A=5ohms}$