Circuit Theory/Resistors

Resistors

Resistors are circuit elements that allow current to pass through them, but restrict the flow according to a specific ratio called "Resistance". Flow that is restricted by resistors is said to be "lost to the resistor". Resistors are commonly used as heating elements, because energy lost to the resistor is frequently dispersed into the surroundings as heat. Every resistor has a given resistance. Resistors that have a variable resistance as a function of position are known as "potentiometers". Resistors that have a variable resistance as a function of temperature are called "thermisters".

Function of Temperature

Resistance also depends on surrounding temperature. It could be defined as

$R_t=R_o(1+aT)$

Where:

R_t is Final resistance,

R_o is Initial resistance,

a is Temperature coefficient,

T is Temperature

For most cases, especially in this book, we will treat resistance as being a constant, and not a function of time and temperature.

Resistance

Resistance is measured in terms of units called "Ohms" (volts per ampere), which is commonly abbreviated with the Greek letter Ω. 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:

$r=\frac{\rho L}{A}$

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":

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

Conductance can be useful to describe resistors in parallel, since the sum of conductances is equal to the equivalent conductance. However, conductance is rarely used in practice, and this wikibook will probably never mention it again.

Ohm's Law

Image 1
A simple circuit digram relating current, voltage, and resistance

Ohm's law is a fundamental tenet of Electrical Engineering, and it is a building block of circuit analysis techniques. Without a knowledge of Ohm's law, the remainder of this wikibook will not be possible.

From image 1, we can relate the values r, v, and i with the following equation:

$v=ir$

In plain English, Ohm's law relates the voltage drop across a resistive element to the current flowing through the element and its resistance. It is important to note that across a resistive element, the voltage drops, whereas across a voltage source, the voltage increases. Sometimes it is important to denote that the voltage in ohm's law is a negative voltage to correspond to the voltage drop, although frequently it is sufficient to remember that this is a drop, and not an increase.

Ohm's law is fundamental and axiomatic. We can accept it without proof.