Electrical Resistance is one of the main characteristics of electrical conductors. When we have a physical element which has a certain resistance value, it is called resistor. In practical applications, resistors can be purchased from commercial stores with various standard values. In Figure 02-01 we see a photo of some examples of resistors that are available on the market.
Resistor is one of the most
common elements in electrical circuits and
because it assumes several functions, from current limiters,
voltage dividers, substance heaters, such as electric shower,
coffee makers, toasters, etc ... Can be made of numerous materials such as coal, metal,
conductive rubber (on computer keyboards), etc ... So to calculate
its value (ohmic) depends on the material used in its manufacture. In function of this
property we use the Electrical Resistivity of the material. Besides that,
the resistance depends on the geometric shape, being important the area and the length.
We can relate the resistance with the following equation, which is mentioned
in some books, such as the Ohm's First Law , or:
eq. 02-01
Where the variables are:
R - Resistance whose unit of measure is ohm
ρ - Resistivity whose unit of measure is ohm.meter
L - Length whose unit of measure is meter
A - Area where the unit of measure is m^{2}
In the Table 02-01 we see some materials and their resistivity.
Table 02-01
Material
Resistivity (ohm. meter)
Copper
1.69 x 10^{-8}
Gold
2.35 x 10^{-8}
Silver
1.62x 10^{-8}
Aluminum
2.75 x 10^{-8}
Platinum
10.6 x 10^{-8}
Iron
9.68 x 10^{-8}
Glass
10^{10} a 10^{14}
Source: Haliday, Vol 3 - p. 148
Whenever the electric current passes through a resistor, a voltage drop is generated
between your terminals. Thus, knowing the current that circulates through a resistance
and the potential difference between its extremes we can calculate the value of the
resistance through the equation below,
mentioned in the books as the Ohm's Second Law , or:
eq. 02-02
About this law we have created a special chapter (See here!) that addresses this topic in more detail.
What characterizes an association SERIES is the fact that we observe
any node of the circuit, we will always have only two components attached to the node.
We can also say that the current
that crosses the considered series circuit must be the same in all the route.
In the Figure 02-02 we see a series association of three resistors. We can replace them by a
single resistance equal to the sum of all resistances
who participate in the circuit. This is the basic feature of a series circuit. We can always
compute the equivalent resistance view of the terminals a-b .
We can extend this concept to "n" resistors that are in series and to
calculate the value of the equivalent resistance view
by the terminals a-b . So, we use the equation below:
What characterizes a PARALLEL association is the fact that all resistances
will be subjected to the same potential difference V_{ab }. Therefore, the current that will
circulate in each resistance will only depend on its ohmic value.
In the Figure 02-03 we see a parallel association of three resistors.
In this case, the current that will circulate through each resistor only depends on its
ohmic value. The total current flowing through the circuit is the sum of the currents in each circuit resistance.
Then to calculate the equivalent resistance of a parallel circuit of "n" resistors
we use the equation below:
eq. 02-04
Observation
In the case of there being only two resistors in parallel in a circuit, we can simplify the eq. 02-04 through a
algebraic manipulation and find the equivalent resistance of the parallel through eq. 02-4a below.
eq. 02-4a
It can be seen that in most of the proposed problems, there are resistors in parallel with the characteristic that their values are multiples of each other. This way, there is a quick way to
calculate the equivalent resistance using an algebraic manipulation. Suppose two resistors in parallel, R_{1} and R_{2}. Let's assume
that R_{2} > R_{1}. So let's define n = R_{2} / R_{1}. Now let's perform an algebraic manipulation on eq. 02-4a as follows:
R_{eq} = R_{2} / (R_{1} / R_{1} + R_{2} / R_{1})
Making possible simplifications and using n = R_{2} / R_{1}, we obtain:
eq. 02-4b
See how much simpler it is to calculate the parallel. To do this, simply calculate how many times R_{2} is greater than R_{1}. This is the value of n.
We add one to this value. Then we divide the value of R_{2} by n + 1. Note that R_{2} is always the resistor with the highest value. Now, we find the value of the equivalent resistance of the parallel. That simple.
Let's see some examples.
6 Ω e 12 Ω ⇒ R_{eq} = 12 / (2 + 1) = 4 Ω
5 Ω e 20 Ω ⇒ R_{eq} = 20 / (4 + 1) = 4 Ω
36 Ω e 12 Ω ⇒ R_{eq} = 36 / (3 + 1) = 9 Ω
40 Ω e 10 Ω ⇒ R_{eq} = 40 / (4 + 1) = 8 Ω
60 Ω e 15 Ω ⇒ R_{eq} = 60 / (4 + 1) = 12 Ω
120 Ω e 24 Ω ⇒ R_{eq} = 120 / (5 + 1) = 20 Ω
See the simplicity of the calculations. And for three or more resistors in parallel, do it two by two until you get the final value.
In the MIX association, as the name itself is saying,
we will have a circuit that contains both parallel and serial association.
As we see Figure 02-04 the resistances R_{1 } and R_{2}
are in parallel, and this set is in series with R_{3}.
To find equivalent resistance, simply find the value of the association parallel of
R_{1} with R_{2} and to this value add the value
of R_{3}.
Conductance is defined as the opposite of the resistance, that is, it is the ease
that the electric current encounters when crossing a certain material. We symbolize the conductance
by letter G
and its unit of measure is siemens (S) .
See in the eq. 02-05 below how to calculate the conductance.
With the advancement of technology, electrical components have become smaller and smaller. With that came the need to identify the components in a more practical way due to their small size. So, for resistors (and for some types of capacitors as well) a color code was created, which would be stamped on the component body. Thus, through a code, the ohmic value of the resistor could be determined, as well as its tolerance.
Table 02-02
See Table 02-02 for the coding used for resistors. Let's learn how to use this table.
See Figure 02-02 for a photo of a real resistor. Let's determine its value. Notice that looking from left to right we have the following color sequence: red, red and brown. By Table 02-02, the color red has value 2. And the color brown has value 1.
So its value will be 22 x 10^{1} = 220 Ω. And on the right side of the resistor we have a golden stripe or
gold and, according to the table, the resistor has a tolerance of 5%. This means that if we measure the ohmic value
of that resistor with an ohmmeter, any value measured between 220 - 5% = 209 Ω and 220 + 5% = 231 Ω , we can consider the resistor within the standards. Values measured below 209 Ω and above 231 Ω place this resistor outside the established standard.
So, reviewing it we realize that the first two bands are the first two numbers that form the resistor value. And the third range indicates what exponent the number 10 will be raised to. This result, which will always be a multiple of ten,
is the multiplier factor. In the case of the resistor above, if the third band were orange, then the multiplier factor would be 10^{3} = 1000. Therefore, the value of the resistor would be equal to 22 x 1000 = 22,000 Ω.