Practical Electronics/Resistors

Resistors are passive devices, meaning that they cannot provide any power gain (amplification). They have a linear IV characteristic, meaning that the voltage across the resistor is directly proportional to the current through the resistor. The coefficient of proportionality is called the resistance (R). This is Ohm's Law:


 * $$V=IR$$

Resistance


Resistors have the symbol on the right. Resistance is defined as the voltage developed across the resistor per ampere of current through the resistor. Resistance has a symbol of R measured in unit called Ohm which has a symbol Ω.


 * $$R = \frac{V}{I}$$

Conductance
Conductance is the reciprocal of resistance (current per unit voltage), and is given the symbol G and the units siemens with symbol S.


 * $$G = \frac{1}{R} = \frac{I}{V}$$

Resistor's Construction
If we assume a resistor to be made of a prismatic (invariant cross-sectional area A along its length) conductor with length l, and conductivity ρ, we can express the conductance as follows:
 * $$G = \rho \frac{A}{l}$$

Form above, conductivity of materials can be calculated by
 * $$\rho = \frac{I}{V} \frac{l}{A}$$

When a resistor is connected with a DC source voltage, the resistance is calculated by Ohm's law:
 * $$R = \frac{V}{I}$$

When a resistor is connected with an AC voltage source, voltage and current in the resistor have zero phase difference. The impedance of a resistor is calculated in the same way as in the DC case:
 * $$Z_R = \frac{V}{I} = R$$

Identification
Resistors used in practical electronics range from one ohm to several million ohms

Often a shorthand is used which means that the &Omega; symbol, which is usually not easily accessed on a computer, is not needed. The shorthand also eliminates the need for decimal points which are sometimes lost or missed off when documents are copied. The shorthand works by replacing a decimal point with the prefix of the resistance (e.g. K for kilo-ohms) or, for resistances in just ohms, R:

This notation is preferred, and will be used in this book. please note that the resistances are still said the same, so the value of a 1K2 resistor is pronounced "one point two kilo-ohms". The shorthand is not used when we are not talking about an individual resistor, for example, when measuring the resistance of a combination of resistors, we then express the value in ohms.

Resistors are not made perfect, and so they each have a tolerance. This is the maximum that a resistor can deviate from its specified value. It is expressed in percent, and the standard tolerance is 10%, which is more than adequate for most of our needs.

Preferred Values
To prevent thousands of different values of resistors clogging everything up, resistors only come in specified values. These values are spread across multiples of ten so that each is a constant multiple of the one beneath. The number of divisions per multiple of ten (decade) depends on the tolerance of the resistors. Standard (10%) resistors belong to the E12 series. The values are spread according to the rounded results of the following rule:


 * $$R_n=R_{n-1} \times \sqrt[12]{10}$$

These numbers are:

1.0 1.2 1.5 1.8 2.2 2.7 3.3 3.9 4.7 5.6 6.8 8.2

After that, the cycle repeats, but a power of ten higher. Therefore, the first three decades are as follows:

1R 1R2  1R5  1R8  2R2  2R7  3R3  3R9  4R7  5R6  6R8  8R2 10R 12R  15R  18R  22R  27R  33R  39R  47R  56R  68R  82R 100R 120R 150R 180R 220R 270R 330R 390R 470R 560R 680R 820R

This pattern continues right up to the top of the resistor values that are available. For more accurate resistors, there exists an E24 (5%), E96 (1%) and an E192 (0.5%) series. There is also an E6 series for 20% resistors. To see a list of all values in these series, see this appendix.

Another way to understand the choice of resistor values in a given series (based on tolerance of the resistors) is as follows: Say there is a 10K resistor (nominal value) of tolerance 10%, then the actual value can be anything between 9K and 11K (approximately). So specifying another resistor value within this range is nonsense. The next available value is actually 12K (nominal), since its actual value can spread from 11K to 13.5K (approx). The following nominal value is 15K (with actual stagger within 13.5K and 16.5K) and so on. Hence the values available in this series are 10K, 12K, 15K, 18K, 22K, 27K, 33K, 39K, 47K, 56K, 68K, 82K and 100K (which starts the next series).

Check:

68K (range (68K - 6K8) to (68K + 6K8) or between 61K to 75K)

82K (range (82K -8K2) to (82K + 8K2) or between 74K to 90K) etc.

Colour Codes
For more in-depth info, read up on this chapter Practical Electronics/Finding component information/Resistors

Power dissipation
Besides resistance, the resistors available in the market also vary according to the power they can dissipate. In general terms, the greater the power dissipation, the larger the resistor. Thus size can be taken as a rough measure of power dissipation.

Resistors in Series and Parallel
Resistors can be connected in series to increase resistance or in parallel to decrease resistance

Series
When resistors are wired in series, the total resistance is the sum of all the individual resistances. For an explanation of this, see this proof. So for the resistor network to the right, the total resistance, Rtot, is:


 * $$R_{tot}=R_1+R_2+\ldots+R_n$$

If n identical resistors are in series, then the total resistance is n times the resistance of those identical resistors. This is useful when you want a simple multiple of the resistance.

When a resistance is added in series with others, the total resistance always increases. The total resistance will therefore be more than the greatest value of resistor present.

Parallel
When resistors are wired in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. For an explanation of this, see this proof. So for the resistor network to the right, the total resistance, Rtot, is:


 * $$\frac{1}{R_{tot}}=\frac{1}{R_1}+\frac{1}{R_2}+\ldots+\frac{1}{R_n}$$

If n identical resistors are in parallel, then the total resistance is 1/n the resistance of those identical resistors. This is useful when you want a simple fraction of the resistance.

When a resistance is added in parallel with others, the total resistance always decreases. The total resistance will therefore be less than the smallest value of resistor present.

Series and Parallel
Resistors can also be combined in a combination of series and parallel. To calculate the total resistance in these cases, simply break it into smaller parts that are basic series/parallel combinations and treat each one as one resistor. Consider the arrangement to the right. This is made up of two resistors in parallel, in series with another resistor.

The resistance of the two in parallel is:


 * $$\frac{1}{R_p}=\frac{1}{100}+\frac{1}{10}$$
 * $$R_p=\frac{100}{11}$$

When combined in series with the other resistance,


 * $$R_{tot}=R_p+3.3\,$$
 * $$R_{tot}=\frac{100}{11}+3.3\,$$
 * $$R_{tot}=\frac{1363}{110}$$
 * $$R_{tot}=12.4 \Omega\,$$

Uses of Resistors
As we saw in a previous chapter, the current flowing in a simple circuit depends only on the voltage and resistance. This means that given a fixed voltage, by changing the resistance, the current can be tuned. The same applies when the voltage is what we want to fix, and the current is constant (or cannot be changed). By altering the resistance, we can set the voltage.

Because they resist flow of charge, resistors are also used to limit the current flowing through sensitive components. For example, a component that has a low resistance but cannot tolerate too much current should be used in series with a resistor, which will limit the current. The exact value of the resistor depends on the specifications of the device being used. A common application like this is when an LED is used. LEDs generally need 2V at 20mA to work properly. Say we have a 9V circuit, we need to lose 7V to operate the LED. Putting this into Ohm's Law, we get:


 * $$R=\frac{V}{I}$$
 * $$R=\frac{7}{20 \times 10^{-3}}$$
 * $$R=350\,$$

Therefore, the resistor should be 350R to give the LED 2V at 20mA. However, this is not in the E12 series, so we find the next best, which is 390R. 330R is closer, but may exceed the LEDs current capacity, damaging it.

2 Port Resistor Network

 * Voltage Divider
 * π Network
 * T Network
 * WheatStone Network

Reference

 * Circuit Theory page on resistive circuit analysis
 * Electronics page on resistors