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Foreword

Aim of This Textbook

The aim of this textbook is to explain the design and function of electronic circuits and components. The text covers electronic circuit components, DC analysis, and AC analysis.

It should be useful to beginner hobbyists as well as beginner engineering students, teaching both theory and practical applications.

It should be thought of as a companion project to the Wikipedia articles about electronics. While the Wikipedia covers many details about the technology used in electronics components and related fields, the Electronics Wikibook covers a lot of the "how-to" aspects that aren't covered in an encyclopedia. The book will focus on how to use the components to build useful circuits.

Prerequisites

Prerequisite Topics

Most important / Required knowledge

Moderately Important / Aids in comprehension

Slightly Important / Related or helpful

Mathematics

Algebra

Calculus

Multivariable Calculus

Geometry

Physics

Physics in Electronics

Electricity and Magnetism

Other Useful Topics

International System of Units

Preface

Importance of Electronics

Electronics is the study and use of devices that control the flow of electrons (or other charged particles). These devices can be used to process information or perform tasks using electromagnetic power.

Electronic circuits can be found in numerous household products, including such items as telephones, computers, and CD players. Electronic devices have also allowed greatly increased precision in scientific measurements.

Interest in the field of electronics increased around 1900 and the advent of radio. Interest reached an all-time high in the 1940s, 50s, 60s, with the invention of transistor radios, the launch of Sputnik, and the science and math educational push to win the space race. Interest in electronics as a hobby in the 1970s led to the advent of the personal computer (PC).

Electronics have since seen a decline in hobbyist interest. Electronics is now generally studied as part of a college-level program in electrical engineering.

This book is an attempt at reviving the hobbyist mentality that made electronics so big in the first place, by making electronics concepts more accessible and giving practical knowledge, as well as providing technical information for the student.

Charge and Coulomb's Law

Two atoms are walking down the street. The first atom says to the second atom "I think I lost an electron!" The second says "Are you sure?" To which the first states "I'm positive!"

Basic Understanding

Conductors: Materials which contain movable charges that can flow with minimal resistance.

Insulators: Materials with few or no movable charges, or with charges which flow with extremely high resistance.

Semiconductors: Materials whose behavior ranges between that of a conductor and that of an insulator under different conditions. Their conducting behavior may be heavily dependent on temperature. They are useful because we are able to change their conducting behavior to be dependent on many other factors.

The Atom: An atom contains a positively charged nucleus and one or more negatively charged electrons. The atom exists in three states: neutral, positively charged, and negatively charged. A neutral atom has the same number of electrons and protons, a positively charged atom has more protons than electrons and a negatively charged atom has more electrons than protons.

(+) and (-) Ions: An ion is an atom that has an unequal number of electrons and protons. Ions are when a neutral atom gains or loses electrons during a chemical reaction. In a battery, the positive side has + ions, meaning there are fewer electrons than protons, giving it an overall positive charge, and - side, more electrons than protons, giving it an overall negative charge. +ve and -ve charge will attract each other, and it is the use of such an attractive force that allows the battery to do work.

Note: Electric current is not the same as electron flow as is widely mistaken. Firstly, the total current has the opposite direction compared to electron flow. This is a lucky "mistake" on our forefathers' part to put it this way. It is also because of this lucky legacy that we are reminded that electricity can flow in materials other than metals alone. For example, in water, it is not electrons that flow, it is ions, and the +ve ions and -ve ions flow in opposite directions, contributing half of the total current each.

Balance of Charge

Atoms, the smallest particles of matter that retain the properties of the matter, are made of protons, electrons, and neutrons. Protons have a positive charge, Electrons have a negative charge that cancels the proton's positive charge. Neutrons are particles that are similar to a proton but have a neutral charge. There are no differences between posit charges except that particles with the same charge repel each other and particles with opposite charges attract each other. If a solitary positive proton and negative electron are placed near each other they will come together to form a hydrogen atom. This repulsion and attraction (force between stationary charged particles) is known as the Electrostatic Force and extends theoretically to infinity, but is diluted as the distance between particles increases.

When an atom has one or more missing electrons it is left with a positive charge, and when an atom has at least one extra electron it has a negative charge. Having a positive or a negative charge makes an atom an ion. Atoms only gain and lose protons and neutrons through fusion, fission, and radioactive decay. Although atoms are made of many particles and objects are made of many atoms, they behave similarly to charged particles in terms of how they repel and attract.

In an atom the protons and neutrons combine to form a tightly bound nucleus. This nucleus is surrounded by a vast cloud of electrons circling it at a distance but held near the protons by electromagnetic attraction (the electrostatic force discussed earlier). The cloud exists as a series of overlapping shells / bands in which the inner valence bands are filled with electrons and are tightly bound to the atom. The outer conduction bands contain no electrons except those that have accelerated to the conduction bands by gaining energy. With enough energy an electron will escape an atom (compare with the escape velocity of a space rocket). When an electron in the conduction band decelerates and falls to another conduction band or the valence band a photon is emitted. This is known as the photoelectric effect.

A laser is formed when electrons travel back and forth between conduction bands emitting synchronized photons.

When the conduction and valence bands overlap, the atom is a conductor and allows for the free movement of electrons. Conductors are metals and can be thought of as a bunch of atomic nuclei surrounded by a churning "sea of electrons".
When there is a large energy level gap between the conduction and valence bands, the atom is an insulator; it traps electrons. Many insulators are non-metals and are good at blocking the flow of electrons.
When there is a small energy level gap between the conduction and valence bands, the atom is a semiconductor. Semiconductors behave like conductors and insulators, and work using the conduction and valence bands. The electrons in the outer valence band are known as holes. They behave like positive charges because of how they flow. In semiconductors electrons collide with the material and their progress is halted. This makes the electrons have an effective mass that is less than their normal mass. In some semiconductors holes have a larger effective mass than the conduction electrons.

Electronic devices are based on the idea of exploiting the differences between conductors, insulators, and semiconductors but also exploit known physical phenomena such as electromagnetism and phosphorescence.

Conductors

In a metal the electrons of an object are free to move from atom to atom. Due to their mutual repulsion (calculable via Coulomb's Law ), the valence electrons are forced from the centre of the object and spread out evenly across its surface in order to be as far apart as possible. This cavity of empty space is known as a Faraday Cage and stops electromagnetic radiation, such as charge, radio waves, and EMPs (Electro-Magnetic Pulses) from entering and leaving the object. If there are holes in the Faraday Cage then radiation can pass.

One of the interesting things to do with conductors is demonstrate the transfer of charge between metal spheres. Start by taking two identical and uncharged metal spheres which are each suspended by insulators (such as a pieces of string). The first step involves putting sphere 1 next to but not touching sphere 2. This causes all the electrons in sphere 2 to travel away from sphere 1 to the far end of sphere 2. So sphere 2 now has a negative end filled with electrons and a positive end lacking electrons. Next sphere 2 is grounded by contact with a conductor connected with the earth and the earth takes its electrons leaving sphere 2 with a positive charge. The positive charge (absence of electrons) spreads evenly across the surface due to its lack of electrons. If suspended by strings, the relatively negatively charged sphere 1 will attract the relatively positively charged sphere 2.

Insulators

In an insulator the charges of a material are stuck and cannot flow. This allows an imbalance of charge to build up on the surface of the object by way of the triboelectric effect. The triboelectric effect (rubbing electricity effect) involves the exchange of electrons when two different insulators such as glass, hard rubber, amber, or even the seat of one's trousers, come into contact. The polarity and strength of the charges produced differ according to the material composition and its surface smoothness. For example, glass rubbed with silk will build up a charge, as will hard rubber rubbed with fur. The effect is greatly enhanced by rubbing materials together.

Van de Graaff Generator: A charge pump (pump for electrons) that generates static electricity. In a Van de Graaff generator, a conveyor belt uses rubbing to pick up electrons, which are then deposited on metal brushes. The end result is a charge difference.

Because the material being rubbed is now charged, contact with an uncharged object or an object with the opposite charge may cause a discharge of the built-up static electricity by way of a spark. A person simply walking across a carpet may build up enough charge to cause a spark to travel over a centimetre. The spark is powerful enough to attract dust particles to cloth, destroy electrical equipment, ignite gas fumes, and create lightning. In extreme cases the spark can destroy factories that deal with gunpowder and explosives. The best way to remove static electricity is by discharging it through grounding. Humid air will also slowly discharge static electricity. This is one reason cells and capacitors lose charge over time.

Note: The concept of an insulator changes depending on the applied voltage. Air looks like an insulator when a low voltage is applied. But it breaks down as an insulator, becomes ionised, at about ten kilovolts per centimetre. A person could put their shoe across the terminals of a car battery and it would look like an insulator. But putting a shoe across a ten kilovolt powerline will cause a short.

Quantity of Charge

Protons and electrons have opposite but equal charge. Because in almost all cases, the charge on protons or electrons is the smallest amount of charge commonly discussed, the quantity of charge of one proton is considered one positive elementary charge and the charge of one electron is one negative elementary charge. Because atoms and such particles are so small, and charge in amounts of multi-trillions of elementary charges are usually discussed, a much larger unit of charge is typically used. The coulomb is a unit of charge, which can be expressed as a positive or negative number, which is equal to approximately 6.2415×10¹⁸ elementary charges. Accordingly, an elementary charge is equal to approximately 1.602×10^-19 coulombs. The commonly used abbreviation for the coulomb is a capital C. The SI definition of a coulomb is the quantity of charge which passes a point over a period of 1 second ( s ) when a current of 1 ampere (A) flows past that point, i.e., C = A·s or A = C/s. You may find it helpful during later lessons to retain this picture in your mind (even though you may not recall the exact number). An ampere is one of the fundamental units in physics from which various other units are defined, such as the coulomb.

Force between Charges: Coulomb's Law

The repulsive or attractive electrostatic force between charges decreases as the charges are located further from each other by the square of the distance between them. An equation called Coulomb's law determines the electrostatic force between two charged objects. The following picture shows a charge q at a certain point with another charge Q at a distance of r away from it. The presence of Q causes an electrostatic force to be exerted on q.

The magnitude of the electrostatic force F, on a charge q, due to another charge Q, equals Coulomb's constant multiplied by the product of the two charges (in coulombs) divided by the square of the distance r, between the charges q and Q. Here a capital Q and small q are scalar quantities used for symbolizing the two charges, but other symbols such as q₁ and q₂ have been used in other sources. These symbols for charge were used for consistency with the electric field article in Wikipedia and are consistent with the Reference below.

$F = \frac {k q Q}{r^2}$

F = magnitude of electrostatic force on charge q due to another charge Q
r = distance (magnitude quantity in above equation) between q and Q
k = Coulomb's constant = 8.9875×10⁹ N·m²/C² in free space

The value of Coulomb's constant given here is such that the preceding Coulomb's Law equation will work if both q and Q are given in units of coulombs, r in metres, and F in newtons and there is no dielectric material between the charges. A dielectric material is one that reduces the electrostatic force when placed between charges. Furthermore, Coulomb's constant can be given by:

$k = \frac {1}{4 \pi \epsilon }$

where $\epsilon \,$ = permittivity. When there is no dielectric material between the charges (for example, in free space or a vacuum),

$\epsilon _0\,$ = 8.85419 × 10^-12 C²/(N·m²).

Air is only very weakly dielectric and the value above for $\epsilon _0\,$ will work well enough with air between the charges. If a dielectric material is present, then

$\epsilon = \kappa \epsilon _0\,$

where $κ$ is the dielectric constant which depends on the dielectric material. In a vacuum (free space), $κ = 1$ and thus $ε = ε 0$ . For air, $κ = 1.0006$ . Typically, solid insulating materials have values of $κ > 1$ and will reduce electric force between charges. The dielectric constant can also be called relative permittivity, symbolized as $ε r$ in Wikipedia.

Highly charged particles close to each other exert heavy forces on each other; if the charges are less or they are farther apart, the force is less. As the charges move far enough apart, their effect on each other becomes negligible.

Any force on an object is a vector quantity. Vector quantities such as forces are characterized by a numerical magnitude (i. e. basically the size of the force) and a direction. A vector is often pictured by an arrow pointing in the direction. In a force vector, the direction is the one in which the force pulls the object. The symbol $\vec {F}$ is used here for the electric force vector. If charges q and Q are either both positive or both negative, then they will repel each other. This means the direction of the electric force $\vec {F}$ on q due to Q is away from Q in exactly the opposite direction, as shown by the red arrow in the preceding diagram. If one of the charges is positive and the other negative, then they will attract each other. This means that the direction of $\vec {F}$ on q due to Q is exactly in the direction towards Q, as shown by the blue arrow in the preceding diagram. The Coulomb's equation shown above will give a magnitude for a repulsive force away from the Q charge. A property of a vector is that if its magnitude is negative, the vector will be equal to a vector with an equivalent but positive magnitude and exactly the opposite direction. So, if the magnitude given by the above equation is negative due to opposite charges, the direction of the resulting force will be directly opposite of away from Q, meaning the force will be towards Q, an attractive force. In other sources, different variations of Coulombs' Law are given, including vector formulas in some cases (see Wikipedia link and reference(s) below).

In many situations, there may be many charges, Q₁, Q₂, Q₃, through Q_n, on the charge q in question. Each of the Q₁ through Q_n charges will exert an electric force on q. The direction of the force depends on the location of the surrounding charges. A Coulomb's Law calculation between q and a corresponding Q_i charge would give the magnitude of the electric force exerted by each of the Q_i charges for i = 1 through n, but the direction of each of the component forces must also be used to determine the individual force vectors, $\vec {F_1}, \vec {F_2}, \vec {F_3}, ...., \vec {F_n}$ . To determine the total electric force on q, the electric force contributions from each of these charges add up as vector quantities, not just like ordinary (or scalar) numbers.

$\vec {F_e} = \vec {F_1} + \vec {F_2} + \vec {F_3} + ..... + \vec {F_n}$

The total electric force on q is additive to any other forces affecting it, but all of the forces are to be added together as vectors to obtain the total force on the charged object q. In many cases, there are billions of electrons or other charges present, so that geometrical distributions of charges are used with equations stemming from Coulomb's Law. Practically speaking, such calculations are usually of more interest to a physicist than an electrician, electrical engineer, or electronics hobbyist, so they will not be discussed much more in this book, except in the section on capacitors.

In addition to the electrostatic forces described here, electromagnetic forces are created when the charges are moving. These will be described later.

Reference(s):

College Physics Volume 2 by Doug Davis, Saunders College Publishing, Orlando, FL, 1994

Wikipedia:Coulomb's Law

Wikipedia:Dielectric constant

Next: Voltage, Current, and Power
Return to: Electronics Outline

Basic Concepts

What is Electronics?

Electronics is the study of flow of electrons in various materials or space subjected to various conditions. In the past, electronics dealt with the study of Vacuum Tubes or Thermionic valves, today it mainly deals with flow of electrons in semiconductors. However, despite these technological differences, the main focus of electronics remains the controlled flow of electrons through a medium. By controlling the flow of electrons, we can make them perform special tasks, such as power an induction motor or heat a resistive coil.

Electricity Basics

To understand electronics, you need to understand electricity and what it is. Basically, electricity is the flow of electrons due to a difference in electrical charge between two points. This difference in charge is created due to a difference in electron density. If you have a point where the electron density is higher than the electron density at another point, the electrons in the area of higher density will want to balance the charge by migrating towards the area with lower density. This migration is referred to as electrical current. Thus, flow in an electrical circuit is induced by putting more electrons on one side of the circuit than the other, forcing them to move through the circuit to balance the charge density.

Plumbing Analogy

A simple way to understand electrical circuits is to think of them as pipes. Let's say you have a simple circuit with a voltage source and a resistor between the positive and negative terminals on the source. When the circuit is powered, electrons will move from the negative terminal, through the resistor, and into the positive terminal. The resistor is basically a path of conduction that resists the movement of electrons. This circuit could also be represented as a plumbing network. In the plumbing network, the resistor would be equivalent to a section of pipe, where the water is forced to move around several barriers to pass through, effectively slowing its flow. If the pipe is level, no water will flow in an organized fashion, since the pressure is equal throughout the pipe. However, if we tilt the pipe to a vertical position (similar to turning on a voltage source), a pressure difference is created (similar to a voltage difference) and the water begins flowing through the pipe. This flow of water is similar to the flow of electrons in a circuit.

Cells

Cells

Cell: Two materials with a voltage difference between them. This causes current to flow, which does work. Electrons travel from the cathode, do some work, and are absorbed by the anode.

Anode: Destination of electrons.
Cathode: Source of electrons.
ions: An atom with an imbalance of electrons.

cell operation: The cell runs and electrons are depleted at the cathode and accumulate at the anode. This creates a reverse voltage which stops the flow of electrons.

irreversible: At some point the voltage difference reactions between the cathode and anode will decrease to a point that the cell in unusable. At this point, in an irreversible cell, the voltage difference is irreplaceably lost, and the cell is of no further use.
reversible: Able to run the cell backwards.
rechargeable: In a rechargeable cell, when the voltage difference between the cathode and anode decreases, the cell can be recharged, thereby increasing the voltage difference to a suitable level to allow continued use.

humid air will discharge cells.

cells are usually made of toxic or corrosive substances, for example lead and sulphuric acid. Such substances have been known to explode.

Electronegativity [1]

What is the relationship between voltage and electronegativity?

Electronegativity is a concept in chemistry used to measure and predict the relative likelihood of a chemical reaction causing electrons to shift from one chemical to another resulting in ions and molecular bonds. A battery cell operates by allowing two chemicals to react and supply ions to the anode and cathode. When the supply of a reactant is consumed, the battery is dead. It no longer produces different electrical potential at the anode and cathode driven by the chemical reaction.

Voltage is the electrical potential of a point due to surrounding measurable electric charge distributions and points as calculated by application Coulomb's Law. Voltage difference between two points connected by a conductor results in electron flow.

Resistors

For basic understanding skip "Construction" and the following "Resistance Example"

Resistors

Some resistors

A resistor is a block of material that limits the flow of current. The greater the resistance, the lower the current will be. Since conductors have an "electron cloud" around the atoms, they behave like a wide pipe filled with water, and have low resistance to a flow of water. Insulators, on the other hand, behave more like a tiny pipe, or a sponge-filled pipe. While they are porous and allow current to flow, a sponge that is more dense and has fewer holes will have a higher resistance and a smaller flow of current, if the pressure pushing on the water is the same.

Resistance can vary from very small to very large. A superconductor has zero resistance, while something like the input to an op-amp can have a resistance near 10¹² Ω, and even higher resistances are possible. For most materials, as temperature increases resistance tends to increase as well. Resistance converts electrical energy into heat. Resistors which dissipate large amounts of power are cooled so that they are not destroyed, typically with finned heatsinks.

Resistors have two leads (points of contact) to which the resistor can be connected to an electrical circuit. A symbol for a resistor used in electrical circuit diagrams is shown below. The endpoints at the left and right sides of the symbol indicate the points of contact for the resistor. The ratio of the voltage to current will always be positive, since a higher voltage on one side of a resistor is a positive voltage, and a current will flow from the positive side to the negative side, resulting in a positive current. If the voltage is reversed, the current is reversed, leading again to a positive resistance.

The ratio of voltage to current is referred to as Ohm's Law, and is one of the most basic laws that govern electronics.

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

(Ohm's law is not necessarily expressed in this way, but does express that an opposition is equivalent to the ratio of a cause to the effect)

Unlike some electrical components, it does not matter which way you plug in resistors; they have no polarity. Also, as most electronics components have internal resistance, this is sometimes shown by putting a resistor in series with the component to take the resistance into account.

Resistance is given in ohms (Ω) where:

$1 \Omega = \frac{1 \mbox{ V}}{1 \mbox{ A}}$

An ohm is the amount of resistance which passes one ampere of current when a one volt potential is placed across it. (The ohm is actually defined as the resistance which dissipates one watt of power when one ampere of current is passed through it.)

Load

Lower valued resistors are sometimes referred to as a load. A resistor dissipates energy as electrons strike the atoms and transfer the energy to the resistor material. A load is defined as the power dissipated between two terminals. Usually, this is an output, and the composition of the load is unknown. This measure is not related to the conductance, which is the inverse of resistance. Conductance is measured in Siemens (S) or sometimes referred to as Mhos (Ω^-1). Used as an adjective, a load has a current with a measurable power draw (expressed in Watts). The power drawn (load) is the amount of voltage across the terminals multiplied by the current through the terminals.

Power: $P = V \cdot I$

After Ohm's law is substituted into the equation:

$P = \frac{V^2}{R} = I^2 R$

For an AC signal or any kind of changing signal, the average power dissipated is related to the RMS value of the voltage, not the peak-to-peak voltage:

$P = V_{RMS} \cdot I_{RMS}$

For a pure sine wave, the relationship between peak-to-peak voltage and RMS voltage is

$V_{RMS} = {1 \over \sqrt {2}} \cdot V_{PP} = 0.707 \cdot V_{PP} \,$

So a resistor with 1 V DC across it and an equal-value resistor with 1 V_RMS = 1.414 V_PP AC sinusoid will both dissipate the same amount of power (heat).

Consider the definition of Power Rating below. When associating resistors to produce an equivalent Load, we can have an equivalent resistance which is larger or smaller than each individual element, depending on the type of association being series or parallel, but the equivalent power rating will always be larger than each individual power ratings. In other words:

Two equal value resistors in parallel will have half the resistance but twice the power rating.

Two equal value resistors in series will have twice the resistance and twice the power rating.

Labeling (See also Identification)

A manufactured resistor is usually labeled with the nominal value (value to be manufactured to) and sometimes a tolerance. Rectangular resistors will usually contain numbers that indicate a resistance and a multiplier. If there are three or four numbers on the resistor, the first numbers are a resistance value, and the last number refers to the number of zeroes in the multiplier. If there is an R in the value, the R takes the place of the decimal point.

Examples

2003 means 200×10³ = 200kΩ

600 means 60×10⁰ = 60Ω

2R5 means 2.5Ω

R01 means 0.01Ω

Cylindrical resistors (axial) usually have colored bands that indicate a number and a multiplier. Resistance bands are next to each other, with a tolerance band slightly farther away from the resistance bands. Starting from the resistance band side of the resistor, each colour represents a number in the same fashion as the number system shown above.

Colour System

Black	Brown	Red	Orange	Yellow	Green	Blue	Violet	Grey	White

0	1	2	3	4	5	6	7	8	9

Clue : B.B.ROY of Great Britain was a Very Good Worker. Additional Colours: A gold band in the multiplier position means 0.1, but means a 5% tolerance in the tolerance position. A silver band in the multiplier position means 0.01, but means 10% in the tolerance position.

Construction

The resistance R of a component is dependent on its physical characteristics and can be calculated using:

$R = {\rho L \over A} \;$

where ρ is the electrical resistivity (resistance to electricity) of the material, L is the length of the material, and A is the cross-sectional area of the material.

If you increase ρ or L you increase the resistance of the material, but if you increase A you decrease the resistance of the material.

ρ: Resistivity of the Material

Every material has its own resistivity, depending on its physical makeup. Most metals are conductors and have very low resistivity; whereas, insulators such as rubber, wood, and air all have very high resistivity. The inverse of resistivity is conductivity, which is measured in units of Siemens/metre (S/m) or, equivalently. mhos/metre.

In the following chart, it is not immediately obvious how the unit ohm-meter is selected. Considering a solid block of the material to be tested, one can readily see that the resistance of the block will decrease as its cross-sectional area increases (thus widening the conceptual "pipe"), and will increase as the length of the block increases (lengthening the "pipe"). Given a fixed length, the resistance will increase as the cross-sectional area decreases; the resistance, multiplied by the area, will be a constant. If the cross-sectional area is held constant, as the length is increased, the resistance increases in proportion, so the resistance divided by the length is similarly a constant. Thus the bulk resistance of a material is typically measured in ohm meters squared per meter, which simplifies to ohm - meter (Ω-m).

Conductors Ω-m (Ohm-meter)
Silver	1.59×10^-8
Copper	1.6×10^-8
Gold	1.7×10^-8
Aluminum	2.82×10^-8
Tungsten	5.6×10^-8
Iron	10×10^-8
Platinum	11×10^-8
Lead	22×10^-8
Nichrome	1.50×10^-6 (A nickel-chromium alloy commonly used in heating elements)
Graphite	~10^-6
Carbon	3.5×10^-5
Semiconductors
Pure Germanium	0.6
Pure Silicon	640
Common purified water	~10³
Ultra-pure water	~10⁵
Pure Gallium Arsenide	~10⁶
Insulators
Diamond	~10¹⁰
Glass	10¹⁰ to 10¹⁴
Mica	9×10¹³
Rubber	10¹³ to 10¹⁶
Organic polymers	~10¹⁴
Sulfur	~10¹⁵
Quartz (fused)	5 to 75×10¹⁶
Air	very high

Silver, copper, gold, and aluminum are popular materials for wires, due to low resistivity. Silicon and germanium are used as semiconductors. Glass, rubber, quartz crystal, and air are popular dielectrics, due to high resistivity.

Many materials, such as air, have a non-linear resistance curve. Normal undisturbed air has a high resistance, but air with a high enough voltage applied will become ionized and conduct very easily.

The resistivity of a material also depends on its temperature. Normally, the hotter an object is, the more resistance it has. At high temperatures, the resistance is proportional to the absolute temperature. At low temperatures, the formula is more complicated, and what counts as a high or low temperature depends on what the resistor is made from. In some materials the resistivity drops to zero below a certain temperature. This is known as superconductivity, and has many useful applications.

(Some materials, such as silicon, have less resistance at higher temperatures.)

For all resistors, the change in resistance for a small increase in temperature is directly proportional to the change in temperature.

Current passing through a resistor will warm it up. Many components have heat sinks to dissipate that heat. The heatsink keeps the component from melting or setting something on fire.

L: Length

The length of an object is directly proportional to its resistance. As shown in the diagram below, 1 unit cubed of material has 1 ohm of resistance. However, when 4 units are stacked lengthwise and a connection is made to the front and back sides respectively, the total resistance is 4 ohms. This is because the length of the unit is 4, whereas the cross-sectional area remains 1. However, if you were to make connections on the sides, the exact opposite would be true: the cross-sectional area would be 4 and the length 1, resulting in 0.25 ohms total resistance.

A: Cross-Sectional Area

Increasing area is the same as having resistors in parallel, so as you increase the area you add more paths for current to take.

The resistance of a material is inversely proportional to its cross-sectional area. This is shown in the diagram below, where 1 unit cubed has one ohm of resistance. However, if 4 units cubed are stacked on top of each other in the fashion such that there is 4 units squared of cross-sectional area, and the electrical connections are made to the front and back such that the connections are on the largest sides, the resultant resistance would be 0.25 ohms.

Additional note: There are two reasons why a small cross-sectional area tends to raise resistance. One is that the electrons, all having the same negative charge, repel each other. Thus there is resistance to many being forced into a small space. The other reason is that they collide, causing "scattering," and therefore they are diverted from their original directions. (More discussion is on page 27 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.)

Resistance Example

For instance, if you wanted to calculate the resistance of a 1 cm high, 1 cm wide, 5 cm deep block of copper, as shown in the diagram below:

You would first need to decide how it's oriented. Suppose you want to use it from front to back (lengthwise), like a piece of wire, with electrical contacts on the front and rear faces. Next you need to find the length, L. As shown, it is 5 cm long (0.05 m). Then, we look up the resistivity of copper on the table, 1.6×10^-8 Ω-meters. Lastly, we calculate the cross-sectional area of the conductor, which is 1 cm × 1 cm = 1 cm² (0.0001 m²). Then, we put it all in the formula, converting cm to m:

${0.05 \ \mbox{m} \cdot 1.6 \times 10^{-8} \ \Omega \cdot \mbox{m} \over 0.0001 \ \mbox{m}^2} = {0.08 \times 10^{-8} \ \Omega \cdot\mbox{m}^2 \over 0.0001\ \mbox{m}^2}$

units m² cancel:

${0.08 \times 10^{-8}\ \Omega \over 0.0001}$

Which, after evaluating, gives you a final value of 8.0×10^-6 Ω, or 8 microohms, a very small resistance. The method shown above included the units to demonstrate how the units cancel out, but the calculation will work as long as you use consistent units.

Internet Hint: Google calculator can do calculations like this for you, automatically converting units. This example can be calculated with this link: [2]

Properties of the material

Wirewound: Used for power resistors, since the power per volume ratio is highest. These usually have the lowest noise.

Carbon Film: These are easy to produce, but usually have lots of noise because of the properties of the material.

Metal Film: These resistors have thermal and voltage noise attributes that are between carbon and wirewound.

Ceramic: Useful for high frequency applications.

Resistor Junctions

Resistors in Series

Resistors in series are equivalent to having one long resistor. If the properties of two resistors are equivalent, except the length, the final resistance will be the sum of the two construction methods:

$R_{EQ} = \frac{\rho L_{EQ}}{A} = \frac{\rho L_1}{A} + \frac{\rho L_2}{A}$

This means that the resistors add when in series.

$R_{EQ} = R_1 + R_2 + \cdots + R_n \,\!$

Christmas tree lights are usually connected in series, with the unfortunate effect that if one light blows, the others will all go out (This happens because the circuit is not complete, if a circuit is not complete then the current cannot flow, hence the light bulbs all go out). However, most modern Christmas light strings have built in shunt resistors in parallel to the bulb, so that current will flow past the blown light bulb.

Resistors in Parallel

In a parallel circuit, current is divided among multiple paths. This means that two resistors in parallel have a lower equivalent resistance than either of the parallel resistors, since both resistors allow current to pass. Two resistors in parallel will be equivalent to a resistor that is twice as wide:

$R_{EQ} = \frac{\rho L}{A_{EQ}} = \frac{\rho L}{A_1} + \frac{\rho L}{A_2}$

Since conductances (the inverse of resistance) add in parallel, you get the following equation:

$\frac{1}{R_{EQ}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$

For example, two 4 Ω resistors in parallel have an equivalent resistance of only 2 Ω.

To simplify mathematical equations, resistances in parallel can be represented with two vertical lines "||" (as in geometry). For two resistors the parallel formula simplifies to:

$R_{EQ} = R_1 \| R_2 = {R_1 R_2 \over R_1 + R_2}$

Combinations of series and parallel

Resistors in parallel are evaluated as if in a mathematical set of "parentheses." The most basic group of resistors in parallel is evaluated first, then the group in series with the new equivalent resistor, then the next group of resistors in parallel, and so on. For example, the above portion would be evaluated as follows:

$R_{EQ} = (R_1 \| R_2) + R_3 = {R_1R_2 \over R_1 + R_2} +R_3$

Resistor variations

Variable Resistor or Potentiometer: Variable resistors are tunable, meaning you can turn a dial or slide a contact and change the resistance. They are used as knobs to control the volume of a stereo, or as a dimmer for a lamp. The term Potentiometer is often abbreviated as 'pot'. It is constructed like a resistor, but has a sliding tap contact. Potentiometers are used as Voltage Dividers. It is rare to find a variable resistor with only two leads. Most are potentiometers with three leads, even if one is not connected to anything.

Rheostat: A variation of the potentiostat with a high current rating, which is used to control the amount of power going through a load, such as a motor.

Thermistor or Photoresistor: Temperature-sensitive resistor, in which the resistance decreases as the temperature rises. They are used in fire alarms, so if things get too hot the current rises and trips a switch that sounds an alarm.

LDR (Light Dependent Resistor): A resistor which changes values depending on the amount of light shining on its surface. The resistance decreases as the amount of light increases. They are used in street lamps, so when it gets dark the current decreases and turns on the street lamp.

Applications

Voltage division / Attenuation: Sometimes a voltage will be too large to measure, so a means to linearly reduce the voltage is required. Placing two resistors in series to ground will provide a point in the middle to tap. Resistor R_A is placed between the input voltage and the output node, and the resistor R_B is placed between the output node and ground. This creates a voltage divider to lessen the output voltage. Typically, the resistors are near the value of ~10kΩ. The Thevenin model of the circuit gives an output resistance R_OUT = R_A||R_B. A larger output resistance will more likely be affected by the input resistance of the measuring circuit (this is a desired effect in the transistor biasing circuits). Placement of the voltage divider should be close to the measuring circuit, to minimize noise (in this arrangement, it will be also lessened Rb/(Ra + Rb) times). The output voltage of the voltage divider is

$V_{output} = \frac{R_B}{R_A + R_B}V_{input}$

Pull-up / Pull-down: If there is nothing to drive a signal node, the node will be left "floating" (for example, such a situation occurs at the trigger input of a car alarm system when the driver has switched off the internal lamp). This may lead to unintended values being measured, or causing side-effects when the voltage is propagated down the remainder of the circuit. To prevent this, a relatively high value resistor (usually ~10kΩ to ~1MΩ) is placed between the node and ground (pull-down) or a high voltage (pull-up) to bring the voltage of the "floating" node near to the voltage it is being pulled. A resistive voltage divider is another example where the upper resistor "pulls" the output point up toward the input voltage while the lower resistor "pulls" the output point toward the ground. This idea is evolved in the circuit of a resistive voltage summer (for example, the resistors R₁ and R₂ of an op-amp inverting amplifier) supplied by two voltages (V_IN and -V_OUT) having opposite polarities. The two voltage sources "pull" the output point in opposite directions; as a result, if R₂/R₁ = -V_OUT/V_IN, the point becomes a virtual ground. Placement of a pull-up or pull-down resistor does not have a significant effect on the performance of the circuit, if they have high resistances.

Current limiting / Isolation: In order to protect circuits from conditions that may cause too much current in a device, a current limiting resistor is inserted in the middle of the circuit. A digital input to a microcontroller may benefit from a current limiting resistor. The inputs to modern microcontrollers have protection circuitry built in that will protect the input from an overvoltage condition, provided that the current is small enough. For instance, a common microcontroller will be capable of withstanding 20mA. If there are 12V nets on a circuit board, or in a system, the digital input will benefit from a 350Ω resistor (refer to calculation below). Usually a slightly larger resistance is used in practice, but too large of a resistor will cause noise, and may prevent the input from being able to read the voltage. It is good practice to place the resistor as close as possible to the microcontroller input, so that an accidental short on the board will mean that the microcontroller input is likely still protected.

$R = \frac{12V}{20mA} = 600\Omega$

Line termination / Impedance matching: The properties of an electric wave propagating through a conductor (such as a wire) create a reflection, which can be viewed as unwanted noise. A reflection can be eliminated by maximizing the power transfer between the conductor and the termination resistor. By matching the resistance (more importantly the impedance ), the wave will not cause a reflection. The voltage of the echo V_r is calculated below in reference to the original signal V_o as a function of the conductor impedance Z_C and the terminator impedance Z_T. As the name implies, the termination resistor goes at the end of the conductor.

$V_r = V_o\frac{Z_C - Z_T}{Z_C + Z_T}$

Current sensing: Measurement of a current cannot be done directly. There are three major ways to measure a current: a resistor, a hall sensor, and an inductor. The hall sensor and inductor use a property of the magnetic field to sense the current through a nearby conductor. According to Ohm's law, if a current I flows through a resistor R, a voltage V = R.I appears across the resistor. Therefore, the resistor can act as a passive current-to-voltage converter. In this arrangement, the resistor should have a very low value (sometimes on the order of ~0.01Ω), so it does not affect the current flow or heat up; however, a smaller value has a lower voltage to read, which means more noise may be introduced. This contradiction is solved in the circuit of an active current-to-voltage converter where the resistor may have a significant resistance as an op-amp compensates the "undesired" voltage drop across it (unfortunately, this remedy may be applied only in low-current measurements). The current sense resistor should be placed as close as possible to where the measurement occurs, in order not to disturb the circuit.

Filtering: Filtering is discussed later, after an introduction to capacitors and inductors. Filters are best placed close to where measurement takes place.

Specifications

Resistors are available as pre-fabricated, real-world components. The behavior of such components deviates from an ideal resistor in certain ways. Therefore, real-world resistors are not only specified by their resistance, but also by other parameters. In order to select a manufactured resistance, the entire range of specifications should be considered. Usually, exact values do not need to be known, but ranges should be determined.

Nominal Resistance

The nominal resistance is the resistance that can be expected when ordering a resistor. Finding a range for the resistance is necessary, especially when operating on signals. Resistors do not come in all of the values that will be necessary. Sometimes resistor values can be manipulated by shaving off parts of a resistor (in industrial environments this is sometimes done with a LASER to adjust a circuit), or by combining several resistors in series and parallel.

Available resistor values typically come with a resistance value from a so called resistor series. Resistor series are sets of standard, predefined resistance values. The values are actually made up from a geometric sequence within each decade. In every decade there are supposed to be $n$ resistance values, with a constant step factor. The standard resistor values within a decade are derived by using the step factor $i$

$i = \sqrt[n]{10}; n = 6, 12, 24, 48$

rounded to a two digit precision. Resistor series are named E $n$ , according to the used value of $n$ in the above formula.

 n Values/Decade  Step factor i  Series
----------------------------------------
        6             1.47         E6
       12             1.21         E12
       24             1.10         E24
       48             1.05         E48

For example, in the E12 series for $n = 12$ , the resistance steps in a decade are, after rounding the following 12 values:

1.00, 1.20, 1.50, 1.80, 2.20, 2.70, 
3.30, 3.90, 4.70, 5.60, 6.80, and 8.20

and actually available resistors from the E12 series are for example resistors with a nominal value of 120Ω or 4.7kΩ.

Tolerances

A manufactured resistor has a certain tolerance to which the resistance may differ from the nominal value. For example, a 2kΩ resistor may have a tolerance of ±5%, leaving a resistor with a value between 1.9kΩ and 2.1kΩ (i.e. 2kΩ±100Ω). The tolerance must be accounted for when designing circuits. A circuit with an absolute voltage of 5V±0.0V in a voltage divider network with two resistors of 2kΩ±5% will have a resultant voltage of 5V±10% (i.e. 5V±0.1V). The final resistor tolerances are found by taking the derivative of the resistor values, and plugging the absolute deviations into the resulting equation.

The above mentioned E-series which are used to provide standardized nominal resistance values, are also coupled to standardized nominal tolerances. The fewer steps within a decade there are, the larger the allowed tolerance of a resistor from such a series is. More precises resistors, outside of the mentioned E-series are also available, e.g. for high-precision measurement equipment. Common tolerances, colors and key characters used to identify them are for example:

 Series  Values/Decade  Tolerance  Color Code  Character Code
--------------------------------------------------------------
   E6          6          ±20%       [none]        [none]
   E12        12          ±10%       silver          K
   E24        24           ±5%       gold            J
   E48        48           ±2%       red             G
    -         -            ±1%       brown           F
    -         -            ±0.5%       -             D
    -         -            ±0.25%      -             C
    -         -            ±0.1%       -             B

Resistor manufacturers can benefit from this standardization. They manufacture resistors first, and afterwards they measure them. If a resistor does not meet the nominal value within the defined tolerance of one E-series, it might still fit into a lower series, and doesn't have to be thrown away, but can be sold as being compliant to that lower E-series standard. Although typically at a lower price.

Series: Resistors that combine in series add the nominal tolerances together.

Derivation:

d R T = d R A + d R B

Example: For two resistors in series R_A = 1.5kΩ±130Ω and R_B = 500Ω±25Ω, the tolerance is 130Ω + 25Ω, resulting in a final resistor value R_T = 2kΩ±155Ω.

Parallel: Resistors that combine in parallel have a combined tolerance that is slightly more complex.

Derivation: $dR_P = \frac{dR_A R_B^2 + A^2 dR_B}{(A+B)^2}$

Example: For two resistors in parallel R_A = 1.5kΩ±130Ω and R_B = 500Ω±25Ω.

Power Rating

Because the purpose of a resistor is to dissipate power in the form of heat, the resistor has a rating (in watts) at which the resistor can continue to dissipate before the temperature overwhelms the resistor and causes it to overheat. When a resistor overheats, the material begins to melt away, which will cause the resistance to increase (usually), until the resistor breaks.

Operating Temperature

Related to power rating, the operating temperature is the temperature that the resistor can continue to operate before being destroyed.

Maximum Voltage

In order to avoid sparkovers or material breakdown a certain maximum voltage over a resistor must not be exceeded. The maximum voltage is part of a resistor's specification, and typically a function of the resistor's physical length, distance of the leads, material and coating.

For example, a resistor with a maximum operating voltage of 1kV can have a length in the area of 2", while a 0.3" resistor can operate under up to several tens of volts, probably up to a hundred volts. When working with dangerous voltages it is essential to check the actual specification of a resistor, instead of only trusting it because of the length.

Temperature Coefficient

This parameter refers to the constant in which the resistance changes per degree Celsius (units in C^-1). The change in temperature is not linear over the entire range of temperatures, but can usually be thought of as linear around a certain range (usually around room temperature). However, the resistance should be characterized over a large range if the resistor is to be used as a thermistor in those ranges. The simplified linearized formula for the affect on temperature to a resistor is expressed in an equation:

R = R 0 [1 + α(T - T 0)]

Capacity and Inductance

Real world resistors not only show the physical property of resistance, but also have a certain capacity and inductance. These properties start to become important, if a resistor is used in some high frequency circuitry. Wire wound resistors, for example, show an inductance which typically make them unusable above 1kHz.

Packaging

Resistors can be packaged in any way possible, but are divided into surface mount, through hole, soldering tag and a few more forms. Surface mount is connected to the same side that the resistor is on. Through hole resistors have leads (wires) that typically go through the circuit board and are soldered to the board on the side opposite the resistor, hence the name. Resistors with leads are also used in point-to-point circuits without circuit boards. Soldering tag resistors have lugs to solder wires or high current connectors onto.

Usual packages for surface mount resistors are rectangular, referenced by a length and a width in mils (thousands of an inch). For instance, an 0805 resistor is a rectangle with length .08" x .05", with contacts (metal that connects to the resistor) on either side. Typical through hole resistors are cylindrical, referenced either by the length (such as 0.300") or by a typical power rating that is common to the length (a 1/4W resistor is typically 0.300"). This length does not include the length of the leads.

Related Wikimedia resources

Wikibooks

Circuit Idea: Passive voltage-to-current converter shows how the bare resistor can act as a simple voltage-to-current converter.

Wikipedia

Voltage-to-current converter builds consecutively the passive and active versions of the voltage-to-current converter.
Current-to-voltage converter is dedicated to the passive and active versions of the inverse current-to-voltage converter.

Capacitors

Capacitors

Model of a capacitor

A capacitor (historically known as a "condenser") is a device that stores energy in an electric field, by accumulating an internal imbalance of electric charge. It is made of two conductors separated by a dielectric (insulator). Using the same analogy of water flowing through a pipe, a capacitor can be thought of as a tank, in which the charge can be thought of as a volume of water in the tank. The tank can "charge" and "discharge" in the same manner as a capacitor does to an electric charge. A mechanical analogy is that of a spring. The spring holds a charge when it is pulled back.

When voltage exists one end of the capacitor is getting drained and the other end is getting filled with charge.This is known as charging. Charging creates a charge imbalance between the two plates and creates a reverse voltage that stops the capacitor from charging. As a result, when capacitors are first connected to voltage, charge flows only to stop as the capacitor becomes charged. When a capacitor is charged, current stops flowing and it becomes an open circuit. It is as if the capacitor gained infinite resistance.

You can also think of a capacitor as a fictional battery in series with a fictional resistance. Starting the charging procedure with the capacitor completely discharged, the applied voltage is not counteracted by the fictional battery, because the fictional battery still has zero voltage, and therefore the charging current is at its maximum. As the charging continues, the voltage of the fictional battery increases, and counteracts the applied voltage, so that the charging current decreases as the fictional battery's voltage increases. Finally the fictional battery's voltage equals the applied voltage, so that no current can flow into, nor out of, the capacitor.

Just as the capacitor charges it can be discharged. Think of the capacitor being a fictional battery that supplies at first a maximum current to the "load", but as the discharging continues the voltage of the fictional battery keeps decreasing, and therefore the discharge current also decreases. Finally the voltage of the fictional battery is zero, and therefore the discharge current also is then zero.

This is not the same as dielectric breakdown where the insulator between the capacitor plates breaks down and discharges the capacitor. That only happens at large voltages and the capacitor is usually destroyed in the process. A spectacular example of dielectric breakdown occurs when the two plates of the capacitor are brought into contact. This causes all the charge that has accumulated on both plates to be discharged at once. Such a system is popular for powering tasers which need lots of energy in a very brief period of time.

Capacitance

The capacitance of a capacitor is a ratio of the amount of charge that will be present in the capacitor when a given potential (voltage) exists between its leads. The unit of capacitance is the farad which is equal to one coulomb per volt. This is a very large capacitance for most practical purposes; typical capacitors have values on the order of microfarads or smaller.
The basic equation for capacitance is.

$C = \frac{Q}{V}$

Where C is the capacitance in farads, V is the potential in volts, and Q is the charge measured in coulombs. Solving this equation for the potential gives:

$V=\frac{Q}{C}$

The impedance of a capacitor at any given angular frequency is given by:

$Z=\frac{1}{j \omega C}$

where j is $\sqrt{-1}$ , $ω$ is the angular frequency and C is the capacitance.

The charge in the capacitor at any given time is the accumulation of all of the current which has flowed through the capacitor. Therefore, the potential as a function of time can be written as:

$v(t)=\frac{1}{C} \int i(t)dt$

Where i(t) is the current flowing through the capacitor as a function of time.

This equation is often used in another form. By differentiating with respect to time:

$i(t)=C\frac{dv(t)}{dt}$

Capacitor Labeling

(http://www.twysted-pair.com/capidcds.htm)

Capacitors are labelled in several different ways.

Ceramic Disc

Sometimes labeled implicitly, usually labeled with number scheme (223 = 22 000 pF, where 3 represents the number of "0" for instance) The letters "mfd" are often used in place of "µF".

Ceramic Dipped

Ceramic-dipped capacitors.

These usually use the number code. In the above example, the smallest one says "104". This means 10 0000 pF = 100,000 pF. M is a tolerance. The middle one is labeled 393. This means 39 000 pF. The last is 223, meaning 22 000 pF. K is the tolerance. It also has a 100 V working voltage labeled.

Red dipped caps are ___

Green dipped caps are ___

Mylar or polyester film?

Resin-potted mylar/polyester

"Box-style" capacitors.

These are usually labeled explicitly, as there is lots of surface area to write on. This one is 4700 pF, 250 V, 5 kV test. The frequency f₀ = 21 MHz is the frequency at which it stops behaving like a capacitor, and more like an inductor.

Electrolytic

Axial 1000 µF capacitor with a maximum voltage rating of 35 V (black), and radial 10 µF capacitor with a maximum voltage rating of 160 V (blue).

This image shows a pair of radial electrolytic capacitors with leads. The industry standard, as shown, is that the positive lead is longer than the negative lead.

Usually electrolytic caps are labeled explicitly, making identification easy.

Electrolytics are available in axial and radial-leaded packages. In axial-leaded parts, the negative terminal is indicated by a minus sign printed on the package, or by a shorter lead.

Radial-lead parts often uses color code like resistors. The polarity is usually indicated by arrows on a stripe pointing to the negative terminal.

Aluminum electrolytic capacitors of the "snap-in" type. The leads are angled so that they may be "snapped" into a printed cirucit board with holes of the correct spacing.

Warning: You should never connect an electrolytic capacitor in such a way that a negative voltage is applied across the terminals from positive to negative. It will explode.

Tantalum

Radial tantalum capacitor.

Tantalum capacitors have high capacitance and low ESR, but low operating voltages. When tantalum capacitors fail, it tends to be "spectacular," they essentially blow up.

Construction

The capacitance of a parallel-plate capacitor constructed of two identical plane electrodes of area A at constant spacing D is approximately equal to the following:

$C = \epsilon_0 \epsilon_r \frac{A}{D}$

where C is the capacitance in farads, ε₀ is the Permittivity of Space, ε_r is the Dielectric Constant, A is the area of the capacitor plates, and D is the distance between them.

A dielectric is the material between the two charged objects. Dielectrics are insulators. They impede the flow of charge in normal operation. Sometimes, when a too large voltage has been reached, charge starts flowing. This is called dielectric breakdown and destroys the capacitor. Beginners sometimes misunderstand this. Timing circuits do measure the rate at which a capacitor charges, but they measure a threshold voltage instead of allowing the voltage to build up until dielectric breakdown. (A device which does function this way is a spark gap.)

No charge should ever flow from one plate to the other. Although a current does flow through the capacitor, charges are not actually moving from one plate to the other. As charges are added to one plate, their electric field displaces like charges off of the other plate. This is called a displacement current.

Materials

Capacitors can be made either polarized or non-polarized. A polarized capacitor requires that the capacitor be hooked up such that the voltage is always biased in one direction. Hooking a polarized capacitor backwards will result in the capacitor exploding, sometimes releasing harmful fumes. Non-polarized capacitors can be biased in either direction without harm to the capacitor. Polarized and non-polarized capacitors have an upper limit of voltage, where the material will break down and the capacitor will no longer function. This can also cause fumes to be released depending on the type of material.

Different materials and their properties.

Ceramic: These are normally low capacitance (between ~1pF to ~1μF). Ceramic capacitors have a very low inductance due to the shape. This means that the capacitance value continues into extremely high frequencies, making them perfect for RF applications. However, ceramic capacitors tend to vary their capacitance with temperature.

- C0G or NP0 - Typical 4.7 pF to 0.047 µF, 5%. High tolerance and temperature performance. Larger and more expensive.
- X7R - Typical 3300 pF to 0.33 µF, 10%. Good for non-critical coupling, timing applications.
- Z5U - Typical 0.01 µF to 2.2 µF, 20%. Good for bypass, coupling applications. Low price and small size.

Polystyrene: Slightly larger than ceramic, but still has small values (usually in the picofarad range).
Polyester: from about 1 nF to 1 μF
Polypropylene: low-loss, high voltage, resistant to breakdown
Tantalum: These are polarized capacitors that are still small enough to be surface mount. Normally the dielectric breakdown voltage is rather low -- typically less than 20 volts -- so the capacitors are not suitable for high voltage applications. Tantalum capacitors have a stable capacitance across varying temperatures, but higher (worse) ESR than any other capacitor material except for electrolytic. Tantalum capacitors have the highest (best) energy density of any material.
Electrolytic: These are also polarized, are much larger than tantalums. The dielectric strength is much higher in these, and so is the capacitance. Capacitance values can range between 1μF and 1mF (sometimes up into the farad range). These are compact capacitors -- higher (best) energy density of any material other than tantalums. Electrolytic capacitors also very lossy -- at high frequencies they have the highest (worst) ESR of any capacitor material. They are useful for smoothing power supplies because of the high capacitance.
Air-gap
Aerogel: These capacitors are more compact than normal electrolytic capacitors, giving capacitance values in the farad range, but normally have an extremely low breakdown voltage.
Super capacitors 2500 F to 5000 F

Reference

Junctions

Capacitors in Series

Capacitors in series are the same as increasing the distance between two capacitor plates. As well, it should be noted that placing two 100 V capacitors in series results in the same as having one capacitor with the total maximum voltage of 200 V. This, however, is not recommended to be done in practice. Especially with capacitors of different values. In a capacitor network in series, all capacitors can have a different voltage over them.

In a series configuration, the capacitance of all the capacitors combined is the reciprocal of the sum of the reciprocals of the capacitance of all the capacitors.

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n}$

Capacitors in Parallel

Capacitors in parallel are the same as increasing the total surface area of the capacitors to create a larger capacitor with more capacitance. In a capacitor network in parallel, all capacitors have the same voltage over them.

In a parallel configuration, the capacitance of the capacitors in parallel is the sum of the capacitance of all the capacitors.

$C_{eq} = C_1 + C_2 + \cdots + C_n \,\!$

RC Circuit

Introduction

An RC circuit is short for 'Resistor-Capacitor' circuit. A capacitor takes a finite amount of time to discharge through a resistor, which varies with the values of the resistor and capacitor. A capacitor acts interestingly in an electronic circuit, practically speaking as a combination of a voltage source and a variable resistor.

Basics

Below is a simple RC Circuit:

There is a capacitor in parallel with the resistor and current probe. The way the capacitor functions is by acting as a very low resistance load when the circuit is initially turned on. This is illustrated below:

Initially, the capacitor has a very low resistance, almost 0. Since electricity takes the path of least resistance, almost all the electricity flows through the capacitor, not the resistor, as the resistor has considerably higher resistance.

As a capacitor charges, its resistance increases as it gains more and more charge. As the resistance of the capacitor climbs, electricity begins to flow not only to the capacitor, but through the resistor as well:

Once the capacitor's voltage equals that of the battery, meaning it is fully charged, it will not allow any current to pass through it. As a capacitor charges its resistance increases and becomes effectively infinite (open connection) and all the electricity flows through the resistor.

Once the voltage source is disconnected, however, the capacitor acts as a voltage source itself:

As time goes on, the capacitor's charge begins to drop, and so does its voltage. This means less current flowing through the resistor:

Once the capacitor is fully discharged, you are back to square one:

If one were to do this with a light and a capacitor connected to a battery, what you would see is the following:

Switch is hit. Light does not light up.
Light gradually becomes brighter and brighter...
Light is at full luminosity.
Switch is released. Light continues to shine.
Light begins to fade...
Light is off.

This is how a capacitor acts. However, what if you changed the values of R1? C1? The voltage of the battery? We will examine the mathematical relationship between the resistor, capacitor, and charging rate below.that's all about it.

The Time Constant

In order to find out how long it takes for a capacitor to fully charge or discharge, or how long it takes for the capacitor to reach a certain voltage, you must know a few things. First, you must know the starting and finishing voltages. Secondly, you must know the time constant of the circuit you have. Time constant is denoted by the Greek letter 'tau' or τ. The formula to calculate this time constant is:

$\tau={R}{C} \,\!$

Great, so what does this mean? The time constant is how long it takes for a capacitor to charge to 63% of its full charge. This time, in seconds, is found by multiplying the resistance in ohms and the capacitance in farads.

According to the formula above, there are two ways to lengthen the amount of time it takes to discharge. One would be to increase the resistance, and the other would be to increase the capacitance of the capacitor. This should make sense. It should be noted that the formula compounds, such that in the second time constant, it charges another 63%, based on the original 63%. This gives you about 86.5% charge in the second time constant. Below is a table.

Time Constant	Charge
1	63%
2	87%
3	95%
4	98%
5	99+%

For all practicality, by the 5th time constant it is considered that the capacitor is fully charged or discharged.

put some stuff in here about how discharging works the same way, and the function for voltage based on time

$v(t)=\frac{1}{C} \int i(t)dt$

Where i(t) is the current flowing through the capacitor as a function of time.

This equation is often used in another form. By differentiating with respect to time:

$i(t)=C\frac{dv(t)}{dt}$

Substituting v/r for i(t) and integrating the above equation gives you an equation used to describe the charging and discharging characteristics of RC circuits. A charging characteristic curve exponentially increases from 0% (0 volts) and approaches 100% full (maximum voltage), similarly, a discharge curve starts at the theoretical 100% (maximum voltage) and exponentially falls back to 0% (0 volts).

Capacitor Specifications

When a capacitor is being discussed, it is referred to with certain "specifications" or characteristics. Capacitors are usually "specified" in the following manner-

they are specified by type (tantalum, electrolytic, etc.)
they are specified by package (axial, radial, as discussed above).
they are specified by how to connect to them, their connection type (such as "snap in" or leaded, or threaded screw holes, or surface mount).
they are specified by capacitance value, e.g. in microfarads (µF).
they are specified by voltage rating (i.e., 30 V). This indicates the maximum voltage over which it is safe to use the referenced capacitor.
some types, such as electrolytic capacitors, are specified by operating temperature (usually 80 or 120 °C), which reflects the maximum temperature that the capacitor can reach before failing. Note- common practice is to use capacitors well below their maximum operating voltage and temperature in order to ensure longevity.
they can be specified by other parameters, including ESR or "equivalent series resistance" (explained above). Also, some capacitors can be specified by UL or other saftey rating. A "X" type capacitor indicates that the capacitor meets certain standards one of which is that it is appropriate to be used with line-level voltages (such as 117 or 220 V) typically found from the wall outlet, as well as that it can withstand surges typically found in power distribution systems.
they are specified in percentage accuracy, i.e., how much they are likely to to deviate from their rated capacitance. Common ratings are + or - 20%.

Capacitor Variations

Variable Capacitor:

Electrolytic Capacitor: See: wikipedia entry on Electrolytic Capacitors

Inductors

Introduction

An inductor is a passive electronic component dependant on frequency, and is used to store electric energy in the form of a magnetic field. An inductor has the symbol:

Inductance is the characteristic of the Inductor to generates a magnetic field for a given current. Inductance has a letter symbol L and measured in units of Henry (H).

$L = \frac{B}{I}$

Important Qualities of Inductors

There are several important properties for an inductor that may need to be considered when choosing one for use in an electronic circuit. The following are the basic properties of a coil inductor. Other factors may be important for other kinds of inductor, but these are outside the scope of this article.

Current carrying capacity is determined by wire thickness and resistivity.
The quality factor, or Q-factor, describes the energy loss in an inductor due to imperfection in the manufacturing.
The inductance of the coil is probably most important, as it is what makes the inductor useful. The inductance is the response of the inductor to a changing current.

The inductance is determined by several factors.

Coil shape: short and squat is best
Core material
The number of turns in the coil. These must be in the same direction, or they will cancel out, and you will have a resistor.
Coil diameter. The larger the diameter (core area) the less induction.

[[:Image:]]==Inductor's Characteristic==

A simple inductor made from a coil of conductor with the magnetic field associated with the current i

Inductance

The following illustrates the properties of inductors using the example of a coil. Let this coil have the following properties:

Area enclosed by each turn of the coil is A
Length of the coil is 'l'
Number of turns in the coil is N
Permeability of the core is μ. μ is given by the permeability of free space, μ₀ multiplied by a factor, the relative permeability, μ_r
The current in the coil is 'i'

The magnetic flux density, B, inside the coil is given by:

$B=\frac{N \mu i}{l}$

We know that the flux linkage in the coil, λ, is given by;

$\lambda=NBA \,$

Thus,

$\lambda=\frac{N^2A\mu}{l}i$

The flux linkage in an inductor is therefore proportional to the current, assuming that A, N, l and μ all stay constant. The constant of proportionality is given the name inductance (measured in Henries) and the symbol L:

$\lambda=Li \,$

Taking the derivative with respect to time, we get:

$\frac{d \lambda}{dt}=L \frac{di}{dt} + i \frac{dL}{dt}$

Since L is time-invariant in nearly all cases, we can write:

$\frac{d \lambda}{dt}=L \frac{di}{dt}$

Now, Faraday's Law of Induction states that:

$-\mathcal{E}=N \frac{d \Phi}{dt}=\frac{d \lambda}{dt}$

We call $-\mathcal{E}$ the electromotive force (emf) of the coil, and this is opposite to the voltage v across the inductor, giving:

$v=L \frac{di}{dt}$

This means that the voltage across an inductor is equal to the rate of change of the current in the inductor multiplied by a factor, the inductance. note that for a constant current, the voltage is zero, and for an instantaneous change in current, the voltage is infinite (or rather, undefined). This applies only to ideal inductors which do not exist in the real world.

This equation implies that

The voltage across an inductor is proportional to the derivative of the current through the inductor.
In inductors, voltage leads current.
Inductors have a high resistance to high frequencies, and a low resistance to low frequencies. This property allows their use in filtering signals.

An inductor works by opposing current change. Whenever an electron is accelerated, some of the energy that goes into "pushing" that electron goes into the electron's kinetic energy, but much of that energy is stored in the magnetic field. Later when that or some other electron is decelerated (or accelerated the opposite direction), energy is pulled back out of the magnetic field.

Inductor's Voltage

$v=L\frac{di}{dt}$

Inductor's Current

$i=\frac{1}{L} \int_{t_1}^{t_2} v(t)\, dt + i \left(t_1 \right)$

Note that usually, the voltage will fall off to zero as the magnetic field is set up, reaching zero at an infinite time after the voltage is applied, so the integral of voltage tends to a finite limit, meaning that the current is also a finite quantity.

Inductor's Impedance

For an ideal lossless inductor the impedance of an inductor at any given angular frequency $ω$ is given by

Z = j ω L

where j is $\sqrt{-1}$ and L is the inductance.

Z = s L

Z = ωL /_90

In practice, all conductor has a resistance hence for non-ideal lossless inductor

Z L = R L + Z

Z = R L + j ω L

where j is $\sqrt{-1}$ and L is the inductance.

Z = R L + s L

Z = R_L/-0 + ωL /_90

Quality factor

Quality factor denoted as Q is defined as the ability to store energy to the sum total of all energy losses within the component

$Q = \frac{X}{R}$

where

Q = quality factor (no units)
R = total resistance associated with energy losses (in ohms)
X = reactance (in ohms). ( $X L = 2π f L$ for inductors; $X_C = \frac{1}{2\pi f C}$ for capacitors; f is the frequency of interest ).

Inductor's Network

Series Connection

$L_{eq}=L_1+L_2+\cdots+L_n$

Parallel Connection

$\frac{1}{L_{eq}}=\frac{1}{L_1}+\frac{1}{L_2}+\cdots+\frac{1}{L_n}$

Other Components

Ideal voltage sources

An ideal voltage source is a fundamental electronics component that creates a constant voltage between two points regardless of whatever else is connected to it. Since it is ideal, some circuit configurations are not allowed, such as short circuits, which would create infinite current. (I = V / 0)

A water analogy would be a pump with pressure sensors on both sides. The difference in pressure between the in port and out port is constantly measured, regardless of the absolute pressure of each side, and the pump speed is adjusted so that the pressure difference stays constant.

Real voltage sources, such as batteries, power supplies, piezoelectric disks, generators, steam turbines, wall outlets, etc. have an internal source impedance (in series with the ideal voltage source), which is very important to understand.

Ideal current sources

An ideal current source is a fundamental electronics component that creates a constant current through a section of circuit, regardless of whatever else is connected to it. Since it is ideal, some circuit configurations are not allowed, such as open circuits, which would create an infinite voltage.

A water analogy would be a pump with a flow meter. It measures the amount of water flowing by per unit time and changes the speed of the pump so that the current flow is constant.

Real current sources, such as batteries, power supplies, piezoelectric disks, generators, etc. have an internal source impedance (in parallel with the source), which is very important to understand.

Real sources generally behave more like voltage sources than current sources, because the internal impedance in series is very low. A current source can be created from a voltage source with a circuit such as a current mirror.

Dependent Sources

A dependent source is either a voltage or a current source which is dependent upon another value within the circuit, usually another voltage or current. Typically, these are used in circuit modeling and analysis.

There are four main types of such sources.

Voltage-controlled voltage source (VCVS)

This is a voltage source whose value is controlled by another voltage elsewhere in the circuit. Its output will typically be given as $V o = A V c$ , where A is a gain term and V_cis a control voltage.

An example of a VCVS may be an idealized amplifier, where A is the gain of the amplifier.

Current-controlled voltage source (CCVS)

This is a voltage source whose value is controlled by a current elsewhere in the circuit. Its output is typically given as $V o = A I c$ , where A is a gain term and I_c is a control current.

Voltage-controlled current source (VCCS)

This is a current source whose value is controlled by a voltage elsewhere in the circuit. Its output is typically given as $I o = A V c$ , where A is a gain term and V_c is a control voltage.

Current-controlled current source (CCCS)

This is a current source whose value is controlled by a current elsewhere in the circuit. Its output is typically given as $I o = A I c$ , where A is a gain term and I_c is a control current.

An example of a CCCS is an idealized bipolar junction transistor, which may be thought of as a small current controlling a larger one. Specifically the base current, I_b is the control and the collector current I_c is the output.

Switch

A switch is a mechanical device that connects or disconnects two parts of a circuit.

A switch is a short circuit when it is on.

And it is a open circuit when it is off.

When you turn a switch on it completes a circuit that allows current to flow. When you turn the switch off it creates an air gap (depending on the type of switch), and since air is an insulator no current flows.

A switch is a device for making or breaking an electric circuit.

Usually the switch has two pieces of metal called contacts that touch to make a circuit, and separate to break the circuit. The contact material is chosen for its resistance to corrosion, because most metals form insulating oxides that would prevent the switch from working. Sometimes the contacts are plated with noble metals. They may be designed to wipe against each other to clean off any contamination. Nonmetallic conductors, such as conductive plastic, are sometimes used. The moving part that applies the operating force to the contacts is called the actuator, and may be a rocker, a toggle or dolly, a push-button or any type of mechanical linkage.

Contact Arrangements

Switches can be classified according to the arrangement of their contacts. Some contacts are normally open until closed by operation of the switch, while normally closed contacts are opened by the switch action. A switch with both types of contact is called a changeover switch.

The terms pole and throw are used to describe switch contacts. A pole is a set of contacts that belong to a single circuit. A throw is one of two or more positions that the switch can adopt. These terms give rise to the following abbreviations.

S (single), D (double).
T (throw), CO (changeover).
CO = DT.

(single|double) pole ((single|double) throw|changeover)

SPST = single pole single throw, a simple on-off switch.
SPDT = single pole double throw, a simple changeover or on-off-on switch.
SPCO = single pole changeover, equivalent to SPDT.
DPST = double pole single throw, equivalent to two SPST switches controlled by a single mechanism.
DPDT = double pole double throw, equivalent to two SPDT switches controlled by a single mechanism.
DPCO = double pole changeover, equivalent to DPDT.

Switches with larger numbers of poles or throws can be described by replacing the "S" or "D" with a number.

Biased Switches

A biased switch is one containing a spring that returns the actuator to a certain position. The "on-off" notation can be modified by placing parentheses around all positions other than the resting position. For example, an (on)-off-(on) switch can be switched on by moving the actuator in either direction away from the centre, but returns to the central off position when the actuator is released.

The momentary push-button switch is a type of biased switch. This device makes contact when the button is pressed and breaks when the button is released.

Special Types

Switches can be designed to respond to any type of mechanical stimulus: for example, vibration (the trembler switch), tilt, air pressure, fluid level (the float switch), the turning of a key (key switch), linear or rotary movement (the limit switch or microswitch).

The mercury tilt switch consists of a blob of mercury inside a glass bulb. The two contacts pass through the glass, and are shorted together when the bulb is tilted to make the mercury roll on to them. The advantage of this type of switch is that the liquid metal flows around particles of dirt and debris that might otherwise prevent the contacts of a conventional switch from closing.

DC Voltage and Current Laws

Ohm's Law

Ohm's law describes the relationship between voltage, current, and resistance.

Voltage and current are proportional at a given temperature:

$V = I \cdot R$

Voltage (V) is measured in volts (V); Current (I) in amperes (A); and resistance (R) in ohms (Ω).

In this example, the current going through any point in the circuit, I, will be equal to the voltage V divided by the resistance R.

In this example, the voltage across the resistor, V, will be equal to the supplied current, I, times the resistance R.

If two of the values (V, I, or R) are known, the other can be calculated using this formula.

Any more complicated circuit has an equivalent resistance that will allow us to calculate the current draw from the voltage source. Equivalent resistance is worked out using the fact that all resistor are either in parallel or series. Similarly, if the circuit only has a current source, the equivalent resistance can be used to calculate the voltage dropped across the current source.

Kirchoff's Voltage Law

Kirchoff's Voltage Law (KVL): The sum of voltage drops around any loop in the circuit that starts and ends at the same place must be zero.

When you have a positive potential on one side of a battery, then there must be a negative potential on the other side of the battery.

(With Kirchoff's law, it's the sum of the voltages around the entire loop -- including the battery -- that equals zero. So, say you just have a 9 Volt battery connected to a resistor: there's 9V across the resistor, and 9V across the battery; the directions work out so that they subtract: 9V − 9V = 0.)

Analogy to elevation: A person is at the bottom of a mountain. They walk up the mountain, down the other side, and around to their starting position. Even though they changed elevation during the walk, they are at the same elevation as when they started.

KVL Example

Insert diagram and example here: A voltage source and two resistors in series. Calculate the voltage across each component using ohm's law and the students knowledge of resistors in series, then start at ground, add the voltage source, and then subtract the drops across each resistor, and show that it comes back to zero.

Voltage as a Physical Quantity

Voltage is the potential difference between two charged objects.

You really should put in something here about voltage being equal to the electric field times distance. It's the analogue equation of the gravitational potential of an object equalling the gravitational field times height.

Yeah. It should go in the Basic Concepts section, though. Analogies are always good.

Another way of thinking about voltage is potential energy per unit of charge. This gives a similar connection to gravitational potential -- to get the potential energy you multiply gravitational potential by mass or multiply the electrical potential by charge. In addition, it gives a connection to p=iv. Power is a change in energy divided by time. Voltage times current is the same as voltage*charge / time, which is energy divided by time.

The nice things about potentials is you can add or subtract them in series to make larger or smaller potentials as is commonly done in batteries.

Electrons flow from areas of high potential to lower potential.

At a given place in a circuit there are numerous paths to ground (what about negative voltages?). Each of them has the same voltage as they have the same potential from ground (why?) (-> because of KVL).

All the components of a circuit have resistance that acts as a potential drop.

Additional note: The following explains why voltage is "analogous" to the pressure of a fluid in a pipe (although, of course, it is only an analogy, not exactly same thing), and it also explains the strange-sounding "dimensions" of voltage. Consider the potential energy of compressed air being pumped into tank. The energy increases with each new increment of air. Pressure is that energy divided by the volume, which we can understand intuitively. Now consider the energy of electric charge (measured in coulombs) being forced into a capacitor. Voltage is that energy per charge, so voltage is analogous to a pressure-like sort of forcefulness. Also, dimensional analysis tells us that voltage ("energy per charge") divides out to be "charge per distance," the distance being between the plates of the capacitor. (More discussion is on page 16 of "Industrial Electronics," by D. J. Shanefield, Noyes Publications, Boston, 2001.)

Kirchoff's Current Law

Kirchoff's Current Law (KCL): The sum of all current entering a node must equal the sum of all currents leaving the node.

Kirchoff's current law can be described with a sentence as "What comes in, must go out". It's that simple.

This means that current is conserved. If you have a current into a junction, the same current must go out of the junction.

Analogy to traffic: The number of cars entering an intersection is equal to the number of cars leaving the intersection.

KCL Example

-I₁ + I₂ + I₃ = 0 ↔ I₁ = I₂ + I₃

I₁ - I₂ - I₃ - I₄ = 0 ↔ I₂ + I₃ + I₄ = I₁

$I_1 = I_2 + I_3 + \cdots + I_n \,\!$

Here is more about Kirchhoff's laws, which can be integrated here

Consequences of KVL and KCL

Voltage Dividers

If two circuit elements are in series, there is a voltage drop across each element, but the current through both must be the same. The voltage at any point in the chain divides according to the resistances. A simple circuit with two (or more) resistors in series with a source is called a voltage divider.

Figure A: Voltage Divider circuit.

Consider the circuit in Figure A. According to KVL the voltage $V i n$ is dropped across resistors $R 1$ and $R 2$ . If a current i flows through the two series resistors then by Ohm's Law.

$i = \frac{V_{in}}{R_1+R_2}$ .

So

$V_{out}=iR_2 \,\!$

Therefore

$V_{out}=\frac{V_{in}R_2}{R_1+R_2}$

Similary if $V R 1$ is the voltage across $R 1$ then

$V_{R1}=\frac{V_{in}R_1}{R_1+R_2}$

In general for n series resistors the voltage dropped across one of them say $R i$ is

$V_{Ri}=\frac{V_{in}R_i}{R_{eq}}$

Where

$R_{eq}=R_1+\cdots+R_n$

Voltage Dividers as References

Clearly voltage dividers can be used as references if you have a 9 volt battery and you want 4.5 volts then connect two equal valued resistors in series and take the reference across the second and ground. There are clearly other concerns though the first concern is current draw and the effect of the source impedance clearly connecting two 100 ohm resistors is a bad idea if the source impedance is say 50 ohms. Then the current draw would be 0.036 mA which is quite large if the battery is rated say 200 milliampere hours. The loading is more annoying with that source impedance too, the reference voltage with that source impedance is $\frac {9(100)}{250}=3.6\mbox{ V}$ . So clearly increasing the order of the resistor to at least 1 k $Ω$ is the way to go to reduce the current draw and the effect of loading. The other problem with these voltage divider references is that the reference cannot be loaded if we put a 100 Ω resistor in parallel with a 10 kΩ resistor, when the voltage divider is made of two 10 kΩ resistors, then the resistance of the reference resistor becomes somewhere near 100 Ω. This clearly means a terrible reference. If a 10 MΩ resistor is used for the reference resistor will still be some where around 10 kΩ but still probably less. The effect of tolerances is also a problem; if the resistors are rated 5% then the resistance of 10 kΩ resistors can vary by ±500 Ω. This means more inaccuracy with this sort of reference.

Current Dividers

If two elements are in parallel, the voltage across them must be the same, but the current divides according to the resistances. A simple circuit with two (or more) resistors in parallel with a source is called a current divider.

Figure B: Parallel Resistors.

If a voltage V appears across the resistors in Figure B with only $R 1$ and $R 2$ for the moment then the current flowing in the circuit, before the division, i is according to Ohms Law.

$i=\frac{V}{R_{eq}}$

Using the equivalent resistance for a parallel combination of resistors is

$i=\frac{V(R_1+R_2)}{R_1R_2}$ (1)

The current through $R 1$ according to Ohms Law is

$i_1=\frac{V}{R_1}$ (2)

Dividing equation (2) by (1)

$i_1=\frac{iR_2}{R_2+R_1}$

Similarly

$i_2=\frac{iR_1}{R_2+R_1}$

In general with n Resistors the current $i x$ is

$i_x=\frac{iR_1R_2\cdots R_n}{ (R_2\cdots R_n+\cdots+ R_1\cdots R_{n-1}) R_x}$

Or possibly more simply

$\frac{i_x}{i}=\frac{R_{eq}}{R_x}$

Where

$R_{eq}=\frac{R_1\cdots R_n}{R_2\cdots R_n+\cdots+ R_1\cdots R_{n-1}}$

Nodal Analysis

Nodes

A node is a section of a circuit which connects components to each other. All of the current entering a node must leave a node, according to Kirchoff's Current Law. Every point on the node is at the same voltage, no matter how close it is to each component, because the connections between components are perfect conductors. This voltage is called the node voltage, and is the voltage difference between the node and an arbitrary reference, the ground point. The ground point is a node which is defined as having zero voltage. The ground node should be chosen carefully for convenience. Note that the ground node does not necessarily represent an actual connection to ground, it is just a device to make the analysis simpler. For example, if a node has a voltage of 5 Volts, then the voltage drop between that node and the ground node will be 5 Volts.

Note that in real circuits, nodes are made up of wires, which are not perfect conductors, and so the voltage is not perfectly the same everywhere on the node. This distinction is only important in demanding applications, such as low noise audio, high speed digital circuits (like modern computers), etc.

Nodal Analysis

Nodal analysis is a formalized procedure based on KCL equations.

Steps:

Identify all nodes.
Choose a reference node. Identify it with reference (ground) symbol. A good choice is the node with the most branches, or a node which can immediately give you another node voltage (e.g., below a voltage source).
Assign voltage variables to the other nodes (these are node voltages.)
Write a KCL equation for each node (sum the currents leaving the node and set equal to zero). Rearrange these equations into the form A*V1+B*V2=C (or similar for equations with more voltage variables.)
Solve the system of equations from step 4. There are a number of techniques that can be used: simple substitution, Cramer's rule, the adjoint matrix method, etc.

Complications in Nodal Analysis

Dependent Current Source

Solution: Write KVL equations for each node. Then express the extra variable (whatever the current source depends on) in terms of node voltages. Rearrange into the form from step 4 above. Solve as in step 5.
Independent Voltage Source

Problem: We know nothing about the current through the voltage source. We cannot write KCL equations for the nodes the voltage source is connected to.

Solution: If the voltage source is between the reference node and any other node, we have been given a 'free' node voltage: the node voltage must be equal to the voltage source value! Otherwise, use a 'super-node', consisting of the source and the nodes it is connected to. Write a KCL equation for all current entering and leaving the super-node. Now we have one equation and two unknowns (the node voltages). Another equation that relates these voltages is the equation provided by the voltage source (V2-V1=source value). This new system of equations can be solved as in Step 5 above.
Dependent Voltage Source

Solution: Same as an independent voltage source, with an extra step. First write a super-node KCL equation. Then write the source controlling quantity (dependence quantity?) in terms of the node voltages. Rearrange the equation to be in the A*V1+B*V2=C form. Solve the system as above.

Example

Given the Circuit below, find the voltages at all nodes.

node 0: $V_0 = 0V\,$ (defined as ground node)
node 1: $V_1 = 9V\,$ (free node voltage)
node 2: $\frac{V_1 - V_2}{1k} = \frac{V_2 - V_0}{3k} + \frac{V_2 - V_3}{2k}$

node 3: $\frac{V_2 - V_3}{2k} = \frac{V_3 - V_0}{2k}$

which results in the following system of linear equations: $\left\{\begin{matrix} +11 V_2 & -3 V_3 & = & 54 \\ +1 V_2 & -2 V_3 & = & 0\end{matrix}\right.$

therefore, the solution is: $\left\{\begin{matrix} V_0 =0.00V \\ V_1 =9.00V \\ V_2 =5.68V \\ V_3 =2.84V\end{matrix}\right.$

Another solution with KCL would be to solve node in terms of node 2;

$\frac{V_2 - 9V}{1k} + \frac{V_2}{3k} + \frac{V_2}{4k} = 0$

Mesh Analysis

Meshes

A 'mesh' (also called a loop) is simply a path through a circuit that starts and ends at the same place. For the purpose of mesh analysis, a mesh is a loop that does not enclose other loops.

Mesh Analysis

Similar to nodal analysis, mesh analysis is a formalized procedure based on KVL equations. A caveat: mesh analysis can only be used on 'planar' circuits (i.e. there are no crossed, but unconnected, wires in the circuit diagram.)

Steps:

1. Draw circuit in planar form (if possible.)

2. Identify meshes and name mesh currents. Mesh currents should be in the clockwise direction. The current in a branch shared by two meshes is the difference of the two mesh currents.

3. Write a KVL equation in terms of mesh currents for each mesh.

4. Solve the resulting system of equations.

Complication in Mesh Analysis

1. Dependent Voltage Sources

Solution: Same procedure, but write the dependency variable in terms of mesh currents.

2. Independent Current Sources

Solution: If current source is not on a shared branch, then we have been given one of the mesh currents! If it is on a shared branch, then use a 'super-mesh' that encircles the problem branch. To make up for the mesh equation you lose by doing this, use the mesh current relationship implied by the current source (i.e. $I 2 - I 1 = 4 m A$ ).

3. Dependent Current Sources

Solution: Same procedure as for an independent current source, but with an extra step to eliminate the dependency variable. Write the dependency variable in terms of mesh currents.

Example

Given the Circuit below, find the currents $I 1$ , $I 2$ .

The circuit has 2 loops indicated on the diagram. Using KVL we get:
Loop1: $0 = 9 - 1000 I 1 - 3000(I 1 - I 2)$
Loop2: $0 = 3000(I 1 - I 2) - 2000 I 2 - 2000 I 2$
Simplifying we get the simultaneous equations:
$0 = 9 - 4000 I 1 + 3000 I 2$
$0 = 0 + 3000 I 1 - 7000 I 2$
solving to get:
$I 1 = 3.32 m A$
$I 2 = 1.42 m A$

Thevenin and Norton Equivalent Circuits

Source Transformation

Any linear time invariant network of impedances can be reduced to one equivalent impedance. In particular, any network of sources and resistors can be reduced to one ideal source and one resistor, in either the Thevenin or Norton configurations. In this way, a complicated network attached to a load resistor can be reduced to a single voltage divider (Thevenin) or current divider (Norton).

Thevenin and Norton equivalents let you replace a Voltage source in series with a resistor by a current source in parallel with a resistor, or vice versa. This is called a source transformation.

The point to be noted is that the block that is replaced with such an equivalent should be linear and time invariant, i.e. a linear change in the electrical source in that block produces a linear change in the equivalent source, and the behavior can be replicated if the initial conditions are replicated. The above shown transformation figures are true only if the circuit contains at least one independent voltage or current source. If the circuit comprises only dependent sources then Thevenin equivalent consists of R_Th alone See This to get a clear idea LINK DEAD

Thevenin Equivalents

The Thevenin equivalent circuit of a (two-terminal) network consists of a voltage source in series with a resistor. The Thevenin equivalent will have the same output voltage and current regardless of what is attached to the terminals.

Techniques For Finding Thevenin Equivalents

Network contains no sources (only resistors): The Thevenin resistance is equal to the equivalent resistance of the network. The Thevenin voltage is zero.

Basic: Works for any network except one with no independent sources. Find the voltage across the terminals (with positive reference at terminal A) when they are open-circuited. Find the current from terminal A to terminal B when they are short-circuited. Then

$R_{th}=\frac{V_{oc}}{I_{sc}}$
The Thevenin voltage source value is equivalent to the open-circuit voltage.

If the network has no dependent sources, the independent sources can be zeroed, and the Thevenin resistance is equal to the equivalent resistance of the network with zeroed sources. Then, find $V o c$ .

Only Dependent Sources:

If the network has only dependent sources, either attach a test voltage source to the terminal points and measure the current that passes from the positive terminal, or attach a test current source to the terminal points and measure the voltage difference across the terminals. In both cases you will have values for $V o c$ and $I s c$ , allowing you to use the $R_{th}=\frac{V_{oc}}{I_{sc}}$ relation to find the Thevenin resistance.

Norton Equivalents

Norton equivalents can be found by performing a source transformation on the Thevenin equivalent. The Norton Equivalent of a Thevenin Equivalent consists of a current source, $I_N= \frac {V_{th}}{R_{th}}$ in parallel with $R t h$ .

Thevenin and Norton Equivalent

Figure 1: Circuit for the determination of Equivalents

The steps for creating the Equivalent are:

1. Remove the load circuit.

2. Calculate the voltage, V, at the output from the original sources.

3. Now replace voltage sources with shorts and current sources with open circuits.

4. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, looking back into the circuit, with the sources removed.

5. The equivalent circuit is a voltage source with voltage V in series with a resistance R in series with the load.

The Thevenin Equivalent is determined with $R 2$ as the load as shown in Figure 1. The first step is to open circuit $R 2$ . Then the voltage v is calculated with $R 2$ open circuited must be calculated. The voltage across $R 2$ is $V 1$ this is because no current flows in the circuit so the voltage across $R 2$ must be $V 1$ by KVL.

Since this circuit does not contain any dependent sources, all that needs to be done is for all the Independent Voltage sources to be shorted and for all Independent Current Sources to be open circuited. This results in the circuit shown in Figure 2.

Figure 2: Circuit for the Equivalent Resistance

Now the Thevenin Resistance is calculated looking into the two nodes. The Thevenin resistance is clearly $R 1 + R 3 | | R 4$ . The Thevenin Equivalent is shown in Figure 3 and $R t h$ and $V t h$ have the values shown below.

$R_{th}=R_1 + R_3 | | R_4 \,$ (1)

$V_{th}=V_1\,$

Figure 3: Equivalent Thevenin Circuit

The Norton Equivalent is created by doing a source transformation using $I_{N}=\frac {V_{th}}{R_{th}} \,$ .(2)

Figure 4: Equivalent Norton Circuit

If $R 2 = R 1 = 1 k Ω$ and $R 3 = R 4 = 2 k Ω$ and $V 1 = 15 V$ then

$R_{th}=1k + 2k | | 2k \,$

$R_{th}=2k \Omega \,$

As a final note if the voltage across $R 2$ is calculate by Voltage Divider Rule using the Thevenin Equivalent circuit in Figure 3.

$V_{R2}=\frac {V_{th}R_2}{R_{th}+R_2}$ (3)

If the value of $R t h$ form equation 1 is substituted into equation 3.

$V_{R2}=\frac {V_{th}R_2}{R_1 + R_3 | | R_4+R_2}$ (4)

Now look at Figure 1 and calcute $V R 2$ by voltage divider rule it has the same value as equation 4. If the current through $R 2$ is calculated in Figure 4 by current divider rule.

$I_{R2}=\frac {I_NR_{th}}{R_{th}+R_2}(5)$

Substituting equation 2 into 5.

$I_{R2}=\frac {V_{th}}{R_{th}+R_2}(4)$

If equation 4 and Ohm's Law are used to get the voltage across $R 2$ equation 3 is reached.

Please note: The "||", a symbol that is used as an operator here, holds higher precidence than the "+" operator. As such, it is evaluated before a sum.

See Norton's theorem and Thevenin's theorem for more examples. - Omegatron 18:22, 4 Jun 2005 (UTC)

Superposition

Superposition Principle

(a)

(b)

(c)

Figure 1: The circuits showing the linearity of resistors.

Most basic electronic circuits are composed of linear elements. Linear elements are circuit elements which follow Ohm’s Law. In Figure 1 (a) with independent voltage source, V₁, and resistor, R, a current i₁ flows. The current i₁ has a value according to Ohm’s Law. Similarly in Figure 1 (b) with independent voltage source, V2, and resistor, R, a current i₂ flows. In Figure 1 (c) with independent voltage sources, V1 and V2, and resistor, R, a current i flows. Using Ohm’s Law equation 1 is reached. If some simple algebra is used then equation 2 is reached. But V₁/R has a value i₁ and the other term is i₂ this gives equation 3. This is basically what the Superposition Theorem states.

$i=\frac {V_1+V_2}{R}$ (1)

$i=\frac {V_1}{R} + \frac {V_2}{R}$ (2)

$i=i_1+i_2 \,$ (3)

The Superposition Theorem states that the effect of all the sources with corresponding stimuli on a circuit of linear elements is equal to the algebraic sum of each individual effect. Each individual effect is calculated by removing all other stimuli by replacing voltage sources with short circuits and current sources with open circuits. Dependent sources can be removed as long as the controlling stimuli is not set to zero. The process of calculating each effect with one stimulus connected at a time is continued until all the effects are calculated. If kth stimulus is denoted sk and the effect created by sk denoted ek.

$\sum_{k=1}^N e_k$ (4)

The steps for using superposition are as follows:

1. Calculate the effect of each source in turn with all other independent voltage sources short circuited and independent current sources open circuited.

2. Sum these effects to get the complete effect.

Note: the removal of each source is often stated differently as: replace each voltage source with its internal resistance and each current with its internal resistance. This is identical to what has been stated above. This is because a real voltage source consists of an independent voltage source in series with its internal resistance and a real current source consists of a independent current source in parallel with its internal resistance.

Superposition Example

Figure 2: The circuit for the example.

Problem: Calculate the voltage, v, across resistor R₁.

Step 1: Short circuit V₂ and solve for v₁. By voltage divider rule.

$v_{1}= \frac {V_1(R_1 | | R_3)}{R_2+(R_1 | | R_3)}$ (5)

Short circuit V₁ and solve for v₂. By voltage divider rule.

$v_{2}= \frac {V_2(R_1 | | R_2)}{R_3+(R_1 | | R_2)}$ (6)

Step 2: Sum the effects.

$v=v_1+v_2 \,$

Using equations 5 and 6.

$v=\frac {V_1(R_1 | | R_3)}{R_2+(R_1 | | R_3)}+\frac {V_2(R_1 | | R_2)}{R_3+(R_1 | | R_2)}$

If $R 1 = R 2 = R 3 = 2 k Ω$ and $V 1 = V 2 = 15 V$ then

$v=\frac {15(2k | | 2k)}{2k+(2k | | 2k)}+\frac {15(2k | | 2k)}{2k+(2k | | 2k)}$

$v=\frac {15(1k)}{3k}+\frac {15(1k)}{3k}$

$v=10V \,$

Diagnostic Equipment

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Diagnostic and Testing Equipment

There is a wide array of devices used to test and diagnose electronic equipment. This chapter will attempt to explain the differences and different types of equipment used by electronics technicians and engineers.

Ammeter

An ammeter measures current.

The ammeter's terminals must be in series with the current being measured. Ammeters have a small resistance (typically 50 ohms) so that they only have a small effect on the current.

Ohmmeter

An ohmmeter measures resistance.

The two terminals of ohmmeter are each placed on a terminal of the resistance being measured. This resistance should be isolated from other effects. (It should be taken out of a circuit, if it is in one.)

Voltmeter

A voltmeter measures voltage.

The voltmeter's terminals must be in parallel with the voltage being measured. Voltmeters have a large resistance (typically 1 megaohm), so that they only have a small effect on the voltage.

Multimeter

A multimeter is a combination device, (usually) capable of measuring current, resistance, or voltage. Most modern models measure all three, and include other features such as a diode tester, which can be used to measure continuity in circuits (emitting a loud 'beep' if there is a short).

A digital multimeter

Oscilloscope

An oscilloscope is the same as a Spectrum Analyzer.

Spectrum Analyzer

Spectrum analyzer is the same as a Oscilloscope.

Logic analyzer

A logic analyzer is, in effect, a specialised oscilloscope. The key difference between an analyzer and an oscilloscope is that the analyzer can only display a digital (on/off) waveform, whereas an oscilloscope can display any voltage (depending on the type of probe connected). The other difference is that logic analyzers tend to have many more signal inputs than oscilloscopes - usually 32 or 64, versus the two channels most oscilloscopes provide. Logic analyzers can be very useful for debugging complex logic circuits, where one signal's state may be affected by many other signals.

Frequency counter

A frequency counter is a relatively simple instrument used to measure the frequency of a signal in Hertz (cycles per second). Most counters work by counting the number of signal cycles that occur in a given time period (usually one second). This count is the frequency of the signal in Hertz, which is displayed on the counter's display.

Electrometer

A voltmeter with extremely high input resistance capable of measuring electrical charge with minimal influence to that charge. Ubiquitous in nucleonics, physics and bio-medical disciplines. Enables the direct verification of charge measured in coulombs according to Q=CV. Additionally, electrometers can generally measure current flows in the femtoampere range, i.e. .000000000000001 ampere.

DC Circuit Analysis

DC Circuit Analysis

In this chapter, capacitors and inductors will be introduced (without considering the effects of AC current.) The big thing to understand about Capacitors and Inductors in DC Circuits is that they have a transient (temporary) response. During the transient period, capacitors build up charge and stop the flow of current (eventually acting like infinite resistors.) Inductors build up energy in the form of magnetic fields, and become more conductive. other words, in the steady-state (long term behavior), capacitors become open circuits and inductors become short circuits. Thus, for DC analysis, you can replace a capacitor with an empty space and an inductor with a wire. The only circuit components that remain are voltage sources, current sources, and resistors.

Capacitors and Inductors at DC

DC steady-state (meaning the circuit has been in the same state for a long time), we've seen that capacitors act like open circuits and inductors act like shorts. The above figures show the process of replacing these circuit devices with their DC equivalents. In this case, all that remains is a voltage source and a lone resistor. (An AC analysis of this circuit can be found in the AC section.)

Resistors

If a circuits contains only resistors possibly in a combination of parallel and series connections then an equivalent resistance is determined. Then Ohm's Law is used to determine the current flowing in the main circuit. A combination of voltage and current divider rules are then used to solve for other required currents and voltages.

Simplify the following:
(a) (b) (c)

Figure 1: Simple circuits series circuits.

The circuit in Figure 1 (a) is very simple if we are given R and V, the voltage of the source, then we use Ohm's Law to solve for the current. In Figure 1 (b) if we are given R1, R2 and V then we combine the resistor into an equivalent resistors noting that are in series. Then we solve for the current as before using Ohm's Law. In Figure 1 (c) if the resistors are labeled clockwise from the top resistor R1, R2 and R3 and the voltage source has the value: V. The analysis procedes as follows.

$R_{eq}=R_1+R_2+R_3 \,$

This is the formula for calculating the equivalent resistance of series resistor. The current is now calculated using Ohm's Law.

$i=\frac {V }{R_{eq}}= \frac {V}{R_1+R_2+R_3}$

If the voltage is required across the third resistor then we can use voltage divider rule.

$V_{R3}=\frac {VR_3}{R_1+R_2+R_3}$

Or alternatively one could use Ohm's Law together with the current just calculated.

$V_{R3}=iR_3= \frac {VR_3}{R_1+R_2+R_3}$

(a) (b)

Figure 2: Simple parallel circuits.

In Figure 2 (a) if the Resistor nearest the voltage source is R1 and the other resistor R2. If we need to solve for the current i. Then we precede as before first we calculate the equivalent resistance then use Ohm's Law to solve for the current. The resistance of a parallel combination is:

$R_{eq}=\frac {R_1R_2}{R_1+R_2}$

So the current, i, flowing in the circuit is, by Ohm's Law:

$i=\frac {V(R_1+R_2)}{R_1R_2}$

If we need to solve for current through R2 then we can use current divider rule.

$i_{R2}=\frac {V(R_1+R_2)(R_1)}{(R_1R_2)(R_1+R_2)}= \frac {V}{R_2}$ (1)

But it would probably have been simpler to have used the fact that V most be dropped across R2. This means that we can simply use Ohm's Law to calculate the current through R2. The equation is just equation 1. In Figure 2 (b) we do exactly the same thing except this time there are three resistors this means that the equivalent resistance will be:

$\frac {1}{R_{eq}}= \frac {1}{R_1}+\frac {1}{R_2}+\frac {1}{R_3}$

$R_{eq} =\frac {R_1R_2R_3}{R_1R_2+R_1R_3+R_2R_3}$

Using this fact we do exactly the same thing.

(a) (b) (c)

Figure 3: Combined parallel and series circuits

In Figure 3 (a), if the three resistors in the outer loop of the circuit are R1, R2 and R3 and the other resistor is R4. It is simpler to see what is going on if we combine R2 and R3 into their series equivalent resistance $R 23$ . It is clear now that the equivalent resistance is R1 in series with the parallel combination of $R 23$ and R4. If we want to calculate the voltage across the parallel combination of R4 and $R 23$ then we just use voltage divider.

$V_{4 | | 23}=\frac {R_4 | |(R_2+R_3)V}{R_4 | |(R_2+R_3) + R_1}$

If we want to calculate the current through R2 and R3 then we can use the voltage across $R 4 | | (R 2 + R 3)$ and Ohm's law.

$i_{23} = \frac {R_4 | |(R_2+R_3)V}{(R_4 | |(R_2+R_3) + R_1)(R_2+R_3)}$

Or we could calculate the current in the main circuit and then use current divider rule to get the current.

In Figure 3 (b) we take the same approach simplifying parallel combinations and series combinations of resistors until we get the equivalent resistance.

In Figure 3 (c) this process doesn't work then because there are resistors connected in a delta this means that there is no way to simplify this beyond transforming them to a star or wye connection.

Note: To calculate the current draw from the source the equivalent resistance always must be calculated. But if we just need the voltage across a series resistor this may be necessary. If we want to calculate the current in parallel combination then we must use either current divider rule or calculate the voltage across the resistor and then use Ohm's law to get the current. The second method will often require less work since the current flowing from the source is required for the use of current divider rule. The use of current divider rule is much simpler in the case when the source is a current source because the value of the current is set by the current source.

File:Star Electronics.png

The above image shows three points 1, 2, and 3 connected with resistors R₁, R₂, and R₃ with a common point. Such a configuration is called a star network or a Y-connection.

The above image shows three points 1, 2, and 3 connected with resistosr R₁₂, R₂₃, and R₃₁. The configuration is called a delta network or delta connection.

We have seen that the series and parallel networks can be reduced by the use of simple equations. Now we will derive similar relations to convert a star network to delta and vice versa. Consider the points 1 and 2. The resistance between them in the star case is simply

R₁ + R₂

For the delta case, we have

R₁₂ || (R₃₁ + R₂₃)

We have similar relations for the points 2, 3 and 3, 1. Making the substitution r₁= R₂₃ etc., we have, simplifying,

$R_1 = \frac{r_2r_3}{r_1 + r_2 + r_3}$

$r_1 = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1}$

in the most general case. If all the resistances are equal, then R = r/3.

Measuring Instruments

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Measuring Instruments

Ammeters

Ammeters measure current. Current in electronics is usually measured in mA which are called milliamperes, which are 1/1000s of an ampere.
..... Basically an ammeter consists of a coil that can rotate inside a magnet, but a spring is trying to push the coil back to zero. The larger the current that flows through the coil, the larger the angle of rotation, the torque (= a rotary force) created by the current being counteracted by the return torque of the spring.
..... Usually ammeters are connected in parallel with various switched resistors that can extend the range of currents that can be measured. Assume, for example, that the basic ammeter is "1000 ohms per volt", which means that to get the full-scale deflection of the pointer a current of 1 mA is needed (1 volt divided by 1000 ohms is 1 mA - see "Ohm's Law").
..... To use that ammeter to read 10 mA full-scale it is shunted with another resistance, so that when 10 mA flows, 9 mA will flow through the shunt, and only 1 mA will flow through the meter. Similarly, to extend the range of the ammeter to 100 mA the shunt will carry 99 mA, and the meter only 1 mA. ==Vo‘«←§»’

Ohmmeters

Ohmmeters are basically ammeters that are connected to an internal battery, with a suitable resistance in series. Assume that the basic ammeter is "1000 ohms per volt", meaning that 1 mA is needed for full-scale deflection. When the external resistance that is connected to its terminals is zero (the leads are connected together at first for calibration), then the internal, variable, resistor in series with the ammeter is adjusted so that 1 mA will flow; that will depend on the voltage of the battery, and as the battery runs down that setting will change. The full scale point is marked as zero resistance. If an external resistance is then connected to the terminals that causes only half of the current to flow (0.5 mA in this example), then the external resistance will equal the internal resistance, and the scale is marked accordingly. When no current flows, the scale will read infinity resistance. The scale of an ohmmeter is NOT linear.Ohmmeters are usually usuful in cheking the short circuit and open circuit in boards.its about sss

Multimeters

A digital multimeter

Multimeters contain Ohmeters, Voltmeters, Ammeters and a variety of capabilities to measure other quantities. AC and DC voltages are most often measurable. Frequency of AC voltages. Multimeters also feature a continuity detector, basically an Ohmmeter with a beeper if the multimeter sees less than 100 Ω then it beeps otherwise it is silent. This is very useful for finding whether components are connected when debugging or testing circuits. Multimeters are also often able to measure capacitance and inductance. This may be achieved using a Wien bridge. A diode tester is also generally onboard, this allows one to determine the anode and cathode of an unknown diode. A LCD display is also provided for easily reading of results.

Wikipedia:Multimeter

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Electronics Laboratory Instruments

Oscilloscope

The instrument is used to view AC waveforms. For better explanation of the oscilloscope.

Wikipedia:Oscilloscope

Spectrum Analyzer

Wikipedia:Spectrum_Analyzer

Signal Generator

This instrument is used to generate low voltage AC signals. Most common signal generators can create sinusoidal(sine), triangular and square waves of various frequencies. They are used in conjunction with the oscilloscope to test analogue circuits.

Wikipedia:Signal_Generator

Logic Probe

This instrument generates high and low logic states to test digital circuits. If a logic probe is not available a square wave through a signal generator can be used. Square waves can also be used to test the response time of a digital circuits.

Wikipedia:Logic Probe

Noise in electronic circuits

Electrical Noise: any unwanted form of energy tending to interfere with the proper and easy reception and reproduction of wanted signals.

Classification

Based on Origin

External noise
1. Atmospheric
2. Extraterrestrial
  1. solar
  2. Cosmic
3. Industrial
Internal noise
1. Thermal Agitation Noise
2. Shot Noise
3. Transit Time Noise
4. Flicker Noise
5. Miscellaneous Sources

Thermal noise

Thermal Agitation Noise: Also known as Johnson noise or White noise.

$P_n \propto\ T\;\delta\!f = k\,T\;\delta\!f$

where k = Boltzmann's constant = 1.38x10^-23J/K

T = absolute temperature, K = 273 + °C

δ f = bandwidth of interest

P_n = maximum noise power output of a resistor

$P_n = \frac{V^2}{R_L} = \frac{\left(\frac{V_n}{2}\right)^2}{R} = \frac{V_n^2}{4R}$
$V_n^2 = 4RP_n = 4RkT\;\delta\!f$
$V_n = \sqrt{4kT\;\delta\!f\;R}$

Shot Noise

Wikipedia has related information at
shot noise#In electronic devices

$i_n = \sqrt{2ei_p\;\delta\!f}$

where i_n = r.m.s. shot-noise current

e = charge of an electron = 1.6x10^-19C

i_p = direct diode current

δ f = bandwidth of system

Noise Calculations

Addition due to several sources

noise voltages:

$V_n1 = \sqrt{4kT\,\delta\!f\,R_1}$ , $V_n2 = \sqrt{4kT\,\delta\!f\,R_2}$ ...and so on, then $V_{n,tot} = \sqrt{{V_n1^2}+{V_n2^2}+...} = \sqrt{4kT\,\delta\!f\,R_{tot}}$

where R_tot = R₁+R₂+...

Addition due to Cascaded Amplifier stages

R_eq = R₁+R'₂
$R_{eq} = R_1 + \frac{R_2}{A_1^2} + \frac{R_3}{{A_1^2}{A_2^2}}$

Analog Noise Models

CMOS

BJT

Noise in digital circuits:

Methods of reducing noise

Differential signaling

Differential signaling is a method of transmitting information electrically by means of two complementary signals sent on two separate wires. The technique can be used for both analogue signaling, as in some audio systems, and digital signaling, as in RS-422, RS-485, PCI Express and USB.

Good grounding

An ideal signal ground maintains zero voltage regardless of how much electrical current flows into ground or out of ground.

References

Kennedy, George 'Electronic Communication Systems' , 3^rd Ed. ISBN 0-07-034054-4

Chapter 2: AC Circuits

AC Circuits

Relationship between Voltage and Current

Resistor

In a resistor, the current is in phase with the voltage always. This means that the peaks and valleys of the two waveforms occur at the same times. Resistors can simply be defined as devices that perform the sole function of inhibiting the flow of current through an electrical circuit. Resistors are commercially available having various standard values, nevertheless variable resistors are also made called potentiometers, or pots for short.

Capacitor

The capacitor is different from the resistor in several ways. First, it consumes no real power. It does however, supply reactive power to the circuit. In a capacitor, as voltage is increasing the capacitor is charging. Thus a large initial current. As the voltage peaks the capacitor is saturated and the current falls to zero. Following the peak the circuit reverses and the charge leaves the capacitor. The next half of the cycle the circuit runs mirroring the first half.

The relationship between voltage and current in a capacitor is: $i(t)=C{d(v(t)) \over dt}$ . This is valid not only in AC but for any function v(t). As a direct consequence we can state that in the real world, the voltage across a capacitor is always a continuous function of the time.

If we apply the above formula to a AC voltage (i.e. $v(t)=V \cdot sin(\omega t + \Phi)$ ), we get for the current a 90° phase shift: $i(t)=V \cdot \omega \cdot cos(\omega t + \Phi)$ .

In an AC circuit, current leads voltage by a quarter phase or 90 degrees. Note that while in DC circuits after the initial charge or discharge no current can flow, in AC circuits a current flows all the time into and out of the capacitor, depending on the impedance in the circuit. This is similar to the resistance in DC circuits, except that the impedance has 2 parts; the resistance included in the circuit, and also the reactance of the capacitor, which depends not only on the size of the capacitor, but also on the frequency of the applied voltage. In a circuit that has DC applied plus a signal, a capacitor can be used to block the DC, while letting the signal continue.

Inductor

In inductors, current is the negative derivative of voltage, meaning that however the voltage changes the current tries to oppose that change. When the voltage is not changing there is no current and no magnetic field.

In an AC Circuit, voltage leads current by a quarter phase or 90 degrees.

Voltage Defined as the derivative of the flux linkage:

$V(t)={d(Ni(t)) \over dt}$

Resonance

A circuit containing resistors, capacitors, and inductors is said to be in resonance when the reactance of the inductor cancels that of the capacitor to leave the resulting total resistance of the circuit to be equal to the value of the component resistor. The resonance state is achieved by fine tuning the frequency of the circuit to a value where the resulting impedance of the capacitor cancels that of the inductor, resulting in a circuit that appears entirely resistive.

Phasors

Sinusoids and Phasors

Sinusoidal signals can be represented as $A\sin({\omega}t+\varphi)$ where A is the amplitude, $ω$ is the frequency in radians per second, and $\varphi$ is the phase angle in radians (phase shift). The signal is completely characterized by A, $ω$ , and $\varphi$ .

Using Euler's formula,

$Ae^{j({\omega}t+\varphi)}=A\cos({\omega}t+\varphi)+jA\sin({\omega}t+\varphi) \,$

so

$A\cos({\omega}t+\varphi)=\mathfrak{Re}(Ae^{j({\omega}t+\varphi)})$

Note: In electrical engineering, the symbol j is used to denote the imaginary unit rather than the symbol i because i is used to denote current, especially small signal current.

A complex exponential can also be expressed as

$Ae^{j({\omega}t+\varphi)}=Ae^{j\varphi}e^{j{\omega}t}$

The quantity $\tilde{A}=Ae^{j\varphi}$ is a phasor. It contains information about the magnitude and phase of a sinusoidal signal, but not the frequency. This simplifies use in circuit analysis, since most of the time, all quantities in the circuit will have the same frequency. (For circuits with sources at different frequencies, the principle of superposition must be used.)

Another phasor notation is $\tilde{A}=A\angle\varphi$ . Note that this is simply a polar form, and can be converted to rectangular notation by:

$\tilde{A}=A\angle\varphi=A\cos\varphi+jA\sin\varphi$

and back again (with care to place the angle in the right quadrant) by:

$\tilde{A}=X+jY=\sqrt{X^2+Y^2}\angle{\tan^{-1}{\frac{Y}{X}}}$

For the moment, consider single-frequency circuits. Every steady state current and voltage will have the same basic form:
$\tilde{A_i}e^{j{\omega}t}$ where $\tilde{A_i}$ is a phasor. So we can "divide through" by $e j ω t$ to get phasor circuit equations. We can solve these equations for some phasor circuit quantity $\tilde{Y}$ , multiply by $e j ω t$ , and convert back to the sinusoidal form to find the time-domain sinusoidal steady-state solution.

This type of phasor circuit analysis requires knowledge of Electronics/Impedance.

Impedance

Definition

Impedance, $\tilde{Z}$ , is the quantity that relates voltage and current in the frequency domain. (The tilde indicates a phasor. An overscore or arrow may also be used.)

$\tilde{Z}=\frac{\tilde{V}}{\tilde{I}}=\frac{V\angle\phi_V}{I\angle\phi_I}=\frac{V}{I}\angle{\phi_V-\phi_I}$ .

In rectangular form,

$\tilde{Z}=R+jX$

where R is the resistance and X is the reactance. Impedance is generally a function of frequency, i.e.

$\tilde{Z}(\omega)=R(\omega)+jX(\omega)$

NOTE: ω = 2 π f

where f is the frequency in cycles per second (f=50 or 60 Hertz usually, depending on the country concerned. Aircraft systems often use 400 Hertz.)

Reactance

Reactance (symbol $X$ ) is the resistance to current flow of a circuit element that can store energy (ie. a capacitor or an inductor), and is measured in ohms.

The reactance of an inductor of inductance $L$ (in Henries), through which an alternating current of angular frequency $w$ flows is given by:

$X L = w L$

The reactance of a capacitor of capacitance $C$ (in Farads) is given similarly:

$X C = 1 / (w C)$

The two formulae for inductive reactance and capacitive reactance create interesting counterpoints. Notice that for inductive reactance, as the frequency of the AC increases, so does the reactance. Hence, higher frequencies result in lower current. The opposite is true of capacitive reactance: The higher the frequency of AC, the less reactance a capacitor will present.

Similarly, a more inductive inductor will present more reactance, while a capacitor with more capacitance will yield less reactance.

Resistors

Resistors have zero reactance, since they do not store energy, so their impedance is simply $\tilde{Z}=R$ .

Capacitors

Capacitors have zero resistance, but do have reactance. Their impedance is

$\tilde{Z}=\frac{1}{j{\omega}C}$

where C is the capacitance in farads. The reactance of one microfarad at 50 Hz is -3183 ohms, and at 60 Hz it is -2653 ohms.

it is store in the power.

Inductors

Like capacitors, inductors have zero resistance, but have reactance. Their impedance is

$\tilde{Z}=j{\omega}L$

where L is the inductance in henries. The reactance of one henry at 50 Hz is 314 ohms, and at 60 Hz it is 377 ohms.

Circuit Analysis Using Impedance

Analysis in the frequency domain proceeds exactly like DC analysis, but all currents and voltages are now phasors (and so have an angle). Impedance is treated exactly like a resistance, but is also a phasor (has an imaginary component/angle depending on the representation.)

(In the case that a circuit contains sources with different frequencies, the principle of superposition must be applied.)

Note that this analysis only applies to the steady state response of circuits. For circuits with transient characteristics, circuits must be analyzed in the Laplace domain, also known as s-domain analysis.

Steady State

That can be said to be the condition of "rest", after all the changes/alterations were made. This may imply, for examples, that nothing at all happens, or that a "steady" current flows, or that a circuit has "settled down" to final values - that is until the next disturbance occurs.

If the input signal is not time invariant, say if is a sinosoid, the steady state wont be invariant either. The response of a system can be considered to be composed of a transient response: the response to a disturbance, and the steady state response, in the absense of disturbance.

The transient part of the response tends to zero as time since a disturbance tends to infinity, so the steady state can be considered to be the response remaining as T -> infinity.

Electronics/Print Version

From Wikibooks, the open-content textbooks collection

Foreword

Aim of This Textbook

Prerequisites

Prerequisite Topics

Preface

Importance of Electronics

Charge and Coulomb's Law

Basic Understanding

Balance of Charge

Conductors

Insulators

Quantity of Charge

Force between Charges: Coulomb's Law

Basic Concepts

What is Electronics?

Electricity Basics

Plumbing Analogy

Cells

Cells

Resistors

Resistors

Load

Labeling (See also Identification)

Construction

ρ: Resistivity of the Material

L: Length

A: Cross-Sectional Area

Resistance Example

Properties of the material

Resistor Junctions

Resistors in Series

Resistors in Parallel

Combinations of series and parallel

Resistor variations

Applications

Specifications

Nominal Resistance

Tolerances

Power Rating

Operating Temperature

Maximum Voltage

Temperature Coefficient

Capacity and Inductance

Packaging

Related Wikimedia resources

Wikibooks

Wikipedia

Capacitors

Capacitors

Capacitance

Capacitor Labeling

Construction

Materials

Junctions

Capacitors in Series

Capacitors in Parallel

RC Circuit

Introduction

Basics

The Time Constant

Capacitor Specifications

Capacitor Variations

Inductors

Introduction

Important Qualities of Inductors

Inductance

Inductor's Voltage

Inductor's Current

Inductor's Impedance

Quality factor

Inductor's Network

Series Connection

Parallel Connection

See Also

Other Components

Ideal voltage sources

Ideal current sources

Dependent Sources