Getting started

A learner asked, "What math and science prerequisites—skills already achieved—might be considered the minimum for this electrical engineering course?"

Good question! Here is a summary of the math and science preparation that will help you have the best experience learning the electrical engineering topics taught on Khan Academy.

Math

Math fundamentals

• Equation of a line, slope, y-intercept
• Quadratic formula
• Exponents
• Radians
• Vectors, vector addition
• Solving simultaneous equations
• Complex numbers

Trigonometry

• Definitions of sine, cosine, and tangent from the sides of a triangle [[☃ explanation 3]]
• For further review, see trigonometry.

A few concepts from calculus

• Limits
• Derivative
• Derivative notation

These math fundamentals, and a little bit of terminology from calculus will get you all the way through resistor networks and circuit analysis methods.

Calculus

As we move beyond resistor circuits and start to include capacitors and inductors, we need calculus to understand how they work. Think of calculus as a corequisite in parallel with electrical engineering. You don't need to have a complete calculus background to get started, but it is helpful before too long. Many students learn calculus at the same time as introductory electrical engineering classes.

These are the calculus concepts we use in electrical engineering at Khan Academy:

• Derivative of simple functions (xⁿ where n is an integer )
• Derivative of eˣ
• Notation for integral
• Integral of xⁿ
• Evaluating definite integrals
• Derivative of sine and cosine
• Line integral

Calculus kicks in when we get to circuits involving time—circuits with capacitors or inductors. We have to use calculus to get a meaningful solution.

Differential equations:
When we solve first-order differential equations, we walk through the solution step by step (example: RC). The most advanced problems involve second-order differential equations, and again, we go through the solution step by step.

Electrostatics:
The electrostatics section has the most advanced topics we cover in electrical engineering. This sequence develops precise definitions of electric field and voltage. My goal is to have you appreciate (but not recreate) the derivations of voltage, and the field equations for point, line, and plane of charge.

Physics

• High school physics: force, energy, and work
• Force problems using vectors: here, and here.
• Scientific notation
• Significant figures

Chemistry

• High school chemistry: elements, atoms, the electron and nucleus, mole

Classics

Engineering equations make more sense if you recognize the Greek alphabet. [[☃ explanation 4]]

Summary

Welcome to electrical engineering on Khan Academy.
Good luck in your studies!

[[☃ explanation 1]]

Voltage and current are the cornerstone concepts in electricity. We will create our first mental models for these basic electrical quantities. We will also talk about power, which is what happens when voltage and current act together.

Charge

The concept of electricity arises from an observation of nature. We observe a force between objects, that, like gravity, acts at a distance. The source of this force has been given the name charge. A very noticeable thing about electric force is that it is large, far greater than the force of gravity. Unlike gravity, however, there are two types of electric charge. Opposite types of charge attract, and like types of charge repel. Gravity has only one type: it only attracts, never repels.

Conductors and insulators

Conductors are made of atoms whose outer, or valence, electrons have relatively weak bonds to their nuclei, as shown in this fanciful image of a copper atom. When a bunch of metal atoms are together, they gladly share their outer electrons with each other, creating a "swarm" of electrons not associated with a particular nucleus. A very small electric force can make the electron swarm move. Copper, gold, silver, and aluminum are good conductors. So is saltwater.

[[☃ image 2]]

There are also poor conductors. Tungsten—a metal used for light bulb filaments—and carbon—in diamond form—are relatively poor conductors because their electrons are less prone to move.

Insulators are materials whose outer electrons are tightly bound to their nuclei. Modest electric forces are not able to pull these electrons free. When an electric force is applied, the electron clouds around the atom stretch and deform in response to the force, but the electrons do not depart. Glass, plastic, stone, and air are insulators. Even for insulators, though, electric force can always be turned up high enough to rip electrons away—this is called breakdown. That's what is happening to air molecules when you see a spark.

Semiconductor materials fall between insulators and conductors. They usually act like insulators, but we can make them act like conductors under certain circumstances. The most well-known semiconductor material is Silicon (atomic number $14$). Our ability to finely control the insulating and conducting properties of silicon allows us to create modern marvels like computers and mobile phones. The atomic-level details of how semiconductor devices work are governed by the theories of quantum mechanics.

Current

Current is the flow of charge.

Charge flows in a current. [[☃ explanation 1]]

Current is reported as the number of charges per unit time passing through a boundary. Visualize placing a boundary all the way through a wire. Station yourself near the boundary and count the number of charges passing by. Report how much charge passed through the boundary in one second. We assign a positive sign to current corresponding to the direction a positive charge would be moving.

[[☃ image 1]]

Since current is the amount of charge passing through a boundary in a fixed amount of time, it can be expressed mathematically using the following equation:

$i = \dfrac{dq}{dt}$

[[☃ explanation 2]]

That's current in a nutshell.

A few remarks on current

What carries current in metal? Since electrons are free to move about in metals, moving electrons are what makes up the current in metals. The positive nuclei in metal atoms are fixed in place and do not contribute to current. Even though electrons have a negative charge and do almost all the work in most electric circuits, we still define a positive current as the direction a positive charge would move. This is a very old historical convention.

Can current be carried by positive charges? Yes. There are lots of examples. Current is carried by both positive and negative charges in saltwater: If we put ordinary table salt in water, it becomes a good conductor. Table salt is sodium chloride, NaCl. The salt dissolves in water, into free-floating Na$^+$ and Cl$^-$ ions. Both ions respond to electric force and move through the saltwater solution, in opposite directions. In this case, the current is composed of moving atoms, both positive and negative ions, not just loose electrons. Inside our bodies, electrical currents are moving ions, both positive and negative. The same definition of current works: count the number of charges passing by in a fixed amount of time.

What causes current? Charged objects move in response to electric and magnetic forces. These forces come from electric and magnetic fields, which in turn come from the position and motion of other charges.

What is the speed of current? We don't talk very often about the speed of current. Answering the question, "How fast is the current flowing?" requires understanding of a complex physical phenomenon and is not often relevant. Current usually isn't about meters per second, it's about charge per second. More often, we answer the question "How much current is flowing?" all the time.

How do we talk about current? When discussing current, terms like through and in make a lot of sense. Current flows through a resistor; current flows in a wire. If you hear, "the current across ...", it should sound odd.

Voltage

To get our initial toehold on the concept of voltage, let's look at an analogy:

Voltage resembles gravity

For a mass $m$, a change of height $h$ corresponds to a change in potential energy, $\Delta U = mg\Delta h$.

For a charged particle $q$, a voltage $V$ corresponds to a change in potential energy, $\Delta U = qV$.

Voltage in an electric circuit is analogous to the product of $g\cdot \Delta h$. Where $g$ is the acceleration due to gravity and $\Delta h$ is the change of height.

A ball at the top of the hill rolls down. When it is halfway down, it has given up half of its potential energy.

An electron at the top of a voltage "hill" travels "downhill" through wires and elements of a circuit. It gives up its potential energy, doing work along the way. When the electron is halfway down the hill, it has given up, or "dropped", half of its potential energy.

[[☃ image 2]]

For both the ball and the electron, the trip down the hill happens spontaneously. The ball and electron move towards a lower energy state all by themselves. On the trip down, there can be things in the way of the ball, like trees or bears to bounce off. For electrons, we can guide electrons using wires and make them flow through electronic components —circuit design— and do interesting things along the way.

[[☃ explanation 3]]

[[☃ explanation 5]]

[[☃ explanation 4]]

We can express the voltage between two points mathematically as the change of energy experienced by a charge:

$V = \dfrac{\Delta U}{q}$

That's an intuitive description of voltage in a nutshell.

Power

Power is defined as the rate energy ($\text U$) is transformed or transferred over time. We measure power in units of joules/second, also known as watts.

($1 ,\text{watt} = 1,\text{joule}/\text{second}$)

$\text{power} = \dfrac{\text dU}{\text dt}$

An electric circuit is capable of transferring power.
Current is the rate of flow of charge, and voltage measures the energy transferred per unit of charge. We can insert these definitions into the equation for power:

$\text{power} = \dfrac{\text dU}{\text dt} = \dfrac{\text dU}{\text dq} \cdot \dfrac{\text dq}{\text dt} = v ,i $

Electrical power is the product of voltage times current. in units of watts.

Summary

These mental models for current and voltage will get us started on all sorts of interesting electric circuits.

If you want to reach beyond this intuitive description of voltage you can read this more formal mathematical description of electric potential and voltage.

[[☃ explanation 1]]

Electrical engineers come across very large and very small numbers compared to everyday experience. This article gives you an initial exposure to large and small numbers and has examples of how they show up in engineering applications.

Engineering numbers are written in engineering notation, similar to scientific notation. It helps to get comfortable with engineering notation and the wide, dynamic range of numbers that engineers deal with on a regular basis.

Scientific notation

If you've studied math or science, you have probably run across scientific notation. You can brush up on scientific notation with this video. To express a number in scientific notation, you rewrite it as a number $\ge1$ and $\lt10$, multiplied by a power of $10$. It might make more sense if we look at some examples:

• Avogadro's Number looks like this in scientific notation:
  $6.02214082 \times 10^{23}$. You may see the same number in computer syntax like this: $6.02214082\text { E}23$, where "E" for "Exponent" stands in for "$\times10 \text{ to the ...}$".

• The speed of light is $299792458$ meters per second. This is expressed in scientific notation as $2.99792458 \times 10^{8} ,\text{m/s}$ and may be rounded to fewer digits like this: $3.00 \times 10^8,\text{m/s}$.

• The charge on an electron is a tiny, unwieldy number. $0.00000000000000000016021766208$ coulombs. Rather than writing all those zeros—and getting it wrong most of the time—we can use scientific notation to write the number more simply: $1.6021766208 \times 10^{−19}$ coulombs. [[☃ explanation 1]]

Engineering notation

The habit in engineering is to use a slightly modified scientific notation. Engineers like exponents in multiples of three. This means the digits to the left of the decimal point fall in the range of one to 999. Our minds do a pretty good job relating to numbers in this range.

Engineering notation is only slightly different than scientific notation.

It takes light 0.0000333564095 seconds to travel 10 kilometers in a vacuum. Let's convert this small number into engineering notation:

• Find the decimal point.
• Hop over three digits at a time, going right, until you hop over one, two, or three nonzero digits. In this case, take two hops to the right, until you hop over $33$.
• Write down $33$.
• Add a decimal point: $33.$
• Write down the remaining digits: $33.3564095$.
• Because we hopped right, finish by writing $10$ raised to the negative number of hops times three: $-2 ,\text{hops} \times 3 = -6$.

$33.3564095 \times10^{-6}$ seconds is the time it takes for light to travel 10 kilometers in a vacuum, in engineering notation.

A few more examples of engineering notation:

• Speed of light: $300 \times 10^6,\text{m/s}$
• A blink of an eye: ${\sim} 350 \times 10^{-3},\text s$

The number format rules are not rigid. As long as the point you are trying to make is clear and unambiguous, you may make exceptions. The blink of an eye may be clearly written 0.350 seconds if you intend the reader to compare the value to one second.

One flaw in engineering notation is that it can mislead about the number of significant figures. Engineers generally deal with wide tolerances of manufactured components, so the number of significant figures in circuit designs is usually small: two to three. If the tolerance is important, it is common to write it next to the number, as shown in this example:

A large resistance value: $33.3 \times 10^6 ,\Omega::\pm1%$.

Over time, you will develop a feel for numerical accuracy and rounding in different situations. When done appropriately, rounding to a few digits is not a sign of laziness, but a realization that real-world components are not all the same—and yet your design still has to work every time. There are other instances, such as during long calculations using computer arithmetic, where even tiny rounding errors are important to anticipate and control. It all depends on the situation. This is the engineering art.

Prefixes

Many numbers have names derived from Greek or Latin. Engineers and scientists use Système International d'Unités (SI) number prefixes.

Some of the most common prefixes in engineering are listed below. Notice that the exponents are multiples of three. These prefixes are shorter and easier to say or abbreviate than the numerical equivalent: "$\times10 \text{ to the ...}$".

Do engineers really deal with numbers this large and small?

Yes! Below are examples of large, medium and small numbers used in real-world electrical systems. These examples are common occurrences, and you can always find greater extremes.

Frequency: Frequency counts the number of times something happens per second—or another unit of time. The SI unit for frequency is hertz (Hz), which is the same as $1/s$. You could also say "reciprocal second" or "per second". The internal clock of a modern personal computer runs at a frequency of around 3 GHz $(3 \times 10 ^{9},\text{Hz})$. This corresponds to a clock period—the amount of time between clock ticks—of $1/(3 \times 10^9)$ or 333 ps $(333 \times 10^{-12} ,\text{s})$. A human heart beats about one time per second (1 Hz), as detected by an electrocardiogram (ECG) machine.

Resistance: Resistance is measured in units of ohms $(\Omega)$. The resistance of a wire is often much less than one ohm. Resistance up to tens of megohms $(10\times10^{6}, \Omega)$ is not unusual.

Voltage: The unit of electrical potential is the volt (V). A flashlight battery is 1.5 volts. You can hold this battery in your hand without fear of electric shock. Inside a computer, the chips usually operate with 3 to 5 volts. A car battery is 12 volts. A wall socket is 110 or 220 volts, depending on where you live. This voltage can be fatal if you touch it with bare hands. High-tension power lines overhead are hundreds of thousands of volts—tension is the French word for voltage. As for tiny voltages, wireless signals are measured in microvolts $(10^{-6},\text{V})$ when detected by a radio or mobile phone receiver.

Current: Currents are measured in amperes (A). One ampere is a large current. Car batteries momentarily supply 100 amperes or more to start a car. A house may consume 150 amperes if everything is turned on. Currents can also be absurdly small. There are situations where 1 femtoamp $(1 \times 10^{-15} ,\text{A})$ matters.

Time: Electrical circuits are capable of working at very short time scales. Time intervals in electronics range from 1 second, for the heartbeat example above, down to 1 picosecond $(1\times10^{-12},\text{s})$.

Capacitance: Capacitance has units of farads (F). A farad is defined as a coulomb per volt. Since a coulomb is such a large amount of charge, a farad is a large unit of capacitance. As a result, capacitance values you come across are tiny numbers. 100 microfarads is a large capacitance. If you twist two 1-inch (2- centimeter) pieces of ordinary insulated hookup wire together, those wires have a capacitance around one picofarad $(1 \times 10^{-12} ,\text F)$. [[☃ explanation 2]]

Distance and Length: Distance and length have units of meters. We deal with huge distances and tiny lengths on a regular basis. Nature gives us some staggering distances—light travels at $300 \times 10^{6}$ meters (300 million meters) in one second. Modern microelectronics blesses us with astoundingly small dimensions inside integrated circuits. Today's most aggressive—and expensive—integrated-circuit processes have dimensions as small as 15 nanometers $(15 \times 10^{-9},\text m)$. That's 15 billionths of a meter!

Unit grammar

These are the grammatical guidelines for writing unit names and symbols.

• Names of all units start with a lowercase letter, even if the unit is named after a person.
• Symbols for units are uppercase if the unit is named after a person, otherwise lowercase.

Symbol name | example names | Symbol| example symbols | Named after
:-:|-|:-:|-|
second | 1 millisecond | $\text {s}$ | $2 \text { ns}$ |
meter | 300 kilometer | $\text {m}$ | $35 \text { nm}$ |
hertz | 10 kilohertz | $\text {Hz}$ | $100 \text { MHz}$ | Hertz
ohm | 2 megohm | $\Omega$ | $47 \text { k}\Omega$ | Ohm
farad | 10 picofarad | $\text {F}$ | $220 \text { pF}$ | Faraday
ampere | 35 microamp | $\text {A}$ | $65 \text { mA}$ | Ampère
volt | 11 kilovolt | $\text {V}$ | $5 ,\mu \text {V}$ | Volta
|

Isn't it cool. Ohm gets a Greek symbol: $\Omega$, "Ohm ega."

The short form "amp" is a perfectly acceptable way to abbreviate "ampere".

The numbers you encounter as you study electrical engineering span a huge range. You will come across these numbers on Khan Academy, in textbooks, and in real-world electronic systems.

[[☃ explanation 1]]

Electrical units can be described in a formal manner, and that's what we do here. The standard electrical units are defined in a specific order. The ampere is defined first. It is an SI base unit, the only electrical unit derived from the outcome of an experiment.

Next up after the ampere comes the coulomb and charge on an electron. Then we derive the rest of our favorites, the watt, the volt, and the ohm. These derived electrical units are defined in terms of the ampere and other SI base units (meter, kilogram, second). [[☃ explanation 2]]

Ampere

The definition of the SI unit of current, the ampere, comes from the study of magnetism. Electric currents in wires give rise to magnetic fields (Biot–Savart Law, 1820). Those magnetic fields in turn give rise to magnetic forces on the wires (Ampere's Force Law, 1825). Two parallel wires carrying current exert a force on each other. The official SI definition of the ampere is:

The ampere is that constant current which—if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum—would produce between these conductors a force equal to $2 \times 10^{-7}$ newtons per meter of length.

The definition of the ampere comes from the outcome of an experiment. To create a standard $1$ ampere, you perform some version of the following experiment. Set up two 1-meter-long wires in parallel, and arrange for a way to measure the force on the wires (a strain gauge).

[[☃ image 6]]

Apply the same current to both wires, flowing in the same direction. Adjust the currents in the wires up or down while measuring the force on the wires. When the force is $2 \times 10^{-7}$ newtons, the current is 1 ampere, by definition.

The ampere was redefined in 2019.

[[☃ explanation 1]]

Coulomb

The coulomb is the SI unit of charge. The size of a coulomb is derived from the ampere. One coulomb is defined as the amount of charge flowing when the current is 1 ampere.

$1 ,\text{ampere} = 1,\text{coulomb}/\text{second}$

or equivalently,

$1 ,\text{coulomb} = 1,\text{ampere} \cdot ,\text{second}$

Electron charge

In 1897, J.J. Thomson proved the existence of the electron. Twelve years later, starting in 1909, Robert Millikan performed his oil drop experiments to establish the charge of the electron.

The charge on an electron can be expressed in coulombs as $e = -1.602176565 \times 10^{-19} ,\text {coulomb}$.

If we invert this expression, we see that the coulomb can be stated in terms of number of electron charges:

$1 ,\text{coulomb} = 6.241509343 \times 10^{18} ,\text{electrons}$

Concept check

** How many electrons in 1 ampere?**
[[☃ explanation 1]]

** How many coulombs in 1 mole of electrons?**
One mole of electrons is $6.02214 \times 10^{23}$ electrons — Avogadro's Number.
[[☃ explanation 2]]

Watt

The watt is the unit of power. Power is the amount of energy transferred or consumed per unit of time; equivalently, power is the rate of doing work. In standard-speak, the watt is the power which in one second gives rise to energy of 1 joule.

$1 ,\text{watt} = 1 ,\text{joule}/,\text{second}$

Volt

The volt is the unit of electric potential difference—electric potential difference is also known as voltage. The size of 1 volt is officially defined as the potential difference between two points of a wire carrying a current of 1 ampere when the power dissipated in the wire is 1 watt.

$1 ,\text{volt} = 1,\text{watt} / \text{ampere}$

The volt can also be expressed in terms of energy and charge as,

$1,\text{volt} = 1 ,\text{joule}/\text{coulomb}$

You can find an intuitive description of voltage in the introductory article on basic electrical quantities. Also, there is a formal derivation of the meaning of voltage in the electrostatics section.

Ohm

The ohm is the electrical unit of resistance. One ohm is defined as the resistance between two points of a conductor when 1 volt is applied and a current of 1 ampere is flowing.

$1 ,\text{ohm} = 1,\text{volt}/\text{ampere}$

We've now defined, in order, a basic set of our favorite electrical units.

Systems of Units

Over the last 200 years, there have been three main systems of scientific units:

• SI
• MKS
• cgs

** SI ** is the International System of Units—in French, Système International d'Unités. It is the modern form of the metric system and is the most widely used system of measurement. The system was published in 1960 as the result of discussions that started in 1948. SI is based on the metre-kilogram-second system (MKS). In the United States, the SI is used in science, medicine, government, technology, and engineering.

MKS is based on measuring lengths in meters, mass in kilograms, and time in seconds. MKS is generally used in engineering and beginning physics. It was proposed in 1901. The most familiar units of electricity and magnetism—ohm, farad, coulomb, etc.—are MKS units.

cgs is based on measuring lengths in centimeters, mass in grams, and time in seconds. It was introduced in 1874. The cgs system is commonly used in theoretical physics. The difference between the SI and cgs systems goes much deeper than a simple scaling of the units for length and mass.

There are seven SI base units.

SI base units

One SI base unit comes from electricity: the ampere. The ampere has the same lofty status as the meter, kilogram, and second. It is defined as its own thing, not in terms of other units.

[[☃ explanation 1]]

SI derived units used in electricity

The remaining electrical units are SI derived units, formed by combinations of the base units. If the ampere is the "first" electrical unit, these derived electrical units follow close behind.

[[☃ explanation 2]]

[[☃ explanation 1]]