Appearance
1.1 — Charge, Current, Voltage and Power
Rub a balloon on your hair and it sticks to a wall. Walk across a carpet in dry weather and you get a spark off a doorknob. Both are the same fact: some material has ended up with more electrons than it started with, and some has ended up with fewer, and the imbalance wants to be undone. Everything in this volume is a controlled version of that spark.
Nothing here assumes any earlier electronics. What it does assume is arithmetic with units, which Volume IV Chapter 1.1 sets out, and the idea of a rate of change, which is Volume II Chapter 5.2 — current is going to turn out to be a rate, and so is power.
1. Charge — the quantity that everything else is built from
Charge is a property that some particles carry, the way they carry mass. There are two kinds, and the names are historical accidents. Benjamin Franklin, around 1750, guessed that electricity was a single fluid that flowed from a surplus to a deficit, and he labelled the surplus "positive". He had a fifty-fifty chance and he got it backwards. The particle that actually moves in a wire is the electron, and it turned out to carry what Franklin's convention calls negative charge. We have been living with that inherited mistake ever since, and it is the source of one of the most persistent confusions in electronics, which we will settle in section 3.
The symbol for charge is Q (or q for a charge that changes with time), and the unit is the coulomb, written C, named after Charles-Augustin de Coulomb, who measured the force between charges in the 1780s using a torsion balance delicate enough to detect the twist of a silk thread.
One electron carries a charge of
e = 1.602 \times 10^{-19}\ \text{C}
Read aloud: "one point six oh two times ten to the minus nineteen coulombs." That exponent is worth pausing on. 10^{-19} means a decimal point followed by eighteen zeros and then the digits. To collect one single coulomb of charge you need
\frac{1}{1.602\times10^{-19}} \approx 6.24 \times 10^{18} \ \text{electrons}
which is 6.24 billion billion of them. A coulomb is an enormous number of electrons, and that is why the currents in ordinary circuits, which are a few coulombs per second at most, correspond to unimaginable numbers of particles in motion.
Charge is conserved. It is never created or destroyed, only moved. This is not a rule of thumb; it is one of the deepest conservation laws in physics, and it is the direct reason Kirchhoff's current law in Chapter 1.2 has to be true.
2. Current — charge in motion
Current is the rate at which charge passes a point. If Q coulombs flow past a cross-section of wire in t seconds at a steady rate,
I = \frac{Q}{t}
and when the rate is not steady, the honest definition is the derivative:
i(t) = \frac{dq}{dt}
Read aloud: "the current at time t equals dee q by dee t," meaning the instantaneous rate of change of charge with respect to time. If you have not met derivatives, the plain meaning is all you need: the current right now is how fast charge is passing right now.
The unit is the ampere, written A, after André-Marie Ampère. One ampere is one coulomb per second. So a current of 1 A through a wire means roughly 6.24\times10^{18} electrons cross every cross-section of that wire every second.
Turning the definition around, the total charge delivered between two times is the accumulation of current:
Q = \int_{t_1}^{t_2} i(t)\,dt
Read aloud: "Q equals the integral from t-one to t-two of i of t, dee t." In plain words: add up the current over every instant in the interval. This is the equation behind a battery rating. A phone battery marked 4000 mAh (milliamp-hours) is telling you it can supply 4000 milliamps for one hour, or 400 milliamps for ten hours, or 4 milliamps for a thousand hours — the product is what is fixed, because the product is charge.
Converting that to coulombs makes the size vivid:
4000\ \text{mAh} = 4\ \text{A} \times 3600\ \text{s} = 14400\ \text{C}
Nearly fifteen thousand coulombs sitting in your pocket, which is about 9\times10^{22} electrons waiting their turn.
How fast do the electrons actually move?
Slowly. Embarrassingly slowly. In a typical copper wire carrying a typical current, the average electron drift velocity is under a millimetre per second — you could out-walk it. The formula is
v_d = \frac{I}{nAq}
where n is the number of free electrons per cubic metre, A is the wire's cross-sectional area, and q is the electron charge. Copper has about 8.5\times10^{28} free electrons per cubic metre, an absurd density, and it is that density in the denominator that makes the velocity so small.
So why does the light come on the instant you flick the switch? Because the wire is already full of electrons everywhere along its length. You are not waiting for an electron to travel from the switch to the bulb. You are pushing on one end of a pipe that is already full of water; the push travels at nearly the speed of light even though the water crawls. The signal is the electric field propagating through the wire, at roughly two-thirds the speed of light in typical cable, and the electrons everywhere start shuffling almost simultaneously.
This distinction — the push travels fast, the carriers travel slowly — comes back in Chapter 7.7 when transmission lines make the propagation delay large enough to matter.
3. Conventional current versus electron flow, settled once
Here is the confusion that Franklin left us, stated plainly:
- Conventional current flows from the positive terminal, through the external circuit, to the negative terminal. This is what every circuit diagram, every arrow, every formula in this volume and every piece of engineering practice uses.
- Electrons actually drift the opposite way, from negative to positive, because they are negatively charged and are attracted to the positive terminal.
Both statements are true. They are not in conflict, because a negative charge moving left is electrically identical to a positive charge moving right. Every measurement you can make gives the same answer either way.
The practical rule: always use conventional current. Draw the arrow from plus to minus in the external circuit and never think about it again. The only place electron flow matters is inside semiconductor devices in Part 2, where we care about which kind of carrier is moving, and there we will name them explicitly.
4. Voltage — the push
Current needs a reason to flow. That reason is voltage, also called potential difference or electromotive force depending on the context, and it is the single most misunderstood quantity in the subject because people talk about voltage "at" a point when voltage is always between two points.
The definition is energetic. Voltage between two points is the energy per unit charge needed to move charge from one to the other:
V = \frac{W}{Q}
where W is work in joules and Q is charge in coulombs. The unit is the volt, written V, after Alessandro Volta, who built the first battery around 1800 by stacking discs of zinc and copper separated by brine-soaked cloth — the "voltaic pile". One volt is one joule per coulomb.
So a 9-volt battery is making a specific promise: every coulomb of charge that travels from its negative terminal to its positive terminal inside the battery gains 9 joules of energy, and it will give those 9 joules back to whatever circuit it travels through on the way round.
The gravity analogy, and where it breaks
Height is the standard analogy and it is a good one. Voltage is like height, current is like the flow rate of water, and a resistor is like a narrow pipe. Water at the top of a hill has potential energy; let it flow down and the energy is released.
The analogy earns its keep on three specific points:
- Height is only meaningful as a difference. "This point is 3 metres up" is meaningless until you say up from what. The same is true of voltage, which is why every circuit needs a reference point, called ground or the common node, arbitrarily declared to be 0 V. Ground is not magic and it is not the earth. It is a choice.
- Going round a closed loop returns you to the same height. That is Kirchhoff's voltage law, Chapter 1.2.
- The water that leaves a junction equals the water that arrives. That is Kirchhoff's current law.
Where it breaks: water can pile up in a tank, but charge in a circuit essentially cannot, and a battery is not a pump that raises water so much as a chemical reaction that separates charge. Push the analogy past those three points and it will mislead you.
5. Power — the rate of energy transfer
Combine the definitions. Voltage is joules per coulomb. Current is coulombs per second. Multiply them:
\frac{\text{joules}}{\text{coulomb}} \times \frac{\text{coulombs}}{\text{second}} = \frac{\text{joules}}{\text{second}}
The coulombs cancel and what is left is joules per second, which is the definition of power. The unit is the watt, written W, after James Watt of steam-engine fame.
\boxed{P = VI}
This is the most-used formula in all of electrical engineering, and it fell straight out of the units without any physics being added. Power is voltage times current. That is why a 230 V mains kettle drawing 9 A consumes 230\times9 = 2070 W, and why the same kettle on a 115 V supply would take twice the current for the same power.
Energy is power accumulated over time:
E = \int P\,dt = \int vi\,dt
and for constant power it is simply E = Pt. The electricity meter on your house measures energy, not power, in kilowatt-hours: one kilowatt sustained for one hour, which is 1000 \times 3600 = 3.6 million joules.
Sign conventions — who is supplying and who is absorbing
You need a rule to tell a source from a load, and the rule is the passive sign convention:
If current enters the terminal you have marked as positive, the element is absorbing power, and P = VI comes out positive. If current leaves the positive terminal, the element is supplying power and P comes out negative.
A resistor always absorbs. A battery being discharged supplies, so its computed power is negative. A battery being charged absorbs, so it is positive. This single convention keeps the bookkeeping honest in every circuit you will ever analyse, and it is why a correct analysis always has the supplied powers and absorbed powers summing to zero. That sum is conservation of energy, and it is the best available check that your arithmetic is right.
6. Resistance and Ohm's law
Push charge through a material and the material pushes back. The carriers collide with the vibrating atoms of the lattice, losing energy to heat. The measure of that opposition is resistance, symbol R, unit the ohm, written \Omega (the Greek capital omega), after Georg Ohm.
Ohm published in 1827 the observation that for many materials, at a fixed temperature, the current is directly proportional to the applied voltage:
\boxed{V = IR}
Read: the voltage across a resistor equals the current through it multiplied by its resistance. Rearranged, I = V/R and R = V/I.
Note carefully what kind of statement this is. Ohm's law is not a law of nature. It is a description of a class of materials — the ones we call ohmic — that happen to have a straight-line relationship between voltage and current. Metals at constant temperature are ohmic. A diode is not, a filament lamp is not once it heats up, and a transistor is emphatically not. Part 2 is almost entirely about non-ohmic devices, and the whole reason they are interesting is that they break this straight line.
Ohm was ridiculed for the result. His colleagues thought a relationship so simple could not describe something as mysterious as electricity, and he lost his teaching post over it. It took about fifteen years for the work to be accepted; the Royal Society gave him the Copley Medal in 1841.
What resistance depends on
For a uniform piece of material of length \ell and cross-sectional area A:
R = \rho\,\frac{\ell}{A}
where \rho (Greek letter rho) is the resistivity, a property of the material itself, in ohm-metres. Long and thin means high resistance; short and fat means low. Exactly like a pipe.
Copper's resistivity is about $1.68\times10^{-8}\ \Omega!\cdot!$m. Silicon's, pure, is around 2.3\times10^{3} — eleven orders of magnitude higher. That gap is what the word semiconductor is pointing at, and Part 2 shows how doping moves silicon anywhere you like along that scale, which is the entire foundation of the electronics industry.
Resistivity changes with temperature. In metals it rises as things get hot, because the atoms vibrate more and the electrons collide more often. In semiconductors it falls with temperature, because heat frees more carriers. That opposite sign is the reason a filament bulb draws a big surge of current at switch-on when it is still cold, and it is also the reason a power transistor can destroy itself through thermal runaway, which Chapter 2.3 will make concrete.
The three power formulas
Substituting Ohm's law into P=VI gives two more forms, and all three are worth having ready:
P = VI = I^2R = \frac{V^2}{R}
The middle one, P = I^2R, is the important one for anything involving wires and heat. Power lost in a cable rises as the square of the current. Double the current, quadruple the heat. This one fact decided how the entire world's electricity is distributed: to move power at low loss you want low current, and since P=VI fixes the product, low current means high voltage. That is why transmission lines run at hundreds of kilovolts and why the transformer, Chapter 9.1, made alternating current win the argument against Edison's direct current.
7. Sources — where the energy comes in
A circuit needs something driving it. Circuit theory idealises the drivers into four kinds, and drawing the distinction properly now saves confusion in every chapter that follows.
An ideal voltage source maintains a fixed voltage across its terminals no matter what current is drawn. Its symbol is a circle with plus and minus signs, or a battery's long-and-short line pair. If you short-circuit an ideal voltage source, the current becomes infinite, which tells you immediately that it is a fiction.
An ideal current source forces a fixed current through itself no matter what voltage that requires. Its symbol is a circle with an arrow inside. If you open-circuit an ideal current source, the voltage becomes infinite. Also a fiction, but a useful one — a transistor in its active region behaves remarkably like one, which is why Part 2 leans on it.
Below, the four source symbols you will meet on every schematic in this volume, plus what happens to a real source when you load it.
Read the figure left to right. The blue circle is an ideal voltage source, holding its voltage regardless of load. The purple circle is an ideal current source, pushing its current regardless of the voltage required. The middle group is the honest model of a real source: an ideal source in series with a small internal resistance R_s. The graph on the right is what that internal resistance does — as you draw more current, more voltage is lost inside the source itself, so the voltage available at the terminals droops along a straight line whose slope is -R_s.
This is why a car struggles to start with a weak battery while its headlights still look fine. The headlights draw a few amps and the sag is small. The starter motor draws two hundred, and even 0.02\ \Omega of internal resistance costs 200 \times 0.02 = 4 V, which drags a 12 V battery down to 8 V and the motor turns over slowly. The battery has not lost its voltage; it has gained internal resistance as it aged.
Dependent sources
The fourth kind is the dependent source (or controlled source), drawn as a diamond rather than a circle. Its value is set by a voltage or current elsewhere in the circuit, not by a fixed number. There are four flavours, named by what controls what: voltage-controlled voltage source, voltage-controlled current source, current-controlled voltage source, current-controlled current source.
They look like an abstraction invented to torture students, and then Chapter 2.3 reveals what they are for: a transistor's small-signal model is a dependent source. A bipolar transistor's collector current is controlled by its base current, which is exactly a current-controlled current source. A MOSFET's drain current is controlled by its gate voltage, which is a voltage-controlled current source. Every amplifier analysis you will do in Part 2 comes down to solving a circuit that contains one of these diamonds.
8. The five passive elements, previewed
Everything in Part 1 is built from a short list. Three are the classical passives, and it is worth seeing their defining equations side by side before meeting them properly, because the pattern is the point.
| Element | Defining relation | Stores | Unit |
|---|---|---|---|
| Resistor | v = iR | nothing — dissipates | ohm, \Omega |
| Capacitor | i = C\,\dfrac{dv}{dt} | energy in electric field | farad, F |
| Inductor | v = L\,\dfrac{di}{dt} | energy in magnetic field | henry, H |
The resistor's relation involves no derivative, so it has no memory: the current now depends only on the voltage now. The other two involve a derivative, which means they have memory, and memory is what makes a circuit able to delay, filter, oscillate and ring. Chapter 1.5 develops both fully.
Notice also the symmetry between the last two rows. Swap v with i and C with L and the two equations turn into each other. That is not a coincidence; it is the duality that runs through all of circuit theory, and it means every result you prove about capacitors has a mirror-image twin about inductors that you get for free.
9. The unit prefixes you must be able to read instantly
Electronics spans an absurd range of magnitudes — picofarads to farads is twelve orders of magnitude — so the prefixes are not optional vocabulary.
| Prefix | Symbol | Multiplier | Typical use |
|---|---|---|---|
| pico | p | 10^{-12} | capacitance, stray effects |
| nano | n | 10^{-9} | capacitance, gate delays |
| micro | µ | 10^{-6} | capacitance, currents |
| milli | m | 10^{-3} | currents, volts |
| kilo | k | 10^{3} | resistance |
| mega | M | 10^{6} | resistance, frequency |
| giga | G | 10^{9} | frequency |
Two traps. First, capital M is mega and lowercase m is milli, differing by a factor of a billion, and writing one for the other is the classic student error that produces answers off by nine orders of magnitude. Second, in resistor markings the prefix is often written in place of the decimal point to survive smudged printing: 4k7 means 4.7 kΩ, 1R2 means 1.2 Ω, 2M2 means 2.2 MΩ.
Everything so far has described single elements in isolation. The moment you wire two of them together, you need rules for what happens at the junctions and around the loops — and those two rules turn out to be enough to solve any circuit ever built. That is Chapter 1.2.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
The base definitions
Q = It \qquad\text{more honestly}\qquad i = \frac{dq}{dt}, \qquad Q = \int i\,dt
Current is the rate charge passes a point. The integral form is the definition; the product form is what it becomes when the current is constant.
V = \frac{W}{Q}
Voltage is energy per unit charge. One volt is one joule per coulomb. Everything else about voltage follows from this one line.
P = \frac{dW}{dt} = \frac{dW}{dQ}\cdot\frac{dQ}{dt} = VI
Derivation of P = VI: power is energy per second; voltage is energy per coulomb; current is coulombs per second. Multiply the last two and the coulombs cancel.
E = \int P\,dt = Pt \ \text{(constant power)}
Ohm's law and resistance
V = IR, \qquad I = \frac{V}{R}, \qquad R = \frac{V}{I}
An empirical description of ohmic materials, not a law of nature.
R = \rho\frac{\ell}{A}
Where it comes from: double the length and you double the number of collisions a carrier must survive, so double the resistance. Double the cross-section and you provide two parallel paths, so halve it. Resistance is therefore proportional to \ell and inversely proportional to A, and \rho is the constant of proportionality that carries the material's identity.
R_T = R_0\left[1 + \alpha(T-T_0)\right]
The linear temperature model, with \alpha positive for metals and negative for semiconductors and thermistors.
The three power forms
P = VI
P = (IR)I = I^2R \quad\text{(substituting } V = IR)
P = V\cdot\frac{V}{R} = \frac{V^2}{R} \quad\text{(substituting } I = V/R)
Use I^2R when you know the current — cables, fuses, transistor dissipation. Use V^2/R when the voltage is fixed — heaters and lamps on a mains supply.
What the next chapter fixes
Everything so far has been a single component with a single voltage across it. Real circuits are meshes of them, and the moment there is more than one loop you need a rule for how the currents divide and how the voltages add up around a path. Chapter 1.2 gives the two laws that answer both questions, and they are the only two laws the whole of circuit analysis rests on.