1. The Cast of Characters

Five basic elements show up in nearly every electronic schematic. Get comfortable with each one's water-flow analog and you will already understand most of what circuits do.

1.1 The resistor: friction in the wire

A resistor drops voltage in proportion to the current passing through it:

$V = IR$

where $V$ is in volts, $I$ in amperes, and $R$ in ohms (Ω). This is Ohm's law, named after the German physicist Georg Ohm who measured it in 1827. It is the most-quoted equation in electronics.

Water-pipe analogy. Think of voltage as water pressure, high pressure at one end, lower at the other, water "wanting" to flow from high to low. Think of current as the actual flow rate (gallons per minute). A resistor is a pinch in the pipe, friction. Narrower pipe, more friction, less flow at the same pressure.

Push the analogy farther: a 1 Ω resistor at 1 V produces 1 A, a generous flow at modest pressure. Crank the pressure to 10 V and the flow becomes 10 A. Same pipe, ten times the pressure, ten times the flow. Linear. That is what makes the resistor easy to think about.

The product of voltage across and current through a resistor, the power dissipated, comes out as heat:

$P = VI = I^2 R = V^2/R$

All three forms (each from substituting Ohm's law into $P = VI$) are interchangeable. The $I^2 R$ form matters most: power grows as the square of current. Twice the current, four times the heat. This is why long-distance power transmission runs at hundreds of kilovolts: at fixed power, doubling voltage halves current and quarters resistive losses.

Real resistors come in standard values (E12 series), standard tolerances (5%, 1%, 0.1%), and various power ratings and materials (carbon film, metal film, wire-wound, thin film), trading cost, accuracy, temperature stability, and parasitic inductance. A modern smartphone has thousands of resistors.

Real-world examples of resistors at work:

• Current-limiting an LED. With 5 V supply and a red LED that wants 10 mA at 2 V, the resistor handles 3 V at 10 mA: $R = 300$ Ω, so use 330 Ω. Forget the resistor and the LED dies.
• Pull-up/pull-down resistors on logic-level signals. A floating MCU pin reads garbage; a 10 kΩ pull-up to VDD ties it high when nothing drives it. Buttons, I²C buses, open-drain outputs all rely on pulls.
• Voltage divider. Two series resistors with output at the midpoint. Used to attenuate a sensor for an ADC, set transistor bias, or scale signals before precision measurement.
• Sense resistor. A small mΩ resistor in series with a load turns load current into voltage. Found in every battery monitor, motor controller, and load tester. The same architecture is exactly what an attacker uses to capture a chip's power trace: a tiny shunt in the ground return turns "instantaneous current" into "voltage we can sample with a scope."

1.2 Series and parallel resistors

When resistors are wired end-to-end (series), the current must flow through both, and the voltages add. By Ohm's law:

$R_{series} = R_1 + R_2 + R_3 + \cdots$

When wired side-by-side (parallel), the voltage across both is the same and the currents add:

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

For just two resistors in parallel, this rearranges to the famous "product over sum":

$R_{parallel} = \frac{R_1 R_2}{R_1 + R_2}$

Two pipes in parallel. Imagine two pipes side-by-side, each with its own friction, both connecting the same two reservoirs. Water can take either pipe. The combined flow is more than either pipe alone (more cross-section means less effective friction). So parallel resistance is smaller than either individual resistance. It is a counterintuitive direction at first, but the math forces it.

Why does adding more resistors lower the resistance? Because each parallel branch is another path for current to leave. Two paths conduct twice as easily as one. Conductance (the reciprocal of resistance) adds in parallel, so resistance shrinks.

Useful number sense: two equal parallel resistors equal half of one. A 1 kΩ in parallel with a 1 MΩ is essentially 1 kΩ; a 1 kΩ in parallel with a 1 Ω is essentially 1 Ω. The smaller resistor dominates.

1.3 The capacitor: energy stored in an electric field

A capacitor is two conductive plates separated by an insulator (the dielectric). Apply voltage; charge accumulates on the plates, positive on one, negative on the other, separated by the dielectric. The accumulated charge is proportional to the applied voltage:

$Q = CV$

where $C$ is the capacitance, in farads (F). One farad equals one coulomb per volt. Practical capacitors range from femtofarads (fF, 10⁻¹⁵, parasitic on PCB traces) to farads or even kilofarads (supercapacitors that store enough energy to start a small car).

Differentiate: $i = C \, dv/dt$. The current through a capacitor is proportional to the rate of change of voltage across it. This is the defining behavior. Two crucial consequences:

1. DC: no current flows through a capacitor. $dv/dt = 0$ in steady state, so $i = 0$. The cap looks like an open circuit.
2. AC: the cap conducts more readily at higher frequencies. The faster the voltage changes, the bigger the current for a given voltage swing. So a cap looks like a low impedance to high-frequency signals and a high impedance to low frequencies.

Stretchy-balloon analogy. Imagine a balloon stretched across a pipe. Push water against the balloon and it stretches, storing pressure in the rubber tension. No water flows through the balloon. Water just accumulates on one side as the balloon distends. Pump in the opposite direction and the balloon bulges the other way, momentary flow as it changes shape, then no flow once it has stretched to match the pressure.

Now imagine you push and pull rapidly: the balloon bulges back and forth, water sloshes through it as if the balloon were not there. Slowly: water just sits, no flow. Faster: flow happens. Same as a cap: high frequency means high effective conductance.

The energy stored in a capacitor:

$E = \frac{1}{2}CV^2$

A 1 mF cap charged to 10 V holds 50 mJ, enough energy to make a satisfying spark across a screwdriver, and enough to deliver a noticeable shock if you grab the leads with damp hands. Big bulk caps are dangerous; always discharge them through a resistor before working on a circuit.

Series and parallel:

$\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots, \qquad C_{parallel} = C_1 + C_2 + \cdots$

Note: capacitors combine opposite to resistors. Two equal capacitors in parallel give double the capacitance; in series, half. That is because parallel caps share the voltage and add their charge storage, while series caps share the charge and add their voltage drops.

Real-world examples:

• Decoupling capacitors next to every IC's VDD pin (typically 100 nF + 1 µF + 10 µF). Logic gates draw amperes-for-nanoseconds transients at clock edges; without local caps, the inductive supply path makes VDD dip. The decoupling cap is a local energy reservoir that holds VDD stable.
• Bulk capacitors at the output of a rectifier, smoothing lumpy DC into clean DC. (Chapter 1.)
• Coupling caps between amplifier stages: short to AC, open to DC, so signal passes but bias does not.
• Touch screens. Your finger forms a small capacitor with each grid line; the controller measures intersection capacitance to find where you touched.
• Quartz crystal oscillators include load caps that, with the crystal, set the resonant frequency.
• Class-D audio amplifiers use output caps + inductors to integrate PWM into clean audio.
• Camera flashes and defibrillators slowly charge a high-voltage cap, then dump it into a bulb (or a heart) in milliseconds.

1.4 The inductor: energy stored in a magnetic field

An inductor is a coil of wire (sometimes wound around an iron or ferrite core). When current flows through the coil, it creates a magnetic field that stores energy. From Faraday's law, changes in the current produce a voltage across the coil:

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

where $L$ is the inductance in henries (H). The math is the dual of the capacitor's $i = C \, dv/dt$: where the cap resists changes in voltage, the inductor resists changes in current.

Heavy-turbine analogy. Imagine a heavy turbine in a pipe. To start the water flowing, you have to push hard against the turbine's inertia, but once it is up to speed, the water flows easily. To stop it, you have to absorb the kinetic energy of the spinning turbine. Now apply a step in pressure (voltage): water (current) takes time to ramp up because the turbine has inertia. Same with the inductor's magnetic field: it stores energy proportional to current squared.

Energy stored:

$E = \frac{1}{2}LI^2$

Series and parallel: combine like resistors (same formulas), unlike caps. So series inductors add; parallel inductors combine reciprocally.

The inductor's killer feature is its absolute hatred of changing current. Try to abruptly interrupt the current to an inductor (say by opening a switch) and you get a giant voltage spike. $v = L \, di/dt$, and if $di/dt$ is enormous (current dropping to zero in nanoseconds), $v$ is enormous too. This is how spark plugs work (an ignition coil, basically a big inductor, has its current interrupted, the resulting voltage spike jumps a gap in the spark plug). It is also how relays kill MCU pins if you forget the flyback diode: the diode safely catches the spike before it can fry the driver.

Real-world examples:

• Switch-mode power supplies store energy in an inductor briefly, then release it at a different voltage. Laptop chargers shrunk from 2 kg of iron transformer to 100 g of switch-mode supply with a tiny ferrite inductor.
• Wireless charging pads are coils on each side, not in contact but magnetically coupled.
• AM radio tuning circuits use a tank ($L$ parallel with variable $C$) resonating at $\omega = 1/\sqrt{LC}$ to pick one station from the air.
• Ferrite beads on cables (those black cylinders) are small inductors suppressing high-frequency interference.
• Speakers are voice coils in a permanent magnetic field; drive current and Lorentz force pushes the cone.

1.5 Voltage and current sources

We also need sources, the elements that supply energy.

An ideal voltage source maintains its voltage regardless of current drawn. A real battery has internal resistance (an ideal source in series with a small $R$); a 9 V battery's ~1 Ω of internal resistance makes it sag to 8 V or less under heavy load.

An ideal current source maintains its current regardless of voltage. Photodiodes, BJTs in active region, and MOSFETs in saturation behave like current sources over wide voltage ranges.

Beyond independent sources, circuits use dependent sources whose value depends on another voltage or current in the circuit:

• VCVS (voltage-controlled voltage source): the classic op-amp model.
• VCCS: models a MOSFET's drain current as a function of $V_{GS}$.
• CCVS (current-controlled voltage source).
• CCCS: models a BJT's collector current as $\beta I_B$.

Op-amps, amplifiers, and transistors all hide dependent sources inside their small-signal models.

1.6 Source transformation

Here is a useful trick: any voltage source $V$ in series with a resistor $R$ is electrically equivalent to a current source $V/R$ in parallel with the same resistor $R$. They behave identically when measured from the outside.

   ─[V]──[R]──    ⇔    ──[I = V/R]──[R]──
                                    │
                                   GND

This is just Ohm's law restated, but it is incredibly useful: sometimes a problem is easier to solve in one form than the other. Mesh analysis prefers voltage sources; nodal analysis prefers current sources. The transformation lets you flip between them without changing the answer.

A subtle but important point: the transformation is valid only as seen from outside the box. Internally, the current through the resistor differs between the two representations (in the current form the source circulates $V/R$ even with no load attached). From the load's perspective the two forms are indistinguishable. This "outside equivalence" is exactly the intuition that makes Thevenin's theorem (Section 5) feel natural.