Electric circuit fundamentals

An electric circuit is a closed loop (or network of loops) through which electric current can flow, built from interconnected components: passive (resistors, capacitors, inductors) and active (sources, transistors, integrated circuits). The circuit’s behavior — its currents and voltages — is determined by the components’ properties and the topology.

Components

A component is the basic building block of a circuit. Each has well-defined electrical properties:

• Two terminals (most components — resistors, capacitors, inductors, diodes) or three or more (transistors, op-amps, ICs).
• A constitutive equation relating its current to its voltage.

The five core passive elements:

Resistor

• Symbol: rectangular box (US convention) or zigzag (older convention).
• Constitutive equation: $V=IR$ (Ohm’s law).
• Stores: nothing — dissipates energy as heat.
• Units: ohms ($\Omega$).

Capacitor

• Symbol: two parallel plates.
• Constitutive equation: $i(t)=C\frac{dv}{dt}$, equivalent to $v(t)=\frac{1}{C}\int idt$.
• Stores: energy in an electric field between plates.
• Units: farads (F), typically µF or pF in practice.

Inductor

• Symbol: coiled wire.
• Constitutive equation: $v(t)=L\frac{di}{dt}$.
• Stores: energy in a magnetic field around the coil.
• Units: henries (H), typically mH or µH.

Diode

• Symbol: triangle pointing to a line.
• Constitutive equation: nonlinear, approximated as “conducts when forward-biased above $\sim 0.7$ V, blocks when reverse-biased.”
• Use: rectification, voltage clamping.

Transistor

• Symbol: varies (BJT, NMOS, PMOS).
• Constitutive equation: nonlinear; depends on type.
• Use: amplification, switching. The building block of all digital and analog ICs.

Sources

Active components that supply energy:

• Independent voltage source: produces a fixed voltage regardless of load. Symbol: circle with $+$ and $-$.
• Independent current source: produces a fixed current regardless of load. Symbol: circle with arrow.
• Dependent sources: voltage or current as a function of another circuit variable. Used in modeling transistors and op-amps.

DC sources produce constant values; AC sources produce sinusoids — see AC electric circuits.

Kirchhoff’s laws

The two fundamental laws governing all circuits, from the simplest battery-and-bulb to the most complex IC:

Kirchhoff’s current law (KCL)

The sum of currents entering any node equals the sum of currents leaving:

${\sum }_{\text{into node}}i={\sum }_{\text{out of node}}i$

Equivalently, the net current at any node is zero. Conservation of charge.

Kirchhoff’s voltage law (KVL)

The sum of voltage drops around any closed loop equals zero:

${\sum }_{\text{loop}}v=0$

Conservation of energy (going around any loop, you must return to the same potential).

Series and parallel

Two basic combinations:

Series

Components connected end-to-end, sharing the same current:

• Resistors: $R_{\text{eq}}=R_1+R_2+\dots$
• Capacitors: $\frac{1}{C_{\text{eq}}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots$
• Inductors: $L_{\text{eq}}=L_1+L_2+\dots$

Parallel

Components connected to the same two nodes, sharing the same voltage:

• Resistors: $\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots$
• Capacitors: $C_{\text{eq}}=C_1+C_2+\dots$
• Inductors: $\frac{1}{L_{\text{eq}}}=\frac{1}{L_1}+\frac{1}{L_2}+\dots$

Notice resistors and inductors combine the same way; capacitors are inverted (because their constitutive equations are inverted).

Analysis techniques

For circuits more complex than simple series/parallel:

• Mesh analysis: assign currents to each loop, apply KVL, solve linear system.
• Node analysis: assign voltages to each node, apply KCL, solve linear system.
• Superposition: for circuits with multiple sources, analyze the response to each source separately and sum.
• Thévenin / Norton equivalent: replace any sub-circuit with one source + one impedance.

For AC circuits, use these same techniques with Phasor representations and impedances — see AC electric circuits and Phasor relationships for circuit elements.

Application areas

Circuit analysis underpins:

• Power systems: transmitting and distributing electricity.
• Electronics: amplifiers, oscillators, filters.
• Communications: modems, transceivers.
• Computing: every digital chip is built from circuits.
• Sensors and instrumentation: from thermometers to LIDAR.

For AC-specific analysis (frequency domain, impedance), see AC electric circuits and Phasor.