Electric Circuits

Circuit basics

An electric circuit is a closed loop containing:

• Source — battery, generator, or power supply providing voltage V.
• Conductors — wires (typically negligible resistance).
• Components — resistors (R), capacitors (C), inductors (L), diodes, transistors, etc.
• Switch (often) — opens/closes the circuit.

Charge flows around the loop driven by the voltage source. Current I = charge/time = dQ/dt (units: amperes, A).

Series vs parallel

Property	Series	Parallel
Topology	Components on one path	Components on multiple paths
Current	Same through all	Divides among components
Voltage	Adds across components	Same across all
Resistors	R_total = ΣR_i	1/R_total = Σ(1/R_i)
Capacitors	1/C_total = Σ(1/C_i)	C_total = ΣC_i
Inductors	L_total = ΣL_i	1/L_total = Σ(1/L_i)

Key laws

Ohm's Law:

V = I · R

Kirchhoff's Current Law (KCL): Sum of currents into a node equals sum out. (Charge conservation at junctions.)

Kirchhoff's Voltage Law (KVL): Sum of voltage drops around any closed loop equals 0. (Energy conservation around the loop.)

Common components

Component	Symbol	Behavior
Resistor	—/\/\/\—	V = IR; dissipates power as heat
Capacitor	—||—	Stores charge; I = C·dV/dt
Inductor	—UUUU—	Stores energy in B-field; V = L·dI/dt
Diode	—|>|—	One-way valve; passes current in one direction
Transistor	(complex)	Amplifier or switch
Voltage source	—|+ −|—	Constant V regardless of current
Current source	—(↑)—	Constant I regardless of voltage
Switch	—/_/—	Opens/closes the path

Common configurations

Circuit	Behavior
Simple R	I = V/R; voltage drives current through resistor
RC charging	V_C = V·(1 − e^(−t/RC)); approaches V exponentially
RC discharging	V_C = V_0·e^(−t/RC); decays exponentially
RL circuit	I = (V/R)·(1 − e^(−tR/L)); inductor opposes change
RLC resonance	Resonance at f = 1/(2π√(LC))
Voltage divider	V_out = V_in · R₂/(R₁ + R₂)
Current divider	I_branch = I_total · (other_R)/(R₁ + R₂)

JavaScript — circuit calculations

// Series resistors
function seriesResistance(R_array) {
  return R_array.reduce((a, b) => a + b, 0);
}

// Parallel resistors
function parallelResistance(R_array) {
  return 1 / R_array.reduce((sum, R) => sum + 1/R, 0);
}

console.log(`100Ω, 200Ω series: ${seriesResistance([100, 200])} Ω`);  // 300
console.log(`100Ω, 200Ω parallel: ${parallelResistance([100, 200]).toFixed(2)} Ω`); // 66.67
console.log(`Two 100Ω parallel: ${parallelResistance([100, 100])} Ω`); // 50 (half)

// Current and voltage for simple circuit
function ohms(V, R) { return V / R; }    // Current
function dropV(I, R) { return I * R; }   // Voltage drop

// Voltage divider
function voltageDivider(V_in, R1, R2) {
  // V_out across R2: V_out = V_in · R2/(R1+R2)
  return V_in * R2 / (R1 + R2);
}

console.log(`9V divided by 1k+2k: ${voltageDivider(9, 1000, 2000).toFixed(1)} V`);  // 6.0

// Current divider
function currentDivider(I_total, R_branch, R_other) {
  return I_total * R_other / (R_branch + R_other);
}

// RC charging
function rcChargingVoltage(V_source, R, C, t) {
  const tau = R * C;
  return V_source * (1 - Math.exp(-t / tau));
}

// 9V battery, 1kΩ resistor, 100µF capacitor
console.log(`After 0.1 sec: ${rcChargingVoltage(9, 1000, 100e-6, 0.1).toFixed(2)} V`);
// τ = RC = 0.1 sec; after 1τ, V = 9·(1 - 1/e) = 5.69V

// RC discharging
function rcDischarging(V_initial, R, C, t) {
  return V_initial * Math.exp(-t / (R * C));
}

// RLC resonant frequency
function lcResonance(L, C) {
  return 1 / (2 * Math.PI * Math.sqrt(L * C));
}

console.log(`LC: 1mH + 1µF: ${(lcResonance(1e-3, 1e-6) / 1000).toFixed(2)} kHz`);  // ~5.03 kHz

// Power in resistor
function powerInResistor(V, I) {
  // P = VI = I²R = V²/R
  return V * I;
}

console.log(`9V, 0.05A: ${powerInResistor(9, 0.05)} W`);  // 0.45 W

// Bridge circuit (Wheatstone) — used for sensitive resistance measurements
function wheatstoneBridge(R1, R2, R3, R4) {
  // Balanced when R1/R2 = R3/R4
  return Math.abs(R1/R2 - R3/R4) < 1e-9;
}

// Battery voltage with internal resistance r
function loadedVoltage(V_emf, r_internal, R_load) {
  // V = V_emf - I·r where I = V_emf/(r + R_load)
  // V_terminal = V_emf · R_load/(r + R_load)
  return V_emf * R_load / (r_internal + R_load);
}

console.log(`9V battery, 0.5Ω internal, 1Ω load: ${loadedVoltage(9, 0.5, 1).toFixed(2)} V`); // 6.0

Where circuits matter

• Electronics — everywhere. Phones, computers, TVs, IoT, medical devices.
• Power systems. Generation, transmission, distribution; AC and DC grids.
• Control systems. Sensors, actuators, automation.
• Communications. Radio, TV, internet, fiber optics — all involve circuits at endpoints.
• Robotics. Motor drivers, sensor circuits, microcontrollers.
• Energy storage. Battery management, DC-DC converters, charging circuits.
• Education. Foundation of EE; intro circuits taught universally.

Common mistakes

• Confusing series and parallel. Series — same current. Parallel — same voltage. Resistor formulas opposite. Capacitor formulas reversed from resistors.
• Forgetting Kirchhoff signs. Voltage drops in direction of current through resistor; rise from − to + through battery. Sign errors common.
• Using DC analysis on AC. Capacitor blocks DC but passes AC; inductor passes DC, blocks AC. Don't analyze RLC with simple DC-only methods.
• Ignoring polarity (capacitors). Electrolytic caps are polarized; reversed connection damages them. Ceramic caps are not polarized.
• Forgetting wire resistance/heat. Real wires have small but nonzero R. High currents in thin wires → significant voltage drop, heat.
• Open vs closed switch confusion. Open = breaks circuit, no current. Closed = complete loop, current flows.