Electrical Power

Power formulas

General:

P = V · I  (any element)

For ohmic resistor (using V = IR):

P = I² · R = V² / R

All three forms equivalent for ohmic load.

For AC (sinusoidal):

P_avg = V_rms · I_rms · cos(φ)

where φ is the phase angle between V and I, cos(φ) is the "power factor."

Units and energy

Unit	Equivalent	Use
Watt (W)	1 J/s = 1 V·A	SI base unit
kilowatt (kW)	1000 W	Appliances, vehicles
megawatt (MW)	10⁶ W	Power plants, large industry
gigawatt (GW)	10⁹ W	National power grid scales
horsepower (hp)	~746 W	Engines, motors (USA)
kilowatt-hour (kWh)	3.6 MJ	Utility billing

Power consumption

Device	Typical Power
LED bulb	~9-12 W
Incandescent bulb	40-100 W
Refrigerator (avg)	~150 W (3-4 kWh/day)
Microwave oven	800-1500 W
Toaster	800-1500 W
Hair dryer	1500-1875 W
Electric kettle	1500-3000 W
Air conditioner (window)	500-1500 W
Electric vehicle (cruising)	20-30 kW
Tesla Supercharger	250 kW
Average US home (peak)	~6-8 kW
1 MW power plant	1,000,000 W
Hoover Dam	~2 GW

JavaScript — power calculations

// Basic power
function power(V, I) { return V * I; }
function powerFromVR(V, R) { return V * V / R; }
function powerFromIR(I, R) { return I * I * R; }

// 100 W bulb at 120 V — find R and I
const P_bulb = 100;
const V_bulb = 120;
const I_bulb = P_bulb / V_bulb;  // 0.833 A
const R_bulb = V_bulb / I_bulb;   // 144 Ω
console.log(`Bulb: ${I_bulb.toFixed(3)} A, ${R_bulb.toFixed(0)} Ω`);

// AC RMS values
function ACPower(V_rms, I_rms, powerFactor = 1) {
  return V_rms * I_rms * powerFactor;
}

console.log(`120V/15A circuit: max ${ACPower(120, 15, 1)} W`);  // 1800 W

// Energy from power and time
function energyFromPower(P, t_hours) {
  return P * t_hours / 1000;  // kWh
}

console.log(`100 W bulb 24 hours: ${energyFromPower(100, 24)} kWh`);
console.log(`@$0.15/kWh: $${(energyFromPower(100, 24) * 0.15).toFixed(2)}`);

// Cost of running an appliance
function costToRun(P_W, hours_per_day, days, price_per_kWh = 0.15) {
  const energy_kWh = P_W * hours_per_day * days / 1000;
  return energy_kWh * price_per_kWh;
}

console.log(`Refrigerator (150W, 24/7, 30 days): $${costToRun(150, 24, 30).toFixed(2)}`);
console.log(`AC (1500W, 8h, 30 days): $${costToRun(1500, 8, 30).toFixed(2)}`);

// Transmission line losses
function transmissionLoss(P_load, V_line, R_line) {
  const I = P_load / V_line;
  return I * I * R_line;
}

// Compare transmission at low vs high V
const P_total = 1e7;  // 10 MW
const R_wire = 5;  // 5 Ω wire
console.log(`At 1 kV: ${(transmissionLoss(P_total, 1000, R_wire) / 1e6).toFixed(0)} MW lost`);
console.log(`At 230 kV: ${(transmissionLoss(P_total, 230e3, R_wire) / 1e3).toFixed(0)} kW lost`);

// Heating element selection
function powerForHeating(V, R) { return V * V / R; }

// Heater for 1500 W at 120 V
const R_heater = 120 * 120 / 1500;  // 9.6 Ω
console.log(`1500W heater needs R = ${R_heater.toFixed(2)} Ω`);

// EV efficiency
function evRange(batteryCapacity_kWh, energy_per_km_kWh = 0.2) {
  return batteryCapacity_kWh / energy_per_km_kWh;
}

console.log(`75 kWh battery: ${evRange(75)} km range`);  // 375 km

// Solar panel output
function solarPanel(area_m2, efficiency = 0.20, irradiance_W_per_m2 = 1000) {
  return area_m2 * efficiency * irradiance_W_per_m2;
}

// 20 m² of 20% solar
console.log(`Solar: ${solarPanel(20)} W peak`);  // 4000 W
console.log(`Daily: ${solarPanel(20) * 5 / 1000} kWh (5 h equiv)`);  // 20 kWh

Where electrical power matters

• Generation and transmission. Power plants rated in MW; grids carry GW; minimizing losses critical.
• Appliances. Wattage labels tell consumers operating cost.
• Energy storage. Batteries rated in kWh; EV ranges depend on it.
• Power factor correction. Industrial loads need PF correction (capacitor banks) for efficient operation.
• Solar/wind. Renewable systems sized in W or kW.
• Energy efficiency. LED replaces incandescent — 80% reduction in electrical power for same light.
• Industrial design. Heat sinks, cooling, thermal management based on dissipated power.

Common mistakes

• Confusing power and energy. Power (W) is rate; energy (J or kWh) is total. 100 W bulb on 1 hour = 100 Wh = 0.1 kWh.
• Forgetting power factor in AC. Reactive loads (motors) have power factor < 1; apparent power (V·I) exceeds real power (P).
• Wrong RMS vs peak in AC. Power formulas use RMS values. Peak voltage is √2 times RMS.
• Ignoring efficiency. Real devices waste energy as heat. Solar panel input ≠ electrical output. Motors aren't 100% efficient.
• Treating I²R as universal. True for resistive elements only. Inductors and capacitors store and release energy without dissipating.
• Confusing W with W·h. Watt is rate (J/s); watt-hour is energy (= 3600 J).