At a glance
What this guide walks through
Electrical engineering builds up in layers: Ohm's law describes a single resistor, series and parallel rules combine many of them, power ties voltage and current together, AC adds the complication of alternating direction and phase, three-phase scales it up for real power delivery, and transformers move it efficiently across the grid. This guide follows that exact build-up, in order.
The single relationship underneath almost every DC circuit calculation.
From phasor geometry: the difference between equal phase voltages 120° apart.
Cycles per second of the periodic AC waveform; each full cycle includes both polarities.
Set purely by the turns ratio between windings.
01 // Ohm's law: the foundation
Almost every DC circuit calculation in electrical engineering traces back to one relationship between voltage, current and resistance.
| Quantity | Formula | Unit |
|---|---|---|
| Voltage | V = I × R | volts (V) |
| Current | I = V / R | amperes (A) |
| Resistance | R = V / I | ohms (Ω) |
These are the same relationship rearranged three ways: voltage is what pushes current through a resistance, current is how much charge flows per second, and resistance is how much a component opposes that flow. Everything from a simple LED resistor to a full power system's fault-current analysis eventually reduces to some form of this relationship, often combined with the power formulas in section 03.
02 // Interactive: Ohm's law solver
Enter any two of voltage, current and resistance to solve for the third — leave the value you want solved blank.
Fill in exactly two fields and leave the third blank to solve for it. If all three are filled, the calculator checks whether they're consistent.
03 // Series and parallel resistance
Real circuits combine multiple resistive elements, and the two basic combination rules cover almost every DC network when applied repeatedly.
Series
R_total = R₁ + R₂ + …Components share the same current, and voltage divides across them proportionally to each resistance. Total resistance is always larger than the largest individual resistor.
Parallel
1/R_total = 1/R₁ + 1/R₂ + …Components share the same voltage, and current divides between them inversely to each resistance. Total resistance is always smaller than the smallest individual resistor.
A useful intuition: adding a resistor in series always increases total resistance (more material for current to fight through), while adding a resistor in parallel always decreases total resistance (another path current can take), even if the added resistor's own value is large.
04 // Interactive: series & parallel resistance
Enter two resistor values to see the combined resistance both ways.
For more than two resistors, apply the parallel formula repeatedly, combining two at a time.
05 // Electrical power and energy
Power is the rate at which electrical energy is converted — into heat, light, motion, or anything else a circuit does useful work with.
| Formula | When to use it |
|---|---|
| P = V × I | DC, or AC when voltage and current are in phase; for sinusoidal single-phase AC in general use P = VrmsIrmscosφ |
| P = I² × R | Resistive dissipation when current through the resistor is known — power grows with the square of current |
| P = V² / R | Resistive dissipation when voltage across the resistor is known |
| E = P × t | Energy for constant (or average) power over a time interval — electricity meters accumulate energy, commonly in kWh |
The I² relationship is worth sitting with: doubling current through a fixed resistance quadruples the power dissipated as heat — not doubles it. This is the same relationship behind why transmission lines are operated at high voltage and low current (see the companion Learn Power Systems guide) and why a short-circuit fault, which drives current very high, can release damaging amounts of heat almost instantly.
06 // AC fundamentals: why alternating current is different
Direct current (DC) flows one way, like from a battery. Alternating current (AC) — used for virtually all grid power — periodically reverses direction, which introduces a few concepts DC circuits don't need.
RMS value
AC voltage and current constantly change, so "120 V" or "230 V" mains refers to an RMS value. For a sinusoidal waveform, the peak magnitude is √2 times the RMS magnitude; the √2 relationship is not universal for non-sinusoidal waveforms.
Frequency
How many complete waveform cycles occur each second — commonly 50 Hz or 60 Hz depending on the grid. Current changes sign twice during each sinusoidal cycle; frequency counts complete cycles, not individual direction changes.
Phase
Voltage and current don't have to peak at the same instant. Reactive components (inductors and capacitors) shift current's timing relative to voltage, which is where reactance, impedance and power factor (below) all come from.
Reactance is the imaginary part of impedance: inductive reactance (XL = 2πfL) rises with frequency, while capacitive reactance (XC = 1/(2πfC)) falls with frequency. For a simple series circuit with resistance R and net reactance X, Z = R + jX and |Z| = √(R² + X²); more complicated networks are handled with the same complex-impedance rules.
For sinusoidal single-phase AC using RMS values, apparent power S = VI, real power P = VI cosφ, and reactive power Q = VI sinφ. Power factor is P/S (equal to cosφ for a purely sinusoidal voltage/current pair); harmonics can make true power factor differ from displacement cosφ.
07 // Three-phase power
Almost all bulk electricity generation and industrial power delivery uses three-phase AC rather than single-phase — for good structural reasons.
A three-phase supply is three single-phase voltages, each shifted 120° apart in time, delivered together on (typically) three or four conductors. Two practical advantages fall out of this: a three-phase motor produces smooth, continuous torque rather than the pulsating torque a single-phase motor produces, and three-phase delivers more power for a given amount of conductor material than three separate single-phase circuits would.
| Quantity | Formula | Note |
|---|---|---|
| Three-phase real power | P = √3 × VL × IL × cosφ | VL = line-to-line voltage, IL = line current |
| Line vs phase voltage (star/wye) | VL = √3 × Vphase | Line-to-line voltage is √3 times each phase's voltage to neutral |
The √3 (≈1.732) factor is phasor geometry. In a balanced wye system, line-to-line voltage is the vector difference between two equal phase-to-neutral voltages 120° apart, giving VL = √3Vphase. Combining the three equal phase powers yields the familiar balanced-system relation P = √3VLILcosφ.
08 // Transformers: changing voltage with no moving parts
In a conventional isolated two-winding transformer, primary and secondary windings share a magnetic core but are not electrically connected: alternating current in the primary creates changing magnetic flux that induces voltage in the secondary. Autotransformers are an important exception because part of the winding is electrically common to both sides.
| Relationship | What it means |
|---|---|
| Vp/Vs = Np/Ns | Voltage ratio equals the turns ratio between primary and secondary windings |
| Is/Ip = Np/Ns | Current ratio is the inverse — step voltage up, and current steps down proportionally |
That inverse current relationship is central to high-voltage power delivery: in the ideal model Pp = Ps, so stepping voltage up steps current down. Real transformers have winding, core and stray losses plus voltage regulation under load; efficiency varies with rating, design and loading, so manufacturer test data is the right source for a real unit.
09 // Interactive: transformer turns ratio
Enter primary voltage, primary turns and secondary turns to find secondary voltage and the current ratio.
Ideal transformer assumption (no losses, unity efficiency) — real transformers have small additional voltage regulation effects under load.
10 // From generation to the wall socket
Putting the ideas together: generators feed AC networks, transformers raise voltage for efficient long-distance transmission, and substations step it down through regional and local distribution levels before service transformers supply end users. Exact voltage levels vary substantially by country and utility, so the spectrum above is illustrative rather than a universal hierarchy.
For the generation, balance and frequency side of that same journey, see the companion Learn Power Systems guide.
11 // Background, FAQ, references and limitations
Expand for deeper context and the sources behind the figures used here.
Ohm's law (1827) and the circuit-combination rules came first, describing how current, voltage and resistance behave in simple DC networks. As electrical power systems developed through the late 19th century, the "War of the Currents" between Edison's DC and Westinghouse/Tesla's AC systems settled decisively in favor of AC for grid-scale power — precisely because transformers make voltage easy to change, which section 05's I²R relationship shows is essential for transmitting power efficiently over distance.
Three-phase AC became the standard for generation and bulk delivery because of the smoother motor torque and better conductor utilization described in section 07, and that three-phase, transformer-stepped architecture is still the backbone of virtually every modern grid.
What is the difference between series and parallel circuits? In series, components share the same current and total resistance adds. In parallel, components share the same voltage, current divides between them, and total resistance (found from the reciprocal sum) is always less than the smallest individual resistor.
Why is AC power expressed as real, reactive and apparent power? Reactive loads shift current out of phase with voltage, so not all apparent power (V × I) does useful work. Real power is the in-phase portion that does; reactive power is the out-of-phase portion; power factor describes what fraction of apparent power is actually useful.
Why does three-phase power use the square root of 3? In a balanced wye system, line voltage is the phasor difference between two equal phase-to-neutral voltages 120° apart, which has magnitude √3 times one phase voltage. Combining the three equal phase powers gives the same √3 factor in P = √3VLILcosφ.
How does a transformer change voltage without moving parts? Alternating current in the primary winding creates a changing magnetic field in a shared core, which induces a voltage in the secondary winding. The voltage ratio equals the turns ratio between windings.
What is the difference between AC and DC? DC maintains one polarity/direction in the basic idealization. AC varies periodically and reverses polarity; grid frequency is commonly 50 or 60 cycles per second. AC is widely used because transformers make voltage conversion straightforward, while HVDC is also important for selected long-distance and cable links.
- NIST — SI Units: Electric Current — formal definitions of the ampere, volt and ohm referenced throughout this guide.
- IEEE — the professional body behind most electrical and power-engineering standards referenced across EngineerHub's electrical tools.
- U.S. Energy Information Administration — Delivery to consumers — official overview of transmission, substations, transformers and local distribution.
Numeric values were reviewed in August 2026. They remain rounded educational comparisons and may differ from a specific jurisdiction's voltage standards or equipment.
This page is a simplified educational overview, not an electrical design reference. All calculators use ideal-component assumptions (zero source resistance, ideal transformers, unity efficiency where relevant) and are meant to illustrate the underlying relationships, not to size real equipment or wiring. Voltage-level figures in the visual anchor are broadly representative but vary significantly by country and utility. Nothing on this page is validated for real electrical installation, protection design, or safety-critical work.
Electricity at mains voltage and above can be lethal. Always rely on a qualified, licensed electrician or electrical engineer, current wiring regulations, and proper safety procedures for any real electrical work.