Electrical Theory Foundations

Power Atlas assumes you know the fundamentals — what voltage is, why we use AC, what reactive power means physically. This page covers all of it. After reading, §01's formulas will make sense, not just be memorized.

The Three Quantities — Voltage, Current, Resistance

Ohm's Law — The Foundational Equation

Voltage equals current times resistance. Memorize the triangle: cover what you want to find, the formula appears.

Kirchhoff's Laws — How Current and Voltage Behave in Circuits

DC vs AC — Why We Use Alternating Current

The Sinusoid — What "AC" Actually Looks Like

Frequency — Why 60 Hz?

60 Hz is the US/North America standard. 50 Hz is most of Europe + Asia. Higher frequency = smaller transformer cores (good) but more transmission line losses (bad). 60 Hz was Westinghouse's choice; 50 Hz was AEG's. Both work; the world settled on regional standards by ~1920.

Why Three-Phase Is the Standard

Three sinusoids, equal magnitude, 120° apart. Why this beats single-phase and 2-phase:

The Magic of Transformers — Why AC Won

Resistance + Inductance + Capacitance — The Three Passive Elements

Why this matters: Reactive Power Explained Physically

An inductor (motor coil) doesn't dissipate energy — it stores energy in its magnetic field, then releases it. The current flowing back and forth creates this storage/release cycle. The current is real (you have to size wires for it), but no NET energy gets used.

This is what reactive power (Q, in kVAR) is — the apparent flow of power that just sloshes back and forth between source and load. It uses no fuel at the power plant but does use copper in the wires.

Impedance — Generalized Resistance for AC

For AC, the opposition to current isn't just resistance. It includes inductive reactance (X_L) and capacitive reactance (X_C) too.

Phasors — Why Engineers Draw Triangles

Each AC voltage or current is a sinusoid with magnitude AND phase angle. Adding two sinusoids of different phases is messy with trigonometry. Phasors represent each sinusoid as an arrow (length = magnitude, direction = phase angle), then you add the arrows like vectors.

Systematic Methods — Mesh and Nodal Analysis

KVL and KCL by themselves are great for two- or three-element circuits. Once a network gets bigger, you want a recipe — a systematic way to write equations and solve. Two methods, dual to each other.

Mesh recipe: (1) Label each independent loop with an assumed direction (clockwise convention). (2) For each loop, write KVL: assumed-direction source rises − sum of (loop current × resistance) drops − contributions from neighboring loops sharing a resistor = 0. (3) Solve the resulting linear system for the loop currents. Branch currents are then sums or differences of adjacent loop currents.

Nodal recipe: (1) Pick one node as reference (ground = 0 V). (2) Label every other node with an unknown voltage V₁, V₂, … (3) For each labeled node, write KCL — current leaving through each connected resistor = (V_{this node} − V_{other end}) / R, summed to zero (with current sources added). (4) Solve the linear system for node voltages. Branch currents come from V/R afterward.

Why this matters in power systems: matrix-based mesh and nodal analysis is the foundation of power-flow software (Newton-Raphson on Y-bus admittance matrices is just nodal analysis at scale). It's also how relay coordination tools compute fault currents across an interconnected network — they reduce the system to a Thevenin equivalent at each candidate fault point, which is mesh/nodal applied recursively.

Series + Parallel Circuits

Why this matters in power systems: branch circuits in a panel are in PARALLEL with each other (all share the bus voltage; current splits per load). Conductors in PARALLEL feeders share current proportionally to inverse impedance — equal-length matched conductors share equally; mismatched ones don't. NEC 310.10(H) requires identical termination + length for paralleled conductors precisely because of this. (When one of three "elevators" is slow, the other two overload — same physics as paralleled feeders with mismatched impedance.)

Superposition Principle

For any LINEAR circuit with multiple sources, the response (voltage or current) at any point equals the SUM of responses caused by each source individually (with all other sources turned off — voltage sources shorted, current sources opened).

Thevenin + Norton Equivalents

Any complex network of voltage sources, current sources, and resistors can be reduced to a single equivalent source with a single equivalent impedance, when viewed from any pair of terminals.

Symmetrical Components — Intro

Real power systems aren't perfectly balanced. Single-phase faults, ground faults, broken conductors all create UNBALANCED 3-phase conditions. Analyzing them with regular phasors is messy. Symmetrical components decompose any unbalanced 3-phase set into THREE balanced sets:

Each sequence has its own equivalent impedance for any piece of equipment. Faults are then analyzed by combining sequence networks. Full treatment in §12 Short Circuit.

Power = Energy / Time — kW vs kWh

The most common confusion in EE literacy. Let's nail it:

A 100W lightbulb running for 10 hours uses 1 kWh of energy at a power rate of 0.1 kW. Same lightbulb running for 1 hour: still 0.1 kW power, but only 0.1 kWh energy.

The Hill Metaphor — Voltage, Current, Energy

The plumbing analogy gets you started. The HILL metaphor goes deeper. It explains why voltage can exist without current, why batteries differ from capacitors, and why inductors and capacitors mirror each other.

Voltage Exists Without Current

Voltage requires only an imbalance. Current requires a closed path. A battery without a circuit has voltage but zero current. The ball needs somewhere to roll TO before it falls.

Battery vs Capacitor — Chemistry vs Storage

The Electric Field — Where Is the Hill?

The electric field is created by charge separation. Move a positive charge somewhere it doesn't naturally want to be — separate it from a negative charge — and the resulting field is the "hill" between them.

• Battery: chemistry creates and maintains the separation continuously.
• Capacitor: external voltage forced charges apart; the cap holds them in place.
• Open switch with charged cable: the cable is "lifted" by stored charge; voltage measured across the gap.
• Static electricity: rubbing creates separation. Spark = the charges finally finding a path back across the gap.

Voltage isn't a "thing." It's the height difference. The field is the gravity. Charge separation is what builds the hill.

Inductor — The Flywheel

If a capacitor is "rock at height," an inductor is "spinning flywheel." They're not just different — they're opposite in every dimension.

The Flywheel Intuition for Inductors

Imagine a heavy flywheel on bearings:

Charging + Discharging — Time Curves

Phase Relationships — Why Voltage Leads / Lags

For a Pure Capacitor: Current LEADS Voltage

Apply AC voltage to a cap. The cap can't have voltage without charge on its plates. Charges must arrive BEFORE the voltage builds up. So:

1. Current flows TOWARD the cap (charging up).
2. As charge accumulates, voltage rises BEHIND the current.
3. When voltage peaks, current has already started to reverse (charge is starting to leave).
4. Current LEADS voltage by 90° (a quarter cycle).

Memory aid: "ICE" — In a Capacitor, current (I) leads voltage (E).

For a Pure Inductor: Voltage LEADS Current

Apply AC voltage to an inductor. The inductor "fights" the rising current with a back-EMF. Current can't rise until voltage has been pushing for a while:

1. Voltage rises across the coil.
2. Current builds slowly behind it.
3. When current peaks, voltage has already started decreasing.
4. Voltage LEADS current by 90°.

Memory aid: "ELI" — in an inductor (L), voltage (E) leads current (I). Together: "ELI the ICE man" covers both.

Why Capacitors Are Used in Real Circuits

Why Inductors Are Used in Real Circuits

Other Useful Metaphors

Power Triangle = Beer Glass

Reactive Power = Dancers Spinning vs Walking

Real power = dancers walking forward (actually getting somewhere). Reactive power = dancers spinning in place (using energy, not progressing). Both look like effort. Only one delivers useful work to the destination.

Resonance = Pushing a Swing

Push at the wrong rhythm: nothing happens, swing barely moves. Push at the natural frequency: swing climbs higher and higher. Same with electrical resonance — when source frequency matches LC natural frequency, current builds dramatically. Why power-factor caps + system inductance can blow up if not detuned (§15).

Transformers = Gear Ratios

Step-up transformer: like a gearbox in low gear. High torque (high current) on input, fast spin (high voltage) on output. Step-down: opposite. Conservation of energy = conservation of (V × I) = conservation of "horsepower" through the gearbox.

3-Phase = 3 Horses Pulling at Even Spacing

Three horses harnessed to a wagon at 120° angular spacing. Each pulls in a different direction. Combined force is constant in magnitude — wagon experiences smooth pull. Single-phase = one horse — pull pulses from full to zero each step. Why 3φ motors are smooth, 1φ motors vibrate.

Grounding = Dump Valve in a Fish Tank

Your fish tank has water at a specific level. Excess pressure (voltage) needs to escape somewhere. The dump valve = ground. Connects tank to the outside (earth). Earth is so big it can absorb anything without changing level. Why we ground every metal enclosure to earth.

Selective Coordination = Falling Dominoes Backwards

Tip the closest domino — it falls. None of the dominoes upstream fall. That's selectivity: only the device closest to the fault opens. The trick is getting the timing right so upstream doesn't beat downstream to falling.

Arc Flash = Fireworks at Point Blank

A small ignition (electrical fault) becomes massive plasma in microseconds. The energy was already there in the system (fault current); the fault just released it. Hence the equation: I × t = energy. Reduce trip time t (mitigation) and you reduce the explosion.

Power Factor = Pulling a Sled at an Angle

Imagine pulling a sled with a rope. Pull straight ahead: 100% of effort goes to motion (PF = 1.0). Pull at 45° angle: half your effort lifts the sled (wasted motion = reactive), half pulls it forward (real). Power factor = cos(angle) of your pull.

Harmonics = Musical Overtones

A pure sine wave is one note. Add overtones (3rd, 5th, 7th harmonics) and the wave gets distorted — instead of a clean note, you get a chord-like buzzy waveform. VFDs, LEDs, server PSUs all generate harmonics. Equipment hates them: motors overheat from negative-sequence currents (5th harmonic), transformers see neutral overload (3rd harmonic = triplens).

Δ Configuration = Triangle Marketplace

3 sellers in a triangle, each connected to the other two. No central authority. Three transactions in three pairs. No neutral exists.

Y (Wye) Configuration = Spoke Wheel

3 sellers all connected to one central hub (the neutral). Each can sell directly to the center, or to neighbors via the center. Neutral provides a reference point — single-phase voltages can be measured to neutral.

Impedance = Walking Through Different Mediums

Walking through air = low resistance + low reactance = low impedance. Walking through water = higher resistance. Walking through a strong wind blowing in your face = reactance (you fight against it) — and the harder the wind blows (the higher the frequency), the more you fight. Capacitors are like walking with a tailwind that pushes you faster than you'd walk in still air.

Fault Current = Flash Flood

Normally water (current) trickles through pipes (conductors) at a measured rate. A burst pipe (short circuit) creates a flash flood — water everywhere, finding the path of least resistance, doing damage. The "flood gates" (breakers) need to slam shut FAST.

Magnetic Field Basics — How Motors Work

1. Current creates magnetic field. A wire carrying current is surrounded by a circular magnetic field (right-hand rule). Coil the wire → magnetic field becomes a "bar magnet" with N + S poles.
2. Magnetic field exerts force on a current-carrying wire. If you put a current-carrying wire in another magnetic field, the two fields push the wire perpendicular to both (F = I × L × B).
3. Combine the two: a stator (stationary winding) creates a magnetic field. A rotor (moving winding or magnet) sits inside the stator and feels force → rotor spins. This is a motor.
4. Spin a rotor in reverse — get a generator. Mechanical input → magnetic field changes around rotor → induces voltage in stator → output power.
5. 3-phase magic: three windings 120° apart in space, fed by 3-phase currents 120° apart in time, create a smoothly rotating magnetic field that drags the rotor along. This is why 3-phase induction motors are self-starting.

The Power System Hierarchy

Generator → Wall Outlet — One Joule's Journey

1. Coal/gas/nuclear/water turns a turbine. Turbine spins a 60 Hz synchronous generator at 13.8 kV.
2. Step-up transformer brings it to 230 kV for transmission.
3. Power flows hundreds of miles via overhead transmission lines.
4. Substation step-down: 230 kV → 69 kV sub-transmission.
5. Distribution substation: 69 kV → 12.47 kV primary distribution.
6. Neighborhood pole-mount transformer: 12.47 kV → 120/240V single-phase to your house. (Or → 480Y/277V three-phase to a commercial building.)
7. Inside the building: panelboard → branch circuit → wall outlet.
8. Your laptop charger: 120V AC → AC-DC → DC-DC → 19V DC at the laptop.

Total efficiency from fuel to laptop: about 30%. (Most loss at the power plant — thermodynamic limits.) Transmission + distribution losses combined are typically only 5-10%.

Where to Go Next

Now that the theory is in place:

• If you're new to design: read §01 (Conversions) → §02 (How Design Starts) → §03 (Load Analysis). The formulas will make sense, not just be memorized.
• If you have an EE degree but it's been a while: skim §01, focus on §04 (Branch Circuits) onward.
• If you're studying for the PE Power exam: the parent has exam-focused practice with the same notation conventions.

Theory primer · The "why" behind the "what" · Skip if you have an EE background