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
| Quantity | Symbol | Unit | Plumbing analogy | What it physically is |
|---|---|---|---|---|
| Voltage | V | Volt (V) | Water pressure (PSI) | Electrical potential difference between two points. The "push" that wants to move charge. Measured between two points. |
| Current | I | Ampere (A) | Water flow rate (gal/min) | Charge flowing per second. Coulombs per second. Measured by clamp meter around a single conductor. |
| Resistance | R | Ohm (Ω) | Pipe friction | Opposition to current flow. A property of the conductor (size, length, material). |
| Power | P | Watt (W) | Water power (PSI × GPM) | Rate of energy transfer. Voltage × Current. |
| Energy | E | Watt-hour (Wh) | Total water moved | Power × time. What the utility bills you for (kWh). |
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
| Law | Statement | What it means in practice |
|---|---|---|
| KVL — Kirchhoff's Voltage Law | Sum of voltages around any closed loop = 0 | Voltage drops across components in a loop add up to the source voltage. Why we calculate voltage drop along a feeder + branch + load = source voltage. |
| KCL — Kirchhoff's Current Law | Sum of currents entering a node = sum of currents leaving | Current splits at a junction in proportion to inverse impedance. Why parallel branches share current. Why neutral current is the unbalanced sum of phase currents. |
DC vs AC — Why We Use Alternating Current
| DC (Direct Current) | AC (Alternating Current) | |
|---|---|---|
| Direction of flow | One direction, constant | Reverses 60 times per second (US) or 50 (Europe) |
| Voltage transformation | Hard — requires DC-DC converters (inefficient at scale) | Easy — passive transformer steps up/down with ~ 99% efficiency |
| Long-distance transmission | Lossy at low voltage; requires HVDC for long distance (specialized + expensive) | Easy — step up to MV/HV, transmit, step down. I²R losses minimized. |
| Where used | Batteries, electronics, telecom, EV traction | Everything between the power plant and the wall outlet |
| Why it won (1880s) | — | The transformer made AC scalable. Tesla + Westinghouse beat Edison's DC for distribution. |
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:
| Property | Why 3-phase wins |
|---|---|
| Constant total instantaneous power | P1(t) + P2(t) + P3(t) = constant. (For 1-phase, total power pulses at 120 Hz. For 2-phase, it pulses too.) Constant power = smooth motor torque, no vibration. |
| Minimum copper for transmission | 3 wires for the same kW vs 2 wires for 1-phase = ~ 25% LESS copper per kW transmitted. (4 wires for 2-phase = WORSE.) This is why utilities use 3-phase everywhere. |
| Self-starting motors | 3-phase produces a uniform rotating magnetic field. Motor starts on its own. (1-phase motors need start capacitors or shaded poles.) |
| Neutral can be omitted | Balanced 3φ has zero current in the neutral — can use 3 wires only. (Unbalanced loads need a neutral.) |
The Magic of Transformers — Why AC Won
Faraday's Law (1831) — a changing magnetic field induces voltage in a nearby conductor. AC's constant change makes this practical:
- AC current in primary winding creates a changing magnetic flux in the iron core.
- The changing flux induces a voltage in the secondary winding.
- The voltage ratio = the turns ratio: V₁/V₂ = N₁/N₂.
- Power is conserved (minus tiny losses): V₁ × I₁ = V₂ × I₂. Step up voltage → step down current.
Why this matters: at the power plant, generate at 13.8 kV. Step up to 230 kV for transmission (low current = low I²R losses). Step down to 12.47 kV for distribution. Step down to 480V at the building. Step down to 120V at the outlet. Five voltage levels, five transformers, one path.
Resistance + Inductance + Capacitance — The Three Passive Elements
| Element | Symbol | What it does | How it acts in AC | Phase relationship |
|---|---|---|---|---|
| Resistor | R | Dissipates energy as heat. Pure friction. | Voltage and current rise/fall together. No energy storage. | V and I IN PHASE (PF = 1.0) |
| Inductor (L) — coil of wire, motor windings, transformer primary | L | Stores energy in magnetic field. Resists changes in current. | Voltage LEADS current by 90°. Current lags. | I lags V by 90° (PF = 0) |
| Capacitor (C) — two metal plates separated by insulator | C | Stores energy in electric field. Resists changes in voltage. | Voltage LAGS current by 90°. Current leads. | I leads V by 90° (PF = 0) |
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 (XL) and capacitive reactance (XC) too.
| Reactance | Formula | Notes |
|---|---|---|
| Inductive (XL) | XL = 2π × f × L | Higher frequency → more reactance. (Why high-frequency harmonics get blocked by inductors.) |
| Capacitive (XC) | XC = 1 / (2π × f × C) | Higher frequency → LESS reactance. (Why capacitors short out high-frequency noise.) |
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.
Series + Parallel Circuits
| Configuration | Resistance | Voltage | Current |
|---|---|---|---|
| Series (one path) | Rtotal = R1 + R2 + ... | V splits across each R | Same I through every R |
| Parallel (multiple paths) | 1/Rtotal = 1/R1 + 1/R2 + ... | Same V across every R | I splits inversely to R |
| Two parallel resistors (special case) | Rtotal = (R1 × R2) / (R1 + R2) | — | — |
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.
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).
- Multiple-source fault analysis: contribution from utility + on-site generation + motor back-feed. Sum each source's contribution to total fault current.
- Power flow with co-generation: utility provides part of load, on-site PV provides rest. Solve each separately, sum.
- Harmonic analysis: each harmonic frequency is solved separately, then summed (since circuit elements respond differently at each frequency).
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.
| Equivalent | What it is | How to find |
|---|---|---|
| Thevenin equivalent (Vth, Rth) | Voltage source Vth in SERIES with resistance Rth | Vth = open-circuit voltage at the terminals. Rth = resistance looking into the terminals with all sources zeroed. |
| Norton equivalent (In, Rn) | Current source In in PARALLEL with resistance Rn | In = short-circuit current at the terminals. Rn = Rth (same as Thevenin). |
| Conversion | — | Vth = In × Rn · In = Vth / Rth |
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:
| Sequence | Symbol | What it is | When it exists |
|---|---|---|---|
| Positive sequence | V1, I1 | Balanced 3φ rotating ABC | Always present in normal operation |
| Negative sequence | V2, I2 | Balanced 3φ rotating ACB (reverse) | Created by phase-phase faults, single-phasing of motors, unbalanced loads |
| Zero sequence | V0, I0 | 3 in-phase quantities (no rotation) | Created by ground faults; flows in neutral; only exists in 4-wire systems |
Each sequence has its own equivalent impedance for any piece of equipment. Faults are then analyzed by combining sequence networks. Full deep dive in §12 Short Circuit.
Power = Energy / Time — kW vs kWh
The most common confusion in EE literacy. Let's nail it:
| Power (kW) | Energy (kWh) | |
|---|---|---|
| What it is | Rate of energy transfer (instantaneous) | Total energy moved over time |
| Unit | Watt = Joule/sec; kW = 1000 W | Watt-hour = 3600 J; kWh = 1000 Wh |
| Plumbing analogy | Flow rate (gal/min) | Total water moved (gallons) |
| Math | — | kWh = kW × hours |
| What you're billed for | Demand charge ($/kW) | Energy charge ($/kWh) |
| Typical magnitudes | House peak: 5-15 kW. Office: 50-500 kW. Atlas DC1: 5,000 kW. | House annual: 10,000 kWh. Office annual: 1M kWh. Atlas DC1 annual: 44M kWh. |
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.
Magnetic Field Basics — How Motors Work
- 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.
- 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).
- 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.
- Spin a rotor in reverse — get a generator. Mechanical input → magnetic field changes around rotor → induces voltage in stator → output power.
- 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
| Level | Voltage | What happens here | Where in Atlas DC1 |
|---|---|---|---|
| Generation | 13.8 kV typical (at the generator) | Hydro, gas, nuclear, wind, solar generators produce electricity | The utility's plants — not on Atlas DC1 site |
| Transmission | 69 - 765 kV | Long-distance bulk power transfer. Step up to high voltage to minimize I²R losses over hundreds of miles. | — |
| Sub-transmission | 34.5 - 69 kV | Intermediate distribution from transmission substations to local distribution substations | — |
| Primary distribution | 4.16 - 34.5 kV (12.47 kV most common) | From distribution substation to neighborhoods/customers. Customer's MV service feeders. | Atlas DC1 utility service: 12.47 kV |
| Secondary distribution | 120/240V (residential), 480Y/277V (commercial) | Service transformer step-down to utilization voltage | Atlas DC1 480Y/277V main bus |
| Utilization | 120, 208, 240, 277, 415, 480V | The voltage your equipment runs on | Lighting (277V), receptacles (120V), motors (480V), IT (415Y/240V) |
Generator → Wall Outlet — One Joule's Journey
- Coal/gas/nuclear/water turns a turbine. Turbine spins a 60 Hz synchronous generator at 13.8 kV.
- Step-up transformer brings it to 230 kV for transmission.
- Power flows hundreds of miles via overhead transmission lines.
- Substation step-down: 230 kV → 69 kV sub-transmission.
- Distribution substation: 69 kV → 12.47 kV primary distribution.
- Neighborhood pole-mount transformer: 12.47 kV → 120/240V single-phase to your house. (Or → 480Y/277V three-phase to a commercial building.)
- Inside the building: panelboard → branch circuit → wall outlet.
- 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 PE Power site has exam-focused practice with the same notation conventions.
Theory primer · The "why" behind the "what" · Skip if you have an EE background