Electrical Theory
Why This Matters
You can wire a light to a battery using trial and error, but you cannot build a generator, a transformer, or a radio without understanding what electricity actually is. This article gives you the theoretical foundation to design, troubleshoot, and scale electrical systems. Every watt of power your community generates depends on the principles covered here.
What Electricity Actually Is
Forget the water analogy for now. You need to understand what is physically happening inside a wire.
Atoms and Electrons
Every material is made of atoms. Each atom has a nucleus (protons + neutrons) surrounded by orbiting electrons. The outermost electrons are called valence electrons. In metals like copper, these valence electrons are loosely bound. They drift freely between atoms even without an external force.
When you connect a battery to a wire, the battery creates an electric field across the wire. This field pushes free electrons in one direction. That movement of electrons is electric current.
Key numbers:
- A single electron carries a tiny charge: 1.6 x 10^-19 coulombs
- One amp of current = 6.24 x 10^18 electrons passing a point per second
- Electrons actually move slowly (millimeters per second) but the push travels at near light speed, like a tube full of marbles: push one in, one pops out the other end instantly
Conductors vs Insulators
| Material | Behavior | Why |
|---|---|---|
| Copper | Excellent conductor | 1 loosely-held valence electron per atom |
| Aluminum | Good conductor | Similar atomic structure, lighter but higher resistance |
| Iron/Steel | Fair conductor | More tightly bound electrons, higher resistance |
| Water (impure) | Weak conductor | Dissolved ions carry charge |
| Wood | Insulator | No free electrons |
| Glass | Insulator | Electrons tightly bound |
| Rubber | Insulator | No free electrons, used for wire insulation |
| Air | Insulator (usually) | Breaks down at very high voltage (lightning) |
Tip
The distinction is not absolute. Wet wood conducts. Air conducts during lightning. Even glass conducts if heated enough. In practical terms, treat any dry non-metal as an insulator and any metal as a conductor.
Conventional vs Electron Flow
Historical accident: Benjamin Franklin guessed that current flows from positive to negative. He was wrong. Electrons actually flow from negative to positive. But by the time anyone figured this out, all the conventions were set.
Convention used in all circuit diagrams: Current flows from (+) to (-). This is called conventional current flow. Every schematic, every textbook, every engineer uses this convention. Stick with it. The physics does not matter for practical work.
Ohm’s Law: Going Deeper
You already know V = IR from Basic Electrical Circuits. Now let us add power to the equation.
The Power Equation
Power is the rate of energy use, measured in watts (W).
P = V x I (Power = Voltage x Current)
Combined with Ohm's Law:
P = I² x R (useful when you know current and resistance)
P = V² / R (useful when you know voltage and resistance)
Practical Calculations
Example 1: Wire heating You have 50 meters of 18 AWG copper wire (resistance approximately 0.021 ohms/meter, so 1.05 ohms total) carrying 5A.
Power lost in wire = I² x R = 5² x 1.05 = 26.25 watts
That is 26 watts of heat generated in the wire itself. On a 12V system at 5A (60W total), you are losing 44% of your power to heat in the wire alone. This is why wire gauge matters.
Example 2: Choosing a resistor You want to limit an LED (rated 20mA, 2V forward voltage) on a 12V supply.
Voltage across resistor: 12V - 2V = 10V
Resistance needed: R = V/I = 10/0.02 = 500 ohms
Power dissipated by resistor: P = V x I = 10 x 0.02 = 0.2 watts
A quarter-watt resistor will work, but it will run warm. A half-watt resistor is safer.
Warning
Power dissipation creates heat. A 1-watt resistor in free air gets hot enough to burn skin. A 5-watt resistor can ignite paper. Always consider the power rating of every component, not just the resistance value.
Kirchhoff’s Laws
These two laws let you analyze any circuit, no matter how complex. They are not difficult. They simply formalize common sense.
Kirchhoff’s Voltage Law (KVL)
Statement: The sum of all voltages around any closed loop in a circuit equals zero.
In plain language: the battery pushes voltage up, the components drop voltage down, and it all balances out.
Example: A 12V battery with two resistors in series (R1 = 4 ohms, R2 = 8 ohms), total current I = 12/12 = 1A.
Battery: +12V
Drop across R1: -(1A x 4Ω) = -4V
Drop across R2: -(1A x 8Ω) = -8V
Sum: +12 - 4 - 8 = 0 ✓
Why this matters practically: If you measure the voltage across each component in a series circuit and they do not add up to the supply voltage, you have a bad connection, a hidden resistance, or a faulty component somewhere.
Kirchhoff’s Current Law (KCL)
Statement: The total current entering any junction (node) equals the total current leaving that junction.
In plain language: electrons do not appear or disappear. What flows in must flow out.
Example: A wire splits into two branches. Branch 1 has a 12-ohm load, Branch 2 has a 6-ohm load, both on 12V.
Branch 1 current: I = 12V / 12Ω = 1A
Branch 2 current: I = 12V / 6Ω = 2A
Total entering junction: 3A
Total leaving junction: 1A + 2A = 3A ✓
Tip
Use KVL to find unknown voltages. Use KCL to find unknown currents. Together, they solve any circuit. In practice, you will use these instinctively once you understand the pattern: voltages in a loop add to zero, currents at a junction add to zero.
Voltage Dividers
A voltage divider is two resistors in series used to produce a specific lower voltage from a higher source. This is one of the most common and useful circuits.
12V ----[R1: 8Ω]----+----[R2: 4Ω]---- GND
|
Output: 4V
Formula:
V_out = V_in x (R2 / (R1 + R2))
V_out = 12 x (4 / (8 + 4)) = 12 x 0.333 = 4V
Practical uses:
- Reducing voltage for sensors or low-voltage circuits
- Creating reference voltages
- Biasing transistor circuits
Warning
A voltage divider only works correctly under light load (very little current drawn from the output). If you connect a heavy load to the output, it acts as a parallel resistance with R2 and changes the output voltage. For powering devices, use a voltage regulator instead.
Capacitance
A capacitor stores energy in an electric field between two conductive plates separated by an insulator (dielectric).
How Capacitors Work
When you connect a capacitor to a battery:
- Electrons pile up on one plate (making it negative)
- Electrons are pulled away from the other plate (making it positive)
- The voltage across the capacitor rises until it equals the battery voltage
- Current stops flowing
When you disconnect the battery and connect a load, the capacitor discharges through the load, releasing its stored energy.
Capacitance Value
Capacitance (C) is measured in farads (F). One farad is enormous. Practical capacitors are measured in:
- Microfarads (uF) = one millionth of a farad
- Nanofarads (nF) = one billionth
- Picofarads (pF) = one trillionth
What affects capacitance:
- Larger plate area = more capacitance
- Plates closer together = more capacitance
- Better dielectric material = more capacitance
Time Constants
A capacitor does not charge or discharge instantly. The time it takes depends on the capacitance and the resistance in the circuit.
Time constant (τ) = R x C
After 1τ: capacitor reaches 63% charge
After 3τ: reaches 95%
After 5τ: effectively fully charged (99%)
Example: 100uF capacitor through a 10K-ohm resistor:
τ = 10,000 x 0.0001 = 1 second
Full charge: approximately 5 seconds
Inductance
An inductor stores energy in a magnetic field created by current flowing through a coil of wire.
How Inductors Work
When you apply voltage to an inductor:
- Current starts to flow and builds up gradually
- The rising current creates an expanding magnetic field around the coil
- This expanding field opposes the change in current (Lenz’s law)
- Current slowly rises until it reaches its maximum (limited by wire resistance)
When you disconnect the voltage source:
- The magnetic field collapses
- The collapsing field pushes current, trying to maintain the flow
- This can create very high voltage spikes
Inductance Value
Inductance (L) is measured in henries (H). Practical inductors are millihenries (mH) or microhenries (uH).
What affects inductance:
- More turns of wire = more inductance (proportional to turns squared)
- Larger coil area = more inductance
- Iron core instead of air core = much more inductance
- Shorter coil length = more inductance
Warning
When you break the circuit to an inductor suddenly (opening a switch), the collapsing magnetic field generates a voltage spike that can be many times the supply voltage. This spike can arc across switch contacts, damage components, and shock you. Always use a diode across inductor-driven loads (like relay coils) to absorb this spike.
AC vs DC: The Full Story
What Is AC?
Direct current (DC) flows in one direction. Alternating current (AC) reverses direction many times per second, following a sine wave pattern.
DC: ___________________________ (steady line)
AC: /\ /\ /\ /\
/ \ / \ / \ / \ (sine wave)
\/ \/ \/ \/
Key AC Concepts
Frequency: How many complete cycles per second, measured in Hertz (Hz).
- North American standard: 60 Hz (60 complete cycles per second)
- European/most world standard: 50 Hz
- For a rebuilding community, the exact frequency matters less than consistency
Peak vs RMS Voltage: AC voltage swings from positive peak to negative peak. The “effective” voltage that does the same work as an equivalent DC voltage is called RMS (root mean square).
V_RMS = V_peak x 0.707
V_peak = V_RMS x 1.414
Example: 120V AC (RMS) actually peaks at 170V
Example: 240V AC (RMS) actually peaks at 340V
Phase: The timing relationship between two AC signals. Two generators must be “in phase” (peaks aligned) to work together. Out-of-phase generators fight each other and waste energy or cause damage.
Why AC Won the War of Currents
Thomas Edison championed DC. Nikola Tesla and George Westinghouse championed AC. AC won for one devastating reason: transformers.
A transformer can step AC voltage up or down with almost no energy loss. DC cannot be transformed (without complex electronics).
Why this matters for transmission:
Power loss in a wire = I squared times R. If you need to transmit 1,000 watts:
| Voltage | Current (P=VI) | Loss in 1Ω wire (I²R) | % Lost |
|---|---|---|---|
| 100V | 10A | 100W | 10% |
| 1,000V | 1A | 1W | 0.1% |
| 10,000V | 0.1A | 0.01W | 0.001% |
Step voltage up by 10x, current drops by 10x, losses drop by 100x. This is why power lines carry electricity at thousands of volts, then step it down near your home. Without transformers (which only work with AC), long-distance power transmission is impractical.
Tip
For a small community with generators close to loads (under 100 meters), DC works fine. Once you need to send power more than a few hundred meters, you need AC and transformers. Plan your electrical infrastructure accordingly.
Electromagnetism
This is the bridge between electricity and mechanical motion. Every generator, motor, relay, speaker, and transformer depends on electromagnetism.
Current Creates Magnetic Fields
When current flows through a wire, it creates a circular magnetic field around the wire. The right-hand rule tells you the direction: point your right thumb in the direction of conventional current flow, and your fingers curl in the direction of the magnetic field.
Coils Concentrate the Field
A single wire creates a weak field. Winding the wire into a coil concentrates the field dramatically. An iron core inside the coil concentrates it further (by 100x to 10,000x depending on the iron).
Single wire: weak circular field
10-turn coil: 10x stronger field
10-turn coil 100-1000x stronger field
+ iron core:
This is how you build electromagnets, relay coils, transformer cores, and generator field windings.
Faraday’s Law of Electromagnetic Induction
The fundamental law behind all generators:
When a magnetic field through a coil changes, a voltage is induced in the coil. The faster the field changes, the higher the voltage. More turns of wire produce more voltage.
There are three ways to change the field through a coil:
- Move a magnet near the coil (or move the coil near a magnet)
- Rotate a coil inside a magnetic field (this is how generators work)
- Change the current in a nearby coil (this is how transformers work)
Lenz’s Law
The induced voltage always opposes the change that created it. Push a magnet toward a coil, and the induced current creates a magnetic field that pushes the magnet away. Pull it away, and the induced field tries to pull it back.
This is not just a theoretical curiosity. It is why generators require mechanical force to turn. The electrical load creates a magnetic force that resists rotation. More electrical load = harder to turn = more mechanical input needed. Energy is conserved.
Impedance: AC Resistance
In DC circuits, only resistance (R) limits current flow. In AC circuits, capacitors and inductors also limit current flow, but in a frequency-dependent way.
Capacitive Reactance
A capacitor resists changes in voltage. In AC, voltage changes constantly, so a capacitor partially blocks current flow. The opposition is called capacitive reactance (Xc).
Xc = 1 / (2π x f x C)
Higher frequency → less opposition (capacitor passes more AC)
Higher capacitance → less opposition
This is why capacitors are used as filters: they block DC (zero frequency, infinite reactance) while passing AC.
Inductive Reactance
An inductor resists changes in current. In AC, current changes constantly, so an inductor partially blocks current flow. The opposition is called inductive reactance (XL).
XL = 2π x f x L
Higher frequency → more opposition (inductor blocks more AC)
Higher inductance → more opposition
Inductors do the opposite of capacitors: they pass DC freely while blocking high-frequency AC.
Impedance
Impedance (Z) is the total opposition to current flow in an AC circuit, combining resistance, capacitive reactance, and inductive reactance.
Z = √(R² + (XL - Xc)²)
AC version of Ohm's Law: V = I x Z
Tip
You do not need to calculate impedance for basic power systems. It becomes critical when building radios, filters, and matching generators to loads. For now, understand the concept: capacitors and inductors behave differently in AC versus DC, and frequency matters.
Practical Circuit Analysis
When faced with an unfamiliar circuit, follow this systematic approach:
Step 1: Identify the Source
Find the power supply. Note voltage, AC or DC, and current capacity.
Step 2: Trace the Current Path
Follow the wire from positive terminal through every component back to negative. Draw it out if needed. Every component is either in series (same path) or parallel (branching paths).
Step 3: Simplify
Combine series resistances (add them). Combine parallel resistances (1/Rtotal = 1/R1 + 1/R2). Repeat until you have one equivalent resistance.
Step 4: Calculate
Use Ohm’s law (V = IR) and power equation (P = VI) to find voltages, currents, and power at each point.
Step 5: Check with Kirchhoff
Verify: voltages around each loop sum to zero. Currents at each junction sum to zero. If they do not, you made an error.
Example: Mixed series-parallel circuit
A 12V battery feeds two parallel branches. Branch A has a 6-ohm resistor. Branch B has a 3-ohm resistor. Both branches connect through a single 2-ohm resistor in series.
Step 1: Parallel combination: 1/Rp = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 → Rp = 2Ω
Step 2: Total resistance: 2Ω (series) + 2Ω (parallel) = 4Ω
Step 3: Total current: I = 12V / 4Ω = 3A
Step 4: Voltage across series resistor: 3A x 2Ω = 6V
Step 5: Voltage across parallel section: 12V - 6V = 6V
Step 6: Current in Branch A: 6V / 6Ω = 1A
Step 7: Current in Branch B: 6V / 3Ω = 2A
Check: 1A + 2A = 3A ✓ (KCL at junction)
Electrical Safety
Electricity kills through two mechanisms: cardiac arrest (current through the heart disrupts its rhythm) and burns (current generates heat in tissue).
Lethal Current Thresholds
| Current (mA) | Effect on Human Body |
|---|---|
| 1 mA | Barely perceptible tingle |
| 5 mA | Painful shock, can let go |
| 10-20 mA | Muscular contraction, cannot let go (“let-go threshold”) |
| 50-100 mA | Ventricular fibrillation (heart stops pumping) |
| 100-200 mA | Certain cardiac arrest |
| Over 200 mA | Severe burns, cardiac arrest |
Critical insight: It takes very little current to kill. The reason 12V batteries are relatively safe is not because the current is low, but because 12V cannot push enough current through the high resistance of dry human skin (typically 10,000-100,000 ohms).
At 12V through 10,000Ω skin: I = 12/10,000 = 1.2 mA (tingle)
At 120V through 10,000Ω skin: I = 120/10,000 = 12 mA (cannot let go)
At 240V through 10,000Ω skin: I = 240/10,000 = 24 mA (potentially lethal)
Warning
Wet skin drops resistance to 1,000 ohms or less. At 120V through wet skin: I = 120/1,000 = 120 mA. Instantly lethal. Never work on AC systems with wet hands, in rain, or while standing in water.
Grounding Saves Lives
A proper ground connection provides a low-resistance path for fault current to flow to earth instead of through a person. When a hot wire contacts a grounded metal enclosure, massive current flows through the ground wire, tripping the fuse or breaker and de-energizing the circuit in milliseconds.
Without grounding, that enclosure sits at line voltage waiting for someone to touch it and provide a path to ground through their body.
Safety Rules for Higher Voltages
Once your community builds generators and transformers producing voltages above 50V:
- One-hand rule: Work with one hand only. Keep the other in your pocket. This prevents current from flowing arm-to-arm through your heart.
- De-energize first. Disconnect and verify zero voltage before touching anything.
- Lock out, tag out. Physically prevent someone from re-energizing a circuit while you are working on it.
- Insulate your feet. Stand on dry wood, rubber mat, or dry ground.
- Never work alone. Someone nearby should know how to disconnect power and perform rescue breathing.
What’s Next
With theoretical understanding, you are ready to build real power systems:
- Generators and Motors — convert mechanical energy to electricity and back
- Power Transmission — transformers, wiring, and distribution grids
- Telegraph — apply electromagnetic principles to long-distance communication
Electrical Theory — At a Glance
Core equations:
- Ohm’s Law: V = IR
- Power: P = VI = I²R = V²/R
- Kirchhoff’s Voltage Law: voltages around a loop = 0
- Kirchhoff’s Current Law: currents at a junction = 0
Concept Key Point Electrons Negative charges moving through conductor; conventional current is opposite direction Voltage divider V_out = V_in x (R2 / (R1 + R2)) Capacitors Store energy in electric field, block DC, pass AC Inductors Store energy in magnetic field, pass DC, block AC AC advantage Transformable — step up for transmission, step down for use Faraday’s law Changing magnetic field induces voltage in a coil Impedance Z = total AC opposition (resistance + reactance) Safety thresholds:
- 10 mA: cannot let go
- 50 mA: heart fibrillation
- Wet skin drops resistance 10x — 120V wet = lethal
The one rule: It is current that kills, not voltage. But voltage determines how much current your body’s resistance allows through.