The Alternating Machine

NODE 01

DC vs AC

• DC: Maintains one direction.
• AC: Reverses direction periodically.

NODE 02

Sinusoidal AC

• v(t) = V₀ sin(ωt + φ).
• Amplitude, peak value, instantaneous value.

NODE 03

Period & Freq

T = 1/fω = 2πf

• Frequency determines cycles per second.
• Mains systems (50Hz / 60Hz).

NODE 04

RMS Value

• DC-equivalent value for same average power.
• V_rms = V₀/√2. I_rms = I₀/√2.

NODE 05

Inst Value

• Positive/negative half-cycles, zero crossings.
• Time dependence.

NODE 06

Phase

• In phase, out of phase, lead, lag.
• Phase angle difference between v and i.

NODE 07

Resistor in AC

• Voltage and current are IN PHASE (φ = 0).
• P_avg = V_rms * I_rms.

NODE 08

Capacitor in AC

• Current LEADS voltage by 90°.
• Capacitive reactance: Xc = 1/(ωC).

NODE 09

Capacitive Current

I = V / Xcf ↑ → Xc ↓

• Higher freq → lower Xc → more current passes.

NODE 10

Inductor in AC

• Current LAGS voltage by 90°.
• Inductive reactance: Xₗ = ωL.

NODE 11

Inductive Current

I = V / Xₗf ↑ → Xₗ ↑

• Higher freq → higher Xₗ → opposes changing current more.

NODE 12

Three Components

IN PHASE

LAGS 90°

LEADS 90°

• R: in phase. L: current lags. C: current leads.
• Foundation of LCR circuits.

NODE 13

Reactance (X)

Xₗ = ωLXc = 1/ωC

• Opposition associated with energy-storage.
• Does NOT dissipate energy as heat (stores/returns it).

NODE 14

Impedance (Z)

• Total opposition to sinusoidal current. Unit: Ω.
• Incorporates both resistance + reactance.

NODE 15

AC Ohm's Law

V=IZ

• V_rms = I_rms * Z.
• The AC analogue of V = IR.

NODE 16

Phasor Concept

• Rotating-vector representation of sinusoidal quantities.
• Makes LCR analysis visual rather than purely algebraic.

NODE 17

Phasor Diagram

• V_R aligned with I. V_L leads 90°. V_C lags 90°.
• V_net = √[V_R² + (V_L - V_C)²].

NODE 18

Series RLC Circuit

• R + L + C in series.
• Exhibits resistance, reactances, phase difference, impedance.

NODE 19

Series RLC Z

• Z = √[R² + (ωL - 1/ωC)²].
• I = V/Z. Major mathematical equation of Realm 20.

NODE 20

Phase Angle

• tanφ = (Xₗ - Xc)/R.
• Xₗ > Xc: Inductive. Xc > Xₗ: Capacitive.

NODE 21

AC Resonance

Xₗ = Xcω₀ = 1/√LC

• Occurs when inductive reactance = capacitive reactance.
• The central chamber of the realm.

NODE 22

At Resonance

Z = RI = MAXφ = 0

• Impedance is minimum (Z = R).
• Current is maximum. V and I are in phase.

NODE 23

Resonance Curve

• Current vs frequency.
• Peak at resonance. Connects to mechanical resonance.

NODE 24

Quality Factor (Q)

• Measure of resonance sharpness.
• Higher Q = sharper peak, lower relative damping.

NODE 25

Bandwidth

• Δf = f₂ - f₁.
• Q ≈ f₀/Δf. Connects resonance to frequency selectivity.

NODE 26

Selectivity

• LCR circuit can preferentially respond to frequencies near resonance.
• Radio tuning, filters, comms (Realm 29).

NODE 27

AC Power

P_avg = V_rms I_rms cosφcosφ = POWER FACTOR

• Instantaneous power: p(t) = v(t)i(t).
• Average power depends on the power factor.

NODE 28

Power Factor

RESISTORcosφ=1

INDUCTORcosφ=0

CAPACITORcosφ=0

• Real systems have both resistive and reactive parts (0 < cosφ < 1).

NODE 29

True vs Reactive

• Real (P = VI cosφ), Reactive (Q = VI sinφ).
• Apparent (S = VI). S² = P² + Q².

NODE 30

PF Correction

• Low PF increases current for given real power.
• Add capacitor banks to compensate.
• Improves transmission efficiency.

NODE 31

AC Circuit Energy

• Resistor: Dissipated as heat.
• Inductor: Stored magnetically.
• Capacitor: Stored electrically.

NODE 32

Generator

• Converts mechanical energy → electrical energy.
• Through electromagnetic induction (Realm 19).

NODE 33

AC Gen Structure

• Armature/coil, magnetic field, rotor, stator.
• Slip rings, brushes.

NODE 34

Rotating Coil

• Coil rotates → flux changes → EMF changes.
• Produces alternating voltage.

NODE 35

Gen Frequency

MECH ROTATION
↓
ELEC FREQ

• Electrical frequency depends on mechanical rotational speed and pole configuration.

NODE 36

Motor-Gen Symm

ELEC→ MOTOR →MECH

MECH→ GEN →ELEC

• Operating principles are deeply related.
• A wonderful civilization-scale Physics connection.

NODE 37

Transformer

• Integrate into AC power systems.
• Ideal: Vs/Vp = Ns/Np.

NODE 38

Step-Up Trans

• Ns > Np → Vs > Vp.
• High-voltage transmission (reduces current/resistive losses).

NODE 39

Step-Down Trans

• Ns < Np → Vs < Vp.
• Distribution, power supplies, equipment.

NODE 40

AC Transmission

• Station → Step-up → HV Lines → Step-down → Consumer.
• Transforms abstract AC math into civilization infrastructure.

NODE 41

Transmission Loss

P_loss =I²R

• Higher transmission voltage permits lower current.
• Reduces I²R loss for same transmitted power.

NODE 42

Three-Phase

• Three sinusoidal voltages, equal frequency, phase separated.
• Foundational to modern electric-power machinery.

NODE 43

3-Phase Power

• P = √3 VL IL cosφ.
• Distinguish line voltage/current from phase voltage/current.

NODE 44

AC Filters

• Low-pass, High-pass, Band-pass, Band-stop.
• Prepares Realm 29 (Signals).

NODE 45

RC Transient

• Capacitor charging/discharging.
• Time constant τ = RC.
• Bridge between steady-state AC and transient circuits.

NODE 46

RL Transient

• Inductors resist rapid current change.
• Time constant τ = L/R.

NODE 47

Energy Oscillation

• Capacitor energy ↔ Inductor energy.
• LC oscillation: ω₀ = 1/√LC.
• Full circle with oscillation Physics.

NODE 48

Experiment Lab

V_rms

RC

RL

LC_res

φ

• Measure AC waveform, peak, RMS, freq.
• RC/RL charging, LCR resonance, Transformer ratios.

NODE 49

Graph Chamber

• AC voltage vs time, Current vs freq (resonance).
• Impedance vs freq, Xc/Xₗ vs freq.

NODE 50

Misconceptions

X

• "AC means current is constantly zero" → No.
• "Capacitor stores current" → Stores energy.
• "Reactance consumes energy" → Temporarily stores it.

NODE 51

Solving Engine

• Find ω → ID R,L,C → Calc Xₗ, Xc → Find Z.
• Find I, Phase, Power. Check resonance ω₀.

NODE 52

Master Map

AC VOLTAGE → RMS/PHASER/X/Z → RLC → RESONANCEGEN → TRANS → TRANSMISSION

• The entire realm compressed.

NODE 53

Deep Synthesis

MECH OSCILLATION (R13)↕ELEC OSCILLATION (R20)

• Energy oscillates between forms.
• Mechanical: Kinetic ↔ Potential.
• LC circuit: Electric field ↔ Magnetic field.