§ 46. AC Transformer & Mutual Electromagnetic Induction (Grade 9–11)
Mutual electromagnetic induction via shared alternating magnetic flux Φ(t) in a ferromagnetic core and turns ratio U₂/U₁ = N₂/N₁. An alternating magnetic flux confined within a closed iron core transfers electrical power contactlessly between circuits while scaling voltage.
Theory Summary & Scientific Mechanism
An AC transformer steps alternating voltage up or down at constant frequency f via mutual electromagnetic induction. A primary winding of N₁ turns and a secondary winding of N₂ turns are wound around a closed laminated ferromagnetic core of high permeability (μ sim 10³–10⁴), which channels the alternating magnetic flux Φ(t) = Φₘsin (2π f t) through every turn of both coils.
By Faraday’s law of induction, each turn experiences the same induced EMF -(dΦ)/(dt). Consequently, the ratio of secondary to primary RMS voltages equals the turns ratio: (U₂)/(U₁) = (N₂)/(N₁) = k. For N₂ > N₁ (k > 1) it acts as a step-up transformer; for N₂ < N₁ (k < 1) as a step-down transformer.
By conservation of energy with core efficiency η (typically 95–99%, using silicon-steel laminations to suppress eddy currents and hysteresis), output power satisfies P₂ = η P₁ ⇒ U₂ I₂ = η U₁ I₁. Thus stepping up voltage proportionally steps down current: (I₂)/(I₁) = η(N₁)/(N₂). High-voltage power grids (110–750 kV) exploit this to slash transmission line Joule losses Pₗoss = I² Rₗine ∝ 1/U²!
Everyday Analogy
Your phone charger and neighborhood power-grid substations both rely on transformers! Alternating current in the primary coil sends a pulsing magnetic flux through the steel core, inducing voltage in the secondary coil without the wires ever touching: more turns mean higher voltage!
Interactive Question & Derivation
Why can a conventional transformer not step up or step down a steady DC voltage from a battery (ω = 0), even when its primary winding has hundreds of turns?
- A: Because induced EMF in the secondary winding ℰ₂ = -N₂ (dΦ)/(dt) arises only from a time-varying magnetic flux; direct current produces a constant flux ((dΦ)/(dt) = 0), so U₂ = 0
- B: Because a ferromagnetic core cannot conduct a constant magnetic field
- C: Because under direct current U₂ always equals U₁
Study Hint: From the turns ratio (U₂)/(U₁) = (N₂)/(N₁) and power conservation U₂ I₂ ≈ U₁ I₁: when N₂ > N₁ steps voltage up, what happens to secondary current I₂?
Proof & Explanation: A transformer operates by Faraday's law of electromagnetic induction: each turn links the common core flux Φ(t) = Φ₀ sin (ω t), inducing an EMF e(t) = -(dΦ)/(dt). Consequently, the ratio of RMS voltages equals the turns ratio (U₂)/(U₁) = (N₂)/(N₁).
Key Equations & 3D Simulation Parameters
(U₂)/(U₁) = (N₂)/(N₁), ℰ₂(t) = -N₂ ω Φ₀ cos (ω t)
- Primary RMS voltage U₁: 50 … 220 V (default: 110 V)
- Primary winding turns N₁: 10 … 50 (default: 20)
- Secondary winding turns N₂: 10 … 80 (default: 40)
- AC frequency f (scaled): 1 … 5 Hz (default: 2 Hz)