Transformer

How transformers work

Two coils share a magnetic core (typically iron):

• Primary (N₁ turns) — connected to AC source.
• Secondary (N₂ turns) — provides AC output.
• Iron core couples them magnetically.

AC current in primary creates oscillating flux Φ in the core. By Faraday's law, this flux through secondary induces voltage:

V₁ = -N₁ · dΦ/dt
V₂ = -N₂ · dΦ/dt

Ratio:

V₂ / V₁ = N₂ / N₁

Voltage and current ratios

Quantity	Ratio
Voltage	V₂/V₁ = N₂/N₁ (turn ratio)
Current (ideal)	I₂/I₁ = N₁/N₂ (inverse)
Power	V₁·I₁ = V₂·I₂ (conserved, ideal)
Impedance	Z₂/Z₁ = (N₂/N₁)² (squared)

Real-world examples

Use	V₁	V₂	Turn ratio
USA power generation	22 kV	765 kV (transmission)	1:35 step-up
Power plant to grid	765 kV	765 kV (already up)	1:1 (just isolation)
Transmission to distribution	765 kV	13.8 kV	55:1 step-down
Pole-mounted (US)	13.8 kV	240 V (split phase)	57:1 step-down
Phone charger (transformer-based)	120 V	5 V (after rectification)	~24:1
Doorbell transformer	120 V	16-24 V	~5-7:1
Tube guitar amp output	~4 kV	16, 8, or 4 Ω matching	~hundreds:1

JavaScript — transformer calculations

// Voltage ratio
function transformer(V_primary, N_primary, N_secondary) {
  return V_primary * N_secondary / N_primary;
}

// 120V primary, 100 turns, secondary 1000 turns
console.log(`Step-up to: ${transformer(120, 100, 1000)} V`);  // 1200 V

// 12000V primary, 100 turns, secondary 10 turns
console.log(`Step-down to: ${transformer(12000, 100, 10)} V`);  // 1200 V

// Current ratio (ideal)
function transformerCurrent(I_primary, N_primary, N_secondary) {
  return I_primary * N_primary / N_secondary;
}

// Power conservation check
function transformerPowerCheck(V1, I1, V2, I2) {
  return Math.abs(V1*I1 - V2*I2) < 0.01 * V1*I1;  // within 1%
}

// Impedance transformation
function transformerImpedance(Z_load, N_primary, N_secondary) {
  // Reflected impedance on primary side: Z_p = Z_load · (N_p/N_s)²
  return Z_load * (N_primary / N_secondary) * (N_primary / N_secondary);
}

// 8 Ω speaker, 4kV primary tube amp, with 100:8 transformer
console.log(`8Ω reflected: ${transformerImpedance(8, 100, 8)} Ω`); // 1250 Ω

// Transmission loss reduction via voltage step-up
function transmissionLoss(P_load, V, R) {
  return Math.pow(P_load / V, 2) * R;
}

// 1 MW over 100 km of wire (R ~ 10 Ω)
const loss_120 = transmissionLoss(1e6, 120, 10);
const loss_12k = transmissionLoss(1e6, 12000, 10);
const loss_120k = transmissionLoss(1e6, 120000, 10);

console.log(`@120V: ${(loss_120/1e6).toFixed(0)} MW lost (${(loss_120/1e4).toFixed(0)}× more than load!)`);
console.log(`@12kV: ${(loss_12k/1000).toFixed(0)} kW`);   // Reasonable
console.log(`@120kV: ${(loss_120k/1000).toFixed(2)} kW`); // Tiny

// Real transformer efficiency
function realPowerOut(V_in, I_in, efficiency) {
  return V_in * I_in * efficiency;
}

// 99% efficient 100 W transformer
console.log(`100W input, 99% eff: ${realPowerOut(120, 100/120, 0.99).toFixed(0)} W out`); // 99 W

// LRC time constant for primary current rise
function transformerStartup(L, R) {
  return L / R;  // τ for current to reach ~63% of final
}

Where transformers matter

• Power grid. Multiple stages of step-up at generation; step-down at distribution and end use.
• Power supplies. Convert 120/240V wall power to lower DC voltages for electronics.
• Audio. Tube amplifiers use output transformers to match tube impedance to speaker impedance.
• Welding. Welding transformers step down voltage and step up current dramatically (low V, high I produces arc).
• Isolation. Some transformers exist to isolate circuits (1:1 ratio) — for safety, noise reduction.
• RF and communications. Impedance matching, balanced-to-unbalanced (balun).
• Voltage measurement. Measurement transformers (CT, PT) step down high-voltage signals to safe levels for meters.

Common mistakes

• Trying with DC. Doesn't work — needs alternating flux. Use DC-DC converters for DC voltage change.
• Forgetting current ratio is INVERSE. Step-up voltage → step-DOWN current. Easy to flip.
• Treating as 100% efficient. Real losses ~1-5% in modern transformers. Significant for large industrial units.
• Mismatched current rating. Each side has max current rating; exceed and the transformer overheats.
• Ignoring impedance transformation. Z scales as turn ratio SQUARED. Important for audio amp output stages and RF impedance matching.
• Confusing turn ratio with voltage ratio in non-ideal cases. For ideal transformer, V₂/V₁ = N₂/N₁ exactly. Real transformers have small deviations under load.