Transformer (Mutual Induction)

How a transformer works

Two coils share a closed iron core. Apply an AC voltage v_p(t) to the primary winding (N_p turns). The current sets up a time-varying flux Φ(t) in the core. Because the core links both windings, that same flux passes through the secondary (N_s turns). By Faraday's law of induction, the secondary winding sees an induced voltage:

            dΦ                    dΦ
   v_p = N_p ──            v_s = N_s ──
            dt                    dt

   ratio: v_s / v_p = N_s / N_p     (the turns ratio)

If the transformer were lossless, conservation of energy gives the matching current relationship: V_p · I_p = V_s · I_s, so step up the voltage and you step down the current by the same ratio. That second relationship is why long-distance transmission runs at hundreds of kilovolts — at 400 kV instead of 230 V, the same power flows at 1/1700 the current, and resistive losses (I²R) drop by a factor of 1700².

                ┌───┐  iron core   ┌───┐
                │   │              │   │
                │   ╱╱╲╲══════════╱╱╲╲ │
                │  ╱       Φ(t)        ╲│
       v_p ────●  ╱        ──→         ╲ ●──── v_s
       (N_p)  │  ╲                     ╱│  (N_s)
              │   ╲╲╱╱══════════════╲╲╱╱│
              │   │                  │   │
              └───┘                  └───┘
                primary           secondary

Worked example: 240 V → 24 V at 100 W

Take a small bench-power-supply transformer rated 100 W with 240 V mains primary and 24 V secondary.

   turns ratio  n = N_p / N_s = V_p / V_s = 240 / 24 = 10:1

   secondary current at full load:
      I_s = 100 W / 24 V = 4.17 A

   primary current (ideal):
      I_p = I_s / n = 4.17 / 10 = 0.417 A

   primary current (with 95% efficiency):
      I_p_real = 100 / (240 × 0.95) ≈ 0.439 A

The same physics scales massively. A grid step-up transformer at a 1 GW power plant might convert 22 kV generator output to 400 kV transmission. The turns ratio is 22:400, or about 1:18. At 1 GW the primary current is roughly 26,000 A and the secondary is 1,400 A. That is why the 22 kV bus uses copper bars the size of car bumpers while the 400 kV side uses just a few thick conductors.

Where does the lost 1% go?

Real transformers lose energy in two distinct ways:

• Copper losses (I²R) in the windings. Proportional to load current squared. Zero at no load, maximum at rated load. Reduced by using more copper (lower R) at the cost of higher capital expense.
• Iron losses (core losses) — hysteresis + eddy currents. Roughly constant from no-load to full-load, because they depend on the flux swing in the core, not the load current. Reduced by using grain-oriented silicon steel (lower hysteresis) and laminating the core to thin sheets (suppresses eddy currents).

A well-designed power transformer has copper losses about equal to iron losses at full rated load — that's the optimum point for total annual energy loss given a typical load profile. Modern large units achieve no-load losses of 0.05% and full-load losses of 0.4–0.5%, for a peak efficiency of 99.5% or better.

Transformer variants compared

Type	Turns ratio	Galvanic isolation?	Cooling	Typical kVA range	Where used
Step-up	n < 1 (more secondary turns)	Yes	Oil-immersed	10 MVA – 1.5 GVA	Generator → transmission
Step-down	n > 1 (fewer secondary turns)	Yes	Oil or dry	5 kVA – 50 MVA	Distribution → consumer
Autotransformer	Close to 1:1, single tapped winding	No (shared neutral)	Oil or dry	1 kVA – 1 GVA	Voltage matching, motor starting, HVDC interconnects
Isolation	1:1	Yes — explicit safety barrier	Dry	0.1 kVA – 100 kVA	Medical, lab equipment, audio
Three-phase	3 phases on shared 3 or 5-leg core	Yes	Oil-immersed	500 kVA – 1.5 GVA	Transmission, large industrial
Instrument (CT/PT)	1:N high or 1:N low	Yes	Dry, encapsulated	VA range	Metering, protection relaying
Switch-mode (high-frequency)	Various	Yes	Forced air or natural	1 W – 50 kW	SMPSs, EV chargers, USB-PD bricks

Construction families

• Oil-immersed. The core and windings are submerged in mineral or ester oil that both insulates and conducts heat to a corrugated tank wall. The standard for outdoor transformers above about 1 MVA. Oil also breaks down dielectrically before the cellulose insulation does, giving an early-warning fault path. Internal conservator tanks accommodate thermal expansion.
• Dry-type (cast-resin). Windings encapsulated in epoxy resin. No oil to leak or catch fire — safe for indoor and basement installation. Limited to roughly 36 kV, 30 MVA. Standard in commercial buildings, data centres, hospitals.
• Vacuum-cast resin (VPI). Windings wound, then dipped in epoxy under vacuum to drive out air voids. Slightly cheaper than full cast-resin, less robust. Used in industrial transformers and traction.
• Open-ventilated dry. Air-cooled with no encapsulation. Cheap and serviceable but vulnerable to dust and humidity. Common in older buildings and lower-voltage commercial installs.
• Toroidal. Ring-shaped tape-wound core, windings spread around the ring. Lower stray flux, lower hum, slightly more expensive to wind. Common in audio amplifiers and medical isolation transformers.

Real-world specs and milestones

• ASEA HVDC Pacific Intertie, 1965. The first 800 kV-class high-voltage DC transformer set, built by ASEA (now ABB) for the Sylmar converter station. 800 kV bushings stood three storeys tall. The 1965 line ran 1,360 km from the Pacific Northwest to Los Angeles at 1,440 MW.
• Generator step-up at a typical 660 MW unit. A 750 MVA, three-phase, 18.5/420 kV transformer, oil-immersed, 360 tonnes. Tap changer adjusts voltage in 1.25% steps under load. Efficiency 99.5% at full load.
• Distribution pole transformer. 25 kVA, single-phase, 11 kV / 240 V, oil-immersed, 250 kg, hangs from a wooden pole. The omnipresent suburban transformer can.
• USB-C power-delivery brick. 65 W output, 100 kHz switching frequency, ferrite-core flyback transformer the size of a fingernail. Operates near 1.6 T peak flux on the ferrite core. Efficiency ~92%.
• Audio output transformer (vintage tube amp). 25 W, 4–8 Ω secondary, 5 kΩ primary load. Frequency response 20 Hz to 20 kHz, ±0.5 dB. Hand-wound interleaved primary-secondary windings to control leakage inductance.

Common failure modes

• Core saturation from DC offset. Geomagnetically induced currents (during solar storms), HVDC-converter half-cycle saturation, or asymmetric inrush during energisation can drive the core past 1.8 T. Magnetising current spikes 10–50×, audible groan rises, harmonics flood the system. The 1989 Quebec blackout was triggered partly by GIC-driven saturation in step-up transformers.
• Inrush current at energisation. Closing the breaker at the wrong point of the AC waveform can momentarily double flux and saturate the core. The first half-cycle of magnetising current can reach 10× nameplate. Modern relays use second-harmonic restraint to distinguish inrush from a real internal fault.
• Insulation breakdown / interturn shorts. Moisture in oil, partial-discharge erosion of cellulose, or an overvoltage transient that exceeds basic insulation level (BIL) creates a turn-to-turn short. Once started, the local arc carbonises insulation and propagates over seconds. Buchholz relays detect the gas evolution and trip the unit.
• Tap-changer arcing. On-load tap changers shift between winding taps under load. Their make-before-break contacts wear and can stick. A tap-changer fault is the single most common cause of large-transformer failure (about 40% of unscheduled events on units over 50 MVA).
• Through-fault mechanical damage. A close-up secondary-side short pushes huge currents through the windings; magnetic forces on adjacent turns scale with I² and can reach hundreds of tonnes radial force per metre. Cumulative through-faults loosen winding clamps; subsequent inrush events can then move conductors and short turns.

Historical context

Faraday discovered induction in 1831. Joseph Henry, working independently in the US, showed mutual induction between coils the same year. The first practical transformer-like device — Lucien Gaulard and John Gibbs's 1881 "secondary generator" — used an open core and was a poor power-transfer device. The breakthrough was the 1885 ZBD transformer (Károly Zipernowsky, Ottó Bláthy, Miksa Déri at the Ganz works in Budapest), which closed the iron core and made the turns ratio set the voltage ratio cleanly. That single design made the alternating-current grid economically viable.

Westinghouse licensed the ZBD design and demonstrated it at scale in 1886 in Great Barrington, Massachusetts. The "war of the currents" (1888–1893) ended decisively in AC's favour at the 1893 Chicago World's Fair, after which DC transmission essentially vanished for nearly a century. Modern HVDC came back in 1954 (ASEA's 100 kV Gotland link) — but it still relies on transformers at each converter station, doing the original ZBD job at one of the largest scales in human engineering.