Transformers

Transformers change AC voltage and current with ~98–99 % efficiency using mutual induction. Learn the turns ratio, EMF equation, equivalent circuit, losses, efficiency, regulation, three-phase connections and testing.

A transformer is two coils wound on a shared iron core. AC in the primary creates a changing magnetic flux; that flux links the secondary and induces a voltage in it. With no moving parts it can step voltage up or down while conserving power.

Transformers are why the grid works: generators produce ~20 kV, transformers raise it to 400 kV for transmission, lower it in stages for distribution, and finally to 400/230 V at your street. Inside electronics, small and high-frequency transformers provide isolation and voltage conversion.

An analogy

A transformer is like a gearbox for electricity. A gearbox trades speed for torque; a transformer trades voltage for current. Power in ≈ power out.

Step-up and step-down

More turns on the secondary → higher secondary voltage (step-up) and lower current. Fewer turns → step-down. Power is (almost) conserved: V₁I₁ ≈ V₂I₂. Transformers only work with changing current — DC through a transformer produces no secondary voltage and just overheats the primary.

Isolation

Primary and secondary are connected only magnetically. That galvanic isolation keeps mains-referenced faults away from users — the reason phone chargers and medical equipment use transformers even when voltage conversion could be done otherwise.

Losses and efficiency

Core loss (hysteresis + eddy currents) is roughly constant whenever the transformer is energised. Copper loss (I²R) varies with the square of load. Efficiency peaks where the two are equal — distribution transformers are designed to peak around 40–60 % load because they spend most of their life there.

Equivalent circuit and tests

A real transformer is modelled as an ideal transformer plus series winding resistance and leakage reactance, and a shunt magnetising branch. The open-circuit test (rated voltage, no load) measures core loss and the magnetising branch; the short-circuit test (rated current, low voltage) measures copper loss and the series impedance.

Voltage regulation and per-unit impedance

Regulation = (V_no-load − V_full-load)/V_full-load. It depends on the per-unit impedance (typically 4–10 % for distribution units) and the load power factor. A higher impedance also limits the fault current a transformer can deliver: I_sc = I_rated / Z_pu.

Three-phase connections and inrush

Δ–Y (delta primary, star secondary) is standard for distribution: it provides a neutral, blocks triplen harmonics and introduces a 30° phase shift (vector group Dyn11). On energisation, core saturation can draw an inrush current of 8–12× rated for a few cycles, which differential protection must learn to ignore (2nd-harmonic restraint).

Formulas

• Ideal ratios: V₁/V₂ = N₁/N₂ = I₂/I₁ = a
• EMF equation: E = 4.44 · f · N · φ_m
• Impedance reflection: Z₁' = a² · Z₂
• Efficiency at load fraction x: η = x·S·pf / (x·S·pf + P_i + x²·P_cu)
• Maximum efficiency: x = √(P_i / P_cu) (copper loss = core loss)

Worked example: Small step-down transformer

Given: V₁ = 230 V; V₂ = 12 V; S = 60 VA

Find: Turns ratio and full-load currents.

• a = 230 / 12 = 19.17
• I₂ = 60 / 12 = 5 A
• I₁ = 60 / 230 = 0.26 A (= I₂/a ✓)

Answer: a ≈ 19.2, I₂ = 5 A, I₁ ≈ 0.26 A

Worked example: Efficiency at full load and maximum efficiency

Given: S = 100 kVA; P_i = 1 kW; P_cu(FL) = 1.5 kW; pf = 0.8

Find: Full-load efficiency, the load for maximum efficiency and that efficiency.

• Full load: output = 80 kW → η = 80 / (80 + 1 + 1.5) = 96.97 %
• Max η when x²·1.5 = 1 → x = √(1/1.5) = 0.816 → 81.6 kVA
• Output = 0.816 × 80 = 65.3 kW; losses = 1 + 1 = 2 kW
• η_max = 65.3 / 67.3 = 97.03 %

Answer: η_FL = 97.0 %, η_max = 97.03 % at 81.6 kVA

Worked example: EMF equation

Given: f = 50 Hz; N = 200 turns; φ_m = 0.01 Wb

Find: RMS induced voltage.

• E = 4.44 f N φ_m
• = 4.44 × 50 × 200 × 0.01
• = 444 V

Answer: 444 V RMS

Common mistakes

• Wrong: Open-circuiting the secondary of a current transformer while current flows. — Right: A CT secondary must always be shorted or loaded; open-circuit it and a dangerous high voltage appears.
• Wrong: Assuming a transformer rated in kW. — Right: Transformers are rated in kVA — the heating depends on current regardless of power factor.

Applications

• Power grid: Generator step-up, transmission, sub-transmission and distribution transformers change voltage at every stage.
• Switch-mode supplies: A 100 kHz ferrite transformer the size of a sugar cube replaces a 50 Hz iron transformer weighing a kilogram, because core size scales inversely with frequency (E = 4.44fNφ).
• Measurement: Current transformers (CTs) and voltage transformers (VTs) scale grid quantities down to 1 A / 5 A and 110 V for meters and protection relays.

Key takeaways

• V₁/V₂ = N₁/N₂ = I₂/I₁; impedance scales by a².
• E = 4.44 f N φ_m links voltage, frequency, turns and core size.
• Core loss is constant, copper loss ∝ load²; max η when they're equal.
• Rate transformers in kVA and respect CT and inrush hazards.

Going further

• Core materials: Grain-oriented silicon steel laminations cut eddy currents in power transformers; amorphous metal cores cut no-load loss by ~70 %; ferrites serve high-frequency converters. Lamination thickness is chosen relative to skin depth at the operating frequency.
• Tap changers: On-load tap changers switch between winding taps under load to regulate voltage ±10 % in 1–2 % steps, coordinated by automatic voltage control relays at every substation.
• Solid-state transformers: A solid-state transformer rectifies, converts to medium-frequency AC, uses a small high-frequency transformer, then re-synthesises the output. It adds control, DC ports and power-quality functions at the cost of complexity — an active research area for smart grids.