Transformer

 

Transformer Study Guide

Working Principle of Transformers

Transformers are static electrical devices that transfer energy between circuits by electromagnetic induction:contentReference[oaicite:0]{index=0}. An alternating voltage on the primary winding causes an alternating current, creating a time-varying magnetic flux in the core. This flux links the secondary winding and induces an electromotive force (EMF) in it (Faraday’s law):contentReference[oaicite:1]{index=1}:contentReference[oaicite:2]{index=2}. The windings are magnetically coupled but not electrically connected. If the primary has N₁ turns and the secondary has N₂ turns, the induced voltages are related by the turns ratio:

V_1 / V_2 = N_1 / N_2

This means an ideal transformer steps up or down voltage proportional to the turns ratio. (Conversely, the current ratio is I₁/I₂ = N₂/N₁ under ideal conditions.) Transformers only work with AC; DC produces no changing flux and thus no induced voltage:contentReference[oaicite:3]{index=3}.

EMF Equation of a Transformer

The RMS EMF induced in a winding of N turns is given by Faraday’s law. For sinusoidal excitation, the induced EMF (in volts) is:

E = 4.44 \, f \, N \, \Phi_{\max}

Here f is frequency (Hz) and \Phi_{\max} is the peak flux (weber). This equation shows E ∝ f N \Phi_{\max}:contentReference[oaicite:4]{index=4}. It follows that for two windings on the same core, their voltages relate by turn counts (V₁/V₂ = N₁/N₂), consistent with the turns ratio.

Losses in a Transformer

Real transformers have losses that reduce output power. These are split into core (iron) losses and copper (winding) losses:contentReference[oaicite:5]{index=5}. Core losses occur in the magnetic core when it is energized (even with no load), while copper losses occur in the windings when current flows (i.e. under load).

Core (Iron) Losses

Core losses consist of:

• Hysteresis Loss: Energy used in magnetizing and demagnetizing the core each cycle. It depends on flux density and frequency (Steinmetz’s law: P_h = K_h f B_{\max}^n, with n≈1.6).
• Eddy Current Loss: Currents induced in the core by changing flux cause resistive heating. It scales roughly as P_e = K_e (B_{\max} f)^2 t^2, where t is lamination thickness. Thin laminations minimize eddy losses.

The total iron (core) loss is P_{iron} = P_h + P_e:contentReference[oaicite:6]{index=6}.

Copper Losses

Copper losses (I²R losses) occur in the windings when current flows. If I_1, I_2 are the currents and R_1, R_2 the winding resistances, then:

P_{copper} = I_1^2 R_1 + I_2^2 R_2

These losses increase with the square of the load current. Using low-resistance conductors (e.g. copper) and multiple parallel paths reduces copper losses.

Efficiency of a Transformer

Efficiency η is the ratio of output power to input power:contentReference[oaicite:7]{index=7}:

η = \frac{P_{out}}{P_{in}} \times 100\%

Since P_{in} = P_{out} + P_{losses}, equivalently:

η = \frac{P_{out}}{P_{out} + P_{losses}} \times 100\%

High-quality transformers often reach efficiencies of 95–99%. Efficiency varies with load: at very light load (core losses dominate) and at very heavy load (copper losses dominate), efficiency is lower. Maximum efficiency occurs at an intermediate load. In fact, maximum efficiency is obtained when copper losses equal core losses:contentReference[oaicite:8]{index=8}. (All-day efficiency – energy delivered over 24h – is another metric sometimes used but beyond this scope.)