5.3 Transformer principles and equivalent circuits

Transformer Basics

A transformer transfers electrical energy between two or more circuits through electromagnetic induction, without any direct electrical connection between them. This is the core device that makes AC power distribution practical: by stepping voltage up for long-distance transmission (reducing $I^2R$ losses in the lines) and stepping it back down for safe use at the load.

Core Components and Operation

A transformer has two or more coils (windings) wound around a shared magnetic core. The primary winding receives input power and generates a time-varying magnetic flux in the core. The secondary winding picks up that changing flux and delivers power to the load.

The turns ratio $a = N_s / N_p$ defines how voltages and currents scale between windings. For an ideal transformer:

• Voltage ratio: $V_s / V_p = N_s / N_p$
• Current ratio: $I_s / I_p = N_p / N_s$

So voltage scales directly with the turns ratio, while current scales inversely. Power is conserved in the ideal case: $V_p I_p = V_s I_s$.

Electromagnetic Principles

Transformer operation rests on Faraday's law of electromagnetic induction. A time-varying current in the primary creates a time-varying magnetic flux $\Phi$ in the core. That changing flux induces a voltage in the secondary according to:

$e = -N \frac{d\Phi}{dt}$

The mutual inductance $M$ between the windings quantifies how effectively flux produced by one coil links the other. The core is made from ferromagnetic materials (typically silicon steel laminations) to provide a low-reluctance path that maximizes this flux linkage.

Two common core geometries show up in practice:

• Core-type: Windings are placed on separate limbs of the core. Easier to insulate for high-voltage applications.
• Shell-type: The core surrounds the windings. Provides better magnetic shielding and is common in lower-voltage, higher-current designs.

Transformer Losses

Real transformers dissipate some energy as heat. These losses fall into two categories, and understanding both is essential for building accurate equivalent circuits.

Core Losses (Iron Losses)

Core losses are present whenever the transformer is energized, regardless of load. They have two components:

• Hysteresis loss: Energy spent cyclically realigning magnetic domains in the core material each AC cycle. Proportional to frequency and to the area of the B-H loop of the core material.
• Eddy current loss: Circulating currents induced directly in the conductive core by the changing flux. These currents dissipate power as $I^2R$ heating within the core itself.

Both losses increase with operating frequency and peak flux density. They're minimized by using laminated core construction (thin insulated sheets reduce eddy current paths) and selecting high-resistivity materials like silicon steel.

Because core losses depend on voltage and frequency rather than load current, they're also called no-load losses.

Copper Losses and Efficiency

Copper losses (also called $I^2R$ losses or load losses) come from the resistance of the winding conductors:

$P_{copper} = I_p^2 R_p + I_s^2 R_s$

These losses scale with the square of the current, so they increase rapidly as load grows.

Transformer efficiency is:

$\eta = \frac{P_{out}}{P_{out} + P_{core} + P_{copper}} \times 100\%$

Large power transformers routinely achieve 95% to 99% efficiency. Maximum efficiency occurs at the load level where copper losses equal core losses, a useful design and analysis fact.

Transformer Equivalent Circuit

The equivalent circuit lets you analyze a real transformer using standard circuit techniques, capturing all the non-ideal behaviors in a single lumped-element model.

Building the Model, Step by Step

Here's how each element maps to a physical effect:

1. Winding resistances $R_p$ and $R_s$: Placed in series with each winding to model copper losses.
2. Leakage reactances $X_{lp}$ and $X_{ls}$: Also in series with each winding. These represent the portion of flux produced by each coil that does not link the other coil. Leakage flux stores energy but doesn't contribute to power transfer.
3. Magnetizing reactance $X_m$: A shunt element (in parallel) that models the current needed to establish the core flux. This is the reactive component of the no-load current.
4. Core-loss resistance $R_c$: In parallel with $X_m$, modeling the real power dissipated by hysteresis and eddy currents.
5. Ideal transformer (turns ratio $a = N_s / N_p$): Sits between the primary-side and secondary-side circuit elements to handle voltage/current scaling.

The resulting referred equivalent circuit has $R_{eq} = R_p + R_s'$ and $X_{eq} = X_{lp} + X_{ls}'$ in series, with the shunt branch ($R_c \parallel X_m$) connected at the input. For heavy-load analysis, the shunt branch is sometimes neglected entirely (the approximate equivalent circuit), since magnetizing current is small compared to load current.

Impedance Transformation

A transformer reflects impedances across its windings. A load impedance $Z_L$ on the secondary appears at the primary terminals as:

$Z_L' = \left(\frac{N_p}{N_s}\right)^2 Z_L$

This property is critical for impedance matching, where you choose a turns ratio so the source sees a load impedance equal to its own internal impedance, maximizing power transfer. Applications range from audio amplifier output stages to RF antenna matching networks.

Transformer Applications

Voltage Manipulation and Power Distribution

• Step-up transformers ($N_s > N_p$) increase voltage at generating stations for long-distance transmission, reducing line current and therefore $I^2R$ transmission losses.
• Step-down transformers ($N_s < N_p$) reduce voltage at substations and distribution points for safe delivery to loads.
• Distribution transformers handle the final conversion from medium voltage (e.g., 11 kV) to low voltage (e.g., 120/240 V) for residential and commercial use.
• Instrument transformers (CTs and PTs) provide scaled-down, isolated replicas of line currents and voltages for metering and protective relaying.

Specialized Transformer Types

• Autotransformers use a single tapped winding shared between primary and secondary. They're smaller and more efficient than two-winding transformers for modest voltage ratios, but they don't provide electrical isolation.
• Isolation transformers (1:1 turns ratio) provide galvanic isolation between circuits for safety and to break ground loops that cause noise.
• Three-phase transformers handle three-phase power in a single unit (or as a bank of three single-phase units) and are standard in industrial and utility-scale systems.
• Pulse transformers are optimized for transmitting fast, short-duration pulses with minimal distortion, used in gate drive circuits for power electronics and in digital communication.