The Ideal Transformer

In single-phase two-winding transformers, two windings are coiled around a magnetic core characterized by cross-sectional area A and magnetic permeability μ. A phasor current i₁ enters the left winding while i₂ exits the right winding, establishing the fundamental working of the transformer through electromagnetic principles.

Ampere's Law forms the basis of understanding the magnetic field within the transformer. It states that the integral of the magnetic field intensity's tangential component along a closed path equals the net enclosed current. For transformers, the mean length of the magnetic circuit is used as this closed path. This implies that the magnetic field intensity, denoted as H, is constant along this path. Mathematically, it can be expressed as:

where N₁ and N₂ are the number of turns in the primary and secondary windings, respectively.

The relationship between the magnetic flux density (Β) and core flux (Φ) is crucial in understanding the magnetic circuit's behavior. By substituting these values into Ampere's law, we derive Ohm’s law for magnetic circuits:

Here, F represents the magnetomotive force (MMF), R denotes the reluctance of the magnetic core, and Φ is the magnetic flux. For an ideal transformer with infinite core permeability (μ→∞), the core reluctance R approaches zero, leading to an ideal scenario where the phase currents in both windings are in phase.

Faraday's Law is instrumental in determining the voltage induced in the windings due to time-varying magnetic flux. Under the assumption of sinusoidal steady-state flux confined entirely within the core and operating at a constant frequency, the induced voltage in a winding is given by:

The turn ratio (N=N₁/N₂) plays a vital role in relating the voltages and currents in the primary and secondary windings. By substituting this ratio into the primary equations, one can derive the relationship between the input and output voltages and currents. The complex power entering the primary winding and leaving the secondary winding can be calculated as:

For an ideal transformer, S₁=S₂, indicating that the power entering the transformer equals the power leaving it, implies no power loss within the transformer. This principle of power conservation is foundational in transformer operation, ensuring efficiency and performance in electrical systems.