Introduction to Single Phase Transformers | Electrical Machines - Electrical Engineering (EE) PDF Download

What are Transformers?

• The first electrical machine we study is the transformer. It is not an electromechanical energy converter; rather it converts the voltage level of an alternating current (AC) supply with minimal energy conversion losses.
• Large modern power stations are built on the principle of economy of scale - the larger the plant, the lower the unit cost of generated electrical energy. Such plants are usually located far from load centres because of constraints like land, pollution, fuel and water availability.
• Consequently, electrical power must be transmitted efficiently over long distances (hundreds of kilometres) and in large quantities (thousands of megawatts).
• To achieve efficient long-distance transmission, the voltage level at which power is sent must be high. Transformers make it possible to change voltage levels at high efficiency, and that is why they are central to AC power systems.

How the voltage level of transmission affects the efficiency of power transmission?

• Consider a load that needs a power P at a voltage V and is situated a distance L from the generating station.
• The load current is given by:

I = P / V

• The transmission line is usually designed for a specified current density J, so the conductor cross-sectional area is:

a = I / J

• The resistance of the transmission conductor of length L and cross-section a is:

R = ρ L / a

Power loss in the transmission line

• The line power loss is:

P_L = I² R

• Substitute I = P / V and R = ρ L / a to get:

P_L = (P / V)² ρ L / a

• For fixed P, L and conductor material, the transmission loss varies approximately as:

P_L ∝ (1 / V²)

• Thus, a larger voltage V gives a much smaller transmission loss. This is why power is transmitted at very high voltages (commonly 400 kV or higher on long-distance high-voltage networks).
• Generating stations typically produce at moderate voltages (for example, around 11 kV) because making the generator insulation for hundreds of kilovolts is impractical. At the receiving end, loads require safer, lower voltages. Therefore, voltage stepping-up at the generation end and stepping-down at the consumer end are necessary.
• Transformers are used to change voltage levels with high efficiency (transformer efficiencies are often over 98% for large power transformers), making high-voltage AC transmission practical and economical.

Transformer used both at Generating end and at load end

Generating Station feeding Transformer

Why not use a potential divider or resistive methods to change voltage?

• A resistive potential divider can step down voltage but wastes power continuously in the resistances and cannot raise voltage above the supply. It is therefore inefficient for power transfer.
• An inductive divider reduces resistive loss in steady state but the arrangement becomes more than a simple divider; magnetic coupling between sections becomes important.
• An autotransformer (single winding with several taps) can provide step-up or step-down in voltage, and it transfers power partly conductively and partly by magnetic induction. Autotransformers are compact and economical when the change in voltage is small and the same neutral is shared.
• A two-winding transformer (primary and secondary electrically isolated but magnetically coupled) removes the direct conductive connection. It is preferred when large voltage transformation ratios are required or when electrical isolation is needed because it reduces the chance of catastrophic overvoltage at the lower-voltage side during faults.

Magnetic coupling and the magnetic circuit

Transformers operate by magnetic coupling. To understand this, we recap magnetic-circuit concepts used in transformer analysis.

• A typical magnetic circuit comprises a ferromagnetic core of regular geometry with a winding around part of the core.
• When current i flows in the winding of N turns, the product N i is called the magneto-motive force (MMF). MMF drives magnetic flux around the closed magnetic path of the core.
• Magnetic quantities form an analogy with electrical circuits: MMF (N i) ≈ voltage (electrical), magnetic flux (Φ) ≈ current (electrical), and reluctance (ℜ) ≈ resistance (electrical).
• The reluctance of a uniform magnetic path of length l, cross-sectional area A, and permeability μ = μ0 μr is:

ℜ = l / (μ0 μr A)

• The flux produced by MMF is:

Φ = (N i) / ℜ

Magnetic field basics and laws

• A current-carrying conductor produces a magnetic field near it. The magnetic field is described by the magnetic field intensity H and the magnetic flux density B.
• The relation between them is:

B = μ0 μr H

• Here μ0 is the permeability of free space (μ0 = 4π × 10^-7 H/m) and μr is the relative permeability of the material (for ferromagnetic materials μr can be several thousand).
• The Biot-Savart law gives the differential contribution to the magnetic flux density from a small current element:

dB = (μ0 μr / 4π) · (i dl × r) / r³

• In practice, integrating the Biot-Savart expression is difficult for complex geometries. For magnetic circuits with high-permeability cores, almost all flux is confined inside the core and Ampère's circuital law is more convenient.
• Ampère's law (integral form):

∮ H · dl = I_enclosed

• Using Ampère's law for a uniform closed path inside a core gives a practical expression for H in terms of MMF and the path length:

H = N i / l

Flux, reluctance and nonlinearity (B-H curve)

• For a linear magnetic material, Φ = B A = μ0 μr H A = (μ0 μr A / l) · N i. The magnetic circuit is analogous to a voltage source N i driving flux through reluctance ℜ.
• Ferromagnetic materials are nonlinear. The relationship between B and H is given by the material's B-H characteristic. Initially B increases approximately linearly with H, but beyond a region it tends to saturate - further increases in H produce only small increases in B.
• The region where the curve bends is called the knee point. Above the knee the effective relative permeability μr decreases markedly and reluctance increases with flux.
• Because core reluctance depends on operating flux, magnetic-circuit analysis is more complicated than linear electrical-circuit analysis and often requires graphical or iterative methods.

Magnetic circuits with air gaps

• Many practical magnetic circuits include an air gap. Air has μr ≈ 1, so the air-gap reluctance is much larger than that of the ferromagnetic parts and is approximately linear (B-H for air is linear).
• When a magnetic circuit contains different sections (core and gap), the total MMF is the sum of MMFs across each section:

N i = H_core ℓ_core + H_gap ℓ_gap

• Since flux Φ is common in series magnetic paths, and B_core A_core = B_gap A_gap, one can write relations between B and H in the core and gap and solve the equations. The core's B-H characteristic and the linear relation for the gap together determine the operating point graphically: the intersection of the material B-H curve and the circuit load line gives the operating flux density and field intensity.

Induced voltage in a coil - Faraday's law

A transformer uses induced voltage in windings when the magnetic flux linking those windings changes with time. Faraday's law describes the induced emf.

• If a winding of N turns links a time-varying flux Φ(t), the instantaneous induced emf (Faraday's law) is:

e(t) = - N (dΦ / dt)

• The negative sign indicates that the induced emf has a polarity that opposes the change of flux (Lenz's law).
• If the flux varies sinusoidally: Φ(t) = Φ_max sin(ω t), then:

e(t) = - N ω Φ_max cos(ω t)

• The rms value of the induced voltage is:

E_rms = 4.44 · f · N · Φ_max

• In phasor representation, the induced emf phasor E leads the flux phasor by 90° (since emf ∝ dΦ/dt).
• Some texts refer to the induced voltage as the counter-emf because it opposes the applied voltage. Authors differ in sign convention; the core relation e = - N dΦ/dt is the standard physical law, and in circuit equations the sign is used consistently to reflect whether the induced emf subtracts from or adds to applied voltages.

• For a coil with alternating flux the practical design goal is to keep the maximum flux density Φ_max below saturation so the core operates in the near-linear region of the B-H curve.
• Designers choose number of turns N, frequency f and core cross-section to satisfy E_rms = 4.44 f N Φ_max for the desired voltage level without saturating the core.

Two-winding transformer versus autotransformer

• An autotransformer uses a single continuous winding with taps. It is economical and smaller for certain step-up/step-down ratios, but primary and secondary are electrically connected - there is no isolation.
• A two-winding transformer uses separate primary and secondary windings magnetically coupled through the core. It provides electrical isolation and is safer when voltage levels differ widely or when faults must be constrained.
• In a severe fault that shorts the low-voltage side, an autotransformer can transfer the full high voltage to the low-voltage terminals (depending on the fault location and connection), potentially causing catastrophic damage. A two-winding transformer reduces this risk because both windings would have to be simultaneously shorted to transfer the full high-side voltage directly.

Practical notes on transformer use in power systems

• Power generation is normally at moderate voltages (e.g., ~11 kV); step-up transformers raise the voltage to transmission levels (e.g., 132 kV, 220 kV, 400 kV or higher) to reduce I²R losses.
• At substations near load centres, step-down transformers reduce voltage to distribution and consumer levels for safety and equipment compatibility.
• Transformers are highly efficient stationary devices with core and copper losses. Large power transformers are designed for efficiencies above 98%; nevertheless, their losses are included in economic decisions about transmission voltage and tower cost.
• Transformer design must account for core material B-H characteristics, flux density limits (to avoid saturation), cooling, insulation, short-circuit withstand, and maintenance concerns.

Summary

The transformer is the fundamental device that enables efficient AC power transmission by changing voltage levels with high efficiency using magnetic coupling. Understanding magnetic circuits, MMF, reluctance, the nonlinear B-H behaviour of ferromagnetic cores, and Faraday's law for induced emf is essential to analyse and design transformers. Practical transformer choice (autotransformer vs two-winding) depends on required voltage ratio, isolation needs and safety considerations.