Transformers Knowledge-Based Study

Study notes

1. What is a Transformer?

   A transformer is a static electrical device that transfers electrical energy between two or more circuits through electromagnetic induction. It is primarily used to increase (step up) or decrease (step down) alternating voltages while keeping frequency constant.

   Core Components

   • Primary Coil: The winding connected to the alternating voltage input source.
   • Secondary Coil: The winding connected to the load or output circuit.
   • Ferromagnetic Core: Typically made of laminated soft iron or silicon steel, providing a low-reluctance magnetic path that links both coils.

   Principle of Operation

   Transformers operate on the principle of mutual induction:

   1. An alternating current ($I_p$) flows through the primary winding, producing an alternating magnetic flux ($\Phi$) in the iron core.
   2. This continuously changing magnetic flux links with the turns of the secondary winding through the core.
   3. According to Faraday's law of electromagnetic induction, a changing magnetic flux induces an alternating electromotive force (emf) across the terminals of the secondary winding.

   Note: Transformers cannot operate on steady direct current (DC) because constant current creates a stationary magnetic field, which yields zero rate of change of magnetic flux ($\frac{d\Phi}{dt} = 0$), inducing no secondary voltage.

   The Voltage Ratio

   According to Faraday's law of induction, the induced electromotive force in each winding is proportional to its number of turns:

   $$\mathcal{E}_p = -N_p \frac{d\Phi}{dt} \quad \text{and} \quad \mathcal{E}_s = -N_s \frac{d\Phi}{dt}$$

   Assuming an ideal transformer with zero flux leakage and negligible coil resistance, the terminal voltage equals the induced emf. Dividing the two relationships gives the turns-voltage ratio:

   $$\frac{V_s}{V_p} = \frac{N_s}{N_p}$$

   • $V_p$: Primary voltage
   • $V_s$: Secondary voltage
   • $N_p$: Number of turns in the primary winding
   • $N_s$: Number of turns in the secondary winding

   Ideal Transformer and Current Relationship

   An ideal transformer experiences no power loss, meaning input power equals output power:

   $$P_{\text{in}} = P_{\text{out}} \implies V_p I_p = V_s I_s$$

   Combining this with the turns ratio yields:

   $$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

   Notice that current is inversely proportional to voltage and turns. When voltage is stepped up, current is stepped down by the same factor.

   Step-Up vs. Step-Down Transformers

   • Step-Up Transformer: Has more secondary turns than primary turns ($N_s > N_p$). Consequently, $V_s > V_p$ and $I_s < I_p$.
   • Step-Down Transformer: Has fewer secondary turns than primary turns ($N_s < N_p$). Consequently, $V_s < V_p$ and $I_s > I_p$.

   Energy Losses in Real Transformers

   Real transformers achieve very high efficiency (often over 95%), but they are subject to several loss mechanisms:

   • Joule (Copper) Losses: Resistive heating in the copper windings given by $P_{\text{loss}} = I^2 R$. Minimized by using thick wires with low resistivity.
   • Eddy Current Losses: Circulating currents induced in the conductive iron core by the alternating magnetic flux, causing heating. Minimized by constructing the core from thin, insulated laminations instead of a solid block.
   • Hysteresis Losses: Energy dissipated as heat during repeated magnetization and demagnetization cycles of the core material. Minimized by selecting soft magnetic materials with narrow hysteresis loops, such as silicon steel.
   • Flux Leakage: A fraction of the magnetic flux fails to link both windings. Minimized by interleaving primary and secondary windings or using shell-type core geometries.

   Electrical Grid Power Transmission

   Transformers play an indispensable role in national power grids:

   1. Generation and Stepping Up: Power stations generate electricity at moderate voltages (e.g., 11 kV to 25 kV). Step-up transformers raise this potential to very high voltages (e.g., 110 kV to 400 kV) for long-distance transmission lines.
   2. Minimizing Transmission Losses: Electrical power transmitted is given by $P = V I$. For a given power level, transmitting at a higher voltage requires a proportionally smaller current ($I = \frac{P}{V}$). Because Joule heating in the cables depends on the square of current ($P_{\text{cable}} = I^2 R$), stepping up the voltage significantly decreases energy dissipation over long distances.
   3. Stepping Down for Distribution: Step-down transformers progressively reduce the transmission voltage at regional substations and neighborhood distribution points to safe, usable domestic levels (e.g., 230 V or 120 V).

   Efficiency Calculation

   The efficiency ($\eta$) of a real transformer is defined as the ratio of useful output power to total input power:

   $$\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% = \frac{V_s I_s \cos\phi_s}{V_p I_p \cos\phi_p} \times 100\%$$

   Where $\cos\phi$ represents the power factor of the respective circuit.