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A transformer is an essential electrical device used to change the voltage level of alternating current (AC) power without altering its frequency. Transformers play a vital role in electrical power systems, especially in power generation, transmission, and distribution. For example, in India, transformers enable effective transmission of electrical energy by stepping up voltages to high levels for long-distance travel, thereby reducing losses, and stepping down voltages for safe use in homes and industries.
At its core, a transformer operates on fundamental principles from the field of electromagnetism, specifically electromagnetic induction. Because transformers work only with alternating current, understanding why this is so involves reviewing key ideas such as magnetic flux and Faraday's laws of induction.
The working of a transformer is based on one of the most important laws of electromagnetism: Faraday's Law of Electromagnetic Induction. This law states that a voltage (also called electromotive force or EMF) is induced in a coil when there is a change in magnetic flux linkage through it.
In indoor terms:
This process is mutual induction - where one coil induces voltage in another coil through their shared magnetic field.
A transformer mainly consists of two coils: the primary winding and the secondary winding, both wound on a common core made of iron or other magnetic material. The core serves to guide and concentrate magnetic flux between windings to maximize efficiency.
The core is usually made of silicon steel sheets laminated together rather than a solid piece. Lamination is done to reduce eddy currents, which are induced currents inside the core that cause energy loss and heating.
Primary winding is connected to the input AC supply, and the secondary winding delivers the output voltage to the load. The number of turns (loops) of wire in each winding determines the voltage transformation.
One of the key relationships in transformers involves the number of turns in the primary and secondary windings, and how this determines voltage and current on each side.
Turns Ratio: The ratio of the number of turns in the primary winding (\( N_p \)) to that in the secondary winding (\( N_s \)) is called the turns ratio.
For an ideal transformer (one without losses), the voltage induced across the windings relates directly to their number of turns:
Similarly, because power in an ideal transformer is conserved (neglecting losses):
Using this, current is inversely proportional to the turns, given by:
| Parameter | Primary Side | Secondary Side |
|---|---|---|
| Number of Turns | \( N_p \) | \( N_s \) |
| Voltage (V) | \( V_p \) | \( V_s \) |
| Current (I) | \( I_p \) | \( I_s \) |
| Power (P) | \( P_p = V_p I_p \) | \( P_s = V_s I_s \) |
While ideal transformers assume no energy loss, real transformers experience various losses that reduce their efficiency.
Efficiency (\( \eta \)) is the ratio of output power to input power, usually expressed as a percentage:
Voltage regulation indicates how much the output voltage changes when the transformer goes from no load to full load. Good regulation means little voltage drop under load.
graph TD A[Input Power (Pin)] B[Copper Loss] C[Core Loss] D[Stray Loss] E[Output Power (Pout)] A -->|Power In| F[Transformer] F --> E F --> B F --> C F --> D style B fill:#f96,stroke:#333,stroke-width:2px style C fill:#f96,stroke:#333,stroke-width:2px style D fill:#f96,stroke:#333,stroke-width:2px style E fill:#6f6,stroke:#333,stroke-width:2px click B "https://en.wikipedia.org/wiki/Copper_loss" "More about copper loss" click C "https://en.wikipedia.org/wiki/Iron_loss" "More about iron/core loss" click D "https://en.wikipedia.org/wiki/Eddy_current" "More about stray losses"
Step 1: Identify the known values:
Step 2: Use the voltage transformation ratio formula:
\[ \frac{V_p}{V_s} = \frac{N_p}{N_s} \] Rearranged to find \( V_s \): \[ V_s = V_p \cdot \frac{N_s}{N_p} \]
Step 3: Substitute the values:
\[ V_s = 230 \times \frac{1000}{500} = 230 \times 2 = 460\, \text{V} \]
Answer: The secondary voltage is 460 V AC.
Step 1: Calculate total losses:
\[ \text{Total Loss} = \text{Copper Loss} + \text{Core Loss} = 150 + 100 = 250\, \text{W} \]
Step 2: Calculate output power \( P_{out} \):
\[ P_{out} = P_{in} - \text{Losses} = 5000 - 250 = 4750\, \text{W} \]
Step 3: Calculate efficiency:
\[ \eta = \frac{P_{out}}{P_{in}} \times 100 = \frac{4750}{5000} \times 100 = 95\% \]
Answer: The transformer efficiency is 95%.
Step 1: Identify values:
Step 2: Apply the voltage regulation formula:
\[ \text{Voltage Regulation} = \frac{V_{no-load} - V_{full-load}}{V_{full-load}} \times 100 = \frac{220 - 210}{210} \times 100 \]
Step 3: Calculate:
\[ = \frac{10}{210} \times 100 = 4.76\% \]
Answer: The voltage regulation is 4.76%.
Step 1: Open circuit test gives core loss component and magnetizing current. Calculate core loss resistance \( R_c \):
\( R_c = \frac{V_{oc}^2}{P_{oc}} = \frac{220^2}{110} = \frac{48400}{110} = 440\, \Omega \)
Step 2: Calculate copper loss resistance \( R_{eq} \) from short circuit test:
\( R_{eq} = \frac{P_{sc}}{I_{sc}^2} = \frac{800}{20^2} = \frac{800}{400} = 2\, \Omega \)
Step 3: Calculate impedance from short circuit voltage and current:
\( Z_{eq} = \frac{V_{sc}}{I_{sc}} = \frac{50}{20} = 2.5\, \Omega \)
Step 4: Calculate reactance \( X_{eq} \) using:
\[ X_{eq} = \sqrt{Z_{eq}^2 - R_{eq}^2} = \sqrt{2.5^2 - 2^2} = \sqrt{6.25 - 4} = \sqrt{2.25} = 1.5\, \Omega \]
Answer: Equivalent resistance \( R_{eq} = 2\, \Omega \), equivalent reactance \( X_{eq} = 1.5\, \Omega \).
Step 1: Calculate energy lost per day:
\[ \text{Energy Loss} = \text{Power Loss} \times \text{Time} = 5\, \text{kW} \times 10\, \text{h} = 50\, \text{kWh} \]
Step 2: Calculate cost of losses:
\[ \text{Cost} = \text{Energy Loss} \times \text{Cost per kWh} = 50 \times 7.5 = Rs.375 \]
Answer: The daily cost due to transformer losses is Rs.375.
When to use: During numerical problems involving voltage and current ratios.
When to use: At the start of any example to avoid unit mismatch errors.
When to use: When given partial transformer parameters in questions.
When to use: To avoid sign confusion and wrong regression in percentage values.
When to use: In advanced numerical problems related to transformer performance.
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