Problem 12 - Applied Physics

Question

\(V_{P}=25 \overline{0} \mathrm{~V}\) \(N_{P}=73 \overline{0}\) turns \(N_{S}=275\) turns Find \(V_{S}\).

Step-by-Step Solution

Answer

The secondary voltage \(V_{S}\) is approximately 9.42 V.

Step 1: Understanding the Problem

In this exercise, we are given the primary voltage \(V_{P}\), and the number of turns in the primary \(N_{P}\) and secondary \(N_{S}\) coils of a transformer. We need to find the secondary voltage \(V_{S}\). This can be done using the transformer equation, which relates the number of turns in each coil to their respective voltages: \( \frac{V_{P}}{V_{S}} = \frac{N_{P}}{N_{S}} \).

Step 2: Rearranging the Transformer Equation

To find the secondary voltage \(V_{S}\), we rearrange the transformer equation: \( V_{S} = V_{P} \times \frac{N_{S}}{N_{P}} \). This formula allows us to find \(V_{S}\) using the known values of \(V_{P}\), \(N_{P}\), and \(N_{S}\).

Step 3: Substitute Known Values

Substitute the given values into the rearranged equation: \[ V_{S} = 25 imes \frac{275}{730} \].

Step 4: Calculate \(V_{S}\)

Perform the calculation:\[ V_{S} = 25 imes \frac{275}{730} = 25 \times 0.3767 \approx 9.4175 \]. Therefore, the secondary voltage \(V_{S}\) is approximately 9.42 V.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is a fundamental principle behind the operation of transformers. This phenomenon occurs when a change in the magnetic field around a conductor induces an electromotive force (EMF) in the conductor. In simple terms, it means producing voltage by changing the magnetic environment of a coil. The key concept here is Faraday's Law of Induction, which states that the induced voltage in a coil is proportional to the rate of change of magnetic flux through the coil. This principle is what enables transformers to transfer electrical energy between circuits. In a transformer, AC (alternating current) flows through the primary coil, creating a changing magnetic field. This changing magnetic field induces a voltage in the secondary coil due to electromagnetic induction. Both the primary and secondary coils are wound around a common magnetic core, enhancing the transfer efficiency.

Primary and Secondary Coils

In the context of transformers, primary and secondary coils are crucial components. The primary coil is the one connected to the input voltage source, while the secondary coil delivers the altered voltage to the output. Each coil is made of wire turns wrapped around a magnetic core. The main purpose of the coils is to facilitate the process of electromagnetic induction, which allows the voltage transformation from the primary to the secondary coil.

• The primary coil receives the input voltage and generates a magnetic field.
• The secondary coil picks up the induced voltage from the primary coil’s magnetic field.
  This coil's attributes determine if the voltage is stepped up or down based on the turns ratio.

The efficiency and functioning of a transformer depend significantly on the design and number of turns in these coils.

Voltage Transformation

Voltage transformation is the process through which a transformer's primary and secondary coils either increase or decrease the electrical voltage. This transformation depends on the turns ratio, which dictates how the primary coil’s voltage is converted.Using the transformer equation \[ \frac{V_{P}}{V_{S}} = \frac{N_{P}}{N_{S}} \] where:

• \( V_{P} \) is the primary voltage,
• \( V_{S} \) is the secondary voltage,
• \( N_{P} \) is the number of turns in the primary coil,
• \( N_{S} \) is the number of turns in the secondary coil.

It determines whether voltage increases or decreases as it moves through the transformer. For instance, a higher turns ratio (\( \frac{N_{S}}{N_{P}} \) greater than 1) results in an increased output voltage, known as stepping up the voltage. Conversely, if the ratio is less than 1, the output voltage is reduced, stepping it down. These calculations are pivotal in designing transformers suited for specific applications.

Turns Ratio

The turns ratio, in a transformer, is the ratio of the number of turns in the secondary coil (\( N_{S} \)) to the number of turns in the primary coil (\( N_{P} \)). It plays a crucial role in determining how the input voltage is transformed. This ratio is essential as it defines the magnitude of voltage transformation, relying solely on the coil's design.

• A turns ratio greater than one implies that the transformer is a step-up transformer, increasing the voltage.
• A turns ratio less than one indicates a step-down transformer, reducing the voltage.

For example, in our exercise, with \( N_{S} = 275 \) turns, and \( N_{P} = 730 \) turns, the turns ratio is \( \frac{275}{730} \approx 0.3767 \), indicating a step-down transformation. Understanding this concept is fundamental for efficiently designing and utilizing transformers in electrical circuits.