Problem 23 - Applied Physics

Question

The primary coil of a step-down transformer has \(75 \overline{0} 0\) turns, and the secondary coil has 125 turns. The voltage across the primary is \(72 \overline{0} 0 \mathrm{~V}\). (a) Find the voltage across the secondary. (b) If the current in the secondary coil is \(36.0 \mathrm{~A}\), find the primary current.

Step-by-Step Solution

Answer

(a) 120 V, (b) 0.6 A.

Step 1: Understanding Transformer Basics

A step-down transformer decreases voltage from the primary coil to the secondary coil. The voltage ratio is given by the ratio of the number of turns on the primary coil to the number of turns on the secondary coil.

Step 2: Setting Up the Voltage Ratio

The voltage ratio is set by the equation \( \frac{V_p}{V_s} = \frac{N_p}{N_s} \). Here, \( V_p = 7200 \ \mathrm{V} \), \( N_p = 7500 \), and \( N_s = 125 \). Solving for \( V_s \) (voltage across the secondary coil), we get: \[ V_s = V_p \times \frac{N_s}{N_p} = 7200 \times \frac{125}{7500}. \]

Step 3: Calculating Secondary Voltage

Carry out the calculation: \[ V_s = 7200 \times \frac{125}{7500} = 120 \ \mathrm{V}. \] Thus, the voltage across the secondary coil is 120 V.

Step 4: Relating Current and Turns in a Transformer

Transformers also relate current by the equation \( \frac{I_s}{I_p} = \frac{N_p}{N_s} \). Here, \( I_s = 36.0 \ \mathrm{A} \),\( N_p = 7500 \), and \( N_s = 125 \). Solving for \( I_p \) (current in the primary coil), we use: \[ I_p = I_s \times \frac{N_s}{N_p}. \]

Step 5: Calculating Primary Current

By substituting the values into the equation, calculate: \[ I_p = 36.0 \times \frac{125}{7500} \approx 0.6 \ \mathrm{A}. \] Therefore, the primary current is approximately 0.6 A.

Key Concepts

Understanding a Step-down Transformer

A step-down transformer is an electrical device that reduces a high voltage to a lower voltage. It plays a crucial role in the distribution of electrical energy. The primary purpose is to safely decrease the voltage to advisable levels for home appliances.

These transformers have two coils: a primary and a secondary coil. The primary coil receives the input voltage, and the secondary coil delivers the reduced output voltage. This reduction is essential for compatibility with household devices and electrical systems, which operate at lower voltages for safety and efficiency.

Some key applications of step-down transformers include:

• Transforming voltage levels in power supply lines.
• Providing safe voltage levels for electronic devices and household appliances.
• Facilitating efficient long-distance power transmission by initially stepping up voltage then stepping it down.

Step-down transformers are fundamental components in the modern electrical grid, enabling the safe and efficient distribution of electric power.

Voltage Ratio in Transformers

The voltage ratio is the relationship between the voltage in the primary coil and the voltage in the secondary coil. This ratio is vital for understanding how a transformer manipulates electrical energy.

Mathematically, the voltage ratio is expressed as \( \frac{V_p}{V_s} = \frac{N_p}{N_s} \). Here:

• \( V_p \) is the voltage on the primary coil.
• \( V_s \) is the voltage on the secondary coil.
• \( N_p \) is the number of turns on the primary coil.
• \( N_s \) is the number of turns on the secondary coil.

The equation shows that the voltage change is directly proportional to the turns ratio. In our exercise, if the primary voltage is high and we need a lower secondary voltage, the secondary coil should have fewer turns. This ensures the transformer fulfills its role of stepping down the voltage as required.

Current Calculation in Transformers

In transformers, not only is the voltage transformed, but current levels are as well. Understanding current calculations is crucial for designing and using transformers.

The relationship between the current in the primary and secondary coils is given by \( \frac{I_s}{I_p} = \frac{N_p}{N_s} \). This equation implies that increasing the number of turns on the primary compared to the secondary will decrease the current in the primary coil while maintaining the same power level (disregarding losses).

For the problem at hand:

• \( I_s = 36.0 \ \, \mathrm{A} \) is the current in the secondary coil.
• By applying the formula, we find \( I_p \), the primary current, which is important for safely managing the input power.

The concept of current transformation is paramount in applications where specific current levels are needed to match the electrical device requirements and design specifications.

Exploring Turns Ratio

The turns ratio is a simple yet powerful concept that dictates the functionality of transformers. The turns ratio is the ratio of the number of turns in the primary coil to the number of turns in the secondary coil.

This concept is crucial because it determines how the transformer will adjust both voltage and current. The equation \( \frac{N_p}{N_s} \) captures this ratio, showing how it directly impacts the voltage and current ratios.

In practical terms:

• A higher turns ratio implies a greater change between the input and output voltage.
• Engineers use this ratio to design transformers tailored for specific applications, ensuring the correct voltage levels are achieved.
• The turns ratio directly influences transformer efficiency and performance, making it an essential part of design considerations.

Understanding and optimizing the turns ratio is key in achieving the desired transformation properties in any device that relies on transformers.