Problem 38 - College Physics

Question

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is \(25.0 \mathrm{V},\) (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a \(125 . \Omega\) resistor?

Step-by-Step Solution

Answer

(a) 75.8 V across the secondary coil; (b) effective load resistance: 12.08 Ω.

Step 1: Understanding the Transformer Equation

For transformers, the relationship between primary and secondary windings and voltages is given by the equation \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \), where \( V_s \) and \( V_p \) are the voltages across the secondary and primary coils, and \( N_s \) and \( N_p \) are the number of windings for the secondary and primary coils, respectively.

Step 2: Calculate the Voltage Across the Secondary Coil

Given that \( N_p = 275 \), \( N_s = 834 \), and \( V_p = 25.0 \, \mathrm{V} \), we can solve for \( V_s \):\[V_s = V_p \times \frac{N_s}{N_p} = 25.0 \, \text{V} \times \frac{834}{275} = 75.8 \, \mathrm{V}.\]So, the voltage across the secondary coil is 75.8 V.

Step 3: Understanding Effective Load Resistance in Transformers

The effective load resistance on the secondary coil \( R_s \) can be calculated using the formula \( R_s = R_L \times \left(\frac{N_p}{N_s}\right)^2 \), where \( R_L \) is the load resistance connected across the secondary coil.

Step 4: Calculate the Effective Load Resistance

Given that \( R_L = 125 \, \Omega \), we can substitute into the formula:\[R_s = 125 \, \Omega \times \left(\frac{275}{834}\right)^2 \approx 12.08 \, \Omega.\]Therefore, the effective load resistance of the secondary coil is approximately 12.08 \( \Omega \).

Key Concepts

Voltage Calculation

When dealing with electrical transformers, one important task is calculating voltages across coils. This involves understanding the relationship between primary and secondary windings. For any transformer, the voltage ratio can be derived from the number of windings. Typically, this uses the equation: \[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \]where:

• \( V_s \) is the secondary voltage,
• \( V_p \) is the primary voltage,
• \( N_s \) is the number of secondary windings, and
• \( N_p \) is the number of primary windings.

This means that the voltage across the secondary coil is proportional to the turns ratio of the coils. For example, if you know the primary voltage and the number of windings for both the primary and secondary, you can easily calculate the secondary voltage. Remember, the formula allows you to adjust based on the primary settings for a desired secondary output.

Load Resistance

Load resistance in transformers plays a crucial role in determining how efficiently the device operates. It's important to comprehend the difference between the actual load resistance \(R_L\) attached and the effective load resistance \(R_s\) perceived by the transformer. The transformation of load resistance is affected by the winding ratio, where the effective load resistance is calculated as:\[ R_s = R_L \times \left( \frac{N_p}{N_s} \right)^2 \]

• \( R_s \) is the effective load resistance on the secondary,
• \( R_L \) is the actual load resistance,
• \( N_p \) and \( N_s \) are the primary and secondary windings, respectively.

This means that the effective load experienced by the transformer tends to be significantly less than the actual load due to the square of the winding ratio. Understanding this can aid in setting the right load resistance, ensuring optimal performance and avoiding overloading your system.

Transformer Equations

The essence of transformer function revolves around several key equations that govern their operation. At the heart are two equations: one that connects voltages and turns, and another that links load resistance. The primary equation:\[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \]helps determine how input voltage converts to output voltage, given the number of turns. The secondary equation:\[ R_s = R_L \times \left( \frac{N_p}{N_s} \right)^2 \]relates to the load resistance transformation. These equations underline the fundamental physics of transformers, showcasing how energy loss is minimized and allowing efficient conversion of electrical energy from one circuit to another. They serve as guides in designing and analyzing transformer systems, ensuring correct voltage conversion and appropriate load handling.