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Chapter 21: Problem 38

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?

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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 \( \Omega \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A 1.50 \(\mathrm{mH}\) inductor is connected in series with a dc battery of negligible internal resistance, a 0.750 \(\mathrm{k} \Omega\) resistor, and an open switch. How long after the switch is closed will it take for (a) the current in the circuit to reach half of its maximum value, (b) the energy stored in the inductor to reach half of its maximum value? (Hint: You will have to solve an exponential equation.)

A very thin 15.0 \(\mathrm{cm}\) copper bar is aligned horizontally along the east-west direction. If it moves horizontally from south to north at 11.5 \(\mathrm{m} / \mathrm{s}\) in a vertically upward magnetic field of \(1.22 \mathrm{T},\) (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

A 12.0\(\mu \mathrm{F}\) capacitor and a 5.25 \(\mathrm{mH}\) inductor are connected in series with an open switch. The capacitor is initially charged to 6.20\(\mu \mathrm{C}\) . What is the angular frequency of the charge oscillations in the capacitor after the switch is closed?

You're driving at 95 \(\mathrm{km} / \mathrm{h}\) in a direction \(35^{\circ}\) east of north, in a region where the earth's magnetic field of \(5.5 \times 10^{-5} \mathrm{T}\) is horizontal and points due north. If your car measures 1.5 \(\mathrm{m}\) from its underbody to its roof, calculate the induced emf between roof and underbody. (You can assume the sides of the car are straight and vertical.) Is the roof of the car at a higher or lower potential than the underbody?

A coil of wire with 200 circular turns of radius 3.00 \(\mathrm{cm}\) is in a uniform magnetic field along the axis of the coil. The coil has \(R=40.0 \Omega\) . At what rate, in teslas per second, must the magnetic field be changing to induce a current of 0.150 \(\mathrm{A}\) in the coil?
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