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Chapter 31: Problem 71

31.71. Finding an Unknown Inductance. Your boss gives you an inductor and asks you to measure its inductance. You have available a resistor, an ac voltmeter of high impedance, a capacitor, and an ac source. Explain how you might use these to determine the inductance, and cite any other piece of equipment you may need. Be sure to explain clearly how to use the equipment and what you need to measure to find the unknown inductance.

Short Answer

Expert verified
Set up a series RLC circuit, measure resonant frequency with an AC voltmeter, and calculate inductance using the resonance formula.

Step by step solution

01

Assemble the RLC Circuit

First, construct a series RLC circuit by connecting the resistor (R), unknown inductor (L), and capacitor (C) in series. Connect the AC source across this series circuit. This setup will allow you to investigate the resonant properties of the circuit.
02

Set Up and Measure AC Voltmeter

Connect the high-impedance AC voltmeter across the combined resistor-inductor-capacitor circuit. The voltmeter will measure the voltage across the circuit, which helps in determining the resonance frequency.
03

Adjust AC Source Frequency

Adjust the frequency of the AC source. Gradually change the frequency while observing the voltmeter. At resonance, the frequency at which the voltmeter reading reaches its maximum, the impedance of the circuit is at its minimum.
04

Calculate Resonance Frequency

Locate the frequency at which the maximal voltage is observed on the voltmeter. This frequency is the resonance frequency (\( f_0 \)). Record this frequency value for further calculations.
05

Use Resonance Formula

Use the resonance frequency formula for an RLC circuit: \[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]Rearrange to solve for the inductance \( L \):\[ L = \frac{1}{(2\pi f_0)^2 C} \]You will need the capacitance \( C \) value and the resonance frequency \( f_0 \) recorded.
06

Measure Capacitance

Use a capacitance meter to measure the capacitance \( C \) of the capacitor used in the circuit. This value is crucial for calculating the inductance value.
07

Compute Inductance Value

Substitute the measured values of \( C \) and \( f_0 \) into the rearranged formula from Step 5 to calculate the inductance \( L \) of the inductor.

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Key Concepts

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

RLC circuit analysis
RLC circuit analysis involves studying circuits composed of resistors, inductors, and capacitors connected in series or parallel. These circuits are fundamental for understanding how electrical components interact with alternating current (AC). To analyze an RLC circuit, it's crucial to understand that:
  • Resistors impede current flow due to resistance.
  • Inductors store energy in magnetic fields and resist changes in current.
  • Capacitors store energy in electric fields and resist changes in voltage.
By connecting these components in series or parallel with an AC source, you can examine how they behave over a range of frequencies. In our exercise, creating a series RLC circuit is essential for determining the circuit's resonance frequency. Understanding the behavior at this frequency allows us to extract valuable electrical properties such as inductance.
resonance frequency determination
Resonance frequency is a critical concept in RLC circuits. It's the frequency at which the circuit's impedance is at its minimum and the voltage is the highest across the components. To determine this frequency:
  • Set up the RLC circuit with the AC source and adjust the source frequency.
  • Use an AC voltmeter to track voltage changes across the components.
  • The resonance frequency is found when the voltmeter reads its maximum value.
At this point, the energy transfer between the inductor and capacitor is most efficient. This frequency is crucial since it's used in the equation to solve for unknowns like inductance or capacitance in the circuit.
measuring capacitance
Measuring capacitance accurately is essential for resolving other unknown quantities in RLC analysis. In our exercise, the capacitance of the capacitor was measured to find the unknown inductance. Various tools can be employed, such as a capacitance meter, which measures capacitance by charging a capacitor and observing the time constant:
  • A capacitance meter directly measures by applying a known charge.
  • It then determines capacitance based on the voltage observed over time.
This precise value is vital for calculations that use the resonance formula, influencing the accuracy of the final inductance results.
inductance calculation
Inductance calculation defines how much electrical energy is stored in an inductor in an RLC circuit. With the resonance frequency known, you can find the inductance using the resonance formula:\[ L = \frac{1}{(2\pi f_0)^2 C} \]In this formula:
  • \( f_0 \) is the resonance frequency obtained from the voltmeter readings.
  • \( C \) is the capacitance measured with the capacitance meter.
Substitute these values into the formula to calculate the inductance \( L \). This computed value helps understand how the inductor will behave in the circuit, particularly in dynamic AC environments.
AC circuits
AC circuits are electrical systems powered by alternating current, as opposed to direct current (DC). In these circuits, the current periodically reverses direction, making the behavior of components like resistors, inductors, and capacitors frequency-dependent. For practical purposes:
  • AC sources are adjustable, allowing frequencies to be tweaked for specific needs.
  • Components like inductors and capacitors respond differently than in DC circuits, affecting impedance and power distribution.
Analysing AC circuits involves measuring changes over these alternating periods and predicting the behavior of the circuit across various frequencies. Understanding these aspects is crucial when designing systems that rely on precise electrical behavior, such as tuning radios or managing power grids.

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Most popular questions from this chapter

31.41. A coil has a resistance of 48.0\(\Omega\) . At a frequency of 80.0 \(\mathrm{Hz}\) the voltage across the coil leads the current in it by \(52.3^{\circ} .\) Determine the inductance of the coil.

31.14. You have a \(200-\Omega\) resistor, a \(0.400-\mathrm{H}\) inductor, and a \(6.00-\mu \mathrm{F}\) capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 \(\mathrm{V}\) and an angular frequency of 250 \(\mathrm{rad} / \mathrm{s}\) . (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle \(\phi\) of the source voltage with respect to the current? Does the source voltage lag or lead the conrent? (e) Construct the phasor diagram.

31.12. A \(250-\Omega\) resistor is connected in series with a \(4.80-\mu \mathrm{F}\) capacitor. The voltage across the capacitor is \(v_{c}=\) \((7.60 \mathrm{V}) \sin [(120 \mathrm{rad} / \mathrm{s}) t] .\) (a) Determine the capacitive reactance of the capacitor. (b) Derive an expression for the voltage \(v_{R}\) across the resistor.

31.35. A series circuit consists of an ac source of variable frequency, a \(115-\Omega\) resistor, a \(1.25-\mu F\) capacitor, and a \(4.50-\mathrm{mH}\) inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency; (b) twice the resonance angular frequency; (c) half the resonance angular frequency.

31.62. A series circuit consists of a \(1.50-\mathrm{mH}\) inductor, a \(125-\Omega\) resistor, and a \(25.0-\mathrm{nF}\) capacitor connected across an ac source having an rms voltage of 35.0 \(\mathrm{V}\) and variable frequency. (a) At what angular frequency will the current amplitude be equal to \(\frac{1}{3}\) of its maximum possible value? (b) At the frequency in part (a) what are the current amplitude and the voltage amplitude across each of the circuit elements (including the ac source)?
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