An R L C circuit has a resistance R=25 Ωand an inductance L=160 mH , and is connected to an ac g

An R L C circuit has a resistance R=25 Ωand an inductance...

An $R L C$ circuit has a resistance $R=25 \Omega$ and an inductance $L=160 \mathrm{mH}$ , and is connected to an ac generator with a frequency of 55 $\mathrm{Hz}$ . The phasor diagram for this circuit is shown in Figure $24-39 .$ Find (a) the impedance, $Z,$ and (b) the capacitance, $C,$ for this circuit. (c) If the value of $C$ is decreased, will the impedance of the circuit increase, decrease, or stay the same? Explain.

Key Concepts

Impedance in AC Circuits

Impedance is a fundamental concept in AC circuit analysis that combines both resistance and reactance into a single complex quantity, typically written as Z = R + jX, where j is the imaginary unit. It represents the total opposition a circuit presents to alternating current, impacting both the amplitude and phase of the current relative to the voltage.

Reactance

Reactance is the component of impedance that arises due to capacitors and inductors in the circuit. Inductive reactance (XL) is given by XL = ?L and opposes changes in current, while capacitive reactance (XC) is given by XC = 1/(?C) and opposes changes in voltage. The net reactance (XL - XC) determines the extent to which the circuit’s behavior is dominated by inductive or capacitive effects.

Phasor Diagrams

Phasor diagrams are graphical representations that depict the magnitude and phase relationships between different sinusoidal quantities like voltage and current in AC circuits. They visually illustrate how the resistive and reactive components combine to form the overall impedance, making it easier to analyze phase angles and relative contributions of different circuit elements.

Effect of Component Variation on Impedance

In RLC circuits, changes to components such as capacitance directly affect the capacitor's reactance (XC = 1/(?C)). Altering the capacitance will change the balance between capacitive and inductive reactance, thus modifying the overall impedance. Understanding this relationship is crucial for predicting how circuit behavior will change in response to component adjustments.

Transcript

00:01 For the first item, remember that the cosine of the phase is equal to the resistance divided by the impedance.

00:10 Then the impedance is equal to the resistance divided by the cosine of the phase.

00:18 The resistance is equal to 25 oms.

00:22 The cosine of the phase is equal to the square root of 3 divided by 2.

00:26 This is the cosine of 30 degrees.

00:28 And this is approximately 29 oms.

00:34 For the second item, remember that the impedance is equal to the square root of the resistance squared plus the difference between the capacity reactances squared.

00:49 Then we can solve this equation for the capacity reactance to get the following.

00:56 We first square both sides to get z squared is equal to r squared plus xl minus xc squared.

01:06 Then we do this to get z squared minus r squared is equal to xl minus xc squared.

01:21 Then we take the square root of both sides to get plus or minus the square root of z squared.

01:28 Squared minus r squared is equals to xl minus xc finally xc the capacity reactants is equal to xl plus or minus the square root of z squared minus r squared then we can calculate using the data given in the problem xl is equal to 2 pi times the frequency times 160 times 10 to the minus 3 which is the inductance plus or minus the square root of z squared so to keep the precision let's do like this 25 divided by the square root of 3 divided by 2 squared which is the impedance squared minus 25 squared then we get a reactive capacitance of approximately 69 .726 oms or 40 .858 oms then we have to choose one of those values how can we do that well note that the circuit is such that the voltage is leading the current...

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