Book edition 1st Edition

Author(s) Paul Peter Urone

Pages 1272 pages

ISBN 9781938168000

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Short Answer

An RLC series circuit has a $200 Ω$ resistor and a $25.0 m H$ inductor. At $8000 H z$ , the phase angle is $45 . 0^{\circ}$ . (a) What is the impedance? (b) Find the circuit's capacitance. (c) If $V_{r m s} = 408 V$ is applied, what is the average power supplied?

The impedance is $282.9 Ω$ .
The circuit's capacitance two possible solutions are $18.8 n F$ or $13.6 n F$ .
The average power supplied is $416.2 W$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Concept introduction

In an electronic circuit, a resistor is an electrical component that controls or regulates the passage of electrical current. Resistors can also be used to supply a set voltage to an active device like a transistor.

Step 2: Given data

The resistance of the RLC circuit is $R = 200 Ω$

The capacitance of the RLC circuit is $C = 25.0 m F (\frac{10^{- 3} F}{1 m F}) = 2.50 \times 10^{- 2} F$

The frequency of the circuit is $f = 8000 H z$

The phase angle is $ϕ = 45^{\circ}$

Step 2: Write the formula for the capacitor

The reactance of an RLC circuit can be expressed as,

$Z = \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$ ………………(1)

This means we'll need to figure out the capacitor and inductor's active resistances, which can be given as,

$X_{C} = \frac{1}{2 π f C}$ ………………….(2)

$X_{L} = 2 π L$ …………………..(3)

The phase angle can be expressed as,

$\cos ϕ = \frac{R}{Z}$ ………………….(4)

Step 2: Find the impedance(a)

Using the given data and rearranging equation (4), we get,

$Z = \frac{R}{\cos ϕ} = \frac{200 Ω}{0.707} = 282.9 Ω$

Therefore, the impedance is $282.9 Ω$

Step 3: Calculate the circuit's capacitance(b)

We may rearrange the concept of impedance to determine the capacitive reactance from equation (1), we get

${(X_{L} - X_{C})}^{2} = Z^{2} - R^{2}$

This means that the difference between inductive and capacitive reactances will have an absolute value of

$X_{L} - X_{C} = \sqrt{Z^{2} - R^{2}}$ …………………(5)

The inductive reactance can be found using equation (3), such that

$X_{L} = 2 π \times 8000 \times (2.50 \times 10^{- 2} F) = 1256.6 Ω$

Therefore using the value of $X_{L}$ in equation (5), we obtain that the value of capacitive reactance can be $X_{C 1} = 1057 Ω$ or $X_{C 2} = 1457 Ω$ is the capacitive reactance.

In these situations, we can look for capacity as follows using equation (2), such that,

$C = \frac{1}{2 π f X_{C}}$

Using the given data in the above expression we get,

For , $X_{C 1} = 1057 Ω, C_{1} = \frac{1}{2 π \times 8000 \times 1057 Ω} = 18.8 \times 10^{- 9} F (\frac{1 n F}{10^{- 9} F}) = 18.8 n F$

And for $X_{C 2} = 1457 Ω$ , we have

$C_{2} = \frac{1}{2 π \times 8000 \times 1457 Ω} = 13.6 \times 10^{- 9} F (\frac{1 n F}{10^{- 9} F}) = 13.6 n F$

Therefore, regarding the circuit's capacitance, two possible solutions are 18.8 nF 13.6 nF.

Step 4: Find the average power supplied(d)

The average power will be calculated by using the formula:

$P_{a v g} = V_{r m s} I_{r m s} = \frac{V_{r m s}^{2}}{Z}$

Substituting the given data in the above equation, we get,

$P_{a v g} = \frac{{(408 V)}^{2}}{282.9 Ω} = 416.2 W$

Therefore, the average power supplied is 416.2 W.

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College Physics (Urone)

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Short Answer

An RLC series circuit has a $200 Ω$ resistor and a $25.0 m H$ inductor. At $8000 H z$ , the phase angle is $45 . 0^{\circ}$ . (a) What is the impedance? (b) Find the circuit's capacitance. (c) If $V_{r m s} = 408 V$ is applied, what is the average power supplied?

Step by Step Solution

Step 1: Concept introduction

Step 2: Given data

Step 2: Write the formula for the capacitor

Step 2: Find the impedance(a)

Step 3: Calculate the circuit's capacitance(b)

Step 4: Find the average power supplied(d)

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College Physics (Urone)

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Short Answer

An RLC series circuit has a 200Ω resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is 45.0∘ . (a) What is the impedance? (b) Find the circuit's capacitance. (c) If Vrms=408 V is applied, what is the average power supplied?

Step by Step Solution

Step 1: Concept introduction

Step 2: Given data

Step 2: Write the formula for the capacitor

Step 2: Find the impedance(a)

Step 3: Calculate the circuit's capacitance(b)

Step 4: Find the average power supplied(d)

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An RLC series circuit has a $200 Ω$ resistor and a $25.0 m H$ inductor. At $8000 H z$ , the phase angle is $45 . 0^{\circ}$ . (a) What is the impedance? (b) Find the circuit's capacitance. (c) If $V_{r m s} = 408 V$ is applied, what is the average power supplied?