Assignment 23: AC 2, Power, and Resonance

A Resistor and a Capacitor in a Series AC Circuit

A resistor with resistance and a capacitor with capacitance are connected in series to an AC voltage source. The

time-dependent voltage across the capacitor is given by .

Part A

What is the amplitude of the total current in the circuit?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Applying Ohm's law to a capacitor

Hint not displayed

Hint A.3 The reactance of a capacitor

Hint not displayed

Express your answer in terms of any or all of , , , and .

ANSWER:

=

Correct

Part B

What is the amplitude of the voltage across the resistor?

Hint B.1 Relating to

Hint not displayed

Express your answer in terms of , , , and .

ANSWER:

=

Correct

Part C

If , , , and , what is ?

Hint C.1 Calculating the answer

Hint not displayed

Express your answer numerically, in millivolts, to the nearest integer.

ANSWER:

=

3

Correct mV

± A Voltage-Driven Parallel L-C Circuit

An AC source that provides a voltage drives an inductor having inductance and a capacitor having

capacitance , all connected in parallel.

Part A

Recall that the currents and through the inductor and capacitor are not in phase with their respective

voltages and . In particular, for a sinusoidal voltage driver, which of the following statements is true?

ANSWER:

and both lag their respective voltages.

and both lead their respective voltages.

leads and lags .

lags and leads .

Correct

The phase angle between voltage and current for inductors and capacitors is radians. Among other things, this

means that no power is dissipated in either, since the time-averaged product of current and voltage (= power) is

zero: .

Part B

What is the amplitude of the current through the voltage source? In other words, if the time-dependent current is

, what is the value of ?

Hint B.1 Voltage in a parallel circuit

Hint not displayed

Hint B.2 Current through the capacitor

Hint not displayed

Hint B.3 The current through the inductor

Hint not displayed

Hint B.4 Total current

Hint not displayed

Express the amplitude of the current in terms of , , , and .

ANSWER:

=

Correct

Part C

With and representing the respective magnitudes of the currents through the inductor and capacitor,

which of the following statements is true?

ANSWER:

At very high frequency and at very low frequency .

At very high frequency and at very low frequency .

for all frequencies.

for all frequencies.

and are about the same at all frequencies.

Correct

Part D

The L-C circuit is one example of a system that can exhibit resonance behavior. The resonant frequency is

. At this frequency, what is the amplitude of the current supplied by the voltage source?

Express your answer in terms of and any other need terms from the problem introduction.

ANSWER:

= 0

Correct

Part E

Which of the following statements best describes the implications of the result that when ?

ANSWER:

The currents ; ; are zero at all times.

The current is zero at all times although and are not.

The current is zero only when the voltage is zero.

The current is zero only when the amplitude of the voltage is not changing.

Correct

At the resonant frequency of the circut, the voltage source can supply voltage easily because the parallel L-C

circuit draws no current at all! (For this reason, a parallel L-C circuit can be used as a filter to pass signals at the

resonance frequency.) Of course, the same current flows in the inductor and the capacitor as the current that

flows when they are driven separately by the voltage source. However, the currents are exactly out of phase, so

that the net effect is a current circulating around the loop made by the capacitor and the inductor without passing

into or out of the voltage source.

Alternating Current, LC circuit

A capacitor with capacitance is connected in parallel to two inductors: inductor 1 with inductance , and inductor

2 with inductance , as shown in the figure. The capacitor is charged up to a voltage , at which point it has a

charge . There is no current in the inductors. Then the switch is closed.

Part A

Since the two inductors are in parallel, the voltage across them is the same at any time. Hence, ,

where and are the reactances of inductors 1 and 2, and and are the currents through them. Use this

equality to express in terms of .

Hint A.1 The reactance of an inductor

Hint not displayed

Express your answer in terms of .

ANSWER:

=

Correct

Part B

What is the effective inductance of the inductors 1 and 2 in the circuit?

Hint B.1 Formulas for effective inductance

Hint not displayed

Express your answer in terms of .

ANSWER:

=

Correct

Part C

Find the maximum current through inductor 1.

Hint C.1 Use conservation of energy to find the total maximum current

Hint not displayed

Express your answer in terms of , , and .

ANSWER:

=

Correct

Problem 31.40

Five infinite-impedance voltmeters, calibrated to read rms values, are connected as shown in figure

. Let 200 0.400 , 6.00 and 30.0 .

Part A

What is the reading of the voltmeter if 200 ?

ANSWER:

=

5.44

Correct

Part B

What is the reading of the voltmeter if 200 ?

ANSWER:

=

2.18

Correct

Part C

What is the reading of the voltmeter if 200 ?

ANSWER:

=

22.7

Correct

Part D

What is the reading of the voltmeter if 200 ?

ANSWER:

=

20.5

Correct

Part E

What is the reading of the voltmeter if 200 ?

ANSWER:

=

21.2

Correct

Part F

What is the reading of the voltmeter if 1000 ?

ANSWER:

=

13.8

Correct

Part G

What is the reading of the voltmeter if 1000 ?

ANSWER:

=

27.6

Correct

Part H

What is the reading of the voltmeter if 1000 ?

ANSWER:

=

11.5

Correct

Part I

What is the reading of the voltmeter if 1000 ?

ANSWER:

=

16.1

Correct

Part J

What is the reading of the voltmeter if 1000 ?

ANSWER:

=

21.2

Correct

Exercise 31.23

An L-R-C series circuit = 0.123 , = 244 , and = 7.33 carries an rms current of 0.448 with a

frequency of 402 .

Part A

What is the phase angle?

ANSWER:

=

0.811

Correct radians

Part B

What is the power factor for this circuit?

ANSWER:

0.689

Correct

Part C

What is the impedance of the circuit?

ANSWER:

=

354

Correct

Part D

What is the rms voltage of the source?

ANSWER:

=

159

Correct

Part E

What average power is delivered by the source?

ANSWER:

=

49.0

Correct

Part F

What is the average rate at which electrical energy is converted to thermal energy in the resistor?

ANSWER: = 49.0

Correct

Part G

What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor?

ANSWER:

0

Correct

Part H

What is the average rate at which electrical energy is dissipated (converted to other forms) in the inductor?

ANSWER:

0

Correct

Resonance in an R-L-C Circuit

In an L-R-C series circuit, the resistance is 320 ohms, the inductance is 0.400 henrys, and the capacitance is

1.60×10−2

microfarads.

Part A

What is the resonance angular frequency of the circuit?

Hint A.1 Definition of the resonance angular frequency

Hint not displayed

Hint A.2 Relationship between current and voltage amplitudes

Hint not displayed

Hint A.3 What is an expression for impedance?

Hint not displayed

Hint A.4 Finding the formula for the resonant frequency

Hint not displayed

Express your answer in radians per second to three significant figures.

ANSWER:

=

1.25×104

Correct

Part B

The capacitor can withstand a peak voltage of 560 volts. If the voltage source operates at the resonance

frequency, what maximum voltage amplitude can the source have if the maximum capacitor voltage is not

exceeded?

Hint B.1 Voltage across a capacitor

Hint not displayed

Hint B.2 Current at the resonance frequency

Hint not displayed

Express your answer in volts to three significant figures.

ANSWER:

=

35.8

Correct

Problem 31.68

A resistance , capacitance , and inductance are connected in series to a voltage source with amplitude and

variable angular frequency .

Part A

If , the resonance angular frequency, find the maximum current in the resistor.

Express your answer in terms of the variables , , , and .

ANSWER:

=

Correct

Part B

Find the maximum voltage across the capacitor.

Express your answer in terms of the variables , , , and .

ANSWER:

=

Correct

Part C

Find the maximum voltage across the inductor.

Express your answer in terms of the variables , , , and .

ANSWER:

=

Correct

Part D

Find the maximum energy stored in the capacitor.

Express your answer in terms of the variables , , , and .

ANSWER:

=

Correct

Part E

Find the maximum energy stored in the inductor.

Express your answer in terms of the variables , , , and .

ANSWER:

=

Correct