Alternating Current

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Summary

Chapter Seven: Alternating Current

Summary

Alternating current (AC) is a type of electrical current that changes direction periodically.
AC voltage varies with time, typically in a sine wave pattern, and is commonly used in electrical devices.
AC can be easily transformed to different voltages using transformers, making it efficient for long-distance transmission.
The relationship between voltage and current in AC circuits can be represented using phasors.
Key characteristics of AC circuits include:
- RMS Voltage (V): Effective voltage, calculated as V = √2 * Vₘ.
- RMS Current (I): Effective current, calculated as I = √2 * Iₘ.
- Inductive Reactance (Xₗ): Xₗ = ωL, where ω is the angular frequency and L is inductance.
- Capacitive Reactance (Xₐ): Xₐ = 1/(ωC), where C is capacitance.
- Impedance (Z): Total opposition to current flow in an AC circuit.
- Power Factor (cosΦ): Ratio of real power to apparent power in the circuit.
In a series RLC circuit, the average power loss is given by P = VIcosΦ.
Transformers can step up or step down voltages based on the number of turns in the primary and secondary coils.

Important Formulas and Definitions

Physical Quantity	Symbol	Formula/Definition	Unit	Remarks
RMS Voltage	V	V = √2 * Vₘ	V	Vₘ is the amplitude of the AC voltage.
RMS Current	I	I = √2 * Iₘ	A	Iₘ is the amplitude of the AC current.
Inductive Reactance	Xₗ	Xₗ = ωL	[ML²T⁻³]	ω is the angular frequency.
Capacitive Reactance	Xₐ	Xₐ = 1/(ωC)	[ML²T⁻³]	C is the capacitance.
Impedance	Z	Depends on circuit elements	[ML²T⁻³]	-
Resonant Frequency	ω₀	ω₀ = 1/√(LC)	Hz	For a series RLC circuit.
Quality Factor	Q	Q = ω₀/Δω	-	Dimensionless, relates to bandwidth.
Power Factor	cosΦ	cosΦ = P/(VI)	-	Φ is the phase difference between voltage and current.

Common Mistakes and Exam Tips

Mistake: Confusing RMS values with peak values. Always remember that RMS values are used for calculations in AC circuits.
Tip: When dealing with transformers, remember the relationship between primary and secondary voltages and currents based on the turns ratio.
Mistake: Forgetting that the average power in a purely inductive or capacitive circuit is zero.
Tip: Use phasor diagrams to visualize relationships between voltage and current in AC circuits, especially in RLC circuits.

Learning Objectives

Understand the concept of alternating current (AC) and its significance in electrical systems.
Describe the characteristics of AC voltage and current, including phase relationships.
Explain the operation of inductors and capacitors in AC circuits.
Analyze the behavior of series RLC circuits under AC voltage.
Calculate the root mean square (rms) values for voltage and current in AC circuits.
Define and calculate impedance, reactance, and power factor in AC circuits.
Understand the principles of transformers and their applications in voltage transformation.

Detailed Notes

Chapter Seven: Alternating Current

7.1 Introduction

Direct Current (DC): Currents that do not change direction with time.
Alternating Current (AC): Currents that vary with time, commonly represented as a sine function.
Importance of AC: Most electrical devices require AC voltage due to its efficient transmission and transformation.

7.2 Key Concepts

AC Voltage: Voltage that varies with time, denoted as v = vₘ sin(wt).
AC Current: Current driven by AC voltage, denoted as i = iₘ sin(wt + Φ).
Peak Values:
- Peak current: iₘ = √2 * I_rms
- Peak voltage: vₘ = √2 * V_rms

7.3 Circuit Behavior

Inductive Reactance (Xₗ): Xₗ = wL, where w is the angular frequency and L is the inductance.
Capacitive Reactance (Xₗ): X_C = 1/(wC), where C is the capacitance.
Impedance (Z): Z = √(R² + (Xₗ - X_C)²).
Power Factor: cos(Φ) = P/(VI), where P is the average power.

7.4 Transformers

Transformer Basics: A device that transforms AC voltage from one level to another using mutual induction.
Primary and Secondary Coils:
- Primary coil (Nₚ turns) connected to AC source.
- Secondary coil (Nₛ turns) induces voltage based on turns ratio.
Voltage Relationship: Vₛ/Vₚ = Nₛ/Nₚ.
Current Relationship: Iₛ/Iₚ = Nₚ/Nₛ.

7.5 Example Problem

Example 7.5: Inserting an iron rod into an inductor increases inductance, causing the light bulb's glow to decrease due to increased inductive reactance.

7.6 Important Formulas

Physical Quantity	Symbol	Formula	Remarks
RMS Voltage	V	V = √2 * vₘ	Amplitude of AC voltage
RMS Current	I	I = √2 * iₘ	Amplitude of AC current
Inductive Reactance	Xₗ	Xₗ = wL	Depends on frequency and inductance
Capacitive Reactance	X_C	X_C = 1/(wC)	Depends on frequency and capacitance
Impedance	Z	Z = √(R² + (Xₗ - X_C)²)	Total opposition to AC
Power Factor	-	cos(Φ) = P/(VI)	Efficiency of power usage

7.7 Points to Ponder

RMS values are typically used for AC voltage and current measurements.
Average power in AC circuits is calculated using RMS values.
In purely inductive or capacitive circuits, average power is zero despite current flow.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding AC and DC: Students often confuse alternating current (AC) with direct current (DC). Remember that AC changes direction periodically, while DC flows in one direction.
Ignoring RMS Values: When given voltage or current values, they are typically root mean square (RMS) values. Failing to recognize this can lead to incorrect calculations.
Phase Relationships: Students may forget that in inductive circuits, the current lags the voltage by π/2, while in capacitive circuits, the current leads the voltage by π/2.
Power Factor Confusion: The power factor, defined as cos(Φ), is crucial for understanding power loss in AC circuits. Not accounting for it can lead to errors in power calculations.

Tips for Success

Practice Phasor Diagrams: Familiarize yourself with phasor diagrams to visualize the relationships between voltage and current in AC circuits.
Understand Transformers: Know the difference between step-up and step-down transformers and how the number of turns in the coils affects voltage and current.
Review Key Formulas: Make sure to memorize key formulas related to AC circuits, such as those for impedance (Z), inductive reactance (Xₗ), and capacitive reactance (Xᶜ).
Work Through Examples: Solve example problems, especially those involving LCR circuits and transformers, to solidify your understanding of the concepts.

Practice & Assessment

Multiple Choice Questions

Resistor

Inductor

Capacitor

AC Voltage Source

Correct Answer: B

Solution:

In the diagram of a series LCR circuit, the inductor is depicted as a coil.

The circuit is purely resistive

The circuit is purely inductive

The circuit is purely capacitive

The circuit is at resonance

Correct Answer: A

Solution:

A power factor of 1 indicates that the voltage and current are in phase, which is characteristic of a purely resistive circuit.

Current leads the voltage by 90 degrees

Current lags the voltage by 90 degrees

Current is in phase with the voltage

Current leads the voltage by 180 degrees

Correct Answer: A

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees.

They are aligned in the same direction.

They are perpendicular to each other.

They are in opposite directions.

They form a 45-degree angle with each other.

Correct Answer: C

Solution:

At resonance, the inductive reactance

X_L

equals the capacitive reactance

X_C

, and the voltage phasors

V_L

and

V_C

are equal in magnitude but opposite in direction, canceling each other out.

Current decreases

Current increases

Current remains the same

Current becomes zero

Correct Answer: B

Solution:

In an AC circuit with a capacitor, increasing the frequency decreases the capacitive reactance, which increases the current.

5 Ω

8 Ω

4 Ω

10 Ω

Correct Answer: A

Solution:

The impedance

Z

of the circuit is calculated using

Z = \sqrt{R^2 + (X_L - X_C)^2}

. Given

X_L = 8 \text{ Ω}

and

X_C = 4 \text{ Ω}

Z = \sqrt{3^2 + (8 - 4)^2} = 5 \text{ Ω}

Resistor

Inductor

Capacitor

Both Inductor and Capacitor

Correct Answer: D

Solution:

Both the inductor and capacitor introduce a phase difference between voltage and current in an AC circuit.

It increases.

It decreases.

It remains unchanged.

It becomes infinite.

Correct Answer: B

Solution:

The resonant frequency is given by

\omega_0 = \frac{1}{\sqrt{LC}}

. Increasing

L

results in a decrease in

\omega_0

0.5

Infinity

Correct Answer: C

Solution:

The power factor of a purely resistive AC circuit is 1 because the voltage and current are in phase.

It acts as a power source.

It acts as a resistance.

It acts as a voltage source.

It acts as an inductor.

Correct Answer: B

Solution:

In a purely capacitive AC circuit, capacitive reactance acts as a resistance, limiting the current flow.

It remains the same.

It doubles.

It is halved.

It decreases by a factor of

\sqrt{2}

Correct Answer: D

Solution:

The resonant frequency

\omega_0

for a series RLC circuit is given by

\omega_0 = \frac{1}{\sqrt{LC}}

. If the inductance

L

is doubled, the resonant frequency becomes

\omega_0 = \frac{1}{\sqrt{2LC}}

, which is

\frac{1}{\sqrt{2}}

times the original frequency.

X_L > X_C

X_L < X_C

X_L = X_C

None of the above

Correct Answer: C

Solution:

At resonance in a series LCR circuit, the inductive reactance

X_L

equals the capacitive reactance

X_C

7200 W

4800 W

9600 W

3600 W

Correct Answer: B

Solution:

The power dissipated in the circuit is given by

P = I^2 R = (40)^2 \times 3 = 4800 \text{ W}

. The power factor does not affect the calculation of power dissipated in terms of resistance.

The impedance of the circuit decreases.

The impedance of the circuit increases.

The frequency of the circuit changes.

The phase difference between voltage and current becomes zero.

Correct Answer: A

Solution:

When a metal object is detected, it changes the impedance of the circuit, typically decreasing it, which results in a significant change in current, triggering the detector.

It becomes maximum

It becomes minimum

It remains constant

It becomes infinite

Correct Answer: B

Solution:

At the resonant frequency, the impedance of a series LCR circuit is minimum, allowing maximum current to flow through the circuit.

The phase difference is zero.

The phase difference is 53.1 degrees.

The phase difference is 60 degrees.

The phase difference is 30 degrees.

Correct Answer: B

Solution:

The power factor is given by

\cos \phi

, where

\phi

is the phase difference. If the power factor is 0.6, then

\phi = \cos^{-1}(0.6) = 53.1

degrees.

It increases.

It decreases.

It remains constant.

It becomes zero.

Correct Answer: B

Solution:

The capacitive reactance

X_C

is given by

X_C = \frac{1}{\omega C}

, where

\omega

is the angular frequency and

C

is the capacitance. As the frequency increases,

\omega

increases, leading to a decrease in

X_C

The current leads the voltage by 90 degrees.

The current lags the voltage by 90 degrees.

The current is in phase with the voltage.

The current is zero.

Correct Answer: B

Solution:

In a purely inductive AC circuit, the current lags the voltage by 90 degrees due to the nature of inductive reactance.

The current continues to flow steadily.

The current falls to zero.

The current reverses direction.

The current increases.

Correct Answer: B

Solution:

In a purely capacitive AC circuit, when the capacitor is fully charged, the current falls to zero as the voltage across the capacitor opposes the current.

Ohm

Henry

Farad

Watt

Correct Answer: A

Solution:

The unit of impedance in an AC circuit is Ohm, similar to resistance.

It doubles.

It halves.

It remains the same.

It becomes zero.

Correct Answer: B

Solution:

The capacitive reactance

X_C

is given by

X_C = \frac{1}{\omega C}

, where

\omega

is the angular frequency. If the frequency is doubled,

\omega

is doubled, and thus

X_C

is halved.

The resonant frequency increases.

The resonant frequency decreases.

The resonant frequency remains unchanged.

The resonant frequency becomes zero.

Correct Answer: C

Solution:

The resonant frequency is determined by the inductance and capacitance, and is not affected by the resistance.

It increases the impedance

It decreases the impedance

It cancels the inductive reactance

It has no effect

Correct Answer: C

Solution:

At resonance, the capacitive reactance cancels the inductive reactance, minimizing the impedance.

The impedance is minimum and equals the resistance

R

The impedance is maximum and equals the sum of

R

X_L

, and

X_C

The impedance is zero.

The impedance is infinite.

Correct Answer: A

Solution:

At resonance in a series RLC circuit, the inductive reactance

X_L

equals the capacitive reactance

X_C

, making the impedance purely resistive and equal to the resistance

R

50 \ \Omega

60 \ \Omega

70 \ \Omega

80 \ \Omega

Correct Answer: B

Solution:

The impedance

Z

is calculated as

Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{40^2 + (50 - 30)^2} = \sqrt{1600 + 400} = \sqrt{2000} = 60 \ \Omega

They are equal in magnitude and in phase.

They are equal in magnitude but out of phase by

\pi

The voltage across the inductor is zero.

The voltage across the capacitor is zero.

Correct Answer: B

Solution:

At resonance in a series RLC circuit, the inductive reactance

X_L

equals the capacitive reactance

X_C

, making the voltages across the inductor and capacitor equal in magnitude but out of phase by

\pi

(180 degrees).

It completely blocks the current

It allows current to flow continuously

It limits or regulates the current

It amplifies the current

Correct Answer: C

Solution:

In a purely capacitive AC circuit, the capacitor limits or regulates the current, but does not completely prevent the flow of charge.

0 \ \Omega

10 \ \Omega

20 \ \Omega

30 \ \Omega

Correct Answer: B

Solution:

At resonance, the impedance

Z

is equal to the resistance

R

since

X_L = X_C

. Therefore,

Z = R = 10 \ \Omega

0.5

Infinity

Correct Answer: B

Solution:

The power factor is given by

\cos(\Phi)

, where

\Phi

is the phase difference. For

\Phi = 90^\circ

\cos(90^\circ) = 0

. Therefore, the power factor is 0.

It doubles.

It halves.

It remains the same.

It becomes zero.

Correct Answer: B

Solution:

The quality factor

Q

is given by

Q = \frac{1}{R} \sqrt{\frac{L}{C}}

. If

R

is doubled,

Q

becomes half of its original value, since

Q

is inversely proportional to

R

0 degrees

90 degrees

180 degrees

270 degrees

Correct Answer: A

Solution:

In a purely resistive AC circuit, the voltage and current are in phase, meaning the phase difference is 0 degrees.

Both are in phase with the current phasor.

Both are out of phase with the current phasor.

The inductor voltage phasor leads the current phasor, and the capacitor voltage phasor lags the current phasor.

The inductor voltage phasor lags the current phasor, and the capacitor voltage phasor leads the current phasor.

Correct Answer: C

Solution:

In a series LCR circuit, the inductor voltage phasor leads the current phasor by 90 degrees, and the capacitor voltage phasor lags the current phasor by 90 degrees.

It decreases by a factor of 2.

It decreases by a factor of 4.

It increases by a factor of 2.

It remains the same.

Correct Answer: A

Solution:

The resonant frequency

\omega_0 = \frac{1}{\sqrt{LC}}

is inversely proportional to the square root of the inductance

L

. If

L

is increased by a factor of 4, then

\omega_0

decreases by a factor of

\sqrt{4} = 2

X_L > X_C

X_L < X_C

X_L = X_C

X_L + X_C = 0

Correct Answer: C

Solution:

At resonance in a series RLC circuit, the inductive reactance

X_L

equals the capacitive reactance

X_C

, i.e.,

X_L = X_C

. This ensures that their effects cancel each other out.

The phase angle increases.

The phase angle decreases.

The phase angle remains the same.

The phase angle becomes zero.

Correct Answer: B

Solution:

The phase angle

\phi

in an RL circuit is given by

\phi = \tan^{-1}(\frac{X_L}{R})

, where

X_L

is the inductive reactance. Increasing the resistance

R

decreases the value of

\tan^{-1}(\frac{X_L}{R})

, thus decreasing the phase angle.

The amplitude of the voltage

The amplitude of the current

The phase difference between voltage and current

The frequency of the AC source

Correct Answer: C

Solution:

In a phasor diagram, the angle between the voltage and current phasors represents the phase difference between them.

X_C = \frac{1}{wC}

X_C = wC

X_C = \frac{w}{C}

X_C = Cw

Correct Answer: A

Solution:

The capacitive reactance

X_C

is given by

X_C = \frac{1}{wC}

, where

w

is the angular frequency and

C

is the capacitance.

The circuit is in resonance.

The circuit has maximum impedance.

The circuit has zero impedance.

The circuit is purely resistive.

Correct Answer: A

Solution:

When the inductive reactance

X_L

equals the capacitive reactance

X_C

in a series RLC circuit, the circuit is at resonance. At this point, the impedance is minimum and equals the resistance

R

, allowing maximum current to flow.

It decreases.

It increases.

It remains the same.

It becomes zero.

Correct Answer: B

Solution:

The inductive reactance

X_L = \omega L

is directly proportional to the frequency

\omega

. Therefore, increasing the frequency increases the inductive reactance.

The frequency of the AC source

The phase angle between voltage and current

The amplitude or peak value of the oscillating quantity

The average value of the oscillating quantity

Correct Answer: C

Solution:

In a phasor diagram, the length of a phasor represents the amplitude or peak value of the oscillating quantity.

It becomes maximum.

It becomes zero.

It becomes minimum.

It becomes infinite.

Correct Answer: C

Solution:

At resonant frequency in a series RLC circuit, the impedance becomes minimum, allowing maximum current to flow.

Resistor

Inductor

Capacitor

AC Source

Correct Answer: B

Solution:

In a series LCR circuit, the inductor is responsible for storing energy in the form of a magnetic field.

It is the ratio of the inductive reactance to the capacitive reactance.

It is the cosine of the phase difference between voltage and current.

It is always equal to one.

It is the sine of the phase difference between voltage and current.

Correct Answer: B

Solution:

The power factor is the cosine of the phase difference between the voltage applied and the current in the circuit.

0.5

0.866

Correct Answer: A

Solution:

The power factor is given by

\cos(\phi)

, where

\phi

is the phase difference between the voltage and the current. For a phase difference of 60 degrees,

\cos(60^\circ) = 0.5

36.87^\circ

45^\circ

53.13^\circ

60^\circ

Correct Answer: A

Solution:

The power factor is

\cos \Phi = 0.8

. Therefore,

\Phi = \cos^{-1}(0.8) = 36.87^\circ

1.00 Mrad/s

2.00 Mrad/s

0.50 Mrad/s

1.50 Mrad/s

Correct Answer: A

Solution:

The resonant frequency

\omega_0

for a series RLC circuit is given by

\omega_0 = \frac{1}{\sqrt{LC}}

. Substituting

L = 1.00 \times 10^{-3}

H and

C = 1.00 \times 10^{-9}

F, we get

\omega_0 = \frac{1}{\sqrt{1.00 \times 10^{-3} \times 1.00 \times 10^{-9}}} = 1.00 \times 10^6

rad/s or 1.00 Mrad/s.

It remains the same.

It doubles.

It halves.

It becomes zero.

Correct Answer: A

Solution:

The resonant frequency of a series RLC circuit is given by

\omega_0 = \frac{1}{\sqrt{LC}}

. It is independent of the resistance

R

, so doubling

R

does not affect the resonant frequency.

It completely blocks the current.

It allows current to flow continuously without opposition.

It regulates the current by charging and discharging.

It acts as a short circuit.

Correct Answer: C

Solution:

In an AC circuit, a capacitor alternately charges and discharges as the current reverses each half cycle, regulating the current.

5 Ω

10 Ω

3 Ω

8 Ω

Correct Answer: A

Solution:

To find the impedance

Z

, we calculate

X_L = 2\pi f L = 8 \Omega

and

X_C = \frac{1}{2\pi f C} = 4 \Omega

. The impedance

Z

is given by

Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{3^2 + (8 - 4)^2} = 5 \Omega

It increases.

It decreases.

It remains the same.

It becomes zero.

Correct Answer: B

Solution:

The quality factor

Q

is given by

Q = \frac{1}{R} \sqrt{\frac{L}{C}}

. Increasing the resistance

R

decreases the quality factor.

Resistance

Inductive Reactance

Capacitive Reactance

Impedance

Correct Answer: C

Solution:

The opposition to current flow provided by a capacitor in an AC circuit is called capacitive reactance.

The power dissipated is maximum.

The power dissipated is zero.

The power dissipated is half of the maximum.

The power dissipated is negative.

Correct Answer: B

Solution:

When the power factor is zero, the power dissipated in the circuit is zero because the current is entirely reactive.

0 degrees

90 degrees

180 degrees

270 degrees

Correct Answer: B

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees.

The current leads the voltage by 90 degrees.

The voltage leads the current by 90 degrees.

The voltage and current are in phase.

The voltage and current are out of phase by 180 degrees.

Correct Answer: C

Solution:

In a purely resistive AC circuit, the voltage and current are in phase with each other.

Resistor

Inductor

Capacitor

Both Inductor and Capacitor

Correct Answer: D

Solution:

In a series LCR circuit, both the inductor and capacitor contribute to the phase difference between the voltage and current.

The impedance decreases.

The impedance increases.

The impedance remains constant.

The impedance becomes zero.

Correct Answer: B

Solution:

When a person carrying metal walks through a metal detector, the impedance of the circuit changes due to the metal altering the inductance or capacitance, causing a significant change in current, which is detected by the system. Typically, this results in an increase in impedance.

Voltage leads current by 90 degrees.

Voltage lags current by 90 degrees.

Voltage and current are in phase.

Voltage leads current by 180 degrees.

Correct Answer: C

Solution:

In a purely resistive AC circuit, the voltage and current are in phase with each other.

It remains the same.

It doubles.

It halves.

It increases by a factor of

\sqrt{2}

Correct Answer: D

Solution:

The resonant frequency

\omega_0

is given by

\omega_0 = \frac{1}{\sqrt{LC}}

. If

L

is doubled and

C

is halved, the new resonant frequency

\omega_0' = \frac{1}{\sqrt{2L \cdot \frac{C}{2}}} = \frac{1}{\sqrt{LC/2}} = \sqrt{2} \cdot \frac{1}{\sqrt{LC}} = \sqrt{2} \cdot \omega_0$$. Therefore, the resonant frequency increases by a factor of

\sqrt{2}$.

The impedance is equal to the resistance.

The impedance is greater than the resistance.

The impedance is less than the resistance.

The impedance is zero.

Correct Answer: B

Solution:

At half the resonant frequency, the capacitive reactance

X_C

is greater than the inductive reactance

X_L

, resulting in a net capacitive reactance. This increases the overall impedance, making it greater than the resistance.

Voltage leads current by

\pi/2

Voltage lags current by

\pi/2

Voltage and current are in phase.

Voltage and current are out of phase by

\pi

Correct Answer: B

Solution:

In a capacitive circuit, the voltage across the capacitor lags the current by

\pi/2

, as shown in the phasor diagram.

In phase with the voltage phasor.

90 degrees ahead of the voltage phasor.

90 degrees behind the voltage phasor.

180 degrees out of phase with the voltage phasor.

Correct Answer: B

Solution:

In a series RC circuit, the current phasor leads the voltage phasor across the capacitor by 90 degrees, as the current reaches its maximum before the voltage does.

i_m = \omega C v_m

i_m = \frac{v_m}{R}

i_m = \frac{v_m}{\omega C}

i_m = \omega L v_m

Correct Answer: A

Solution:

In a purely capacitive AC circuit, the peak current

i_m

is given by

i_m = \omega C v_m

, where

\omega

is the angular frequency and

C

is the capacitance.

It becomes zero.

It becomes infinite.

It equals the resistance.

It equals the reactance.

Correct Answer: C

Solution:

At resonance, the impedance of a series RLC circuit equals the resistance because the inductive and capacitive reactances cancel each other out.

The circuit is purely resistive.

The circuit is purely inductive or capacitive.

The circuit has maximum power dissipation.

The circuit has no power dissipation.

Correct Answer: B

Solution:

A power factor of zero indicates that the voltage and current are 90 degrees out of phase, which occurs in purely inductive or capacitive circuits where no real power is dissipated.

It increases the impedance.

It decreases the impedance.

It cancels the effect of the capacitor.

It has no role at resonance.

Correct Answer: C

Solution:

At resonance in a series LCR circuit, the inductor cancels the effect of the capacitor, resulting in minimum impedance.

The circuit is purely resistive.

The circuit is purely inductive.

The circuit is purely capacitive.

The circuit has maximum impedance.

Correct Answer: A

Solution:

A power factor of 1 indicates that the voltage and current are in phase, meaning the circuit is purely resistive, and all the power is being used effectively.

Ohm

Hertz

Farad

Henry

Correct Answer: B

Solution:

The unit of resonant frequency in a series RLC circuit is Hertz (Hz).

The circuit behaves predominantly as a capacitor

The circuit behaves predominantly as an inductor

The circuit behaves predominantly as a resistor

The circuit behaves predominantly as a short circuit

Correct Answer: B

Solution:

At frequencies much higher than the resonant frequency, the inductive reactance

X_L = \omega L

dominates over the capacitive reactance

X_C = \frac{1}{\omega C}

, making the circuit behave predominantly as an inductor.

5 \ \Omega

6 \ \Omega

7 \ \Omega

8 \ \Omega

Correct Answer: A

Solution:

The impedance

Z

is calculated using

Z = \sqrt{R^2 + (X_L - X_C)^2}

. First, calculate

X_L = 2\pi fL = 2 \times 3.14 \times 50 \times 25.48 \times 10^{-3} = 8 \ \Omega

and

X_C = \frac{1}{2\pi fC} = \frac{1}{2 \times 3.14 \times 50 \times 796 \times 10^{-6}} = 4 \ \Omega

. Hence,

Z = \sqrt{3^2 + (8 - 4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \ \Omega

Both are in phase with the current phasor.

The inductor voltage phasor leads the current phasor by 90°, and the capacitor voltage phasor lags the current phasor by 90°.

Both lag the current phasor by 90°.

Both lead the current phasor by 90°.

Correct Answer: B

Solution:

In a phasor diagram, the voltage across the inductor leads the current by 90°, while the voltage across the capacitor lags the current by 90°. This is due to the phase relationships in inductive and capacitive reactances.

Voltage phasor is perpendicular to current phasor.

Voltage phasor leads current phasor by 90 degrees.

Voltage and current phasors are in the same direction.

Voltage phasor lags current phasor by 90 degrees.

Correct Answer: C

Solution:

In a phasor diagram for a resistor, the voltage and current phasors are in the same direction, indicating they are in phase.

It acts as resistance to the current flow.

It increases the current flow.

It stores energy in the form of a magnetic field.

It converts AC to DC.

Correct Answer: A

Solution:

In a purely capacitive AC circuit, the capacitive reactance

X_c = \frac{1}{\omega C}

acts as resistance to the current flow, limiting the current.

It doubles.

It halves.

It remains the same.

It becomes zero.

Correct Answer: A

Solution:

The capacitive reactance is given by

X_C = \frac{1}{\omega C}

. Halving the frequency

\omega

doubles the reactance.

Ohm

Farad

Henry

Volt

Correct Answer: A

Solution:

Reactance, like resistance, is measured in Ohms.

200 W

400 W

500 W

600 W

Correct Answer: B

Solution:

The average power consumed in an AC circuit is given by

P = VI\cos\phi

, where

V

is the rms voltage,

I

is the rms current, and

\cos\phi

is the power factor. The rms voltage

V = \frac{V_m}{\sqrt{2}} = \frac{100}{\sqrt{2}} \approx 70.7

V. Thus,

P = 70.7 \times 5 \times 0.8 = 400

It becomes zero.

It reaches its maximum value.

It becomes infinite.

It equals the capacitive reactance.

Correct Answer: B

Solution:

At resonance, the current amplitude in a series RLC circuit reaches its maximum value.

Voltage leads the current by 90 degrees

Voltage lags the current by 90 degrees

Voltage is in phase with the current

Voltage leads the current by 180 degrees

Correct Answer: A

Solution:

In a series LCR circuit, the voltage across the inductor leads the current by 90 degrees.

0 degrees

90 degrees

180 degrees

270 degrees

Correct Answer: B

Solution:

In a purely inductive AC circuit, the current lags the voltage by 90 degrees.

Impedance increases, reducing current.

Impedance decreases, increasing current.

Impedance remains unchanged.

Impedance becomes zero.

Correct Answer: B

Solution:

When a metal object is introduced, it alters the inductance and capacitance, leading to a decrease in impedance, which increases the current, triggering the alarm.

True or False

Correct Answer: True

Solution:

In a purely inductive AC circuit, the voltage leads the current by 90 degrees, meaning the current lags the voltage by 90 degrees.

Correct Answer: False

Solution:

Resonance in a circuit requires both an inductor and a capacitor to be present. It cannot occur in a circuit that only contains a resistor and an inductor.

Correct Answer: True

Solution:

At the resonant frequency, the impedance of the circuit is minimized, resulting in the maximum current amplitude.

Correct Answer: True

Solution:

In a series LCR circuit, the voltage across the inductor is ahead of the current by 90 degrees due to the inductive reactance.

Correct Answer: True

Solution:

In a series RLC circuit, the impedance is minimum at the resonant frequency because the inductive and capacitive reactances cancel each other out.

Correct Answer: True

Solution:

In a purely resistive AC circuit, the voltage and current are in phase, resulting in a phase angle of zero.

Correct Answer: False

Solution:

The power consumed in an AC circuit is never negative. It is always a positive value as power represents energy consumption.

Correct Answer: True

Solution:

In a purely resistive AC circuit, the voltage and current reach their maximum and minimum values at the same time, indicating that they are in phase.

Correct Answer: True

Solution:

In a series RLC circuit, the voltage across the resistor is parallel to the current phasor, meaning they are in phase.

Correct Answer: True

Solution:

In a series LCR circuit, the voltage across the resistor is in phase with the current because the resistor does not introduce any phase shift between voltage and current.

Correct Answer: True

Solution:

In a purely resistive AC circuit, the voltage and current reach their maximum and minimum values simultaneously, indicating they are in phase.

Correct Answer: True

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees due to the nature of the capacitive reactance.

Correct Answer: False

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees.

Correct Answer: True

Solution:

At the resonant frequency, the inductive reactance

X_L

equals the capacitive reactance

X_C

, resulting in the minimum impedance, which is equal to the resistance

R

Correct Answer: False

Solution:

In a DC circuit, a capacitor will initially allow current to flow as it charges, but once fully charged, it will oppose and stop the current flow.

Correct Answer: False

Solution:

In an AC circuit, the average power consumed is not zero; it is given by

P = VI \cos \phi

, where

\phi

is the phase difference.

Correct Answer: True

Solution:

The quality factor of a series RLC circuit is dimensionless, as it is a ratio of reactance to resistance.

Correct Answer: True

Solution:

The power factor, defined as the cosine of the phase angle between voltage and current, ranges from 0 to 1.

Correct Answer: True

Solution:

The quality factor, which measures the sharpness of resonance, is a dimensionless quantity.

Correct Answer: False

Solution:

In a purely resistive AC circuit, the power factor is 1 because the voltage and current are in phase.

Correct Answer: False

Solution:

In an AC circuit, the voltage and current are not always in phase. For example, in a purely inductive or capacitive circuit, there is a phase difference of 90 degrees between them.

Correct Answer: True

Solution:

At resonance, the inductive reactance and capacitive reactance cancel each other out, making the impedance equal to the resistance.

Correct Answer: False

Solution:

In a purely resistive AC circuit, the power factor is 1 because the voltage and current are in phase.

Correct Answer: False

Solution:

Resonance in an AC circuit requires both an inductor and a capacitor. It cannot occur in a circuit with only a resistor and an inductor.

Correct Answer: False

Solution:

The impedance of a series RLC circuit is equal to the resistance only at the resonant frequency. At other frequencies, the impedance is affected by the inductive and capacitive reactances.

Correct Answer: False

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees.

Correct Answer: False

Solution:

The average power consumed in an AC circuit is not zero; it is determined by the power factor and the rms values of voltage and current.

Correct Answer: True

Solution:

In a series LCR circuit, all components are connected in series, so the same current flows through each component.

Correct Answer: False

Solution:

In a purely resistive AC circuit, the voltage and current are in phase, meaning they reach their maximum and minimum values simultaneously.

Correct Answer: True

Solution:

Metal detectors operate on the principle of resonance in AC circuits, detecting changes in impedance when metal is present.

Correct Answer: False

Solution:

In a purely resistive AC circuit, the voltage and current are in phase with each other.

Correct Answer: False

Solution:

In a series LCR circuit, the voltage across the capacitor lags the current by 90 degrees.

Correct Answer: False

Solution:

The power factor in an AC circuit is the cosine of the phase angle between the voltage and current, which ranges from 0 to 1.

Correct Answer: True

Solution:

In an AC circuit, the current alternates direction, resulting in an average current of zero over one complete cycle.

Correct Answer: True

Solution:

The rms (root mean square) value of an AC voltage is calculated as the peak value divided by the square root of 2, making it always less than the peak value.

Correct Answer: True

Solution:

Metal detectors operate based on the principle of resonance in AC circuits. The presence of metal changes the impedance, causing a detectable change in current.

Correct Answer: True

Solution:

At resonance, the impedance of a series RLC circuit is purely resistive and equal to the resistance, as the inductive and capacitive reactances cancel each other.

Correct Answer: True

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees, making the power factor, which is the cosine of the phase difference, equal to zero.

Correct Answer: True

Solution:

The capacitive reactance, denoted as

X_C = \frac{1}{\omega C}

, plays the role of resistance in a purely capacitive AC circuit, determining the current amplitude.

Correct Answer: True

Solution:

If the power factor is zero, it means that the current is entirely reactive (wattless), resulting in zero average power consumption.

Correct Answer: True

Solution:

The power factor is indeed defined as the cosine of the phase difference between the applied voltage and the current in the circuit.

Correct Answer: True

Solution:

At the resonant frequency, the impedance is minimum, leading to a maximum current amplitude in a series RLC circuit.

Correct Answer: True

Solution:

At resonance, the impedance of a series RLC circuit is purely resistive and equal to the resistance.

Correct Answer: False

Solution:

In a series LCR circuit, the voltage across the capacitor lags the current by 90 degrees, so they are not in phase.

Correct Answer: True

Solution:

Impedance in an AC circuit takes into account both resistance and reactance, and is therefore always greater than or equal to the resistance alone.

Correct Answer: True

Solution:

Resonance occurs in an RLC circuit when the inductive reactance

X_L

equals the capacitive reactance

X_C

, minimizing the impedance.

Correct Answer: True

Solution:

The power factor in an AC circuit is indeed defined as the cosine of the phase difference between the voltage and the current.

Correct Answer: False

Solution:

Resonance requires both inductance (L) and capacitance (C) in the circuit. It cannot occur in an RL or RC circuit alone.

Correct Answer: True

Solution:

In a purely capacitive AC circuit, the current leads the voltage by 90 degrees due to the phase difference introduced by the capacitor.

Correct Answer: True

Solution:

At the resonant frequency, the impedance is minimized, and thus the current amplitude is maximized in a series RLC circuit.

Correct Answer: False

Solution:

The power consumed in an AC circuit is never negative. It represents the average power, which is always a positive value.

Correct Answer: True

Solution:

In a purely inductive AC circuit, the current lags the voltage by 90 degrees due to the inductive reactance.

Correct Answer: True

Solution:

Resonance in a series RLC circuit occurs when the inductive reactance (

X_L

) equals the capacitive reactance (

X_C

), minimizing the impedance.

Correct Answer: False

Solution:

At resonance, the impedance of a series RLC circuit is equal to the resistance, not the inductive reactance.

Correct Answer: False

Solution:

The rms value of voltage in an AC circuit is not the same as the peak value. It is given by

V = \frac{V_m}{\sqrt{2}}

, where

V_m

is the peak value.

Correct Answer: True

Solution:

At the resonant frequency, the impedance is minimum, and the current amplitude is maximum, given by

i_m = \frac{v_m}{R}

Alternating Current

AI Learning Assistant

Summary

Chapter Seven: Alternating Current

Summary

Important Formulas and Definitions

Common Mistakes and Exam Tips

Learning Objectives

Learning Objectives

Detailed Notes

Chapter Seven: Alternating Current

7.1 Introduction

7.2 Key Concepts

7.3 Circuit Behavior

7.4 Transformers

7.5 Example Problem

7.6 Important Formulas

7.7 Points to Ponder

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.In a series LCR circuit connected to an AC source, which component is represented by a coil in the diagram?

Correct Answer: B

Q2.In an AC circuit, the power factor is defined as the cosine of the phase angle between voltage and current. If the power factor is 1, what can be inferred about the circuit?

Correct Answer: A

Q3.In a purely capacitive AC circuit, how is the current related to the voltage?

Correct Answer: A

Q4.In a phasor diagram for a series RLC circuit at resonance, how are the voltage phasors across the inductor VLV_LVL​ and the capacitor VCV_CVC​ oriented relative to each other?

Correct Answer: C

Q5.In an AC circuit with a capacitor, what happens to the current when the frequency of the source is increased?

Correct Answer: B

Q6.A series RLC circuit has a resistance of 3 Ω, an inductance of 25.48 mH, and a capacitance of 796 µF. If a sinusoidal voltage of peak value 283 V and frequency 50 Hz is applied, what is the impedance of the circuit?

Correct Answer: A

Q7.Which component in an AC circuit is responsible for the phase difference between voltage and current?

Correct Answer: D

Q8.For a series RLC circuit at resonance, if the inductance LLL is increased while keeping the capacitance CCC constant, what happens to the resonant frequency ω0\omega_0ω0​?

Correct Answer: B

Q9.What is the power factor of a purely resistive AC circuit?

Correct Answer: C

Q10.What is the role of capacitive reactance in a purely capacitive AC circuit?

Correct Answer: B

Q11.In a series RLC circuit with an AC source, if the inductance is doubled while keeping the capacitance constant, what happens to the resonant frequency?

Correct Answer: D

Q12.In a series LCR circuit at resonance, what is the relationship between the inductive reactance XLX_LXL​ and the capacitive reactance XCX_CXC​?

Correct Answer: C

Q13.In an AC circuit, if the power factor is 0.6 and the rms current is 40 A, what is the power dissipated in the circuit if the resistance is 3 Ω?

Correct Answer: B

Q14.A metal detector at an airport works on the principle of resonance in AC circuits. What change occurs in the circuit when a metal object is detected?

Correct Answer: A

Q15.In a series LCR circuit connected to an AC source, what happens to the impedance at the resonant frequency?

Correct Answer: B

Q16.A series RLC circuit is connected to an AC source. If the power factor of the circuit is 0.6, what does this indicate about the phase difference between the voltage and the current?

Correct Answer: B

Q17.In a purely capacitive AC circuit, if the frequency of the AC source is increased, what happens to the capacitive reactance XCX_CXC​?

Correct Answer: B

Q18.In a purely inductive AC circuit, how is the current related to the voltage?

Correct Answer: B

Q19.In a purely capacitive AC circuit, what happens to the current when the capacitor is fully charged?

Correct Answer: B

Q20.What is the unit of impedance in an AC circuit?

Correct Answer: A

Q21.In a purely capacitive AC circuit, if the frequency of the AC source is doubled, what happens to the capacitive reactance XCX_CXC​?

Correct Answer: B

Q22.What is the effect of increasing the resistance in a series RLC circuit on the resonant frequency?

Correct Answer: C

Q23.In a series LCR circuit, what is the role of the capacitor at resonance?

Correct Answer: C

Q24.In a series RLC circuit, if the inductive reactance XLX_LXL​ is equal to the capacitive reactance XCX_CXC​, what can be said about the impedance ZZZ of the circuit at this condition?

Correct Answer: A

Correct Answer: B

Q26.In a series RLC circuit at resonance, what is the relationship between the voltage across the inductor VLV_LVL​ and the voltage across the capacitor VCV_CVC​?

Correct Answer: B

Q27.What is the role of a capacitor in a purely capacitive AC circuit?

Correct Answer: C

Q28.In a series RLC circuit at resonance, the impedance is purely resistive. If the resistance R=10 ΩR = 10 \ \OmegaR=10 Ω, what is the impedance at resonance?

Correct Answer: B

Q29.What is the power factor of an AC circuit if the phase difference between the voltage and current is 90 degrees?