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7.4 - AC VOLTAGE APPLIED TO AN INDUCTOR

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Inductive Reactance

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Teacher
Teacher

Today we're diving into inductive reactance, which is crucial for understanding how inductors behave in AC circuits. Inductive reactance is defined as X_L = ωL. Can anyone tell me what ω represents?

Student 1
Student 1

Isn’t ω the angular frequency?

Teacher
Teacher

Correct! It's the rate of oscillation of the AC voltage. Now, when we increase the frequency, what happens to the inductive reactance?

Student 2
Student 2

It increases as well!

Teacher
Teacher

Exactly! So, in a purely inductive circuit, the reactance affects the current flowing. Can someone remind me about the relationship between voltage and current for an inductor?

Student 3
Student 3

The current lags behind the voltage by 90 degrees.

Teacher
Teacher

Yes! That's a crucial concept when analyzing AC circuits. Remember the mnemonic 'Lagging Larry' to help you recall that current lags voltage in inductors. Let's sum up this session: inductive reactance increases with frequency, and current lags voltage by 90 degrees.

Phase Difference

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Teacher
Teacher

Now, let's delve deeper into the phase difference in an inductor. Who can explain why the current lags the voltage?

Student 4
Student 4

It lags because inductors resist changes in current, right?

Teacher
Teacher

Exactly! This resistance to change in current is due to the stored magnetic energy in the inductor. Can anyone visualize how this is represented in a phasor diagram?

Student 1
Student 1

The voltage phasor would be pointing upwards and the current phasor would be at a right angle to the left?

Teacher
Teacher

Well put! The current phasor's position shows it's 90 degrees behind the voltage. Remember, 'Voltage leads, current lags.' Please note this because it often comes up in questions. To summarize, current lags voltage in an inductor due to the energy storage.

Power Dissipation

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Teacher
Teacher

Let's discuss the concept of power dissipation in inductors. What happens to the average power consumed by a pure inductor?

Student 2
Student 2

It’s zero, right? Because the current and voltage alternate.

Teacher
Teacher

Correct again! The instant power varies, but over a complete cycle, the average is zero. This differentiates inductors from resistive loads where power is always consumed. How does this fact help when analyzing circuit behavior?

Student 3
Student 3

It shows that while energy is temporarily stored, it doesn’t get converted into power like with resistors.

Teacher
Teacher

Well said! Using the phrase 'No power loss with L' can help you recall that inductors don’t dissipate power. In summary, even though the voltage and current vary, the average power is zero.

Key Equations

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Teacher
Teacher

Now, let’s look at the key equations related to inductors in AC circuits. The amplitude of current can be represented as i_m = v_m / X_L. Why is it essential to know the peak voltage?

Student 4
Student 4

The peak voltage tells us the maximum value of voltage applied over a cycle.

Teacher
Teacher

Exactly! This is critical when calculating current amplitude. Now, who can express the relationship between current and voltage phasors?

Student 1
Student 1

Voltage leads the current phasor by 90 degrees.

Teacher
Teacher

Good job! Always visualize the vectors. To wrap up, remember the equations for analyzing current in a purely inductive AC circuit, along with their phase relationship.

Practical Significance

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Teacher
Teacher

Finally, let's discuss the practical applications of inductors. Where have you seen inductors used in everyday technology?

Student 2
Student 2

They are in transformers and power supplies, right?

Teacher
Teacher

Absolutely! Inductors are indispensable in electrical engineering for controlling current and managing energy storage. Can someone remind me why they are essential in AC systems compared to DC?

Student 3
Student 3

Because they can limit current flow without dissipating power.

Teacher
Teacher

Exactly! Remember, they play a critical part in electrical applications like motors and filters as well. To summarize, inductors are vital for numerous technologies due to their energy storage capabilities without power loss.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explores how an alternating current (AC) voltage affects an inductor, emphasizing the concepts of inductive reactance and the phase difference between current and voltage.

Standard

The section elaborates on the behavior of an inductor in an AC circuit, demonstrating that the current lags the voltage by π/2 (90 degrees) due to inductive reactance. It provides key equations that relate voltage, current, and inductance in these scenarios, reinforcing the understanding of power dissipation.

Detailed

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Audio Book

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Introduction to AC Voltage in Inductors

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Figure 7.5 shows an ac source connected to an inductor. Usually, inductors have appreciable resistance in their windings, but we shall assume that this inductor has negligible resistance. Thus, the circuit is a purely inductive ac circuit. Let the voltage across the source be v = v sinωt. Using the Kirchhoff’s loop rule, (cid:229) ε(t)= 0, and since there is no resistor in the circuit, di
v - L = 0 (7.10)
dt
where the second term is the self-induced Faraday emf in the inductor; and L is the self-inductance of the inductor. The negative sign follows from Lenz’s law (Chapter 6).

Detailed Explanation

This chunk introduces the basic setup of an inductive circuit with an AC voltage source. It describes how an inductor is typically connected to an AC source and highlights its characteristic of having negligible resistance. The voltage is expressed as a sinusoidal function, which is common in AC circuits. The Kirchhoff's loop rule is applied to describe how voltage and current interact in the circuit. The self-induced emf is mentioned as crucial for understanding how current behaves in an inductor.

Examples & Analogies

Think of an inductive circuit like a large water tank with a valve (the inductor), where you are trying to fill it with water (the AC voltage). When the water flows in, it creates pressure (self-induction) that affects how much water can come out the other end (the current), showing how even in an ideally designed system, the flow dynamics can be complex.

Current Equation in an Inductive Circuit

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Combining Eqs. (7.1) and (7.10), we have
di v
= = m sinωt (7.11)
dt L L
Equation (7.11) implies that the equation for i(t), the current as a function of time, must be such that its slope di/dt is a sinusoidally varying quantity, with the same phase as the source voltage and an amplitude given by v /L. To obtain the current, we integrate di/dt with respect to time:
∫ di
dt = vm∫ sin(ωt)dt
L
and get,
v
i = -m cosω(t + π/2) + constant
L
The integration constant has the dimension of current and is time-independent. Since the source has an emf which oscillates symmetrically about zero, the current it sustains also oscillates symmetrically about zero, so that no constant or time-independent component of the current exists. Therefore, the integration constant is zero.

Detailed Explanation

This chunk explains how to derive the current through an inductor when subjected to an AC voltage. The key equation shows how the current's rate of change (slope) relates to the voltage across the inductor. When calculating the current, we use integration to determine how the current varies over time. The cosine function describes the lagging nature of the current against the voltage, signifying that the current reaches its peak after the voltage does.

Examples & Analogies

Imagine riding a bike uphill (voltage) while the bike takes its time to gain speed (current). Just as you reach your maximum effort after the bike has started moving, the current in the inductor peaks after the voltage reaches its maximum, showcasing a delay in response to the input.

Inductive Reactance and Current Amplitude

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Using
ωL
i = vm (7.12)
L
where i = m is the amplitude of the current. The quantity ωL is analogous to the resistance and is called inductive reactance, denoted by XL:
X = ωL (7.13)
L
The amplitude of the current is, then
v
i = m (7.14)
m XL
The dimension of inductive reactance is the same as that of resistance and its SI unit is ohm (Ω). The inductive reactance limits the current in a purely inductive circuit in the same way as the resistance limits the current in a purely resistive circuit. The inductive reactance is directly proportional to the inductance and to the frequency of the current.

Detailed Explanation

This section introduces the concept of inductive reactance, which acts like resistance but specifically applies to AC circuits with inductors. The equations derived show how the amplitude of the current relates to the voltage and the inductive reactance. Key points are that inductive reactance increases with both the inductance and the frequency, influencing the current flowing in the circuit.

Examples & Analogies

Consider a water hose connected to a faucet (inductive circuit). When you turn it on (apply voltage), the wider the hose (higher inductance) and the stronger the water pressure (frequency), the more water (current) flows out. Inductive reactance controls how much water gets through, akin to a valve that restricts flow based on certain conditions.

Phase Relationship Between Voltage and Current

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A comparison of Eqs. (7.1) and (7.12) for the source voltage and the current in an inductor shows that the current lags the voltage by π/2 or one-quarter (1/4) cycle. Figure 7.6 (a) shows the voltage and the current phasors in the present case at instant t1. The current phasor I is π/2 behind the voltage phasor V. When rotated with frequency ω counter-clockwise, they generate the voltage and current given by Eqs. (7.1) and (7.12), respectively and as shown in Fig. 7.6(b).

Detailed Explanation

This chunk explains the phase difference between voltage and current in an inductor circuit. Specifically, it points out that the current does not peak at the same time as the voltage; it lags behind by a quarter of a cycle, indicated by π/2. This lag can be visualized using phasor diagrams, which illustrate the relationship between these alternating quantities effectively.

Examples & Analogies

Think of a concert band where the drummer (voltage) strikes the beat first, and the rest of the musicians (current) join in a moment later. The delay in their response mimics the phase lag between voltage and current in an inductive circuit, highlighting how timing impacts the collective performance.

Average Power Supplied to Inductor

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We see that the instantaneous power supplied to the inductor is
p = iv = i sin(ωt - π/2) × v sin(ωt)
m m
= -i v cos(ωt) sin(ωt)
= -ivm sin(2ωt)
2
So, the average power over a complete cycle is
P = < -ivm sin(2ωt) > = 0,
Since the average of sin(2ωt) over a complete cycle is zero. Thus, the average power supplied to an inductor over one complete cycle is zero.

Detailed Explanation

In this section, the discussion revolves around how power behaves in an inductive circuit. It explains that while power can be momentarily supplied to the inductor, over a complete cycle, the average power dissipated is zero. This outcome occurs because the inductor merely stores energy without converting it into usable power like a resistive element would.

Examples & Analogies

Imagine a rollercoaster ride (instantaneous power) where the thrill of the climb (energy storage) is followed by a drop back down to its original position (average power of zero). The exhilaration is temporary; hence, over the entire ride, you didn't really gain anything that lasts, similar to how inductors manage electrical energy.

Example of Reactance and RMS Current

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Example 7.2 A pure inductor of 25.0 mH is connected to a source of 220 V. Find the inductive reactance and rms current in the circuit if the frequency of the source is 50 Hz. Solution The inductive reactance,
XL = 2πnL=2·π·50·25·10⁻³ = 7.85Ω
The rms current in the circuit is
I = V/X = 220V/7.85Ω = 28A.

Detailed Explanation

This example illustrates the application of previously learned formulas to calculate the inductive reactance and the RMS current in an AC circuit with a given inductance and voltage. It effectively converts theoretical concepts into practical outcomes by applying numerical values.

Examples & Analogies

Imagine measuring out exact ingredients for a recipe (inductive reactance) to create the perfect dish (RMS current). Just as accurate quantities lead to a successful outcome, utilizing the correct calculations in an electrical circuit ensures efficient functioning.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Inductive Reactance: The property of an inductor opposing changes in current flow, given by X_L = ωL.

  • Phase Difference: The angle by which the current lags behind the voltage in an inductor, typically 90 degrees in AC circuits.

  • Power Dissipation: Average power consumed in a purely inductive circuit is zero over a complete cycle.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • When an AC voltage of 220V at 50Hz is applied to a 25mH inductor, the inductive reactance can be calculated as X_L = ωL, leading to a specific current amplitude based on the frequency.

  • In practical applications, inductors are used in filters within power supplies to manage alternating currents without dissipating energy.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • With inductance high and UI low, the current lags that we know.

📖 Fascinating Stories

  • Farmer Joe had a field filled with magnetic waves, and these waves influenced his cows on how the currents would behave, always lagging behind the charge arriving from the rave.

🧠 Other Memory Gems

  • Remember 'Lags in Inductors' to recall that 'Current Lags Voltage' in AC circuits with inductors.

🎯 Super Acronyms

Use 'PAC', for 'Power Average is Constant' to remember that inductors average zero power over a cycle.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Inductive Reactance

    Definition:

    The opposition that an inductor presents to the change in current, defined by the formula X_L = ωL.

  • Term: Phase Difference

    Definition:

    The difference in phase between the voltage and current in an AC circuit, which describes how much the current lags or leads the voltage.

  • Term: Average Power

    Definition:

    The mean power consumed over a complete cycle in an AC circuit, calculated as the integral of instantaneous power.