Inductors and RL Circuits Study Notes

Chapter 11: Inductors

Overview

• Topics Covered:

  • Inductors

  • Inductance

  • Inductors in DC

  • Inductors in AC

  • RL Circuits and Filters

  • Inductor Applications

  • Capacitor vs Inductor

Inductors

• Definition: When a length of wire is formed into a coil, it operates as a basic inductor.

• Formation of Magnetic Field:

  • When current flows in the inductor, it generates magnetic lines of force around each loop of the coil.

  • These lines combine with the adjoining loops, creating a strong electromagnetic field within and around the coil.

• Induced Voltage:

  • A change in current causes the magnetic field to change, inducing a voltage across the inductor that opposes the change in current.

• Types of Inductors:

  • Basic Inductor

  • Encapsulated Inductor

  • Toroidal Inductor

  • Variable Inductor

Inductance

• Definition: Inductance, denoted as L and measured in henries (H), describes the coil's ability to establish an induced voltage due to current changes, referred to as self-inductance.

• Henry Definition:

  • One henry (H) is the inductance of a coil when a current changing at a rate of one ampere per second induces one volt across the coil.

  • Typically, coils are much smaller than 1 H; millihenries (mH) and microhenries (μH) are commonly used units.

• Factors Affecting Inductance:

  • Increased number of turns in the coil.

  • Use of a magnetic material core (air, iron, ferrite).

• Formulas:

  • The inductance can be calculated using the formula:
    $L = rac{N^2 imes \boldsymbol{ ext{μ}} imes A}{l}$

  • Where:

  • L = Inductance in henries

  • N = Number of turns

  • μ = Permeability in H/m (Wb/At-m) of core material

  • A = Cross-sectional area of the coil in square meters

  • l = Length of the coil in meters

  • Energy Stored in an Inductor:
    $E = rac{1}{2}L I^2$

Example Problem

• Inductance Calculation:

  • Given:

  • Length l = 2 cm

  • Turns N = 150

  • Diameter = 0.5 cm (radius r = 0.25 cm = 0.0025 m)

  • Permeability of low carbon steel μ = $2.5 imes 10^{-4}$ H/m

  • Area Calculation:
    $A = \boldsymbol{ ext{π}} r^2 = \boldsymbol{ ext{π}} (0.0025)^2 = 7.85 imes 10^{-5} ext{ m}^2$

  • Inductance Calculation:
    $L = rac{(150)^2 imes (2.5 imes 10^{-4}) imes (7.85 imes 10^{-5})}{0.02} = 22 ext{ mH}$

  • Energy Calculation:

  • If current I = 1 A:
    $E = rac{1}{2} imes 22 imes 10^{-3} ext{ H} imes (1)^2$

  • Energy stored = 0.011 J

Series and Parallel Inductors

Series Inductors

• Total Inductance Formula:
  $L<em>T = L</em>1 + L<em>2 + … + L</em>n$

• Example Problem:

  • When a 1.5 H inductor is connected in series with a 680 mH inductor,
    $L_T = 1.5 H + 0.680 H = 2.18 H$

Parallel Inductors

• Total Inductance Formula:

  • The reciprocal of the total inductance equals the sum of the reciprocals of the individual inductances.
    $rac{1}{L<em>T} = rac{1}{L</em>1} + rac{1}{L<em>2} + … + rac{1}{L</em>n}$

• Example Problem:

  • For a 1.5 H inductor connected in parallel with a 680 mH inductor:
    $rac{1}{L_T} = rac{1}{1.5 H} + rac{1}{0.680 H} \ ext{Calculated Total Inductance} = 468 ext{ mH}$

Inductors in DC Circuits

Series RL Circuit

Current Behavior

• At 5τ (tau) time, all source voltage drops across the resistor (none across inductor).

• Current at 5τ is maximum, equal to $I = rac{V_s}{R}$.

• Instantaneously upon switch closure, inductor acts open, source voltage across it and initial current = 0.

• Current build-up is exponential while the induced coil voltage decreases.

• Voltage across the resistor increases with current increase.

RL Time Constant

• Definition: Determines the rate of current change, established by the ratio of inductance to resistance.

• Formula:
  $au = rac{L}{R}$

• Current Increase: - In one time constant (τ), current reaches approx. 63% of its full value after the switch is closed.

• Final Current: Reaching its full value takes approximately 5τ seconds.

• Behavior with DC: Inductor acts as a short circuit under constant DC conditions, causing no induced voltage.

Current Equations in RL Circuits

Exponential Relationships

• Rising Exponential: $i = I<em>F + (I</em>i - I_F)e^{- rac{Rt}{L}}$

  • Where:

  • $I_F$ = final current value

  • $I_i$ = initial current value

  • $i$ = instantaneous current value.

• Falling Exponential:

  • The final current decreases below the initial current during magnetic field collapse.

General Current Formula

• Example: For RL circuits experiencing increasing and decaying current:

  • Current values might represent percent of the final value across time constants (t).

• Graph: Plotting across time against percentage of the final current.

Inductors in AC Circuits

Inductive Reactance

• Definition: Inductive reactance is the opposition to sinusoidal current; it directly varies with frequency.

• Formula:
  $X_L = \boldsymbol{ ext{j}} imes \boldsymbol{ ext{ω}} L = 2 \boldsymbol{ ext{π}} f L$

• Phase Shift: There is a phase shift between voltage and current; voltage leads current by 90 degrees when a sine wave is applied to an inductor.

• Graphical Representation:

  • When drawing the reactance, it is important to factor in this phase shift for accurate predictions and actions.

Series Inductive Reactance Examples

• Example:

  • Reactance of a 33 mH inductor at 550 kHz:
    $X_L = 2 \boldsymbol{ ext{π}} imes (550 ext{ kHz}) imes (33 ext{ mH}) = 114 ext{ kΩ}$

• Further Problem:

  • If three 220μH inductors are in series with a 455 kHz ac source, calculate total reactance, yielding 1.89 kΩ.

Parallel Inductive Reactance

• Example: Reactance when the same individual inductors are placed in parallel.

  • Each inductor (at 455 kHz) has an individual reactance calculated.

  • Total reactance for three inductors in parallel leads to a calculated result of 210 Ω.

Impedance of Series RL Circuits

• Components:

  • Total impedance Z is from phasor sum of resistance R and inductive reactance X_L.

• Impedance Triangle:

  • Visual representation plotting R along the x-axis and X_L along the y-axis.

• Magnitude of Impedance:
  $Z = rac{R^2 + X_L^2}$

• Phase Angle:
  $heta = an^{-1} rac{X_L}{R}$

• Problem Example:

  • R = 1.2 kΩ; (X_L = 960Ω) leads to Z = 1.33 kΩ, and an angle of approx 39°.

Filters and RL Circuits

Phase Angle Variation

• Reactance phasors depict frequency-dependent behavior.

• Changes in frequency influence output voltage or gain in reactive filters.

• Frequency Response Equation:
  $Gain = rac{V<em>{out}}{V</em>{in}}$

Low Pass Filters

• Characteristics:

  • Passes frequencies ≤ $oldsymbol{f_c}$ and rejects all others.

  • Output V_out is taken across the resistor.

  • Formula for cutoff frequency:
    $f_c = rac{R}{2 \boldsymbol{ ext{π}} L}$

• Example Data:

  • At 1 kHz, output yields 8.46 V rms from a 10 V input under 10 mH and 100Ω resistance. Data illustration trends over selected frequencies.

High Pass Filters

• Characteristics:

  • Output across the inductor increases with input frequencies ≥ $oldsymbol{f_c}$ and rejects lower frequencies.

  • Basic lag network with output leading input voltage.

  • Similar formula for cutoff frequency.

• Example Data: At varying frequencies, voltage outputs adjust accordingly.

Applications of Inductors

• Traffic Sensors: Utilize the inductance change to detect vehicle presence above buried coils.

• Tuned Circuits and Filters: Essential in electronic communication systems to manage frequencies.

• RF Chokes: Suppress conducted and radiated noise in high-frequency applications.

Capacitor vs. Inductor

Similarities & Differences

• Both are energy storage devices with diverse functions across electrical circuits.

  • Capacitor: Comprises two conducting surfaces separated by a dielectric. Stores energy in an electric field, opposes voltage changes, and has current leading voltage by 90 degrees.

  • Inductor: A coiled conductor with magnetic properties. Stores energy in a magnetic field, opposes current changes, leads to voltage lagging by 90 degrees.

• Overall Function: Both provide reactive control in circuits, essential for filter designs, resonant arrangements, and support high-frequency signal management.

Faraday’s Law and Lenz’s Law

• Faraday’s Law: Voltage induced in a coil is proportional to the rate of change of magnetic field.

• Lenz’s Law: States that the direction of induced voltage acts against the change in current through a coil, emphasizing conservation of energy principles.

Reference

• Electronics Fundamentals: Circuits, Devices & Applications

  • 9th Edition, by Thomas L Floyd, David M. Buchla, Gary D. Snyder

  • Published by Pearson © 2022

  • Print ISBN: 9780135583739

  • eText ISBN: 9780135583845