CHAPTER 13: INDUCTORS
Page 1: Introduction to Inductors in AC Circuits
Title: Inductors in AC Circuits
Overview of inductors in alternating current (AC) circuits.
Page 2: Learning Objectives
After completing this chapter, you will be able to:
Analyse an AC Inductive Circuit.
Calculate the power in an AC Inductive Circuit.
Page 3: Basic Characteristics of Inductors
An inductor:
Acts as a short circuit in direct current (DC) circuits, becoming "invisible."
Generates voltage in AC circuits.
Behavior in AC and DC:
Voltage across the inductor, V_L, varies with AC and is a function of time.
For AC, V_L is not nil, but in DC (V_D), it behaves differently.
Page 4: Role of an Inductor in Different Circuits
Diagram showing:
In a DC source, an inductor acts like a short circuit.
In an AC source, it behaves like a voltage source.
Page 5: Current and Voltage Relationship in Inductors
When sinusoidal AC voltage is applied:
A sinusoidal AC current is generated.
Voltage (V) and current (I) are not in phase.
V leads I by 90 degrees, meaning I lags V by 90 degrees.
Page 6: Phase Relationships in Resistors
In a purely resistive circuit:
Voltage (V) is in phase with current (I).
Both variables rise and fall together.
Page 7: Phase Relationships in Inductive Circuits
In a purely inductive circuit:
Voltage leads current by 90 degrees.
Current lags voltage by 90 degrees.
The phase difference is crucial for analyzing inductive behavior.
Page 8: Visualizing Phase Relationships
Diagram of the phase relationship between voltage (V_L) and current (I_L) in an inductor.
Consistent lags of 90 degrees in purely inductive circuits, causing voltage to lead current.
Page 9: Mathematical Representation
Considered equations:
If i(t) = I sin(ωt), then:
v(t) = L (di/dt) -> v(t) = LI cos(ωt), indicating leading current by 90 degrees.
Page 10: Polar Form of Circuit Current
If voltage is referenced with an angle of 0 degrees, circuit current in polar form becomes:
I = I_A ∟ -90 degrees.
Page 11: Reference Angles in Circuits
If circuit current is assigned an angle of 0 degrees:
In a purely inductive circuit:
Current (I) lags voltage (V) by 90 degrees.
Page 12: Inductive Reactance and Impedance
An inductor opposes AC currents, known as Inductive Reactance (X_L):
Formula: X_L = ωL = 2πfL.
Impedance (Z_L) relates to both inductance and frequency.
Page 13: Application of Ohm's Law in Inductive Circuits
In inductive circuits, Ohm’s law applies with replaced resistance (R) by inductive reactance (X_L).
Expressed in complex numbers due to phase angles.
Page 14: Example - Calculating Inductive Reactance
Given: 1 kHz signal applied to a coil, L = 5 mH.
Calculate X_L:
X_L = 2π(1000)(5x10^-3) = 31.4 Ω.
Page 15: Example - Calculating Current in an Inductive Circuit
Given: Vs = 5 V, f = 10 kHz, L = 100 mH.
Find current I using relevant formulas and substitutions.
Page 16: Related Problem - Current Calculation
Calculate current I with a different scenario:
Voltage Vs = 12 V, f = 4.9 kHz, and L = 680 µH.
Page 17: Types of Power in an AC Inductive Circuit
Three types of power are defined:
Instantaneous Power.
True Power.
Reactive Power.
Page 18: Instantaneous Power Analysis
Instantaneous power varies constantly:
P(t) = v(t) * i(t) (unit: Watts).
Page 19: Observations on Instantaneous Power
Power can be:
Positive (when supplied by source).
Negative (when returned to the source).
Page 20: True Power and Reactive Power
True Power (P):
Represents net consumption; zero in purely inductive circuits.
Reactive Power (Q):
Stored within the inductor; measured in volt-amperes reactive (VAR).
Page 21: Calculating Reactive Power
Reactive Power (Q) formulation:
Q = V_rms * I_rms * sin(φ).
Specifically for inductive circuits, sin(90°) simplifies calculations.
Page 22: Example - Calculating Reactive Power
For a 10V, 1kHz signal applied to a 10mH coil:
Q = 1.59 VAR
True power: 0 W.
Page 23: Effect of Frequency on Reactive Power
Reactive power decreases with increased frequency:
Relationship influences impedance.
Page 24: Summary of Key Concepts
In purely inductive AC circuits:
Voltage leads current by 90°.
True power consumed is zero (no energy loss).
Page 25: Conclusion
End of Chapter 13.
Copyright © 2005 Christopher Teoh, Tan HJ & Wong WY Singapore Polytechnic.