Electrical Fundamentals II (CASA B-03b) 2024 PDF

MODULE 03 Category B1 and B2 Licences CASA B-03b Electrical Fundamentals II Copyright © 2020 Aviation Australia All rights reserved. No part of this document may be reproduced, transferred, sold or otherwise disposed of, without the written permission of Aviation Australia. CONTROLLED DOCUMENT 2024-...

MODULE 03 Category B1 and B2 Licences CASA B-03b Electrical Fundamentals II Copyright © 2020 Aviation Australia All rights reserved. No part of this document may be reproduced, transferred, sold or otherwise disposed of, without the written permission of Aviation Australia. CONTROLLED DOCUMENT 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 2 of 284 Knowledge Levels Category A, B1, B2 and C Aircraft Maintenance Licence Basic knowledge for categories A, B1 and B2 are indicated by the allocation of knowledge levels indicators (1, 2 or 3) against each applicable subject. Category C applicants must meet either the category B1 or the category B2 basic knowledge levels. The knowledge level indicators are defined as follows: LEVEL 1 Objectives: The applicant should be familiar with the basic elements of the subject. The applicant should be able to give a simple description of the whole subject, using common words and examples. The applicant should be able to use typical terms. LEVEL 2 A general knowledge of the theoretical and practical aspects of the subject. An ability to apply that knowledge. Objectives: The applicant should be able to understand the theoretical fundamentals of the subject. The applicant should be able to give a general description of the subject using, as appropriate, typical examples. The applicant should be able to use mathematical formulae in conjunction with physical laws describing the subject. The applicant should be able to read and understand sketches, drawings and schematics describing the subject. The applicant should be able to apply his knowledge in a practical manner using detailed procedures. LEVEL 3 A detailed knowledge of the theoretical and practical aspects of the subject. A capacity to combine and apply the separate elements of knowledge in a logical and comprehensive manner. Objectives: The applicant should know the theory of the subject and interrelationships with other subjects. The applicant should be able to give a detailed description of the subject using theoretical fundamentals and specific examples. The applicant should understand and be able to use mathematical formulae related to the subject. The applicant should be able to read, understand and prepare sketches, simple drawings and schematics describing the subject. The applicant should be able to apply his knowledge in a practical manner using manufacturer's instructions. The applicant should be able to interpret results from various sources and measurements and apply corrective action where appropriate. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 3 of 284 Table of Contents Inductance and Inductor I (3.11) 13 Learning Objectives 13 Faraday’s Law 14 Electromotive Force 14 Variations of Faraday’s Law 14 Lenz's Law 16 Induced Electromotive Force Polarity Electromotive Force 16 17 Electromagnetic Induction 17 Magnetic Field of Current (Electromagnetism) 17 Electromagnetic Induction 18 Self-Inductance 19 Back Electromotive Force Inductance and Inductor II (3.11) 20 23 Learning Objectives 23 Inductance 24 Induction Fundamentals 24 Inductor Types 27 Factors Affecting Coil Inductance Induction Factors 29 29 Mutual Inductance 34 Mutual Inductance Principles 34 Primary Current Affecting Induced Voltage 35 Coefficient of Coupling 37 Magnetic Saturation 37 Series and Parallel Inductors 41 Calculating Inductance of Series and Parallel Connected Inductors 41 Voltage/Current Relationship 41 LR Time Constant AC Theory (3.13) Learning Objectives 43 Sinusoidal Waveform 48 2024-02-15 47 47 Sine Wave Generation 48 Simple AC Generator 50 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 4 of 284 Sine Wave Terminology 53 AC Cycle and Alternation 53 AC Frequency 53 Sine Wave Period 54 Root, Mean, Square Voltage or Current 56 Average AC Voltage or Current 57 Multimeter Measurement Values 58 Phase 59 Sine Waves in Phase 61 Sine Waves Out of Phase 62 Three-Phase AC Power 63 Various AC Signal Types 65 AC Signals 65 Square Wave 65 Sawtooth Wave 66 Triangular Wave 66 LCR Circuits I (3.14) Learning Objectives 68 68 Alternating Current Circuits 69 AC and Resistive Circuits 69 Lead and Lag 70 AC and Inductive Circuits 71 AC Phase Relationships in Inductive, Capacitive, Resistive Circuits 73 Inductive Circuit Phase Angle 74 AC and Capacitive Circuits 75 AC Phase Angle in LCR Circuits 77 Inductive Reactance 78 Factors Affecting Inductive Reactance 79 Calculating Inductive Reactance 80 Capacitive Reactance 81 Calculating Capacitive Reactance 83 Impedance 2024-02-15 84 Definition of Impedance 84 Impedance in Series LCR Circuits 84 Determining Impedance of Series LCR Circuits 85 Determining Total Circuit Current of Series LCR Circuits 88 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 5 of 284 Determining Component Voltage Drops of Series LCR Circuits 88 Circuit Phase Relationships of Series LCR Circuits 91 Summary of Series LCR Circuits 93 Parallel LCR Circuits 94 Determining Reactance in Parallel LCR Circuits 95 Determining Branch Currents in Parallel LCR Circuits 96 Determining Total Circuit Current in Parallel LCR Circuits 97 Current Phase Relationships in Parallel LCR Circuits 98 Summary of Parallel LCR Circuit Analysis LCR Circuits II (3.14) Learning Objectives Power 102 102 103 True Power and Reactive Power 103 Power in Pure Resistive Circuits 103 Instantaneous Power 104 True Power 105 Power in a Series Parallel Resistor Circuit 106 Series Parallel Resistor Power Calculation Example 106 Series Parallel Resistance Circuit 106 Series Parallel Resistance Circuit - True Power 107 Series Parallel Resistance Circuit - Current 107 Series Parallel Resistance Circuit - Component Power 107 Series Parallel Resistance Circuit - Voltage 108 Series Parallel Resistance Circuit - Total Power 108 Series Parallel Resistance Circuit - Power Summary 109 Reactive Power in Inductive Circuits 111 Pure Inductive AC Circuits 111 Pure Inductive AC Circuits - Instantaneous Power 111 Pure Inductive Circuits - Reactive Power 113 Series Parallel Inductor Circuit - Reactive Power 113 Series Parallel Inductive Circuit - Total Inductive Reactance 114 Series Parallel Inductive Circuit - Circuit Current 115 Series Parallel Inductive Circuit - Total Reactive Power 115 Reactive Power in Capacitive Circuits 2024-02-15 100 117 Pure Capacitive Circuits - Reactive Power 117 Pure Capacitive Circuits - Instantaneous Power 117 Pure Capacitive Circuits - Power Consumption 118 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 6 of 284 Series Parallel Capacitor Circuit - Reactive Power 119 Series Parallel Capacitive Circuit - Capacitive Reactance 120 Series Parallel Capacitive Circuit - Total Current 121 Series Parallel Capacitive Circuit - Reactive Power 121 Series Parallel Capacitive Circuit - Total Reactive Power 122 Reactive Power Summary 122 Apparent Power 124 Apparent Power Description 124 Apparent Power - Inductive Resistive Circuits 124 Formula for Apparent Power 125 Apparent Power - Phasor Diagrams LCR Circuits III (3.14) Learning Objectives 126 Power Factor 2024-02-15 129 129 130 Power Factor Description 130 Apparent Power Triangle for AC Inductive Circuits 130 Example Calculations of TP, RP and AP in a Series Inductor Resistor Circuit 132 Determining the Power Factor in a Series LR Circuit 133 Determining the Phase Angle in a Series LR Circuit 133 Apparent Power in Parallel LR Circuits 134 Deriving the Apparent Power Formula in a Parallel LR Circuit 134 Apparent Power Triangle for a Parallel LR Circuit 135 Example Calculations of TP, RP and AP in a Parallel LR Circuit 136 Determining True Power in a Parallel LR Circuit 136 Determining Reactive Power in a Parallel LR Circuit 137 Determining the Power Triangle of a Parallel LR Circuit 137 Determining the Phase Angle of a Parallel LR Circuit 137 Determining Power Factor of a Parallel LR Circuit 138 Determining Apparent Power of a Parallel LR Circuit 138 Apparent Power in Series Capacitive Resistive Circuits 139 Instantaneous Power Curve in a Series CR Circuit 140 Apparent Power Formulas for a Series CR Circuit 141 Power Phasor Diagram of a Series CR Circuit 142 Power Factor in a Series CR Circuit 144 Power Triangle of a Series CR Circuit 144 Example Calculations of TP, RP and AP in a Series CR Circuit 145 Determining AP of a Series CR Circuit 146 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 7 of 284 Determining TP of a Series CR Circuit 146 Determining RP of a Series CR Circuit 147 Determining Capacitor Values in a Series CR Circuit 147 Example Calculations of TP, RP and AP in a Parallel CR Circuit 148 Apparent Power Formulas in a Parallel CR Circuit 149 Power Factor Formula in a Parallel CR Circuit 149 Power Triangle of a Parallel CR Circuit 149 Example Calculations of TP, RP and AP in a Parallel CR Circuit 150 Determining AP of a Parallel CR Circuit 150 Determining RPC of a Parallel CR Circuit 151 Determining TP of a Parallel CR Circuit 151 Determining Power Factor of a Parallel CR Circuit 152 Determining Phase Angle of a Parallel CR Circuit 152 Compare Power Triangles and Phasor Diagrams Inductive Circuit Power Triangles and Phasor Diagrams 154 Capacitive Circuit Power Triangles and Phasor Diagrams 154 Power in LCR Circuits 2024-02-15 154 156 Power in Series LCR Circuits 156 Power Triangle of a Series LCR Circuit 157 Determining Formulas for Power in LCR Circuits 158 Example Calculations of TP, RP and AP in a Series LCR Circuit 158 Determining Reactance in a Series LCR Circuit 159 Determining Impedance in a Series LCR Circuit 159 Determining Total Current in a Series LCR Circuit 160 Determining Reactive Power in a Series LCR Circuit 161 Determining True Power in a Series LCR Circuit 161 Determining Apparent Power in a Series LCR Circuit 162 Determining Power Factor of a Series LCR Circuit 162 Determining Phase Angle of a Series LCR Circuit 162 Power in Parallel LCR Circuits 163 Example Calculations of TP, RP and AP in a Parallel LCR Circuit 164 Determining TP in a Parallel LCR Circuit 165 Determining RPC in a Parallel LCR Circuit 165 Determining RPL in a Parallel LCR Circuit 165 Determining RP(Eq) in a Parallel LCR Circuit 166 Determining AP in a Parallel LCR Circuit 166 Determining PF in a Parallel LCR Circuit 166 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 8 of 284 Determining Phase Angle in a Parallel LCR Circuit 167 Summary of the Types of Power 167 Transformers I (3.15) Learning Objectives 170 170 Components of a Transformer Transformer Construction Principles 171 Core Characteristics 172 Transformer Types 174 Hollow-Core Transformers 174 Shell-Core Transformers 174 Transformer Windings 175 Transformer Symbols 177 Transformer Operation 178 Transformers and Alternating Current 178 No-Load Condition 178 Counter-EMF 181 Production of Counter-EMF 181 Primary and Secondary Phase Relationship 181 Coefficient of Coupling 184 Turns and Voltage Ratios 185 Effect of a Load on Transformer Operation 188 Mutual Flux 189 Turns and Current Ratios 190 Primary and Secondary Winding Power Transformers II (3.15) Learning Objectives Transformer Losses 192 195 195 196 Power Transformers 196 Copper Losses 196 Hysteresis Losses 196 Eddy Current Losses 197 Transformer Efficiency 198 Transformer Efficiency Calculations Impedance Matching 198 199 Maximum Power Transfer 199 Transformer Ratings 204 Auto-Transformers 2024-02-15 171 206 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 9 of 284 Auto-Transformer Configuration Filters (3.16) 206 208 Learning Objectives 208 Filter Characteristics 209 Filters 209 Typical Use of Filters on Aircraft – Vibration Sensor 211 Basic Filter Operation 214 Filter Operation 214 Practical Filters 215 RC Low-Pass Filter 217 RL Low-Pass Filter 217 Summary of Low-Pass Filters 218 High-Pass Filter Operation 219 Introduction to High-Pass Filter Operation 219 RC High-Pass Filter 219 RL High-Pass Filter 220 Summary of High-Pass Filters 220 Series and Parallel Resonance 224 Series Resonance 224 Circuit Operation 224 Parallel Resonance 225 Tuned Circuits 226 Resonant Circuits as Filters 227 Band-Pass Filter 229 Introduction to Band-Pass Filter 229 Simple Band-Pass Filter 229 Band-Stop Filter 232 Introduction to Band-Stop Filter Simple Band Stop Filters AC Generators (3.17) Learning Objectives 232 232 235 235 AC Generator Theory 236 The AC Generator 236 Alternator Types 237 Revolving-Armature Alternator 238 Rotating-Field Alternators 239 Permanent Magnet Generators 240 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 10 of 284 Brushless Alternators 241 Sinusoidal Sine Wave 243 Alternator Phase Types 246 Single-Phase Alternators 246 Two-Phase Alternators 246 Two-Phase Three-Wire Alternator 248 Three-Phase Alternators 249 Formula for Calculating Power in 3-Ø Circuits 251 Phase Sequence 252 Advantages and Applications of Three-Phase Connections 253 Alternator Frequency Control 254 AC Generator Aircraft Connections 256 Constant Speed Drive 256 Integrated Drive Generator 256 AC Motors (3.18) Learning Objectives 258 258 AC Motor Theory I 259 Advantages of AC Motors 259 Three-Phase Rotating Fields 259 Speed of AC Motors 262 AC Motor Speed Control 263 Direction of Rotation 264 Induction Motors 266 The Induction Motor 266 Squirrel-Cage Rotor 267 Slanted Rotor 269 Wound Rotor 269 Induction Motor Operation 270 Slip 272 Synchronous Motors 273 The Synchronous Electric Motor 273 Synchronous Motor Operation 273 AC Motor Theory II 2024-02-15 278 Star and Delta Windings 278 Single-Phase Induction Motors 278 Split-Phase Induction Motors 280 Capacitor-Start Split-Phase Motors 280 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 11 of 284 2024-02-15 Permanent-Split Capacitor Motors 281 Resistance-Start Motors 281 Shaded-Pole Induction Motors 282 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 12 of 284 Inductance and Inductor I (3.11) Learning Objectives 3.11.1 Describe Faraday's Law (Level 2). 3.11.2 Describe the action of inducing a voltage in a conductor moving in a magnetic field (Level 2). 3.11.3 Describe the principles of induction (Level 2). 3.11.4.1 Describe the effects of the following on the magnitude of an induced voltage: magnetic field strength (Level 2). 3.11.4.2 Describe the effects of the following on the magnitude of an induced voltage: rate of change of flux (Level 2). 3.11.4.3 Describe the effects of the following on the magnitude of an induced voltage: number of conductor turns (Level 2). 3.11.8 Describe Lenz's Law and polarity determining rules (Level 2). 3.11.9 Describe back electromotive force and self-induction (Level 2). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 13 of 284 Faraday’s Law Electromotive Force Michael Faraday found experimentally that the magnitude of the induced Electromotive Force (EMF) in a circuit is proportional to the rate at which the magnetic flux changed. If a circuit contains N tightly wound loops and the flux through each loop changes by ΔФ (Phi) during an interval Δt (time), the average EMF induced is given by Faraday’s law: ΔB E = −N Δt Ε = EMF - = the effect of Lenz’s law N = number of turns ΔB = rate of magnetic flux change Δt = time interval. Note: All rules in this section relate to electron flow unless otherwise stated. Faraday’s law practical example using permanent magnet movement and a coil of wire 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 14 of 284 Variations of Faraday’s Law The concept of Faraday's law is that any change in the magnetic environment of a coil of wire will cause a voltage (EMF) to be 'induced' in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet towards or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, and so on. The voltage generated when a coil is moved into a magnetic field is sometimes called motional EMF and is proportional to the speed with which the coil is moved into the magnetic field. That speed can be expressed in terms of the rate of change of the area which is in the magnetic field. © Aviation Australia Relative motion and direction of induced current 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 15 of 284 Lenz's Law Induced Electromotive Force Polarity When an EMF is generated by a change in magnetic flux according to Faraday's law, the polarity of the induced EMF is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the illustration examples, if the B (flux) field is increasing, the induced field acts in opposition to it. If B is decreasing, the induced field acts in the direction of the applied field to try to keep it constant. ∆B B ∆B B N N In V Induced emf B Out Induced emf Induced B V Induced Aviation Australia Lenz’s Law practical example relative motion and induced EMF 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 16 of 284 Electromotive Force Electromagnetic Induction An electromotive force is developed whenever there is relative motion between a magnetic field and a conductor. EMF is a difference of potential or voltage which exists between two points in an electrical circuit. In generators and inductors, the EMF is developed by the action between the magnetic field and the electrons in a conductor. Electromagnetic induction occurs with relative movement between a conductor and a magnetic field. The amount of induced EMF can be increased by: Increasing the magnetic field strength Increasing the rate of change of flux (or rate of movement of the conductor) Increasing the number of conductor turns that are cut by magnetic flux. Induced EMF is directly proportional to the length of the conductor in the magnetic field. Left hand rule for determining direction of current flow 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 17 of 284 Magnetic Field of Current (Electromagnetism) We know the magnetic field lines around a long wire which carries an electric current form concentric circles around the wire. Magnetic field strength is proportional to current. The direction of the magnetic field is perpendicular to the wire and is in the direction the fingers of your left hand would curl if you wrapped them around the wire with your thumb in the direction of the current (electron flow – from negative to positive). Aviation Australia Direction of the magnetic field in relation to current flow There is a simple rule for remembering the direction of the magnetic field around a conductor, called the left-hand rule. If a person grasps a conductor in their left hand with the thumb pointing in the direction of the current, their fingers will circle the conductor in the direction of the magnetic field. A word of caution about the left-hand rule: For the left-hand rule to work, it is essential to remember that for the direction of current flow, electron flow must be used. If conventional current flow is preferred, merely use the right hand. Everything is the same for the right hand as long as conventional current flow is always used. Left hand grasp rule for determining the direction of the magnetic field around a wire 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 18 of 284 Electromagnetic Induction If we switch the current to the coil on or off, the magnetic field will build up or collapse, creating movement between it and another conductor placed next to it. The coil which is producing the magnetic field and the coil into which the voltage is induced are both stationary. The movement of the magnetic field is produced by varying the strength of the current in the coil which is producing the field. When the switch is closed, an expanding magnetic field is produced around the current-carrying primary conductor. As this field expands, the lines of force cut across the adjacent secondary conductor and induce a voltage into this conductor. Therefore, in accordance with Lenz’s law, this voltage will be in a direction opposite the magnetic field produced by the current in the primary conductor. This induced voltage is indicated for only a fraction of a second because the field becomes stationary as soon as the maximum current is flowing through the primary conductor. Electromagnetic induction principles 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 19 of 284 Self-Inductance Even a perfectly straight length of conductor has some inductance. Current in a conductor produces a magnetic field surrounding the conductor. When the current changes, the magnetic field changes. © Jeppesen Magnetic field around a conductor This causes relative motion between the magnetic field and the conductor, and an EMF is induced in the conductor. This EMF is called a self-induced EMF because it is induced in the conductor carrying the current. The EMF produced by this moving magnetic field is also referred to as counterelectromotive force (CEMF). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 20 of 284 Back Electromotive Force The polarity of the counter-electromotive force is in the opposite direction of the applied voltage of the conductor. The overall effect is to oppose a change in current magnitude. CEMF may also be known as back EMF. This effect is summarised by Lenz's law, which states that the induced EMF in any circuit always moves in the direction opposite the effect that produced it. If the shape of the conductor is changed to form a loop, then the electromagnetic field around each portion of the conductor cuts across some other portion of the same conductor. Suppose a length of conductor is looped so that two portions of the conductor lie next to each other, Conductor 1 and Conductor 2. Laws for induced electromotive force When the switch is closed, current (electron flow) in the conductor produces a magnetic field around all portions of the conductor. The magnetic field (expanding lines of flux) is present in a single plane that is perpendicular to both conductors. Although the expanding field of flux originates at the same time in both conductors, it is considered as originating in Conductor 1 and its effect on Conductor 2 will be explained. With increasing current, the flux field expands outward from Conductor 1, cutting across a portion of Conductor 2. This results in an induced EMF in Conductor 2. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 21 of 284 Note: The induced EMF is in the opposite direction of (in opposition to) the battery current and voltage, as stated in Lenz's law. The direction of this induced voltage may be determined by applying the left-hand rule for generators. This rule states that if you point the thumb of your left hand in the direction of relative motion of the conductor and your index finger in the direction of the magnetic field, your middle finger, when extended, indicates the direction of the induced current which will generate the induced voltage (CEMF). After the switch has been opened, the flux field collapses. Applying the left-hand rule in this case shows that the reversal of flux movement causes a reversal in the direction of the induced voltage. Flemming’s left-hand rule The induced voltage then moves in the same direction as the battery voltage. The most important thing for you to note is that the self-induced voltage opposes both changes in current. That is, when the switch is closed, this voltage delays the initial build-up of current by opposing the battery voltage. When the switch is opened, it keeps the current flowing in the same direction by aiding the battery voltage. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 22 of 284 Inductance and Inductor II (3.11) Learning Objectives 3.11 Describe series and parallel inductors and the voltage/current relationship (S). 3.11.5 Describe mutual induction (Level 2). 3.11.6 Describe the effect the rate of change of primary current and mutual inductance has on induced voltage (Level 2). 3.11.7.1 Describe factors affecting mutual inductance: number of turns in coil (Level 2). 3.11.7.2 Describe factors affecting mutual inductance: physical size of coil (Level 2). 3.11.7.3 Describe factors affecting mutual inductance: permeability of coil (Level 2). 3.11.7.4 Describe factors affecting mutual inductance: position of coils with respect to each other (Level 2). 3.11.10 Describe saturation point (Level 2). 3.11.11 Describe the principal uses of inductors (Level 2). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 23 of 284 Inductance Induction Fundamentals Inductance is the characteristic of an electrical circuit that opposes the start, stop or change in value of a current. This statement is so important to the study of inductance that it bears repeating. Inductance is the characteristic of an electrical conductor that opposes change in current. The symbol for inductance is L and the basic unit of inductance is the henry (H). One henry is equal to the inductance required to induce one volt in an inductor by a change of current of one ampere per second. You do not have to look far to find a physical analogy of inductance. Anyone who has ever had to push a heavy load (wheelbarrow, car, etc.) is aware that it takes more work to start the load moving than it does to keep it moving. Once the load is moving, it is easier to keep the load moving than to stop it again. This is because the load possesses the property of inertia. Inertia is the characteristic of mass which opposes a change in velocity. Inductance has the same effect on current in an electrical circuit as inertia has on the movement of a mechanical object. It requires more energy to start or stop current than it does to keep it flowing. © Aviation Australia Induction symbols and terms 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 24 of 284 Inductors Inductance is the electrical property which opposes any change in the magnitude of current in a circuit. Devices which are used to provide the inductance in a circuit are called inductors. Inductors are also known as chokes, reactors and coils. These three names describe the way inductance behaves in a circuit. Inductance, and thus an inductor, ‘chokes off’ and restricts sudden changes in current. Inductors react against (resists) changes, either increases or decreases, in current. Inductors are usually coils of wire. Inductance is the result of a voltage being induced in a conductor. The magnetic field that induces the voltage in the conductor is produced by the conductor itself. Inductance Applications Radio antenna – Radio waves are electromagnetic. The oscillating magnetic field of the electromagnetic waves induce an EMF in the coil. Aviation Australia Antenna use inductance to detect very weak electromagnetic waves Induction stove – A changing flux through the bottom of the metal pot generates an EMF, which causes current to circulate around the bottom of the pot. Power (I2R), as heat, is dissipated in the metal pot, but not in the glass pot or the stovetop because they are insulators. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 25 of 284 Uses of inductance Toroid choke – A sudden surge in current is partially choked off or restricted by the CEMF induced when the magnetic flux through the loop suddenly changes. The flux change is multiplied by the presence of the soft iron cylinder surrounding the wire. Toroidal choke Inductance can be defined as the property of an electrical circuit which opposes any change in current in that circuit. If the current increases, the induced current tries to stop or delay that increase. If the current decreases, then the induced current delays the decrease by trying to maintain the current flow. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 26 of 284 Inductance in an AC circuit To increase the property of inductance, the conductor can be formed into a loop or coil. A coil is also called an inductor. The image below shows a conductor formed into a coil. Current through one loop produces a magnetic field that encircles the loop in one direction. As current increases, the magnetic field expands and cuts all the loops. The current in each loop affects all other loops. The field cutting the other loop has the effect of increasing the opposition to a current change. Aviation Australia Magnetic fields in a coil of wire 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 27 of 284 Inductor Types Inductors are classified according to core type. The core is the centre of the inductor, just as the core of an apple is the centre of an apple. The inductor is made by forming a coil of wire around a core. The core material is normally one of two basic types: soft iron or air. The schematic symbol for an iron-core conductor is represented with lines across the top to indicate the presence of an iron core. The air-core inductor may be nothing more than a coil of wire, but it is usually a coil formed around a hollow form of some nonmagnetic material, such as cardboard. This material serves no purpose other than to hold the shape of the coil. Different types of inductor cores - air and soft iron 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 28 of 284 Factors Affecting Coil Inductance Induction Factors Several physical factors affect the inductance of a coil. They include the number of turns in the coil, the diameter of the coil, the coil length, the type of material used in the core and the number of layers of winding in the coils. Inductance depends entirely on the physical construction of the circuit and can be measured only with special laboratory instruments. Number of Turns Of the factors mentioned, consider first how the number of turns affects the inductance of a coil. Suppose you have two coils. Coil A has two turns and Coil B has four turns. In Coil A, the flux field set up by one loop cuts one other loop. In Coil B, the flux field set up by one loop cuts three other loops. Aviation Australia Increasing the number of turns increases inductance Doubling the number of turns in the coil will produce a field twice as strong if the same current is used. A field twice as strong, cutting twice the number of turns, will induce four times the voltage. Therefore, it can be said that the inductance varies with the square of the number of turns. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 29 of 284 Diameter of Coil The second factor is the coil diameter. Physically, it requires more wire to construct a coil of large diameter than one of small diameter with an equal number of turns. Therefore, more lines of force exist to induce a CEMF in the coil with the larger diameter. Actually, the inductance of a coil increases directly as the cross-sectional area of the core increases. Recall the formula for the area of a circle: A = πr2. Doubling the radius of a coil increases the inductance by a factor of four. Aviation Australia Increasing the diameter of a coil increases it's inductance 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 30 of 284 Coil Length The third factor affecting the inductance of a coil is the length of the coil. Suppose Coil A has three turns, rather widely spaced, making a relatively long coil. A coil of this type has few flux linkages due to the greater distance between each turn. Therefore, Coil A has a relatively low inductance. Coil B has closely spaced turns, making a relatively short coil. This close spacing increases the flux linkage, increasing the inductance of the coil. Doubling the length of a coil while keeping the same number of turns halves the value of inductance. Aviation Australia Varying the coil length varies the inductance 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 31 of 284 Core Material The fourth physical factor is the type of core material used with the coil. For example, imagine two coils: Coil A with an air core and Coil B with a soft iron core. The magnetic core of Coil B is a better path for magnetic lines of force than is the nonmagnetic core of Coil A. The soft-iron magnetic core's high permeability has less reluctance to the magnetic flux, resulting in more magnetic lines of force. This increase in the magnetic lines of force increases the number of lines of force cutting each loop of the coil, thus increasing the inductance of the coil. It should now be apparent that the inductance of a coil increases directly as the permeability of the core material increases. Aviation Australia The permeability of the core material also determines the strength of the magnetic field 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 32 of 284 Number of Coil Layers Another way of increasing the inductance is to wind the coil in layers. We may imagine the construction of three cores with different amounts of layering. Coil A is a poor inductor compared to the others because its turns are widely spaced and there is no layering. The flux movement does not link effectively because there is only one layer of turns. Coil B is a more inductive coil with turns that are closely spaced and a wire wound in two layers. The two layers link each other with a greater number of flux loops during all flux movements. With nearly all the turns next to four other turns, the flux linkage is increased. Coil C is still more inductive because it is wound in three layers. The increased number of layers improves flux linkage even more. Some of its turns lie directly next to six other turns. In practice, layering can continue on through many more layers. The important fact to remember is that the inductance of the coil increases with each layer added. The more inductor coil layers the greater the inductance of the coil 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 33 of 284 Mutual Inductance Mutual Inductance Principles When a conductor’s magnetic flux induces voltage in another electrically isolated conductor, it is called mutual inductance. With mutual inductance, circuits that are electrically separated can be magnetically coupled together. A transformer uses the mutual inductance principle. Mutual inductance depends on: Number of turns in each coil Physical size of each coil Permeability of each coil Position of coils with respect to each other. Aviation Australia Mutual inductance example Whenever two coils are located so that the flux from one coil links with the turns of the other coil, a change of flux in one coil causes an EMF to be induced in the other coil. This allows the energy from one coil to be transferred or coupled to the other coil. The two coils are said to be coupled or linked by the property of mutual inductance (M). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 34 of 284 Aviation Australia Mutual induction The amount of mutual inductance depends on the relative positions of the two coils. If the coils are separated by a considerable distance, the amount of flux common to both coils is small and the mutual inductance is low. Conversely, if the coils are close together so that nearly all the flux of one coil links the turns of the other, the mutual inductance is high. Mutual inductance can be increased greatly by mounting the coils on a common iron core. Increasing the following elements will increase the induced EMF: Magnetic field strength Number of conductor turns Rate of change of flux (increasing frequency) Permeability of core(s). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 35 of 284 Primary Current Affecting Induced Voltage The theory of mutual induction leads on to transformer operating principles (3.15 Transformers) which state that if we supply power to a large (primary) wire wrapped around a ferromagnetic core and have a corresponding smaller (secondary) wire unpowered wrapped around the same core, the change in current in the primary will induce a voltage in the secondary. We have seen that a changing magnetic field from one coil can induce a voltage in the second coil. Faraday found the rate of decreasing magnetic flux affects the induced voltage, and by combining this fact with the knowledge that the magnetomotive force (MMF) is the number of turns multiplied by the current, we know that any increase in the MMF will increase the voltage generated in the second conductor. To increase the MMF, we can increase the number of turns. However, doing that would also increase the CEMF, reducing the current through the coil to an even lower level than it was at the start. This reduces the MMF to a lower level, reducing the induced voltage in the second coil. To increase the current flow in the primary wire, we would have to apply a larger voltage to overcome the CEMF. Thus current rises and CEMF rises, so we must increase the voltage to overcome the new CEMF. In other words, to double the voltage in the primary wire, the applied voltage may have to triple. Each rise in primary current increases the voltage in the second conductor. A transformer is connected to an AC power source, but the secondary conductor is not connected to a circuit, so no current is flowing. AC in PRIMARY AC out SECONDARY Aviation Australia A basic transformer showing primary and secondary windings The situation changes once a load is added. Now current flows in the secondary circuit and, just as current is changing in the primary circuit and inducing a voltage in the secondary circuit, the current in the secondary circuit now has its own magnetic field. However, it opposes the primary magnetic field, weakening it. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 36 of 284 The reduced total flux in the primary circuit means a lower CEMF in it, therefore the current and flux increase until the CEMF and EMF are once again in balance with the voltage and current in the secondary circuit. Coefficient of Coupling The coefficient of coupling between two coils equals the ratio of the flux cutting one coil to the flux originated in the other coil. If the two coils are positioned with respect to each other so that all of the flux of one coil cuts all of the turns of the other, the coils are said to have a unity coefficient of coupling. It is never exactly equal to unity, but it approaches this value in certain types of coupling devices. If all of the flux produced by one coil cuts only half the turns of the other coil, the coefficient of coupling is 0.5. The coefficient of coupling is designated by the letter K. M = K √L 1 × L 2 Where: M = mutual inductance in henries K = coefficient of coupling L1 and L2 = inductance of the coils in henries. The mutual inductance between two coils L1 and L2 is expressed in terms of the inductance of each coil and the coefficient of coupling K. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 37 of 284 Magnetic Saturation When a material is magnetically saturated, no additional amount of external magnetisation force will cause an increase in its internal level of magnetisation. Therefore, after saturation, no increase in the amount of current through the primary circuit will increase the amount of mutual induction that is induced EMF in the secondary circuit. Hysteresis Loop The hysteresis loop is generated by measuring the magnetic flux of a ferromagnetic material while the magnetising force is changed. For a ferromagnetic material that has never been previously magnetised or has been thoroughly demagnetised, the greater the amount of current applied, the stronger the magnetic field in the component. When almost all of the magnetic domains are aligned, an additional increase in the magnetising force will produce very little increase in magnetic flux. The material has reached the point of magnetic saturation. Aviation Australia Coil with a metal core connected to a battery and variable resistor When the current is reduced to zero, some magnetic flux remains in the material even though the magnetising force is zero. This is referred to as the point of retentivity and indicates the remanence or level of residual magnetism in the material (some of the magnetic domains remain aligned but some have lost their alignment). As the magnetising force is reversed, the flux is reduced to zero. This is called the point of coercivity (the reversed magnetising force has flipped enough of the domains so that the net flux within the material is zero). The force required to remove the residual magnetism from the material is called the coercive force or coercivity of the material. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 38 of 284 As the magnetising force is increased in the negative direction, the material again becomes magnetically saturated, but in the opposite direction. When the current is reduced to zero, the material has a level of residual magnetism equal to that achieved in the other direction. Increasing the current back in the positive direction will return the magnetic field strength to zero. The current, however, does not return to zero here because some force is required to remove the residual magnetism. The curve will take a different path back to the saturation point, where it completes the loop. From the hysteresis loop, a number of primary magnetic properties of a material can be determined. Aviation Australia Hysteresis loop terminology 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 39 of 284 1. Retentivity – a measure of the residual flux density corresponding to the saturation induction of a magnetic material. In other words, it is a material's ability to retain a certain amount of residual magnetic field when the magnetising force is removed after achieving saturation. (The value of “B” at point b on the hysteresis curve.) 2. Residual magnetism or residual flux – the magnetic flux density that remains in a material when the magnetising force is zero. Note that residual magnetism and retentivity are the same when the material has been magnetised to the saturation point. However, the level of residual magnetism may be lower than the retentivity value when the magnetising force did not reach the saturation level. 3. Coercive force – the amount of reverse magnetic field which must be applied to a magnetic material to make the magnetic flux return to zero. (The value of “H” at point f on the hysteresis curve.) 4. Permeability (m) – a property of a material that describes the ease with which a magnetic flux is established in the component. 5. Reluctance – the opposition that a ferromagnetic material shows to the establishment of a magnetic field. Reluctance is analogous to the resistance in an electrical circuit. Aviation Australia Hysteresis loop explanation 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 40 of 284 Series and Parallel Inductors Calculating Inductance of Series and Parallel Connected Inductors Inductors can be arranged in series or in parallel to achieve the desired results. Induction formulas are (basically) the same as those for calculating resistance. Aviation Australia Series and parallel inductor inductance calculations 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 41 of 284 Voltage/Current Relationship What effects does an inductor have on a DC power source? Suppose we have an inductor and a resistor connected in series to a DC power source by a switch. Aviation Australia Inductor resistor (LR) circuit charging The very instant that switch is closed, there will be no current flow in our circuit. This is because an inductor will oppose any change in current, and (just for an initial instant) the rate of change of current is maximum and the inductor has a voltage (Back EMF) induced equal and opposite to the supply voltage. However the inductor cannot maintain this voltage therefore as the Back EMF reduces the current will increase. Kirchhoff's voltage law will still be true as at any instance in time the VR plus VL will still add up to the supply voltage. The induced voltage across the inductor will decrease until, after five time constants, the induced voltage across the inductor will be near zero and the current flowing in the circuit will depend on the size of the resistor and the resistance of the circuit wiring. If we toggle the switch to disconnect the inductor from the power source, the inductor will again try to oppose this change by inducing a voltage across the inductor the magnitude of the voltage will be many times larges than the supply voltage due to the speed of collapse of the magnetic field. The polarity will be the same as the original power source keeping the current flowing through the inductor in the same direction as the supply. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 42 of 284 Aviation Australia Inductor resistor (LR) circuit discharging This causes the inductor to act as a power source, supplying a current which flows through the resistor in the same direction the normal circuit current was flowing. After five time constants, however, the energy stored in the inductor is dissipated by the resistor as heat. As the inductor has no energy remaining, the current falls to zero. The time constant depends on the amount of inductance (L) or the amount of resistance (R) in the circuit. That is, if the circuit resistance is high, the time elapsed for one time constant will be short, while if the inductance is high and the resistance is low, the time constant will be long. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 43 of 284 LR Time Constant The LR Time Constant (TC) is a valuable tool for determining the time required for current in an inductor to reach a specific value. L T ime Constant (T C) = R As illustrated by the LR Time Constants (TC) graph, one L (henry)/R (ohms) time constant is the time required for the current in an inductor to increase to 63% (actually 63.2%) of the maximum current. Each TC is equal to the time required for the current to increase by 63.2% of the difference in value between the current flowing in the inductor and the maximum current. Maximum current flows in the inductor after five L/R time constants are completed. The following example should clear up any confusion about TCs. Assume that maximum current in an LR circuit is 10 A. As you know, when the circuit is energised, it takes time for the current to go from 0 to 10 A. When the first TC is completed, the current in the circuit is equal to 63.2% of 10 A. Thus the amplitude of current at the end of 1 TC is 6.32 A. Aviation Australia LR time constant graph During the second TC, current again increases by 63.2% (0.632) of the difference in value between the current flowing in the inductor and the maximum current. This difference is 10 A minus 6.32 A and equals 3.68 A; further, 63.2% of 3.68 A is 2.33 A. This increase in current during the second TC is added to that of the first. Thus, upon completion of the second TC, the amount of current in the LR circuit is 6.32 A + 2.33 A = 8.65 A. To really see the reason most inductors are used the inductor has to be connected to an alternating current (AC) supply. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 44 of 284 Aviation Australia Diagram showing phase difference between I and V in an inductive circuit In the graphs next to the inductive circuit note that the largest rate of change of supply voltage gives the largest rate of change of current. This can be found by drawing the tangent lines on the Sine wave. Due to the operation of the inductor largest rate of change of current produces the largest Back EMF (Counter EMF). As can be seen from the rate of change of voltage that the back EMF is at maximum when the rate of change of current is maximum and the EMF is at minimum when the rate of change of current is minimum. In a purely inductive circuit, current lags voltage by 90°. Aviation Australia For inductors - current lags voltage In an inductor, the voltage leads the current or the current lags the voltage, as a way of remembering it we use the cord CIVIL. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 45 of 284 For Capacitors (C) current (I) leads voltage (V) and voltage leads current (I) in an Inductive (L) circuit. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 46 of 284 AC Theory (3.13) Learning Objectives 3.13.1.1 Describe phase, period and frequency characteristics of a sinusoidal waveform (Level 2). 3.13.1.2 Describe cycle characteristics of a sinusoidal waveform (Level 2). 3.13.2.1 Describe the term 'instantaneous' and perform calculations with regards to instantaneous voltage, current and/or power (Level 2). 3.13.2.2 Describe the term 'average AC' and perform calculations with regards to AC voltage and current (Level 2). 3.13.2.3 Describe the term 'RMS or root mean square' and perform calculations with regards to instantaneous voltage, current and/or power (Level 2). 3.13.2.4 Describe the term 'peak' and perform calculations with regards to instantaneous voltage, current and/or power (Level 2). 3.13.2.5 Describe the term 'peak to peak' and perform calculations with regards to instantaneous voltage, current and/or power (Level 2). 3.13.3.1 Describe characteristics of triangular AC signal waveform (Level 2). 3.13.3.2 Describe characteristics of square AC signal waveform (Level 2). 3.13.4.1 Explain the principles governing the generation of single phase AC power (Level 2). 3.13.4.2 Explain the principles governing the generation of three phase AC power (Level 2). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 47 of 284 Sinusoidal Waveform Sine Wave Generation Alternating current is electric current whose flow alternates in direction; the duration of flow in one direction is a half-period, and the duration of all half-periods is the same. Alternating current is abbreviated to AC. An AC waveform may be sinusoidal or non-sinusoidal. It can be generated by a mechanical rotating generator (alternator) or by an electronic inverter or oscillator. The output of an alternator is sinusoidal. Electronic circuits can produce sinusoidal or non-sinusoidal waveforms. The simplest and cheapest type of electronic oscillator produces a (non-sinusoidal) square wave. A sine wave is a symmetrical waveform that varies equally around a fixed level and can be a representation of either voltage or current. The sine wave is an alternating (swings both positive and negative) waveform and is the most commonly encountered AC waveform. A sinusoidal waveform bears a direct relationship to the rotary motion of an alternator. An alternating quantity is described as sinusoidal when its trace, plotted against a linear time base is a sine wave. The time base may be expressed in units of time or in degrees of rotation. In practical terms, this means the amplitude of the voltage or current at any instant is proportional to the sine value that corresponds to the angle of rotation at that instant, i.e. 0.500 × maximum amplitude at 30° of rotation, 0.707 × max. amplitude at 45° of rotation, 0.866 × max. amplitude at 60°, 1.000 × max. amplitude at 90°, etc. Sinusoidal waveform With AC, electrons flow first in one direction, then in the other. Both current and voltage vary continuously. The graphic representation for AC is a sine wave, which can represent current or voltage. Two axes are used to depict a sine wave. The vertical axis represents the magnitude and direction of current or voltage. The horizontal axis represents time or angle of rotation in degrees. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 48 of 284 When the waveform is above the time axis, current is said to be flowing in a positive direction. When the waveform is below the time axis, current is said to be flowing in a negative direction. A complete cycle occurs in 360°, half of which is positive and half negative. The sinusoidal sine wave shows the value of induced EMF at each instant of time during a 360° rotation of the loop. A sinusoidal sine wave is a representation of induced EMF for a single coil rotated through a uniform magnetic field at a constant speed. AC, unlike direct current (DC), flows first in one direction and then in the opposite direction. Current amperage is the function of time. DC amperage is constant. The most common AC waveform is a sine (or sinusoidal) waveform. When a conductor is cutting lines of flux quickly, it produces a greater force to drive electrons and hence a greater potential difference. This is represented by the sine wave peaking when the wires are moving directly across (perpendicular to) the face of the magnetic field. Each cycle of the sine wave consists of two identically shaped variations in voltage. The variations that occur during the time considered the positive alternation indicate current movement in one direction. The direction of current movement is determined by the generated terminal voltage polarities. The variations that occur during the time considered the negative alternation indicate current movement in the opposite direction because the generated voltage terminal polarities have reversed. The distance from zero to the maximum value of each alternation is the amplitude. The amplitude of the positive alternation and the amplitude of the negative alternation are the same. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 49 of 284 360 degree cycle of a sine wave 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 50 of 284 Simple AC Generator The sinusoidal sine wave shows the value of induced EMF at each instant of time during 360° rotation of the loop. A sinusoidal sine wave is a representation of induced EMF for a single coil rotated through a uniform magnetic field at a constant speed. AC, unlike DC, flows first in one direction and then in the opposite direction. Current amperage is the function of time. DC amperage is constant. The most common AC waveform is a sine (or sinusoidal) waveform. When a conductor is cutting lines of flux quickly, it produces a greater force to drive electrons and hence a greater potential difference. This is represented by the sine wave peaking when the wires are moving directly across (perpendicular to) the face of the magnetic field. As the armature rotates through the magnetic field, at the initial position of 0°, the armature conductors are moving parallel to the magnetic field. They are not cutting through any magnetic lines of flux. Therefore, no voltage is induced. As the armature rotates from 0° to 90°, the conductors cut through more and more lines of flux. Induced voltage builds to a maximum in the positive direction. Sinusoidal wave produced as armature is rotated through 360 degrees 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 51 of 284 As the generator continues to rotate from 90° to 180°, the armature cuts fewer and fewer lines of flux. The induced voltage decreases from a maximum positive value to zero. The armature continues to rotate from 180° to 270°. The conductors cut more and more lines of flux, but in the opposite direction. The voltage is induced in the negative direction, building up to a maximum at 270°. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 52 of 284 Sine Wave Terminology AC Cycle and Alternation A cycle is one complete sequence of voltage or current change from zero through a positive peak, back to zero, through a negative peak and back to zero again. An alternation is one half of an AC cycle, in which the voltage or current rises or falls from zero to a peak and back to zero again. One cycle of a sine wave 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 53 of 284 AC Frequency The frequency of AC is the number of cycles completed in one second. Frequency is expressed in hertz (Hz), with one hertz equal to one cycle per second. If the loop makes one complete revolution each second, the generator produces one complete cycle of AC during each second (1 Hz). Increasing the number of revolutions to two per second produces two cycles of AC per second (2 Hz). The number of complete cycles of AC or voltage completed each second is the frequency. Frequency is always measured and expressed in hertz. Frequency = number of cycles per second 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 54 of 284 Sine Wave Period The time required to complete one cycle of a waveform is the period. An individual cycle of any sine wave represents a definite amount of time. If two cycles occur each second, one cycle must require one half second of time. The period for this example is 0.5 s. We can easily calculate period from frequency. If the frequency is 50 Hz, the period is 1 s ÷ 50 = 0.02 s. If the frequency is 400 Hz, the period is 1 s ÷ 400 = 0.0025 s. So, the faster the coil is spun, the higher the frequency. Because the loop is spinning faster, the sine waves are produced more quickly and the period time is reduced as rpm increases. The period of a sine wave is expressed in time, usually as a fraction (or decimal) of a second. Sine wave period = time taken for one cycle An increase in the number of poles causes a corresponding increase in the number of cycles completed in a revolution. A two-pole generator completes one cycle per revolution and a four-pole generator completes two cycles per revolution. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 55 of 284 Aviation Australia Two cycles for one rotation Peak Voltage or Current The peak value of a sine wave occurs twice each cycle: once at the positive maximum value and once at the negative maximum value. Peak-to-Peak Voltage or Current The magnitude of the voltage or current between the peak positive and peak negative values is called the peak-to-peak value. Instantaneous Voltage or Current The instantaneous value of voltage or current is the value at any time on the sine wave. It can be anywhere from zero to peak value. This example illustrates instantaneous values at 90o, 150o, and 240o. The peak voltage in this example is 100 volts. The instantaneous value of the sine wave is found by the following formula: Ø V I N ST = V P K × sin Ø is the angle of rotation of the cycle. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 56 of 284 Root, Mean, Square Voltage or Current Most books define Root Mean Square (RMS) as the 'amount of AC power that produces the same heating effect as an equivalent DC power'. The RMS value is more than just that. It is the square root of the mean, or average, value of the squared function of the instantaneous values. The symbols used for defining an RMS value are VRMS or IRMS. This is known as the RMS value. For example, if the voltage is said to be 120 V, this is the RMS value. For a sine wave, the RMS value is 0.707 times the peak value. Voltmeters measure in average but indicate AC as RMS. RMS voltage or current = 0.707 of peak value 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 57 of 284 Average AC Voltage or Current The average value of an alternating current or voltage is the average of all the instantaneous values during one alternation. Since the voltage increases from zero to peak value and decreases back to zero during one alternation, the average value must be some value between those two limits. The average value can be determined by adding together a series of instantaneous values of the alternation (between 0° and 180°) and then dividing the sum by the number of instantaneous values used. The computation shows that one alternation of a sine wave has an average value equal to 0.637 times the peak value. Do not confuse the above definition of an average value with that of the average value of a complete cycle. Because the voltage is positive during one alternation and negative during the other alternation, the average value of the voltage values occurring during the complete cycle is 0. Aviation Australia AC sine wave peak to peak labelled 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 58 of 284 Multimeter Measurement Values Most multimeters measure in average but display in RMS as RMS is the accepted and most-used value of a sine wave. There are some True Value RMS meters available, but they are quite expensive. denklim/stock.adobe.com used with permission Multimeters display in RMS values 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 59 of 284 Phase Phase is a frequently used term with AC. It is applied to the periodic changes of some quantities, such as the voltage or current in an AC circuit. Electrical phase is measured in degrees, with 360° corresponding to a complete cycle. A sinusoidal voltage is proportional to the cosine or sine of the phase. When the voltage and current sine waves cross the zero line at the same time, they are said to be ‘in phase’. This is also true if there is more than one AC power source and they are perfectly paralleled together (for example, if they are in phase). In-phase sine waves Out of phase sine waves 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 60 of 284 Aviation Australia Sine waves that are 180 degrees out of phase 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 61 of 284 Sine Waves in Phase When a voltage sine wave is applied to a resistance in a circuit, the resulting current is also a sine wave. This follows Ohm's law: current is directly proportional to the applied voltage. Notice in the illustration that the sine wave of voltage and the resulting sine wave of current are superimposed on the same time axis. Notice also that as voltage increases in a positive direction, current increases along with it, and when voltage reverses direction, current also reverses direction. When two sine waves are precisely in step with one another, they are said to be in phase. To be in phase, the two sine waves must go through their maximum and minimum points at the same time and in the same direction. In some circuits, several sine waves can be in phase with each other. Thus, it is possible to have two or more voltage drops in phase with each other and also in phase with circuit current. Aviation Australia Sine waves in-phase 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 62 of 284 Sine Waves Out of Phase The illustration shows voltage wave for Phase A which is considered to start at 0° position. As Phase A voltage reaches its positive peak at 90° rotation, Phase B voltage transitions through 0V toward the maximum positive peak. Since these voltage waves do not go through their maximum and minimum points at the same instant of time, a phase difference exists between them. The two waves are said to be 'out of phase'. Sine waves out of phase 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 63 of 284 Three-Phase AC Power A three-phase (3Ø) generator has three conductors and therefore three phases of AC power (120° apart). Aviation Australia Three phase AC generator 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 64 of 284 Various AC Signal Types AC Signals AC voltage levels change with time and alternate between positive values (above the t [time]-axis) and negative values (below the t-axis). Signals with repeated shapes are called waveforms and include sine waves, square waves, triangular waves and sawtooth waves. A distinguishing feature of alternating waves is that equal areas are enclosed above and below the t-axis. Different waveforms (a) sine, (b) square, (c) triangular, and (d) sawtooth There are many different ways of generating a variety of wave shapes, the following examples are a small sample of those available. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 65 of 284 Square Wave The easiest and simplest square wave generator is a switch: on and off. For low-speed (frequency) operations this is adequate. Current computer processors use a PC clock, which is just a very fast square wave generator. Current CPU clocks can operate at over 4.0 GHz, i.e. 4,000,000,000 clock pulses per second. This speed is not possible using a simple switch. Square wave Sawtooth Wave A simple sawtooth wave can be generated using a DC circuit that measures the voltage at a capacitor that is is gradually charged and rapidly discharged once the peak voltage is reached. An identifying feature of a sawtooth wave is the different time taken for the rise or increase in values in comparison to the fall or decrease of values. Aviation Australia Sawtooth wave shape 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 66 of 284 Triangular Wave The difference between a triangular and a sawtooth wave is the timing associated with the rise and fall of the wave shape. While a sawtooth wave has unequal rise and fall times the triangular wave has equal rise and fall time periods. There are many methods of generating triangular waves, but a square wave which has been processed by an integrator produces a triangular wave. Integration is a mathematical function in which one function is affected by another. For example, if acceleration is integrated over time, it gives velocity. In mathematics it is achieved through the use of calculus, but in electronics it is achieved by charging and discharging capacitors. Triangular wave generator 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 67 of 284 LCR Circuits I (3.14) Learning Objectives 3.14.1.1 Describe the phase relationship of voltage and current in L, C, R circuits (Level 2). 3.14.1.2 Describe the phase relationship of voltage and current in L, C and R parallel circuits (Level 2). 3.14.1.3 Describe the phase relationship of voltage and current in L, C and R series circuits (Level 2). 3.14.1.4 Describe the phase relationship of voltage and current in L, C and R series parallel circuits (Level 2). 3.14.3.1 Describe impedance and methods of calculating impedance (Level 2). 3.14.3.2 Describe phase angle and methods of calculating phase angle (Level 2). 3.14.3.4 Describe methods of calculating current (Level 2). 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 68 of 284 Alternating Current Circuits AC and Resistive Circuits If an Alternating Current (AC) sinusoidal voltage is applied across a resistor, a sine wave current will flow. Aviation Australia AC voltage and current in a resistive circuit When the sinusoidal instantaneous AC voltage is zero, the current flow through the circuit will also be zero, and when the instantaneous AC voltage is at maximum, the current flow will be at maximum. The same is true for polarity, that is, when the voltage polarity changes direction, the current flow’s direction will reverse. As the instantaneous values of AC current and voltage conform to Ohm's law the Peak and RMS values can also be used. We can represent the magnitude and phase angle of the sine waves by constructing a phasor diagram. © Aviation Australia Phasor diagram of AC current and voltage phases in a resistive circuit 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 69 of 284 Although the voltage and current phasors are of different magnitude, they will have a phase angle relationship of 0°. As a result, current and voltage in a purely resistive AC circuit are in phase with each other. Summary Current and voltage in a purely resistive AC circuit are in phase with each other. Ohm’s and Kirchhoff's laws are valid for purely resistive AC circuits. When calculating circuit values in a resistive AC circuit, ensure the circuit parameters are expressed in the same terms before you start your calculations. To understand the operation of inductive and capacitive AC circuits, you must first understand the various rates of change a sine wave represents. The rate at which the sinusoidal waveform is changing varies along the sine wave’s curve; this can be seen as the 'Gradient' of the curve. At the point where the sine wave crosses the horizontal axis (zero crossing), the curve is changing at a faster rate than anywhere else on the curve (use a Tangent Line along the curve and you will be able to easily see the varying gradients). Aviation Australia Line showing the point of maximum rate of change in a sinewave 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 70 of 284 Lead and Lag The terms 'phase shift', 'lead' and 'lag' are commonly used throughout the topic of LCR circuits. Aviation Australia Phase shift - lead and lag AC and Inductive Circuits Before addressing inductance in AC circuits, we will revisit inductance in DC circuits. The inductor (L), resistor (R) and DC voltage source are connected in series with a switch (S1). Aviation Australia DC current in an inductive resistive (LR) circuit - charging 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 71 of 284 The very instant that switch (S1) is closed (A-B), there will be no current flow in our circuit. This is because an inductor will oppose any change in current, and (just for an initial instant) the inductor is completely successful in doing this. This means at the instant the switch is closed, the rapid rate of change of current in the circuit is sufficient to induce a voltage across the inductor that is equal in magnitude but opposite in polarity to the applied voltage. The induced voltage across the inductor will decrease until, after five time constants, the induced voltage across the inductor will be near zero and the current flowing in the circuit will be maximum. When the switch is changed to position B-C the inductor is disconnected from the power source, the inductor will again try to oppose this change by inducing a voltage across the inductor of the same polarity as the original power source. Aviation Australia DC current in an inductive resistive (LR) circuit - discharging This causes the inductor to act as a power source, supplying a current which flows through the resistor via contacts 'B-C' of the switch in the same direction the normal circuit current was flowing. After five time constants, however, the energy stored in the inductor is dissipated by the resistor as heat. As the inductor has no energy remaining, the current falls to zero. To show the phase relationship between voltage and current, we must examine the effects of an inductor on an AC circuit. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 72 of 284 Aviation Australia Diagram showing phase difference between current (I) and voltage (V) 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 73 of 284 AC Phase Relationships in Inductive, Capacitive, Resistive Circuits We know the amount of voltage induced across an inductor (coil) is directly proportional to the rate at which the current is changing (Faraday’s law, DC theory). The coil voltage will be at maximum (peak) when the current sine wave has its greatest rate of change. That point occurs where it crosses the zero reference line, and when the current is changing at its minimum rate (zero at the peak), the voltage will be at its minimum value (zero). By comparing two sine waves, you can see that the current peak occurs one quarter cycle after the voltage peak. Therefore it can be said that in a purely inductive AC circuit, the current will lag the voltage by 90°, or we can say that the voltage leads the current by 90°. In an LR, circuit the relationship between the current and voltage is altered. The acronym CIVIL is a helpful mnemonic device. In a purely inductive circuit, current lags voltage by 90°. When using 'CIVIL', V (Voltage) is the midpoint. C represents capacitive loads or circuits and L represents inductive loads or circuits. For capacitive loads, I (current) comes before V, that is, current leads voltage. For inductive loads, I comes after V, that is, current lags voltage. Aviation Australia CIVIL is a very helpful mnemonic for determining current and voltage phase relationships 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 74 of 284 Inductive Circuit Phase Angle The phasor representing the current (I) is drawn 90° clockwise from the phasor representing the voltage (V). Aviation Australia Inductive circuit phase relationship with the current lagging the voltage by 90° AC and Capacitive Circuits Before addressing capacitance in AC circuits, we will review capacitance in DC circuits. A capacitor (C), resistor (R) and a DC voltage source are connected in series with a switch. The very instant that the switch (S1) is closed (A-B) the circuit current will be maximum. This is because the capacitor will start to charge and will therefore appear as a short circuit. The amount of current is limited only by the size of the resistor and the amount of resistance in the circuit wiring. Aviation Australia DC current in capacitive restive (CR) circuits - charge/discharge cycles At the instant the switch is switched to position B-C, current through the resistor will be maximum but in the opposite direction of original current flow. This is because the capacitor will be fully charged when the switch is initially opened, and as time passes, the capacitor will discharge through the resistor, decreasing the voltage stored in the capacitor as well as the amount of current. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 75 of 284 After five time constants, the capacitor will be fully discharged and at this time no current will flow through the resistor. Aviation Australia Time constant graph for a discharging capacitor Therefore, the rate at which the charge is moved from one plate to the other (Q/t = current, Coulomb's law, DC theory), determines the rate at which the voltage changes. When the current is changing at its maximum rate (at the zero crossing), the voltage is at its maximum value (peak). When the current is changing at its minimum rate (zero at the peak), the voltage is at its minimum value (zero). Aviation Australia Phase relationship showing I leading V by 90° 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 76 of 284 By comparing the two sine waves, we see that the current peaks at one quarter cycle before the voltage sine peaks. Therefore it can be said that in a purely capacitive AC circuit, the current leads the voltage by 90°, or we can say that the voltage lags the current by 90°. In an CR circuit, the relationship between the current and voltage is altered. Using CIVIL, in a purely capacitive circuit, current leads voltage by 90°. The phasor representing current (I) is drawn 90° anti-clockwise from the phasor representing voltage (V). This represents the phase relationship of the current leading the voltage by 90°. Aviation Australia Phase relationship showing I leading V by 90° 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 77 of 284 AC Phase Angle in LCR Circuits In AC, in which the values are constantly changing, certain circuit components cause a phase shift between the voltage and the current. The amount of shift is referred to as the phase angle. For example, some electrical components cause the current to reach its maximum value 90° before the voltage. In this situation, there is a 90° phase angle between current and voltage. If the load is purely capacitive, the current will lead the voltage by 90°. If the load is purely inductive, the current will lag the voltage by 90°. © Aviation Australia AC phase angles between V and I in inductive and capacitive circuits 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 78 of 284 Inductive Reactance When an alternating current is connected across an inductor, the inductor has an effect on the circuit similar to that of a resistor. You will recall that the opposition to current flow in a resistive circuit is given by the relationship: Aviation Australia Ohm's law diagram Similarly, the opposition to current provided by pure inductance in an AC circuit is given by the relationship VL /IL. Although this relationship of voltage to current must be expressed in ohms (Ω), it cannot be called resistance (a pure inductance has no resistance.). Instead, the term inductive reactance is used, represented by the letter abbreviation XL. The following formula is used to calculate inductive reactance: XL = 2024-02-15 VL IL B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 79 of 284 Factors Affecting Inductive Reactance Inductive reactance is the opposition to current flow due to an inductance. Inductance is the property of an inductor whereby a change in current causes the inductor to produce an opposing voltage (EMF). That is, if the value of an inductor remains constant and the current through the inductor increases, then the voltage across the inductor will increase, opposing the current flowing through it. Anything that increases the voltage (back EMF, or CEMF) must also increase the inductive reactance. V L = LΔI L Remember, inductive reactance is the opposition to current flow. Now the voltage across the inductor can be found from the current flowing through it: L = the inductance in henrys ∆IL = the rate of change in current through the inductor. From the above two formulas we can see that an increase in L must increase VL and any increase in VL will cause a corresponding increase in the inductive reactance (XL ). XL is proportional to L. Any change in the size of the inductor (L) will change the inductive reactance XL. The other factor affecting inductive reactance is frequency. ΔIL , the rate of change in the current through the inductor, is directly related to frequency. The higher the frequency, the faster the current changes. A higher frequency means the peak value of the current is reached in a shorter time, so ΔIL must be higher. If inductance is held constant and the frequency increases, the inductive reactance also increases. XL is proportional to frequency. Any change in the frequency will change the inductive reactance (XL ). So not only is the inductive reactance directly proportional to inductance, but it is also directly proportional to frequency. XL is proportional to fL. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 80 of 284 Calculating Inductive Reactance Combining two effects of frequency and inductance, the formula for XL is obtained: © Aviation Australia Inductive reactance formula Where: XL is the inductive reactance in ohms. 2π (6.28) is a constant related to the sine waves derivation from a circle. f is the frequency in hertz. L is the inductance in henrys. This equation for inductive reactance is valid for sine wave applications only. Note that XL does not depend on the amplitude of the applied signal. For example: assuming that the value of inductance and frequency remain constant, XL remains constant even if the amplitude of the applied signal changes. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 81 of 284 Capacitive Reactance If the AC circuit is assumed to be purely capacitive, its only opposition to current flow is capacitive reactance (XC). Like inductive reactance in the inductive circuit, capacitive reactance is the opposition to current flow due to capacitance. However, there is a major difference in the effects of inductance and the effects of capacitance in an AC circuit. Aviation Australia Measuring XC When the capacitance value of a capacitor increases, the amount of charge the capacitor can hold will increase and therefore the current through the capacitor will also increase. By applying Ohm’s law, we can state that: XC = VC IC From this relationship, we can see that if VC is held constant and IC increases due to an increase in the size of the capacitor, XC must then decrease. 1 X C is proportional to C Any change in the size of the capacitor (C) will change the capacitive reactance XC. In a capacitive circuit, if the rate (frequency) at which the voltage is changing increases, the amount of charge moving through the circuit in a given period of time must also increase. An increase in the amount of current for a fixed amount of voltage indicates that opposition to the current has decreased. In other words, the opposition to current (XC) varies inversely with frequency. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 82 of 284 When capacitance is constant: At high frequencies, XC approaches a short circuit (very low reactance). At low frequencies, XC approaches an open circuit (very high reactance). So, not only is capacitive reactance inversely proportional to capacitance, but it is also inversely proportional to frequency. 1 X C is proportional to fC Calculating Capacitive Reactance Combining two effects of frequency and capacitance, the formula for XC is obtained: 1 XC = 2πf C Where: XC is the capacitive reactance in ohms. 2π (6.28) is a constant related to the sine waves derivation from a circle. f is the frequency in hertz. C is the capacitance in farads. A decrease in capacitance causes an increase in capacitive reactance. Again, as with XL , XC is independent of the amplitude of the applied signal. Increasing or decreasing the amplitude has no effect on capacitive reactance. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 83 of 284 Impedance Definition of Impedance Impedance is the total opposition to current flow in an AC circuit. The abbreviation for impedance is Z. Impedance is expressed in ohms (Ω). In a purely resistive circuit, the impedance is simply equal to the total circuit resistance. In a purely inductive circuit, the impedance is equal to the total inductive reactance. In the case of a purely capacitive circuit, the impedance is equal to the total capacitive reactance. V Z = I Aviation Australia Components of impedance As most electrical circuits are not purely inductive or capacitive but contain some resistance, we need to be able to determine the circuit impedance for a combination of resistors, capacitors and inductors. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 84 of 284 Impedance in Series LCR Circuits The AC source supplies voltage to the circuit at a particular frequency. Varying the frequency causes both the inductive and capacitive reactance to change. This changes not only the total circuit impedance (Z), but also the phase angle between the total circuit current (IT) and the applied voltage (VGEN). At some frequencies, the circuit will act capacitively and the current will lead the applied voltage. At other frequencies, the circuit will act inductively and the current will lag the applied voltage. Frequency and impedance in a series LCR circuit In a series circuit, a single path for current flow exists, i.e. the same current will flow through all three components. However, each component will have a different voltage dropped across it. These voltage drops will have various phase relationships to the circuit current, depending on the component across which they appear. Knowing the current and the impedance of each component, we can use Ohm's law to determine the voltage drop across the individual circuit components. From these circuit parameters, we can construct a phasor diagram that shows the phase relationships among the components. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 85 of 284 Determining Impedance of Series LCR Circuits Suppose we have a typical series LCR circuit connected to an AC source. R 1 kΩ XL=750Ω 15 V 20 kHz XC=1500Ω Aviation Australia Reference impedance series LCR circuit Using the component values, we can calculate the reactance of the inductor and capacitor at the applied input frequency. The reactance values can be shown graphically on a phasor diagram. Y Y XL=750Ω -X X -X X R=1kΩ R=1kΩ Xeq= 750 Ω XC=1.5kΩ -Y -Y Aviation Australia Phasor diagrams of the reference series LCR circuit 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 86 of 284 The total equivalent reactance (Xeq) is: X EQ = X C − X L X EQ = 1500 − 750 X EQ = 750 Ω XL and XC are of opposite phase (CIVIL), they are subtracted to give a single value which represents the equivalent reactance. The overall effect of reactance on the circuit is capacitive. A large capacitive reactance cancels out a smaller inductive reactance, resulting in a circuit which acts capacitively. That is, the current leads the voltage (CIVIL). If the inductive reactance is larger than the capacitive reactance, the circuit acts as a series LR circuit. To determine if a series LCR circuit acts inductively or capacitively, simply note which reactance is larger. Using a phasor diagram, we can derive an equivalent circuit representing each phasor. The phasor diagram shows the circuit acting like a 1k Ω resistor connected in series with an equivalent capacitor having a reactance of 750 Ω. The equivalent circuit is shown below. Aviation Australia Resultant resistor capacitor (RC) circuit and phasor diagram of the reference series LCR circuit Note: Any change in the frequency will cause both the inductive and capacitive reactance to change and hence the equivalent circuit will no longer be valid. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 87 of 284 The total circuit impedance (Z) is the vector addition of the reactance and resistance phasors. Knowing the reactance and resistance, we can determine the magnitude of the total circuit impedance (Z): Z = √R Z = √ 2 + X 1 000 2 2 EQ + 750 2 Z = √ 1 000 000 + 562 500 Z = √ 1 562 500 Z = 1250 Ω Determining Total Circuit Current of Series LCR Circuits This continues from the example in the previous section. Since the same current flows through all components, the total circuit current (IT) can be calculated using Ohm's law. As we have determined the magnitude of the impedance (Z) as 1250 Ω and we know the applied voltage (VGEN) is 15V, we can determine IT by: IT = V GEN Z 15 IT = 1250 I T = 12 mA Determining Component Voltage Drops of Series LCR Circuits This continues from the example in the previous section. Once the total circuit current (IT) has been determined, the voltage drops across each component can also be found by using Ohm's law: Resistive Component: V R1 = I T × R V R1 = 12 × 10 −3 × 1000 V R1 = 12 V 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 88 of 284 Inductive Component: VL = IT × XL V L = 12 × 10 −3 × 750 VL = 9 V Capacitive Component: VC = IT × XC V C = 12 × 10 −3 × 1500 V C = 18 V Another way to tell if a series LCR circuit is inductive or capacitive is to measure the voltage drops across the inductor and capacitor. The component with the highest voltage drop determines the total reactive effect of that component. If VC is higher than VL the circuit is capacitive. Y VR=12V -X IT=12mA X -Y Aviation Australia Resistor component voltage phasor and the current reference phasor for the reference series LCR circuit In a series circuit, the current is the same through all the components. We can therefore use the total circuit current (IT) as the reference phasor. The voltage across the resistor is in phase with the current and is drawn parallel to it. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 89 of 284 Y -X VL=9 V IT=12mA X VR=12 V VC=18 V -Y Aviation Australia Reference series LCR circuit voltage phasor diagram Recalling the mnemonic CIVIL, we can determine the phase relationships for the voltage across the capacitor and the inductor. For the capacitor, the voltage lags the current by 90°. For the inductor, the voltage leads the current by 90°. If the two reactances are opposite in their effect, the larger capacitive reactance cancels the inductive reactance. The total reactive voltage effect in the circuit is capacitive because the capacitive reactance is larger than the inductive reactance. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 90 of 284 -X Y IT= 12mA X VR= 12 V VC-VL= 9 V -Y Aviation Australia Reference series LCR circuit resultant voltage capacitor resistor (CR) phasor diagram In a series LCR circuit, the voltage across each component cannot be added directly because the voltages are out of phase with one another. They must be added vectorally. The net circuit reactance is capacitive; therefore, the current leads the applied voltage (VGEN). To check the results, we can calculate the value of VGEN by the phasor sum of the equivalent capacitor voltage (VC - VL) and the resistor voltage (VR). V GEN = √ V 2 R + (V C − V L ) 2 2 2 V GEN = √ 12 + 9 V GEN = √ 225 V GEN = 15 V If the calculated value of VGEN equals the applied voltage, our calculations are correct. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 91 of 284 Circuit Phase Relationships of Series LCR Circuits This continues from the example in the previous section. With all circuit voltages known, the phase angle of the applied voltage can be determined. SOH-CAH-TOA can be used for any vector addition. Opp SOH ⟹ Sin θ = H yp Adj CAH ⟹ Cos θ = H yp Opp T OA ⟹ T an θ = Adj Note the polarity of the Y-axis voltage in the phasor diagram – this determines whether the angle is leading or lagging. Reference is always to the X-axis of the phasor diagram. The phase angle is calculated: Opp T an θ = Adj θ = T an −1 ( VC − VL ) VR θ = T an −1 9 ( ) 12 θ = T an -X −1 (0.75) = 36.85° Y IT= 12mA 36.850 X VR= 12 V VC-VL= 9 V -Y 2024-02-15 Aviation Australia B-03b Electrical Fundamentals Page 92 of 284 CASA Part 66 - Training Materials Only series LCR circuit Resultant voltage phasor angle of the reference Once all circuit parameters are known, the complete phasor diagram can be drawn. Notice that the circuit current (IT), which is in phase with the resistor voltage (VR), leads the applied voltage by 36.85 degrees. An important fact to remember is that the circuit values depend on the frequency of the applied voltage. The reactance values and the total impedance are only valid at a frequency of 20 kHz. Changing the frequency will change the reactance values and other factors in the circuit. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 93 of 284 Summary of Series LCR Circuits We can determine the parameters for any series LCR circuit by following these steps: Determining Impedance of Series LCR Circuits Calculate the reactances (XL and XC) using: X L = 2πf L 1 XC = 2πf C Calculate the total equivalent reactance XEQ (difference between XL and XC). Calculate the total impedance (Z). 2 Z = √R + X 2 EQ Determining the Total Circuit Current of Series LCR Circuits Calculate the total circuit current (IT) using: V IT = Z Determining Component Voltage Drops in Series LCR Circuits VR = IT × R VL = IT × XL VC = IT × XC Determining Phase Angle of Series LCR Circuits Calculate the phase angle between the applied voltage and the total circuit current using: θ = tan −1 VC − VL VR Subtract the smallest from the largest VC-VL or VL-VC. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 94 of 284 Parallel LCR Circuits The parameters required for a parallel LCR circuit are similar to those of a series circuit. The difference is mainly in the fact that in a series circuit, the current is the same for all components and the voltage for each component needs to be calculated separately. However, in a parallel circuit the voltage is the same for all components and the branch currents need to be calculated for each component. For a parallel LCR circuit, the four main parameters required to analyse the circuit are: 1. Reactance (both inductive and capacitive) 2. Branch currents 3. The total circuit current 4. The phase angle between the applied voltage and the branch currents. Aviation Australia Parallel LCR circuit current paths In a parallel LCR circuit, the supply voltage appears across each of the components. At the frequency of operation, the inductor and capacitor have specific reactance values. Using Ohm's law, you can determine the current through each of the three circuit branches. Knowing the three branch currents, you can calculate the total current (IT) drawn from the generator. With total circuit current, you can determine the total circuit impedance by applying Ohm's law. Once the branch currents are known, a phasor diagram can be constructed to determine whether the circuit is inductive or capacitive. The diagram also helps you to determine the phase relationship between the total circuit current and the applied voltage. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 95 of 284 Determining Reactance in Parallel LCR Circuits In a parallel LCR circuit, the applied voltage appears across each component as shown. IT VGEN XL=750Ω R=1kΩ XC=1.5kΩ 15 V 20KhZ IC IL IR Aviation Australia Reference parallel LCR circuit reactance parameters Using these values, we can calculate the inductive reactance and capacitive reactance as follows: XL = 2πf L, 1 XC = 2024-02-15 2πf C B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 96 of 284 Determining Branch Currents in Parallel LCR Circuits In a parallel circuit the current through each branch will be different depending on the reactance or resistance of each component. The individual branch currents can be found by using Ohm's law: Resistive Component IR = V GEN R 15 V IR = 1000 Ω I R = 15 mA Capacitive Component IC = V GEN XC 15 V IC = 1500 Ω I C = 10 mA Inductive Component IL = V GEN XL 15 V IL = 750 Ω I L = 20 mA 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 97 of 284 Determining Total Circuit Current in Parallel LCR Circuits The sum of the individual branch currents in a parallel LCR circuit is equal to the total current drawn from the source (IT). Because IL and IC are of opposite phase, they can be added to give a single value of reactive current. The reactive currents in the circuit are vector quantities, so they cannot be added directly to the resistive current. To add the reactive and resistive currents, we will use Pythagoras’ equation for a right angle triangle: I T = √I 2 R + (I C − I L ) 2 −3 2 −3 2 I T = √ (15 × 10 ) + (10 × 10 ) I T = √ (225 × 10 −6 ) + (100 × 10 −6 ) I T = 18 mA The resulting current (IT) is capacitive because at this frequency the inductor current is larger than the capacitor current. If IL is greater than IC , the total reactance will be inductive and the circuit will act as a parallel LR circuit. If IC is greater than IL , the total reactance will be capacitive and the circuit will act as a parallel CR circuit. If XC is equal to XL, a special case exists known as resonance. When XC is equal to XL, the resulting branch currents IC and IL are equal in magnitude and are always 180° out of phase with each other. As a result, the two currents cancel and the resulting phase angle of the resonant circuit is zero. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 98 of 284 Current Phase Relationships in Parallel LCR Circuits When different phase shifts occur in the parallel circuit, the individual branch currents cannot be added directly. They must be added using phasors. Y IC= 10mA -X IR= 15mA θ Ieq= 10mA X VGen= 15 V IT= 15mA IL= 20mA -Y Aviation Australia Current phase relationship in the reference parallel LCR circuit An applied voltage that is common to all components is used as the horizontal reference for the phasor diagram. The current through a resistor is always in phase with the voltage across it, so the current through the resistor must be in phase with the applied voltage. If the current through the inductor lags the applied voltage by 90°, its phasor is drawn 90° behind the applied voltage. If the current through the capacitor leads the applied voltage by 90°, its phasor is drawn 90° ahead of the applied voltage. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 99 of 284 If the circuit is capacitive, the current leads the applied voltage (VGEN ) by an angle between 0° and 90°. This angle can be determined from the current amplitudes in the phasor diagram. The phase angle is: θ = tan −1 IL − IC IR θ = tan 10 × 10 −1 −3 1.5 × 10 θ = tan −1 −3 (0.666) º θ = 33.6 Varying the frequency will change the reactances, branch currents, total current, impedance and phase angles. At the lower frequencies, XL will be lower than XC, so the circuit will be inductive. At the higher frequencies, XC will be lower than XL, so the circuit will be capacitive. 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 100 of 284 Summary of Parallel LCR Circuit Analysis We can determine the parameters for any parallel LCR circuit by the following sequence: Determining Reactance of Parallel LCR Circuits Calculate the reactances (XL and XC) using: XL = 2πf L 1 XC = 2πf C Determining Total Circuit Current of Parallel LCR Circuits Calculate the current through each component (IR, IC and IL). IR = IC = IL = V GEN R V GEN XC V GEN IL Calculate the equivalent reactive current (difference between IC and IL). Calculate the total circuit current (IT). 2 2 I T = √I + (I C − I L ) R Determining the Phase Angle of Parallel LCR Circuits Calculate the total current pahse angle using SOH...CAH...TOA. tan θ = ( IL − IC ) IR 2024-02-15 B-03b Electrical Fundamentals CASA Part 66 - Training Materials Only Page 101 of 284 LCR Circuits II (3.14) Learning Objectives 3.14.2 Explain power dissipation, causes and methods of calculating energy losses (Level 2). 3.14.4.1 Describe true power and methods of calculating true power (Level 2). 3.14.4.2 Describe apparent