ueeneeg048b solve problems in complex multi-path … · dc networks ... conventions for solution by...

UEE07 Electrotechnology

Training Package

Learner guide

Version 4

Training and Education Support

Industry Skills Unit

Meadowbank

Product Code: 5526

UEENEEG048B

Solve problems in complex multi-path power circuits

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© TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank)

2011

UEENEEG048B Solve problems in complex multi-path power circuits V4

AcknowledgmentsThe TAFE NSW Training and Education Support Industry Skills Unit, Meadowbank would like to acknowledge the support and assistance of the following people in the production of this learner resource guide:

Writer:David ArnoldWestern InstituteTAFE NSW

Reviewers:Greg BellTAFE NSW

Project Manager:Steve ParkinsonKerry BarlowTAFE NSW

EnquiriesEnquiries about this and other publications can be made to:

Training and Education Support Industry Skills Unit, Meadowbank Meadowbank TAFE Level 3, Building J See Street MEADOWBANK NSW 2114

Tel: 02-9942 3200 Fax: 02-9942 3257

© TAFE NSW (Training and Education Support, Industry Skills Unit Meadowbank) 2011

Copyright of this material is reserved to TAFE NSW Training and Education Support, Industry Skills Unit Meadowbank. Reproduction or transmittal in whole or in part, other than for the purposes of private study or research, and subject to the provisions of the Copyright Act, is prohibited without the written authority of TAFE NSW Training and Education Support, Industry Skills Unit Meadowbank.

ISBN 978-1-74236-270-0

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TABLE OF CONTENTS

Introduction ......................................................................................... 71. General introduction ................................................................................ 7

2. Using this learner guide .......................................................................... 7

3. Prior knowledge and experience ................................................................ 9

4. Unit of competency overview ................................................................... 9

5. Assessment .......................................................................................... 12

Section 1 Voltage/Current Sources, Kirchhoff’s Law for DC Linear Circuits ..15Voltage sources ......................................................................................... 16

Current sources ......................................................................................... 24

Conversion between sources ....................................................................... 28

Kirchhoff’s Voltage law ................................................................................ 31

Answers to student exercises ...................................................................... 34

Section 2 Superposition principles for D.C. Linear Circuits ........................37DC Networks ............................................................................................. 38

Two-source networks with voltage sources .................................................... 38

Two-source networks with current sources .................................................... 41

Networks with three sources and three meshes .............................................. 44


Section 3 Mesh and Nodal Analysis for D.C. Linear Circuits ......................55Mesh Analysis ........................................................................................... 56

Nodal analysis ........................................................................................... 65


Section 4 Thevenin’s principles for D.C. linear circuits .............................89Thevenin’s Theorem ................................................................................... 90

Two-mesh circuits ...................................................................................... 92

Three-mesh circuits ................................................................................... 94


Section 5 Norton’s principles for D.C. linear circuits ..............................103Norton’s Theorem .....................................................................................104

Three-mesh circuits ..................................................................................107

Source Conversion ....................................................................................108

Circuit simplifi cation by source conversion ....................................................110

Answers to student exercises .....................................................................115

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Section 6 Phasor Analysis ................................................................. 119Alternating sinusoidal waveforms, angular frequency and units of measurement 120

Peak voltage and frequency ........................................................................ 129

Review Summary ...................................................................................... 131

Supplementary notes ................................................................................ 132

Answers to student exercises ..................................................................... 137

Section 7 Complex impedance ........................................................... 141The impedance triangle ............................................................................. 142

Resistance and Reactance .......................................................................... 142

Admittance, susceptance and conductance ................................................... 146

Real components equivalent series circuit ..................................................... 150

Element voltage drops ............................................................................... 155

Real component equivalent parallel circuits ................................................... 159


Section 8 Series and parallel A.C. linear circuits ................................... 177Series equivalent impedance ...................................................................... 178

Parallel Equivalent Impedance .................................................................... 181

The voltage divider and current splitter ........................................................ 184

Series Parallel AC Circuits .......................................................................... 187


Section 9 Superposition principles and Kirchhoff’s Laws applied to A.C. linear circuits ............................................................................. 197

Voltage drops and voltage rise .................................................................... 198

Conventions for solution by Kirchhoff’s Laws ................................................. 198

Conventions for solution by superposition ..................................................... 198

Solving equations derived by Kirchhoff’s Laws ............................................... 199

Current divider using admittances ............................................................... 202

Solving equations by superposition .............................................................. 204


Section 10 Mesh and Nodal analysis for A.C. linear circuits ...................... 211Currents and mesh analysis ....................................................................... 212

Mesh analysis using determinants ............................................................... 212

Voltages and nodal analysis ........................................................................ 218


Section 11 Thevenin and Norton theorems applied to A.C. linear circuits ... 231Thevenin’s equivalent circuit ....................................................................... 232

Norton’s equivalent circuit .......................................................................... 234

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Thevenin/Norton source conversion ............................................................. 238


Section 12 Complex A.C. power and maximum power transfer theorem .... 245True power .............................................................................................. 246

Reactive power ......................................................................................... 248

Apparent, reactive and real power ............................................................... 251

Power factor ............................................................................................. 252

Power triangles ......................................................................................... 253

Maximum-power transfer ........................................................................... 255

Proportion of power consumed by a source ................................................... 255

Answers to student exercise ....................................................................... 257

Section 13 Series resonance ............................................................... 261Resistance, reactance, impedance and frequency .......................................... 262

Resonant frequency .................................................................................. 264

Resonant series impedance and power factor ................................................ 264

Voltage magnifi cation factor ....................................................................... 265

Q factor ................................................................................................... 266

Selectivity ................................................................................................ 266

Bandwidth ............................................................................................... 267

Half power (3dB) Points and Powerfactor ...................................................... 268

Practical applications of resonant circuits ...................................................... 271

Problem Resonance ................................................................................... 272


Section 14 Parallel Resonance ............................................................. 277Resistance, reactance and impedance vs. frequency ...................................... 278

Selectivity ................................................................................................ 278

Bandwidth ............................................................................................... 279

Q-Factor in parallel resonant circuits ............................................................ 281

Current amplifi cation ................................................................................. 283

Impedance vs. frequency ........................................................................... 284

Frequency of Maximum Impedance ............................................................. 285

Frequency of Unity power factor ................................................................. 285

Loading of Parallel Resonant Circuits ............................................................ 286

High Q Factor Conditions ............................................................................ 288

Half Power (3dB) Points and Power Factor .................................................... 289

Uses of Resonant Circuits ........................................................................... 289

Problem Resonance ................................................................................... 290

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Section 15 Transients ......................................................................... 295Transients in R-C circuits ............................................................................ 296

Growth and decay ..................................................................................... 297

Transients in L–R circuits ............................................................................ 301


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© TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011 Page 13 of 310


Section 1

Voltage/current sources, Kirchhoff’s law

for DC linear circuits

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Section 1 Voltage/Current Sources, Kirchhoff’s Law for DC Linear Circuits

Contents

• Voltage sources

• Current sources

• Conversion between sources

• Kirchhoff’s Voltage Law

• Kirchhoff’s Current Law

• Conventions

• Establishing equations

• Solving equations

Learning Objectives

Learners should be able to meet the following learning objectives:

a. Calculate the effect of the internal resistance on terminal voltage and current delivered for practical voltage sources and current sources.

b. Calculate current and voltage in any DC network of up to two loops and three sources.

c. Calculate current and voltage in any AC network of up to two loops and two sources.

d. Describe the function and operation of an electronics circuit simulation program.

e. Enter given circuit specifi cations into an electronic circuit simulation program to determine circuit currents and voltagesSAMPLE



Voltage sourcesAn ideal voltage source provides a constant-output voltage independent of the value of the load current. This simply means that the voltage across the terminals (ie the terminal voltage) of an ideal voltage source is constant, no matter what current is drawn from it.

The model of an ideal dc voltage source of E volts is shown in Figure 1.1a given below, with the box in broken lines representing such a source.

The V-I characteristic of an ideal dc voltage source is shown in Figure 1.1b given below. You will note that the V-I characteristic is a horizontal line, parallel to the current axis and intercepting the voltage axis at the value E.

Figure 1.1 a. Ideal dc voltage source b. characteristic of an ideal dc voltage source

Note that the dc voltage source may be a power supply, battery, photovoltaic cell, or any other source of dc power.

Practical voltage sources

In practical dc voltage sources, as the current IL drawn from the source is increased, the terminal voltage across the source (and load), V, decreases. This drop in the source terminal voltage from its ideal value E is due to a resistance within the source. This resistance is called internal resistance, ri, of the source.

The magnitude of the internal resistance varies between sources and depends on the construction, size and type of the source.

The model of a practical dc voltage source of E volts and internal resistance ri ohms is shown in Figure 1.2a below, with the box in broken lines representing such a source.

The V-I characteristic of a practical dc voltage source is shown in Figure 1.2b.

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Figure 1.2 a. Practical dc voltage b. V-I characteristic of a practical dc voltage source

The voltage drops across resistors ri and RL sum together and equal the applied voltage E.

Hence:

E = V + ILri 1 Rearranging

V = E - ILri 2

where

E = open-circuit voltage of the source (i.e. the voltage at the source terminals when the load is disconnected, IL= 0)

V = the terminal voltage of the source

IL = load current (ie current delivered to the external circuit)

ri = internal resistance of the source

Applying Ohm’s law to the load, we get this equation:

V = ILRL 3

From equations 1 and 3 above we have:

E = IL(RL + ri)

Therefore:

IL = ERL + ri

4

Note: The internal resistance of an ideal voltage source is zero, so that the terminal voltage of the source on load is equal to the open-circuit voltage of the source.

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The short circuit current ISC a voltage source may be obtained by making the load zero (i.e. RL = 0). This is shown in Figure 1.3 below.

Figure 1.3 Short circuit current

From equation 4 above:

Isc = IL = E0+ ri

Isc = Eri 5

Work through Examples 1 and 2 given below. Solve the examples yourself before going through the worked solutions. These examples will show you how to apply equations 1, 2, 3, 4 and 5.

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Example 1.1

A practical voltage source delivers a current of 0.5 A to a load of 280 Ω, and a current of 3.75 A to a load of 20 Ω.

a. Determine the internal resistance of the voltage source.

b. Determine the open-circuit voltage of the voltage source.

c. Draw the circuit model of the voltage source.

d. Determine the short-circuit current of the voltage source.

e. Determine the source’s terminal voltage and the current delivered to a load of 30 Ω.

Solution

Using equations 1 and 3 , along with the two data points given in the problem gives two equations with two unknowns.

Solving these equations gives the solution.

(a)

Ω

Figure 1.4

RL 2

Ω

Figure 1.5

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ueeneeg048b solve problems in complex multi-path … · dc networks ... conventions for solution by...

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