UEE07 Electrotechnology Training Package UEENEEG048B Solve problems in complex multi-path power circuits Learner guide Version 4 Training and Education Support Industry Skills Unit Meadowbank Product Code: 5526
Acknowledgments The 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 Arnold Western Institute TAFE NSW Reviewers: Greg Bell TAFE NSW Project Manager: Steve Parkinson Kerry Barlow TAFE NSW Enquiries Enquiries 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 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011
TABLE OF CONTENTS Introduction... 7 1. 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..15 Voltage 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...37 DC Networks... 38 Two-source networks with voltage sources... 38 Two-source networks with current sources... 41 Networks with three sources and three meshes... 44 Answers to student exercises... 47 Section 3 Mesh and Nodal Analysis for D.C. Linear Circuits...55 Mesh Analysis... 56 Nodal analysis... 65 Answers to student exercises... 78 Section 4 Thevenin s principles for D.C. linear circuits...89 Thevenin s Theorem... 90 Two-mesh circuits... 92 Three-mesh circuits... 94 Answers to student exercises... 99 Section 5 Norton s principles for D.C. linear circuits...103 Norton s Theorem...104 Three-mesh circuits...107 Source Conversion...108 Circuit simplification by source conversion...110 Answers to student exercises...115 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011
Section 6 Phasor Analysis... 119 Alternating 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... 141 The 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 Answers to student exercises... 169 Section 8 Series and parallel A.C. linear circuits... 177 Series equivalent impedance... 178 Parallel Equivalent Impedance... 181 The voltage divider and current splitter... 184 Series Parallel AC Circuits... 187 Answers to student exercises... 191 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 Answers to student exercises... 208 Section 10 Mesh and Nodal analysis for A.C. linear circuits... 211 Currents and mesh analysis... 212 Mesh analysis using determinants... 212 Voltages and nodal analysis... 218 Answers to student exercises... 224 Section 11 Thevenin and Norton theorems applied to A.C. linear circuits... 231 Thevenin s equivalent circuit... 232 Norton s equivalent circuit... 234 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011
Thevenin/Norton source conversion... 238 Answers to student exercises... 241 Section 12 Complex A.C. power and maximum power transfer theorem... 245 True 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... 261 Resistance, reactance, impedance and frequency... 262 Resonant frequency... 264 Resonant series impedance and power factor... 264 Voltage magnification 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 Answers to student exercises... 273 Section 14 Parallel Resonance... 277 Resistance, reactance and impedance vs. frequency... 278 Selectivity... 278 Bandwidth... 279 Q-Factor in parallel resonant circuits... 281 Current amplification... 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 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011
Answers to student exercises... 291 Section 15 Transients... 295 Transients in R-C circuits... 296 Growth and decay... 297 Transients in L R circuits... 301 Answers to student exercises... 305 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011
Section 1 Voltage/current sources, Kirchhoff s law for DC linear circuits 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 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 specifications into an electronic circuit simulation program to determine circuit currents and voltages TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011 Page 15 of 310
Voltage sources An 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 I L 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, r i, 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 r i 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. Page 16 of 310 TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011
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: where E = V + I L r i 1 Rearranging V = E - I L r i 2 E = V = I L = r i = open-circuit voltage of the source (i.e. the voltage at the source terminals when the load is disconnected, I L = 0) the terminal voltage of the source load current (ie current delivered to the external circuit) internal resistance of the source Applying Ohm s law to the load, we get this equation: V = I L R L 3 From equations 1 and 3 above we have: E = I L (R L + r i ) Therefore: E I L = 4 R L + r i 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. TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011 Page 17 of 310
The short circuit current I SC a voltage source may be obtained by making the load zero (i.e. R L = 0). This is shown in Figure 1.3 below. Figure 1.3 Short circuit current From equation 4 above: I sc = I L = E 0+ ri I sc = E 5 ri 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. Page 18 of 310 TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011
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) Figure 1.4 Ω R L 2 Ω Figure 1.5 TAFE NSW (Training & Education Support, Industry Skills Unit Meadowbank) 2011 Page 19 of 310