UEENEEG048B Solve problems in complex multi-path power circuits SAMPLE. Version 4. Training and Education Support Industry Skills Unit Meadowbank

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1 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

2 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: Fax: 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 TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011

3 TABLE OF CONTENTS Introduction General introduction Using this learner guide Prior knowledge and experience Unit of competency overview Assessment Section 1 Voltage/Current Sources, Kirchhoff s Law for DC Linear Circuits..15 Voltage sources Current sources Conversion between sources Kirchhoff s Voltage law Answers to student exercises Section 2 Superposition principles for D.C. Linear Circuits...37 DC Networks Two-source networks with voltage sources Two-source networks with current sources Networks with three sources and three meshes Answers to student exercises Section 3 Mesh and Nodal Analysis for D.C. Linear Circuits...55 Mesh Analysis Nodal analysis Answers to student exercises Section 4 Thevenin s principles for D.C. linear circuits...89 Thevenin s Theorem Two-mesh circuits Three-mesh circuits Answers to student exercises Section 5 Norton s principles for D.C. linear circuits Norton s Theorem Three-mesh circuits Source Conversion Circuit simplification by source conversion Answers to student exercises TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011

4 Section 6 Phasor Analysis Alternating sinusoidal waveforms, angular frequency and units of measurement 120 Peak voltage and frequency Review Summary Supplementary notes Answers to student exercises Section 7 Complex impedance The impedance triangle Resistance and Reactance Admittance, susceptance and conductance Real components equivalent series circuit Element voltage drops Real component equivalent parallel circuits Answers to student exercises Section 8 Series and parallel A.C. linear circuits Series equivalent impedance Parallel Equivalent Impedance The voltage divider and current splitter Series Parallel AC Circuits Answers to student exercises Section 9 Superposition principles and Kirchhoff s Laws applied to A.C. linear circuits Voltage drops and voltage rise Conventions for solution by Kirchhoff s Laws Conventions for solution by superposition Solving equations derived by Kirchhoff s Laws Current divider using admittances Solving equations by superposition Answers to student exercises Section 10 Mesh and Nodal analysis for A.C. linear circuits Currents and mesh analysis Mesh analysis using determinants Voltages and nodal analysis Answers to student exercises Section 11 Thevenin and Norton theorems applied to A.C. linear circuits Thevenin s equivalent circuit Norton s equivalent circuit TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011

5 Thevenin/Norton source conversion Answers to student exercises Section 12 Complex A.C. power and maximum power transfer theorem True power Reactive power Apparent, reactive and real power Power factor Power triangles Maximum-power transfer Proportion of power consumed by a source Answers to student exercise Section 13 Series resonance Resistance, reactance, impedance and frequency Resonant frequency Resonant series impedance and power factor Voltage magnification factor Q factor Selectivity Bandwidth Half power (3dB) Points and Powerfactor Practical applications of resonant circuits Problem Resonance Answers to student exercises Section 14 Parallel Resonance Resistance, reactance and impedance vs. frequency Selectivity Bandwidth Q-Factor in parallel resonant circuits Current amplification Impedance vs. frequency Frequency of Maximum Impedance Frequency of Unity power factor Loading of Parallel Resonant Circuits High Q Factor Conditions Half Power (3dB) Points and Power Factor Uses of Resonant Circuits Problem Resonance TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011

6 Answers to student exercises Section 15 Transients Transients in R-C circuits Growth and decay Transients in L R circuits Answers to student exercises TAFE NSW (Training & Education Support Industry Skills Unit, Meadowbank) 2011

7 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

8 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

9 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

10 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

11 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

12 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