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Engineering review on AC circuit steady state analysis. Presentation lecture for energy engineering class.
Course MS in Renewable Energy Engineering, Oregon institute of technology

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• Practically, most of the time, we’re only interested in the steady state response of a system. i.e. the system has been running for quite a long time. This is the normal operation mode of the system. Ex: When you turn on the light, it take sometime to go to the normal operation mode. When you turn on the motor it will take sometime to go to the normal mode.
• ### Ac steady state analysis

1. 1. AC Power Engineering Review Presented by Brenden Motte, Alissa Olson & Long Pham
2. 2. Why not just use DC? New York City utility lines, 1980
3. 3. AC vs DC • DC was first use for electricity transmission. • Then “The war of current.” NY, 1880s: ▫ AC Vs DC ~ Testla Vs Edison ~ Employee Vs Boss. ▫ Edison’s least favorite of Tesla’s “impractical” ideas was the concept of using alternating current (AC) to bring electricity to the people. Edison insisted that his own direct current (DC) system was superior, in that it maintained a lower voltage from power station to consumer, and was, therefore, safer. “Direct current is like a river flowing peacefully to sea, while alternating current is like a torrent rushing violently over a precipice.” - Tom ▫ Tesla insisted that he could increase the efficiency of Edison’s prototypical dynamos (by changing from DC to AC generator). Edison promised him \$50,000 if he succeeded. Tesla worked around the clock for several months and made a great deal of progress. When he demanded his reward, Edison claimed the offer was a joke, saying, “When you become a full-fledged American, you will appreciate an American joke.” Ever prideful, Tesla quit, and spent the next few months picking up odd jobs across New York City. Nikola Tesla: ditch digger. ▫ Tesla eventually raised enough money to found the Tesla Electric Light Company, where he developed several successful patents including AC generators, wires, transformers, lights, and a 100 horsepower AC motor. The most significant contribution to the early success of ac was his patenting of the poly-phase ac motor in 1888. ▫ Always more of a visionary than a businessman, Tesla ended up selling most of his patents (for the healthy but finite sum of \$1 million) to George Westinghouse, an inventor, entrepreneur, and engineer who had himself been feuding with Edison for years. Their partnership, made the eventual popularizing of AC that much more bitter for Edison. • AC electrical energy has been the most convenient form of energy to be generated, transmitted, and distributed.
4. 4. DC vs AC • While AC was perfectly adequate for the conditions of much of the 19th & 20th century, the needs of the 21st century are showing its limits. • We are facing a revolution in the way our electricity is produced and used. More and more electricity is being generated from renewable sources of energy in remote areas: hydropower plants (mountains far from urban centers), wind farms (people tend not to live in windy areas), offshore wind farms (have higher capacity factor, better alignment with peak demand), etc. DC is the only technology that allows power to be transmitted economically over very long distances, and DC is the type of power produced by photovoltaic panels. • On the consumer side, more and more equipment runs on DC: computers, cell phones, LED lights, CFLs, high efficiency motor drives which found in new HVAC systems, industrial, etc. ABB estimates the savings from using DC instead of AC in buildings could be in the order of 10 to 20 percent. Reference: http://www.abb.com/cawp/seitp202/c646c16ae1512f8ec1257934004fa545.aspx
5. 5. AC Power: Today’s topics • Part 1: AC Circuits Analysis ▫ Steady-state sinusoidal response, Impedance Model. • Part 2: AC Power Analysis ▫ Power in AC circuits, Power Factor, Power factor corrections, Poly-Phase Circuits. • Part 3: Elements of the AC Power Systems ▫ Basics elements comprising the AC grid: Power Generator, Transformers, Capacitor Banks, Transmission circuits, etc.
6. 6. Part 1: AC Circuits • Complex Number: A Quick Review • Resistor, capacitor, Inductor: A practical explanation • Wave form of a signal & characteristic of Sinusoids signal ▫ Period, frequency, radian frequency (or angular frequency), phase leading & lagging, etc. • • • • • Sinusoidal response of RC Network: usual approach Phasor domain analysis or Frequency domain analysis Sinusoidal response of RC Network: Impedance model Using Impedance Model to solve AC circuits Examples
7. 7. Complex Number: A Quick Review Rectangular form Polar form Exponential form Im  A b |A|  0 Re a Note: Most calculator can do complex arithmetic & convert between Rectangular form to/from polar form
8. 8. Complex Number Example: • Evaluate the following complex numbers into rectangular form: a. [(5  j2)( 1  j4)  5 60 o ] b. 10  j5  340 o  10 30 o  3  j4 Solution: a. –15.5 + j13.67 b. 8.293 + j2.2
9. 9. Using complex number Solving Trigonometric nightmare • Problem: v1  t   20cos t  45  V v2  t   10sin t  60  V Find vs  v1  v2  ? • Solution: V1  20  45 V V2  10  30 V    Vs  V1  V2 Why can we do this?  20  45  10  30  14.14  j14.14  8.660  j5  23.06  j19.14  29.97  39.7 V Complex number doesn’t make life more complicated but more simple!
10. 10. Resistors • Purely resistive loads are almost non-existent in AC networks. AC network loads consist primarily out of inductive and to lesser extent capacitive loads, both in combination with resistive loads. It is therefore VERY important to understand the characteristics of capacitors and inductors in an AC environment. • Inductors and capacitors are energy storage elements. The difference lies in how and the type of energy that is stored by each.
11. 11. Inductors • When current flows, a magnetic field is created. • Energy provided by the current is stored in the magnetic field. The stronger the current or higher the number of coils (higher inductance), the greater the stored amount of energy will be. • Inductor does however have a limit as to the amount of energy it can store and the rate at which it can store the energy. (~Saturated) • Energy stored in inductor’s magnetic field can be retrieved. • Inductance is represented by: L and measured in Henry: (H) It is clear that an inductor stores energy in the form of a magnetic field created by current flowing through the inductor coil.
12. 12. Inductors Large 50 MVAR three-phase iron-core loading inductor at German utility substation
13. 13. Inductors characteristics • When you pushing a car, it will slowly increase it’s velocity (~kinetic energy). Similarly, when you apply a voltage across an inductor, it will slowly increase it’s current (~magnetic energy). Inductor’s current cannot change instantly just as car’s velocity. (voltage can change instantly) • When you stop pushing, the car will continue to run. Similarly, when you stop applying voltage, the inductor’s current will continue to flow. • What happens if something block the car from running? What happens if something block the inductor’s current from flowing? • The car will try its best to run until it can’t run anymore; all the kinetic energy stored in the car will dissipate to the blocker. A big force will created by the car’s effort to try to maintain it’s velocity. • Similarly, the inductor will try its best to flow the current until it can’t run anymore; all the magnetic energy store in the field will dissipate to the blocker. A big voltage is created by the inductor in its effort to try to maintain it’s current. • Mathematically, the relation between inductor’s current & voltage is the same as the relation between car’s velocity & force:
14. 14. Capacitors • When apply a voltage to a capacitor, a electrical field is created between the capacitor’s plates. • Energy provided by a voltage difference is stored in the electric field. The higher the voltage or bigger capacitor (higher capacitance), the stronger the electric field and more energy is stored inside the capacitor. • Capacitor does however have a limit as to the amount of energy it can store. • Energy stored in capacitor’s magnetic field can be retrieved. • Capacitance is represented by: C and measured in Farad: (F) It is clear that a capacitor stores energy in the form of a electric field created by potential difference across the capacitor plates.
15. 15. Capacitors
16. 16. Waveform of a signal • A waveform is the shape or form of a signal against time, physical medium or an abstract representation. • Common periodic waveforms are: Sine, Square, Triangle, etc. Example: ▫ ▫ ▫ ▫ ▫ ▫ Electrocardiogram: 1Hz Main power: 50Hz Audio signal: 20Hz Wifi signal: 2.4Ghz GPS signal L1 band: 1575.42Mhz GLONASS L1 signal: 1602Mhz • Sinusoids’ important because signals can be represented as a sum of sinusoids. Response to sinusoids of various frequencies -- aka frequency response -- tells s a lot about the system.
17. 17. Sinusoidal signal: •
18. 18. Sinusoidal response of RC Network
19. 19. Usual approach Current flow out of node i equal 0: That was easy!
20. 20. How to solve DE • Solving differential equations is actually quite easy –commonly guesswork and applying patterns
21. 21. • (Homogeneous solution for any linear constantcoefficient ODE is always of this form.)
22. 22. • It WORK! But trig. Nightmare! (is there an easier way to find Vp?)
23. 23. 3. The total solution:
24. 24. Sinusoidal Steady State (SSS) In AC Power, we are usually interested only in the particular solution for sinusoids, i.e. after transients have died.
25. 25. Phasor Analysis “I have found the equation that will enable us to transmit electricity through alternating current over thousands of miles. I have reduced it to a simple problem in algebra.” Charles Proteus Steinmetz, 1893 The use of complex numbers to solve ac circuit problems was first done by German-Austrian mathematician and electrical engineer Charles Proteus Steinmetz in a paper presented in 1893. He is noted also for the laws of hysteresis and for his work in manufactured lighting. Steinmetz was born in Breslau, Germany, the son of a government railway worker. He was deformed from birth and lost his mother when he was 1 year old, but this did not keep him from becoming a scientific genius. Just as his work on hysteresis later attracted the attention of the scientific community, his political activities while he was at the University at Breslau attracted the police. He was forced to flee the country just as he had finished the work for his doctorate, which he never received. He did electrical research in the United States, primarily with the General Electric Company. His paper on complex numbers revolutionized the analysis of ac circuits, although it was said at the time that no one but Steinmetz understood the method. In 1897 he also published the first book to reduce ac calculations to a science.
26. 26. Phasor: A useful way to think about sine wave • A phase vector, or phasor is a complex number that represents the amplitude and phase of a sinusoidal signal whose amplitude, frequency, and phase are timeinvariant. • Frequency is common to all signal in our analysis. Phasor allow this common to be factored out, leaving just the amplitude and phase features. The result is that trigonometry reduces to algebra, and linear differential equations become algebraic ones. • Phasor is a complex number. So it can be represent in one of the three forms of complex number: Rectangular, Polar, Exponential. A phasor can be considered a vector rotating about the origin in a complex plane The sum of phasors as addition of rotating vectors
27. 27. Time Domain Vs Phasor Domain representation of sinusoidal signals Time domain Phasor domain
28. 28. Example: Convert SS between Time Domain & Phasor Domain Time domain Phasor domain
29. 29. Sinusoidal Signals: Arithmetic Comparisons Time domain ( )= − Adding & Scaling trigonometry Derivative and Integral of sin &cos Phasor domain = + Adding & scaling complex numbers
30. 30. Example: Solving DE using Phasor Domain Find i(t): di 4i  8 idt  3  50 cos(2t  75) dt - Convert to Phasor domain - Solve algebraic equation - Convert back to Time domain Answer: i(t) = 4.642cos(2t + 143.2o) A
31. 31. Voltage / Current relationships Time domain R L C Complicated! Phasor domain
32. 32. The Impedance Model in Phasor domain
33. 33. Back to the RC example… Time domain DONE! Phasor domain Calculator
34. 34. Example Time domain Phasor domain
35. 35. Another example of Sinusoidal Steady state response (SSS) Remember, we want only the Steady-state response to sinusoid: SSS SSS: Sinusoidal Steady-state response Calculator
36. 36. Solving AC Circuits The Easy Way
37. 37. Impedance and Admittance • The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms Ω. V VM  v VM Z   ( v   i ) | Z |  z  R  jX I I M  i I M where R = Re(Z) is the resistance and X = Im(Z) is the reactance. Positive X is for L and negative X is for C. • The admittance Y is the reciprocal of impedance, measured in siemens (S). 1 I Y  Z V
38. 38. Series Impedances
39. 39. Parallel Impedance
40. 40. Wye – Delta conversion
41. 41. Impedance example
42. 42. Impedance example
43. 43. Impedance example
44. 44. Basic circuit example
45. 45. Basic example
46. 46. AC Equivalent Circuits Thévenin and Norton equivalent circuits apply in AC analysis • Equivalent voltage/current is complex and frequency dependent I Thévenin Equivalent Source + V – Load Norton Equivalent I I VT(jω) + – ZT IN(jω) + V – Load + ZN V – Load
47. 47. Thevenin and Norton Transformation Thevenin transform Norton transform
48. 48. Computation of Thévenin and Norton Impedances: 1. 2. Remove the load (open circuit at load terminal) Zero all independent sources   3. Voltage sources Current sources short circuit (v = 0) open circuit (i = 0) Compute equivalent impedance across load terminals (with load removed) Z3 Z1 + – Vs(jω) ZL Z2 Z3 Z1 a Z4 a ZT Z2 Z4 b NB: same procedure as equivalent resistance b
49. 49. Computing Thévenin voltage: 1. 2. 3.  4. Remove the load (open circuit at load terminals) Define the open-circuit voltage (Voc) across the load terminals Chose a network analysis method to find Voc node, mesh, superposition, etc. Thévenin voltage VT = Voc Z3 Z1 + – Vs(jω) + – Vs(jω) Z2 Z4 b Z3 Z1 a Z2 Z4 a + VT – b
50. 50. Computing Norton current: 1. 2. 3.  4. Replace the load with a short circuit Define the short-circuit current (Isc) across the load terminals Chose a network analysis method to find Isc node, mesh, superposition, etc. Norton current IN = Isc Z3 Z1 + – Vs(jω) + – Vs(jω) Z2 Z4 b Z3 Z1 a a IN Z2 Z4 b
51. 51. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF Rs vs(t) + ~ – L C RL + vL –
52. 52. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF Rs 1. L Note frequencies of AC sources Only one AC source: ω = 103 rad/s vs(t) + ~ – C RL + vL –
53. 53. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF Rs vs(t) + ~ – 1. 2. L Note frequencies of AC sources Convert to phasor domain ZL Zs C RL + vL – Vs(jω) + ~ – ZC ZLD
54. 54. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF ZL ZL Zs Vs(jω) + ~ – Zs ZC ZLD 1. 2. 3. Note frequencies of AC sources Convert to phasor domain Find ZT • Remove load & zero sources Z T  Z S  Z C || Z L ZC  RS  ( jL)(1 / jC ) ( jL)  (1 / jC )  RS  j L 1   2 LC  50  j 65.414  82.330.9182
55. 55. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF ZL ZL Zs Vs(jω) + ~ – 4. Zs Vs(jω) ZC ZLD + ~ – ZC 1. 2. 3. + VT(jω) – Note frequencies of AC sources Convert to phasor domain Find ZT • Remove load & zero sources Find VT(jω) • Remove load NB: Since no current flows in the circuit once the load is removed: VT  VS ZT  82 .330.9182
56. 56. • Example: find the Thévenin equivalent • ω = 103 rads/s, Rs = 50Ω, RL = 50Ω, L = 10mH, C = 0.1uF ZT ZL Zs VT(jω) Vs(jω) + ~ – + ~ – ZC ZLD ZLD VT  VS ZT  82 .330.9182
57. 57. Thevenin and Norton Equivalent Circuits
58. 58. Thevenin and Norton Equivalent Circuits
59. 59. Thevenin and Norton Equivalent Circuits -j1 + + j2 AC 60 V -j1 -j1 + VOC - AC 20 A 60 V - - 20 A I0 VOC  6  2(1  j )  8  j 2 -j1 8  j2 8  j 2 8.246  14.04 I0    1  j1  j 2  2 3  j1 3.16218.43 I 0  2.608  32.47 + 8  j2 j2 AC - I0 ZTH  1  j1
60. 60. Obtain current Io using Norton’s theorem at terminals a-b.
61. 61. Obtain the Thevenin equivalent circuit at terminals a-b:
62. 62. Source Transformation
63. 63. Source Transformation
64. 64. Mesh Analysis
65. 65. Nodal Analysis
66. 66. Super position for AC Circuits • Usual procedures for DC circuits apply. • However, phasor transformation must be carefully carried out if the circuit has sources operating at different frequencies • A different phasor circuit for each source frequency because impedance is a frequency-dependent quantity. • For sources of different frequencies, the total response must be added in the time domain • DO NOT ADD INDIVIDUAL RESPONSES IN THE PHASOR DOMAIN IF THE SOURCES HAVE DIFFERENT FREQUENCIES.
67. 67. Superposition Find I0.
68. 68. Superposition
69. 69. Circuit example Time domain Phasor domain
70. 70. Circuit example
71. 71. Thank you! End of part 1