* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Example - s3.amazonaws.com
Immunity-aware programming wikipedia , lookup
Electronic engineering wikipedia , lookup
Electromagnetic compatibility wikipedia , lookup
Topology (electrical circuits) wikipedia , lookup
Power inverter wikipedia , lookup
Three-phase electric power wikipedia , lookup
History of electric power transmission wikipedia , lookup
Signal-flow graph wikipedia , lookup
Distribution management system wikipedia , lookup
Schmitt trigger wikipedia , lookup
Voltage regulator wikipedia , lookup
Power electronics wikipedia , lookup
Electrical substation wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Integrated circuit wikipedia , lookup
Flexible electronics wikipedia , lookup
Voltage optimisation wikipedia , lookup
Buck converter wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Stray voltage wikipedia , lookup
Power MOSFET wikipedia , lookup
Surge protector wikipedia , lookup
Alternating current wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Current source wikipedia , lookup
Mains electricity wikipedia , lookup
EGR 272 – Circuit Theory II Lecture #6 Read: Chapter 9 and Appendix B in Electric Circuits, 6th Edition by Nilsson Using Phasors to Add Sinusoids Sinusoidal voltages or currents could be added using various trigonometric identities; however, they are more easily combined using phasors. Example: If v1 = 10cos(200t + 15), v2 = 15cos(200t + -30), and v3 = 8sin(200t), find v4. + V2 _ + + V1 V3 _ _ + V4 _ 1 Lecture #6 EGR 272 – Circuit Theory II sin(wt) or cos(wt)? Since phasor analysis is used to calculate results that are relative to the sources, it generally doesn’t matter whether the sinusoidal source is expressed using sin(wt) or cos(wt). If the sources are expressed using sin(wt), then the results will also be in terms of sin(wt) and if the sources are expressed using cos(wt), then the results will also be in terms of cos(wt). However, the approach for a given circuit containing multiple sources must be consistent – either using cos(wt) or sin(wt). Review of DC Circuit Analysis Techniques Analyzing AC circuits is very similar to analyzing DC resistive circuits. Several examples are presented below which will also serve to review many DC analysis techniques, including: • Source transformations • Mesh equations • Node equations • Superposition • Thevenin’s and Norton’s theorems • Maximum Power Transfer theorem 2 Lecture #6 EGR 272 – Circuit Theory II Source transformations A phasor voltage source with a series impedance may be transformed into a phasor current source with a parallel impedance as illustrated below. The two sources are identical as far as the load is concerned. Notes: • Not all sources can be transformed. Discuss. • The two sources are not equivalent internally. For example, the voltage across Zs is not equivalent to the voltage across Zp. • Dependent sources can be transformed. I Zs Vs + _ I + V + Load Ip Zp _ Converting a real current source to a real voltage source: V s I p Z p and Zs Z p V Load _ Converting a real voltage source to a real current source: Ip Vs and Zs Z p Zs 3 EGR 272 – Circuit Theory II Lecture #6 Example: Solve for the voltage V using source transformations. + + 100cos(20t) V 8H V _ 100 240 500 uF 3sin(20t) _ 4 EGR 272 – Circuit Theory II Lecture #6 Mesh equations: Example: Solve for the voltage V using mesh equations. + 50cos(400t) V _ 10 50 mH 100 uF 30 + 50 uF V _ 5 Lecture #6 EGR 272 – Circuit Theory II Node equations: Example: Solve for the current i(t) using node equations. + 3sin(4t) V1 _ 6 1 F 8 4 i(t) 2H + _ V1 2 6 Lecture #6 EGR 272 – Circuit Theory II Superposition: Superposition can be used to analyze multiple-source AC circuits in a manner very similar to analyzing DC circuits. However, there are two special cases where it is highly recommended that superposition be used: • Circuits that include sources at two or more different frequencies • Circuits that include both DC and AC sources (Note: you could think of DC sources as acting like AC sources with w = 0.) Example 1 (sources with different frequencies): Solve for the voltage V using superposition. + + 2cos(4t) V V _ 4 4 _ 1 F 8 + 3cos(2t) V _ 7 Lecture #6 EGR 272 – Circuit Theory II Example 2 (AC and DC sources): Solve for the voltage V using superposition. + + 6cos(5t) V V _ 2 3 0.2 F 0.8 H + _ 10 V _ 8