1 / 22

ECE 3336 Introduction to Circuits & Electronics

ECE 3336 Introduction to Circuits & Electronics. Note Set #10 Phasors Analysis. Fall 2012, TUE&TH 4:00-5:30 pm Dr. Wanda Wosik. Review of Phasor Analysis and Circuit Solutions. A phasor is a transformation of a sinusoidal voltage or current.

hyacinth-patton
Download Presentation

ECE 3336 Introduction to Circuits & Electronics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ECE 3336 Introduction to Circuits & Electronics Note Set #10 Phasors Analysis Fall 2012, TUE&TH 4:00-5:30 pm Dr. Wanda Wosik

  2. Review of Phasor Analysis and Circuit Solutions A phasor is a transformation of a sinusoidal voltage or current. Using phasor analysis, we can solve for the steady-state solution for circuits that have sinusoidal sources. It means that the frequency will be the same so we only need to find the amplitude and phase. All previous techniques used for DC conditions will be applicable to phasors: • Ohms law • Kirchhoff laws • Node Voltage Method • Mesh Currents • Theveninand Norton equivalents.

  3. Table of Phasor Transforms Do not mix the domains! Notation important! Impedance for passive elements! The phasor transforms can be summarized to be used in circuits analyzes. We will use “static” phasors so the frequency will be remembered for V and I.

  4. Graphical Correlation Between Time Dependent Signals and Their Phasors Rotation of the phasor (voltage vector) V with the angular frequency   In general the vector’s length is r (it is the amplitude) so V=a+jb in the rectangular form: Or in the polar form: V=rejr/  V=rcos(t+)+jrsin(t+) At t=0 t=0 Corresponds to the time dependent voltage changes

  5. Impedances Represented by Complex Numbers Current lagging voltage by 90° Current leading voltage by 90°

  6. Inductance Capacitance Current lagging voltage by 90° Current leading voltage by 90° For resistance R both vectors VR(jt) and IR (jt) are the same and there is no phase shift!

  7. Solve Circuits in the PhasorDomain Use Known Methods Replace all elements defined in the time domain v(t), i(t), R, C, and L by the corresponding elements in the frequency domain V(j), I(j), R, ZC, and ZL.

  8. Equivalent Circuits; Thevenin and Norton Equivalent All elements here are in the Phasor Domain

  9. Use Equivalent Admittances for Parallel Connections and Impedances for the Series Connections.

  10. Previous Example Solution: find i(t) The circuit here has a sinusoidal source. What is the steady state value for the current i(t)? Phasor Domain diagram. Solution: We use the phasor analysis technique. The first step is to transform the problem into thephasor domain. Note time does not appear in this diagram.

  11. Previous Example Solution Phasors Used in the Transformed Circuit Still looking for the steady state value for the current i(t)? Phasor Domain diagram. We replace the phasors with their complex numbers, where Im and q are the values we want, specifically, the magnitude and phase of the current.

  12. Previous Example Solution Phasors Bring Back Ohm’s Law Two impedances are in series. We can combine them in the same way we would combine resistances. We can then write the complex version of Ohm’s Law, Magnitudes and phases of both sides have to be equal. where Im and q are the unknowns.

  13. Finding Phasor Magnitude and Phase To find thesteady state current i(t) we inverse transform the current phasor. We find the amplitude(Im) and the phaseof i(t). reminder Magnitude Phase

  14. Previous Example Solution Inverse Transformation of Phasor Gives i(t) The complete expression for the steady state value of the current i(t) will now be calculated as iss(t). The inverse phasor transformgive us the solution in time domain

  15. Another Example What is the steady state value for the voltage vX(t)?

  16. Phasor Transformation of all Components All components have been transformed to the phasor domain, including the current, iX, that the dependent source depends on.

  17. Numerical Solution: Use NVM There are onlytwo essential nodes, so we will write the node-voltage equations,

  18. Solution in the Phasor Domain Now, we can substitute Ix,m back into this equation, and we get

  19. Solve Equations We can solve. We collect terms on each side, and get

  20. Now, we need to solve for Va,m. We get Simplify the Circuit (Still in the Phasor Domain) Next, we note that we can get Vx,m from Va,m by using the complex version of the voltage divider rule, since ZR2 and ZC2 are in series.

  21. Using the complex version of the voltage divider rule, we have Find the Complete Phasor Note: there is no information about the frequency in this expression. But we remember!

  22. Inverse Transformation Gives v(t) The final step is to inverse transform. We need to remember that the frequency was 50[rad/s], and we can write,

More Related