1. 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. 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 Thevenin and Norton equivalents.

3. 3 Table of Phasor Transforms 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. ComponentValue (time domain – NO j’s) Transform (phasor domain – NO t’s) Voltages Currents Resistors Inductors Capacitors Do not mix the domains! Notation important! Impedance for passive elements!

4. 4 t=0 Corresponds to the time dependent voltage changes Graphical Correlation Between Time Dependent Signals and Their Phasors At t=0  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=re j   r/  V=rcos(  t+  )+jrsin(  t+  )

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

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

7. 7 Solve Circuits in the Phasor Domain 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, Z C, and Z L.

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

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

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

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

12. 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, where I m and  are the unknowns. Magnitudes and phases of both sides have to be equal.

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

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

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

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

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

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

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

20. 20 Simplify the Circuit (Still in the Phasor Domain) Now, we need to solve for V a,m. We get Next, we note that we can get V x,m from V a,m by using the complex version of the voltage divider rule, since Z R2 and Z C2 are in series.

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

22. 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,