Leading & lagging phases Circuit element phasor relations Series & parallel impedance Examples Lecture 19. AC Steady State Analysis 1

2 Leading and Lagging Phase x 1 (t) leads x 2 (t) by  -  x 2 (t) lags x 1 (t) by  -  On the following plot, which signals lead and which signals lag?

3 Phase Green leads blue and red Red lags blue and green

4 Circuit Element Phasor Relations ElementV/I RelationPhasor RelationPhase CapacitorI = C dV/dtI = j ω C V = ω C V  90° I leads V by 90° InductorV = L dI/dtV = j ω L I = ω L I  90° V leads I by 90° ResistorV = I RV = R I = R I  0° V and I are in-phase

5 Z eq Series Impedance Z1Z1 Z3Z3 Z2Z2

6 Parallel Impedance Z3Z3 Z1Z1 Z2Z2 Z eq

7 Example - 1F1F VCVC + - 2mA  40  1k  VRVR V I = 2mA  40  V R = 2V  40  V C = 5.31V  -50  V = 5.67V  

8 Phasor Diagram Real Axis Imaginary Axis VRVR VCVC V

9 Example 20k  + - 1F1F10V  0  VCVC + -  = 377, Find V C

10 Example (cont’d) Now use the voltage divider to find V C : Z c = 2.65k  -90 

11 What happens when  changes? 20k  + - 1F1F10V  0  VCVC + -  = 10 Find V C

12 Example: Low Pass Filter Find v(t) for  =2  k  0.1  F 5mA  0  + - V

13 Class Examples P8-6, P8-9, P8-8