1. Lesson 19 Impedance

2. Learning Objectives For purely resistive, inductive and capacitive elements define the voltage and current phase differences. Define inductive reactance. Understand the variation of inductive reactance as a function of frequency. Define capacitive reactance. Understand the variation of capacitive reactance as a function of frequency. Define impedance. Graph impedances of purely resistive, inductive and capacitive elements as a function of phase.

3. R, L and C circuits with Sinusoidal Excitation R, L, C have very different voltage-current relationships Sinusoidal (ac) sources are a special case Review

4. The Impedance Concept Impedance ( Z ) is the opposition that a circuit element presents to current in the phasor domain. It is defined Ohm’s law for ac circuits

5. Impedance Impedance is a complex quantity that can be made up of resistance (real part) and reactance (imaginary part). Unit of impedance is ohms (  ).  R X Z

6. Resistance and Sinusoidal AC For a purely resistive circuit, current and voltage are in phase.

7. Resistors For resistors, voltage and current are in phase.

8. Example Problem 1 Two resistors R 1 =10 kΩ and R 2 =12.5 kΩ are in series. If i(t) = 14.7 sin (ωt + 39˚) mA a) Compute V R1 and V R2 b) Compute V T =V R1 + V R2 c) Calculate Z T d) Compare V T to the results of V T =IZ T

9. Inductance and Sinusoidal AC Voltage-Current relationship for an inductor It should be noted that for a purely inductive circuit voltage leads current by 90º.

10. Inductive Impedance Impedance can be written as a complex number (in rectangular or polar form): Since an ideal inductor has no real resistive component, this means the reactance of an inductor is the pure imaginary part:

11. Inductance and Sinusoidal AC Voltage leads current by 90˚

12. Inductance For inductors, voltage leads current by 90º.

13. Since X L =  L = 2  fL, inductive reactance is directly proportional to frequency. Extreme case f = 0 Hz (DC): inductor looks like a short circuit! Variation with Frequency

14. Impedance and AC Circuits Solution technique 1. Transform time domain currents and voltages into phasors 2. Calculate impedances for circuit elements 3. Perform all calculations using complex math 4. Transform resulting phasors back to time domain (if reqd)

15. Example Problem 2 For the inductive circuit: v L = 40 sin (ωt + 30˚) V f = 26.53 kHz L = 2 mH Determine V L and I L Graph v L and i L

16. Example Problem 2 solution v L = 40 sin (ωt + 30˚) V i L = 120 sin (ωt - 60˚) mA vLvL iLiL Notice 90°phase difference!

17. Example Problem 3 For the inductive circuit: v L = 40 sin (ωt + Ө) V i L = 250 sin (ωt + 40˚) μA f = 500 kHz What is L and Ө?

18. Capacitance and Sinusoidal AC Current-voltage relationship for an capacitor It should be noted that, for a purely capacitive circuit current leads voltage by 90º.

19. Capacitive Impedance Impedance can be written as a complex number (in rectangular or polar form): Since a capacitor has no real resistive component, this means the reactance of a capacitor is the pure imaginary part:

20. Capacitance and Sinusoidal AC

21. Capacitance For capacitors, voltage lags current by 90º.

22. Variation with Frequency Since, capacitive reactance is inversely proportional to frequency. Extreme case f = 0 Hz (DC): capacitor looks like an open circuit!

23. Example Problem 4 For the capacitive circuit: v C = 3.6 sin (ωt-50°) V f = 12 kHz C=1.29 uF Determine V C and I C

24. Example Problem 5 For the capacitive circuit: v C = 362 sin (ωt - 33˚) V i C = 94 sin (ωt + 57˚) mA C = 2.2 μF Determine the frequency

25. ELI the ICE man E leads I I leads E When voltage is applied to an inductor, it resists the change of current. The current builds up more slowly, lagging in time and phase. Since the voltage on a capacitor is directly proportional to the charge on it, the current must lead the voltage in time and phase to conduct charge to the capacitor plate and raise the voltage Voltage Inductance Current Voltage Capacitance Current

26. Frequency dependency Inductors Capacitors