Lesson 19 Impedance

1. Lesson 19Impedance

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

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

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 R1=10 kΩ and R2=12.5 kΩ are in series. If i(t) = 14.7 sin (ωt + 39˚) mA • Compute VR1 and VR2 • Compute VT=VR1 + VR2 • Calculate ZT • Compare VT to the results of VT=IZT

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

14. Example Problem 2 For the inductive circuit: vL = 40 sin (ωt + 30˚) V f = 26.53 kHz L = 2 mH Determine VL and IL Graph vL and iL

15. Example Problem 2 solution vL = 40 sin (ωt + 30˚) V iL = 120 sin (ωt - 60˚) mA iL Notice 90°phase difference! vL

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

17. 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.

18. 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:

19. Capacitance and Sinusoidal AC

20. Capacitance • For capacitors, voltage lags current by 90º.

21. Example Problem 4 For the capacitive circuit: vC = 3.6 sin (ωt-50°) V f = 12 kHz C=1.29 uF Determine VC and IC

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

23. ELI the ICE man Current Current Voltage Voltage Inductance Capacitance I leads E E leads I 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 When voltage is applied to an inductor, it resists the change of current. The current builds up more slowly, lagging in time and phase.