Lesson 19 impedance
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Lesson 19 Impedance. 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.

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Lesson 19 impedance

Lesson 19Impedance


Learning objectives
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.


R l and c circuits with sinusoidal excitation

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


The impedance concept
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


Impedance

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 ().


Resistance and sinusoidal ac
Resistance and Sinusoidal AC

  • For a purely resistive circuit, current and voltage are in phase.


Resistors
Resistors

  • For resistors, voltage and current are in phase.


Example problem 1
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


Inductance and sinusoidal ac
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.


Inductive impedance
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:


Inductance and sinusoidal ac1
Inductance and Sinusoidal AC

  • Voltage leads current by 90˚


Inductance
Inductance

  • For inductors, voltage leads current by 90º.


Impedance and ac circuits
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)


Example problem 2
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


Example problem 2 solution
Example Problem 2 solution

vL = 40 sin (ωt + 30˚) V

iL = 120 sin (ωt - 60˚) mA

iL

Notice 90°phase difference!

vL


Example problem 3
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?


Capacitance and sinusoidal ac
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.


Capacitive impedance
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:



Capacitance
Capacitance

  • For capacitors, voltage lags current by 90º.


Example problem 4
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


Example problem 5
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


Eli the ice man
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.


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