complex impedances sinusoidal steady state analysis elec 308 elements of electrical engineering dr....

Complex ImpedancesSinusoidal Steady State Analysis

ELEC 308Elements of Electrical Engineering

Dr. Ron Hayne

Images Courtesy of Allan Hambley and Prentice-Hall

Complex Impedances

ELEC 308 2

Inductance and Capacitance represented as Complex Numbers

Inductance

ELEC 308 3

Consider an inductance in which the current is a sinusoid given by

iL t Im sin t IL Im 90o The voltage across an inductance is

vL t LdiL t

dtLIm cos t VL LIm Vm

Note : The current LAGS the voltage for a pure inductance.The voltage can be written as

VL LIm L90o Im 90o jL IL So we have Ohm's Law in phasor form: VL ZLIL

where ZL jL L90o is the impedance of the inductance.

Inductance

ELEC 308 4

Impedance

Ohm’s Law in phasor form: Phasor voltage equals impedances times the

phasor current Impedance is COMPLEX, in general

Can be strictly REAL Impedance = Resistance

Can be strictly IMAGINARY Impedance = ReactanceBoth inductances and capacitances

ELEC 308 5

Capacitance

ELEC 308 6

Consider an capacitance where the voltage across it is given by

vC t Vm sin t VC Vm 90o The current through the capacitance is

iC t CdvC t

dtCVm cos t IC CVm Im

Note : The current LEADS the voltage for a pure capacitance.The voltage can be written as

VC Im

C 90o Im

C90o IC

jC j 1

CIC

So we have Ohm's Law in phasor form: VC ZCIC

where ZC 1

jC

1C

90o is the impedance of the capacitance.

Capacitance

ELEC 308 7

Resistance The phasors are related by

VR = RIR

ELEC 308 8

Exercise 5.7

ELEC 308 9

diagram.phasor theDraw

tage.phasor vol current,phasor e,capacitanc theof impedance theFind

e.capacitanc F-100 a to applied is 200cos100 A voltage

ttvC

Exercise 5.8

ELEC 308 10

diagram.phasor theDraw

tage.phasor vol current,phasor ,resistance theof impedance theFind

.resistance -50 a to applied is 200cos100 A voltage

ttvR

Steady-State Circuit Analysis

Circuit Analysis Using Phasors and Impedances1. Replace the time descriptions of voltage and

current sources with corresponding phasors.All of the sources must have the SAME frequency!

2. Replace inductances, capacitances, and resistances with their corresponding impedances.

3. Analyze the circuit using any of the techniques from Chapters 1 and 2 by performing the calculations with complex arithmetic.

ELEC 308 11

Example 5.4

Find the steady-state current for the circuit shown below. Also, find the phasor voltage across each element and construct a phasor diagram.

ELEC 308 12

Phasor Diagram

ELEC 308 13

Example 5.5 Series and Parallel Combinations of Complex Impedances Find the voltage vc(t) in steady state. Find the phasor current

through each element, and construct a phasor diagram showing the currents and source voltage.

ELEC 308 14

Phasor Diagram

ELEC 308 15

Exercise 5.9

Find i(t) in the circuit below. What is the phase relationship between vs(t) and i(t)?

ELEC 308 16

Exercise 5.9

Find i(t) in the circuit below. What is the phase relationship between vs(t) and i(t)?

ELEC 308 17

Exercise 5.10

Find the phasor voltage and current for each circuit element.

ELEC 308 18

Summary

Complex Impedances Inductance Capacitance Resistance

Sinusoidal Steady State Analysis Ohm’s Law KVL (Mesh-Current Analysis) KCL (Node-Voltage Analysis)

ELEC 308 19