Complex ImpedancesSinusoidal Steady State Analysis

ELEC 308Elements of Electrical Engineering

Dr. Ron Hayne

Images Courtesy of Allan Hambley and Prentice-Hall

Complex Impedances

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Inductance and Capacitance represented as Complex Numbers

Inductance

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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

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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

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Capacitance

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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

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Resistance The phasors are related by

VR = RIR

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Exercise 5.7

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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

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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.

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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.

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Phasor Diagram

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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.

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Phasor Diagram

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Exercise 5.9

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

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Exercise 5.9

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

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Exercise 5.10

Find the phasor voltage and current for each circuit element.

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Summary

Complex Impedances Inductance Capacitance Resistance

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

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