Power in AC Circuits

ELEC 308

Elements of Electrical Engineering

Dr. Ron Hayne

Images Courtesy of Allan Hambley and Prentice-Hall

Power Delivered to a Load

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I V

Z

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ZIm where Im

Vm

Z

RESISTIVE Load

Pure Resistance Current in Phase

with Voltage

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tIti

tVtv

RZ

mm

m

m

R

2cos

cos

cos

0

INDUCTIVE Load

Pure Inductance Current Lags Voltage Reactive power flows from

source to load; Pavg = ______

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tItIti

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LLjZ

mm

mm

m

L

sincos

sin90cos

cos

90

CAPACITIVE Load

Pure Capacitance Current Leads Voltage Reactive power flows from

source to load; Pavg = ______

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tItIti

tVtv

CCjZ

mm

mm

m

C

sincos

sin90cos

cos

9011

Importance of Reactive Power

No average power is consumed by a pure energy-storage element

Reactive power still important Transmission lines, transformers, fuses, etc. must be

able to withstand the current associated with reactive power

Possible to have loads that draw LARGE currents, even though little average power is consumed

Electric power companies STILL charge for reactive power (at a lower rate), as well as total energy delivered

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Real Power

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tVtv

ZZ

m

m

RLCRLC

cos

cos

(W). in watts are power (REAL) average of Units

cos

cos22

cos2

P

IV

IVIVP

rmsrms

mmmm

Power Factor

The term cos(θ) is called the power factor: PF = cos(θ)

The power angle θ is taken as the phase of the voltage θv minus the phase of the current θi

θ = θv-θi

Current lags voltage = POSITIVE power angle Current leads voltage = NEGATIVE power angle

Sometimes stated as a PERCENTAGE e.g. 90% lagging => cos(θ) = 0.9 and Curr. lags Volt.

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Reactive Power

Capacitance voltage increasing/decreasing Energy flowing into/out of capacitance

Inductance current increasing/decreasing Energy flowing into/out of inductance

Instantaneous power can be VERY large Average power (and net energy) is still _________

Reactive power is peak instantaneous power associated with energy-storage elements Q = VrmsIrmssin(θ) Units are VARs (Volt Amperes Reactive)

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Apparent Power

Apparent power is the product of the effective voltage and effect current S = VrmsIrms

Units are volt-amperes (VA) Can be determined from real and reactive

powers:

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22 QPS

Units

Units indicate whether quantity is power, reactive power, or apparent power 5-kW load means that P = 5 kW 5-kVA load means that VrmsIrms = 5 kVA 5-kVAR load means that Q = 5 kVAR

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Power Triangle

Demonstrates relationships between Real power, P Reactive power, Q Apparent power VrmsIrms Power angle, θ

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Additional Power Relationships

X

VQ

XIQ

R

VP

RIP

rms

rms

rms

rms

2

2

2

2

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Using Power Triangles

Find the power, reactive power, and power factor for the source in the circuit below.

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Using Power Triangles

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Using Power Triangles

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Power Factor Correction

In heavy industry Many loads are partly inductive = large amounts of

reactive power flow Causes higher current in transmission system

Energy rates charged to industry depend on the power factor Higher charges for energy delivered at lower power

factors Advantageous to choose loads that operate at near

unity power factor Common approach is to place capacitors in parallel

with an inductive load to increase the power factor

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Thévenin and Norton

Use same techniques for circuits with impedances as we did for resistive circuits

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Maximum Power Transfer

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Maximum Power Transfer

Maximum power transferred achieved by maximizing the current

First case: Load is complex impedance Load impedance for max. power transfer is

Reactance of load CANCELS internal reactance of two-terminal circuit

Second case: Load is pure resistance Load impedance for max. power transfer is

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Z load Zt

Z load Rload Z t

Maximum Power Transfer

Determine the maximum power that can be delivered to a load by the two-terminal resistance below if The load can be any complex impedance The load must be a pure resistance

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Summary

AC Power Real Power Reactive Power Apparent Power Power Factor

Thevenin and Norton Maximum Power Transfer

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