Transcript Introduction

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
V Vm0o
Vm
I 
 Im   where Im 
Z
Z 
Z
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RESISTIVE Load
 Pure Resistance

Current in Phase
with Voltage
Z R  R0
vt   Vm cost 
it   I m cost 
pt   vt it   Vm I m cos2 t 
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INDUCTIVE Load
 Pure Inductance


Current Lags Voltage
Reactive power flows from
source to load; Pavg = ______
Z L  jL  L90
vt   Vm cost 


i t   I m cos t  90  I m sin t 
pt   vt i t   Vm I m cost sin t 
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CAPACITIVE Load
 Pure Capacitance


Current Leads Voltage
Reactive power flows from
source to load; Pavg = ______
1
1
ZC 

  90
jC  C
vt   Vm cost 


i t   I m cos t  90   I m sin t 
p t   vt i t   Vm I m cost sin t 
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Importance of Reactive Power
 No average power is consumed by a pure
energy-storage element
 Reactive power still important

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
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
Z RLC  Z RLC 
vt   Vm cost 
i t   I m cost   
Vm I m
V I
cos   m m cos 
2
2 2
 Vrms I rms cos 
P
Unitsof average(REAL)power P are in watts(W).
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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
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
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
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
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S = VrmsIrms
Units are volt-amperes (VA)
Can be determined from real and reactive
powers:
S  P Q
2
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Units
 Units indicate whether quantity is power,
reactive power, or apparent power
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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

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
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Real power, P
Reactive power, Q
Apparent power VrmsIrms
Power angle, θ
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Additional Power Relationships
PI
2
rms
R
2
rms
V
P
R
2
Q  I rms
X
2
Vrms
Q
X
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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


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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
Zload  Zt
Reactance of load CANCELS internal reactance of twoterminal circuit
 Second 
case: Load is pure resistance

Load impedance for max. power transfer is
Zload  Rload  Zt
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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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