Chapter 17
Active Power
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Instantaneous power to a load is p = v
• i |
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In an ac circuit |
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p may be positive sometimes and
negative other times |
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Average value of the power, P |
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Real power |
Active Power
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Average value of instantaneous
power, real power, active power, and
average power mean the same thing |
Reactive Power
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During times when p is negative, power
is being returned from load |
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This can happen for inductive or
capacitive loads |
Reactive Power
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Power that flows into these loads and
back out is called the reactive power |
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Average value of reactive power is zero |
Power to a Resistive Load
Power to a Resistive Load
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p is always positive (except when zero) |
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Power flows only from source to load |
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Power is absorbed by the load |
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Power to a pure resistance consists of
active power only |
Average Power
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Average value of power is halfway
between zero and peak value of VmIm |
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P = VmIm/2 |
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If V and I are in RMS values |
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Then P = VI |
Average Power
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Also, P = I2R and P = V2/R |
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Active power relationships for
resistive circuits are the same for ac as for dc |
Power to an Inductive
Load
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Voltage and current of an inductor are
90°out of phase |
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Average power to an inductance over a
full cycle is zero |
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There are no power losses associated
with a pure inductance |
Power to an Inductive
Load
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Power that flows into and out of a pure
inductance is reactive power only |
Power to an Inductive
Load
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pL = VI sin 2wt (V and I rms values) |
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Product VI is the reactive power, QL |
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QL = VI = I2XL
= V2/XL |
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Units are VARs |
Power to an Inductive
Load
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VAR means Volt-Amperes-Reactive |
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Inductive reactive power is represented
as positive |
Power to a Capacitive
Load
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Voltage and current are 90°out of phase |
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Average power over one complete cycle
is equal to zero |
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There are no power losses associated
with a pure capacitance |
Power to a Capacitive
Load
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Power that flows into and out of a pure
capacitance is reactive power only |
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This power cycle is 180°out of phase
with the inductive cycle |
Power to a Capacitive
Load
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pC = –VI sin 2wt |
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QC = VI |
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QC = I2XC
= V2/XC |
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Capacitive reactive power is
represented as negative |
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Units are VARs |
Power in More Complex
Circuits
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It does not matter how a circuit or
system is connected |
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Sum of the power is found by summing
individual powers |
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Total real power P is found by summing
each of the individual real powers |
Power in More Complex
Circuits
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Total Reactive power Q is found by
summing individual Q’s |
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Inductive powers are positive |
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Capacitive powers are negative |
Apparent Power
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Power to a load is VI |
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If load has both resistance and
reactance |
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Product is neither the real power nor
the reactive power, but a combination of both |
Apparent Power
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This is called the apparent power, S |
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S = VI = I2Z = V2/Z |
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Units are volt-amperes (VA) |
Relationship Between P,Q,
and S
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P, Q, and S are related by the “power
triangle” |
Active and Reactive Power
Equations
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P = VI cos q = S
cos q |
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Q = VI
sin q = S sin q |
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V and I
are RMS values |
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q is
the phase angle between V and I |
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Q is positive for inductive circuits
and negative for capacitive circuits |
Power Factor
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Ratio of real power to apparent power
is called the power factor, Fp |
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Fp = P/S = cos q |
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Angle q is angle between voltage and current |
Power Factor
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For pure resistance q = 0° |
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For inductance, q = 90° |
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For capacitance, q = -90° |
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For a circuit containing a mixture, q is somewhere between 0° and ±90° |
Power Factor
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Unity power factor |
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For a purely resistive circuit, the
power factor will be one |
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For load containing resistance and
inductance |
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Power factor will be less than one and
lagging |
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Current lags the voltage |
Power Factor
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For a circuit containing resistance and
capacitance |
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Fp is less than one and is
leading |
Why Equipment Is Rated in
VA
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A highly reactive load |
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May seem to require a small amount of
power while requiring a large current |
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Equipment is rated in VA to prevent
overloading the circuit |
Why Equipment Is Rated in
VA
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Size of electrical apparatus required
by a load |
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Governed by its VA requirements |
Power Factor Correction
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A load with a small power factor can
draw a large current |
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Can be alleviated by |
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Cancelling some or all reactive
components of power by adding reactance of opposite type to the circuit |
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This is power factor correction |
Power Factor Correction
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Industrial customers may pay a penalty
for low power factors due to large currents required for highly reactive
loads |
AC Power Measurement
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To measure power in an ac circuit you
need a wattmeter |
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Meter consists of |
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Current-sensing circuit |
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Voltage-sensing circuit |
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Multiplier circuit |
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Averaging circuit |
AC Power Measurement
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This will measure load voltage and
current and find the product and the angle between these |
Effective Resistance
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At high frequencies |
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Resistance of a circuit may change |
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Reff = P/I2 |
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Anything that affects P will affect
resistance |
Effective Resistance
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Changing magnetic fields may set up
eddy currents in conductors |
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These cause power losses that
affect effective resistance |
Effective Resistance
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Ferromagnetic materials |
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Power losses due to hysteresis effects |
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Magnetically induced voltages created
by a changing magnetic field cause a non-uniform current called a skin effect |
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Causes an increase in resistance |
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Energy escapes due to radiation
resistance |