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Power in AC CircuitsChapter 17
Active PowerIn dc circuits, for example, the only power
relationship you encounter is P =VI watts.
This is referred to as real power or active power and is the power that does useful work such as light a lamp, power a heater, run an electric motor, and so on.
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Reactive Power
In ac circuits, you also encounter this type of power. For ac circuits that contain reactive elements however, (i.e., inductance or capacitance), a second component of power also exists.
This component, termed reactive power, represents energy that oscillates back and forth throughout the system.
For example, during the buildup of current in an inductance, energy flows from the power source to the inductance to create its magnetic field. When the magnetic field collapses, this energy is returned to the circuit.
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Reactive Power This movement of energy in and out of the inductance
constitutes a flow of power. However, since it flows first in one direction, then in the other, it contributes nothing to the average flow of power from the source to the load.
For this reason, reactive power is sometimes referred to as wattless power. (A similar situation exists regarding power flow to and from the electric field of a capacitor.)
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Apparent PowerFor a circuit that contains resistive as
well as reactive elements, some energy is dissipated while the remainder is shuttled back and forth as described above; thus, both active and reactive components of power are present.
This combination of real and reactive power is termed apparent power.
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Active Power
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Instantaneous power to a load is p = v • iIn an ac circuit
p may be positive sometimes and negative other times
Average value of the power, PReal power
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Active Power
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A positive value for p means that power transfer is in the direction of the reference arrow, while a negative value means that it is in the opposite direction.
Thus, during positive parts of the power cycle, power flows from the source to the load, while during negative parts, it flows out of the load back into the circuit.
Active Power Since p represents the power flowing to the load, its average will
be the average power to the load. Denote this average by the letter P.
If P is positive, then, on average, more power flows to the load than is returned from it. (If P is zero, all power sent to the load is returned.)
Thus, if P has a positive value, it represents the power that is really dissipated by the load. For this reason, P is called real power. In modern terminology, real power is also called active power.
Thus, active power is the average value of the instantaneous power, and the terms real power, active power, and average power mean the same thing.
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Active Power
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Average value of instantaneous power, real power, active power, and average power mean the same thing
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Reactive PowerConsider again Figure 17–2. During the
intervals that p is negative, power is being returned from the load. (This can only happen if the load contains reactive elements: L or C.)
The portion of power that flows into the load then back out is called reactive power. Since it first flows one way then the other, its average value is zero; thus, reactive power contributes nothing to the average power to the load.
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Reactive Power Although reactive power does no useful work, it cannot be
ignored.
Extra current is required to create reactive power, and this current must be supplied by the source; this also means that conductors, circuit breakers, switches, transformers, and other equipment must be made physically larger to handle the extra current.
This increases the cost of a system. At this point, it should be noted that real power and reactive power do not exist as separate entities. Rather, they are components of the power waveform shown in Figure 17–2.
However, as you will see, we are able to conceptually separate them for purposes of analysis.
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Reactive Power
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During times when p is negative, power is being returned from load
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
Average value of reactive power is zero
Power to a Resistive Load
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Power to a Resistive Load
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tIVp
tIVp
tItVvip
mm
mm
mm
212
2
cos
sin
sinsin
- Current is in phase with voltage
Power to a Resistive Load
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p is always positive (except when zero)Power flows only from source to load
Power is absorbed by the loadPower 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
P = VmIm/2If V and I are in RMS values
Then P = VI
Average Power
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Also, P = I2R and P = V2/RActive 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 phaseAverage power to an inductance over a full
cycle is zeroThere are no power losses associated with
a pure inductance
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Power to an Inductive Load
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Power that flows into and out of a pure inductance is reactive power only
Note that during the first quarter-cycle, p is positive and hence power flows to the inductance, while during the second quarter-cycle, p is negative and all power transferred to the inductance during the first quarter-cycle flows back out.
Similarly for the third and fourth quarter-cycles. Thus, the average power to an inductance over a full cycle is zero, i.e., there are no power losses associated with a pure inductance. Consequently, PL 0 W and the only power flowing in the circuit is reactive power.
Power to an Inductive Load
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pL = VI sin 2t (V and I rms values)
Product VI is the reactive power, QL
QL = VI = I2XL = V2/XL
Units are VARs
Power to an Inductive Load
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VAR means Volt-Amperes-Reactive Inductive reactive power is represented as positive
Because it represents “power” that alternately flows into, then out of the inductance, QL contributes nothing to the average power to the load and, as noted earlier, is sometimes referred to as wattless power.
As you will soon see, however, reactive power is of major concern in the operation of electrical power systems.
Power to a Capacitive Load
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Voltage and current are 90°out of phaseAverage power over one complete cycle is
equal to zeroThere are no power losses associated with a
pure capacitance
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Power to a Capacitive Load
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Power that flows into and out of a pure capacitance is reactive power only
This power cycle is 180°out of phase with the inductive cycle
Power to a Capacitive Load
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pC = –VI sin 2t
QC = VI
QC = I2XC = V2/XC
Capacitive reactive power is represented as negative
Units are VARs
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Power in More Complex Circuits
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It does not matter how a circuit or system is connectedSum of the power is found by summing
individual powersTotal 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’sInductive powers are positiveCapacitive powers are negative
Example
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Example
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Example
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Example…contd…
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Apparent Power
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Power to a load is VIIf load has both resistance and reactance
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, SS = VI = I2Z = V2/ZUnits are volt-amperes (VA)
Apparent Power
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Relationship Between P,Q, and S
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P, Q, and S are related by the “power triangle”
22 QPS
SQ
P
Active and Reactive Power Equations
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P = VI cos = S cos Q = VI sin = S sin V and I are RMS values is the phase angle between V and IQ 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
Fp = P/S = cos Angle is angle between voltage and current
Power Factor
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For pure resistance = 0°For inductance, = 90°For capacitance, = -90°For a circuit containing a mixture, is
somewhere between 0° and 90°
Power Factor
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Unity power factorFor a purely resistive circuit, the power factor
will be oneFor load containing resistance and inductance
Power factor will be less than one and laggingCurrent lags the voltage
Power Factor
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For a circuit containing resistance and capacitanceFp is less than one and is leading
Why Equipment Is Rated in VA
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A highly reactive loadMay seem to require a small amount of
power while requiring a large currentEquipment is rated in VA to prevent
overloading the circuit
VA Ratings We now examine why electrical apparatus is rated in VA instead of
watts. Consider Figure 17–18.
Assume that the generator is rated at 600V, 120 kVA. This means that it is capable of supplying I 120 kVA/600V 200 A. In (a), the generator is supplying a purely resistive load with 120 kW.
Since S, P for a purely resistive load, S 120 kVA and the generator is supplying its rated kVA. In (b), the generator is supplying a load with P 120 kW as before, but Q 160 kVAR. Its apparent power is therefore S 200 kVA, which means that the generator current is I 200 kVA/600 V 333.3 A.
Even though it is supplying the same power as in (a), the generator is now greatly overloaded, and damage may result as indicated in (b).
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Why Equipment Is Rated in VA
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Size of electrical apparatus required by a loadGoverned by its VA requirements
Power Factor Correction
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A load with a small power factor can draw a large current
Can be alleviated byCancelling some or all reactive components of
power by adding reactance of opposite type to the circuitThis is power factor correction
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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
Meter consists of Current-sensing circuitVoltage-sensing circuitMultiplier circuitAveraging 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 frequenciesResistance of a circuit may change
Reff = P/I2
Anything that affects P will affect resistance
Effective Resistance
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Changing magnetic fields may set up eddy currents in conductorsThese cause power losses that affect
effective resistance
Effective Resistance
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Ferromagnetic materialsPower losses due to hysteresis effects
Magnetically induced voltages created by a changing magnetic field cause a non-uniform current called a skin effectCauses an increase in resistanceEnergy escapes due to radiation resistance
