Reactive Power
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Reactive Power Compensation
Reactive Power can best be described as the quantity of “unused” power that is developed by reactive
components, such as inductors or capacitors in an AC circuit or system. In a DC circuit, the product of
“volts x amps” gives the power consumed in watts by the circuit. However, while this formula is also true
for purely resistive AC circuits, the situation is slightly more complex in an AC circuits containing reactive
components as this volt-amp product can change with frequency.
In an AC circuit, the product of voltage and current is expressed as volt-amperes (VA) or kilo volt-amperes (kVA)
and is known as Apparent power, symbol S. In a non-inductive purely resistive circuit such as heaters, irons,
kettles and filament bulbs, etc. their reactance is practically zero, and the impedance of the circuit is composed
almost entirely of just resistance.
For an AC resistive circuit, the current and voltage are in-phase and the power at any instant can be found by
multiplying the voltage by the current at that instant, and because of this “in-phase” relationship, the rms values
can be used to find the equivalent DC power or heating effect.
However, if the circuit contains reactive components, the voltage and current waveforms will be “out-of-phase”
by some amount determined by the circuits phase angle. If the phase angle between the voltage and the current is
at its maximum of 90o, the volt-amp product will have equal positive and negative values.
In other words, the reactive circuit returns as much power to the supply as it consumes resulting in the average
power consumed by the circuit being zero, as the same amount of energy keeps flowing alternately from source
to the load and back from load to source.
Since we have a voltage and a current but no power dissipated, the expression of P = IV (rms) is no longer valid
and it therefore follows that the volt-amp product in an AC circuit does not necessarily give the power consumed.
Then in order to determine the “real power”, also called Active power, symbol P consumed by an AC circuit, we
need to account for not only the volt-amp product but also the phase angle difference between the voltage and the
current waveforms given by the equation: VI.cosΦ.
Then we can write the relationship between the apparent power and active or real power as:
Note that power factor (PF) is defined as the ratio between the active power in watts and the apparent power in
volt-amperes and indicates how effectively electrical power is being used. In a non-inductive resistive AC circuit,
the active power will be equal to the apparent power as the fraction of P/S becomes equal to one or unity. A
circuits power factor can be expressed either as a decimal value or as a percentage.
But as well as the active and apparent powers in AC circuits, there is also another power component that is present
whenever there is a phase angle. This component is called Reactive Power (sometimes referred to as imaginary
power) and is expressed in a unit called “volt-amperes reactive”, (VAr), symbol Q and is given by the equation:
VI.sinΦ.
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Reactive power, or VAr, is not really power at all but represents the product of volts and amperes that are out-ofphase with each other. The amount of reactive power present in an AC circuit will depend upon the phase shift or
phase angle between the voltage and the current and just like active power, reactive power is positive when it is
“supplied” and negative when it is “consumed”.
The relationship of the three elements of power, active power, (watts) apparent power, (VA) and reactive power,
(VAr) in an AC circuit can be represented by the three sides of right-angled triangle. This representation is called
a Power Triangle as shown:
Power in an AC Circuit
From the above power triangle we can see that AC circuits supply or consume two kinds of power: active power
and reactive power. Also, active power is never negative, whereas reactive power can be either positive or negative
in value so it is always advantageous to reduce reactive power in order to improve system efficiency.
The main advantage of using AC electrical power distribution is that the supply voltage level can be changed
using transformers, but transformers and induction motors of household appliances, air conditioners and industrial
equipment all consume reactive power which takes up space on the transmission lines since larger conductors and
transformers are required to handle the larger currents which you need to pay for.
Reactive Power Analogy with Beer
In many ways, reactive power can be thought of like the foam head on a pint or glass of beer. You pay the barman
for a full glass of beer but only drink the actual liquid beer which is always less than a full glass.
This is because the head (or froth) of the beer takes up additional wasted space in the glass leaving less room for
the real beer that you consume, and the same idea is true for reactive power.
But for many industrial power applications, reactive power is often useful for an electrical circuit to have. While
the real or active power is the energy supplied to run a motor, heat a home, or illuminate an electric light bulb,
reactive power provides the important function of regulating the voltage thereby helping to move power
effectively through the utility grid and transmission lines to where it is required by the load.
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While reducing reactive power to help improve the power factor and system efficiency is a good thing, one of the
disadvantages of reactive power is that a sufficient quantity of it is required to control the voltage and overcome
the losses in a transmission network. This is because if the electrical network voltage is not high enough, active
power cannot be supplied. But having too much reactive power flowing around in the network can cause excess
heating (I2R losses) and undesirable voltage drops and loss of power along the transmission lines.
Power Factor Correction of Reactive Power
One way to avoid reactive power charges, is to install power factor correction capacitors. Normally residential
customers are charged only for the active power consumed in kilo-watt hours (kWhr) because nearly all residential
and single phase power factor values are essentially the same due to power factor correction capacitors being built
into most domestic appliances by the manufacturer.
Industrial customers, on the other hand, which use 3-phase supplies have widely different power factors, and for
this reason, the electrical utility may have to take the power factors of these industrial customers into account
paying a penalty if their power factor drops below a prescribed value because it costs the utility companies more
to supply industrial customers since larger conductors, larger transformers, larger switchgear, etc, is required to
handle the larger currents.
Generally, for a load with a power factor of less than 0.95 more reactive power is required. For a load with a
power factor value higher than 0.95 is considered good as the power is being consumed more effectively, and a
load with a power factor of 1.0 or unity is considered perfect and does not use any reactive power.
Then we have seen that “apparent power” is a combination of both “reactive power” and “active power”. Active
or real power is a result of a circuit containing resistive components only, while reactive power results from a
circuit containing either capacitive and inductive components. Almost all AC circuits will contain a combination
of these R, L and C components.
Since reactive power takes away from the active power, it must be considered in an electrical system to ensure
that the apparent power supplied is sufficient to supply the load. This is a critical aspect of understanding AC
power sources because the power source must be capable of supplying the necessary volt-amp (VA) power for
any given load.
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Harmonics and Harmonic Frequency
In an AC circuit, a resistance behaves in exactly the same way as it does in a DC circuit. That is, the current
flowing through the resistance is proportional to the voltage across it. This is because a resistor is a linear
device and if the voltage applied to it is a sine wave, the current flowing through it is also a sine wave.
Generally when dealing with alternating voltages and currents in electrical circuits it is assumed that they are pure
and sinusoidal in shape with only one frequency value, called the “fundamental frequency” being present, but this
is not always the case.
In an electrical or electronic device or circuit that has a voltage-current characteristic which is not linear, that is,
the current flowing through it is not proportional to the applied voltage. The alternating waveforms associated
with the device will be different to a greater or lesser extent to those of an ideal sinusoidal waveform. These types
of waveforms are commonly referred to as non-sinusoidal or complex waveforms.
Complex waveforms are generated by common electrical devices such as iron-cored inductors, switching
transformers, electronic ballasts in fluorescent lights and other such heavily inductive loads as well as the output
voltage and current waveforms of AC alternators, generators and other such electrical machines. The result is that
the current waveform may not be sinusoidal even though the voltage waveform is.
Also most electronic power supply switching circuits such as rectifiers, silicon controlled rectifier (SCR’s), power
transistors, power converters and other such solid state switches which cut and chop the power supplies sinusoidal
waveform to control motor power, or to convert the sinusoidal AC supply to DC. Theses switching circuits tend
to draw current only at the peak values of the AC supply and since the switching current waveform is nonsinusoidal the resulting load current is said to contain Harmonics.
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Non-sinusoidal complex waveforms are constructed by “adding” together a series of sine wave frequencies known
as “Harmonics”. Harmonics is the generalised term used to describe the distortion of a sinusoidal waveform by
waveforms of different frequencies.
Then whatever its shape, a complex waveform can be split up mathematically into its individual components
called the fundamental frequency and a number of “harmonic frequencies”. But what do we mean by a
“fundamental frequency”.
Fundamental Frequency
A Fundamental Waveform (or first harmonic) is the sinusoidal waveform that has the supply frequency. The
fundamental is the lowest or base frequency, ƒ on which the complex waveform is built and as such the periodic
time, Τ of the resulting complex waveform will be equal to the periodic time of the fundamental frequency.
Let’s consider the basic fundamental or 1st harmonic AC waveform as shown.
Where: Vmax is the peak value in volts and ƒ is the waveforms frequency in Hertz (Hz).
We can see that a sinusoidal waveform is an alternating voltage (or current), which varies as a sine function of
angle, 2πƒ. The waveforms frequency, ƒ is determined by the number of cycles per second. In the United Kingdom
this fundamental frequency is set at 50Hz while in the United States it is 60Hz.
Harmonics are voltages or currents that operate at a frequency that is an integer (whole-number) multiple of the
fundamental frequency. So given a 50Hz fundamental waveform, this means a 2nd harmonic frequency would be
100Hz (2 x 50Hz), a 3rd harmonic would be 150Hz (3 x 50Hz), a 5th at 250Hz, a 7th at 350Hz and so on. Likewise,
given a 60Hz fundamental waveform, the 2nd, 3rd, 4th and 5th harmonic frequencies would be at 120Hz, 180Hz,
240Hz and 300Hz respectively.
So in other words, we can say that “harmonics” are multiples of the fundamental frequency and can therefore be
expressed as: 2ƒ, 3ƒ, 4ƒ, etc. as shown.
Complex Waveforms Due To Harmonics
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Note that the red waveforms above, are the actual shapes of the waveforms as seen by a load due to the harmonic
content being added to the fundamental frequency.
The fundamental waveform can also be called a 1 st harmonics waveform. Therefore, a second harmonic has a
frequency twice that of the fundamental, the third harmonic has a frequency three times the fundamental and a
fourth harmonic has one four times the fundamental as shown in the left hand side column.
The right hand side column shows the complex wave shape generated as a result of the effect between the addition
of the fundamental waveform and the harmonic waveforms at different harmonic frequencies. Note that the shape
of the resulting complex waveform will depend not only on the number and amplitude of the harmonic frequencies
present, but also on the phase relationship between the fundamental or base frequency and the individual harmonic
frequencies.
We can see that a complex wave is made up of a fundamental waveform plus harmonics, each with its own peak
value and phase angle. For example, if the fundamental frequency is given as; E = Vmax(2πƒt), the values of the
harmonics will be given as:
For a second harmonic:
E2 = V2max(2×2πƒt) = V2max(4πƒt), = V2max(2ωt)
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For a third harmonic:
E3 = V3max(3×2πƒt) = V3max(6πƒt), = V3max(3ωt)
For a fourth harmonic:
E4 = V4max(4×2πƒt) = V4max(8πƒt), = V4max(4ωt)
and so on.
Then the equation given for the value of a complex waveform will be:
Harmonics are generally classified by their name and frequency, for example, a 2 nd harmonic of the fundamental
frequency at 100 Hz, and also by their sequence. Harmonic sequence refers to the phasor rotation of the harmonic
voltages and currents with respect to the fundamental waveform in a balanced, 3-phase 4-wire system.
A positive sequence harmonic ( 4th, 7th, 10th, …) would rotate in the same direction (forward) as the fundamental
frequency. Where as a negative sequence harmonic ( 2nd, 5th, 8th, …) rotates in the opposite direction (reverse)
of the fundamental frequency.
Generally, positive sequence harmonics are undesirable because they are responsible for overheating of
conductors, power lines and transformers due to the addition of the waveforms.
Negative sequence harmonics on the other hand circulate between the phases creating additional problems with
motors as the opposite phasor rotation weakens the rotating magnetic field require by motors, and especially
induction motors, causing them to produce less mechanical torque.
Another set of special harmonics called “triplens” (multiple of three) have a zero rotational sequence. Triplens are
the odd multiples of the third harmonic ( 3rd, 6th, 9th, …), etc, hence their name, and are therefore displaced by
zero degrees. Zero sequence harmonics circulate between the phase and neutral or ground.
Unlike the positive and negative sequence harmonic currents that cancel each other out, third order or triplen
harmonics do not cancel out. Instead add up arithmetically in the common neutral wire which is subjected to
currents from all three phases.
The result is that current amplitude in the neutral wire due to these triplen harmonics could be up to 3 times the
amplitude of the phase current at the fundamental frequency causing it to become less efficient and overheat.
Then we can summarise the sequence effects as multiples of the fundamental frequency of 50Hz as:
Harmonic Sequencing
Name
Fund. 2nd
3rd
4th
5th
6th
7th
8th
9th
Frequency, Hz
50
100
150
200
250
300
350
400
450
Sequence
+
–
0
+
–
0
+
–
0
Note that the same harmonic sequence also applies to 60Hz fundamental waveforms.
Sequence
Rotation
Harmonic Effect
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+
Forward
Excessive Heating Effect
–
Reverse
Motor Torque Problems
0
None
Adds Voltages and/or Currents in Neutral Wire causing Heating
Harmonics Summary
Harmonics have only been around in sufficient quantities over the last few decades since the introduction of
electronic drives for motors, fans and pumps, power supply switching circuits such as rectifiers, power converters
and thyristor power controllers as well as most non-linear electronic phase controlled loads and high frequency
(energy saving) fluorescent lights. This is due mainly to the fact that the controlled current drawn by the load does
not faithfully follow the sinusoidal supply waveforms as in the case of rectifiers or power semiconductor switching
circuits.
Harmonics in the electrical power distribution system combine with the fundamental frequency (50Hz or 60Hz)
supply to create distortion of the voltage and/or current waveforms. This distortion creates a complex waveform
made up from a number of harmonic frequencies which can have an adverse effect on electrical equipment and
power lines.
The amount of waveform distortion present giving a complex waveform its distinctive shape is directly related to
the frequencies and magnitudes of the most dominant harmonic components whose harmonic frequency is
multiples (whole integers) of the fundamental frequency. The most dominant harmonic components are the low
order harmonics from 2nd to the 19th with the triplens being the worst.
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Passive Components in AC Circuits
Electrical and electronic circuits consist of connecting together many different components to form a
complete and closed circuit. The three main passive components used in any circuit are the: Resistor, the
Capacitor and the Inductor. All three of these passive components have one thing in common, they limit
the flow of electrical current through a circuit but in very different ways.
Electrical current can flow through a circuit in either of two ways. If it flows in one steady direction only it is
classed as direct current, (DC). If the electrical current alternates in both directions back and forth it is classed as
alternating current, (AC). Although they present an impedance within a circuit, passive components in AC circuits
behave very differently to those in DC circuits.
Passive components consume electrical energy and therefore can not increase or amplify the power of any
electrical signals applied to them, simply because they are passive and as such will always have a gain of less than
one. Passive components used in electrical and electronic circuits can be connected in an infinite number of ways
as shown below, with the operation of these circuits depending on the interaction between their different electrical
properties.
Passive Components in AC Circuits
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Where: R is resistance, C is capacitance and L is inductance.
Resistors whether used in DC or AC circuits will always have the same value of resistance no matter what the
supply frequency. This is because resistors are classed to be pure having parasitic properties such as infinite
capacitance C = ∞ and zero inductance L = 0. Also for a resistive circuit the voltage and current are always inphase so the power consumed at any instant can be found by multiplying the voltage by the current at that instant.
Capacitors and inductors on the other hand, have a different type of AC resistance known as reactance, ( XL , XC ).
Reactance also impedes the flow of current, but the amount of reactance is not a fixed quantity for one inductor
or capacitor in the same way that a resistor has a fixed value of resistance. The reactance value of an inductor or
a capacitor depends upon the frequency of the supply current as well as on the DC value of the component itself.
The following is a list of passive components
commonly used in AC circuits along with their corresponding equations which can be used to find their value or
circuit current. Note that a theoretically perfect (pure) capacitor or inductor does not have any resistance.
However in the real world they will always have some resistive value no matter how small.
Purely Resistive Circuit
Resistor – Resistors regulate, impede or set the flow of current through a particular path or impose a voltage
reduction in an electrical circuit as a result of this current flow. Resistors have a form of impedance which is
simply termed resistance, ( R ) with the resistive value of a resistor being measured in Ohms, Ω. Resistors can be
of either a fixed value or a variable value (potentiometers).
Purely Capacitive Circuit
Capacitor – The capacitor is a component which has the ability or “capacity” to store energy in the form of an
electrical charge like a small battery. The capacitance value of a capacitor is measured in Farads, F. At DC a
capacitor has infinite (open-circuit) impedance, ( XC ) while at very high frequencies a capacitor has zero
impedance (short-circuit).
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Purely Inductive Circuit
Inductor – An inductor is a coil of wire that induces a magnetic field within itself or within a central core as a
direct result of current passing through the coil. The inductance value of an inductor is measured in Henries, H.
At DC an inductor has zero impedance (short-circuit), while at high frequencies an inductor has infinite (opencircuit) impedance, ( XL ).
Series AC Circuits
Passive components in AC circuits can be connected together in series combinations to form RC, RL and LC
circuits as shown.
Series RC Circuit
Series RL Circuit
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Series LC Circuit
Parallel AC Circuits
Passive components in AC circuits can also be connected together in parallel combinations to form RC, RL and
LC circuits as shown.
Parallel RC Circuit
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Parallel RL Circuit
Parallel LC Circuit
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Passive RLC Circuits
All three passive components in AC circuits can also be connected together in both series RLC and parallel RLC
combinations as shown below.
Series RLC Circuit
Parallel RLC Circuit
We have seen above that passive components in AC circuits behave very differently than when connected in a
DC circuit due to the influence of frequency, ( ƒ ). In a purely resistive circuit, the current is in-phase with the
voltage. In a purely capacitive circuit the current in the capacitor leads the voltage by 90 o and in a purely inductive
circuit the current lags the voltage by 90o.
The opposition to current flow through a passive component in an AC circuit is called: resistance, R for a resistor,
capacitive reactance, XC for a capacitor and inductive reactance, XL for an inductor. The combination of
resistance and reactance is called Impedance.
In a series circuit, the phasor sum of the voltages across the circuits components is equal to the supply voltage,
VS. In a parallel circuit, the phasor sum of the currents flowing in each branch and therefore through each of the
circuits components is equal to the supply current, IS.
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For both parallel and series connected RLC circuits, when the supply current is “in-phase” with the supply voltage
the circuit resonance occurs as XL = XC. A Series Resonance Circuit is known as an Acceptor Circuit. A Parallel
Resonance Circuit is known as a Rejector Circuit.
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