Download TEMPUS ENERGY: POWER QUALITY, HARMONICS 1: The use of

TEMPUS ENERGY: POWER QUALITY, HARMONICS
1: The use of a frequency spectrum and the THD
By using Fourier analysis, every periodic signal can be considered as the sum of a sinusoidal first
order harmonic and sinusoidal harmonics of higher order. The first order harmonic has the same
frequency as the original periodic signal. An n-th order harmonic is a sine whose frequency equals 𝑛
times the frequency of the original signal.
In case 𝑖(𝑡) has a period 𝑇 (with 𝜔 = 2𝜋⁄𝑇), 𝑖(𝑡) can be expressed as
∞
𝑖(𝑡) = 𝐼0 + ∑ 𝐼𝑛 𝑠𝑖𝑛(𝑛𝜔𝑡 + 𝜑𝐼𝑛 )
𝑛=1
with appropriate values for 𝐼0 , 𝐼𝑛 and 𝜑𝐼𝑛 (for 𝑛 = 1,2,3,4, … ).
When considering a single phase rectifier bridge having a sufficiently large inductor at DC side and
assuming the rectifier diodes are ideal, the current extracted from the grid is a square wave having a
frequency of 50 𝐻𝑧. Suppose the square wave current has an amplitude 𝐵. When applying the
Fourier analysis to this square wave, 𝐼0 = 0 (the current does not contain a DC component), 𝐼𝑛 = 0
for even valued 𝑛 = 2,4, 6, .. and for odd valued 𝑛 = 1,3, 5, ..
𝐼𝑛 =
1 4𝐵
.
𝑛 𝜋
The higher the order 𝑛 of the harmonic, the smaller its amplitude (which is commonly the case). In
theory, an infinite number of harmonics are needed to obtain the original signal 𝑖(𝑡) but in reality by
adding a finite number of harmonics a good approximation is obtained. The sum of these harmonics
up to the seventh order approximates the square wave as visualized in Figure 1. By increasing the
number of harmonics which are added, the better the approximation of the square wave.
Figure 1: Harmonic analysis of a square wave
Instead of visualizing the harmonic contents of the square wave current in the time domain as has
been done in Figure 1, it is also possible to visualize this contents in the frequency domain by using a
frequency spectrum. Figure 2 visualizes the frequency spectrum of the square wave current showing
there is no DC component, there is a first order harmonic having a 50 𝐻𝑧 frequency, there is a third
order harmonic having a 150 𝐻𝑧 frequency, there is a fifth order harmonic having a 250 𝐻𝑧
frequency, there is a seventh order harmonic having a 350 𝐻𝑧 frequency…
Figure 2: Frequency spectrum of a square wave
The Total Harmonic Distortion (THD) indicates the distortion in comparison with the ideal sinusoidal
shape. More precisely
𝑇𝐻𝐷𝐼 =
2
√∑∞
𝑛=2 𝐼𝑛
.
𝐼1
The smaller this number, the better the sinusoidal shape is approximated.
The grid can be modeled by its Thévenin equivalent circuit as visualized in Figure 3. Due to the
current extracted from the grid, a voltage drop appears across this grid impedance implying the
voltage 𝑢𝑔𝑟𝑖𝑑 (𝑡) does not equal 𝑢𝑓1 (𝑡). In case the current 𝑖𝑓1 (𝑡) extracted from the grid is not a
sine, the voltage drop across the grid impedance is also not a sine implying also 𝑢𝑔𝑟𝑖𝑑 (𝑡) is not a sine
anymore (although 𝑢𝑓1 (𝑡) is assumed to be a sine).
Figure 3: Thévenin equivalent circuit of the grid
This means 𝑢𝑔𝑟𝑖𝑑 (𝑡) can be expressed as
∞
𝑢𝑔𝑟𝑖𝑑 (𝑡) = 𝑈0 + ∑ 𝑈𝑛 𝑠𝑖𝑛(𝑛𝜔𝑡 + 𝜑𝑈𝑛 )
𝑛=1
with appropriate values for 𝑈0 , 𝑈𝑛 and 𝜑𝑈𝑛 (for 𝑛 = 1,2,3,4, … ). The Total Harmonic Distortion
(THD) indicates the distortion in comparison with the ideal sinusoidal shape. More precisely
2
√∑∞
𝑛=2 𝑈𝑛
𝑇𝐻𝐷𝑈 =
.
𝑈1
In general, 𝑇𝐻𝐷𝑈 is much smaller than 𝑇𝐻𝐷𝐼 since the distortion of the grid voltage is much smaller
than the distortion of the current.
2: Non linear loads: examples
The number of non linear loads is countless, but a few examples will give an idea.
2.1: Transformer
Due to the non linear behavior of the ferromagnetic material in a transformer, the transformer
consumes a non sinusoidal magnetization current (also when the transformer is supplied with a
sinusoidal voltage). This magnetization current is the current consumed by the primary winding of
the transformer in case the secondary winding is open. The primary winding of the transformer
behaves as a series RL circuit.
Since this magnetization current is small, the voltage drop across the resistance of this series RL
circuit can be neglected. This implies the applied grid voltage 𝑢(𝑡) (which is assumed to be a sine
here) stands across this inductance of the series RL circuit. This causes the magnetization current
implying there is a magnetic flux 𝜙(𝑡). In case the primary winding has 𝑁1 turns, due to Faraday’s
Law
𝑢(𝑡) = − 𝑁1
𝑑 𝜙(𝑡)
.
𝑑𝑡
Since 𝑢(𝑡) is a sine, also the magnetic flux 𝜙(𝑡) is a sine. This implies also the magnetic induction
𝐵(𝑡) =
𝜙(𝑡)
𝐴𝐶
in the core material is a sine (𝐴𝐶 is the area of the magnetic core material).
Although the magnetic induction is a sine, due to saturation effects, the magnetic field intensity 𝐻
and therefore also the magnetization current are not sinusoidal. This is visualized in Figure 4. As the
amplitude of the supply voltage 𝑢(𝑡) increases, the ferromagnetic core material saturates and more
distortion of the magnetization current is obtained. In the linear part of the magnetization curve (BH
curve), the current remains (almost) sinusoidal. The magnetization current in Figure 4 contains a first
order harmonic, a third order harmonic, a fifth order harmonic, …
Figure 4: Magnetization curve of a transformer
When the hysteresis phenomenon is taken into account, the zero crossing of the magnetization
current will be shifted in comparison with the zero crossing of the flux 𝜙(𝑡) and the induction 𝐵(𝑡).
The positive and the negative alternation of the current have a different shape. Even inside the
positive and the negative alternation of the current there is a non-symmetry. Due to this nonsymmetry, the spectrum does not only contain odd harmonics. The spectrum also contains even
harmonics.
Figure 5: Magnetization current of a transformer
2.2: Six pulse rectifier bridge
As already mentioned, power electronic devices are an important source of harmonics in the current.
This is not only the case when considering single phase rectifier bridges, also three phase rectifier
bridges are important sources of harmonics. Figure 6 visualizes a three phase rectifier bridge where
due to an inductor at DC side (smoothing reactor) the DC current is (almost) constant. The rectifier
bridge of Figure 6 is a six pulse rectifier bridge. Each 20 𝑚𝑠 (period of the grid voltage), the output
voltage contains six ripples.
Figure 6: Three phase rectifier bridge with smoothing reactor
In case the diodes or the thyristors of the rectifier bridge are idealized, the current consumed by the
rectifier bridge is visualized in Figure 7. The corresponding spectrum is visualized in Figure 8. The
current of such a six pulse rectifier bridge contains a first order harmonic of 50 𝐻𝑧 , a fifth order
harmonic of 250 𝐻𝑧, a seventh order harmonic of 350 𝐻𝑧, an eleventh order harmonic of 550 𝐻𝑧…
Figure 6: Grid current of a three phase rectifier bridge
Figure 7: Spectrum of the grid current of a three phase rectifier bridge
In case of a six pulse rectifier bridge, the current contains harmonics of order
6𝑘 ±1
for all integer values 𝑘 ≥ 0.
Especially when considering lower powers, the smoothing reactor in Figure 6 is often omitted
(reactors are heavy and they are rather expensive). In Figure 8, a capacitor is placed in parallel with
the DC output of the rectifier to obtain a constant DC voltage. In such a situation, the DC current is
not constant implying the shape of the grid current changes as visualized in Figure 9. Figure 10
visualizes the spectrum of the grid current visualized in Figure 9. Notice the THD of the grid current in
Figure 9 and Figure 10 is higher than the THD of the grid current in Figure 6 and Figure 7.
Figure 8: Three phase rectifier bridge with a capacitor at DC side
Figure 9: Grid current of a three phase rectifier bridge with capacitor at DC side
Figure 10: Spectrum of the grid current of a three phase rectifier bridge with capacitor at DC side
2.3: Twelve pulse rectifier bridge
Figure 11 visualizes a twelve pulse rectifier bridge. A twelve pulse rectifier bridge contains twice a six
pulse rectifier bridge. The three phase rectifiers are fed using a three phase transformer having one
primary winding and two secondary windings. The first secondary winding has delta connected
windings and the second secondary winding has wye connected windings. By combining these delta
and wye connected windings, phase differences of 30° are obtained between the secondary voltages
of these windings.
Figure 11: Twelve pulse rectifier bridge
By cascading the DC voltages of the six pulse rectifier bridges, the total DC voltage is the sum of the
two six pulse rectified DC voltages. Since these DC voltages are shifted over an angle of 30°, the total
voltage contains twelve ripples each 20 𝑚𝑠. This implies indeed a twelve pulse rectifier is obtained.
Both six pulse rectifiers require the same current as visualized in Figure 12 but these currents are
shifted over an angle of 30° . Figure 12 visualizes both individual currents and the total current which
flows at primary side of the transformer. Notice this total current is a much better approximation of a
sine implying its THD is much lower.
Figure 12: Current consumed by a twelve pulse three phase rectifier bridge
Figure 13 visualizes the spectrum of the total current consumed by the twelve pulse rectifier bridge.
This shows the current contains a first order harmonic, eleventh and thirteenth order harmonics, …
In comparison with Figure 7, the fifth and the seventh order harmonics have disappeared (the THD of
the current in Figure 12 and Figure 13 is lower than the THD of the current in Figure 6 and Figure 7).
In case of a twelve pulse rectifier bridge, the current contains harmonics of order
12 𝑘 ± 1
for all integer values 𝑘 ≥ 0.
Figure 13: Spectrum of the current consumed by a twelve pulse rectifier bridge
2.4: Electric lamps
Also electric lamps are important sources of harmonic pollution in the electrical grid. Traditionally
incandescent lamps are commonly used and they are linear loads i.e. when fed by a sinusoidal
voltage also the current is sinusoidal. Figure 14 visualizes the current consumed by an incandescent
lamp. As visualized in Figure 15, also a halogen lamp consumes a current which is almost a sine when
fed by the public electricity grid.
Figure 14: Current consumed by an incandescent lamp
Figure 15: Current consumed by a halogen lamp
Fluorescence lamps are frequently used since they produce approximately 60 lumens per watt which
is approximately four times the efficiency of a typical incandescent lamp. Figure 16 visualizes the
current consumed by a fluorescence lamp in case a classical inductive ballast has been used. Figure
17 visualizes the current consumed by a fluorescence lamp in case an electronic ballast has been
used.
Figure 16: Current consumed by a fluorescence lamp having an inductive ballast
Figure 17: Current consumed by a fluorescence lamp having an electronic ballast
A lot of countries promote the use of low energy light bulbs (compact fluorescent lamps) to replace
the classical incandescent lamp. In a lot of countries, the use of an incandescent lamp is even banned
in order to reduce electrical power consumption (which reduces the dependency on fossil fuels and
CO2 exhaust). Figure 18 visualizes the current consumed by a low energy light bulb. It is clear this
current has a non sinusoidal shape i.e. the current contains a lot of harmonics causing harmonic
pollution in the grid. Figure 19 visualizes the current consumed by a LED light which is also non
sinusoidal and causes harmonic pollution of the grid.
Figure 18: Current consumed by a low energy light bulb
Figure 19: Current consumed by a LED lamp
3: Consequences of harmonic pollution in an electrical grid
When considering harmonics in an electrical grid, Figure 3 shows that harmonics in the current imply
harmonics in the grid voltage. At the other hand, if the grid voltage contains harmonics this implies
harmonics in the current even when the load is linear. This implies current harmonics and voltage
harmonics are related with each other and for this reason the present section will sometimes focus
on current harmonics and sometimes on voltage harmonics.
3.1: Skin and proximity effect
An alternating current has a larger current density near the surface of the conductor than at its core.
The electric current tends to flow at the “skin” of the conductor which is called skin effect (see Figure
20). The skin effect causes the effective resistance of the conductor to increase (according to the law
of Pouillet). Due to this increased resistance, there will be more heat dissipation in the conductor.
When the frequency increases, the influence of the skin effect will be larger.
Figure 20: Current density due to the skin effect
The increase of the resistance depends on the radius 𝑟 of the conductor, the specific resistance ρ of
the conductor material, the permeability μ of the conductor material and the frequency 𝑓 (with
pulsation 𝜔 = 2𝜋𝑓). When 𝑅0 is the DC resistance, than 𝑅 is the resistance with pulsation 𝜔 given by
2
4
6
𝑅
1 𝜇𝜔𝑟 2
1
𝜇𝜔𝑟 2
11
𝜇𝜔𝑟 2
=1+
(
) −
(
) +
(
) − …
𝑅0
12
4𝜌
180
4𝜌
26880
4𝜌
in case
𝜇𝜔𝑟 2
< 2.
4𝜌
This expression allows to calculate 𝑅⁄𝑅0 in case of a 50 𝐻𝑧 and a 250 𝐻𝑧 current as a function of the
cross section of the conductor as visualized in Figure 21. The skin effect is more important when the
cross section of the conductor increases and when the frequency increases. The higher order
harmonics “see” a larger resistance which increases the heat dissipation in the conductors of the
cable. With an increasing order of the harmonic, the more important the skin effect.
Figure 21: Skin effect in a conductor
The proximity effect describes that the current in a first conductor influences the current densities in
a second conductor. Only a part of the cable cross section is used implying the resistance and the
joule losses increase. As the frequency increases, the importance of the proximity effect increases
too. With an increasing order of the harmonic, the more important the proximity effect. Figure 22
visualizes the proximity effect for two parallel conductors. On the left the situation where the two
currents are flowing in the opposite direction and on the right the situation where the two currents
are flowing in the same direction.
Figure 22: Current density due to the proximity effect
3.2: The neutral conductor
Consider a three phase grid containing phase conductors 𝐿1 , 𝐿2 , 𝐿3 and a neutral conductor 𝑁. There
are three identical single phase non linear loads 𝑍1 , 𝑍2 and 𝑍3 as visualized in Figure 23. Suppose
these single phase non linear loads are diode rectifier bridges having a smoothing inductor at DC side
implying they consume (in the idealized case) a square wave current (𝐼𝑓1 (𝑡), 𝐼𝑓2 (𝑡), 𝐼𝑓3 (𝑡)). As
already studied, such a square wave current contains a first order harmonic, a third order harmonic,
a fifth order harmonic, a seventh order harmonic, …
Figure 23: Three phase grid with neutral conductor
Figure 24 visualizes these phase currents 𝐼𝑓1 (𝑡), 𝐼𝑓2 (𝑡) and 𝐼𝑓3 (𝑡) in case only the first order
harmonic, the third order harmonic, the fifth order harmonic and the seventh order harmonic are
taken into consideration. Figure 24 also visualizes the current in the neutral conductor which equals
𝐼𝑁 (𝑡) = 𝐼𝑓1 (𝑡) + 𝐼𝑓2 (𝑡) + 𝐼𝑓3 (𝑡).
Notice the first order harmonic, the fifth order harmonic and the seventh order harmonic do not
appear in the current of the neutral conductor due to the phase shifts between these harmonics in
the phase currents. But the third order harmonics in the three phase conductors have the same
phase i.e. they are homopolar. The sum of these third order harmonics in the neutral conductor is
three times the third order harmonic in one single phase conductor. The same homopolar behavior
applies for all harmonics having an order which is a multiple of three.
Figure 24: Currents in a three phase grid with neutral conductor
Due to this current in the neutral conductor, additional heat dissipation occurs in the grid cable.
There can be a considerable heat dissipation in the neutral conductor since:
-
the cross-section of the neutral conductor is often smaller than the cross section of the
phase conductors,
the current in the neutral conductor can be quite large,
the 150 𝐻𝑧 frequency (or higher) implies a noticeable skin effect.
Traditionally, the cross section of the neutral conductor is smaller than the cross section of the phase
conductors. Indeed, when considering a symmetric three phase grid with sinusoidal currents the
current in the neutral conductor equals zero. In general, the non symmetry and the harmonic
pollution in the grid is limited implying the current in the neutral conductor is sufficiently small to
make the cross section of the neutral conductor smaller (often the cross section of the neutral
conductor is half the cross section of a phase conductor).
In case the loads consume a large third order harmonic (or another homopolar harmonic), a large
current is obtained in the neutral conductor. To limit the heat dissipation in the neutral conductor, it
can be necessary to give the neutral conductor the same cross section as the phase conductors.
Sometimes, it can even be necessary to give the neutral conductor a larger cross section than the
phase conductors. Of course, instead of realizing a neutral conductor with an extra large cross
section the question arises whether it is possible to avoid the harmonic pollution of the grid i.e. to
reduce the harmonic contents of the currents.
3.3: Transformers
When feeding a transformer with a non sinusoidal voltage, the current harmonics in the primary and
secondary windings will cause additional copper losses (skin effect and proximity effect has to be
taken into account). Due to the harmonics, also additional iron losses will occur. A distinction can be
made between hysteresis losses and eddy current losses. Suppose a harmonic current of the n-th
order
𝑖𝑛 (𝑡) = √2 𝐼𝑛 𝑠𝑖𝑛(𝑛𝜔𝑡)
is flowing (𝜔 = 314 𝑟𝑎𝑑/𝑠). In case 𝑛 is increasing, and 𝐼𝑛 does not depend on 𝑛, the hysteresis
losses are proportional with 𝑛 i.e. with the frequency of the harmonic. The eddy current losses will
be proportional with 𝑛2 i.e. with the square of the frequency of the harmonic.
When feeding the transformer with a non sinusoidal voltage 𝑢(𝑡) containing harmonics, not only
current harmonics will flow. The frequencies of the harmonics will also be present in the magnetic
flux of the core (and the magnetic induction). Indeed, due to Faraday’s Law
𝑢(𝑡) = − 𝑁1
𝑑 𝜙(𝑡)
𝑑𝑡
the flux 𝜙(𝑡) contains the same frequencies as 𝑢(𝑡). A flux change with a frequency 𝑓 causes
magnetostriction of the core material causing noise having a frequency 2𝑓. In case there is only a
50 𝐻𝑧 voltage only a 100 𝐻𝑧 hum will appear but a fifth order harmonic will cause a 500 𝐻𝑧 noise, a
seventh order harmonic will cause a 700 𝐻𝑧 noise…
When considering a single phase transformer, all voltage harmonics will be transformed from the
primary winding to the secondary winding (whether you like it or not). When considering a three
phase transformer the non homopolar harmonics will be transformed from the primary winding to
the secondary winding. When considering the homopolar harmonics, it depends on the winding
configurations (delta, wye, with or without neutral conductor) whether these homopolar voltage
harmonics will be transformed from the primary winding to the secondary winding.
3.4: Induction motors
Suppose an induction motor is fed by a polluted grid i.e. by a grid voltage containing harmonics. Due
to these harmonics, there are additional copper losses and additional iron losses similar with the
additional losses of a transformer. The harmonics also cause additional noise due to the
magnetostriction effect of the ferromagnetic material. Moreover, these harmonics have an influence
on the motor torque.
In order to prove the effect on the mechanical torque of the induction machine, it is important that
the harmonics in the applied voltage cause harmonics in the currents. The first order harmonic
generates a magnetic rotating field having a speed (expressed in rpm) 60𝑓⁄𝑝 where 𝑓 is the grid
frequency and 𝑝 is the number of pole pairs. In general, an n-th order harmonic generates a
magnetic rotating field having as speed
60 𝑛 𝑓
.
𝑝
These rotating magnetic fields have a direction which depends on 𝑛. Suppose a three phase voltage
contains a first order harmonic and a seventh order harmonic both generating a magnetic rotating
field in the same direction. This implies there are two torque-speed characteristics. The first torque
speed characteristic has a synchronous speed 𝑁𝑆 = 60 𝑓⁄𝑝 and the second torque speed
characteristic has a synchronous speed 7𝑁𝑆 as visualized in Figure 25.
Figure 25: Torque speed characteristics due to first and seventh order harmonics
As visualized in Figure 25, the rotor rotates with a speed 𝑁 which is somewhat smaller than 𝑁𝑆 . The
total motor torque is the sum of the torques generated due to the first and the second torque speed
characteristic. When comparing with 𝑁𝑆 , the rotor has a slip
𝑠1 =
𝑁𝑆 − 𝑁
𝑁𝑆
and when comparing with 7𝑁𝑆 , the rotor has a slip
𝑠7 =
7𝑁𝑆 − 𝑁
.
7 𝑁𝑆
There are two stator rotating magnetic fields, the first one has the synchronous speed 𝑁𝑆 and the
second one has the synchronous speed 7𝑁𝑆 . This first stator rotating magnetic field generates
voltages and currents having a frequency 𝑠1 50 𝐻𝑧. This generates a rotor rotating magnetic field
having a speed 𝑠1 𝑁𝑆 with respect to the rotor. Since the rotor has a speed (1 − 𝑠1 ) 𝑁𝑆 , the rotor
rotating field has an absolute speed 𝑁𝑆 .
The second stator rotating magnetic field generates voltages and currents having a frequency
𝑠7 350 𝐻𝑧. This generates a rotor rotating magnetic field having a speed 𝑠7 7𝑁𝑆 with respect to the
rotor. Since the rotor has a speed (1 − 𝑠7 )(7𝑁𝑆 ) = (1 − 𝑠1 )𝑁𝑆 , the rotor rotating field has an
absolute speed 7𝑁𝑆 .
This means there are four rotating magnetic fields:
-
Rotating field 1: stator field with speed 𝑁𝑆
Rotating field 2: stator field with speed 7𝑁𝑆
Rotating field 3: rotor field with speed 𝑁𝑆
Rotating field 4: rotor field with speed 7𝑁𝑆 .
The interaction of the rotating fields 1 and 3 gives a constant torque. This is the desired useful torque
(coming from the first torque speed characteristic). The interaction of the rotating fields 2 and 4 also
gives a constant torque. This is the torque coming from the second torque speed characteristic. This
additional torque is not a problem but it causes a lot of heat losses since the slip 𝑠7 is close to unity.
The interaction between the rotating fields 1 and 4 gives a pulsating torque (which is on average
zero). The interaction between the rotating fields 2 and 3 gives a pulsating torque (which is on
average zero). Due to these pulsating torques, the torque of the induction motor is no longer
constant.
3.5: Miscellaneous
Harmonics do not only affect the conductors, transformers and electrical machines. Actually the
presence of harmonics has much more effects on all kind of grid components. In the present
paragraph, just a few additional examples are given i.e. we consider capacitors, protective devices
and measuring devices.
Capacitors placed in parallel with the grid voltage in order to improve the power factor have an
impedance which is decreasing as the order 𝑛 increases. Even if the n-th order voltage harmonic is
rather small, the n-th order harmonic current in the capacitor can be relatively large.
The harmonics in the current increase the RMS-value of this current. Due to this increased RMSvalue, the joule losses in the fuse increase and the melting time decreases. When using an electronic
protection device instead of a classical fuse, quite often the peak value 𝐼𝑝𝑒𝑎𝑘 of the current is
detected using analog electronics. When the current is a sine, its RMS-value equals
𝐼𝑅𝑀𝑆 =
𝐼𝑝𝑒𝑎𝑘
√2
.
Due to harmonics, 𝐼𝑝𝑒𝑎𝑘 can be large. This implies using 𝐼𝑅𝑀𝑆 = 𝐼𝑝𝑒𝑎𝑘 /√2, the estimate of 𝐼𝑅𝑀𝑆 can
be too large. Using digital electronics, it is possible to sample the current signal and a true RMS
measurement can be obtained.
An analog multimeter (D’Arsonval moving coil meter) physically measures the mean value of the
current or voltage. The scale is calibrated for 1.11 𝑢𝑚𝑒𝑎𝑛 or 1.11 𝑖𝑚𝑒𝑎𝑛 which is the RMS-value when
the voltage or current is a sine. In case the voltage/current is not a sine, one still measures
1.11 𝑢𝑚𝑒𝑎𝑛 or 1.11 𝑖𝑚𝑒𝑎𝑛 but this is not the RMS value.
When using power meters to measure active power, reactive power and apparent power in the
presence of harmonics, it is always important to pay attention. Different (digital) measurement
devices can give different results. Especially when considering reactive and apparent power, different
theoretical definitions are used which give different results. In case these measurements are used to
bill the electrical power/energy consumption, discussions about the theoretical definitions and
appropriate measurement techniques are relevant.
4: Reducing harmonic pollution in an electrical grid
4.1: Rectifiers and frequency converter
By using power electronic devices which generate less current harmonics, the harmonic pollution in
the grid can be reduced. It can be appropriate to use a twelve pulse rectifier bridge instead of a six
pulse rectifier bridge. Notice however a twelve pulse rectifier bridge is much more expensive than a
six pulse rectifier bridge implying it will only be used for large powers. For very large powers, it might
even be a good practice to use a 24 pulse rectifier bridge as visualized in Figure 26 in order to obtain
a further reduction of the THD of the consumed grid current (in Figure 26 there is also an inverter
implying a frequency converter is obtained).
Figure 26: 24 pulse rectifier bridge
A 24 pulse rectifier bridge contains four times a six pulse rectifier bridge. Each six pulse rectifier
bridge is fed by a three phase voltage and due to the transformers there are phase differences of 15°
between these voltages. By cascading the DC voltages of the six pulse rectifier bridges, the total DC
voltage is the sum of the four rectified DC voltages. Since these DC voltages are shifted over an angle
of 15°, the total voltage contains 24 ripples each 20 𝑚𝑠. This implies indeed a 24 pulse rectifier is
obtained.
Instead of using diode or thyristor rectifier bridges, it is also possible to use an Active Front End (AFE).
Figure 27 visualizes a frequency converter containing an AFE, a capacitor C and an IGBT inverter
feeding an induction motor. Such an AFE contains six IGBT switches (S1, S2, S3, S4, S5 and S6),
freewheeling diodes (D1, D2, D3, D4, D5 and D6) and an inductor in each phase. These inductors 𝐿1 , 𝐿2
and 𝐿3 are crucial in the operation of the AFE.
Based on a switching frequency in the 𝑘𝐻𝑧 range, the phase currents are compared with the desired
sinusoidal current. When a current is too small, due to an appropriate switching of the IGBTs a
positive voltage 𝑈𝐿 appears over the inductor. Based on
𝑈𝐿 (𝑡) = 𝐿
𝑑 𝑖(𝑡)
,
𝑑𝑡
the current increases. When a current is too large, due to an appropriate switching of the IGBTs a
negative voltage 𝑈𝐿 appears over the inductor implying the current decreases. By using a switching
frequency in the 𝑘𝐻𝑧 range, it is possible to obtain a good approximation of the desired 50 𝐻𝑧
current. Figure 27 visualizes the current consumed by an AFE. Notice the sine shape is well
approximated but also notice the effect of the switching frequency in the 𝑘𝐻𝑧 range.
Figure 27: AFE current
Figure 28 visualizes the spectrum of the AFE current of Figure 27. Notice the first order harmonic
(50 𝐻𝑧) and small harmonics having frequencies below 1 𝑘𝐻𝑧. Notice the impact of the switching
frequency in the spectrum around 3 𝑘𝐻𝑧 and especially 6 𝑘𝐻𝑧.
Figure 28: Frequency spectrum of the AFE current
Figure 29 shows a close-up of the frequency spectrum visualized in Figure 28. Notice the first order
harmonic (50 𝐻𝑧) and very small harmonics giving a small THD when only harmonics up to the 40-th
order are taken into consideration (IEC-61000-3-12). Actually only the seventh order harmonic has
some importance.
2
√∑40
𝑛=2 𝐼𝑛
𝑇𝐻𝐷𝐼 =
𝐼1
.
Figure 29: Frequency spectrum of the AFE current
4.2: The use of passive filters