φ 1, φ 2, φ 3 are the phase shifts of the respective sine wave components

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1 Harmonic Currents Sources, Problems and Solutions Kevin Gaughan 5 November 005 Outline: Mathematical Background Source of Harmonic Currents Total Harmonic Distortion Problems caused by harmonic currents Odd and even Harmonics Solutions to harmonic current problems. Mathematical Background: Using a Fourier Series Expansion any repetitive current waveform can be expressed as the sum of a fundamental sine wave and a number of harmonic components at multiples of the fundamental frequency. i ( t) = I 0 + I sin( ωt + φ ) + I sin(ωt + φ ) + I 3 sin(3ω t + φ) +... I 0 is the DC component of the current I is the RMS of the fundamental component at frequency ω (radians/second) I is the second harmonic component at frequency ω I 3 is the third harmonic component at frequency 3ω and so on. φ, φ, φ 3 are the phase shifts of the respective sine wave components Note: ω = πf where f is the fundamental frequency in Hz. In Ireland the fundamental power frequency f is 50Hz. Example: Fourier Series expansion of a square wave reveals that the harmonics as a percentage of the fundamental are: I 0 =0, I =00%, I =0, I 3 =33.3%, I 4 =0, I 5 =0%, I 6 =0, I 7 =4.3% In general for a square wave I n =0 if n is even and I n =00%/n if n is odd. This is shown graphically in Figure

2 Figure shows the fundamental, third, fifth and seventh harmonics of a square wave and the sum of these. The sum is like a quite square wave with a bit of ripple on top. If we added more harmonics then we would get closer to an ideal square wave. Total Harmonic Distortion Figure Harmonic components of a square wave The total harmonic distortion (THD) is a measure of how badly a waveform is distorted by harmonics. THD = I + I 3 + I I 4 + I % I + I + I + I... is the combined RMS of all the harmonic components I i is the RMS of the fundamental A pure sine wave has no harmonics and has a THD of 0%. In general harmonics are undesirable and we would like the THD to be as close to zero as possible. A square wave has a THD of 48%. In extreme cased the THD can exceed 00% indicating that the harmonics are actually bigger than the fundamental.

3 Odd and Even Harmonics A waveform with odd harmonics (Figure Figure A Waveform with Odd Harmonics) is symmetrical in that the negative half cycle is the same as the positive half cycle flipped over the x-axis. A waveform with even harmonics (Figure 3) does not share this symmetry. Even harmonics are rare in practice because most systems respond symmetrically to positive and negative voltages. For example a half wave rectifier (which only conducts on the positive half cycle) will generate even harmonics but this is almost never used. The full wave rectifier is far more common and this draws equal current on positive and negative half cycles generating odd harmonics. Figure A Waveform with Odd Harmonics Figure 3 AWaveform with Even Harmonics

4 How to determine the harmonics of a waveform:. You can work out the harmonics analytically using a Fourier Series Expansion this involves convolution integrals. Please refer to a Mathematics textbook if you need to do this.. You can use a computer program such as Spice or MATLAB to work out the harmonics 3. You can use an instrument called a harmonic analyzer to measure the harmonics in a real life scenario. This is probably the most important approach in practice. Example of Harmonic Analysis using Spice: Bridge Rectifier Circuit Figure 4 Bridge Rectifier Circuit Figure 5 AC input current of Bridge Rectifier Figure 6 Harmonic Analysis of Bridge Rectifier Input Current Figure 5 shows the characteristically peaky input current of a bridge rectifier and Figure 6 shows a harmonic analysis of that current. Note the fundamental at 50Hz (4.5 Amps). Note that the 3 rd harmonic at 50Hz is almost as large as the fundamental (4. Amps). In fact this circuit has a THD of about 77%. There are no even harmonics because this full wave rectifier draws the same current on positive and negative half cycles.

5 Sources of Harmonic Current: Linear Loads consisting of resistors, inductors and capacitors draw sinusoidal current from the ac mains and do not cause harmonic distortion. Noon linear loads do cause harmonics. Examples of non-linear loads which cause harmonics are: Power Electronic Circuits (Rectifiers) Transformer magnetizing current (due to hysterisis of the core iron) Industrial Arc Furnaces Fluorescent Lamps The prevalence of harmonic generating loads is on the increase and the level of harmonic currents being drawn from the mains is becoming ever more problematic. Harmful Effects of Harmonic Currents. Increased losses and reduction of Power Factor Harmonic currents do not in themselves provide useful power to the load yet the cause losses in the wires and equipment of the distribution system. In mathematical terms harmonic currents reduce the power factor of a load. Power factor is defined as TruePower. You probably already know that power factor is.0 for a resistive load VoltAmps but reduces to cos(φ ) if there is a phase shift φ between the voltage and current. In fact this is not the whole story because harmonics can reduce the power factor even further. The complete expression for power factor is actually: TruePower I PowerFactor = VoltAmps = cos( φ ). I rms In this expression cos(φ ) is the power factor you knew from before, this should correctly be called displacement power factor to signify that it arises from phase displacement. I is the RMS of the fundamental current while I rms is the total rms of the current waveform including harmonics. If a waveform has harmonic content then I will be less than I rms and the power factor is reduced accordingly.. Current Harmonics affect neighboring equipment through voltage distortion at the point of common coupling A load which draws current harmonics will cause voltage distortion at the point of common coupling between itself and other loads. This voltage distortion means that other loads connected to the same supply don t see a sine wave voltage and this may cause them to malfunction.

6 To quantify the impact of voltage distortion consider Figure 7. Although the supply is sinusoidal Load B sees a distorted voltage at the point of common coupling (Vpcc) due to the harmonic currents from load A acting on the source impedance. The voltage distortion caused by any harmonic current is just the harmonic current multiplied by the source impedance at that frequency. Taking the most common case of an inductive source impedance we can say that the n th harmonic voltage component at Vpcc is: ( Vpcc) = I n πnfls n I n is the n th harmonic current component caused by load A. πnfl s is the source impedance at the frequency of the n th harmonic. Figure 7 Voltage Distortion caused by non linear loads A key point to note is that the impedance of an inductive line increases with increasing frequency so even small amount of high order harmonics (at high multiples of the line frequency) can cause a large distortion of the voltage. Voltage distortion is undesirable because it can cause malfunction in the harmonic generating equipment itself or in neighboring equipment. Any circuit which depends on the peak value or the zero crossing of the ac voltage can be affected by harmonics. Transformers and Power Factor correction Capacitors are subject to increased losses and reduced working life in the presence of voltage harmonics. Motors may be subject to torque ripple and unwanted vibrations.

7 3. Triplen Harmonics and Neutral Currents It is a tenet of power systems engineering that the three phase current of a balanced three phase load sum to zero leaving little or no neutral current. In three phase three wire systems no wire is even provided for the neutral current. Unfortunately when harmonics are present the balanced three phase currents may not sum to zero. The fundamental components sum to zero and all harmonics except triplen harmonics do but any harmonic which are a multiple of three actually add together resulting in a neutral current that is the sum of the triplen harmonics in each phase. Figure 8 shows a set of balanced three phase currents with 30% third harmonic on each phase. The phase currents are clearly balanced but the neutral current is far from zero. It has a large third harmonic component equal to three times the third harmonic on each phase. Figure 8 Three Phase Currents with 30% Third Harmonic So in a balanced three phase system where each phase has a third harmonic component of 30% of fundamental the neutral will have a third harmonic which is 90% of the phase current fundamental. In fact if the third harmonic is more than 33% then the neutral will have to carry more than 00% of the phase current fundamental. This is a major issue for installations which were designed for low levels of neutral current but have to carry neutral currents than may even exceed the phase current. Computer power supplies can have more than 70% third harmonic and a large computer installation can put a very significant strain on neutral wiring.