Analysis And Elimination Of Third Harmonics

This article discusses the power quality issues and how is the unwanted component (called harmonic) introduced. Consequence of harmonics according to their sequence will be different and most precarious also. Moreover, different strategies according to different cases are discussed. We have discussed four different cases. Whereever necessary, the mathematical analysis is shown with output waveform to prove the arguments... - Paresh Modha, Minesh Joshi

Modern civilization has become reliant on the unremitting supply of clean electrical power. Poor quality of power causes financial losses, and also utility or consumer or distributor has to pay the loss, which we can say bonus burden. The society expects continuous and pure power quality. The higher magnitude of voltage for extremely short duration is called impulses/Transients. Where a steady state voltage rise lasts for several seconds, it is called over voltage, and if it dips then it is called under voltage. The so called harmonics is nothing but the non fundamental frequency components of a distorted power frequency waveform.

  • Causes of Power Quality Problems or Sources of Power System Harmonics:
  1. Modern internal causes like single and three phase converter also inverter, SMPS, PCs etc.
    ii. External causes like single L-G fault and L-L-G fault are the major reasons for unbalanced voltage in the power supply system.
    iii. Conventional devices like rotating machines, reactors, transformers, welding machines, variable speed drives.
    iv. Modern electrical accessories like Air conditioners, Microwave ovens, printers, photocopiers, fluorescent Lamps.

As the harmonics are generated because of the above so called reasons, responsible for malfunction of circuit breaker, large neutral currents, skin effects, overheating of transformer, decrement in power factor, and distortion in voltages and extra losses in rotating machines.

If the harmonics are related to circuit configuration then they are called characteristic harmonics. Whereas if the imbalance of the system occurs in voltage or in impedance and by frequency converter generates non -characteristics harmonics.

Figure 1: Fundamental Component and Third Harmonics Component

3rd Harmonics

The triplen harmonics most savior harmonics shown by 3(2N+1), where N=0, 1, 2…., i.e. 3, 9, 15, 21, etc. the triplen harmonics have zero sequence nature and it accumulates as neutral current.

The third harmonic currents of R,Y and B phases are in phase with each other and hence adds up, without cancellation in the neutral conductor.

Figure 2: Delta Configuration of TCR

Phase Sequence Harmonics

The Details of positive, negative and zero phase sequence harmonics are mention below.

Table 2: Phase Sequence Harmonics

Consequences of Phase Sequence of Harmonics are different to any electrical system according to their sequence.

Positive Sequence: It causes over heating due to ‘Skin Effect’, aids with fundamental component, responsible for moderate heating and it is relatively less harmful.

Negative Sequence: It causes over heating due to ‘Skin Effect’, opposes the fundamental, responsible for excessive heating and most harmful.

Zero Sequence: causes ‘Skin Effect’, accumulates in the neutral, also creates ’hot neutral’, responsible for neutral to earth voltage and open neutral condition.

– CASE 1: A 3phase TCR Comprises Three Single Phase TCRs Connected in Delta
– CASE2: PWM Methods to Eliminate 3rd harmonic
– CASE 3: 3rd harmonic Reduction by Transformer Connection
– CASE 4: 3rd harmonic Injection Method

Case 1: A 3phase TCR Comprises Three Single Phase TCRs Connected in Delta

In the case of 3 phase, 6 pulse TCR compromises three single phase TCRs connected in delta configuration, the inductor is divided in to two halves and having thyristor connected in anti parallel combination. Only odd harmonics will be generated when all 3 phase supply voltages are in balanced condition, all thyristors are fired equally in each phase and also all inductors are identical then only the symmetrical current pulse having positive and negative half cycles are generated.

Figure 3: Harmonic reduction by PWM in Single Phase Inverter

To prevent the 3rd harmonic from getting into transmission line the delta connection of the three single phase TCRs used. If IRn, IYn, IBn be the line current respectively for delta connected TCR and IRYn,IYBn, IBRn be the nth order harmonic phase current respectively in the delta branches, then

Thus IRY3, IYB3, IBR3 all have the same magnitude. So, IRY3= IYB3 = IBR3. Now all the three currents are in phase and circulate to the thyristor delta, forming a zero sequence system. It follows that the third harmonic line currents reduce to zero, as shown below:

IR3= IRY3-IYR3 = 0
Similarly, Iy3 = 0 and IB3=0

In short not only the 3rd harmonic will be canceled out, but also all the harmonic of the order 3N+3, where N=0,1,2,……..(3,9,15,21,27) cannot flow through the lines during balanced operation.

Case 2: PWM Methods to Eliminate 3rd Harmonic

When there are several pulses per half cycle, lower-order harmonics are eliminated. Here, the figure given illustrates output voltage waveform that can be attained from a single phase full-bridge inverter. This waveform can also be obtained from a single phase half-bridge inverter, but then the amplitude of voltage wave would be Vs/2. The waveform of figure needs ten commutations per cycle (=360°) instead of two in an unmodulated wave. The voltage waveform in the figure is symmetrical about π as well as π/2.

Figure 4: Harmonic reduction by transformer connections

As this voltage waveform has quarter-wave symmetry, an=0.

If third and fifth harmonics are to be eliminated, then from the equation:

The above two simultaneous equations can be solved numerically to calculate α1 and α2 under the condition that 0 < α1 <90° and α1< α2<90°. This gives α1=23.62° and α2=33.304°.

Case 3: 3rd Harmonic Reduction by Transformer Connection

Output voltage from two or more inverters can be combined by means of transformers to get a net output voltage with reduced harmonic content. The essential condition of this scheme is that the output voltage waveforms from the inverters must be similar but phase-shifted from each other. Here, figure given illustrates two transformers in series with their output voltages V01 from inverter 1 and V02 from inverter 2. Here V02 waveform is taken to have a phase shift of π/3 radians with respect to V01 waveforms as shown in given figure. The resultant output voltage V0 is obtained by adding the vertical coordinates of V01 and V02. It is seen that V0 has amplitude of 2VS from π/3 to π, 4π/3 to 2π and so on. Here, the shape of V0 is quasi square wave.

The Fourier analysis of waveforms V01 and V02 gives:

The resultant output voltage V0 = V01 +V02

The resultant of V01 and V02 must be √3 times V01 or V02 and also V0 lags V01 by 30°. To remove the 3rd harmonic V02 must be lags V01 by 180°.

Thus the 3rd and other triplen harmonics are eliminated from net output voltage and the fundamental component of output voltage V0 is

Case 4: 3rd Harmonic Injection Method

Figure 5: Waveforms of elimination of 3rd and other triplen harmonics

The PWM technique is the simplest technique to understand the modulation operation, but to utilize the full DC supply voltage for inverting operation 3rd harmonics techniques will be better. To go in depth, let’s consider a fundamental component of sine wave which contained triple frequency component, so it is denoted by:

To get the maximum value of X let’s take its derivative and equate to zero, Thus:

Thus, the minimum and maximum of the waveform therefore occur at:
cos Ø=0
12Acos2Ø – (9A-1)=cos-10=1

Thus, using the equation (13) and (14) the optimum value of A is obtain when the value of X is minimum, by taking differentiation of X with respect to A, we get A= -1/3 and A= -1/6.

By putting the negative value of A makes X greater then unity. So A= 1/6 will be preferable.

And thus the equation of X can be modified as

X= sinØ + 1/6 sin3Ø …………………. (15)

By substituting the value of A= 1/6 we get the value of cosØ= (1/2), i.e. Ø will be π/3, 2π/3, etc. Thus, by injecting third harmonic with the maximum magnitude of one sixth the modulation waveform reduces the peak value of output by a factor of  without making any change in the fundamental. Thus, if we increase the amplitude of the modulating waveform by a factor M, so that the equation can be articulated asAll triplen harmonics pass through zero will have maximum amplitude of Xmax = 

X = M [sinØ + 1/6sin3Ø] ………………. (16)

Thus, the required value of M for a peak value of unity should satisfy the constraint is:

1=M and therefore, M= .

Therefore for VRN=

VYN and VBN lags from VRN by 1200 and 2400 respectively.

By adding the third harmonics we can increase up to 15.5% of the amplitude of the fundamental phase voltage.

Figure 6: Resultant output voltage and phase difference between V01 and V02

Figure 7: Fundamental wave, 3rd harmonic in R-phase, addition of 3rd Harmonics

Figure 8: Improvement in Fundamental wave by injecting 3rd harmonic in R-phase


This article has included the details of harmonic and the sequence also and their hazardous effect in Electrical System. Moreover different techniques also discussed in Power System as well as Power electronics techniques to suppress the argument the mathematical analysis discussed to remove 3rd harmonics.

  • In TCR using Delta configuration we can eliminate 3rd harmonics.
    • Using PWM techniques by providing proper phase shift we can eliminate 3rd and other triplen harmonics.
    • By injecting 3rd harmonic component to the sinusoidal modulating wave, the fundamental amplitude increase by 15.5%, and hence DC power supply utilize very well.

For future works further simulation should be performed for all the technique as mentioned above. Moreover, using different modulating techniques in different level to improve switching frequency of Inverters.

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