Triplen Harmonics – Why are they in phase?

Definition:

Triplen harmonics are the odd integral multiple of third harmonic wave. If the frequency of third harmonic wave is ω then frequency of triplen harmonic wave will be 3nω, where n = 3,5,9,…These waves posses a special characteristics and hence it becomes important for the study and analysis of power system. The amazing property of triplen harmonics is that they are in time phase in the three output phase of alternator.

Why triplen harmonics are in phase?

Let us consider a three phase alternator having identical phase windings i.e. R, Y and B in which harmonics are produced. Let the three phase emf are V_R, V_Y and V_B. These emf can be represented as below.

V_R = E_1msin(ωt+Ø₁) + E_3msin(3ωt+Ø₃) + E_5msin(5ωt+Ø₅) +…….

V_Y = E_1msin(ωt-2π/3+Ø₁) + E_3msin[3(ωt-2π/3)+Ø₃] + E_5msin[5(ωt-2π/3)+Ø₅] +…….

V_B = E_1msin(ωt-4π/3+Ø₁) + E_3msin[3(ωt-4π/3)+Ø₃] + E_5msin[5(ωt-4π/3)+Ø₅] +…….

Let us simplify each of these phase voltages.

V_R = E_1msin(ωt+Ø₁) + E_3msin(3ωt+Ø₃) + E_5msin(5ωt+Ø₅) +…….

V_Y = E_1msin(ωt-2π/3+Ø₁) + E_3msin[3(ωt-2π/3)+Ø₃] + E_5msin[5(ωt-2π/3)+Ø₅] +…….

     = E_1msin(ωt-2π/3+Ø₁) + E_3msin[(3ωt-2π)+Ø₃] + E_5msin[(5ωt-10π/3)+Ø₅] +…….

     = E_1msin(ωt-2π/3+Ø₁) + E_3msin(3ωt+Ø₃) + E_5msin(5ωt-4π/3+Ø₅) +…….

Now,

V_B = E_1msin(ωt-4π/3+Ø₁) + E_3msin[3(ωt-4π/3)+Ø₃] + E_5msin[5(ωt-4π/3)+Ø₅] +…….

     = E_1msin(ωt-4π/3+Ø₁) + E_3msin[(3ωt-4π)+Ø₃] + E_5msin[(5ωt-20π/3)+Ø₅] +…….

     = E_1msin(ωt-4π/3+Ø₁) + E_3msin(3ωt+Ø₃) + E_5msin(5ωt-2π/3+Ø₅) +…….

Carefully observe the above three expressions for individual phase voltages. It can easily be seen that, all the three phase voltages only contain odd harmonics. They do not contain any even harmonics. This is because “alternator phase voltages do not have even harmonics”. For detail please read “Why Output Voltage of Alternator can’t have Even Harmonics?“

Also, the third harmonic component in each of the V_R, V_Y and V_B are same in i.e. E_3msin(3ωt+Ø₃). Thus we can say that, all triplen harmonics in the three phases are in time phase.

The fifth harmonic components in emf wave are below.

5^th Harmonic in V_R = E_5msin(5ωt+Ø₅)

5^th Harmonic in V_Y = E_5msin(5ωt-4π/3+Ø₅)

5^th Harmonic in V_B = E_5msin(5ωt-2π/3+Ø₅)

But the fundamental components in emf wave are

Fundamental Component in V_R = E_1msin(ωt+Ø₁)

Fundamental Component in V_Y = E_1msin(ωt-2π/3+Ø₁)

Fundamental Component in V_B = E_1msin(ωt-4π/3+Ø₁)

By comparison of 5^th harmonic components with fundamental, we can say that phase sequence of 5^th harmonic are opposite to the fundamental. This mean the phase sequence will be R, B and Y for fifth harmonics. Hence, they have negative phase sequence. The waveform is shown below.

It is clear from the above waveform that first R phase gets maximum followed by B and Y phase. Therefore their phase sequence is R, B and Y.

Conclusion:

• All triplen harmonics i.e. 3^rd, 9^th, 15^th etc. are in phase.
• The 7^th, 13^th, 19^th etc. have positive phase rotation of R, Y and B.
• The 5^th, 11^th, 17^th etc. have negative phase sequence of R, B and Y.