**Figure 1.**
Simulated power system together with the passive harmonic filter (PHF) topologies.

**Figure 1.**
Simulated power system together with the passive harmonic filter (PHF) topologies.

**Figure 2.**
PCC voltage and current waveforms (without filter connection) after the thyristor bridge input line reactor inductance (L) increase.

**Figure 2.**
PCC voltage and current waveforms (without filter connection) after the thyristor bridge input line reactor inductance (L) increase.

**Figure 3.**
Grid voltage waveforms and spectrums: (**a**,**c**) when the load is not connected to the PCC and (**b**,**d**) when the load is connected to the PCC.

**Figure 3.**
Grid voltage waveforms and spectrums: (**a**,**c**) when the load is not connected to the PCC and (**b**,**d**) when the load is connected to the PCC.

**Figure 4.**
(**a**) First-order filter, (**b**) impedance versus frequency characteristic of the first-order filter.

**Figure 4.**
(**a**) First-order filter, (**b**) impedance versus frequency characteristic of the first-order filter.

**Figure 5.**
Supply network voltage and current waveforms: before (**a**,**c**) and after (**b**,**d**) the capacitor bank connection.

**Figure 5.**
Supply network voltage and current waveforms: before (**a**,**c**) and after (**b**,**d**) the capacitor bank connection.

**Figure 6.**
Supply network voltage and current spectrums before (**a**,**c**) and (**b**,**d**) after the capacitor bank connection; (**e**) spectrum of capacitor bank current.

**Figure 6.**
Supply network voltage and current spectrums before (**a**,**c**) and (**b**,**d**) after the capacitor bank connection; (**e**) spectrum of capacitor bank current.

**Figure 7.**
Impedance versus frequency characteristics of power system measured at the input of thyristor bridge: (**a**) with the input reactor L; (**b**) without the input reactor L (just to show that the observed series resonance depends on the L presence).

**Figure 7.**
Impedance versus frequency characteristics of power system measured at the input of thyristor bridge: (**a**) with the input reactor L; (**b**) without the input reactor L (just to show that the observed series resonance depends on the L presence).

**Figure 8.**
(**a**) Single-tuned filter, (**b**) expressions used to compute the single-tuned filter parameters. ω_{re}—resonance frequency, n_{re}—harmonic order at the resonance.

**Figure 8.**
(**a**) Single-tuned filter, (**b**) expressions used to compute the single-tuned filter parameters. ω_{re}—resonance frequency, n_{re}—harmonic order at the resonance.

**Figure 9.**
Single-tuned filter impedance versus frequency characteristics: (**a**) when the filter is tuned to the frequencies lower or equal to the frequency of the 5th harmonic; (**b**) and when it is tuned to the frequencies higher or equal to the frequency of the 5th harmonic.

**Figure 9.**
Single-tuned filter impedance versus frequency characteristics: (**a**) when the filter is tuned to the frequencies lower or equal to the frequency of the 5th harmonic; (**b**) and when it is tuned to the frequencies higher or equal to the frequency of the 5th harmonic.

**Figure 10.**
The single-tuned filter is tuned to the frequencies lower or equal to the frequency of the 5th harmonic: (**a**) grid voltage and (**b**) its spectrum; (**c**) grid current and (**d**) its spectrum before and after the filter connection.

**Figure 10.**
The single-tuned filter is tuned to the frequencies lower or equal to the frequency of the 5th harmonic: (**a**) grid voltage and (**b**) its spectrum; (**c**) grid current and (**d**) its spectrum before and after the filter connection.

**Figure 11.**
The single-tuned filter is tuned to the frequencies higher or equal to the frequency of the 5th harmonic: (**a**) grid voltage and (**b**) its spectrum; (**c**) grid current, and (**d**) its spectrum before and after the filter connection.

**Figure 11.**
The single-tuned filter is tuned to the frequencies higher or equal to the frequency of the 5th harmonic: (**a**) grid voltage and (**b**) its spectrum; (**c**) grid current, and (**d**) its spectrum before and after the filter connection.

**Figure 12.**
Impedance frequency characteristics of the power system seen from the rectifier input when the single filter is tuned to the frequencies lower than 250 Hz (**a**) and when it is tuned to the frequency higher than 250 Hz (**b**).

**Figure 12.**
Impedance frequency characteristics of the power system seen from the rectifier input when the single filter is tuned to the frequencies lower than 250 Hz (**a**) and when it is tuned to the frequency higher than 250 Hz (**b**).

**Figure 13.**
(**a**) Second-order filter, (**b**) expressions used to compute the second-order filter parameters. R_{Lf}—reactor resistance.

**Figure 13.**
(**a**) Second-order filter, (**b**) expressions used to compute the second-order filter parameters. R_{Lf}—reactor resistance.

**Figure 14.**
(**a**) Second-order filter impedance versus frequency characteristics for different damping resistance value: (**b**) R = 60 Ω, (**c**) R = 18 Ω, (**d**) R = 8 Ω, (**e**) R = 3 Ω.

**Figure 14.**
(**a**) Second-order filter impedance versus frequency characteristics for different damping resistance value: (**b**) R = 60 Ω, (**c**) R = 18 Ω, (**d**) R = 8 Ω, (**e**) R = 3 Ω.

**Figure 15.**
(**a**) Grid voltage and (**b**) its spectrum (p.u.); (**c**) grid current and (**d**) its spectrum (p.u.) before and after the filter connection.

**Figure 15.**
(**a**) Grid voltage and (**b**) its spectrum (p.u.); (**c**) grid current and (**d**) its spectrum (p.u.) before and after the filter connection.

**Figure 16.**
(**a**) Active power at the PCC (P_{S}) and filter terminals (P_{f}), (**b**) grid voltage THD, (**c**) grid current THD.

**Figure 16.**
(**a**) Active power at the PCC (P_{S}) and filter terminals (P_{f}), (**b**) grid voltage THD, (**c**) grid current THD.

**Figure 17.**
Second-order filter work efficiency in terms of harmonic mitigation.

**Figure 17.**
Second-order filter work efficiency in terms of harmonic mitigation.

**Figure 18.**
Power system impedance versus frequency characteristic observed from the thyristor bridge input terminals for different values of the filter damping resistance.

**Figure 18.**
Power system impedance versus frequency characteristic observed from the thyristor bridge input terminals for different values of the filter damping resistance.

**Figure 19.**
(**a**) Third-order filter, (**b**) expressions used to compute the third-order filter parameters. n_{re1}—harmonic order at the series resonance, n_{re2}—harmonic order at the parallel resonance.

**Figure 19.**
(**a**) Third-order filter, (**b**) expressions used to compute the third-order filter parameters. n_{re1}—harmonic order at the series resonance, n_{re2}—harmonic order at the parallel resonance.

**Figure 20.**
Third-order filter impedance versus frequency characteristics for different damping resistance value: (**a**) R = 0.08 Ω, (**b**) R = 0.25 Ω, (**c**) R = 1.25 Ω, (**d**) R = 8 Ω.

**Figure 20.**
Third-order filter impedance versus frequency characteristics for different damping resistance value: (**a**) R = 0.08 Ω, (**b**) R = 0.25 Ω, (**c**) R = 1.25 Ω, (**d**) R = 8 Ω.

**Figure 21.**
Grid voltage waveforms and spectrums after the 3rd order filter connection with different value of damping resistance.

**Figure 21.**
Grid voltage waveforms and spectrums after the 3rd order filter connection with different value of damping resistance.

**Figure 22.**
Grid current waveforms and spectrums after the 3rd order filter connection with different value of damping resistance.

**Figure 22.**
Grid current waveforms and spectrums after the 3rd order filter connection with different value of damping resistance.

**Figure 23.**
Third-order filter work efficiency.

**Figure 23.**
Third-order filter work efficiency.

**Figure 24.**
(**a**) Active power at the PCC (P_{S}) and filter terminals (P_{f}), (**b**) grid voltage THD, (**c**) grid current THD.

**Figure 24.**
(**a**) Active power at the PCC (P_{S}) and filter terminals (P_{f}), (**b**) grid voltage THD, (**c**) grid current THD.

**Figure 25.**
Power system impedance versus frequency characteristics observed at the thyristor bridge input for different values of the third-order filter damping resistance (R).

**Figure 25.**
Power system impedance versus frequency characteristics observed at the thyristor bridge input for different values of the third-order filter damping resistance (R).

**Figure 26.**
(**a**) C-type filter, (**b**) expressions used to compute its parameters.

**Figure 26.**
(**a**) C-type filter, (**b**) expressions used to compute its parameters.

**Figure 27.**
(**a**) C-type filter impedance versus frequency characteristics for different damping resistance value: (**b**) R = 1.25 Ω, (**c**) R = 8 Ω, (**d**) R = 25 Ω, (**e**) R = 2500 Ω.

**Figure 27.**
(**a**) C-type filter impedance versus frequency characteristics for different damping resistance value: (**b**) R = 1.25 Ω, (**c**) R = 8 Ω, (**d**) R = 25 Ω, (**e**) R = 2500 Ω.

**Figure 28.**
Grid voltage waveforms and spectrums after the C-type filter connection with different value of damping resistance.

**Figure 28.**
Grid voltage waveforms and spectrums after the C-type filter connection with different value of damping resistance.

**Figure 29.**
Grid current waveforms and spectrums after the C-type filter connection with different value of damping resistance.

**Figure 29.**
Grid current waveforms and spectrums after the C-type filter connection with different value of damping resistance.

**Figure 30.**
Active power at the PCC (P_{S}) and filter terminals (P_{f}) (**a**), grid voltage THD (**b**), grid current THD (**c**) obtained before and after the C-type filter connection with different damping resistance value.

**Figure 30.**
Active power at the PCC (P_{S}) and filter terminals (P_{f}) (**a**), grid voltage THD (**b**), grid current THD (**c**) obtained before and after the C-type filter connection with different damping resistance value.

**Figure 31.**
(**a**,**b**) Power system impedance versus frequency characteristics observed at the input of thyristor bridge for different values of the C-type filter resistance.

**Figure 31.**
(**a**,**b**) Power system impedance versus frequency characteristics observed at the input of thyristor bridge for different values of the C-type filter resistance.

**Figure 32.**
C-type filter work efficiency on the harmonic mitigation.

**Figure 32.**
C-type filter work efficiency on the harmonic mitigation.

**Figure 33.**
Compared topologies of PHF.

**Figure 33.**
Compared topologies of PHF.

**Figure 34.**
Comparison spectrums: (**a**,**b**) grid voltage and current 5th harmonic; (**c**,**d**) grid voltage and current THD; (**e**) filter power losses.

**Figure 34.**
Comparison spectrums: (**a**,**b**) grid voltage and current 5th harmonic; (**c**,**d**) grid voltage and current THD; (**e**) filter power losses.

**Figure 35.**
Grid voltage and current spectrums before and after the filter connection. The filter damping resistance has the value of: (**a**) 0.08 Ω, (**b**) 1.5 Ω, (**c**) 8 Ω, (**d**) 25 Ω.

**Figure 35.**
Grid voltage and current spectrums before and after the filter connection. The filter damping resistance has the value of: (**a**) 0.08 Ω, (**b**) 1.5 Ω, (**c**) 8 Ω, (**d**) 25 Ω.

**Figure 36.**
(**a**) Laboratory set-up, (**b**) equivalent circuit of the laboratory set-up.

**Figure 36.**
(**a**) Laboratory set-up, (**b**) equivalent circuit of the laboratory set-up.

**Figure 37.**
Electrical network (grid) equivalent circuit with the parameters.

**Figure 37.**
Electrical network (grid) equivalent circuit with the parameters.

**Figure 38.**
PCC voltage waveforms (**a**) together with the spectrum (**b**) when the filter and load are not connected.

**Figure 38.**
PCC voltage waveforms (**a**) together with the spectrum (**b**) when the filter and load are not connected.

**Figure 39.**
(**a**) Frequency characteristic of the laboratory investigated single-tuned filter, (**b**) simulated impedance versus frequency characteristic seen from the transformer input.

**Figure 39.**
(**a**) Frequency characteristic of the laboratory investigated single-tuned filter, (**b**) simulated impedance versus frequency characteristic seen from the transformer input.

**Figure 40.**
Measured waveforms of: (**a**) the grid voltage and current waveforms, (**b**) transformer input and single-tuned filter terminals.

**Figure 40.**
Measured waveforms of: (**a**) the grid voltage and current waveforms, (**b**) transformer input and single-tuned filter terminals.

**Figure 41.**
Measured waveforms and spectrums of the PCC voltage before (**a**) and after (**b**) the filter connection.

**Figure 41.**
Measured waveforms and spectrums of the PCC voltage before (**a**) and after (**b**) the filter connection.

**Figure 42.**
Measured waveforms and spectrums of the grid current before (**a**) and after (**b**) the filter connection.

**Figure 42.**
Measured waveforms and spectrums of the grid current before (**a**) and after (**b**) the filter connection.

**Figure 43.**
Laboratory equivalent circuit with the capacitors bank connected to the PCC.

**Figure 43.**
Laboratory equivalent circuit with the capacitors bank connected to the PCC.

**Figure 44.**
(**a**) PCC voltage and (**b**) its spectrum, (**c**) grid current waveforms, and (**d**) its spectrum.

**Figure 44.**
(**a**) PCC voltage and (**b**) its spectrum, (**c**) grid current waveforms, and (**d**) its spectrum.

**Table 1.**
Computed equivalent parameters of the simulated electrical grid see from the point of common coupling (PCC) when no load is connected.

**Table 1.**
Computed equivalent parameters of the simulated electrical grid see from the point of common coupling (PCC) when no load is connected.

S_{SC_PCC} [MVA] | I_{SC_PCC} [A] | L_{S} [µH] | R_{S} [mΩ] | Z_{S} [Ω] | Z_{S(5)} [Ω] |
---|

0.33 | 481.12 | 681.82 | 425.5 | 0.480 | 1.15 |

**Table 2.**
First-order filter parameters together with expressions used to compute these parameters.

**Table 2.**
First-order filter parameters together with expressions used to compute these parameters.

${\mathit{C}}_{f}\mathbf{=}\frac{{\mathit{Q}}_{f}}{{\mathit{U}}_{f}^{\mathbf{2}}{\omega}_{\mathbf{\left(}\mathbf{1}\mathbf{\right)}}}{\mathit{Z}}_{f}\mathbf{=}{\mathit{R}}_{f}\mathbf{+}\frac{{\mathit{U}}_{f}^{\mathbf{2}}}{{\mathit{Q}}_{f}}$ |
---|

Q_{f} [Var] | C_{f} [µF] | Z_{f(1)} [Ω] | Z_{f(5)} [Ω] | Z_{f(7)} [Ω] | R_{f} [Ω] |
---|

−2172.5 | 130.72 | 24.35 | 4.87 | 3.48 | 0.25 |

**Table 3.**
Single-tuned filter parameters. The filter is tuned to the frequencies lower or equal to the frequency of the harmonic to be eliminated (5th).

**Table 3.**
Single-tuned filter parameters. The filter is tuned to the frequencies lower or equal to the frequency of the harmonic to be eliminated (5th).

f_{re} [Hz] | n_{re} | C_{f} [µF] | L_{f} [mH] | Z_{f(5)} [Ω] | Z_{f(1)} [Ω] | Q_{f} [Var] |
---|

205 | 4.1 | 122.95 | 4.9 | 2.52 | 24.35 | −2172.5 |

235 | 4.70 | 124.81 | 3.7 | 0.67 |

245.5 | 4.85 | 125.17 | 3.4 | 0.32 |

250 | 5 | 125.49 | 3.2 | 0.00 |

**Table 4.**
Single-tuned filter parameters. The filter is tuned to the frequencies higher or equal to the frequency of the harmonic to be eliminated (5th).

**Table 4.**
Single-tuned filter parameters. The filter is tuned to the frequencies higher or equal to the frequency of the harmonic to be eliminated (5th).

f_{re} [Hz] | n_{re} | C_{f} [µF] | L_{f} [mH] | Z_{f(5)} [Ω] | Z_{f(1)} [Ω] | Q_{f} [Var] |
---|

250 | 5 | 125.49 | 3.2 | 0 | 24.35 | −2172.5 |

285 | 5.70 | 126.70 | 2.5 | 1.16 |

292.5 | 5.85 | 127 | 2.3 | 1.35 |

305 | 6.1 | 127.21 | 2.1 | 1.64 |

**Table 5.**
Influence of the single-tuned filter detuning phenomenon on the grid voltage and current THD.

**Table 5.**
Influence of the single-tuned filter detuning phenomenon on the grid voltage and current THD.

(a) Grid Voltage and Current THD When the Single-Tuned Filter Is Tuned to the Frequencies Lower or Equal to the Frequency of the 5th Harmonic | (b) Grid Voltage and Current THD When the Single-Tuned Filter Is Tuned to the Frequencies Higher or Equal to the Frequency of the 5th Harmonic |
---|

n_{re} | THD_{(Us)} [%] | THD_{(Is)} [%] | n_{re} | THD_{(Us)} [%] | THD_{(Is)} [%] |
---|

Without filter | 4.93 | 36 | Without filter | 4.93 | 36 |

4.1 | 4.24 | 49.7 | 5 | 3.79 | 18.74 |

4.70 | 3.94 | 31.54 | 5.70 | 6.49 | 172.17 |

4.85 | 3.84 | 22.35 | 5.85 | 6.42 | 169.08 |

5 | 3.79 | 18.74 | 6.1 | 5.77 | 146.29 |

**Table 6.**
Parameters of 2nd order filter (Q_{f} = −2172.5Var).

**Table 6.**
Parameters of 2nd order filter (Q_{f} = −2172.5Var).

R [Ω] | Z_{f(5)} [Ω] | Z_{f(1)} [Ω] | R_{f} [mΩ] | C_{f} [µF] | L_{f} [mH] | n_{re} | q′ |
---|

inf | 0.32 | 24.35 | 74.9 | 125.17 | 3.4 | 4.85 | 14.25 |

60 | 0.61 | 24.35 |

18 | 1.55 | 24.36 |

8 | 2.89 | 24.38 |

3 | 4.46 | 24.52 |

**Table 7.**
Parameters of the 3rd order filter (Q_{f} = −2172.5Var).

**Table 7.**
Parameters of the 3rd order filter (Q_{f} = −2172.5Var).

R [mΩ] | Z_{f(5)} [Ω] | Z_{f(1)} [mΩ] | R_{Lf} [mΩ] | C_{f1} [µF] | C_{f2} [µF] | L_{f} [mH] | n_{re1} | n_{re2} |
---|

80 | 1.07 | 24349.8 | 4.3 | 128.74 | 242.70 | 1.2 | 4.85 | 6 |

250 | 1.36 | 24349.8 |

750 | 2.28 | 24349.9 |

1250 | 2.70 | 24350.0 |

8000 | 3.11 | 24352.9 |

**Table 8.**
Parameters of C-type filter (Q_{f} = −2172.5Var).

**Table 8.**
Parameters of C-type filter (Q_{f} = −2172.5Var).

R [Ω] | Z_{f(5)} [mΩ] | Z_{f(1)} [mΩ] | R_{Lf} [mΩ] | C_{fa} [µF] | C_{fb} [µF] | L_{f} [mH] | n_{re} |
---|

1.25 | 4.73 | 24.34 | 12.7 | 130.72 | 2900 | 3.4 | 4.85 |

5 | 3.51 |

8 | 2.67 |

25 | 1.04 |

2500 | 0.32 |

**Table 9.**
Comparison assumptions.

**Table 9.**
Comparison assumptions.

| Q_{f} = −2172.5Var, θ = 57º, n_{re} = 4.85, q’= 85 |
---|

First-Order | Single-Tuned | Second-Order | Third-Order | C-Type |
---|

R [Ω] | - | - | 0.08 | 0.08 | 0.08 |

- | - | 1.25 | 1.25 | 1.25 |

- | - | 8 | 8 | 8 |

- | - | 25 | 25 | 25 |

R_{Lf} [Ω] | - | 0.0127 | 0.0127 | 0.0043 | 0.0127 |

**Table 10.**
Measured parameters of the investigated single-tuned filter.

**Table 10.**
Measured parameters of the investigated single-tuned filter.

Z_{f(5)} [mΩ] | Z_{f(1)} [mΩ] | R_{Lf} [mΩ] | C_{f∆} [µF] | L_{f} [µF] | Q_{f} [kvar] | n_{re} | q′ |
---|

718.5 | 1839 | 1.68 | 600.84 | 232.49 | −27.87 | 4.91 | 43.47 |

**Table 11.**
Measured powers before and after the filter connection.

**Table 11.**
Measured powers before and after the filter connection.

Grid (PCC) | Transformer Input | Filter |
---|

P_{S} [kW] | Q_{S} [kVar] | P_{T} [kW] | Q_{T} [kVar] | P_{f} [kW] | Q_{f} [kVar] |
---|

Before the filter connection |

2.52 | 23.29 | 2.52 | 23.29 | - | - |

After the filter connection |

2.30 | −3.56 | 2.46 | 24.44 | −0.15 | −28.00 |

**Table 12.**
Measured grid voltage and current parameters before the single-tuned filter connection.

**Table 12.**
Measured grid voltage and current parameters before the single-tuned filter connection.

n | U_{S} [V] | I_{S} [A] |
---|

Ampl | [°] | Ampl | [°] |
---|

1st | 223.49 | −90.72 | 104.84 | −174.54 |

3rd | 3.33 | 47.96 | 1.42 | −99.53 |

5th | 6.33 | 85.39 | 20.18 | −160.96 |

7th | 4.86 | −87.65 | 8.36 | 30.13 |

9th | 1.18 | −31.38 | 1.06 | 126.86 |

11th | 1.86 | −63.59 | 6.08 | 47.50 |

13th | 0.26 | −156.65 | 5.24 | −128.10 |

TTHD [%] | 5.74 | 25.18 |

**Table 13.**
Measured grid voltage and current parameters after the single-tuned filter connection.

**Table 13.**
Measured grid voltage and current parameters after the single-tuned filter connection.

n | U_{S} [V] | I_{S} [A] | I_{f} [A] | I_{T} [A] |
---|

Ampl | [°] | Ampl | [°] | Ampl | [°] | Ampl | [°] |
---|

1st | 227.38 | −89.29 | 18.68 | −32.12 | 123.15 | 1.03 | 108.04 | −173.52 |

3rd | 3.77 | 53.88 | 1.91 | −114.61 | 0.86 | 155.63 | 2.09 | −90.97 |

5th | 8.64 | 82.21 | 42.37 | −174.11 | 23.34 | 169.16 | 21.15 | −155.31 |

7th | 10.93 | −116.56 | 49.68 | −19.30 | 46.61 | −28.71 | 8.46 | 44.35 |

9th | 2.23 | −102.75 | 1.89 | 54.78 | 2.65 | 8.55 | 1.90 | 142.74 |

11th | 1.15 | 178.92 | 6.78 | −25.97 | 9.54 | −71.33 | 6.80 | 63.45 |

13th | 0.63 | 28.27 | 3.96 | −126.15 | 1.75 | 104.06 | 5.27 | −111.42 |

TTHD [%] | 6.92 | 356.21 | 44.67 | 25.06 |