Triple-Passive Harmonic Suppression Method for Delta-Connected Rectifier to Reduce the Harmonic Content on the Grid Side

Abstract

With the development of distributed energy sources such as photovoltaic and wind power, power grids have imposed increasingly higher requirements on power quality. As common nonlinear loads in power grids, multi-pulse rectifiers (MPRs) inject significant harmonics into the grid side. To reduce harmonic pollution at the source, this paper proposes a novel triple-passive harmonic suppression method to reduce the input current harmonics of MPRs. The proposed 48-pulse rectifier comprises a main circuit based on delta-connected auto-transformer (DCT) and a triple-passive harmonic suppression circuit (TPHSC). The TPHSC consists of two interphase reactors (IPRs) and eight diodes. Based on Kirchhoff’s Current Law (KCL), the output currents of the main circuit are calculated, and the operating modes of the TPHSC are analyzed. From the main circuit’s output currents and the DCT topology, the rectifier’s input currents are derived, and the optimal turns ratio of the IPRs for minimizing the input current total harmonic distortion (THD) is determined. The total capacity of the IPRs accounts for only 2.3% of the output load power. Experimental results show that the measured input current THD is close to the theoretical value of 3.8%. Overall, the proposed rectifier offers a cost-effective solution with stronger harmonic suppression capability, making it suitable for applications requiring low grid harmonic pollution.

1. Introduction

With the integration of distributed energy sources such as photovoltaic and wind power, power systems are imposing increasingly higher requirements on power quality. As common loads for accommodating new energy sources, a multi-pulse rectifier (MPR) based on a phase-shifting transformer and a diode-bridge offers distinct advantages such as a simplified structure, high reliability, a superior overload capacity, and lower overall costs. These characteristics make them particularly suitable for high-power rectification applications for the accommodation of new energy sources, including hydrogen production, high-voltage direct current transmission, urban rail transit power supply, and metal smelting [1,2,3,4]. However, harmonic pollution resulting from the nonlinear behaviors of rectification devices adversely affect the power quality on both the grid and output sides.

To improve the harmonic suppression abilities of MPRs, DC link harmonic suppression methods are proposed, and these methods are divided into active methods and passive methods. The active methods employ magnetic devices and controllable switches, while the passive methods rely on magnetic devices and diodes. In MPRs, the size and cost of the magnetic components are primarily determined by their required rated apparent power, which is referred to as the capacity of the device. To enable a fair comparison across different designs, this capacity is conventionally normalized and expressed as a percentage of the system’s total output power. A lower percentage indicates a more compact and cost-effective magnetic design.

The active methods in [5] used a two-stage auxiliary circuit at the DC link of the MPRs to reduce the harmonics to an acceptable level. The THD of the input currents was approximately 3.6%, and the capacity of the magnetic device of the auxiliary circuit was approximately 3.1% of the output power. Although effective in reducing THD, this approach does not reduce output voltage ripple and requires complex control circuitry, increasing design complexity and cost, and thus, the passive methods are proposed. The existing passive methods are divided into pulse-doubling methods (single-passive harmonic suppression methods) and pulse-tripling methods (dual-passive harmonic suppression methods). The pulse doubling method in [6] used a double-tap changer and two diodes to modulate the 12-pulse rectifier into a 24-pulse rectifier. The suppression circuits of the pulse-doubling methods in [7,8] used the IPRs with the secondary windings and the diodes (one secondary winding with four diodes or two secondary windings with two diodes) to suppress harmonics. The pulse-tripling methods in [9,10] were the combination of the above two types of pulse-doubling methods (the suppression circuits consisted of a double-tap changer with a secondary winding and six diodes or a double-tap changer with two secondary windings and four diodes). In recent years, a variety of novel harmonic suppression circuit topologies with low loss and low capacity have been proposed [11,12,13,14,15,16,17,18]. However, in theory, existing passive harmonic suppression circuits can only reduce the harmonic content of a rectifier to between one-half and one-third of the original level, resulting in limited performance.

A delta-connected auto-transformer (DCT) offers a significant reduction in system volume and a higher power density with a low capacity [19,20,21]. Owing to these advantages, it has been widely adopted as the phase-shifting transformer in MPRs. Figure 1 shows a typical 12-pulse delta-connected rectifier based on a DCT. It can be seen that the two rectifier bridges are connected in parallel by the IPR to power the load.

This paper proposes a novel triple-passive harmonic suppression method which employs three passive harmonic suppression circuits to significantly enhance the performance of the twelve-pulse rectifier shown in Figure 1. The proposed approach transforms the 12-pulse rectifier into a 48-pulse system with a simplified structure. The proposed triple-passive harmonic suppression circuit (TPHSC) consists of only eight diodes and two compact interphase reactors (IPRs), with a combined capacity of as low as 2.3% of the output power, while effectively quadrupling the pulse number. The experimental results demonstrate that the proposed 48-pulse rectifier achieves an input current THD of 3.99%, significantly lower than the 13.90% THD of the conventional delta-connected rectifier.

Compared with existing passive harmonic suppression methods, the proposed design improves harmonic suppression performance by one-third while simultaneously reducing the total capacity of the magnetic components. The proposed 48-pulse rectifier is suitable for applications involving the accommodation of new energy sources, where it can effectively reduce the grid-side harmonic content.

2. Topology and Characteristic Analysis of the Proposed Parallel-Connected 48-Pulse Rectifier

Figure 2 shows the circuit configuration of the proposed 48-pulse rectifier. It consists of the main circuit based on a DCT with two three-phase diode-bridge rectifiers, the zero sequence blocking transformer (ZSBT), the TPHSC, and the load.

Based on Figure 2, it can be seen that u_L and i_L represent the output voltage and current of the proposed 48-pulse rectifier, respectively; u_L1 and u_L2 represent the output voltages of the main circuit; i_L1 and i_L2 represent the output currents of the main circuit; and i_s represents the output current of the secondary side full-bridge rectifier of the TPHSC₁.

The phasor diagram of the DCT is shown in Figure 3a, and its winding structure is described in Figure 3b. The 12-pulse rectifier requires a 30° phase shift between the two sets of three-phase voltages supplied by the DCT. To meet this requirement, the angle α in Figure 3a is set to 15°. Thus, in Figure 3b, the winding turns ratio of the DCT can be obtained such that $N_1:N_2=\sqrt{3}:\left(2-\sqrt{3}\right)$.

In Figure 2, the three-phase input voltage is shown as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{a}}=\sqrt{2}E\sin (\omega t)\\ u_{\mathrm{b}}=\sqrt{2}E\sin (\omega t-{\displaystyle \frac{2}{3}}\pi )\\ u_{\mathrm{c}}=\sqrt{2}E\sin (\omega t+{\displaystyle \frac{2}{3}}\pi )\end{array}\right.,$$

(1)

where E is the RMS value of the input voltage.

In Figure 3a, the two sets of three-phase output voltage of the DCT are calculated as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{a}1}=2(\sqrt{3}-1)E\sin (\omega t+\pi /12)\\ u_{\mathrm{b}1}=2(\sqrt{3}-1)E\sin (\omega t-7\pi /12)\\ u_{\mathrm{c}1}=2(\sqrt{3}-1)E\sin (\omega t+3\pi /4)\end{array}\right.\begin{array}{c}\end{array}\left\{\begin{array}{l}{u}_{\mathrm{a}2}=2(\sqrt{3}-1)E\sin (\omega t-\pi /12)\\ u_{\mathrm{b}2}=2(\sqrt{3}-1)E\sin (\omega t-3\pi /4)\\ u_{\mathrm{c}2}=2(\sqrt{3}-1)E\sin (\omega t+7\pi /12)\end{array}\right..$$

(2)

According to the KCL and the principle of ampere-turns balance, the input current i_a of the rectifier is obtained as follows:

$$\begin{array}{l}{i}_{\mathrm{a}}=S_{\mathrm{a}1}{i}_{\mathrm{L}1}+S_{\mathrm{a}2}{i}_{\mathrm{L}2}+{\displaystyle \frac{2-\sqrt{3}}{\sqrt{3}}}(S_{\mathrm{c}2}{i}_{\mathrm{L}2}-S_{\mathrm{c}1}{i}_{\mathrm{L}1}+S_{\mathrm{b}1}{i}_{\mathrm{L}1}-S_{\mathrm{b}2}{i}_{\mathrm{L}2})\\ S_{\mathrm{a}1}=\left\{\begin{array}{l}0,\hspace{1em}\omega t\in [0,\pi /12)\\ 1,\hspace{1em}\omega t\in [\pi /12,3\pi /4)\\ 0,\hspace{1em}\omega t\in [3\pi /4,13\pi /12)\\ -1,\omega t\in [13\pi /12,7\pi /4)\\ 0,\hspace{1em}\omega t\in [7\pi /4,2\pi ]\end{array}\right.\begin{array}{c}\end{array}\left\{\begin{array}{l}{S}_{\mathrm{b}1}=S_{\mathrm{a}1}\angle -2\pi /3\\ S_{\mathrm{c}1}=S_{\mathrm{a}1}\angle +2\pi /3\\ S_{\mathrm{a}2}=S_{\mathrm{a}1}\angle -\pi /6\\ S_{\mathrm{b}2}=S_{\mathrm{b}1}\angle -\pi /6\\ S_{\mathrm{c}2}=S_{\mathrm{c}1}\angle -\pi /6\end{array}\right.\end{array},$$

(3)

where S_a1, S_a2, S_b1, S_b2, S_c1, and S_c2 represent the switch functions of phases a₁, a₂, b₁, b₂, c₁, and c₂, respectively.

According to (3), the expression of i_a within a quarter of a period is obtained as follows:

$$i_{\mathrm{a}}=\left\{\begin{array}{ll}{\displaystyle \frac{4\sqrt{3}-6}{3}}(i_{\mathrm{L}2}-i_{\mathrm{L}1}),& \omega t\in [0,\pi /12)\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{L}1}+{\displaystyle \frac{4\sqrt{3}-6}{3}}{i}_{\mathrm{L}2},& \omega t\in [\pi /12,\pi /4)\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{L}1}+{\displaystyle \frac{2\sqrt{3}}{3}}{i}_{\mathrm{L}2},& \omega t\in [\pi /4,5\pi /12)\hfill \\ {\displaystyle \frac{2\sqrt{3}}{3}}(i_{\mathrm{L}2}++i_{\mathrm{L}1}),& \omega t\in [5\pi /12,\pi /2)\hfill \end{array}\right..$$

(4)

In Figure 2, the TPHSC consists of the TPHSC₁ and the TPHSC₂. The TPHSC₁ consists of the IPR₁ with secondary winding and six diodes, while the TPHSC₂ consists of the IPR₂ with two diodes. The winding structures of the IPR₁ and IPR₂ are shown in Figure 4.

As shown in Figure 4, the primary winding (n₁–n₂) of IPR₁ has N_p1 turns with the terminal voltage u_p1, and its secondary winding (s₁–s₂) has N_s turns with the terminal voltage u_s. For IPR_2, its primary winding (n₃–n₄) has N_p2 turns with the terminal voltage u_p2. It is obvious that u_s:u_p1 = N_s:N_p1. The primary tap winding (t₁–t₂) of IPR₁ has 2N_p11 turns, and the primary tap winding (t₃–t₄) of IPR₂ has 2N_p21 turns. Thus, the tap turns-ratio parameters a_m1 and a_m2 and the secondary-to-primary turns ratio m are defined as follows:

$$a_{\mathrm{m}1}={\displaystyle \frac{N_{\mathrm{p}11}}{N_{\mathrm{p}1}}},a_{\mathrm{m}2}={\displaystyle \frac{N_{\mathrm{p}21}}{N_{\mathrm{p}2}}},m={\displaystyle \frac{N_{\mathrm{s}}}{N_{\mathrm{p}1}}}.$$

(5)

Based on Figure 2, when supplied with a three-phase voltage, the rectifier’s main circuit generates two sets of output currents, i_L1 and i_L2, which share the same waveform but are phase-shifted by 30°. According to the difference between i_L1 and i_L2, the four operating modes of the TPHSC are as described in Figure 5.

(1)

Mode1: When i_L1 > i_L2, the TPHSC operates as per Figure 5a. According to the operating principle of the transformer, the KCL, and the principle of ampere-turns balance, it is obtained that:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}1}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}-N_{\mathrm{p}11})=i_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}+N_{\mathrm{p}11})+i_{\mathrm{s}}{N}_{\mathrm{s}}\\ i_{\mathrm{L}1}({\displaystyle \frac{N_{\mathrm{p}2}}{2}}-N_{\mathrm{p}21})=i_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}2}}{2}}+N_{\mathrm{p}21})\\ i_{\mathrm{L}}=i_{\mathrm{L}1}+i_{\mathrm{L}2}+i_{\mathrm{s}}\end{array}\right..$$

(6)

(2)

Mode2: As i_L1 decreases and i_L2 increases, the secondary side full-bridge rectifier of TPHSC₁ turns off, and the TPHSC operates as per Figure 5b. i_L1 and i_L2 can be shown as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}1}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}-N_{\mathrm{p}11})=i_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}+N_{\mathrm{p}11})\\ i_{\mathrm{L}}=i_{\mathrm{L}1}+i_{\mathrm{L}2}\end{array}\right..$$

(7)

(3)

Mode3: Similarly, when i_L2 increases and i_L1 decreases until i_L2 > i_L1, the TPHSC operates as per Figure 5c, and i_L1 and i_L2 are calculated as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}-N_{\mathrm{p}11})=i_{\mathrm{L}1}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}+N_{\mathrm{p}11})\\ i_{\mathrm{L}}=i_{\mathrm{L}1}+i_{\mathrm{L}2}\end{array}\right..$$

(8)

(4)

Mode4: As i_L1 decreases and i_L2 increases, the secondary side full-bridge rectifier of TPHSC₁ turns on and the TPHSC operates as per Figure 5d, and i_L1 and i_L2 can be shown as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}-N_{\mathrm{p}11})=i_{\mathrm{L}1}({\displaystyle \frac{N_{\mathrm{p}1}}{2}}+N_{\mathrm{p}11})+i_{\mathrm{s}}{N}_{\mathrm{s}}\\ i_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}2}}{2}}-N_{\mathrm{p}21})=i_{\mathrm{L}2}({\displaystyle \frac{N_{\mathrm{p}2}}{2}}+N_{\mathrm{p}21})\\ i_{\mathrm{L}}=i_{\mathrm{L}1}+i_{\mathrm{L}2}+i_{\mathrm{s}}\end{array}\right..$$

(9)

Based on Equations (6)–(9), i_L1 and i_L2 are calculated as follows:

$$i_{\mathrm{L}1}=\left\{\begin{array}{l}{i}_{\mathrm{m}1}\mathrm{mode}1\\ i_{\mathrm{m}2}\mathrm{mode}2\\ i_{\mathrm{m}3}\mathrm{mode}3\\ i_{\mathrm{m}4}\mathrm{mode}4\end{array}\right.\begin{array}{c}\end{array}{i}_{\mathrm{L}2}=\left\{\begin{array}{l}{i}_{\mathrm{m}4}\mathrm{mode}1\\ i_{\mathrm{m}3}\mathrm{mode}2\\ i_{\mathrm{m}2}\mathrm{mode}3\\ i_{\mathrm{m}1}\mathrm{mode}4\end{array}\right.\begin{array}{c}\end{array}\left\{\begin{array}{l}{i}_{\mathrm{m}1}={\displaystyle \frac{m(0.5+a_{\mathrm{m}2})}{m+a_{\mathrm{m}2}-a_{\mathrm{m}1}}}{i}_{\mathrm{L}}\\ i_{\mathrm{m}2}=(0.5+a_{\mathrm{m}1})i_{\mathrm{L}}\\ i_{\mathrm{m}3}=(0.5-a_{\mathrm{m}1})i_{\mathrm{L}}\\ i_{\mathrm{m}4}={\displaystyle \frac{0.5-a_{\mathrm{m}2}}{0.5+a_{\mathrm{m}2}}}{i}_{\mathrm{m}1}\end{array}\right..$$

(10)

As can be seen from Equation (10), i_L1 and i_L2 are eight-step waves with a phase difference of 30° and four current levels.

Based on (6)–(9), the output current i_s of the secondary side full-bridge rectifier of IPR₁ is calculated as follows:

$$i_{\mathrm{s}}=\left\{\begin{array}{ll}{\displaystyle \frac{a_{\mathrm{m}2}-a_{\mathrm{m}1}}{m+a_{\mathrm{m}2}-a_{\mathrm{m}1}}}{i}_{\mathrm{L}}& \mathrm{mode}1\hfill \\ 0& \mathrm{mode}2\hfill \\ 0& \mathrm{mode}3\hfill \\ {\displaystyle \frac{a_{\mathrm{m}2}-a_{\mathrm{m}1}}{m+a_{\mathrm{m}2}-a_{\mathrm{m}1}}}{i}_{\mathrm{L}}& \mathrm{mode}4\hfill \end{array}\right..$$

(11)

It is specified that the direction of current flowing from tap t₁ (t₃) to t₂ (t₄) is positive, and thus, the expressions for the circulating currents in the windings between the two taps on the primary sides of IPR₁ and IPR₂ can be obtained as follows:

$$i_{\mathrm{t}1\mathrm{t}2}=\left\{\begin{array}{ll}-i_{\mathrm{m}4}& \mathrm{mode}1\hfill \\ -i_{\mathrm{m}3}& \mathrm{mode}2\hfill \\ i_{\mathrm{m}3}& \mathrm{mode}3\hfill \\ i_{\mathrm{m}4}& \mathrm{mode}4\hfill \end{array}\right.\hspace{1em}{i}_{\mathrm{t}3\mathrm{t}4}=\left\{\begin{array}{ll}{i}_{\mathrm{m}4}& \mathrm{mode}1\hfill \\ {\displaystyle \frac{(0.5-a_{\mathrm{m}2})a_{\mathrm{m}1}}{a_{\mathrm{m}2}}}{i}_{\mathrm{L}}& \mathrm{mode}2\hfill \\ -{\displaystyle \frac{(0.5-a_{\mathrm{m}2})a_{\mathrm{m}1}}{a_{\mathrm{m}2}}}{i}_{\mathrm{L}}& \mathrm{mode}3\hfill \\ -i_{\mathrm{m}4}& \mathrm{mode}4\hfill \end{array}\right..$$

(12)

Based on (4) and (10), the input current i_a of the rectifier within a quarter of the period [0, π/2] is calculated as follows:

$$i_{\mathrm{a}}=\left\{\begin{array}{ll}{\displaystyle \frac{4\sqrt{3}-6}{3}}(i_{\mathrm{m}2}-i_{\mathrm{m}3})& \omega t\in [0,\pi /12-\beta ]\hfill \\ {\displaystyle \frac{4\sqrt{3}-6}{3}}(i_{\mathrm{m}1}-i_{\mathrm{m}4})& \omega t\in [\pi /12-\beta ,\pi /12]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}4}+{\displaystyle \frac{4\sqrt{3}-6}{3}}{i}_{\mathrm{m}1}& \omega t\in [\pi /12,\pi /12+\beta ]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}3}+{\displaystyle \frac{4\sqrt{3}-6}{3}}{i}_{\mathrm{m}2}& \omega t\in [\pi /12+\beta ,\pi /6]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}2}+{\displaystyle \frac{4\sqrt{3}-6}{3}}{i}_{\mathrm{m}3}& \omega t\in [\pi /6,\pi /4-\beta ]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}1}+{\displaystyle \frac{4\sqrt{3}-6}{3}}{i}_{\mathrm{m}4}& \omega t\in [\pi /4-\beta ,\pi /4]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}1}+{\displaystyle \frac{2\sqrt{3}}{3}}{i}_{\mathrm{m}4}& \omega t\in [\pi /4,\pi /4+\beta ]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}2}+{\displaystyle \frac{2\sqrt{3}}{3}}{i}_{\mathrm{m}3}& \omega t\in [\pi /4+\beta ,\pi /3]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}3}+{\displaystyle \frac{2\sqrt{3}}{3}}{i}_{\mathrm{m}2}& \omega t\in [\pi /3,5\pi /12-\beta ]\hfill \\ {\displaystyle \frac{2(3-\sqrt{3})}{3}}{i}_{\mathrm{m}4}+{\displaystyle \frac{2\sqrt{3}}{3}}{i}_{\mathrm{m}1}& \omega t\in [5\pi /12-\beta ,5\pi /12]\hfill \\ {\displaystyle \frac{2\sqrt{3}}{3}}(i_{\mathrm{m}1}+i_{\mathrm{m}4})& \omega t\in [5\pi /12,5\pi /12+\beta ]\hfill \\ {\displaystyle \frac{2\sqrt{3}}{3}}(i_{\mathrm{m}2}+i_{\mathrm{m}3})& \omega t\in [5\pi /12+\beta ,\pi /2]\hfill \end{array}\right..$$

(13)

In Equation (13), the interval width β for mode1 and mode4 satisfy the following constraint conditions:

$$\left\{\begin{array}{l}\beta =\arcsin \left({\displaystyle \frac{-2-\sqrt{3}}{2m}}\right)+{\displaystyle \frac{\pi }{12}}\\ 0<\beta <{\displaystyle \frac{\pi }{12}}\end{array}\right.\Rightarrow m>7.21.$$

(14)

According to (10)–(14), based on symmetry, the operating current waveforms of the proposed 48-pulse rectifier are as described in Figure 6.

Taking the minimum THD of the input current i_a as the optimization objective, the turns-ratio parameters (a_m1, a_m2, and m) of the IPRs are optimized and designed. The input current THD of i_a can be calculated as follows:

$$\left\{\begin{array}{l}{\mathrm{THD}}_{\mathrm{ia}}={\displaystyle \frac{\sqrt{I_{\mathrm{a}}^2-I_{\mathrm{a}1}^2}}{I_{\mathrm{a}1}}}\\ I_{\mathrm{a}}=\sqrt{{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }{i}_{\mathrm{a}}^2(\omega t)\mathrm{d}(\omega t)}}\\ I_{\mathrm{a}1}={\displaystyle \frac{1}{\sqrt{2}\pi }}{\displaystyle {\int }_0^{2\pi }{i}_{\mathrm{a}}(\omega t)\sin (\omega t)\mathrm{d}(\omega t)}\end{array}\right.,$$

(15)

where I_a represents the RMS value of the input current i_a, and I_a1 represents the fundamental RMS value of i_a.

By substituting Equations (10), (13) and (14) into Equation (15) and solving with MATLAB R2020b, the optimal turns-ratio parameters that minimize the THD are obtained. The results are shown in Figure 7. From Figure 7, it can be seen that the THD of the input current i_a reaches the theoretical minimum value 3.78% when the turns-ratio parameters are a_m1 = 0.122, a_m2 = 0.371, and m = 14.3. In this optimal case, according to (14), it can be calculated that β = π/24.

Figure 8 shows the operating voltage waveforms of the proposed 48-pulse rectifier when using the optimal turns ratio. According to Figure 2 and Kirchhoff’s Voltage Law (KVL), it can be obtained that

$$u_{\mathrm{L}1}-u_{\mathrm{L}2}=u_{\mathrm{p}1}-u_{\mathrm{p}2}=u_{\mathrm{p}},$$

(16)

where the linearized waveform of u_p is a triangular wave with a frequency of 300 Hz.

From Figure 8, it can be seen that, based on symmetry, the load voltage u_L has 48 pulses within [0, 2π], which can be seen as a constant value.

3. Capacity Analysis of the Magnetic Devices

3.1. Capacity Analysis of the DCT

In this section, the theoretical capacities of the magnetic components for the proposed rectifier when employing the optimal turns-ratio parameters in Figure 7 are calculated based on this methodology.

Through linearization, the output DC voltage and current, u_L and i_L, can be seen as the constant values U_L and I_L. According to Equation (2) and Figure 8, the value U_L of the output DC voltage is calculated as follows:

$$U_{\mathrm{L}}\approx 2(\sqrt{3}-1)\times \sqrt{3}\times \cos 15°\times E=\sqrt{6}E.$$

(17)

According to Figure 3b and Figure 8, the capacity of the DCT S_DCT is calculated as follows:

$$S_{\mathrm{DCT}}={\displaystyle \frac{S_{\mathrm{in}}+S_{\mathrm{out}}}{2}},$$

(18)

where S_in is the input apparent power of the DCT (the capacity of windings a-b, b-c, and c-a) and S_out is the output apparent power of the DCT (the capacity of windings a-a₁, a-a₂, b-b₁, b-b₂, c-c₁, and c-c₂).

Based on Equation (10) and Figure 3b and Figure 6, according to the symmetry of the windings, the RMS values I_L1 and I_L2 of i_L1 and i_L2 and the winding currents of the DCT are calculated as follows:

$$\left\{\begin{array}{c}{I}_{\mathrm{L}1}=I_{\mathrm{L}2}\approx 0.566I_{\mathrm{L}}\hfill \\ I_{\mathrm{ab}}=I_{\mathrm{bc}}=I_{\mathrm{ca}}=\sqrt{{\displaystyle \frac{1}{2\pi }}{\displaystyle {\int }_0^{2\pi }{i}_{\mathrm{ab}}^2}\mathrm{d}(\omega t)}=0.0785I_{\mathrm{L}}\hfill \\ I_{\mathrm{a}1}=I_{\mathrm{a}2}=I_{\mathrm{b}1}=I_{\mathrm{b}2}=I_{\mathrm{c}1}=I_{\mathrm{c}2}=0.463I_{\mathrm{L}}\hfill \end{array}\right..$$

(19)

According to Equations (17)–(19), the capacity of the DCT is calculated as follows:

$$\begin{array}{cc}\hfill S_{\mathrm{DCT}}& ={\displaystyle \frac{1}{2}}\times 3\times (U_{\mathrm{ab}}{I}_{\mathrm{ab}}+2U_{\mathrm{cc}1}{I}_{\mathrm{c}1})\hfill \\ & ={\displaystyle \frac{3}{2}}\times [\sqrt{3}E{I}_{\mathrm{ab}}+(4-2\sqrt{3})E{I}_{\mathrm{c}1}]\hfill \\ & =0.235U_{\mathrm{L}}{I}_{\mathrm{L}}\hfill \end{array},$$

(20)

where U_ab and U_cc1 represent the RMS values of the winding voltages u_ab and u_cc1, respectively, of the DCT in Figure 3a, and U_LI_L represents the output load power of the rectifier.

3.2. Capacity Analysis of the IPRs

According to Figure 4a, the capacity of IPR₁ S_IPR1 can be calculated as follows:

$$S_{\mathrm{IPR}1}={\displaystyle \frac{1}{2}}\times [2\times ({\displaystyle \frac{1}{2}}-a_{\mathrm{m}1})U_{\mathrm{p}1}{I}_{\mathrm{L}1}+2a_{\mathrm{m}1}{U}_{\mathrm{p}1}{I}_{\mathrm{t}1\mathrm{t}2}+U_{\mathrm{s}}{I}_{\mathrm{s}}],$$

(21)

where U_p1, U_s, I_L1, I_t1t2, and I_s represent the RMS values of u_p1 u_s, i_L1, i_t1t2, and i_s, respectively.

According to Equation (16) and Figure 8, U_p1 and U_s are calculated as follows:

$$\left\{\begin{array}{l}{U}_{\mathrm{p}1}\approx 0.0571U_{\mathrm{L}}\\ U_{\mathrm{s}}=14.3U_{\mathrm{p}1}\approx 0.817I_{\mathrm{L}}\end{array}\right..$$

(22)

According to Equations (11) and (12), I_t1t2 and I_s are calculated as follows:

$$\left\{\begin{array}{l}{I}_{\mathrm{t}1\mathrm{t}2}\approx 0.28I_{\mathrm{L}}\\ I_{\mathrm{s}}\approx 0.012I_{\mathrm{L}}\end{array}\right..$$

(23)

Substituting Equations (19), (22) and (23) into Equation (21), S_IPR1 is calculated as follows:

$$S_{\mathrm{IPR}1}\approx 0.02U_{\mathrm{L}}{I}_{\mathrm{L}}.$$

(24)

According to Figure 4b, the capacity of IPR₂ S_IPR2 can be calculated as follows:

$$S_{\mathrm{IPR}2}={\displaystyle \frac{1}{2}}\times [2\times ({\displaystyle \frac{1}{2}}-a_{\mathrm{m}2})U_{\mathrm{p}2}{I}_{\mathrm{L}1}+2a_{\mathrm{m}2}{U}_{\mathrm{p}2}{I}_{\mathrm{t}3\mathrm{t}4}],$$

(25)

where U_p2 and I_t3t4 represent the RMS values of u_p2 and i_t3t4, respectively.

According to Equations (11) and (12), U_p2 and I_t3t4 are calculated as follows:

$$\left\{\begin{array}{l}{U}_{\mathrm{p}2}\approx 0.028U_{\mathrm{L}}\\ I_{\mathrm{t}3\mathrm{t}4}\approx 0.1I_{\mathrm{L}}\end{array}\right..$$

(26)

Substituting Equations (19) and (26) into Equation (25), S_IPR2 is calculated as follows:

$$S_{\mathrm{IPR}2}\approx 0.003U_{\mathrm{L}}{I}_{\mathrm{L}}.$$

(27)

Therefore, the total capacity of the TPHSC is obtained by

$$S_{\mathrm{TPHSC}}=S_{\mathrm{IPR}1}+S_{\mathrm{IPR}2}\approx 0.023U_{\mathrm{L}}{I}_{\mathrm{L}}.$$

(28)

From the above calculation results, it can be seen that the capacity of the DCT only accounts for 23.5% of the load power, which can reduce the cost of the phase-shifting transformer. In the TPHSC, the total capacity of the IPRs only accounts for 2.3% of the load power, indicating a low cost for harmonic suppression.

4. Simulation and Experimental Results

A 1 kW experimental platform (Harbin Transformer Factory, Harbin, China) was constructed to validate the performance of the proposed 48-pulse rectifier. The overall setup is shown in Figure 9a, with the implemented IPRs prototypes detailed in Figure 9b.

In the experimental setup, the three-phase input voltage was supplied by a Chroma-61702 programmable AC power source (Chroma ATE Inc., Taoyuan City, Taiwan), with waveforms captured by a HIOKI-PW3198 power quality analyzer (HIOKI E.E. Corporation, Ueda City, Japan). For the rectifier bridges, SQL60A1600V diode modules (CHUNZ Co., Ltd., Shanghai, China) were employed, which provided a significant safety margin over the nominal operating voltage to ensure robustness and prevent failures during laboratory testing.

To facilitate the analysis, the theoretical and simulation studies were conducted under idealized conditions, assuming a balanced three-phase operation and neglecting parameters such as transformer leakage inductance and diode voltage drops. The detailed simulation and experimental parameters are provided in

Table 1. Simulation and experimental parameters.

Parameter	Value
The RMS value of the input voltage	71 V
The input voltage frequency	50 Hz
The turns ratio of the DCT	$N_1:N_2=3:(2-\sqrt{3})$
The turns ratio of the IPR1	am1 = 0.122, m = 14.3
The turns ratio of the IPR2	am2 = 0.371
Load resistance	30 Ω

.

Figure 10 shows the input current i_a and its harmonic spectrum for the 12-pulse rectifier without the TPHSC. The simulated waveform in Figure 10a exhibits a clear 12-step profile, approximating a sinusoidal shape. Correspondingly, the simulated spectrum in Figure 10b confirmed the following theoretical performance: the input current THD was 15.15% and the harmonics were concentrated at the characteristic (12k ± 1)th orders (e.g., 11th, 13th, 23rd, and 25th), with no significant low-order or even-order harmonics present. This result aligned with the theory under ideal conditions.

In contrast, the experimental measurement in Figure 10c,d shows a 12-step wave with a THD of 13.90%. Based on Figure 10b,d, Figure 10e shows the harmonic spectrum comparison between the simulation and the experimental results. In Figure 10e, the key difference lies in the appearance of the low-order, non-characteristic harmonics (such as the 2nd, 4th, 5th, and 7th) in the experimental spectrum. This discrepancy arose from the inevitable non-idealities in any physical setup, including slight imbalances in the three-phase supply, manufacturing tolerances and imbalance factors in the transformer and the IPR, and variations in the diode characteristics. These practical factors disturbed the precise current sharing and phase relationships required for perfect harmonic cancellation.

Critically, however, the experimental deviations shown in Figure 10 do not undermine the primary validation objective. The measured THD of 13.90% remains largely consistent in magnitude with the theoretical prediction of 15.15%. More importantly, the dominant harmonics in the experiment are still unequivocally concentrated at the characteristic (12k ± 1)th orders, which validates the fundamental operating principle. The presence of additional low-order harmonics, while indicative of practical non-idealities, represents a secondary effect. Therefore, the experimental results successfully capture the essential harmonic profile and provide a valid baseline for evaluating the proposed TPHSC.

Figure 11 and Figure 12 show the simulation and experimental results of the 48-pulse rectifier after implementing the TPHSC. As seen in Figure 11, the input current was successfully modulated into a 48-step waveform, which was closer to sine. The simulated THD was reduced to 3.71%, while the experimental measurement showed a slightly higher value of 3.99%.

The results confirmed the core theoretical function of the TPHSC: the dominant characteristic harmonics of the original 12-pulse rectifier (e.g., the 11th, 13th, 23rd, and 25th) were effectively suppressed, as evidenced by the experimental spectrum.

Figure 12c shows the comparison between Figure 11b and Figure 12b. As shown in Figure 12c, the perfect cancellation of low-order harmonics was not achieved in the experiment. The residual harmonics stemmed from practical non-idealities inherent to the physical system, including slight imbalances in the three-phase supply and manufacturing tolerances and imbalance factors in the transformer and the TPHSC. These factors introduced slight deviations in current sharing and phase relationships, thereby limiting the complete annihilation of low-order harmonic components as predicted by the ideal simulation.

From Figure 12c it can be obtained that, although the TPHSC was theoretically designed to cancel specific higher-order harmonics (e.g., 11th and 13th), its implementation also led to a significant reduction in low-order non-characteristic harmonics (e.g., 2nd and 4th). This directly resulted from the improved system symmetry introduced by the TPHSC. In MPRs, the presence of even-order harmonics (like the 2nd and 4th) is a classic indicator of system imbalance, originating from asymmetries in three-phase voltages, transformer windings, or diode characteristics. By providing more balanced and interleaved current paths, the TPHSC effectively mitigated these inherent asymmetries. Consequently, the sources of the low-order, non-characteristic harmonics were suppressed, leading to their observed reduction in the input current spectrum.

In Figure 11b, the simulated spectrum shows distinct peaks at the theoretical 47th and 49th harmonics, whereas their magnitudes were negligible in the experiment. This discrepancy was attributed to the frequency-dependent parasitic effects of the real passive components. Although the TPHSC operates by reconstructing a multi-step waveform at the DC side to modulate the AC input current, the practical balancing reactors possess inter-winding capacitance and exhibit a significant increase in equivalent series resistance at high frequencies due to skin and proximity effects. These parasitics alter the network’s high-frequency impedance, causing the physical TPHSC to function as an effective low-pass filter. Consequently, the very high-order harmonic currents crucial for the ideal 48-step waveform are strongly attenuated before reaching the grid side. This effect, combined with the non-ideal switching characteristics of practical diodes, explains the observed deviation.

Ultimately, the deviations between the ideal simulation and the experimental measurements primarily stemmed from the inevitable tolerances and parasitic effects inherent in practical manufacturing processes. However, these practical deviations do not detract from the validation of the proposed method. Firstly, the substantial suppression of the primary target harmonics (11th, 13th, 23rd, and 25th) were unequivocally achieved. Secondly, the final measured THD of 3.99% represents an improvement which fell within the 5% limit of the IEEE 519-2014 standard [22], as visually consolidated in the comparative chart of Figure 13.

Based on the harmonic spectra in Figure 10d and Figure 12b, Figure 13 presents a comparative analysis of the harmonic contents up to the 25th order under three conditions. This range was selected as it aligns with the scope of key industrial standards and represents the dominant low-order harmonics of practical concern. The comparison included the original 12-pulse rectifier without the TPHSC, the proposed 48-pulse rectifier with the TPHSC, and the corresponding limits of IEEE 519-2014 (defined for systems with a short-circuit ratio of 20–50 and a THD limit of 5%).

From Figure 13, it can be obtained that the proposed TPHSC achieved a remarkable performance improvement. The input current THD was reduced from 13.90% to 3.99%. More specifically, the characteristic harmonics of the 12-pulse rectifier (11th and 13th) were suppressed by 82.8% and 87.6%, respectively. Furthermore, the low-order non-characteristic harmonics, which primarily originated from system imbalances (e.g., 2nd and 4th at 4.20% and 1.95% initially), were significantly attenuated to 0.31% and 0.70%, respectively.

A critical observation is that after applying the proposed suppression method, the harmonic contents of all orders from second to twentieth fell below the limits set by IEEE 519-2014. This confirmed that the proposed topology not only enhances power quality theoretically but also ensures compliance with stringent industrial grid codes.

Figure 14 shows the experimental results of the input three-phase phase voltages (u_a, u_b, and u_c) and output voltages (u_L1 and u_L2) of the main circuit. From Figure 14a, it can be seen that the RMS value of the input phase voltage was 71 V. From Figure 14b, it can be seen that u_L1 and u_L2 were six-pulse waves with a 30° phase difference, meeting the design requirements of the main circuit based on the DCT.

Figure 15 shows the simulation and experimental results of the waveforms of the IPRs’ terminals voltages u_p1, u_p2 and u_s. From Figure 15, it can be obtained that u_p1 and u_s were trapezoidal waves with the same phase and m-times difference in amplitude, and u_p2 is a triangular wave with a 0.5 duty cycle, which was complementary to u_p1. The results are consistent with the theoretical analysis.

Figure 16 shows the simulation and experimental results of the waveforms of i_L1, i_L2, and i_s. From Figure 16a, it can be obtained that i_L1 and i_L2 exhibited eight-step waves with a 30° phase difference, and i_s was close to a square wave with a 0.5 duty cycle, which is consistent with the theoretical analysis. Due to the filtering effect of leakage inductance from the transformer, the experimental results of i_L1 and i_s shown in Figure 16b,c were smoother than the simulation results.

Figure 17 shows the simulation and experimental results of the load voltage before using the TPHSC, and Figure 18 shows the simulation and experimental results of the load voltage and current after using the TPHSC. Comparing Figure 17a,b, it can be obtained that after using the TPHSC, the pulse number of the load voltage increased from 12 to 48 pulses within a power cycle, and the voltage ripple was greatly reduced, which indicated that the TPHSC had the function of pulse-number quadrupling. From Figure 18a,b, it can be seen that in the experiment, the output voltage and current of the proposed rectifier were approximately constant.

To evaluate the robustness of the proposed 48-pulse rectifier and its harmonic suppression circuit under dynamic operating conditions, a simulation study on load-step changes was conducted. In the simulation, the load resistance was stepped from 30 Ω to 60 Ω at 0.02 s, and then it was stepped back to 30 Ω at 0.04 s, thereby simulating the typical dynamic processes of output power halving and recovery. Figure 19 shows the load-step simulation waveform of the input current i_a. During the load transients at 0.02 s and 0.04 s, the amplitude of the input currents adjusted smoothly with the power change. Most importantly, the 48-step waveform characteristic of the input currents remained well preserved throughout the entire transient process and in the new steady states, with no observable waveform distortion or significant increase in low-order harmonics. In conclusion, the load-step simulation result demonstrates that the proposed rectifier system not only achieved excellent harmonic suppression performance in a steady state but also ensured highly stable input-current waveform quality and harmonic characteristics when facing load dynamics.

Table 2. Performance comparison between the proposed rectifier and previous studies.

Topology	THD of the Input Current	Diode Loss of the Harmonic Suppression Circuits	Capacity of the IPRs	Pulse Number
This paper	3.99%	2.67ILUon-drop (8 diodes)	2.3%ULIL	48
[7]	approximately 7.6%	0.0618ILUon-drop (2 diodes)	2.71%ULIL	24
[9,10]	approximately 5%	approximately 1.434ILUon-drop (4 diodes)	2.68%ULIL	36
[15,19]	approximately 7.6%	approximately 0.033ILUon-drop (2 diodes)	6.81%ULIL	24
[20]	approximately 3.46%	7 diodes and 2 MOSFETs	2.56%ULIL	48
12-pulse rectifier	15.2%	/	/	12

presents a performance comparison between the proposed 48-pulse rectifier and previous studies. Compared with the rectifier in [20], the proposed topology eliminated the need for additional control, featuring higher reliability, a lower cost, and a more stable reduction in grid-side harmonic contents in the delta-connected rectifiers. Compared with the rectifiers and the harmonic suppression circuits reported in [7,9,10,15] and [19], the proposed TPHSC exhibited significantly enhanced harmonic suppression capability. Although its diode loss increased slightly, the total capacity of the IPRs was substantially reduced, which indicated a low cost for harmonic suppression.

5. Conclusions

This paper has proposed and validated a novel 48-pulse rectifier topology based on a delta-connected transformer (DCT) and a triple-passive harmonic suppression circuit (TPHSC). The core innovation lies in the TPHSC, which has the function of pulse-number quadrupling, effectively reconstructing the input current into a 48-step waveform. This design offers a superior harmonic cancellation capability compared to conventional passive methods while maintaining simplicity and reliability.

A comprehensive theoretical analysis was employed to determine the optimal turns-ratio parameters for the IPRs, minimizing the magnetic component requirements. The calculated total capacity of the magnetic devices accounted for only 26% of the output load power, indicating a compact and potentially cost-effective magnetic design.

The simulation and experimental results conclusively demonstrate the effectiveness of the proposed system:

(1)

The input current waveform was successfully transformed from a 12-step to a 48-step shape.

(2)

A significant reduction in harmonic pollution was achieved. The measured THD of the input current was reduced to 3.99%, a substantial improvement from the 12-pulse benchmark of 13.90%. This final THD value falls below the 5% threshold stipulated by the IEEE 519-2014 standard, demonstrating the practical efficacy of the topology in meeting stringent grid-connection requirements.

A detailed analysis of the discrepancies between ideal simulation and experimental measurements offered valuable engineering insights. The presence of low-order, non-characteristic harmonics in experiments was explained in the context of practical system imbalances. The attenuation of higher-order characteristic harmonics (e.g., 47th and 49th) was reasonably attributed to the parasitic effects of magnetic components at elevated frequencies (a common phenomenon in practical high-order passive filtering). Furthermore, the supplementary simulations verified that the system exhibits a robust dynamic performance under load step-change conditions.

In summary, the proposed 48-pulse rectifier with the TPHSC offers a simple, reliable, and fully passive solution for high-power-quality AC-DC conversion, characterized by superior harmonic suppression and verified to be in compliance with IEEE 519, with a robust operation. These attributes make it a competitive option for industrial applications where grid power quality and cost-effectiveness are critical.