Introduction

The need for power is increasing day-by-day and fossil fuels are reducing, and environmental impact is more due to fossil fuels, so the research is focused on renewable energy sources. The major drawback of renewable energy is power generation that is not done at constant voltage and frequency so power electronic converters are required to convert to the required voltage and frequency. But with power electronic converters, harmonics are introduced. To minimize the harmonics multilevel inverters are one of the options to minimize the harmonics in converting power from DC to AC at the required voltage and frequency.

Multilevel inverters [MLIs] play an important role in converting the power from DC to AC. MLIs share the inverter operating voltage among the switches so low rating switches can be used to convert high power/voltage, which reduces the cost and size of the inverter. As the level of output voltage increases, the total harmonic distortion [THD] will reduce at low frequencies, which tends to reduce the cost and size of filters [1, 2].

The classical topologies of MLIs are neutral point clamped [3], flying capacitor [4], and Cascaded H-Bridge multilevel inverters [5]. The major issue with these conventional topologies is, as the number of levels in the output increases the number of switches also increased, this leads to the complexity in implementation of hardware circuits and switching patterns increases [6,7,8,9,10]. Based on these topologies, many topologies are developed, but in these topologies, some of the switches must withstand the total operating voltage of the inverter [11,12,13,14] (Fig. 1).

Fig. 1
figure 1

Single-phase 15-level cross-H bridge MLI topology

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Cross-H bridge

The connection diagram for cross-H bridge inverter is shown in Fig. 2 having separate DC voltage sources, and switches are connected in a cross manner. This structure can be applied to any number of phases and any number of voltage levels by switching the proper switches required level of output voltage is obtained [15]. The stress on the switches \(S_{1} ,S_{2} ,S_{n - 1} \,{\text{and}}\,S_{n}\) is \(V_{{{\text{DC}}}}\), and it is \(2V_{{{\text{dc}}}}\) for remaining switches and the total stress on the switches is equal to \(2*(V_{L} - 1)*V_{{{\text{dc}}}}\), where \(V_{L}\) is the level of output voltage.

Fig. 2
figure 2

Three-phase 15-level cross-H bridge MLI topology

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Advantages of cross-H bridge:

  • The number of power electronic switches required is less, and the number of switches conduct is less, so the losses and control circuit complexity are reduced.

  • The total standing voltage is equal to CHB MLI.

Disadvantages of cross-H bridge:

  • Structure is not modular

  • Some switches have voltage rating more than Vdc (i.e., 2*Vdc).

Mathematical analysis

Following relations are obtained from the proposed model

$$S_{N} = L_{V} + 1$$
(1)
$$L_{V} = 2*V_{DCN} + 1$$
(2)
$$S_{N} = 2(V_{DCN} + 1)$$
(3)
$$N_{S} = \frac{{L_{V} + 1}}{2}$$
(4)

These equations are used to calculate the number of switches required and the total number of output voltage levels that can be generated.

Switch losses

Power electronic switches contain two major losses: one is due to conduction (Pcond), and another one is due to switching (Psw IGBT) [16,17,18,19]; the IGBT switches are considered in this paper.

The average conduction losses of IGBT (Pcond.IGBT) and diode (Pcond.diode) are given as follows:

$$P_{cond.IGBT} = \int\limits_{tcon} {P(t){\text{d}}t = \frac{1}{T}\int\limits_{0}^{T} {V_{ce} (t)*i_{ce} (t){\text{d}}t} }$$
(5)
$$P_{cond.diode} = \frac{1}{T}\int\limits_{0}^{T} {V_{f} (t)*i_{d} (t){\text{d}}t}$$
(6)

The total conduction losses of the proposed inverter for one cycle

$$P_{cond} = \left( {P_{cond.IGBT} + P_{cond.diode} } \right) \times N_{s}$$
(7)

The switching losses are present during ON and OFF the power electronic switches. The switching losses of IGBT (Psw IGBT) are given as follows:

$$P_{SWIGBT} = \left( {E_{swon} + E_{swoff} } \right) \times f$$
(8)
$$E_{s\_on} = \int\limits_{ton} {V_{ce} (t)*i_{ce} (t){\text{d}}t}$$
(9)
$$E_{s\_off} = \int\limits_{toff} {V_{ce} (t)*i_{ce} (t){\text{d}}t}$$
(10)

Table 1 represents the comparison of losses of converter for various switching techniques. The losses are in mW scale. Total switch loss in three-phase system has been reduced in third harmonic PWM control by 29.40% when the same converter is fed with single pulse PWM technique. Similarly, in single-phase system, the total losses are also reduced by 29.40%. When looking into switching loss third harmonic PWM control is having less loss when compared with single pulse PWM.

Table 1 Switch losses (total) versus O/P voltage levels
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Third harmonic PWM

For a three-phase load with a floating-point neutral point, the third harmonic voltage is absent from the phase voltages and line to line. Therefore, there is no distortion in the phase voltages. We might improve the sufficiency of the yield voltage waveform by adding a third consonant sign to a low-recurrence sinusoidal reference signal [20,21,22,23,24] (Figs. 3, 4; Table 2).

Fig. 3
figure 3

Block diagram of 15-level cross-H bridge multilevel inverter fed a load using third harmonic PWM technique

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Fig. 4
figure 4

a Generation of third harmonic waveform. b Third harmonic pulse width modulation for 15-level cross-H bridge inverter

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Table 2 Switching pattern of cross-H-bridge 15-level MLI
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Results

The simulation of third harmonic PWM controlled 3-Ø 15-level cross-H bridge multilevel inverter is done. The block diagram of third harmonic PWM controlled 3-Ø 15-level cross-H bridge multilevel inverter and circuit representation of 15-level cross-H bridge is represented in Figs. 5 and 6, respectively. The switch IGBT (FGA15N120ANTD) is considered for simulation and for calculating the losses of switches. Voltage, current, and speed analysis are discussed for third harmonic PWM and single pulse PWM [25] controlled 3-Ø 15-level cross-H bridge multilevel inverter fed induction motor drive.

Fig. 5
figure 5

Three-phase 15-level cross-H bridge MLI fed IM block diagram

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Fig. 6
figure 6

Single-phase 15-level cross-H-bridge MLI topology

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Third harmonic PWM controlled 3-Ø 15-level cross-H Bridge fed IM

Figure 7 represents three-phase simulated output voltages of third harmonic PWM controlled 15-level cross-H-bridge multilevel inverter with the peak output voltage of 140v peak to peak. Similarly, the simulated phase current of 3-Ø IM is shown in Fig. 8. Figure 9 represents the speed response of the motor; the current and the speed are oscillated initially and settled to 1480 RPM at 0.19 s. Figure 10 shows the total harmonic distortion of sinusoidal PWM controlled 15-level cross-H-bridge MLI. The value of THD is recorded as 4.04%. Figure 10 shows the switching pulses of all sixteen switches (Fig. 11).

Fig. 7
figure 7

Three-phase output voltages (THI PWM)

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Fig. 8
figure 8

Stator phase current of 3-Ø induction motor (THI PWM)

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Fig. 9
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Speed response of induction motor (THI PWM)

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Fig. 10
figure 10

THD of third harmonic PWM controlled 15-level cross-H-bridge MLI

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Fig. 11
figure 11

Pulses given to 16 switches in THI-PWM

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Single Pulse PWM controlled 3-Ø 15-level cross-H Bridge fed IM

Figure 12 indicates the 3- Ø output voltages with 140 V peak. Figures 13 and 14 represent the 3- Ø induction motor stator current of one of the phases and speed. From Figs. 13 and 14, it can be observed that the settling time of stator current and speed is 0.22 s. Figure 15 shows the total harmonic distortion, which is recorded as 6.64%. From above wave forms, THD with third harmonic PWM is reduced by 39% and settling time of current and speed is reduced by 13% when compared with single pulse PWM. Comparison of stator current, speed and torque characteristics are shown in Figs. 16, 17 and 18, respectively.

Fig. 12
figure 12

Three-phase output voltages (single pulse PWM)

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Fig. 13
figure 13

Stator phase current of three-phase induction motor (single pulse PWM)

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Fig. 14
figure 14

Speed response of induction motor (single pulse PWM)

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Fig. 15
figure 15

THD of single PWM controlled 15-level cross-H-bridge MLI

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Fig. 16
figure 16

Stator current comparison with THI and single pulse PWM

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Fig. 17
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Speed characteristics comparison with THI and single pulse PWM

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Fig. 18
figure 18

Torque characteristics comparison with THI and single pulse PWM

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Conclusion

In this paper, simulation of third harmonic PWM controlled 3-Ø 15-level cross-H bridge and single pulse PWM controlled 3-Ø 15-level cross-H bridge is done. THD, current, and speed are analyzed with both and PWM techniques. From results obtained, it is found that third harmonic PWM THD is reduced by 39% and settling time of speed and current waveforms is reduced by 13% and total losses of switches are also reduced when compared with single pulse PWM. When compared both the PWM techniques, third harmonic PWM technique is more reliable than single pulse PWM technique.