ANN Optimised RPWM Technique for Minimisation of Conducted EMI in Three-Phase Voltage Source Inverters

To reduce acoustic noise in motor drives, various researchers have explored techniques involving randomisation of the switching frequency. Madasamy et al. (2021) introduced randomness by varying the switching frequency through space vector modulation (SVM) using an asymmetric frequency multicarrier. Kirlin et al. (2011) implemented randomised inverter switching around a nominal frequency, while Bech et al. (2000) randomised the switching frequency by altering the distribution of the zero vector within each switching cycle. These methods introduce non-deterministic elements into the output waveform (Lai et al., 2013; Liaw et al., 2000), which effectively convert narrow-band acoustic noise into broad-band noise by dispersing the harmonic spectrum away from the switching frequency (Binojkumar et al., 2013; Mathe et al., 2012). However, this approach tends to increase waveform distortion due to the randomness, and the average inverter switching losses remain comparable to those of deterministic PWM strategies (Mathe et al., 2012).

Further studies have focussed on applying RPWM techniques using the SVM method. (de Oliveira et al., 2005; Khan et al., 2010; Lee et al., 2000) randomly sampled the reference voltage space vector and achieved averaging by switching among its three nearest vectors. In other studies, including George and Baiju, (2013) and Madasamy et al. (2021), the switching frequency was varied randomly within predefined bounds, producing randomly spaced samples without requiring sector identification. George and Baiju (2013) further noted that the reference phasor is mapped into a vector within a two-level inverter, enabling calculation of switching time vectors. These strategies have demonstrated significant reductions in both acoustic noise and EMI. However, as Makhubele and Ogudo (2020) highlighted, such methods are often complex, computationally demanding and expensive to implement.

Owusu et al. (2023) employed ANNs to optimise a sinusoidal PWM-based controller for a three-phase VSI, achieving promising results. Nonetheless, the deterministic nature of the output results in harmonic clustering around the switching frequency, which contributes to increased noise in motor drives.

The modulation principle underlying the RPWM technique has been extensively examined by several researchers (Jadeja et al., 2015; Kim et al., 2004; Lee et al., 2016). In its implementation, three reference signals S_a, S_b and S_c are individually compared with a single randomised triangular carrier signal C, to generate the switching pulses for each leg of the three-phase VSI. Figure 1 illustrates the modulation principle of the RPWM technique.

Figure 1.

Modulating principle of RPWM. RPWM, random pulse width modulation.

In this context, the switching function u, for each phase of the inverter, is determined by the switching period (T), the delay factor (δ), and the duty cycle (d). In practical implementations, the reference signals are deterministic (uniform and phase-shifted by 2π/3), allowing for effective control of the output voltage. Consequently, only the switching period and delay factor are subject to randomisation. This is achieved using a triangular carrier signal defined by its period T and a fall-time delay β.

The randomisation of these parameters is facilitated by a linear congruential generator (LCG), which uses a uniform probability distribution and applies statistical averaging techniques, as described in the studies of Bhattacharya et al. (2015), Boudjerda et al. (2011), and Boudouda et al. (2012).

The random variation of T is implemented between the interval [T_min, T_max], by using the statistical mean (1) T¯=1Fs \bar T = {1 \over {Fs}} where Fs = switching frequency = 3,000 Hz; the randomness level (2) RT=Tmax−TminT¯ {R_T} = \left[ {{{{T_{\max}} - {T_{\min}}} \over {\bar T}}} \right] then, →T∈T¯1−rt2,T¯1+rt2 \to T \in \left[ {\bar T\left( {1 - {{rt} \over 2}} \right),\bar T\left( {1 + {{rt} \over 2}} \right)} \right]

The random variation of the parameter β is implemented within the interval βmin,βmax with βmin≥0 and βmax≤1,the statistical mean β¯=0.5, \left[ {{\beta _{min}},{\beta _{max}}} \right]\;{\rm{with}}\;{\beta _{min }} \ge 0\;{\rm{and}}\;{\beta _{max }} \le 1,{\rm{ the\; statistical\; mean}}\;\bar \beta = 0.5, and the randomness level is calculated as (3) Rβ=βmax−βminβ¯ {R_\beta } = \left[ {{{{\beta _{\max}} - {\beta _{\min}}} \over {\bar \beta }}} \right] with 0 ≤ R_β ≥ 2; then therefore, β∈β¯1−Rβ2,β¯1+Rβ2 \beta \in \left[ {\bar \beta \left( {1 - {{{R_\beta }} \over 2}} \right),\bar \beta \left( {1 + {{{R_\beta }} \over 2}} \right)} \right]

When the parameters T and β are held constant in triangular signal generation, the result is a triangular waveform with fixed characteristics. When this fixed waveform is compared with sinusoidal reference signals, the modulation technique is classified as deterministic or classical SPWM.

Randomising the parameter T results in random carrier frequency M = modulation (RCFM), while randomising β leads to random pulse positioning modulation (RPPM), both in reference to the sinusoidal signals.

When both T and β are randomised simultaneously, the approach is termed dual random pulse width modulation (DRPWM). Compared to RCFM and RPPM individually, DRPWM exhibits superior spectral spreading capabilities, as demonstrated in the studies by Boudjerda et al. (2011) and Boudouda et al. (2012). In this study, a dual-randomised carrier signal c(t) is generated using a MATLAB Function block, with time (t), switching period T and fall-time delay β as inputs. This synthesised signal serves as a core component of the dataset used to train the ANN.

An ANN was developed using a supervised feedforward architecture with backpropagation, trained with known target outputs (Ali and Mashwani, 2023). It comprised 3 layers: an input layer with 4 neurons, a hidden layer with 10 neurons and an output layer with 3 neurons. Inputs combined three sinusoidal reference signals and a random signal from a MATLAB Function block. After training, the network generated three control signals, later combined with reference and triangular carrier signals for modulation.

Figure 2 illustrates the network’s structure, showing connections between layers via weights (‘M’) and biases (‘B’). Weights, initialised randomly between 0 and 1, adjust input signals through the network, while biases shift the activation function’s input. Each hidden layer neuron computes a weighted sum of its inputs, followed by a sigmoid activation function to model non-linear relationships expressed as; (4) W=R1×M1+R2×M11+R3×M21+C×M31+B1 W = \left( {{R_1} \times {M_1}} \right) + \left( {{R_2} \times {M_{11}}} \right) + \left( {{R_3} \times {M_{21}}} \right) + \left( {C \times {M_{31}}} \right) + {B_1}

Figure 2.

ANN architecture. ANN, artificial neural network.

The network was trained using the ‘trainlm’ function, which implements the Levenberg-Marquardt Optimisation (LMO) algorithm. This method was selected for its robustness, efficiency and high accuracy with smaller datasets. LMO is well-suited for moderate-sized feedforward networks, balancing the speed of Gauss-Newton with the stability of gradient descent. During optimisation, the Hessian matrix H is approximated using the Jacobian J. (5) XK+1=XK−JT+μI−1JTe {X_{K + 1}} = {X_K} - {\left[ {{J^T} + \mu I} \right]^{\left( { - 1} \right)}}{J^T}e

The scalar value μ was initially set to 0 to allow the Hessian matrix approximation in Newton’s method. During optimisation, μ was adjusted to control the step size, decreasing μ enabled larger steps and improving the reduction of the performance function. This function was estimated using the Hessian H = J^TJ, while the gradient was computed as follows: (6) g=JTe g = {J^T}e

With J representing the Jacobian matrices and e as the error.

The mean squared error (MSE) method was employed during the training process to evaluate the performance of the network, mathematically expressed as: (7) MSE=∑(y−y^i)2n MSE = {{\sum {{{(y - {{\hat y}_i})}^2}} } \over n} where n is the number of observations, y is the i^th observed value and y^i {\hat y_i} is the corresponding predicted value

The back propagation (BP) learning algorithm was utilised to update the network’s weights and minimise the MSE. The weight update process in BP is mathematically represented as: (8) Wi+1=wi−ŋgi W\left( {i + 1} \right) = w\left( i \right) - {\ng} g\left( i \right) where ŋ represents the learning rate and g(i) represents the gradient vector

During backpropagation, the chain rule was used to compute the gradient vector. For training, dual random carrier signals combined with three sinusoidal reference signals comprised 70% of the dataset, with the remaining 30% reserved for validation. Both feedforwardnet and cascadeforwardnet were tested, showing similar performance. These functions can approximate any finite input-output mapping with sufficient hidden neurons, using a 1 × N vector of hidden layer sizes and the trainlm backpropagation algorithm. The input and output layer sizes are initially set to zero and adjusted automatically during training. The feedforwardnet was ultimately chosen for its simplicity and fast convergence in solving fitting problems. Table 1 summarises the training data, while Figures 3 and 4 display the training results.

Figure 3.

Output regression line after training.

Figure 4.

MSE output curve after training. MSE, mean squared error.

The proposed ANN modifies the random carrier index to minimise both harmonic distortion and conducted EMI. The ANN receives three normalised inputs: modulation index m_a, reference voltage amplitude V_ref and instantaneous current i_o. The output neuron produces an adaptive carrier frequency shift factor Δfc, governing the instantaneous carrier frequency fc(t) = f_base + Δfc. The network is trained using the Levenberg–Marquardt (LM) algorithm to minimise

J = αTHD + βP_EMI; where α and β weigh the trade-off between waveform quality and spectral spread. The LM algorithm iteratively updates weight matrices W and biases b as Wk+1=Wk−JkTJk+μI−1JkTek {W_{k + 1}} = {W_k} - {\left( {J_k^T{J_k} + \mu I} \right)^{ - 1}}J_k^T{e_k} ensuring rapid convergence with low computational overhead. The trained ANN is implemented in the MATLAB/Simulink model through a feed-forward structure requiring approximately 2 × 10⁴ FLOPs per cycle, easily manageable by mid-range DSPs such as TI TMS320F28379D.

The proposed method improves the RPWM strategy, which is preferred over conventional PWM for controlling three-phase VSIs due to its ability to spread harmonic clusters across a wider frequency range (Addo-Yeboah and Owusu, 2022; Sohaib Sajid et al., 2020). RPWM typically combines a sinusoidal reference with a randomly varied carrier to generate switching pulses (Raju et al., 2013). In this approach, an ANN-generated signal is added to the reference and carrier signals to enhance pulse generation. A low-pass LC filter further smooths the output, reducing THD and improving waveform quality, efficiency and overall performance.

A randomly varied triangular signal C(t) is generated by setting two parameters: the period T and the fall-delay time β. The waveform’s first and second halves are defined by Eqs (9)–(16).

For the fixed triangular carrier in FCSM, Eqs (9)–(12) are applied: (9) T=Ts T = Ts (10) B=T2 B = {T \over 2} (11) T=T+Ts T = T + Ts (12) B=T+Ts2 B = {{T + Ts} \over 2}

Whereas for the random triangular carrier in the DRPWM, Eqs (13)–(16) are applied (13) T=Tmin+rand×Tmax−Tmin T = {T_{{{\min}}}} + \left( {rand \times \left( {{T_{{{\max}}}} - {T_{{{\min}}}}} \right)} \right) (14) β=T0×bmin+rand×bmax−bmin \beta = {T_0} \times \left( {{b_{{{\min}}}} + \left( {rand \times \left( {{b_{{{\max}}}} - {b_{{{\min}}}}} \right)} \right)} \right) (15) T=T0×bmin+rand×bmax−bmin T = {T_0} \times \left( {{b_{{{\min}}}} + \left( {rand \times \left( {{b_{{{\max}}}} - {b_{{{\min}}}}} \right)} \right)} \right) (16) β=T2×bmin+rand×bmax−bmin \beta = {T_2} \times \left( {{b_{{{\min}}}} + \left( {rand \times \left( {{b_{{{\max}}}} - {b_{{{\min}}}}} \right)} \right)} \right) where: (17) Tmax=Ts×1+rt2 {T_{{{\max}}}} = {T_s} \times \left( {1 + {{rt} \over 2}} \right) (18) Tmin=Ts×1−rt2 {T_{{{\min}}}} = {T_s} \times \left( {1 - {{rt} \over 2}} \right) where T_max is the random upper period limit and T_min is the random lower period limit, for rt = 0.2 where rt is the period randomness level. Also, (19) bmax=0.5×1+rt2 {b_{{{\max}}}} = 0.5 \times \left( {1 + {{rt} \over 2}} \right) (20) bmin=0.5×1−rt2 {b_{{{\min}}}} = 0.5 \times \left( {1 - {{rt} \over 2}} \right) where b_max is the random upper amplitude limit and b_min is the random lower amplitude limit, for rb = 2 with rb, the amplitude randomness level.

Eventually, C(t) was generated with t as the time input factor for the MATLAB function block and was used as the principal part of the input data of the ANN.

Three sinusoidal signals with a phase shift of 120° were generated for training purposes using the following Eqs (21)–(23) (21) R1t=sin100πt {R_1}\left( t \right) = sin\left( {100\pi t} \right) (22) R2t=sin100πt+2π3 {R_2}\left( t \right) = sin\left( {100\pi t + {{2\pi } \over 3}} \right) (23) R3t=sin100πt−2π3 {R_3}\left( t \right) = sin\left( {100\pi t - {{2\pi } \over 3}} \right)

Subsequently, four signals served as the input signals in the ANN training. (24) Input−Net=R1t,R2t,R3t,Ct Inpu{t_ - }Net = \left[ {{R_1}\left( t \right),{R_2}\left( t \right),{R_3}\left( t \right),C\left( t \right)} \right]

Three target outputs, Y₁, Y₂ and Y_3, were obtained from the ANN supervised training using the four input signals for the proposed method, by applying the mathematical Eqs (25)–(27) for the proposed method. (25) Y1t=R1t−Ct {Y_1}\left( t \right) = \left[ {{R_1}\left( t \right) - C\left( t \right)} \right] (26) Y2t=R2t−Ct {Y_2}\left( t \right) = \left[ {{R_2}\left( t \right) - C\left( t \right)} \right] (27) Y3t=R3t−Ct {Y_3}\left( t \right) = \left[ {{R_3}\left( t \right) - C\left( t \right)} \right]

The controller in Figure 12 was used to generate switching pulses for the proposed ANN-optimised method. The ANN was trained using the dual random carrier signal and three fundamental sinusoidal signals, as defined by Eqs (13)–(20). The resulting ANN signal was then combined with the sinusoidal and carrier signals to produce switching pulses for each leg of the three-phase VSI.

Figure 12.

Pulse generation for ANN incorporated. ANN, artificial neural network.

The control pulses generated for the switches in the first leg of the three-phase VSI are illustrated in Figure 13.

Figure 13.

Pulse generation for the proposed strategy on one leg of three-phase inverter. ANN, artificial neural network; DRPWM, dual random pulse width modulated.

Figures 14–16 show the output voltage waveforms before and after LC filtering for the FCSM, DRPWM and proposed methods, respectively.

Figure 14.

Output voltage waveform with and without an LC filter for FCSM. FCSM, fixed carrier signal modulated.

Figure 15.

Output voltage waveform with and without LC filter for DRPWM. DRPWM, dual random pulse width modulated.

Figure 16.

Output voltage waveforms before and after the LC filter for the proposed method.

The output voltage waveforms, along with their THD and PSD analysis, are presented and discussed in the Section 6.

PSD analysis was performed using Welch’s method in MATLAB. Figures 17–19 show the PSD results for FCSM, DRPWM and the proposed ANN-optimised method, respectively.

Figure 17.

Output voltage PSD of FCSM. FCSM, fixed carrier signal modulated; PSD, power spectral density.

Figure 18.

Output voltage PSD of DRPWM. DRPWM, dual random pulse width modulated; PSD, power spectral density.

Figure 19.

Output voltage PSD of the proposed method. PSD, power spectral density.

The output voltage waveforms were analysed before and after the LC filter for all three methods. Controllers in Figures 7 and 8 implemented the FCSM and DRPWM methods on the three-phase VSI shown in Figure 11. The apparent amplitude rise is due to the filter’s gain characteristics near its corner frequency, combined with the spectral redistribution introduced by random switching. The LC filter amplifies components near the cut-off frequency before full attenuation takes effect. Figures 20a, b show the FCSM method’s THD levels before and after the LC filter, while Figures 21a, b show the same for the DRPWM method.

Figure 20.

THD content of the FCSM. (a) Before and (b) after the LC filter. FCSM, fixed carrier signal modulated; THD, total harmonic distortion.

Figure 21.

THD content of the DRPWM (a). Before and (b). After the LC filter. DRPWM, dual random pulse width modulated; THD, total harmonic distortion.

The controller in Figure 12 was used to regulate the three-phase VSI (Figure 11), and the output voltage THD of the proposed method was evaluated before and after the LC filter, as shown in Figure 22.

Figure 22.

THD content of the proposed method. (a) Before and (b) After the LC filter. THD, total harmonic distortion.

Table 2 gives a summary of the THD content levels of the three simulations under the two conditions of before and after the LC filter connection.

FFT-based harmonic analysis was conducted for the three modulation techniques. Table 3 lists the first six significant harmonics and their relative magnitudes, highlighting that the 5th and 7th harmonics are reduced by approximately 43% and 37%, respectively.

The results confirm that the ANN-RPWM achieves broader harmonic energy dispersion, yielding a flatter PSD and improved EMI attenuation.

Although the study is simulation-based, the ANN-RPWM algorithm is readily deployable in real-time hardware. The feed-forward structure enables deterministic computation with an estimated execution time of ≈ 3 μs per switching cycle on a 200 MHz digital signal processor (DSP).

The controller requires no additional sensors or switching devices. Implementation can be realised using existing PWM modules with minor firmware modification. Future work will focus on hardware-in-loop testing using Software tool (dSPACE) and experimental validation on a 5 kVA three-phase VSI prototype driven by a Texas Instruments TMS320F28379D DSP will be used to reproduce the MATLAB/Simulink conditions equipped with a passive LC output filter and standard voltage and current sensing circuits. Conducted EMI and THD measurements will be performed using a line impedance stabilisation network (LISN) and a spectrum analyser in accordance with CISPR 16-1 standards. This hardware arrangement will enable direct comparison between simulated and experimental results, providing practical verification of the ANN-RPWM controller’s effectiveness in reducing EMI and harmonic distortion.