A 31–300 Hz Frequency Variator Inverter Using Space Vector Pulse Width Modulation Implemented in an 8-Bit Microcontroller

Abstract

With the advancement in power electronics technology, variable-frequency drives have been widely adopted for motor operation due to their inherent benefits: control performance, extending equipment life, and energy savings. The most used technique is Sine Pulse Width Modulation, as it solely requires the modification of the reference signal (sine wave). However, Space Vector Pulse Width Modulation offers lower total harmonic distortion. Therefore, this study presents a technique for the control of induction motors operating in open-loop mode, utilizing a two-level voltage source inverter with a frequency range of 31 to 300 Hz. The proposed control system is modified to encompass between 930 and 1848 switching periods, varying the number of switching periods along with the frequency variation. This approach allows the use of a single LCL filter across the whole frequency spectrum. It is adapted for implementation in an 8-bit microcontroller, which has its inherent limitations, yet it achieves performance levels similar to those found in high-level processors like FPGAs and DSPs. The signals generated by the microcontroller are captured by a DAQ card and input into a MATLAB/Simulink model to observe and analyze the performance of the proposed control system.

1. Introduction

Variable-frequency drives are used in numerous industrial and commercial applications. They control electric motors in fans, air compressors, pumps, CNC machining operations, extruders, air conditioners, etc. [1]. They allow control performance, and as it is estimated that in the United States above 60% of all electrical energy is used on electrical motors, energy savings achieved by controlling their operational frequency have a significant impact. That is why their adoption keeps growing, and several proposals of variable-frequency drives (VFDs) can be found in the literature [2,3,4,5,6,7,8,9,10,11,12,13], most of them using FPGAs or DSPs to facilitate the implementation of efficient control systems. There are also some works developed to implement dc/ac inverters with SVPWM using low-cost microcontrollers [14,15,16], even microcontroller manufacturers have presented application notes with their own 8-bit microcontrollers [17,18,19]. These works are developed on a fixed operational frequency, 50 or 60 Hz. In [20], they present a variable-frequency controller, though only in the range 20–50 Hz with a maximum resolution of 8 bits for every 60°. Another variable-frequency controller in [21] has a range from 0.5 to 50 Hz, though they do not indicate how the times for T₀, T₁, and T₂ (described in Section 2) are implemented in the microcontroller, whether using delay functions or interrupts. There are papers using SVPWM for frequency variators working with frequencies above the nominal 50 or 60 Hz, but they use carrier-based SVPWM. This method is easier to implement but requires the generation of three sine waves, which makes it suitable for parallel computing devices like FPGAs. In sequential processing, SVPWM generated by timing the sections of a switching period is not merely preferred but necessary.

Industry applications of induction motors controlled by VFDs include conveyor systems, blowers [22], mixing and agitating equipment, extrusion machines [23], swimming pool filtration systems, pressure booster pumps, and HVAC systems [24]. In [25], they even describe the speed of the motors in the cement industry: centrifugal pumps with a range of 200–1800 RPM for vertical turbine pumps (VTPs) and 1800–1960 RPM for horizontal centrifugal pumps (CHPs). Most industrial VFDs have a range of 1–400 Hz, though there are some products reaching even 655 Hz [26,27,28].

In recent years, the main objective of research on VFDs is precise frequency control [8,10,11,12,29,30], frequency and torque [31], minimization of torque ripple, and THD [9,13]. All of these papers use SPWM or carrier-based SVPWM. Therefore, this work presents a VFD with a range from 31 to 300 Hz. Although the frequency range is less than the ranges found in industrial VFDs, the authors limited the range due to the constraints imposed by software, in this case, the language code xC8, version 2.5. The microcontroller could handle a higher range if it was programmed using assembly language. But once the system is evaluated, it can be implemented on industrial applications because, as will be shown in Section 3, its performance is comparable to those obtained using FPGAs or DSPs.

The main contributions of this paper are as follows:

(A)

A detailed insight into the theoretical analysis of SVPWM and its implementation on an 8-bit microcontroller.

(B)

A variable number of switching periods, decreasing with increasing frequency to have between 930 and 1838 per cycle. This is being conducted to reduce the switching frequency variation to have one LCL filter for the whole frequency range.

(C)

Three changes to the switching sequences usually implemented in the literature. Although it increases the switching losses (switching more than a bit at a time), it reduces THD by 17%.

Following the introduction, Section 2 describes the theory of SVPWM and its implementation on an 8-bit microcontroller with the modifications necessary for both the limitations of an 8-bit microcontroller and to reach an industrial level performance. Section 3 presents the results from the system which, as indicated previously, are compared to implementations on higher processing devices. Section 4 gives the conclusion of this work.

2. Development of the Proposed Framework

2.1. Motivation

The motivation of this proposal is to provide an efficient SVPWM implementation for a variable-frequency driver based on an 8-bit microcontroller. The program is written in C code to allow its understanding and possible manipulation and improvement.

2.2. Theory

Space vector theory is well developed and implemented in many examples found in the literature [32,33]. Here, we present the basic principles necessary to understand how it works and, in the next subsection, how it is implemented on an 8-bit microcontroller.

Any three-phase electricity system consists of three AC voltages of identical frequency and similar amplitude but with a 120° phase shift from one another, as is represented by Equation (1).

$$\begin{array}{c}{V}_a=V_m\sin \left(\omega t\right)\\ V_b=V_m\sin \left(\omega t+2\pi /3\right)\\ V_c=V_m\sin \left(\omega t-2\pi /3\right)\end{array}$$

(1)

where V_m is the maximum value of the phase voltage, and ω is the angular velocity of the phase voltage. The basic implementation of the Clarke or α–β transformation allows the representation of this three-phase balanced system by one rotating vector generated by two components. The α–β transformation is given by

$$\left[\begin{array}{c}{V}_{\alpha }\\ V_{\beta }\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -1/2& -1/2\\ 0& \sqrt{3}/2& -\sqrt{3}/2\end{array}\right]\left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right]$$

(2)

where V_α and V_β are the horizontal and vertical components of the α and β axes. This transformation brings a 3D space into a 2D space without loss of information. The vector sum of V_α and V_β generates the rotating vector V_f, as shown in Figure 1.

Now, the Park transformation can be applied to vectors V_α and V_β using Equation (3):

$$\left[\begin{array}{c}{V}_d\\ V_q\end{array}\right]=\left[\begin{array}{cc}\cos \left(\theta \right)& \sin \left(\theta \right)\\ -\sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\left[\begin{array}{c}{V}_{\alpha }\\ V_{\beta }\end{array}\right]$$

(3)

where θ is the instantaneous angle of the ω/(2π) frequency. When a synchronous reference frame is selected, that is, θ = ωt, the dq frame rotates with the same angular speed as the system frequency, and it generates constant values for V_d and V_q. If the system is balanced, V_q = V_m. Representing a three-phase system with two dc quantities has several advantages and applications, from making mathematical computations much easier to facilitating inverter-motor control.

Now, to obtain a three-phase balanced system with a specific frequency and each phase at a specific amplitude, the inverse Park and Clarke transformations are applied, and it starts by defining the values of V_s and ω to define a rotating vector V_F by the following equation:

$$V_F=V_s{e}^{j\omega t}$$

(4)

where ω is the angular frequency of the system, and Vs is the maximum voltage of the signal, which will be taken from a DC source. For the implementation of the inverse transformation using a digital system, the simplest method to generate a three-phase voltage source from a DC source is through a two-level voltage source inverter like the one shown in Figure 2.

The operation of the two-level voltage source inverter is performed by activating the switching elements, which usually are IGBTs or MOSFETs. They are ON when receiving a digital signal ‘1’ and OFF when the digital signal is ‘0’. Switches S₀ and S₃ are connected to phase a, S₁ and S₄ to phase b, and S₃ and S₅ to phase c. Because any pair of switches cannot be ON (short circuit) or OFF (no conduction) at the same time, they are operated in opposite states; therefore, the analysis that follows shows only the upper switches, S₀, S₁, and S₂. With two possible states for each switch, there are 8 combinations: 000, 001, 010, 011, 100, 101, 110, and 111. The first and the last combinations are considered non-conductive states. The other six states are commonly arranged in a sequence where only one switch changes state at a time (this is conducted to reduce losses and high-frequency harmonics); but, in this work, the sequence is kept according to the order they appear in Figure 3 and Figure 4 because it brings the THD 17% lower (from 79.06% to 62.25%). This sequence generates the vectors represented by Equation (5).

$$\begin{array}{cc}\begin{array}{c}{V}_{ak}=V_s{e}^{{\displaystyle \frac{j\left(k-1\right)\pi }{3}}}\\ V_{bk}=V_s{e}^{{\displaystyle \frac{j[\left(k-1\right)\pi -2\pi ]}{3}}}\\ V_{ck}=V_s{e}^{{\displaystyle \frac{j[\left(k-1\right)\pi +2\pi ]}{3}}}\end{array}& fork=1,2,3,4,5,6\\ V_{ak},V_{bk},V_{ck}=0& fork=0,7\end{array}$$

(5)

Here, each one of the integer values (1 to 6) represents a 60° section. A graphical representation is shown in Figure 3. It can be observed that the maximum value of V_k is 2/3 V_s where V_s corresponds to the voltage value of the DC source E.

The digital values, or states, necessary to activate the power switches and generate the output waveforms in Figure 3 with the normalized voltage values are presented in

Table 1. Relation between switch states and voltages from Figure 3.

State S2S1S0	Van	Vbn	Vcn
011	Vdc/3	Vdc/3	−2Vdc/3
001	2Vdc/3	−Vdc/3	−Vdc/3
101	Vdc/3	−2Vdc/3	Vdc/3
100	−Vdc/3	−Vdc/3	2Vdc/3
110	−2Vdc/3	Vdc/3	Vdc/3
010	−Vdc/3	2Vdc/3	−Vdc/3
000, 111	0	0	0

.

Table 1 and V_F from Equation (4) can now be represented on a hexagon where each state corresponds to one of the six vectors, and vectors 000 and 111 are at the origin of the coordinate system, as illustrated in Figure 4.

Each vector is composed of its V_α and V_β parts, which are obtained by their switching state through the transformation described by Equation (6).

$$\left[\begin{array}{c}{V}_{\alpha }\\ V_{\beta }\end{array}\right]={\displaystyle \frac{Vdc}{3}}\left[\begin{array}{ccc}2& -1& -1\\ 0& \sqrt{3}& -\sqrt{3}\end{array}\right]\left[\begin{array}{c}{S}_2\\ S_1\\ S_0\end{array}\right]$$

(6)

Their angle and magnitude are derived from Equations (7) and (8):

$$\theta ={\tan }^{-1}\left[{\displaystyle \frac{V_{\alpha }}{V_{\beta }}}\right]$$

(7)

$$V=\sqrt{{V_{\alpha }}^2+{V_{\beta }}^2}$$

(8)

The output voltage waveforms shown in Figure 3 are not desirable for many applications [34,35]; therefore, to generate a purer sine wave, Space Vector Pulse Width Modulation is implemented. The objective is to generate the reference vector V_F from the two vectors forming a sector. Using Sector 1 as an example, V₁ and V₂ are the vectors to be applied. It is performed by placing inside every sector one or more PWM periods, switching between states 100 and 110 and including states 000 and 111 to fill the whole period.

The time the system stays in each state varies as the reference vector V_F moves from V₁ to V₂ and the magnitude wanted for V_F. Figure 5 shows V_F and its components on V₁ and V₂.

From Figure 5, if T₁V₁/T_PWM and T₂V₂/T_PWM are the fractions from V₁ and V₂ used to generate V_F, Equation (9) is obtained:

$$\begin{array}{c}{T}_1={\displaystyle \frac{\sqrt{3}{T}_{PWM}\left|V_F\right|}{V_{dc}}}·\left(\sin \left({\displaystyle \frac{\pi }{3}}-\alpha \right)\right)\\ T_2={\displaystyle \frac{\sqrt{3}{T}_{PWM}\left|V_F\right|}{V_{dc}}}·\left(\sin \left(\alpha \right)\right)\\ T_0=T_{PWM}-\left(T_1+T_2\right)\end{array}$$

(9)

where T_PWM is the period of the PWM, and α is the angle of V_F as it moves from V₁ to V₂. For the other sectors, if the reference vectors are rotated to the side vectors of each sector, the equations will be the same. The relation |V_F|/V_dc is known as the modulation index. |V_F| must be, as maximum, 2/3V_dc, otherwise it is called overmodulation, which is a topic not covered in this work.

Equation (9) specifies the time of each state within a PWM period. Plotting T₁, T₂, and T₀, as α goes from 0 to 60°, and considering |V_F| = 2/3(2/3V_dc) and T_PWM = 1, is shown in Figure 6.

To reduce the computational demands for the microcontroller, T₀, T₁, and T₂ are linearized; therefore, T₁ will decrease and T₂ will increase by fixed values as α goes from 0° to 60°. Therefore, instead of a look-up table requiring several jumps in the memory code, there will be only subtraction and addition operations. T₀ will be left at a fixed value.

Within each sector, there could be one or more switching periods, depending on the selected PWM frequency. There are several methods to generate a switching sequence [33]. The method used in this work is the symmetrically generated SVPVM. This method is not the simplest to implement or with the lowest switching losses but generates lower total harmonic distortion. The plot for one switching period is shown in Figure 7.

When applying a frequency variation to a motor, there are some factors to consider, like torque and voltage, when varying the operational frequency. In this work, a constant torque is applied for frequencies below 60 Hz [36,37] and constant voltage for frequencies above 60 Hz. The constant torque is performed by reducing the voltage while reducing frequency, as seen in Figure 8. It is conducted by increasing T₀ in Equation (9), inversely proportional to the frequency from 60 Hz, to the minimum of 31 Hz.

Now that the description of the theory and outlines of a variable-frequency motor controller have been presented, the implementation is described in the following section.

2.3. Implementation

The algorithm is implemented in an 8-bit microcontroller, the PIC18F45K50. This microcontroller has two 16-bit comparator: one is for counting the time of each sector, and the other counts the time of each switching section, i.e., T₁, T₂ and T₀. The frequency of the microcontroller is 48 MHz, which is its maximum operational frequency. This frequency allows most of the instructions at assembler level to be executed in 0.0833 µS.

SVPWM is performed by calculating the times for T₀, T₁, T₂, and T₇. T₀ and T₇ are defined when selecting the output frequency. T₁ and T₂ vary within each sector as the reference vector moves along such sector. From Equation (8), the sine of α, from 0 to π/3, is needed. This is usually performed by a look-up table; however, it requires multiple instruction jumps so the curves in Figure 5 have been linearized as mentioned previously. With this approach, with the increment of α, T₁ decreases and T₂ increases, both by the same amount, making them just addition and subtraction operations.

When the user selects a new frequency, which is sent via serial communication, the program calculates the following parameters:

Time per sector in microseconds: T_s = 166666/frequency;

Time of a switching period: T_PWM = T_s/N;

10% of T_PWM for T₀ = T_PWM × 0.025 (0.1 divided by 4) when frequency ≥ 60;

T₀ = T_PWM × [0.025 + (60-frequency) × 0.00375] when frequency < 60;

T₁ starts at its maximum value = N × (T_PWM − 4 × T₀)/(2(N + 1));

T₂ starts at its minimum value: T₂ = T₁/N;

T₇ includes both sides of the PWM sequence: T₇ = 2 × T₀;

Decrements to T₁ and increments to T₂: Increments = T₁/N.

N is the number of switching periods according to

Table 2. Number of PWM cycles per sector vs. frequency.

Frequency (Hz)	Number of Switching Periods	Time Range for Every Switching Period (μs)
31–34	9	597–544
35–38	8	595–548
39–44	7	610–541
45–51	6	617–544
52–61	5	641–546
62–77	4	672–541
78–102	3	712–544
103–154	2	809–541
155–300	1	1075–555

. This is implemented to have a frequency range of switching periods between 930 and 1838. The times for T₀, T₁, and T₂ are adjusted to finish the switching periods at the same time as each sector. T₁ and T₂ must have a minimum value of 6 μs to allow the program to go to the next interrupt. A modulation index (M.I.) of 0.9 is the maximum value that allows the program to finish the interrupt functions.

The change from one switching state to another was conducted by interrupts generated by a 16-bit comparator module. A comparator was selected because, instead of generating an interrupt on overflow, like a timer, it is performed when a counter equals a preset value, resetting the counter, and when changing the preset value, the counter already has been counting, generating precise time values for every switching state.

The flow chart of the algorithm implemented to switch states is illustrated in Figure 9.

The flow chart of the algorithm for changing sectors is illustrated in Figure 10.

3. Results

The focus of this section is to demonstrate that the system generates digital signals, as presented in Figure 7, with precise times, and the frequency variation is achieved for the proposed range from 31 to 300 Hz. The first tests were performed in a virtual environment, using Proteus software, version 8.3. This environment allows visualization of the digital signals generated by the microcontroller, therefore, proving the program works correctly, changing frequency as specified by the user, and that timing is right. In Figure 11, a sample of outputs S₂, S₁, and S₀ shows times for each period. Taking, for example, S₂ (yellow line), its period takes approximately 56 μs. According to Table 2, there are five cycles per sector, and six sectors generate a cycle of the output voltage, hence, the frequency is 59.52 ≈ 60 Hz. When analyzing the signals in MATLAB/Simulink, version R2020b, it is shown that an exact value of 60 Hz is achieved.

The changing values of T₁ (decreasing) and T₂ (increasing) are shown in Figure 12. The states included are 000, 100, 110, and 111. Because there is only a 1-bit change between vectors 100 and 110, the only signal showing variation in the plot is S₁ (blue line).

Programming and testing the microcontroller were performed on a board developed by our research team, as shown in Figure 13. This board allows for programming the microcontroller using a toolkit or through the USB port. It was designed to minimize electromagnetic interference (EMI) and to connect a Bluetooth module, LCD screen, and an SD card. A full description of the board and the Gerber files can be found in [38].

The signals, generated by the microcontroller, were captured by a data acquisition board from National Instruments, the DAQ USB-6259, for a time frame of 0.1 s. They were then fed to a virtual system in MATLAB/Simulink. These signals activated the switching MOSFETs, forming a two-level voltage source inverter, like the one presented in Figure 2. The output voltage waveforms from the voltage source inverter are shown on Figure 14.

The performance of the proposed system is analyzed by measuring the percentage of THD and comparing it to other systems found in the literature. The selected operating frequency to measure THD is 50 Hz; although, at 60 Hz, the value is similar.

Table 3. Comparison of % THD for a modulation index of 0.9.

PWM Technique	This Work	[39]	[40]	[41]	[42]
Operating frequency	50 Hz	50 H		60 Hz	50 Hz
SPWM			77.28	80.33	77.74
SVPWM	61.47	57.28	63.85	64.61	52.05

presents the values of THD for a modulation index (M.I.) of 0.9.

In [39], their system works at 50 Hz, and in [40,41], they present values at 1 and 0.8 M.I., but it shows a value at 0.9 M.I. by using the rule of proportionality. In [42], their system also works at 50 Hz. In [40,41,42], they show the performance of Sine Pulse Width Modulation, which presents a THD 21–49% above SVPWM, indicating clearly the better performance of SVPWM for reducing THD.

In this work, as was explained at the end of Section 2.2, the M.I. at 31 Hz is 0.465 and it generates a THD of 128.2.%, which is consistent with the values shown in [39,40] when they decrease the M.I.

For application specifically on induction motors,

Table 4. Comparison of % THD for application on induction motors (M. I. of 0.9).

PWM Technique	This Work	[39]	[43]	[44]
Operating frequency	50 Hz	-	50 H
SPWM			68.64	89.34
SVPWM	61.47	57.28	57.80	60.53

presents works found in the literature: in [39], they do not specify the operating frequency of their proposed system and only present a system using SVPWM, and in [43,44], they compare SPWM and SVPWM using a fixed frequency.

To demonstrate the performance of the system with an LCL filter, one was added to smooth the signals. The values for the inductors and capacitor were derived from [45], and the values calculated are L₁ = 0.024 H, L₂ = 0.0144 H, and C = 15.00 μF. The values of the parameters for the virtual system are presented in

Table 5. Values of parameters of the virtual system.

Parameter	Value	Parameter	Value
Output phase voltage	260 V	Source voltage	370 V
Active power	1.3 kW	Switching frequency	900 Hz

, and the voltage waveforms at the output of the LCL filter are shown in Figure 15.

From Figure 15, two waveform peaks are measured, and it shows that, for 60 Hz, the system generates a precise frequency, as seen in Figure 16.

Next, the system frequency is changed to operate at 31 Hz to observe its behavior as it is shown in Figure 17. The frequency operates at 30.97 Hz, a deviation of 0.097%. The output phase voltage is reduced to 119 V to keep a constant torque as was expected.

Finally, the system frequency is changed to operate at 300 Hz, and its voltage waveforms are shown in Figure 18. The frequency operates at 299.4 Hz, with a deviation of 0.19%. This is the maximum deviation from the nominal frequency in the entire range. The output pick phase voltage is 216.7 V.

The voltage waveforms for 60 Hz, shown in Figure 16, have a THD of 2.43%; for 31 Hz, observed in Figure 17, the THD was 4.02%; and for 300 Hz, presented in Figure 18, the THD was 0.43%.

Table 6. THD after the LCL filter for different frequencies.

	31 Hz	60 Hz	100 Hz	200 Hz	300 Hz
THD	4.02%	2.43%	1.20%	1.47%	0.43%

presents THD values across various frequencies along the entire range the system can generate. These values have been obtained with a lineal load of 1.3 kW. For a non-lineal load with 500 var added to the active power, the THD for 60 Hz increases to 3.68%, showing the system performance. The IEEE 519–2014 Standard [46] recommends a maximum voltage THD below 8% for systems with a voltage of 1000 volts or less; therefore, the generated system voltage output waveforms adhere to this recommendation, showing the attenuation performance of the filter.

According to [47], the resonance frequency of LCL filters is given by

$$f_{res}={\displaystyle \frac{1}{2\pi }}\sqrt{{\displaystyle \frac{1}{r\left(1-r\right)LC}}}$$

where L = L₁ + L₂, and r = L₁/L. The resonance frequency for the LCL filter is 433.16 Hz, therefore, there is no damping requirement.

Finally, a virtual test bench was implemented in MATLAB/Simulink to evaluate the system using an induction motor with parameters identical to an industrial motor from GE, model number 5KAF145SAA202A, catalogue number M9301. The parameters of the motor are described in

Table 7. Induction motor parameters.

Model	5KAF145SAA202A	Unit
Rated power	2	HP (1491.4 watts)
Rated voltage	230	V
Rated current	5.2	A
Maximum speed	1730	RPM at 60 Hz
Maximum speed	2595	RPM at 90 Hz
Speed range const torque	0–1725	RPM
Speed range const hp	1730–2595	RPM
SF 1.0	POLES 4
LOAD%	100	75
% EFF	86.52	86.94
% PF	78.54	68.82
AMPS	2.74	2.35
Torque	8.2321	N-m

. The system worked at a frequency of 60 Hz. Figure 19 shows the rotor speed and current of phase a with 100% load. The behavior of the motor demonstrates that the system can operate an induction motor properly.

4. Conclusions

This paper presented a functional control system for a variable-frequency driver. The system was implemented in an 8-bit microcontroller programmed using language C. The aim was to offer a motor control system to generate the switching signals of a two-level inverter with variable frequency in the range of 31 to 300 Hz. The control includes a voltage reduction, inversely proportional to the frequency reduction for frequencies below 60 Hz to maintain constant torque. The number of switching periods decreased from nine to one with an increase in frequency, as is shown in Table 2, to enable the use of one LCL filter for the entire range of frequencies the system can generate. Two 16-bit comparators from the microcontroller were used for counting every sector and PWM cycles, obtaining a high-precision output frequency. A change in the switching sequence, following the order vectors presented in Figure 4, generated a THD reduction of 16.81%, and doing so allows this system to be used in industrial applications because it obtains a performance comparable to systems developed on high-level systems like FPGAs and DSPs. Besides, it offers power electronics students the opportunity to follow the theory for SVPWM, acquire the description of the implementation, and download the code to program a microcontroller from the PIC18 family with only having to adjust the configuration lines. Or they can adapt the C code to any microcontroller that can be programmed with language C.