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The paper presents a new algorithm for determination of pulse edges of a modulated wave of a PWM voltage inverter which offers a possibility that natural sampling is realized with an arbitrary accuracy without applying an iterative procedure. The basic idea is to express the angles which determine pulse edges of the modulated signal as polynomials of amplitude modulation index. Geometric interpretation of sampling of the polynomial algorithm is identical with the geometric interpretation of natural algorithm, but the transcendental equation whose solution defines pulse edges of the modulated signal is replaced by a simple procedure of finding values of a polynomial whose coefficients are determined in advance by an exact procedure. This approach gives the possibility of digital implementation of polynomial sampling method using the low-cost microprocessor platforms.

Pulse width modulation is an important method of the technique of control of power converters. At the current technological level, three areas within the pulse width modulation possess certain autonomy as regards the accomplished solutions [

The modulating function contains information regarding the desired waveform, whereas the signal carrier contains information concerning the switching frequency. Pulse edges of a modulated signal are determined by crossings of the modulating function and signal carrier, which makes the basis of all analog implementations of the modulating function methods involving natural sampling. Signal at the output of a pulse width modulator defines the switching function of the inverter branch which controls the on-state of the switch within the branch. Synthesis of the ac voltage at inverter's output, regulation of the amplitude, and frequency are accomplished by the switching actions in inverter branches.

The basic PWM method is related to the mode of determination of pulse edges. Determination of pulse edges based on crossing of the modulating and carrying signals belongs to the group of natural sampling techniques.

The basic shortcoming of the natural sampling technique is tied to the transcendental relation which connects the angles determining pulse edges and modulating function. This aggravated the application of natural sampling technique in digital and microprocessor systems applying PWM methods.

Regular sampling method is the basic solution for digital realization of the modulating function method. The regular sampling method is considerably more flexible since the modulating function is specified at discrete points; thus it is possible to calculate its values in advance and interpret them by means of digital words [

Middle of the eighties marks the beginning of the application of the space vector modulation as the vector approach to PWM for three-phase inverters. The basic advantages of SVM are related to the following.

Expanded linear range of modulation without injection of the third harmonics into the sine modulating function [

Lower harmonics content in relation to the regular methods based on the sine modulating function [

Lower switching losses are conditioned by only one change of state [

Simple digital implementation[

On the other side in the papers [

Finally, in the paper [

Taking into account the relevant literature[

General characteristic of this group of methods is that the synthesis of the pulse at the output of the modulator is carried out on the basis of the reference signal and signal carrier.

The reference signal is a periodic function having the frequency equal to the fundamental frequency of the inverter output voltage and amplitude which is proportional to the amplitude of the fundamental harmonic. The modulating function is analytical representation of the reference signal. In general, this is a function of the following general form [

Parameter

In classical PWM methods the carrier signal is an ac signal of triangular half-periods, of frequency

The basic characteristics of the signal at modulator output, irrespective of the method of edge determination, are the binary form and modulated width. They are transferred to the states of switching elements by means of the switching function.

Pulse edges at modulator output are most frequently determined by the natural or regular sampling method on the basis of the reference and carrier signals.

In case of a three-phase inverter modulation function may contain, in addition to the basic component, components of the order 3

In the regular sampling method the reference signal is discretized, and the pulse edges are determined by comparing the carrier signal with the reference signal modified in this way, which implies a digital realization of the method. Therefore, the regular sampling method has the advantage that microprocessor control is possible compared to the natural sampling method where the pulse edges should be determined by iterative solution of the transcendental equation obtained from the condition for crossing of the reference and carrier (triangular) signals. Figure

Synthesis of the output voltage of a half-bridge inverter by applying the modulating function method and regular sampling.

The interval of

The pulse edges centered at

Switching function

The voltage waveform

If the carrier frequency is considerably higher than the reference signal fundamental frequency (

By transforming expressions (

From (

Regulation amplitude factor is defined in [

The basic elements of PWM using the method of modulating function are reference signals which represent mapping of defined phase voltages on three phase loads of inverters and carrier signal which determines switching frequency of inverter’s branch. These elements are necessary for the synthesis of the switching function

The area (Figure

Active voltage space vectors in

The essence of PWM using the method of space vector is in approximation of average value of reference rotating vector inside the switching segment

The way in which the zero states are inserted into the switching segment defines the basic characteristics of the modulation. By the conventional method of space vector the equal parts of both zero states are inside the switching sector. Also, the sequence of successive switching states is formed of contiguous vectors

Although the method of space vector may initially seem radically different from the method of modulating function, it is possible to construct a modulating function starting from the SVM method and vice versa. During the construction of the modulating function, it is first necessary to define the locus of the space vector. If the referent space vector is defined by the formula

Continual vector modulating function which is derived from the classic method of SVM is acquired in the following way.

Inside each sector the switching segment is formed as a symmetrical sequence of switching states formed of consecutive vectors. By applying this principle to the times active consecutive vectors states, the times of active and zero states in

Using the invariance property of time sequence inside the segment

On the other hand, the average value of the switching function which is obtained from the process of regular symmetrical sampling on appropriate segment is

Transition modulating function (

In the approximate algorithm, instead of an iterative procedure for finding the angle of a pulse edge, a procedure of direct determination of an approximate solution is applied. The angle of a pulse edge is considered as function of the amplitude index of modulation, with the pulse number as a parameter, which is realistic since the amplitude index of modulation varies continuously whereas the pulse number remains constant within a single range of fequency control and varies discretely from a higher to a lower integer value. In this method the angle of pulse edge is approximated by a polynomial whose highest degree determines a measure of goodness of the obtained solution compared to the natural sampling and does not influence the applicability of the approximation. The polynomial approximations are without restrictions readily applicable to all known modulating functions and triangular carrier signal.

The reference signal (the modulating function) is specified by expression (

However, the sequences of successive equations (

Speaking in mathematical terms, this fact means that the frequency index

It follows from relation (

The practical application of the derived expressions is possible through the approximation of series (

In what follows the expression “approximate’’ will hereupon be replaced by “polynomial’’.

In practice, determination of pulse edges

In most of the modulating functions sine or cosine segments inside the more complex modulating function exist. This is why we will use the example of a sinusoidal modulating function to illustrate the method we propose in this work.

In this case, the series of successive transcendental equations whose solutions give the pulse edge angles is

The maximum remainder term

In the case of a sinusoidal reference signal, the first four coefficients of power polynomial obtained using relation (

From expressions (

The deviations of angles

Maximum turncation error values upon a change in

6 | 2.0516 | 0.4791 | 0.1284 | 0.0349 |

9 | 0.7845 | 0.1191 | 0.0250 | 0.0029 |

12 | 0.5099 | 0.0631 | 0.0078 | 0.0012 |

15 | 0.3175 | 0.0324 | 0.0032 | 0.0004 |

However, unlike the mentioned numerical method, formulae (

To achieve a higher accuracy of pulse edge with polynomial of lower order, we will apply the economization procedure by using Chebyshev polynomials [

The use of polynomial (

Maximum truncation error values upon a change in

6 | 1.8155 | 0.1297 |

9 | 0.8717 | 0.0351 |

12 | 0.4925 | 0.0161 |

15 | 0.3124 | 0.0078 |

When used as a PWM sampling algorithm, the modified polynomial permits computer time savings and yields a maximum error of 0.1297 degrees, which is comparable to the use of a nonmodified third-degree polynomial.

Because of the odd symmetry of the ac component of the pulse sequence generated by the method proposed, the Fourier expansion of inverter ac voltage contains only sinusoidal components. The amplitude of the nth harmonic of voltage is given by expression:

When the second-degree polynomial algorithms are applied to the sinusoidal modulating function, the results obtained for

Comparison of amplitude characteristic values for natural (SPWM), regular (RSPWM), and derived polynomial algorithmic sampling.

SPWM | RSPWM | |||

0.1 | 0.0500 | 0.0500 | 0.0483 | 0.0497 |

0.2 | 0.1000 | 0.1000 | 0.0966 | 0.0993 |

0.3 | 0.1500 | 0.1501 | 0.1448 | 0.1489 |

0.4 | 0.2000 | 0.2003 | 0.1929 | 0.1985 |

0.5 | 0.2500 | 0.2505 | 0.2410 | 0.2479 |

0.6 | 0.3000 | 0.3009 | 0.2889 | 0.2972 |

0.7 | 0.3500 | 0.3514 | 0.3367 | 0.3464 |

0.8 | 0.4000 | 0.4021 | 0.3843 | 0.3953 |

0.9 | 0.4500 | 0.4530 | 0.4317 | 0.4441 |

1.0 | 0.5000 | 0.5041 | 0.4788 | 0.4927 |

Comparison of distortion factor values for natural (SPWM), regular (RSPWM), and derived polynomial algorithmic sampling.

SPWM | RSPWM | |||

0.1 | 14.0914 | 14.0913 | 15.7644 | 14.0928 |

0.2 | 13.117 | 13.1175 | 14.7981 | 13.0883 |

0.3 | 12.190 | 12.1908 | 13.8556 | 12.1034 |

0.4 | 11.352 | 11.3554 | 12.9719 | 11.1724 |

0.5 | 10.629 | 10.6401 | 12.1601 | 10.3177 |

0.6 | 10.041 | 10.0701 | 11.4392 | 9.55.76 |

0.7 | 9.6108 | 9.6720 | 10.8261 | 8.9166 |

0.8 | 9.3862 | 9.4685 | 10.3405 | 8.4213 |

0.9 | 9.3532 | 9.4733 | 10.0004 | 8.0984 |

1.0 | 9.2787 | 9.6877 | 9.8204 | 7.9690 |

All that has been mentioned above can be applied to the results for a three-phase inverter, since its bridge structure consists of three half-bridge structures. The proposed polynomial algorithms exhibit their major advantages when applied to the vector modulating function as well. At frequency index values

The odd values of frequency index

Maximum amplitudes of the fundamental harmonic of inverter output voltage obtained using the vector modulating function (

The distortion factor DIS(

For higher pulse number values, all algorithms converge toward the same values of amplitude and distortion harmonic indicators (Figures

It is known that the basic characteristics of the digital implementation of the natural sampling are the iterative procedure and the work with the data in the floating point numbers format, which is a consequence of the transcendental nature of the expressions by which the angles of impulses edges are determined. Algebraic nature of the formula for the angles (

By transition to the time domain we find the formulas which determine the position of the

Graphical representation of main elements of impulse synthesis using polynomial algorithm.

In the formula (

The synthesis of impulses on the modulator output is performed by simulation of the process which is shown in the Figure

Main elements of impulse synthesis, using regular symmetrical algorithm.

Modern microcontrollers are equiped with integrated peripheral units designed for the synthesis of the modulated signals which control the state of switching of inverter branch. Intel’s microcontroller 8XC196MC/MD is optimized for implementation of PWM using the method of modulation function [

is the value of the register of maximum frequency of output inverter's voltage;

is the

is the clock frequency on the pin XTAL1 in (MHz).

The times of leading impulses edge are determined in the form of digital words which are calculated using

The values of the coefficient

In this kind of representation all previously calculated coefficients are shown with the accuracy greater than

The coefficients of the leading and trailing edges are placed in the form of pairs of memory tables so that one table corresponds to the leading impulse edges and the other to the trailing impulse edges. The number of tables depends on the polynomial degree and the number of different values of the pulse number. The coefficients which correspond to the different phases are determined from the same table by applying the suitable index addressing of the data from the tables. The basic element which determines the conditions under which a polynomial algorithm can be applied is that the time necessary for realization of the program must be shorter than half of the carrier signal period. Concerning the memory table it is possible to perform certain optimizations using the properties of the symmetry of pulse angles about the half-period. The amplitude modulation index can be obtained also from the memory table which shows the output amplitude dependency on the frequency. In the experiment performed this dependency is linear, and it maps the ratio

The experimental model (Figure

nominal voltage:

rated current:

power factor :

rated power:

rated speed:

rated power:

rated speed:

Experimentally, the possibility of application of the polynomial algorithm as the modification of the regular selection algorithm for the switching frequency of 2,4 kHz is confirmed. For higher switching frequencies, the times needed for online computation of the positions of the impulse edges are critical regarding their asymmetry about the switching segment. The applications of the polynomial algorithm of the first degree for the output frequency of 40 Hz with the pulse number

Oscilogram of line current and phase voltage with vector modulation and polynomial algorithm of the first degree.

The suggested new sampling PWM algorithm by using the method of modulating function offers a possibility that natural sampling can be realized without any iterative procedure. It is shown that the angle of pulse edges can be approximated by a polynomial, the highest degree of the polynomial determining the measure of approximation of the obtained solution compared to natural sampling. The economizing procedure of using Chebyshev polinomials offered the possibility of using polinomials of a lower order, usually of the first- or second-order, still acomplishing a good approximation of the natural sampling. In terms of simplicity, this algorithm is comparable to a version of regular sampling method. As regards the performance concerning the quality of inverter’s output voltage, improvement by application of the suggested algorithm, which retains all good properties of the natural technique of PWM, is confirmed by simulation. In addition, possibility of implementation of polynomial algorithm in the real application is experimentally confirmed.

^{∗∗}Microcontroler-User manual, INTEL 1996