Method for Defining Pulse Width Modulation (PWM) Waveform Patterns to Produce Inverter Outputs Having a Low Harmonic Content

The present invention relates generally to power conversion systems and deals more specifically with a method for selecting pulse width modulation waveform patterns using weighted factors to control a multiple bridgepower inverter to produce a multiple phase AC voltage output signal having low harmonic signal content.

The problem of producing harmonic free AC output power from a power inverter exists for any known means of generation. It is also known that harmonics in the AC output power are generally more severe when the input to the power inverter is from a switched DC voltage source. The use of high power semiconductor inverters has fostered a number of different approaches to reduce harmonics in the AC voltage output; however, these approaches have generally not been satisfactory. The use of passive filters to reduce harmonics is limited due to losses, cost and unsatisfactory dynamic performance.

Although not completely satisfactory, two methods for reducing harmonics in inverter output voltages have been used with some degree of success. One method of harmonic reduction utilizes combinations of phase displaced, three-phase bridges and reference may be made to U.S. Pat. No. 4,975,822 assigned to the same assignee as the present invention and which disclosure is incorporated herein by reference.

A second known method to achieve harmonic reduction includes exciting the power inverter utilizing a pulse width modulated control signal. The improvement in semiconductor switching devices to handle high power levels has permitted the design of power inverters to produce several hundred kilowatts and pulse width modulation techniques have been used to remove a substantial portion of the unwanted harmonics in the power inverter bridge output voltage signal.

It is not easy to define a given pulse width modulated waveform signal which has the desired harmonic reduction because the waveforms are expressed either as a Fourier series of sine and cosine terms or Bessel functions. One commonly known method to define a pulse width modulated waveform uses the sine-triangle method to generate gating signals to control the semiconductor switching. One problem generally associated with the sine-triangle method is that the peak of the sinewave cannot reach the apex of the triangle and the modulation index over the full switching range approximates 80% whereas a high as possible modulation index is desired.

Also, the sine-triangle method generally results in an inverter output voltage signal having a very high magnitude 11th and 13th harmonic. The high harmonics often lead to high overcurrents which trip the circuit breakers associated with each output and also contributes to a "noisy" output voltage signal.

Another factor to be considered in obtaining a suitable PWM waveform pattern is that harmonics which are reduced at one frequency generally always result in increasing other harmonics at different frequencies. Additional problems generally associated with pulse width modulation techniques for reducing harmonics is that each switch pair added to the PWM waveform pattern reduces the available voltage from the inverter.

There are also several practical considerations that limit the use of the pulse width modulation PWM techniques for reduction of harmonic signals and include among them: dwell time when both switch base drives must be in an off-state to compensate for the semiconductor device turnoff delay; limitations in the minimum pulse width obtainable due to the control design and upper frequency limits of the switch devices used. Although PWM techniques have made possible some improvements in the reduction of harmonics, the reductions are not always predictable and the same harmonics are often cancelled more than once.

It is a general aim of the present invention therefore to provide a method to design a set of pulse width modulation (PWM) waveform patterns that reduce only those harmonics necessary for a particular design and which harmonics are not removed by other portions of the power inverter system. Accordingly, the method of the present invention permits the selection of PWM waveform patterns that eliminate or otherwise minimize the undesired harmonics to an acceptable magnitude.

In accordance with the present invention, a method for defining a pulse width modulation (PWM) waveform pattern to control at least one switching bridge in a power inverter system to produce a low harmonic content output voltage signal wherein the PWM waveform pattern has at least one switching point includes the steps of specifying a desired harmonic signal content in a voltage output for the power inverter system for each of the harmonics produced by the switching bridge; defining the pulse switching rate for the switching bridge; assigning a weighting factor to each harmonic produced by the switching bridge in the power inverter system; identifying each harmonic reduced by the power inverter system and selecting harmonics, other than those harmonics reduced by the power inverter system, to be reduced; representing the selected harmonics to be reduced on a one-for-one basis by a corresponding set of equations and solving the set of equations to find each of the switch points in the PWM waveform pattern.

FIG. 1 is a flowchart showing the general steps embodied in the method of the present invention.

FIG. 2 is a simplified schematic functional block diagram showing a representative three-phase, two-bridge inverter power system with which the method of the present invention may be used.

FIG. 3a and FIG. 3b show a table of solutions for ten switch points for one PWM waveform pattern in accordance with the method of the present invention.

FIG. 4a and FIG. 4b show another table of solutions for ten switch points for a PWM waveform pattern in accordance with the method of the present invention.

Prior to considering the method of the present invention in detail, it is beneficial for a better understanding of the invention to consider the following information.

If the voltage output signal of an unmodulated three-phase bridge is examined, it would be seen that the inverter develops an output line-to-line voltage signal having only odd harmonics and those harmonics that are produced are not integral multiples of three (triplens). With the application of the simplest PWM control waveform pattern comprising an edge or center notch pulses for example, voltage control of the fundamental is obtained at the expense of producing increased higher order harmonics. Additional pulses may be added to the pattern to decrease some of the additional harmonics created however other harmonics will be increased. To applicant's knowledge, there is no direct method or procedure to decide which harmonics are reduced, which harmonics are increased and which harmonics are ignored due to the additional pulses added to the PWM waveform pattern.

A PWM waveform pattern may be represented as a Bessel function or as the sum of many phase displaced square waves. Although the present invention is not dependent on the method of the selection of the square waves in the pattern, the preferred method of the invention assumes a quarterwave (90 degrees) symmetry for the square wave in which case only odd harmonics are generated which simplifies DC unbalance, current limiting and power and VAR control.

Based on the foregoing, the PWM Fourier series may be represented by the expression:

The voltage magnitude coefficient can be represented by the following expression:

Where N=harmonic number

Therefore, an independent equation for each switch point per quartercycle defining the harmonic voltages can be written. For example, for one switch point with a frequency of 3 times the fundamental, the equation for the voltage magnitude coefficient: ##EQU1##

The following equation represents two switch points at a frequency of 5 times the fundamental: ##EQU2##

One approach to solve the nonlinear equations defining the harmonics as a function of the PWM patterns is to set the magnitude equal to some value of the fundamental and the magnitude of a harmonic, for example, the fifth harmonic, equal to 0. The result is two equations with two unknown angles which may be solved for by known mathematical procedures such as Newton's or Piccard's methods.

An alternative is to solve only one equation wherein the fifth harmonic is set equal to 0. For this solution, the value of the larger of the two angles is selected resulting in the following equation: ##EQU3##

The equation (5) can be solved for a range of given values of the angle 2 and the results can then be used to calculate the fundamental voltage. Although the above equation in the foregoing example does not require the solution of simultaneous equations, answers for PWM waveform patterns with more than two switch points requires the solution of simultaneous equations.

For example, the following equations for the fundamental, fifth and seventh harmonic can be written for three switches per quarter cycle: ##EQU4##

By setting the equations (6) (7) and (8) equal to some value, the three angles A, B and C may be found. For example, the fundamental could be set equal to 90% and the fifth and seventh harmonics equal to 0% and the nonlinear equations may be solved to find the value of the three angles if the values exist. It will be apparent that this method may be extended to accommodate a large number of switch points.

According to the present invention, solutions are found for a range of fundamental voltages over which the harmonics are reduced to be within acceptable limits for the inverter design in question. In the first approach to the solution of the nonlinear equations wherein the voltage magnitude is set to some value N equations are solved with the fundamental voltage index defined and N angles are found to reduce N-1 harmonics. In the second approach above wherein the value of the larger of the two angles to be found is selected, N-1 equations are solved for N-1 angles. A range of solutions is found by specifying different value angles. The value of the fundamental may then be calculated.

It has been found that using the second approach yields more consistent results since one angle is fixed. Since non-linear equations have multiple answers, the valid answers tend to group the angles within a 60 degree region. Therefore, maintaining one angle fixed tends to group the solution set in the desired region of valid answers.

Typical restrictions on the solution set are:

a) no reversal of angle positions (yields an incorrect solution).

b) calculated angles lie between specific limits for example, 0.5 to 88.5 degrees.

c) no harmonic above a given specified value.

From a practical standpoint, removal of harmonics not otherwise cancelled results in the available range of modulation indices of the fundamental being very narrow. Applicant has determined that the addition of switch points reduces more harmonics in the range of interest and provides a wider range of modulation indices of the fundamental. The additional switch points requires solving higher orders of equations which increases the computation time and makes it difficult to find the starting values from which to solve the equations.

The solution of the nonlinear simultaneous equations typically is done iteratively wherein initial guesses of the answers are made and are subsequently used to obtain successively better guesses or to determine if there is no solution to the equation. The typical test for a given solution is to compute the magnitudes of the harmonics after the angles are found to determine whether the residual values of the reduced harmonic are below specified maximum value. The method of the present invention departs from prior conventional practice wherein it is attempted to determine and obtain solutions to produce residual values of harmonics below the magnitude of the fundamental by at least 1000 to 1. In one given power inverter, for example, a combination of two three-phase bridges using harmonic reduction and a low pass filter permitted the magnitude of many of the harmonics generated by the bridge to be at relatively high levels, up to 20%, without degrading the output signal.

The method of the present invention as illustrated generally by the flowchart shown in FIG. 1, utilizes a weighting curve which sets the acceptable level of all the harmonics which are reduced by a PWM waveform pattern. The advantage of setting the weighting curve to the solution is that only two of the many harmonics have severe magnitude restrictions. For a two-bridge inverter, such as the one illustrated in FIG. 2, only the 11th and 13th harmonic need to be reduced to as low as 0.3%. Therefore, the method of the invention involves examining the total inverter power system to place a weighting factor on the reduction of the harmonics.

Applicant has developed a controller utilizing the following parameters to obtain a low harmonic output from a two-bridge inverter system having minimal filtering. The parameters are: 1) 1/2 degree minimum between switch points; 2) 10 switch points per one quarter cycle (21 times fundamental); 3) reduce the 11th and 13th harmonic to less than 0.3% of fundamental; 4) reduce the 23rd and 25th harmonic less than or equal to 2% of fundamental; 5) reduce the 35th and 37th harmonic less than or equal to 5% of the fundamental; 6) reduce 5th, 7th, 17th, 19th, 29th, 31st, 41st, 43rd, 47th, 49th harmonic less than 10% of fundamental; and 7) select the 7th, 11th, 13th, 19th, 23rd, 25th, 31st, 35th, 37th and 41st harmonics for reduction. It should be noted that the 7th, 19th, 31st and 41st harmonics are also reduced by the two-bridge connection. Although it is possible to determine a PWM waveform pattern to reduce only the harmonics not reduced by other means, the available range of modulation indices of the fundamental is extremely narrow. Accordingly, applicant has found that reducing one from each pair of harmonics also reduced by the magnetics of the bridge connection and/or harmonic signal reduction transformers such as disclosed in the above-referenced U.S. Pat. No. 4,975,822 yields good results.

One suitable method to solve the nonlinear simultaneous equations involves utilization of a multivariable expansion of Newton's method of approximation. An examination of a table of values of an equation that has a value of f(x)=0, will show that as x changes, the function goes from negative to positive or the reverse between successive steps in x, and over a small range of x, the slope of Df(x)/Dx is constant. Newton's method takes advantage of this fact to obtain a successive better solution to the value of x where f(x)=0. When more than one equation is involved, partial derivatives of the equation are used to derive a set of linear simultaneous equations that can be solved to produce a better estimate of the solution.

It will be recognized that the matrix of partial derivatives is known as the Jacobian. This process is initiated by selecting N phase angles to find N-1 angles and one angle, generally the highest angle does not change. The Jacobian is then generated and the values of the harmonics are calculated for each of the selected values. The two matrices, (N-1),N and (N-1), 1 are solved to find delta angle values that would reduce the magnitude of the harmonic values to 0 if the system were linear. The delta's found are added to the previous guess and the harmonics are again calculated. A solution is found when the harmonics are within the desired range. It should be recognized that the solution for the delta angles results in some cases in non-realizable angles, for example, negative angles, angles greater than 90 degrees, angles reversing positions, and angles less than 1/2 degree apart. In other instances, the computed delta angles are too large. In such instances, a variable lambda called a relaxation factor, is used to produce smaller changes.

For a two-bridge inverter, such as in the example discussed above, three of six harmonics were added to the reduction solution and of the six possible combinations, one provided the best result. These are shown as patterns no. 2 and no. 5 shown in FIGS. 3 and 4, respectively. These solutions cover a sufficiently wide range of modulation indices to be suitable for use.

The PWM waveform patterns are found using a computer program, for example, a Pascal program. In the example presented, the program consists of a main program and several subunits and the program is set to find patterns for three to ten switches per half cycle. The program for finding PWM waveform patterns is presented in Table 1. ##SPC1##

Once the PWM waveform switch points have been determined, the patterns are converted to an information format usable by the controller in the power inverter system. Typically, a microprocessor follows an instruction set to carry out the necessary controller operations which are generally well understood by those skilled in the art. A computer program for converting the switch point information to a form that a microprocessor assembly source code uses to define PWM waveform drive patterns is shown in Table 2. ##SPC2##

A method for specifying a group of PWM waveform patterns to reduce only those harmonics necessary to satisfy a given power inverter design has been disclosed above in several embodiments. It will be recognized that the implementation of the method may be varied without departing from the spirit and scope of the invention and therefore the invention has been disclosed by example rather than limitation.