电气类外文文献及其翻译--正弦脉宽调制

24.437 Power Electronics

Sinusoidal Pulse width modulation

The switches in the voltage source inverter(See Fig.1)can be turned on and off as required.In the simplest approach,the top switch is turned on If turned on and off only once in each cycle,a square wave waveform results.However,if turned on several times in a cycle an improved har- monic profile may be achieved.

Fig 1:Simple V oltage Sourced

In the most straightforward implementation,generation of the desired output voltage is achieved

by comparing the desired reference waveform(modulating signal)with a high-frequency triangu- lar…carrier?wave as depicted schematically in Fig.2.Depending on whether the signal voltage is larger or smaller than the carrier waveform,either the positive or negative dc bus voltage is applied at the output.Note that over the period of one triangle wave,the average voltage applied

to the load is proportional to the amplitude of the signal(assumed constant)during this period.

The resulting chopped square waveform contains a replica of the desired waveform in its low fre- quency components,with the higher frequency components being at frequencies of an close to the carrier frequency.Notice that the root mean square value of the ac voltage waveform is still equal to the dc bus voltage,and hence the total harmonic distortion is not affected by the PWM process. The harmonic components are merely shifted into the higher frequency range and are automati- cally filtered due to inductances in the ac system.

When the modulating signal is a sinusoid of amplitude Am,and the amplitude of the triangular carrier is Ac,the ratio m=Am/Ac is known as the modulation index.Note that controlling the modulation index therefor controls the amplitude of the applied output voltage.With a sufficiently high carrier frequency(see Fig.3 drawn for fc/fm=21 and t=L/R=T/3;T=period of funda- mental),the high frequency components do not propagate significantly in the ac network(or load) due the presence of the inductive elements.However,a higher carrier frequency does result in a larger number of switchings per cycle and hence in an increased power loss.Typically switching frequencies in the 2-15 kHz range are considered adequate for power systems applications.Also

in three-phase systems it is advisable to use s so that all three waveforms are symmetric.

Fig 2:Princ ipal of Pulse Width Modulation

Fig.3:SPWM with fc/fm=48,L/R=T/3

Note that the process works well for m ≤1.For m >1,there are periods of the triangle wave in which there is no intersection of the carrier and the signal as in Fig.4.However,a certain amount

of this“overmodulation”is often allowed in the interest of obtaining a larger ac voltage magni- tude even though the spectral content of the voltage is rendered somewhat poorer.

Note that with an odd ratio for fc/fm,the waveform is anti-symmetric over a 360 degree cycle. With an even number,there are harmonics of even order,but in particular also a small dc compo- nent.Hence an even number is not recommended for single phase inverters,particularly for smal ratios of fc/fm.

SPWM Spectra:

Although the SPWM waveform has harmonics of several orders in the phase voltage waveform, the dominant ones other than the fundamental are of order n and n±2 where n=fc/fm.This is evi- dent for the spectrum for n=15 and m=0.8 shown in Fig.5.Note that if the other two phases are identically generated but 120o apart in phase,the line-line voltage will not have any triplen

har-monics.Hence it is advisable to choose,as then the dominant harmonic will

be eliminated.It is evident from Fig 5b,that the dominant 15th harmonic in Fig.5a is effectively eliminated in the line voltage.Choosing a multiple of 3 is also convenient as then the same trian- gular waveform can be used as the carrier in all three phases,leading to some simplification in hardware.

It is readily seen that as the where E is the dc bus voltage,that the rms value

of the output voltage signal is unaffected by the PWM process.This is strictly true for the phase voltage as triplen harmonic orders are cancelled in the line voltage.However,the problematic har- monics are shifted to higher orders,thereby making filtering much easier.Often,the filtering is carried out via the natural high-impedance characteristic of the load.

Fig.5:SPWM Harmonic Spectra:n=15,m=0.

Selective Harmonic Elimination

(also called Optimal PWM)

Notice that in the SPWM strategy developed above,a large number of switchings are required, with the consequent associated switching losses.With the method of Selective Harmonic Elimina- tion,only selected harmonics are eliminated with the smallest number of switchings.This method however can be difficult to implement on-line due to computation and memory requirements. For a two level PWM waveform with odd and halfwave symmetries and n chops per quarter cycle

as shown in Fig 4,the peak magnitude of the harmonic components including the fundamental,

are given byEqn.1:

Here is the magnitude of the harmonic and is the primary switching angle.Even har- monics do not show up because of the half-wave symmetry.

The n chops in the waveform afford n degrees of freedom.Several control options are thus possi- ble.For example n selected harmonics can be eliminated.Another option which is used here is to eliminate n-1 selected harmonics and use the remaining degree of freedom to control the funda- mental frequency ac voltage.To find theα?s required to achieve this objective,it is sufficient to set the corresponding h?s in the above equations to the desired values(0 for the n-1 harmonics to be eliminated and the desired per-unit ac magnitude for the fundamental)and solve for theα?s.

Fig 4:A two-level PWM waveform with odd and halfwave symm

Equation 1 can be readily proved by finding the fourier coefficients of the waveform shown inFig.4.In general,for a periodic waveform with period,the Fourier Cosine and Sine Coeffi- cients are given by:

Because of the half-cycle symmetry of the waveform of Fig.4,only odd order harmonics exist. Also,it is easy to see that the Fourier Cosine coefficients disappear with the choice of coordinate

axes used.Utilizing the quarter cycle symmetry,the Fourier Sine coefficients become:

Substituting the two-valued pwm waveform for,one obtains(see Fig.4):

The following example illustrates the use of three chops per quarter cycle which allow for three degrees of freedom.We may use these to eliminate two harmonics and control the magnitude of the fundamental to any desired value:

Example:

Selective Harmonic Elimination is applied with a view to controlling the fundamental component of voltage to 50V(rms)and eliminating the 3rd and 5th harmonics.The source voltage is 100 V. Calculate the required chopping angles.

As three objectives are to be achieved,we need 3 chops.The fundamental,3rd and 5th harmonic magnitudes are given by:

We require:

翻译

24.437电力电子

正弦脉宽调制

电压源逆变器的开关（见图1）可以按要求打开和关闭。用最简单的方法，顶部的开关打开，如果每个周期打开和关闭，方波的波形结果只有一次。但是，如果改进谐波的数据则在一个周期内可以形成多次打开关闭。

图1：简单的电压源逆变器

在最直接的实现方式，所期望的输出电压生成是通过比较预期的参考波形与高频率三角?载体?波（调制信号）所描述的图2.根据信号电压是否大于或小于载体波波形，无论是正还是负的直流母线电压施加在输出。注意，在此期间一个三角波周期的平均电压加到负载型成正比（假定不变），信号的振幅。注意,经过一段时期一个三角形波,平均电压的负荷是成正比的幅值的信号(假定常数)在这个时期。由此产生的方波包含在它的低频率元件所需波形的复制，具有较高频率分量在一个载波频率接近的频率的福祉。注意，均方根平方的交流电压波形值仍相等的直流母线电压，由于PWM使得总谐波不失真。谐波成分只是转移到更高的频率范围，并且由于电感的交流系统自动地过滤。

当调制信号为正弦波的振幅Am，和三角载波的振幅Ac的比Am/Ac是已知的调制指数。注意，控制调制指数为施加控制输出电压幅值。具有足够高的载波频率（参见图3得出fc/fm=21 and t=L/R=T/3；T=基础时期），由于感性元件的存在高频成分明显不传播到交流网络（或负载）。然而，由于较高的载波频率，开关较多从而在每个周期不增加功率损耗。电力系统的应用通常在2-15kHz的开关频率范围被认为是足够的。此外，在三相系统中，建议

使用使所有三个波形对称。

图2：主要的脉宽调制

图3：SPWM的fc/fm=48，L/R=T/3

注意，这个过程很适合。因为在图4中有三角波其中有没有交际的载体作为信号周期。然而，这种“过调制”在一定量往往是允许获得更大的交流电压，使电压频谱呈现稍差。

注意，fc/fm使用一个额外的比率，波形是反周期超过360度的对称。随着偶数阶谐波，特别小的直流元件。因此一个单相逆变器的不推荐用偶数，特别是fc/fm额外比率。

SPWM的频谱：

虽然SPWM波形已在几个数量相电压波形的谐波，比其他有根本优势，是因为当n=fc/fm，n和n正负2 。这是经济脆弱性，在图5削弱了频谱对n=15和m=0.8 。请注意，如果其他

两个阶段产生的，但除了在相同阶段120o，先电压不会有任何。因此它选择

是明智的。正如当时占主导地位的谐波将被淘汰。它是由图5b自明的，

占主导地位的第15谐波，图5a有效地消除了线路电压。选择3的倍数也方便，则相同的特里安奇异波形可作为承运的所有三个阶段，可以使其在硬件上简化。

这是容易看到的，当E是输出电压信号的均方根，(pwm(θ))^2=E^2是直流母线电压。这是严格的相电压谐波。然而，问题谐波的转移较高，从而使过滤更加容易。通常情况下，进行过滤通过自然高阻抗负载的特点。

图5：SPWM的谐波谱：n=15，m=0.8

选择性谐波消除

（也称为最优脉宽调制）

注意发展SPWM布局，大量的开关需要与相应的高消耗随之而来。随着选择性谐波Elimina-tion的方法的使用，只有选择谐波消除使开关最少。然而，由于计算和内存的需要这种方法可能很难实现。对于奇数的半波对称二级PWM波形，如图4，包括基础谐波成分峰值幅度，由等式1给出：

在波形n负担n个自由度。几个可能的控制选项。例如选择谐波可以被去除。这里使用的另一个选择是n-1个选择谐波和使用剩余的自由度，以控制频率的交流电压。找到a的要求达到这一目标，在上述方程以设置相应的h为所需的值（0为n-1个被淘汰的谐波和所需的每单位的基本交流大小）和解决a。

图4：奇二电平PWM波形和半

公式1可以很容易地证明了在图中显示的波形的傅立叶系数。图4在一般情况下，周期波形，傅立叶余弦和正弦系数的计算公式如下：

由于波形图的半周期对称。图4只有奇次谐波存在。此外，它很容易看到，利用季度周期对称性，傅立叶余弦系数随着使用的坐标轴的选择消除。

代入f（θ）可得二值PWM波形（见图4）：

下面的例子说明了三个周期，每季度的三自由度允许使用条件。我们可能使用这些两个谐波消除和控制的基本幅度作出任何所需的值：

例如：

应用选择性谐波消除以控制基本元件的电压为50V（RMS）和消除谐波的第三和第五级。源电压为100V。计算所需的砍角。

至于三项目标得以实现，我们需要三个条件。最根本的是，第三和第五次谐波幅度的计算公式如下：

这给了我们三个未知数a1，a2和a3的方程。数值求解，我们得到：