VOLTAGE SPECTRUM OF A SINGLE-PHASE INVERTER WITH PULSE-WIDTH MODULATION

M. A. Zhitkov

(Presented by the scientific seminar of the Department of Electric Drive)

The use of pulse-width modulation (PWM) in energy-conversion systems makes it possible to create universal frequency converters that provide separate regulation of the frequency and amplitude of the fundamental harmonic of the output voltage. At the same time, it becomes possible to combine the functions of inversion and voltage regulation in a single inverter, which leads to a reduction in weight, a reduction in dimensions, and an improvement in the economy of the frequency converter. On the other hand, in the spectrum of a pulse-width-modulated voltage the percentage content of higher harmonics is substantially reduced and their amplitude is decreased, especially when sinusoidal modulation is employed. In this case, the smooth component of the voltage and the load current have a form close to sinusoidal.

In works [1, 2, 3], devoted to the development and investigation of AC power pulse amplifiers with PWM, special attention is paid to the application of width modulation of bipolar pulses, since in this case the realization of the power unit and of the control circuit for it appears simpler than when unipolar PWM is implemented. However, it is known that in the voltage spectrum with unipolar PWM the amplitude values of the harmonics and the current pulsations in the load turn out to be smaller, since the voltage on the load is not reversed within the commutation period. In the present work the problem is posed of obtaining a method, satisfactory from the standpoint of engineering calculations, for analyzing a width-modulated voltage in a single-phase inverter.

The analysis is carried out for one-sided modulation of the second kind of the trailing edge of the pulse.

The inverter circuit that realizes this task is shown in Fig. 1. Here the power thyristors \(T_1, T_2\) are controlled alternately, each during half of the modulation period.

The valves are turned off by an independent commutation unit through the connection in series with them of a capacitor \(C\), previously charged to the level of twice the voltage of the supply source.

The voltage waveform at the inverter output and the process of forming width-modulated pulses are shown in Fig. 2.

The width of any pulse in the sequence is determined by the formula

Fig. 1

\[ \Delta_1=\frac{T_{\mathrm{m}}}{2na}\cdot f(t_i+\Delta i), \tag{1} \]

where \(T_{\mathrm{m}}\) is the period of the modulating function,

\(n\) is the number of commutation periods \(T_{\mathrm{k}}\) contained in half the modulation period.

\[ n=\frac{T_{\mathrm{m}}}{2T_{\mathrm{k}}}, \tag{2} \]

Fig. 2

\(a\) is the amplitude value of the sawtooth voltage, \(f(t)=A\sin \omega t\) is the modulating function.

In the general case, the number \(n\) may be an integer or a fraction, which affects the composition of the harmonics in the spectrum of the output voltage of the inverter with PWM. However, in order to obtain simpler and more convenient formulas for engineering calculations, we shall assume that \(n\) is an integer.

In order to reduce the quantitative differences between the spectra for fractional and integer \(n\), it is necessary to satisfy the relation [1]:

\[ n>4 \div 5. \tag{3} \]

Satisfying these conditions, let us consider the function \(\varphi(t)\) in Fig. 2. This function is discontinuous, is extended over the entire axis as an odd function, and can be expanded in a Fourier sine series:

\[ \varphi(t)=\sum_{\kappa=1}^{\infty} b_{\kappa}\sin \frac{2\pi \kappa}{T_M}t, \tag{4} \]

where \(\kappa=1, 2, 3\ldots\) is the ordinal number of the harmonic. The Fourier coefficients in this case are determined according to the expression

\[ b_{\kappa}=\frac{4}{T_M}\int_{0}^{T_M/2}\varphi(t)\cdot \sin \frac{2\kappa \pi t}{T_M}\,dt. \tag{5} \]

Since

\[ \varphi(t)=U,\quad \text{when } t_i \le t \le t_i+\Delta_i, \tag{6} \]

\[ \varphi(t)=0,\quad \text{when } t_i+\Delta_i \le t \le t_{i+1}, \]

then

\[ b_k=\frac{4U}{T_M}\sum_{i=0}^{n-1}\int_{t_i}^{t_i+\Delta_i}\sin \frac{2\kappa \pi t}{T_M}\,dt. \tag{7} \]

After evaluating the integral we obtain

\[ b_k=\frac{2U}{\kappa \pi}\sum_{i=0}^{n-1} \left[ \cos \frac{\kappa \pi i}{n} - \cos \frac{\kappa \pi i}{n} \left( 1+\frac{2n}{iT_M}\Delta_i \right) \right]. \tag{8} \]

If \(n\) is sufficiently large, so that \(f(t_i)\approx f(t_{i+1})\), then the sum in equation (8) can be replaced by an integral over the limits \(\left(0+\frac{T_M}{2}\right)\), for \(\kappa<n\), when

\[ \cos\left(\frac{\kappa \pi i}{n}\right)\approx \cos\left[\frac{\kappa \pi(i+1)}{n}\right]. \]

Taking equation (1) into account, we obtain

\[ b_k \approx -\frac{4Un}{\kappa \pi T_M} \int_{0}^{T_M/2} \cos \frac{2\kappa \pi}{T_M} \left[ t+\frac{T_M}{2na}f(t) \right]dt. \tag{9} \]

Substituting the value

\[ f(t)=A\cdot \sin \frac{2\pi t}{T_y} \]

into equation (9) and performing a change of variable

Table 1

\[ \frac{2\pi t}{T_{\mathrm{m}}} = z, \]

we obtain

\[ b_k = -\frac{2Un}{k\pi^2}\int_0^\pi \cos\left(kz + \frac{Ak\pi}{an}\sin z\right)dz. \tag{10} \]

Let us denote

\[ \frac{A}{a} = \lambda \]
— the modulation depth

\[ b = \frac{k\pi}{n} > 0. \]

According to [4], the Bessel function of the first kind of order “$k$” is defined by the expression

\[ I_{-k}(b) = \frac{1}{\pi}\int_0^\pi \cos(b\sin z + kz)\,dz. \tag{11} \]

Consequently, equation (10) can be written as follows:

\[ b_k \approx (-1)^{k+1}\frac{2U}{b}J_k(\lambda b). \tag{12} \]

Expression (12) makes it possible to calculate the amplitude values of the harmonics fairly simply, using tables of Bessel functions. The spectrum of unipolar pulses modulated according to a sinusoidal law contains the fundamental harmonic at the modulation frequency and higher harmonics, which constitute quantization noise.

Table 1 gives the values of the harmonics of the voltage spectrum, calculated from equation (12) at a switching frequency \(f_k = 600; 1000\) Hz. Analysis shows that, as \(n\) increases, the amplitudes of the higher harmonics decrease. However, increasing the switching frequency is limited by the frequency properties of the power switches, by switching losses, and by distortion of the load-current waveform, especially in the low-frequency region. The magnitude and shape of the load current are determined by the modulation parameters, the time constant of the load circuit, and the nature of the pulsed current.

In conclusion, it should be noted that in each particular case the value of the parameter \(n\) must have an average optimal value that makes it possible to obtain the load voltage and current with the required level of higher harmonics, while minimizing switching losses.

REFERENCES

1. E. M. Usyshkin. Semiconductor frequency converters for electric drives in cinematography. Dissertation, Moscow, 1964.

2. O. A. Kosov. Transistor power amplifiers in switching mode. Energiya Publishing House, 1964.

3. O. I. Khasaev. Transistor voltage and frequency converters. Nauka Publishing House, 1966.

4. N. N. Lebedev. Special functions and their applications, 1963.