2. Double Fourier series expansion

### 2.1. Double Fourier series decomposition for a double variable function

It is well known that a periodic two variable waveform fðx, yÞ can be expressed in the form

$$f(\mathbf{x}, \mathbf{y}) = \frac{A\_{00}}{2} + \sum\_{n=1}^{\infty} \left[ A\_{0n} \cos n\mathbf{y} + B\_{0n} \sin n\mathbf{y} \right] + \sum\_{m=1}^{\infty} \left[ A\_{m0} \cos m\mathbf{x} + B\_{m0} \sin m\mathbf{x} \right] \tag{1}$$

$$+ \sum\_{m=1}^{\infty} \sum\_{\substack{n=-\infty \\ n \neq 0}}^{\infty} \left[ A\_{mn} \cos \left( m\mathbf{x} + n\mathbf{y} \right) + B\_{mn} \sin \left( m\mathbf{x} + n\mathbf{y} \right) \right] \tag{1}$$

where the double Fourier series components can be found in a complex form:

$$\mathbf{C}\_{mn} = A\_{mn} + jB\_{mn} = \frac{1}{2\pi^2} \int\_{-\pi-\pi}^{\pi} \int\_{-\pi}^{\pi} f(\mathbf{x}, y) \, e^{j(m\mathbf{x} + ny)} \, d\mathbf{x} \, dy. \tag{2}$$

The first term in Eq. (1) is the DC offset that should be zero or negligibly small. The second summation term represents the baseband harmonics. The first baseband harmonic, n ¼ 1, is the fundamental harmonic whose magnitude defines the magnitude of the output waveform. Other baseband harmonics, n > 1, represent low-frequency undesired fluctuations about the fundamental output and should preferably be eliminated with the modulation process. The third summation term in Eq. (1) corresponds to the carrier harmonics which are relatively high-frequency components. Finally, the last double summation term in Eq. (1) corresponds to groups of the sideband harmonics of order n located around the mth carrier harmonic component.

### 2.2. Jacobi-Anger expansion and Bessel functions relations

The magnitudes of harmonic components in Eq. (1) are to be determined for each PWM scheme for each particular combination of indexes m and n. The evaluations are based on Jacobi-Anger expansions

$$e^{\pm j\xi\cos\theta} = f\_0(\xi) + 2\sum\_{k=1}^{\infty} j^{\pm k} f\_k(\xi)\cos k\theta = \sum\_{k=-\infty}^{\infty} j^k f\_k(\xi) \ e^{jk\theta} \tag{3}$$

and a number of Bessel function properties: J<sup>−</sup>nðξÞ¼ð−1Þ n JnðξÞ and Jnð−ξÞ¼ð−1Þ n JnðξÞ, that particularly implies J0ð−ξÞ ¼ J0ðξÞ [5].

### 2.3. Parseval's theorem

width modulation (PWM) implementation: diminished harmonics [1], filtered distortion performance factors [2], and the root-mean-square (RMS) harmonic ripple current [3]. In this text, analytical solutions to PWM strategies are used to compare magnitude of various harmonic

Firstly, the conventional method of determining harmonic components of a switched waveform using fast Fourier transform (FFT) of the waveform is sensitive to the time resolution of the simulation and periodicity of the overall waveform. Moreover, it ensures that intrinsic harmonic components of PWMs are not affected by such factors as simulation round off errors,

Secondly, PWM strategies can be compared at exactly the same phase leg switching frequency. And thirdly, the first-order weighted total harmonic distortion (WTHD) is used for a quick comparison of PWMs since it has a physical meaning (the normalized current ripple expected into an inductive load when fed from the switched waveform) and often used performance

The rest of the paper is organized as follows. In Section 2, information on the double Fourier series expansions and necessary relations is given. Essentials on PWM are provided in Section 3. Different voltage inverter topologies and their analytical PWM solutions are presented in Section 4. Harmonic distortion factors of the introduced inverter topologies, different modulation schemes are compared in Section 5, and a summary on the chapter is given in Section 6.

It is well known that a periodic two variable waveform fðx, yÞ can be expressed in the form

∞ m¼1

<sup>½</sup>Amn cos <sup>ð</sup>mx <sup>þ</sup> nyÞ þ Bmn sin <sup>ð</sup>mx <sup>þ</sup> nyÞ� (1)

½Am<sup>0</sup> cos mx þ Bm<sup>0</sup> sin mx�

<sup>j</sup>ðmxþny<sup>Þ</sup> dx dy: (2)

½A0<sup>n</sup> cos ny þ B0<sup>n</sup> sin ny� þ ∑

2π<sup>2</sup> ð π

−π

The first term in Eq. (1) is the DC offset that should be zero or negligibly small. The second summation term represents the baseband harmonics. The first baseband harmonic, n ¼ 1, is the fundamental harmonic whose magnitude defines the magnitude of the output waveform. Other baseband harmonics, n > 1, represent low-frequency undesired fluctuations about

ð π

fðx, yÞ e

−π

components. This approach has a number of advantages [4].

120 Fourier Transforms - High-tech Application and Current Trends

dead time, switch ON-state voltages, DC bus ripple voltages, etc.

2.1. Double Fourier series decomposition for a double variable function

where the double Fourier series components can be found in a complex form:

Cmn <sup>¼</sup> Amn <sup>þ</sup> jBmn <sup>¼</sup> <sup>1</sup>

2. Double Fourier series expansion

<sup>2</sup> <sup>þ</sup> <sup>∑</sup> ∞ n¼1

∑ ∞ <sup>n</sup>¼−<sup>∞</sup> <sup>n</sup>≠<sup>0</sup>

þ ∑ ∞ m¼1

<sup>f</sup>ðx, <sup>y</sup>Þ ¼ <sup>A</sup><sup>00</sup>

indicator.

GivenfðxÞ is a periodic function with the period T, it can be represented by its Fourier series <sup>f</sup>ðxÞ ¼ <sup>a</sup>0=<sup>2</sup> <sup>þ</sup> <sup>∑</sup><sup>∞</sup> <sup>n</sup>¼<sup>1</sup>an cos <sup>n</sup>ω<sup>t</sup> <sup>þ</sup> bn sin <sup>n</sup>ω<sup>t</sup> where <sup>ω</sup> <sup>¼</sup> <sup>2</sup>π=<sup>T</sup> is the fundamental angular frequency. Then, on ½−T=2, T=2�, the Parseval's theorem assumes the form

$$\frac{1}{T}\int\_{-T/2}^{T/2} f^2(\mathbf{x}) \, d\mathbf{x} = \frac{a\_0^2}{4} + \sum\_{n=1}^{\infty} \frac{a\_n^2 + b\_n^2}{2}. \tag{4}$$
