PIERS ONLINE, VOL. 6, NO. 8, 2010 705 Modeling Buck Converter by Using Fourier Analysis Mao Zhang 1, Weiping Zhang 2, and Zheng Zhang 2 1 School of Computing, Engineering and Physical Sciences, University of Central Lancashire, UK 2 North China University of Technology, Shijingshang District, Beijing, China Abstract By employing theory of PAM (pulse-amplitude-modulation), Buck converter (DC- /DC) have been modeled in this paper. The main contributions are as the followings: (1) A DC transformer model has been proposed to analysis of the voltage gain, efficiency and some steady-state properties; (2) Two a.c. small signal models have been put forward in order to get the transfer functions of input to output and control to output. These a.c. small signal models play an important role on predicting the dynamic behaviors and designing its control system. 1. ANALYSIS OF BUCK CONVERTER BY FOURIER TRANSFORMS Buck converter is one of important traditional converter and applied broadly in the industrial. It is showed in Fig. 1, A Buck converter can be broken into three parts, named as switch network, LPF and the load. The equivalent model is showed in Fig. 2 while we apply the signal process theory to analysis it. 1.1. Equivalent Model by PAM Applying the theory of PAM, a multiplier has been introduced to replace the switch network in Fig. 2, therefore, a new equivalent model, called as PAM-recovery model for converter, can be obtained to shown in Fig. 3. The driven signal v gs (t), shown in Fig. 3, is a pulse train with a period T s and width τ. Its exponential form of the Fourier series has been given out by [1] V gs (t) = C k e jkωst, C k = D sin c(kω s τ/2), D = τ/t S (1) where ω s = 2π/T s, sin c(kω s τ/2) = sin(kω s τ/2)/(kω s τ/2). Figure 1: BUCK converter. Figure 2: The equivalent model of applying signal process. Figure 3: PAM-recovery model of the converter and the waveform of v gs (t).
PIERS ONLINE, VOL. 6, NO. 8, 2010 706 Applying the frequency shifting property [1], one can get the frequency response of v gs (t), V gs (ω) = 2πC k δ(ω kω s ), C k = τ T S sin c(kω s τ/2) 1.2. Analyzing Buck Converter by Employing Fourier Transform In time domain, the output v d (t) of multiplier or switch network is that v g (t) multiplied by v gs (t). Applying the convolution property [1], one can obtain the frequency response of the switch network output, v d (t) = v g (t)v gs (t) = C k e jkωst v g (t) (2) V d (ω) = 1 2π V g(ω) V gs (ω) = C k V g (ω kω s ) (3) In general, the input V g is a DC voltage source which can be derived from an alternating mains supply through a rectifier and smooth capacitor, so it has DC component and 100 Hz a.c. ripple. In order to simplify this problem, V g (ω) is supposed a band limited signal, shown in Fig. 4(a), V g (ω) = 0, for ω > ω M. The frequency response V d (ω) and V g (ω) have been shown in Fig. 4. In Fig. 4, we can find out that V d (ω) are one sets of shifting V g (ω) in frequency domain. The shifting rule is as the followings: (1) The shifting interval is sampling frequency ω s, (2) The magnitude is modified by the factor C k. The transform function of LPF, made of L and C in Fig. 1, is specified by 1 H(ω) = LCω 2 + L (4) R ω + 1 Its magnitude frequency response is shown in Fig. 4(d), ω c = 1/ LC. Therefore, the output V (ω) can be expressed by V (ω) = V d (ω)h(ω) = C k V g (ω kω s )H(ω) (5) 2. DC TRANSFORMER MODEL In this section, a DC transformer model has been investigated to model the basic property of Buck DC/DC converter. If the input V g is ideal DC voltage source, the V g (ω) has only DC component, the Equation (5) can be modified as the follows, V (kω s ) = C k V g δ(ω kω s )H(ω) (6) The normalized spectrum has been plotted in Fig. 5. The magnitudes of higher order harmonics are very smaller than DC component, so these components could be neglected when a DC/DC converter are analyzed. (1) In Buck converter, ω s (= 2πf s ) is the switch frequency and ω c (= 2πf c ) is a cut-off frequency of LPF. In a practical converter, the switch frequency is much greater than the cutoff frequency, for example, f s > 10f c, and LPF is second order Butterworth filter. The filter has 40 db attenuation at the switch frequency. (2) The coefficient of Fourier series, shown in Table 1, has be calculated by the formula, C k = D sin c(kω s τ/2), D = τ/t S (7) The DC component of output voltage is calculated by formula (6) V (0) = C 0 V g H(0) = D V g 1 = DV g (8) If k = 1, V (ω s ) = C 1 V g H(ω s ), the normalized magnitude is defined as A 1 = 20 lg V (ω s ) V (0) = 20 lg C 1 H(ω s ) C 0 H(0) = 20 lg C 1 + 20 lg H(ω s ) H(0) C 0
PIERS ONLINE, VOL. 6, NO. 8, 2010 707 If D = 0.6, A 1 = 24.4 40 = 64 db. Hence, the high order components could be neglected while the DC component of output is calculated. According to Equation (8), the DC equivalent circuit corresponding ideal Buck DC/DC converter can be established. This model, shown in Fig. 6, is called as a DC transformer model for ideal DC/DC converter, and means the DC current can pass the transformer. The DC transformer model describes the main function of a DC/DC converter: (1) transformation of DC voltage and current, V = DV g, I g = DI, where I g is averaging input current. (2) The output voltage can be controlled through changing duty cycle D; (3) The model can be modified to account for loss elements such as loss of inductor and power diode and switch to predict the efficiency. (a) (a) (b) (b) (c) (c) (d) (d) (e) Figure 4: Frequency spectra for PAM. Figure 5: Frequency spectra based on the expression (5). Table 1: C k with different D (from D = 0.4 to 0.9). k = 0 k = 1 k = 2 C k (D = 0.4) 0.4 0.0730 0.0507 C k (D = 0.5) 0.5 0.0989 0.0216 C k (D = 0.6) 0.6 0.0361 0.0338 C k (D = 0.7) 0.7 0.0591 0.048 C k (D = 0.8) 0.8 0.1013 0.0038 C k (D = 0.9) 0.9 0.0527 0.0450
PIERS ONLINE, VOL. 6, NO. 8, 2010 708 Figure 6: DC transformer model. 3. AN a.c. SMALL-SIGNAL MODEL 3.1. An a.c. Small-signal Model for Input-output In this section, two a.c. small-signal models will express the dynamic behaviors of Buck converter. There are the some reasons. (1) In general; the input V g is a DC voltage source which is derived from an alternating mains supply through a rectifier and smooth capacitor, so it has DC component and about 100 120 Hz a.c. ripple, which will lead to the disturbance in the output of the converter. Hence, it is required to find out how to affect the output by the input a.c. ripple; (2) the output current will be changed as the load varies, this will result in the output disturbance. According to the above discussion, the input voltage can be expressed by v g (t) = V g + V gm cos ω line t (9) V g is DC component and V gm cos ω line t is the a.c. ripple signal and f line (= ω line /2π) is two time mains frequency. In order to simplify the analysis process and make sense for engineering analysis, the following assumptions have been made, (1). The amplitude of DC signal is greater than that of a.c. signal. V g V gm. (2). The frequency of a.c. signal is much smaller than the switching frequency, f line f s. The above two assumptions imply that the LPF operates in pseudo steady state during a switching period. The net changes of inductor current and capacitor voltage is too small to consider it to affect the behaviors of the converter during one switch period. Based on above assumptions, ˆV gm (t)(= V gm cos ω line t) is called as an a.c. low-frequency- smallsignal. The Fourier transform of the input voltage is (10), and the frequency spectrum ˆV g (ω) are shown in Fig. 7(a). V g (ω) = V gdc (ω) + ˆV g (ω) = 2πV g δ(ω) + πv gm [δ(ω ω line ) + δ(ω + ω line )] (10) The steady state analysis has gotten a useful conclusion which the higher order harmonics can be ignored in analyzing Buck. This conclusion is also suitable for analysis in this section. The output voltage can be expressed V (ω) = C 0 V g (ω)h(ω) ω < ω s /2 Others is equal to zero (11) From Equation (10), the a.c. small-signal input is ˆV g (ω) = V g (ω) V gdc (ω) = πv gm [δ(ω ω line ) + δ(ω + ω line )] (12) From Equation (10), the a.c. small-signal output is V (ω) = V dc (ω) + ˆV = D[V gdc (ω) + ˆV g (ω)] ω < ω s /2 (13) The a.c. small signal model can be plotted in Fig. 8, from which the transform function of input to output can derived. ˆv(s) D ˆv g (s) = DH(s) = ˆd(s)=0 LCs 2 + L let jω = s (14) R s + 1 3.2. An a.c. Small-signal Model for Control-output If v g (t) only have the DC component, and suppose the duty cycle is modulated sinusoidal: d(t) = D + d m cos ω p t (15) v d (t) = v g (t)d(t) = V g (D + d m cos ω p t) (16)
PIERS ONLINE, VOL. 6, NO. 8, 2010 709 (a) (b) (c) (d) Figure 7: The frequency spectrum of ˆv g (ω), ˆv d (ω) and ˆv(ω). Figure 8: a.c. small-signal model of control-tooutput and input-output. The small signal of ˆd(t) can be rewritten in (17) So using Fourier transform, it is obtained (18) ˆd(t) = d m cos ω p t (17) ˆv(ω) = DV g π[δ(ω ω p ) + δ(ω + ω p )]H(ω) = V g ˆd(ω)H(ω) (18) So when ˆv g (t) = 0, the transform function is expressed in (19) ˆv(s) V g = ˆd(s) ˆvg(s)=0 LCs 2 + L (19) R s + 1 4. CONCLUSIONS In this paper, it is utilized the PAM theory and approaches to analyze and model Buck converter, and figured out that it (the signal processing theory and approaches) also is a powerful tool in power electronics field. For example, analysis and model of Buck converter is a problem in power electronics and PAM is a problem which belongs to signal processing. However, this paper has shown that both Buck converter and PAM have a same mathematic model. So, a mature theory and method about PAM has been successfully employed to predict the behaviors of Buck converter, extracting the following conclusions, (1) A DC transformer model, describing the main function of a DC/DC converter, has been proposed to analysis of the voltage gain, average current gain, efficiency and some steady state properties as well as how to controlling the output energy; (2) An a.c. small-signal model for input-output has been developed. According this model, one can easily get the transfer function of input to output and dynamic input resistance as well as output resistance; (3) An a.c. small-signal model for control-output and its transform function have been investigated. This model and transform function are necessary for deigning its control system and for analyzing the stability of the closed loop control.
PIERS ONLINE, VOL. 6, NO. 8, 2010 710 In one word, this paper opened a new window to study DC/DC converter employing the mature theory and approaches in signal process. REFERENCES 1. Phillips, C. L. and J. M. Parr, Signals Systems and Transforms, 2nd Edition, 275 277, 1998. 2. Zhang, W. P., The Control and Modeling of Switch Converter, 5 15, China Power Publishers, 2006. 3. Erickson, R. W. and D. Masksimovic, Fundamentals of Power Electronics, 2nd Edition, Kluwer Academic Publishers, 2001. 4. Ludeman, L. C., Fundamentals of Digital Signal Processing, Happer & Row Publishers, Inc., New York, 1986.