Randomized Carrier PWM with Exponential Frequency Mapping
Randomized Carrier PWM with Exponential Frequency Mapping Alfonso Carlosena, Wing-Yee Chu, Bertan Bakkaloglu, Sayfe Kiaei Arizona State University, WinTECH-Connection One Tempe, AZ, US Abstract—Pulse Width Modulation (PWM) and Sigma Delta Modulation (SDM) have been used in several high linearity, high accuracy power delivery applications such as DC-DC power converters, audio and RF power amplifiers as well as in data converters. In this paper we present a particular class of Carrier Frequency Modulated PWM (CFMPWM), which spreads the tonal content of regular PWM uniformly in frequency and reduces the baseband noise with respect to SDM by up to 15 dB. The new modulation scheme preserves at the same time the high efficiency properties of PWM. An approximate method to estimate the spectrum of CFMPWM signals with both deterministic and random modulation is also presented. Simulation results and experimental data are reported to analyze the spectral spreading properties of the proposed CFMPWM scheme. I.
INTRODUCTION
Pulse Width Modulation (PWM) and Pulse Density Modulation (PDM) - in particular Sigma Delta Modulation (SDM)- have been used in several high linearity and high accuracy switched-mode power delivery applications such as DC-DC power converters, audio and RF power amplifiers[14]. In several applications where conducted and radiated electromagnetic interference (EMI) is a concern, wide-band shaped noise content of SDM is preferred to inherently tonal content of PWM. SDM is also preferred in low frequency applications where acoustic annoyance has to be suppressed. Also, several techniques to reduce the tonal content of PWM have been proposed. Most common technique to achieve this is to modify regular PWM by randomly varying any of its switching parameters such as pulse position, pulse width or pulse period [5-6]. These procedures give rise to a family of Randomized PWM (RPWM) schemes, each one resulting in a particular type of spectral spreading of discrete tonal content [5-7]. In other cases, one or more of the switching parameters are varied deterministically instead of randomly. A large number of publications, regardless of the end application, share the common goal of spreading out the characteristic tonal content of PWM. Comparison of spectral content of RPWM with respect to regular PWM has been analyzed [7]. As for SDM, well developed theory for data converters [8] has been used to estimate the improvements
attainable with respect to regular PWM. A limited number of publications show, mainly from an experimental point of view, the advantages of SDM over plain PWM to solve EMI, power factor and acoustic tones [1,3,9]. Since RPWM methods have been mainly developed for Power Electronics applications, theoretical procedures to estimate the spectral properties of a given scheme are mostly based on the time Probability Density Function (PDF) of the parameter to be randomized (namely, period, width, duty cycle). This analysis is more straightforward for digital microcontroller based implementations, where PWM signal is generated digitally. However, time PDF based analysis poses serious limitations for continuous-time highfrequency, fully integrated implementations where the randomization possibilities are less flexible. Although SDM and RPWM have been compared against plain PWM, spectral content of SDM with respect to RPWM have not been analyzed. Also, the selection of a randomization technique and its associated PDF to obtain a prescribed spectral profile has not been addressed before. Although the problem is not solvable for a general PDF, a statistical approximation is critical for noise shaping applications. This paper is organized as follows: in section II an approximate method to estimate the spectral profile of RPWM is introduced. Analysis will be restricted to the most useful class of RPWM, known as Carrier Frequency Modulated PWM (CFMPWM) [7]. In section III our proposed exponentially mapped CFMPWM technique will be introduced, and its spectral properties analysed. In section IV it will be compared against an alternative SDM approach. Section V reports experimental results associated with spectral spreading of CFMPWM signals for DC level inputs. CFMPWM with exponential mapping achieves significant reduction in the baseband and out-of-band noise content. II.
CFMPWM SPECTRAL PROPERTIES
Consider the scheme of Figure 1 where a PWM signal is generated from a continuous time input signal. This scheme, which is a modification of the well known ramp generator followed by a comparator PWM generator, can be utilized
A. Carlosena has been supported by Direccion General de Universidade (Spain) under grant PR 2004-147
0-7803-9390-2/06/$20.00 ©2006 IEEE
5059
ISCAS 2006
for a continuous time implementation of CFMPWM. Instead of a constant slope ramp in PWM generator, the slope value is updated at each cycle according to a time series. The time series is obtained by adding a random number to a nominal slope value. This added random number is obtained by applying a uniformly generated random number to a suitable mapping function. If the saw-tooth peak value is constant, then slope is proportional to frequency, resulting in a Carrier Frequency Modulated PWM signal. In this approach duty cycle is preserved. This is critical in closed loop applications since loop dynamics such as gain and phase margin is a function of duty cycle and stays unchanged with this approach [5, 7]. Instantaneous Slope (freq)
Uniform RND Number Generation
(f0)
Mapping fnct.
Vmax
φ
Nominal Slope (f0)
(non-unif. noise)
VSAW
Signal
Vin
Σ
Inst. Slope (freq.)
Slope Generator
Vin VSAW
+ -
VPWM
VPWM
1 0 Tk-1
Tk
Tk+1
probability distribution, and ϕ is the mapping function, which modifies the noise PDF and fB is proportional to the modulation index. The range of the mapping function should be normalized to the interval (-1,+1), so that fi is limited to the interval (f0-fB, f0+fB). Also ϕ should be a monotonic function of x. Assuming a random modulation with bandwidth much lower than f0, the power spectrum of the modulated signal can be approximated by [10]: 2
S( f ) ≈
S ( f ) ≈ (2πD ) ⋅ ∑− ∞ snc 2 (nπD ) ⋅ 2
PT ( f ) + 1 + Re 2 2 1 − PT ( f ) (1) S( f ) = (2πf )2 PT2 ( f 2) PT ( f 2) Re − 2 Re 1 − PT ( f ) 1 − PT ( f )
Where S2(f) is the power spectrum of the randomly modulated PWM signal. PT(f) is the spectrum of the Noise Probability Density Function pT(t), that is used to obtain the successive period values for the PWM signal. However in order to utilize this formula effectively, we need to calculate how the uniform PDF is transformed by the mapping function, and how the resulting PDF is transformed from frequency to period. Unfortunately, equation (1) cannot be analytically evaluated for arbitrary PDFs and mapping functions. Alternatively, in this section an approximate method which derives an easier relationship between the mapping function and the resulting spectrum is proposed. Consider a FM modulated single tone at frequency f0 with the following relationship:
f i = f o + f Bφ (x)
(2)
Where fi stands for the resulting instantaneous frequency, x is a continuous time random variable with uniform
• f − f o f B φ φ −1 f B
Utilizing Carson’s Rule, effective noise bandwidth can be estimated as 2·fB. This result can be extended to a PWM signal, which is a switched waveform that contains a number of harmonics at integer multiples of f0. Their relative power depends on the duty cycle of the PWM signal. These harmonics are spread out according to the profile given by equation (3). Asuming a square wave with duty cycle D and frequency f0, the resulting carrier modulated PWM signal can be then approximated by:
Figure 1 Randomized PWM Generation Scheme
Once duty cycle is determined for stationary load conditions, the spectrum of the switched signal could be calculated. As an example, for a 50% duty cycle, the estimated spectrum would be according to [5]:
(3)
1
2
(4)
1
∞
• f − nf o nf B φ φ −1 nf B
In order to correctly use formula (4), each of his terms has to be evaluated in the range nf0+n·fB (bandwidth 2·n·fB). In most applications, where the PWM signal is low-pass filtered, only the first few harmonics are of practical importance. Another implementation issue to consider is that PWM frequency cannot change instantaneously (i.e., ramp slope has to be constant throughout a cycle) as implicitly assumed by equation (4). However, the result given by this expression is still valid if we assume a slow varying frequency with respect to f0. This analysis can also be extended to deterministic modulation case, where modulating signal is periodic. If we utilize a periodic ramp signal between -1 and +1 with frequency fm
INTRODUCTION
Pulse Width Modulation (PWM) and Pulse Density Modulation (PDM) - in particular Sigma Delta Modulation (SDM)- have been used in several high linearity and high accuracy switched-mode power delivery applications such as DC-DC power converters, audio and RF power amplifiers[14]. In several applications where conducted and radiated electromagnetic interference (EMI) is a concern, wide-band shaped noise content of SDM is preferred to inherently tonal content of PWM. SDM is also preferred in low frequency applications where acoustic annoyance has to be suppressed. Also, several techniques to reduce the tonal content of PWM have been proposed. Most common technique to achieve this is to modify regular PWM by randomly varying any of its switching parameters such as pulse position, pulse width or pulse period [5-6]. These procedures give rise to a family of Randomized PWM (RPWM) schemes, each one resulting in a particular type of spectral spreading of discrete tonal content [5-7]. In other cases, one or more of the switching parameters are varied deterministically instead of randomly. A large number of publications, regardless of the end application, share the common goal of spreading out the characteristic tonal content of PWM. Comparison of spectral content of RPWM with respect to regular PWM has been analyzed [7]. As for SDM, well developed theory for data converters [8] has been used to estimate the improvements
attainable with respect to regular PWM. A limited number of publications show, mainly from an experimental point of view, the advantages of SDM over plain PWM to solve EMI, power factor and acoustic tones [1,3,9]. Since RPWM methods have been mainly developed for Power Electronics applications, theoretical procedures to estimate the spectral properties of a given scheme are mostly based on the time Probability Density Function (PDF) of the parameter to be randomized (namely, period, width, duty cycle). This analysis is more straightforward for digital microcontroller based implementations, where PWM signal is generated digitally. However, time PDF based analysis poses serious limitations for continuous-time highfrequency, fully integrated implementations where the randomization possibilities are less flexible. Although SDM and RPWM have been compared against plain PWM, spectral content of SDM with respect to RPWM have not been analyzed. Also, the selection of a randomization technique and its associated PDF to obtain a prescribed spectral profile has not been addressed before. Although the problem is not solvable for a general PDF, a statistical approximation is critical for noise shaping applications. This paper is organized as follows: in section II an approximate method to estimate the spectral profile of RPWM is introduced. Analysis will be restricted to the most useful class of RPWM, known as Carrier Frequency Modulated PWM (CFMPWM) [7]. In section III our proposed exponentially mapped CFMPWM technique will be introduced, and its spectral properties analysed. In section IV it will be compared against an alternative SDM approach. Section V reports experimental results associated with spectral spreading of CFMPWM signals for DC level inputs. CFMPWM with exponential mapping achieves significant reduction in the baseband and out-of-band noise content. II.
CFMPWM SPECTRAL PROPERTIES
Consider the scheme of Figure 1 where a PWM signal is generated from a continuous time input signal. This scheme, which is a modification of the well known ramp generator followed by a comparator PWM generator, can be utilized
A. Carlosena has been supported by Direccion General de Universidade (Spain) under grant PR 2004-147
0-7803-9390-2/06/$20.00 ©2006 IEEE
5059
ISCAS 2006
for a continuous time implementation of CFMPWM. Instead of a constant slope ramp in PWM generator, the slope value is updated at each cycle according to a time series. The time series is obtained by adding a random number to a nominal slope value. This added random number is obtained by applying a uniformly generated random number to a suitable mapping function. If the saw-tooth peak value is constant, then slope is proportional to frequency, resulting in a Carrier Frequency Modulated PWM signal. In this approach duty cycle is preserved. This is critical in closed loop applications since loop dynamics such as gain and phase margin is a function of duty cycle and stays unchanged with this approach [5, 7]. Instantaneous Slope (freq)
Uniform RND Number Generation
(f0)
Mapping fnct.
Vmax
φ
Nominal Slope (f0)
(non-unif. noise)
VSAW
Signal
Vin
Σ
Inst. Slope (freq.)
Slope Generator
Vin VSAW
+ -
VPWM
VPWM
1 0 Tk-1
Tk
Tk+1
probability distribution, and ϕ is the mapping function, which modifies the noise PDF and fB is proportional to the modulation index. The range of the mapping function should be normalized to the interval (-1,+1), so that fi is limited to the interval (f0-fB, f0+fB). Also ϕ should be a monotonic function of x. Assuming a random modulation with bandwidth much lower than f0, the power spectrum of the modulated signal can be approximated by [10]: 2
S( f ) ≈
S ( f ) ≈ (2πD ) ⋅ ∑− ∞ snc 2 (nπD ) ⋅ 2
PT ( f ) + 1 + Re 2 2 1 − PT ( f ) (1) S( f ) = (2πf )2 PT2 ( f 2) PT ( f 2) Re − 2 Re 1 − PT ( f ) 1 − PT ( f )
Where S2(f) is the power spectrum of the randomly modulated PWM signal. PT(f) is the spectrum of the Noise Probability Density Function pT(t), that is used to obtain the successive period values for the PWM signal. However in order to utilize this formula effectively, we need to calculate how the uniform PDF is transformed by the mapping function, and how the resulting PDF is transformed from frequency to period. Unfortunately, equation (1) cannot be analytically evaluated for arbitrary PDFs and mapping functions. Alternatively, in this section an approximate method which derives an easier relationship between the mapping function and the resulting spectrum is proposed. Consider a FM modulated single tone at frequency f0 with the following relationship:
f i = f o + f Bφ (x)
(2)
Where fi stands for the resulting instantaneous frequency, x is a continuous time random variable with uniform
• f − f o f B φ φ −1 f B
Utilizing Carson’s Rule, effective noise bandwidth can be estimated as 2·fB. This result can be extended to a PWM signal, which is a switched waveform that contains a number of harmonics at integer multiples of f0. Their relative power depends on the duty cycle of the PWM signal. These harmonics are spread out according to the profile given by equation (3). Asuming a square wave with duty cycle D and frequency f0, the resulting carrier modulated PWM signal can be then approximated by:
Figure 1 Randomized PWM Generation Scheme
Once duty cycle is determined for stationary load conditions, the spectrum of the switched signal could be calculated. As an example, for a 50% duty cycle, the estimated spectrum would be according to [5]:
(3)
1
2
(4)
1
∞
• f − nf o nf B φ φ −1 nf B
In order to correctly use formula (4), each of his terms has to be evaluated in the range nf0+n·fB (bandwidth 2·n·fB). In most applications, where the PWM signal is low-pass filtered, only the first few harmonics are of practical importance. Another implementation issue to consider is that PWM frequency cannot change instantaneously (i.e., ramp slope has to be constant throughout a cycle) as implicitly assumed by equation (4). However, the result given by this expression is still valid if we assume a slow varying frequency with respect to f0. This analysis can also be extended to deterministic modulation case, where modulating signal is periodic. If we utilize a periodic ramp signal between -1 and +1 with frequency fm
