A Novel Analog-to-digital conversion Technique using nonlinear duty-cycle modulation

(1)

BP 8390, Yaoundé, Cameroun [email protected]

4_{Advanced Teachers' Training College for Technical Education,}

University of Douala, BP 1872, Douala, Cameroun

Abstract- A new type of analog-to-digital conversion technique is presented in this paper. The interfacing hardware is a very simple nonlinear circuit with 1-bit modulated output. As a implication, behind the hardware simplicity retained is hidden a dreadful nonlinear duty-cycle modulation ratio. However, the overall nonlinear behavior embeds a sufficiently wide linear range, for a rigorous digital reconstitution of the analog input signal using a standard linear filter. Simulation and experimental results obtained using a well tested prototyping system, show the feasibility and good quality of the proposed conversion technique.

Keywords: Analog-to-digital conversion, interfacing circuit, nonlinear duty-cycle modulation, digital filer.

1. Introduction

A variety of n-bit A/D (analog-to-digital) conversion structures have been invented over years, in order to meet the specific requirements of instrumentation systems. These A/D conversion structures include ([1], [2]): successive approximation, parallel approximation or flash, simple or multiple slopes, sigma-delta, and pipelined. In addition, the PWM (pulse width modulation technique) usually encountered in power and control electronics has received a great attention of professionals and researchers for solving A/D conversion problems ([3]-[5]). It is important to mention that PWM is a special case of DCM (duty-cycle modulation) providing 1-bit modulated code, under a constant modulating frequency [6].

However, most n-bit A/D converters require complex conversion logic and a great size of the binary code. Even for the PWM approach of A/D conversion with 1-bit modulated code, the related interfacing hardware is greatly complex, and consists of a variety of devices including an oscillator, a triangle wave generator, a comparator, logic gates, digital counters, and more. Thus, the aim of this paper is to present a novel A/D conversion technique, founded on nonlinear DCM, and embedding a wide linear modulating range to be used for a rigorous digital reconstitution of the control input. In Section 2, the principle of A/D conversion via nonlinear DCM is presented. Then, the proposed conversion scheme is analyzed and simulated in Section 3, and the experimental results obtained from a well tested workbench are presented in Section 4.

2. Principle of A/D conversion via nonlinear DCM

The principle of A/D conversion scheme via nonlinear DCM is described in Fig 1. It consists of a

(2)

Figure 1: A/D conversion scheme via nonlinear DCM.

1-bit TTL output signal xm (t)ϵ {0, Vcc}. As an implication, the related duty-cycle

R

m

(x

)

is defined as follows:

R

m

(

x

)

=

T

on

(

x

)

/

T

m

(

x

)

(1)

where Tm(x) = Ton(x)+Tof(x) is the modulation period, Ton(X) and Tof (x) being the durations for which xM(t) switches and remains to +E or –E respectively. Both quantities Ton(x) and Tm(x) simultaneously vary according to the shape of the modulating input x. In addition, the Fourier’s series of xM(t) in Fig. 1(c) obtained after a few straightforward developments, is given at discrete times 0, T, 2T, …, k T, …, by :

(

)

(

)

4

3

4

1

4 4 3

4 4 4

1

terms frequency High n m m term frequency Low y m M

x

T

k

n

x

R

n

E

x

R

T

k

x

f

∑

∞ =





































+

−

=

1 :

(

)

(

2 cos

)

(

sin

4

1 )

(

2 )

(

π

(2)

Then, for a given sampling period T, the discrete wave {xM (0), xM (T), xM (2T), …, xM(kT), …} obtained from “Eq.

(2)”, could be processed using a low-pass filter, in order to recover the numerical value of the modulating input signal x from the estimate of the low frequency term (2 Rm(x) -1) E. In PWM A/D technique, the duty-cycle

R

m

(x

)

in “Eq. (2)”, is a linear function of the modulating input x at the expense of the great complexity of the hardware interfacing circuit. Whereas the novelty of the piecewise linear DCM A/D conversion proposed in this paper, is to resort to a low-cost nonlinear interfacing circuit, while providing around an operating point a sufficiently linear modulating range to be numerically explored, for a rigorous extraction of the input sample using a simple digital linear filter. Thus, in the target

range of interest, the nonlinear function

R

_m

(x

)

is equivalent to a linear approximate

R

~

_m

(

x

)

defined by,

R

_m

(x

)

≅

R

~

_m

(

x

)

=

2

1 +

x

p

_m , (3)

where pm is a design parameter, in which case an ideal digital low-pass filter with static gain 1/(2 E pm ) could be

used for a numerical measure of x from the sampled wave xM(kT) given by “Eq. 2”.

3. Analysis and simulation of a prototyping A/D conversion system via nonlinear DCM

3.1 The overall nonlinear duty-cycle structure and its linear approximation

(3)

observe on Fig. 3 that, in all cases, the relative error due to the linear approximation,

R

m

(x

)

(

)

~

x

R

_m

≅

=

2

1 +

x

p

_m where

     

− + − = =

=

1 1 2 1 2 1

0

1 1 log

) 1 ( )

(

α

E dx

x dR p

x m

m , (5)

indicates that the duty-cycle

R

_m

(x

)

embeds an exact linear behavior in the modulating range [-2 2] volts.

(4)

Fiureg 3: Simulation of Rm(x) &

(

)

~

x

R

m , εRm(x) and εx (x) for different values of α1.

3.2 The linear digital filter

In the linear modulating range of interest, a digital linear filter is then used to extract the control input x from the sampled image of the DCM wave xM. The predicted static gain Kf = 1/(2 pm E) of the filter is computed from “Eq.

(5)” as follows:

_      − + − = 1 1 2 1 2 1 1 1 log 2 ) 1 (

α

f

K (6)

In addition, the discrete model of the filter is computed from that of an analog filter with transfer function

2 2 2 2 ) ( ) ( ) ( n n n f M f c s s K s X s X s F

ω

ξ

ω

+ + =

= (7)

where ξ is the damping coefficient,

ω

_n being the natural frequency. Then, for a sampling period T, the related recursive equation obtained from the transfer function given by “Eq. (7)”, using Tustin’s discretization technique ([8], [9]), is defined at discrete times k T (for k = 0, 1, 2, …) as follows :

∑

(

)

=

−

+

=

2 1

0

(

)

(

)

(

)

(

i i M i

M

k

b

x

k

i

a

y

k

i

x

b

k

y

(8)

where y(k) = 0 for all k < 0, where as y(0) and y(1) are given. The coefficients b0, b1, b2, a1, a2 coefficients resulting

(5)

Figure 4: A/D conversion via NDCM – Simulation results.

In Fig. 4(a), the modulating signal x is plotted in white color within the graph of the related modulated wave xM(t). Fig 4(b) indicates that, the zoomed view of the modulated wave xM (numerically computed from “Eq. (2)” for the first 20th

harmonics is plotted from time 0.1 s to 0.1040 s. The distortions observed on the modulated wave results from the effects of two combined phenomena. In fact, it is due to the numerical computation of the Fourier series “Eq. (2)” over a fine number Nh of harmonics, and to Gibb's phenomenon.

4. Experimentation of a prototyping A/D conversion system via nonlinear DCM

The didactic workbench built for testing the A/D conversion via nonlinear DCM is presented in Fig 5. The instrumentation set (Fig. 5(a)) consists of an integrated signal generator (ICL8038) used as a triangle wave source, a low frequency generator (Philips PM5107) used as a sine signal source, and a Hewlett Packard HP1631 analyzer, used as a reference monitoring instrument. The duty-cycle modulation interface Fig. 5(b) is the same used for previous simulations. Furthermore, a NEC Power Mate Computer (single core, 2 Mhz) with an embedded LPT port is used as a digital processing and monitoring unit. Then, the related input state register from which the TTL modulated signal xm(t) is sampled is

(6)

Figure 5: Workbench for A/D conversion via nonlinear DCM.

Fig 6 shows a sample of results recorded during experiments and run in Matlab environment. In Fig. 6(a), the modulating sine wave x(t) = 2 sin(20 π t) to be converted observed on the screen of the HP1631D analyzer is generated using Philips PM5107 low frequency generator. In Fig. 6(b), the parameters (E, α1, N, fc, , ξ, ) and the related experimental results (modulated signal xM, and DCM-based A/D conversion output y), are observed on the screen of the control panel

of our visual instrumentation software. In Fig. 6(c), the graphs of the DCM-based A/D conversion output y obtained from experiments and that of the pure modulating input x(t) = 2 sin (20 π t) are plotted for the sake of comparison. Finally, Fig. 7 shows a sample of results obtained under a triangle modulating input x generated by the ICL8038 integrated circuit.

For all cases, the experimental results obtained and compared with the simulated behavior show the high quality of the proposed DCM-based A/D conversion technique. Thus, according to our best knowledge, combining a simple nonlinear interfacing hardware and a piece of software, for implementing a rigorous A/D conversion solver, appears to be a novelty in instrumentation engineering.

5. Conclusion

(7)

Figure 6: A/D conversion via NDCM – Experimental results obtained when a sine

(8)

Figure 7: Real time A/D conversion results obtained when the modulating signals are generated by the ICL8038 integrated circuit.

Finally, for the sake of higher performance, an implementation of the proposed A/D conversion scheme using a microchip device with a high precision and stable clock, might be a great challenge in industrial instrumentation engineering. These attractive unsolved instrumentation problems, are new perspectives for short term research works.

REFERENCES

[1] B. Loriferne, "La conversion analogique-numérique et numérique-analogique ", 108 p., Editions de l’usine, 1981.

[3] Nasrudin Abd Rahim, Jeyraj Selvaraj , "A novel PWM multilevel inverter for PV application", Vol. 6, Issue: 15, Pages: 1105-

1111, Krismadinata Chaniago in Electronics, 2009..

[4] Kevin Daugherty, "Pulse width modulation A/D conversion techniques COP800 family microcontrollers - application note 607", ©

National Semiconductor, August 1991.

[5] Altera, Pulse Width Modulation Using MAX II CPLDs, Application note AN-501, Altera Corporation, December 2007.

[6] E. Roza, "Analog-to-digital conversion via duty-cycle modulation", IEEE transactions on circuits and systems II: Analog and digital

signal processing, Vol. 44 , No. 11, pp. 907-914, 1997.

[7] J. MBIHI, F. NDJALI BENG & M. MBOUENDA, "Modelling and simulation of a class of duty-cycle modulators for industrial

instrumentation", Iranian Journal of Electrical and Computer engineering, Vol. 4, No. 2, pp. 121-128, 2005.