Ph Dondon- Enseirb

A SMALL SIGNAL MODEL OF THE DESIGN OF A CLASS D POWER SWITCHING AMPLIFIER

The class D amplifier is not well known in audio applications. Its excellent power ratio (greater than 90%) is the most important advantage. The MOS transistors switching power stage is able to drive a useful power up to 100W to the loud speaker. However, designing such an amplifier is more difficult than designing a classical class A or AB power amplifier. The distortion level is also slightly higher than in AB structures. Despite the abundant knowledge on class D, there was no electronic model available till now. Then, a small signal model is présentented to make the design as simple as possible. The experimental results are given to illustrate the design method.

INTRODUCTION

The Class D amplification [START_REF] Girard | Amplificateur de puissance Mc GrawHill[END_REF] uses MOS power transistors in switching mode. It is able to obtain a power ratio close to 95%. It is then possible to reduce the size and thereby to miniaturize the audio amplifier. The principle is given in figure 1. The audio signal is compared to a high frequency triangular signal. We then obtain a Pulse Width Modulated intermediate signal. This two level signal drives a full bridge MOS power transistor which works in switching mode. Free running diodes are not shown in figure 1. The power supply over 30V, enables it to carry a high current in the load. A low pass ouput filter between the MOS stage and the speaker restores the audio signal. And the feed back active network set the voltage gain of the circuit

DESIGN METHODOLOGY

Principle of Pulse width modulation

An analog signal is compared to a high frequency triangular waveform. The output is a two level pulsed signal as shown in figure 2. Let "m" be the modulation index. It is given by the ratio between the peak level Vbfe of the audio signal and the peak level Vtri of the triangular waweform : m = Vbfe/Vtri with 0<m<1 Let bf be the pulsation of the audio signal and T tri the period of the triangular signal. The modulated pulse duration  can be written :

 (1+m sin ( bf t) ) with  =T tri /2 The "equivalent gain" at low frequencies (ratio between the peak value of the output Low Frequency contribution of the modulated signal and the input peak value Vbfe) is constant. It is given by :

(1) Gpwm = Vd/(2.Vtri ), where Vd is the peak value of PWM signal.

Spectrum analysis of the output PWM signal

Let F tri be the frequency of the triangular signal and F bf the frequency of the audio input signal. As a complete mathematical approach is very difficult, we can simplify it, by assuming that the pwm signal is periodic. Then, we can write : [START_REF]Audio and HIFI Handbook[END_REF]

          2 1 1 2 2 2 1 0
The frequency spectrum of the PWM signal can obviously be approximated with Bessel functions. But, it is easily obtained by numerical simulation. An example of the normalized spectrum is given in figure 3 for two values of m, 5% and 99%. The spectrum of the output signal contains the useful audio signal and also frequencies around n.F tri  k.F bf . The peak value of these contributions depends on the value of m. For its maximum value (m=1 ie Vbfe = Vtri), the peak value of the audio frequency Vmax will be obtained from equation (1), thus :

(3) Vmax=0,5.Vd where Vd is the peak value of the PWM signal.

Taking F tri around 200kHz, (i.e greater than ten times Fbf), the reproduction of the audio signal will be quite easy with a simple ouptut second order lowpass filter.

Starting the design

Let Va to be the bridge power supply and Pu the maximum power transmitted to the load. Assuming that the load is fully resistive in audio bandwidth, (loud speaker 8 or 4 ), we first calculate the peak value of the useful low frequency audio lines of the modulated signal by Pu = Vbfs 2 /R. We initially consider a perfect ouput filter without losses and we assume that the MOS "on state" drop voltage is negligible . As the "equivalent voltage gain" of the full bridge is : 2Va/Vd, and taking m=1 (best case of modulation), we can determine the minimum supply voltage Va of the bridge ; from (3), it yields :

Vbfs max = 0,5.Vd x (2Va/Vd) and :

Vbfs max = Va Knowing the normalized input voltage level, the value of the required voltage gain is easily calculated. The power ratio of the amplifier obviously depends on the switching and the conduction losses of the MOS transistors and also of the output filter losses. Thus, a small "Ron" MOS transistor should be used.

OPEN LOOP DESCRIPTION

Driver stage

The driver circuit (in our example HIP4080 [START_REF]Harris power seminar handbook[END_REF]), located between the PWM modulator and the full bridge, allows the MOS transistors (here IRF 530) to switch under the best conditions by :

-delivering a 12V pulse and a peak current of 3A to improve the switching times.

-generating a dead time to avoid the simultaneous conduction of two transistors at the switching times.

-inhibiting the gates command signal in case of current overload.

Overload protection circuit

The current overload protection consists of one "shunt" resistor, a low pass filter, a threshold comparator, and a MOS transistor driver inhibition circuit (inside the driver circuit).

Triangle generator

The triangle generator is a commercial circuit MAX 038. The output level is ± 1V. A voltage level shift is added to center the signal around 6V. Its frequency can be adjusted between 100kHz and 1Mhz.

Feed back network

The diagram of this active lowpass filter is given in figure 4. The first order transfer function H(p) is :

(4) Vs V2 V1 R3 (R1 R2) 1 (1 C1. R1. R2 R1 R2 p)     
The gain in the audio bandwidth is:

GBP  R3 ( R1

CLOSED LOOP MODEL

Small signal model

From a low frequencies point of view, The PWM modulator and the bridge can be considered as a simple gain K, proportionnal to Gpwm, Va and 1/Vd (see equation ( 3) and ( 4)). K = Gpwm .2Va/Vd soit : K = Va/Vtri Moreover, at the audio frequency, the feedback network may be considered as a real gain. The circuit can be simplified as indicated in figure 5. A classical small signal analysis is then possible. (

5) B(p) R4 R5 R1 R2 R3 1 1 C K R4 R3 (R1 R2)p    
For a given voltage gain Gv, an audio bandwidth BP and a choosen gain K, it comes :

(6) Gv R4 R5 R1 R2 R3   et BP 1 2p C K R4 R3 (R1 R2)  
From these formulas, we can calculate the required capacitor and resistor values.

Stability study

The feed back loop should be taken just before the loud speaker for an optimum effect. Including this filter into the loop would generate a too great phase rotation. And it would be impossible to compensate the system. So, the feed back is done just before the output filter.

OUTPUT FILTER

It is a second order low pass filter with a cutoff frequency of 12KHz. The order choice comes from a compromise between complexity and filtering efficiency. The R,C serial network value depends on the speaker characteristics.

EXPERIMENTAL RESULTS

For safety reasons, the test was performed on the amplifier limited to an output power of 15W.

Frequency response

The experimental open loop transfer function is given in figure 8 and figure 9. The gain and phase margin are large enough to make the amplifier unconditionally stable. The output filter transfer function V HP /(V1-V2), including the speaker load is given in figure 10. The loss of 1dB in the audio bandwidth, comes from the parasitic serial resistor of the coils which are not negligible compared to the speaker resistor value. Figure 12 shows the output spectrum for a 500 Hz sinewave input signal. The second harmonic attenuation is better than 40db. Thus, the distortion rate is smaller than 1% as indicated before.

Time response

DISCUSSION

Sample frequency variation effects

The power ratio obviously depends on the ouput level. In our example it reaches 83% for an output power of 15W and a "sample" frequency F tri around 200 kHz. Increasing the frequency F tri does not improve the distortion level. Moreover, it increases the switching losses and lowers the power ratio. Thus F tri should not exceed ten times the maximum audio frequency.

Bridge power supply variations effects

Assuming that nothing else changes :

-if the supply voltage increases, then the gain K also increases. The PWM modulator input signal decreases making the index m smaller. The HF spectrum lines increase. The distortion and the bandwidth in closed loop increase. The noise to signal ratio is then worse and the power ratio is smaller.

-if the supply voltage decreases, m increases.In the worst case, the LF signal is always greater than the triangular signal. We obtain then, the modulation indicated in figure 13. The amplifier saturates and the output signal is then a squared waveform with a fundamental frequency of Fbf. The harmonic contents 3, 5, 7 can appear in the bandwidth give high harmonic distortion (figure 14).

8.CONCLUSION

In this study, we presented a method to design a power switching class D amplifier as simply as possible. Modelling the closed loop circuit at the audio frequency for small signals was the most difficult step in this method. The experimentals results show a good matching with our theoretical approach. However, this method only enables us to find the most important parameters of the amplifier. The power ratio must be taken into account when the electronic design is started. In our example, it could be easily improved by reducing the power comsumption of the triangle generator and the active feedback filter.

The optimisation of the power ratio will allow the reduction of the size of the heatsinks and probably their elimination for a medium power audio amplifier (i.e 15W). Thus, the full integration of our amplifier will be possible.

Figure 1 :

 1 Figure 1 : class D amplifier principle

Figure 2 :

 2 Figure 2 : PWM modulation

Figure 3 :

 3 PWM signal spectrum

Figure 4 :

 4 Figure 4 : feedback network

Figure 5 :Figure 6 :

 56 Figure 5 : amplifier small signal model Simplifiying the circuit, it comes :

  Figure 7 : output filter

Figure 9 :

 9 Figure 8 : voltage gain response

Figure 10 :

 10 Figure 10 : output filter gain response

Figure 11

 11 Figure11shows the PWM signal modulated at 90%.

Figure 11 :

 11 Figure 11 : PWM signal modulated at 90%

Figure 14 :

 14 Figure 13 : overmodulation


Relationship between structure and some physical properties of a new high-pressure series of perovskite-like oxides

Gérard Demazeau, Michel Pouchard, Paul Hagenmuller


Ph M Dondon-J

Micouleau-P

Kadionik Enseirb

IMPROVING LEARNING EFFICIENCY FOR DIGITAL MODULATION COURSES

Teaching the digital transmission techniques-like other fields of electronic-is not always easy : There is classically theoretical courses and practical lessons. The link between the two is often the most important difficulty that appears for the students. An abundant literature on digital wireless with a high level mathematical of modelling is of course available. Since a few years, theoretical courses are easily illustrated by the 2D computer simulations. But simple descriptions or analysis are rarely found and simulations do not replace a concrete true design. In order to improve the efficiency of our teaching, we present here a practical and pedagogical approach including practical electronic considerations, original and simple "Costas loop" description and design considerations for understanding and designing communication circuits. Based on the psychological Hermann brain model, this method looks like a progressive linear guide. A QPSK modulator/demodulator topology is used to illustrate our approach.

Introduction

Since a few years, we observed some major changes in student's behaviour in our engineer school: They are less interested in theory and "reject" all courses which have no concrete application. So, one of our challenge is how to motivate these new students generation to the fundamental courses. Moreover, even if simulations tools like SIMULINK are helpful and "funny to use", they do not fit perfectly the behaviour of a true circuit. The use of an intuitive and practical approach seems to be more efficient than traditional courses for these students. The job we present here is a try to create a didactical link between theoretical course and practical lesson. The example of a hand made QPSK modulator/demodulator is used to illustrate this approach.

Digital modulation backgrounds

At the beginning of the practical lesson, we first extract from the theory, the fundamental definitions of digital modulation [START_REF] Proakis | Digital communications[END_REF]. So, the beginners can also understand the main concept of transmission even they are not familiar with the full theory. After a first practical lesson on amplitude and frequency modulation, they started the QPSK training lesson. To be as simple as possible, our experimental kit works at an intermediate frequency of 455 kHz with a maximum bit rate of 20kbits/s.

QPSK modulation 3.1 QPSK Modulator design

The modulator is based on the classical structure given in figure 1. [START_REF] De Dieuleveult | Electronique appliquée aux hautes fréquences[END_REF] Thus, the impact of filtering on modulated carrier spectrum and eye diagram can be easily observed (cf section 5). The I, Q data's can also be (or not) synchronised with the carrier just to make the observation of modulated signal easier.

QPSK Demodulator design

The demodulator is based on the classical coherent demodulator structure given in figure 6. -Two analogue multipliers, a error OP amp circuit, and a first order loop filter for the Costas loop.

Even if the Costas loop is well known and classically used in the most digital transmission since almost 40 years, only highly strong mathematical analysis are available to understand the operating mode of this loop.

Between the too much theoretical approach and the too much simple explanations and synoptic we founded in the literature, we developed a quite simple approach to help the students in that way. The incoming signal can be written as: Ve(t) = I cost +Q sint , where  is the carrier frequency.

suppose that the local oscillator (V ol ) of the receiver is locked on the carrier frequency but not exactly in phase. Thus, we can say: V ol = A cos(t+).

To understand the impact of , we have to make a simple trigonometric calculus. At the multiplier output of the "I arm", we obtain:

A

cos(t+ ).[ I cost +Q sint]

After low pass filtering on the "I arm":

A.I/2 .cos -A.Q/2 .sin     
At the multiplier output of the "Q arm", we obtain: A sin(t+ ).[ I cost +Q sint]

After low pass filtering on the "Q arm": -A.I/2 .sin + A.Q/2 .cos    

Assuming  small, and using limited development formulas, it yields:

For the I arm and from (1) :

A/2.I (cos -Q/I. tg ) Thus: A/2.I (1-  -(Q/I).) (3) For the Q arm and from (2) A/2.Q(cos + I/Q. tg ) Thus: A/2.Q(1-  + I/Q. ) (4)
As we can see on (3) and (4), I and Q are fully well recognised if  equal zero.(case of perfect coherent demodulation).

After crossed multiplication between one arm by the sign of the other, the error voltage Vd can be written as :

Vd={sign (arm I) x arm Q} -{sign (arm Q) x arm I} I Q Error voltage Vd +1 -1 A/2. [-I.(1+  -Q.(1-  ] -1 +1 A/2. [+I.(1+  + Q. .(1-  ] +1 +1 A/2.[I.(1-  -Q.(1+  ] -1 -1 A/2.[-I.(1-  +Q.(1+  ] Table 2
After replacing I and Q by there values (+1,-1) in each expression, it yields:

Vd= A/2.(-2.

) whatever the symbol value

The error voltage Vd is thus always, proportional to the phase shift while  remains small (between -/4 et +/4). In fact, we can show that Vd=f() is a /2 periodic function. The loop can be locked with a phase ambiguity of k. /2. Confusion or inversion on I and Q bits can then occur. This well known phenomenon can be visualized on our practical kit. From this shortened analysis, we can understand some basic effects of a non zero phase shift: -When increases, we can guess, from table 1, a change in the amplitude of I and Q : a confusion between the logical level '1' et '0' can occur.

-These changes can also be simulated using Excel or "Matlab Simulink" software, for example. -At least, the effect of the phase shift can be observed looking at the I,Q constellation :  introduces a rotation compared to the ideal position.

Then, the amplitude changes can also be interpreted by comparing the vector projection on I and Q axis. (cf figure 7) In order to characterize the loop behaviour as simply as possible, we add on our demodulator card, some extra functionalities as indicated in figure 8.

Electronic design considerations

We spontaneously designed our QPSK kit using classical discrete components instead of fully integrated ones. This allows a better understanding of the individual functionalities and a full test and characterization of each block included in the design.

Teaching strategy 4.1 Hermann model short description

Ned Hermann, past manager of the formation at General electric company developed a technique making it possible to easily know the cerebral preferences of an individual. This universally recognised method is intended for better including/understanding the preferences of an individual like his approach to particular problems. It results in a representation of the brain into four quadrants (figure 9) which correspond to four preferences of operation. Quadrant A (blue) shows the preference of the individual for logic, modelling, (typical profile : mathematician, data processing specialist …)B (green) its aptitude for the practice and planning (real time production scheduler, administrative profile…). The quadrant D (yellow) shows the preference for the risk and projection in the future (typical profile : "start-up" manager, risk manager, artist…), and C (red) for the relational one, emotion (typical profile : social and commercial workers.

Planification, concrete

As said previously, the most important for the students is to link the theory of digital modulation to a practical design as simply as possible:

The theoretical classical course on digital communications is chained with the practical training as soon as possible to maximize the efficiency of teaching.

According to the Hermann model the four quadrant of the brain model are swept, starting with the right C quadrant. And the practical lesson looks like a linear progressive tutorial guide with important "mile stones" : -Frequency and phase locking ability after a cut of modulator/demodulator connection.

-I,Q demodulated stream observation e) Results debriefing (interactive work with the teacher) ("A" quadrant solicitation)

Then, this lesson can be (or not) followed by an other one's, much more detailed, depending on the optional courses chosen by students (in the third year school). A vector signal analyser is then used for refining and completing measurements like jitter, phase error, and other complex parameters like bit error rate.

Experimental

As example, we give some experimental results obtained during this practical lesson. The figure 11 shows the initial modulated signal spectrum without filtering. And figure 10 shows the effect of I,Q Bessel filtering with a cut-off frequency half of the bit rate :

-Reduction of useful bandwidth to the main lobe (secondary lobes suppression) -Proportionality between bit rate (20 kbits/s here) and the main lobe width (20kHz). (center frequency 459kHz)

Conclusion

This paper is a digest of our integrated pedagogical approach. The link between theory and practical lessons has been exposed. A more detailed paper is obviously given to our engineer students. this full integrated approach is now used in our engineer school to teach the digital circuit's application and design. As we can see since a few years, the classical and conventional teaching method reached their limits. Including psychological aspects, we have shown that it is possible to improve the efficiency of our teaching.

Even if it is always difficult to "measure" the impact of a teaching strategy, this one seems to be more attractive and efficient than before, looking at the results of annual "student satisfaction report": The last one shows that the satisfaction rate raised from 45% up to 65%. One other important point is that the students stay until they finish all the lesson even it is late: they do not leave anymore before the end, like before. This is an encouraging sign of a true motivation for this practical approach.

Of course, these first improvements will have to be obviously confirmed in the future.

Fig 1 :

 1 Fig 1 : QPSK modulator

Fig 2 :

 2 Fig 2 : I,Q data generator schematic

Fig 4 :

 4 Fig 4 : Cosine raised filter response

Fig 5 :

 5 Fig 5 : modulator block diagram

Fig 6 :

 6 Fig 6 : QPSK Costas loop demodulator Our hand made demodulator kit includes: -A voltage controlled oscillator at 4 x 455 kHz and a divider down to 455 kHz, -A sinusoidal carrier quadrature generator with 455 kHz ceramic filter -Two analogue multipliers for coherent demodulation -Two simple low pass filters (I, Q filtering) -Two fast comparators for sign detection

Fig 7 :

 7 Fig 7 : effect of phase shift

Fig 8 :

 8 Fig 8 : demodulator block diagram

Fig. 9 :

 9 Fig. 9 : Herman brain model

Fig. 10 :

 10 Fig. 10 : optimal use of brain capacities According to Hermann conclusion, the best way to optimize the motivation of our students is to sweep all the quadrants starting from the right parts during an integrated lesson. (figure 10)

  a) Analysis of schematic diagram: (interactive work with the teacher) ("C" and "D" quadrant solicitation) -Functional analysis -Electrical analysis -Validation of Components choices, -Main components data sheet readings -Effect of current and voltage static offset of the integrated circuits, -Signal level choices and dynamic range justification, -Pseudo random generator analysis (clock, NRZ, I and Q signal observation (analog/digital mixed oscilloscope) -Impact of the clock rate and of the random sequence length on the NRZ spectrum. (Oscilloscope FFT) b) Test of the QPSK modulator : (individual student's work) ("B" quadrant solicitation) -Local quadrature oscillator frequency and distortion measurement (oscilloscope, time and FFT) -QPSK modulated signal observation (time and frequency domain, spectrum analyser) -I,Q eye diagram (constellation analyser) [5] -Bessel and cosine filter characterization (frequency and time responses drawings) -Effect of I,Q filtering (secondary lobe suppression, spectrum analyser) -Effect of the bit rate on useful bandwidth (manual sweep of the data clock, spectrum analyser) c) Test of the demodulator: (individual student's work) -Transfer curve Vd=f() drawing ( using the programmable integrated delay line, and a DC voltmeter) -Effect of carrier phase shift on I, Q. Visualization of the constellation rotation. (Delay line and X,Y oscilloscope) -Open loop test (lock range adjustment) -Open loop gain transfer curve drawing -Loop filter cut off frequency determination (graphical method) -Closed loop test -Constellation and eye diagram observation when locked (oscilloscope) d) Test of modulator/demodulator (individual student's work) ("B" quadrant solicitation) -Phase ambiguity carrier recovery observation. (Powering on and off the power supply) -Costas loop lock range measurement. (Manual frequency sweep of the modulator carrier)

Fig 11 :

 11 Fig 11 : Modulated carrier spectrum without I,Q filtering

Fig 14 :

 14 Fig 14: QPSK constellation rotation

Fig 15 :

 15 Fig 15: eye diagram

Fig 15 :

 15 Fig 15 : QPSK test bench