On Instantaneous Power Dissipation in Class B Amplifier

On Instantaneous Power Dissipation in Class B Amplifier Hristo Zhivomirov1, Ekaterinoslav Sirakov1 1 Technical University - Varna, Str. Studentska 1, Varna, Bulgaria Abstract – The present paper describes the analysis of the instantaneous power dissipation by the two active components in a class B power amplifier. Attention is paid to restrictions of the instantaneous power dissipation relations in reference literature, and the consequences of their misuse. A new generalized equation that takes into account the power dissipated by the two active devices is proposed. The theoretical statement is substantiated by Matlab® numeric computation and visualization, Cadence OrCAD® simulations and measurements of a real-world audio power amplifier performed by NI USB-6211 measurement complex. Keywords – Instantaneous power, Power dissipation, Class B Amplifier, Generalized equation. For 0 ≤ t ≤ T/2, is active the transistor connected to a positive supply rail U cc+ . Similarly for Т/2 ≤ t ≤ T, current flows through the lower arm of the amplifier [2],[4],[5]. Therefore the active regime of each transistor is determined by the period at which it is conducting [6],[10],[11]. Fig. 2.describes the process of amplification of the upper arm of the amplifier. IC U cc IC PC ma x π 2 π 3π 2 2π α ωt π π iCT1 T2 RL uCET2 uout π Ucc UL 2 π 3π 2 Clipping headroom 2π ωt Figure. 1. Functional circuit of the power amplifier stage working in class B The output voltage and current of the amplifier operating with sine-wave signal is described as [2]: = uout U= U out m sin α out m sin ω t RL UCsat Uout m Figure. 2. Output dynamic characteristics of the upper arm transistor and the process of amplification iCT2 sin ωt Clipping headroom 0 Um, Im – voltage and current amplitude ICmax, UCEmax – maximum collector-emitter voltage and collector current ICsat,UCsat – collector saturation voltage and current out iout U out m 2 ωt Ucc – power supply voltage uCET1 = = iout UCE 2 2π T1 U cc− Ucc Uout m 3π + cc in UCsat UCE UCEmax Ucc UCsat β 0 The functional circuit of power amplifier working in class B is given in Fig. 1 [1]. U Hyperbola of the maximum power losses RL ICsat ICsat ICm 0 1. Introduction Load line tg α = 1/RL Saturation line tg β = 1/Rsat ICmax U out m RL sin α (1) (2) where U outm is the amplitude of the output voltage; R L – resistance of the load; ω – angular frequency; t – time; Т – period of the signal; α =ωt – phase of the signal [3]. TEM Journal – Volume 3 / Number 2 / 2014. www.temjournal.com One should note that for the designer of audio power amplifiers (as opposed to sonar or RF amplifiers) the main design parameter is not efficiency, but rather inefficiency, i.e. the power dissipated by output active device [6]. This paper aims the instantaneous power dissipation calculation, which directly effects the calculation of average power dissipation, SOA-limitation and the selection of active components and thermal heat sinks. 2. Instantaneous power dissipation operating with resistive load 2.1. Basic relations The power dissipated by each transistor can be represented as a product of the collector-emitter voltage drop u CE and collector current i C : 101 P= uCE (t ) ⋅ iC (t ) . D ( inst ) (3) For the upper arm transistor connected to the positive supply rail U cc+ [3, 4, 5, 7]: uCET 1 = U cc+ − uout , for 0 ≤ α ≤ 2π π 0 2π uout (5) α 0 uCET1 and for the transistor connected to the negative rail U cc− : uCET 2 = U cc− − uout , for 0 ≤ α ≤ 2π (6) for 0 ≤ α ≤ π 0 . iCT 2 =  = ≤ sin≤α for π α 2π iout I outm (7) One can assume U cc = U cc+ = −U cc− [2]. The signs in Eqs. (4) ÷ (7) are in accordance to the established rules in electrical engineering, e.g. currents entering the node – plus sign, for voltages from high potential to low potential – plus sign, and vice-versa. Table 1.shows some parameters of a class B amplifier operating with resistive load. Ucc 0 uCET2 α 0 α -Ucc iCT1 α 0 iCT2 α 0 iout α 0 Table 1.Some parameters of the circuit from Fig. 1. PD(inst)T1 0≤α≤π π≤ α ≤ 2π Parameter T1 > U cc− T1 > U cc+ T2 < U cc− 0 PD(inst)T2 uCE < U cc+ T2 iC P D(inst) = i out >0 =0 =0 =0 =0 = i out >0 0 PD(inst) α α It can be shown (Fig. 2, Fig. 3) that: ≤ 2U cc − U CEsat uCE  ≥ U CEsat i= iCT 1 + iCT 2 . out α 0 (8) whereU CEsat is the collector – emitter saturation voltage of the transistors. The relation between the output current of the amplifier and the collector current of each transistor is: 102 α (4) = I outm ≤ sin≤α for 0 α π i iCT 1 =  out for π ≤ α ≤ 2π 0 α uin (9) Figure 3. Graphical representation of the voltages, currents and instantaneous dissipated power of the class B amplifier By means of substitution of Eqs. (4) ÷ (7) into (3) for the instantaneous power dissipation of the amplifier can be written: - for the upper arm of the amplifier [2], [6]: ( ) PD(inst)T1 = U cc − U out m sin α ⋅ iCT 1 = ( −= U cc U out m⋅sin α ) U out m RL sin α (10) TEM Journal – Volume 3 / Number 2 / 2014. www.temjournal.com for 0 ≤ α ≤ π; P D(inst)T1 = 0 for π≤ α ≤ 2π. - for the lower arm: ( ) PD(inst)T2 =−U cc − U out m sin α ⋅ iCT 2 ( ) =−U cc − U out m sin α ⋅ U out m RL When the amplifier operates with resistive load and sine-wave signal: sin α (11) U out m PD(inst) = sgn ( sin α )U cc − U out m sin α  ⋅ sin α (18) RL for 0 ≤ α ≤ 2π. If the coefficient of effective use of supply forπ≤ α ≤ 2π; P D(inst)T2 = 0 for 0 ≤ α ≤ π. voltage is assumed to be ξ = 2.2. Generalized equation U out m [4], then Eq. (18) U cc can be rewritten as: In terms of more compact representation of Eqs. (10) and (11), Eqs. (5) and (7) are transformed as U2 PD ( inst ) − cc sgn ( sin α ) ξ sin α ξ 2 sin 2 α  .(19) = follows: RL - for the upper arm of the amplifier [4, 8]: The results of the analysis of Eqs. (10), (14) and (18) I outm sin ωt + I outm sin ωt (12) are represented in Fig. 4. for 0 α 2 RL 2π P D(inst) vs. α, Eq. (10) 20 for 0 ≤ α ≤ 2π. - for the lower arm: ( PD(inst)T2 −= U cc U out m⋅sin α D(inst) -40 0 50 100 150 P D(inst) 200 250 α, ° vs. α, Eq. (14) 300 350 300 350 300 350 ) U out m 2 RL +( sin α sin α ) (14) ,W 5 D(inst) ( PD(inst)T1 −= U cc U out m⋅sin α -20 0 P Eqs. (10) and (11) become: - for the upper arm: 2π . (13) P I outm sin ωt − I outm sin ωt ≤ ≤ for 0 α 2 RL 0 -5 0 50 100 150 P D(inst) 200 250 α, ° vs. α, Eq. (18) 6 ) U out m 2 RL −( sin α sin α ) (15) for 0 ≤ α ≤ 2π. To summarize, using Eqs. (3) and (9) and with reference to Fig. 3, a new generalized relation is proposed about the total instantaneous power dissipated by the amplifier (by the two arms): PD(inst) = PD(inst)T1 + PD(inst)T2 = uCET 1 ⋅ iCT 1 + uCET 2 ⋅ iCT 2 = = (U cc − uout ) ⋅ iCT 1 + ( −U cc − uout ) ⋅ iCT 12 = = sgn ( iout )U cc − uout  ⋅ iout (16) where the signum function is used [9]:  − 1 for x < 0  sgn( x) = 0 for x 0 . = +1 for x > 0  TEM Journal – Volume 3 / Number 2 / 2014. www.temjournal.com (17) ,W iCT 2 ,W - for the lower arm: D(inst) ≤ P ≤ iCT 1 4 2 0 0 50 100 150 200 α, ° 250 Figure. 4. Graphical representation of the analysis of Eqs. (10) and (14), as well the author’s Eq. (18) for 0 ≤ α ≤ 2π and U cc = 12 V, R L = 8 Ω, U m = 9 V. 3. Amplifier Simulation Computer simulation using Cadence®OrCAD® software is provided in order to verify the theoretical statement and to illustrate the process of amplification. Simulation test circuit is shown in Fig. 5. Attention is paid to the collector-emitter voltage u CE and the collector current i C of the transistors (blue probes for the upper-arm and red probes for the lower-arm). Several periods of the signals is shown, so that one can track the changes of the signals in time domain[12],[13]. In Fig. 6 the simulation results are illustrated for the upper and lower arm of the amplifier. One can 103 note that u CE of each transistor is oscillating about its power supply voltage and the minimum of u CE corresponds to maximum of u out and vice-versa. In Fig. 6 (bottom) the results for instantaneous power dissipation are shown, that correspond to these obtained by the author’s Eqs. (16) and (18). The measurements are made by measurement complex NI USB-6211 and author’s Matlab®software, according to measurement setup shown in Fig. 8 (where measurement connection is shown only for the upper arm of the amplifier), similar to this in [14]. Figure. 5. Test circuit for simulation of class B push-pull amplifier Fig. 7. Results from the measurement of real-world class B push-pull solid state audio power amplifier Figure. 6. Results from the simulation of class B push-pull amplifier (i CT1 , i CT2 , u CET1 , u CET2 , P D(inst)T1 , P D(inst)T2 ) 4. Amplifier measurements A measurement of real-world class B amplifier has been made and the results fully correspond to the Eq. 18. The measured amplifier is based on integral circuit LM3886 (Hi-Fi audio power amplifier) and has the following parameters (one and the same as the simulation circuit): - Power supply voltage U cc = 12 V; - Resistive load R L = 8 Ω; - Test signal: sine-wave (f=1 kHz, U m = 9 V). The results of the measurement are plotted in Fig. 7. One should note there is no synchronization between simulation and measurement results, since the last begin in arbitrary moment. 104 Figure. 8. Measurement setup for measurement of realworld class B push-pull solid state audio power amplifier 5. Conclusion It should be noted that the results in Fig. 4 obtained by Eq. (10) for π≤ α ≤ 2π are erroneous. Hence references [2],[6] must be used very carefully, as far as there is no explanation on what are the restrictions of Eq. (10), e.g. it holds true only for 0 ≤ α ≤ π. TEM Journal – Volume 3 / Number 2 / 2014. www.temjournal.com Results in Fig. 4 obtained by Eq. (14) are true for 0 ≤ α ≤ 2π, but concern the operation only of the upper arm of the amplifier, and Eq. (15) – only the lower arm, respectively. The proposed Eq. (18) describe the power dissipation of the two active components of the amplifier, e.g. the whole class B amplifier device for the full cycle of operation (0 ≤ α ≤ 2π) as shown in Fig. 4. This is of great significance, especially for integrated solid state power amplifiers. The comparison of graphical representation of Eq. (18) in Fig. 4, with simulation and measurement results about P D(inst)T1 , P D(inst)T2 in Fig. 6 and Fig. 7 confirms not only the qualitative aspect of the proposed expression but also the quantitative one. The generalized Eq. (16) gives a new look on the instantaneous power dissipation of the class B power amplifiers and is conductive to further examination of the subject. Further considerations concerning a mathematical modeling of power parameters of class B amplifier will be given in future work, including the operation of amplifiers with stochastic signals as music or speech. Acknowledgements The authors would like to acknowledge the financial support from the Project № BG051PO001-3.3.06-0005 “Development of potential of PhD students, young scientists and graduate students of engineering at the Technical University of Varna and its contribution to developing an economy based on knowledge”. [2]. Marshall Leach W., Jr. (2003). 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TEM Journal – Volume 3 / Number 2 / 2014. www.temjournal.com Corresponding author: Hristo Zhivomirov Institution: Technical University – Varna, Bulgaria E-mail: hristo_car@abv.bg 105