Simulation of a Vacuum-Tube Push-Pull Guitar Power Amplifier

Proc. of the 14 th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011SIMULATION OF A VACUUM-TUBE PUSH-PULL GUITAR POWER AMPLIFIERJaromir MacakDept. of Telecommunications,FEEC, Brno University of TechnologyBrno, Czech Republicjaromir.macak@phd.feec.vutbr.czJiri SchimmelDept. of Telecommunications,FEEC, Brno University of TechnologyBrno, Czech Republicschimmel@feec.vutbr.czABSTRACTPower amplifiers play an important role in producing of guitarsound. Therefore, the modeling of guitar amplifiers must alsoinclude a power amplifier. In this paper, a push-pull guitar tubepower amplifier, including an output transformer and influence of aloudspeaker, is simulated in different levels of complexity in orderto find a simplified model of an amplifier with regards to accuracyand computational efficiency.1. INTRODUCTIONGuitar power amplifiers have a big effect for guitar sound. Powervacuum-tubes have characteristic nonlinear distortion [1]. Furthermore,compared to semiconductor amplifiers, tube power amplifiershave different circuit topology that brings other effects, suchas typical frequency response and signal compression when a largeinput signal is supplied [1]. Moreover, output transformers play animportant role in the output signal generation. The output signalis distorted by nonlinear hysteresis of a transformer core and frequencyresponse of the output transformer [2].Two types of topologies are used in tube power amplifier construction.Single ended power amplifiers consist mainly of onepower tube, an output transformer, and resistors and capacitors.They provide asymmetrical limiting of the output signal that createsforemost even order harmonic components in the output signalspectrum. Push-pull power amplifiers have more complex topology.They consist of a phase splitter, two or four power tubes thatprocesses opposite half-waves of the signal and output transformerthat sums contributions from opposite power tubes. The push-pullamplifiers offer a symmetrical transfer function that creates oddorder harmonic components in the output signal spectrum [3].A single-ended power amplifier is simulated in [4] using wavedigital filters. The model of the power amplifier contains a triode,a linearized model of the output transformer and a linearizedmodel of a loudspeaker recomputed to its electrical equivalent.This model was improved in [5] where the complex nonlinear WDFmodel of the output transformer is used. A single-ended poweramplifier with a pentode tube is discussed in [6]. This model alsocontains a linearized model of the output transformer but the loudspeakeris replaced by a constant load. Nevertheless, guitar tubeamplifiers mostly contain a push-pull power amplifiers with twoor four tubes (pentodes EL34 or beam tetrodes 6L6) that work inclass AB or B [2]. However, the circuit topology and the circuitequations are the same for both classes and they differ only in biaspoint. A simulation of a push-pull amplifier is described in [1].The complete circuit is divided into a phase splitter, a pentode circuitand a feedback circuit and they are connected using a modifiedblockwise method [1]. However, this simulation uses a model ofthe ideal output transformer and the constant load.In this paper, the pentode circuit is simulated in different levelof complexity. At first, the pentode circuit with an ideal transformerand a constant load is solved, then the circuit with an idealtransformer and a linearized model of a loudspeaker. Finally, thecircuit with a nonlinear model of the output transformer is simulated.2. BASE CIRCUIT AND DEVICE MODELSA typical pentode push-pull power amplifier contains two or fourpentodes, the output transformer with connected loudspeaker andsome resistors and capacitors (see figure 1). The circuit schematicalso contains input units built by resistors R g, R b , capacitors C gand bias voltage V b . The input units are independent from the restof the circuit because there is no coupling via the tube’s cathode intopology with fixed bias. Their simulation is described in [1], andtherefore the simulation of this part is omitted and there is only afocus on the output part of the amplifier. The typical values forcircuit elements are listed in table 1.V s1V in1N 1V DR s2R dRN 1 s1C d V psV a1 V a2V s1 V LVN 2 in2R g1R g2R b1 R b2VC bC g2g1 Vinp Vinp2Figure 1: Circuit schematic of a push-pull tube amplifier.Table 1: Values for circuit elements from figure 1.R s[Ω] R d [Ω] C d [µF] N 1 N 2 V ps[V] V b [V]500 500 100 1560 60 500 −50DAFX-1DAFx-59

Proc. of the 14 th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 20112.1. Pentode ModelThe Koren’s model of the pentode [7] was chosen as the pentodemodel. The pentode plate current is in form I a(V ak , V gk1 , V gk2 )where V ak is plate-to-cathode voltage, V g1k is the grid-to-cathodevoltage and U g2k is the screen-to-cathode voltage. The screen currentis given in form I s(V gk1 , V gk2 ). The description of functionsI a and I s is omitted here and is available in [7]. Frequency propertiesof the tube (e.g. Miller capacitance) are not considered becauseit should be included in the simulation of input unit.2.2. Output Transformer ModelAn ideal output transformer is considered to be an impedance dividerthat transforms input voltages V p and currents I p to outputV s and I s according to2.3. Loudspeaker ModelLoudspeakers play a very important role in the output signal generationvia its frequency response. When considering linearizedloudspeakers, one can model the frequency response with measuredimpulse responses with good results [1]. However, it is importantto simulate the interaction between the tube amplifier andloudspeaker because the loudspeaker impedance is frequency dependent.The impedance can be modeled using the circuit schematicin figure 2 [12]. The values are derived from the added massmethod and Thiele/Small parameters of a Celestion Vintage 30loudspeaker placed in an Engl combo. The transformer leakageinductance and resistance can be modeled by modifying inductorL sp1 and resistor R sp1 values.V sV p= N2N 1= IpI s(1)where N 1, N 2 are numbers of windings of the transformer. However,a real transformer is far away from the ideal one. For an accuratesimulation, losses caused by hysteresis and core saturationhave to be considered. Nonlinear behavior of the real transformeris described in numerous literature, e.g. [8, 9]. According to Amper’slaw, the magnetizing force H isHl mag = N 1I p − N 2I s (2)where l mag is the length of the induction path. The flux density Bis computed from Farraday’s law∂B∂t =Vs(3)N 2Swhere S is the transformer-core cross-section. The well-knownnonlinear relation B = µH can be implemented according to theFrolich equation [9] given byB =Hc + b |H|where c and b are constants derived from material properties. However,this model simulates only the core saturation. When simulatinghysteresis, one can use e.g. Jiles-Atherton model [10] modifiedin [8] in order to remove nonphysical behavior of minor hysteresisloops. Magnetization of the core is obtained from∂M∂H = M an − MδM + c ∂Mankδ ∂Hwhere M an is anhysteretic curve given by( H + αMM an = M s(cotha)−)aH + αMand δ =sign(∂H/∂t). Parameters M s, α, a, c and k are derivedfrom material properties and their identification can be found e.g.in [11]. Parameter δ M =0when the nonphysical minor loop is goingto be generated (anhysteric magnezation has lower value thanthe irreversible magnezation) alternatively δ M =1[8] . Flux densityis then obtained from(4)(5)(6)B = µ 0 (M + H) . (7)Figure 2: Simplified loudspeaker model – electric equivalent.The loudspeaker impedance given by voltage V L and currentI L can be expressed as the solution of the set of equationsVL[n] − IL[n]Rsp1 − V3[n]I L[n] =I L[n − 1] +L sp1f sV3[n]Gsp2 − IL[n] − IL2[n]V 3[n] =V 3[n − 1] + (8)C sp1f sI L2[n] =I L2[n − 1] + V3[n]L sp2f swhere I L[n − 1], V 3[n − 1], I L2[n − 1] are state variables andf s is a sampling frequency. The equations were obtained usingnodal analysis of the circuit in figure 2 and then discretized usingBackward Euler formula.Because the set of equations (8) is linear, it can be simplifiedinto one linear equationI L[n] =−c 1V L[n]+I tmp (9)where I tmp is a linear combination of state variables given byI tmp = −c 2I L[n − 1] + c 3V 3[n − 1] − c 4I L2[n − 1]. (10)The new state variable values are then computed fromV 3[n] =−c 5V 3[n − 1] − c 6I L[n]+c 7I L2[n − 1] (11)andI L2[n] =I L2[n − 1] + c 8V 3[n]. (12)Coefficients c 1−8 are derived from (8).3. SIMULATION OF THE AMPLIFIERIn the simplest case, the load is considered to be constant. Usingnodal analysis and discretization by Euler method, one can obtainthe set of circuit equationsDAFX-2DAFx-60

Proc. of the 14 th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 20114. SIMULATION RESULTS0=− VL[n] + N1(I a1 − I a2)R L N 20=−V a1[n]+V D[n] − N1V L[n]N 20=−V a2[n]+V D[n]+ N1V L[n]N 2VD[n] − Vs1[n]0=I s1 +0=I s2 +R s1VD[n] − Vs2[n]R s2,(13)where V D is voltage on the power supply, capacitor C d is from thecircuit schematic in figure 1 and I a1 = I a(V in1[n],V a1[n],V s1[n]),I s = I s (V in1[n],V s1[n]) and similarly I s2 = I s (V in2[n],V s2[n])and I a2 = I a(V in2[n],V a2[n],V s2[n]). The equations (13) aresolvable for given value V D and the solution can be implementedusing a static waveshaper. It can be precomputed for different valuesof V D voltage and stored in a look-up table. Then, during thesimulation, the proper static waveshaper is chosen according to theV D voltage. The V D voltage is then actualized usingV D[n] = −Ia1 − Ia2 − Is1 − Is2 + (V PS−V D [n−1 ])R D+ V D[n − 1].C 1f fs(14)3.1. Circuit with Loudspeaker ModelIf the loudspeaker is connected to the power amplifier, the loadimpedance is no longer given explicitly, but it is determined by V L,I L relation implicitly given by equations (8). In order to excludethe loudspeaker impedance R L, the first equation from system (13)is modified toI L[n] = N1(I a1 − I a2) (15)N 2and then solution of the modified equations (13) with (15) is expressedas a function I L[n] =f IL (V in1[n],V in2[n],V D[n],V L[n]).Finally, the solution of the whole system using (9) is given by− c 1V L[n]+I tmp = f IL (16)for the unknown variable V L[n] and state variables I tmp and V D[n].3.2. Circuit with Nonlinear Transformer ModelThe simulation of a circuit with a nonlinear transformer model isbased on (2), (3) and the nonlinear core model. The system isdescribed using0=−H(B[n])l mag − (c 1V L[n]+I tmp)N 2 + f ILN 20=B[n − 1] − B[n]+VL(17)N 2Sf swhere term f ILN 2 is recomputed current I L to primary winding,B[n − 1] is the flux density in the previous sampling period andH(B) is the core model derived from (4) or from (7) if the hysteresisis considered.All the simulations from section 3 were implemented in Matlabenvironment using Mex files and C language. The Newton methodwas used for solving implicit nonlinear equations. The derivationof function or the Jacobian matrix were obtained using finite differenceformula, the maximal number of iterations was 100 and thenumerical error was chosen as 0.0001. The values for the transformermodel were chosen experimentally: M s = 1.11 × 10 6 ,3a =8.56, α =8.82×10 −5 , c =0.14 and k =51.65. However,they can be computed from a measured hysteresis loop data [11].The parameters for the Frohlich core model were c = 113.38,b =0.71 and the dimensions of the transformer were chosen asS =0.003 m 2 and l mag =0.2 m.The transformer-core cross-section S together with the numberof windings determines the low cutoff frequency of the transformer.The simulation of the output part of the power amplifierwas appended with a phase splitter, input pentode unit and feedbackaccording to [1] and the results were compared to the measuredEngl combo. The data was obtained from a measured voltageon a parallel loudspeaker output with connected soundcard. Thefrequency dependence of the first harmonic content of the voltageoutput signal, obtained using sweep sine signal, is shown infigure 3. The simulation with a constant load provides the worstresults – there is no resonance around 45 Hz that appears in allof the other simulations and measured data. The simulation withhysteresis loop has similar behavior as a measured amplifier in thearea of low frequencies due to the core losses. In the area of midfrequencies,the simulation and measured data differ because ofthe other resonances and mechanical properties of the loudspeakerdiaphragm, which are not considered in the simulation but can beincluded in the simulation by improving the loudspeaker modellike in [13]. The nonlinear distortion was investigated as well. Theoutput spectrum for an input sinewave signal with an amplitude of2 V is shown in figures 4. The individual spectra are measured forthe same frequency but they are shifted in the graph. All the algorithmsexcluding the variant with constant load give very similarresults. The nonlinear distortion caused by transformer hysteresismanifests very slightly and only at frequencies below cca 150 Hzand it is very dependent on transformer parameters. The spectrogramof simulation using JA-model is shown in figure 5.The table 2 shows the hypothetical computational complexitythat was determined using measurement of time duration of simulationfor input guitar riff signal with a length of 18 s and an amplitudeof 30 V. The results showed that the algorithm is capable ofworking in real-time. Sound examples of all the algorithm variantsand detailed graphs and other spectrograms are available on theweb page www.utko.feec.vutbr.cz/~macak/DAFx11/.Table 2: Normalized computational complexity.Constant load Loudspeaker Frohlich J-A model0.04 % 0.05 % 0.12 % 0.27 %5. CONCLUSIONSThe simulation of a push-pull tube amplifier was discussed in thispaper. The impact of the loudspeaker and output transformer toDAFX-3DAFx-61

Proc. of the 14 th Int. Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011Proc. of the 14th International Conference on Digital Audio Effects (DAFx-11), Paris, France, September 19-23, 2011the amplifier properties was investigated. The loudspeaker wasmodeled using simplified electric circuit and Frohlich and Jiles-Atherton transformer models were used. The results showed thatthe output transformer had a minor impact on the simulation results.It only manifested at very low frequencies while the computationalcomplexity was increased significantly. Both transformermodels provided very similar simulation results and therefore, theFrohlich model is more efficient for real-time simulations.f [Hz]100001000100M [dB]40353025201510MeasuredJ−A modelFrohlichLoudspeakerConstant load100 0.5 1 1.5 2 2.5 3t [s]Figure 5: Spectrogram of output signal for simulation with JAmodel of transformer.50−5−1010 0 10 2 10 4f [Hz]Figure 3: Frequency dependence of the first harmonic content ofthe output signal.M [dB]3020100−10−20J−A modelFrohlichLoudspeakerConstant loadMeasured−300 100 200 300 400 500 600 700 800 900f [Hz]Figure 4: Output spectrum for input 90 Hz sinewave.6. ACKNOWLEDGMENTSThis paper was supported by the Fund of the Council of HigherEducation Institutions of the Czech Republic under project no.2704/2011 and project no. FR-TI1/495 of the Ministry of Industryand Trade of the Czech Republic.7. REFERENCES[1] J. Macak and J. Schimmel, “Real-time guitar tube amplifiersimulation using approximation of differential equations,” inProceedings of the 13th International Conference on DigitalAudio Effects DAFx10, Graz, Austria, Sep. 6-10, 2010.[2] D. Self et al, Audio Engineering, Elsevier, Burlington, MA,USA, 1 st edition, 2009.[3] U. Zölzer, DAFX - Digital Audio Effects, J. Wiley & Sons,Ltd, 1 st edition, 2002.[4] J. Pakarinen, M. Tikander, and M. Karjalainen, “Wave digitalmodeling the output chain of a vacuum-tube amplifier,”in Proc. Intl. Conf. on Acoustics, Speech, and Signal Proc.,Como, Italy, Sept. 1-4, 2009, pp. 55–59.[5] R. C. D. de Paiva, J. Pakarinen, V. Välimäki, and M. Tikander,“Real-time audio transformer emulation for virtual tubeamplifiers,” EURASIP Journal on Advances in Signal Processing,vol. 2011, pp. 15, 2011.[6] I. Cohen and T. Helie, “Real-time simulation of a guitarpower amplifier,” in Proceedings of the 13th InternationalConference on Digital Audio Effects DAFx10, Graz, Austria,Sep. 6-10, 2010.[7] N. Koren, “Improved vacuum tube models for SPICEsimulations,” Available at http://www.normankoren.com/Audio/Tubemodspice_article.html, 2003.[8] P. Kis, Jiles-Atherton Model Implementation to Edge FiniteElement Method, Ph.D. thesis, Budapest University of Technologyand Economics, 2010.[9] S. E. Zocholl, A. Guzman, and D. Hou, “Transformermodeling as applied to differential protection,” Tech. Rep.,Schweitzer Engineering Laboratories, Inc. Pullman, Washington,1999.[10] D. C. Jiles and D. L. Atherton, “Ferromagnetic hysteresis,”IEEE Transactions on Magnetics, , no. 5, pp. 2183–2185,1983.[11] D. C. Jiles, J. B. Thoelke, and M. K. Devine, “Numericaldetermination of hysteresis parameters for the modeling ofmagnetic properties using the theory of ferromagnetic hysteresis,”IEEE Transactions on Magnetics, vol. 28, no. 1, pp.27–35, 1992.[12] R. Elliot, “Measuring Thiele / Small Loudspeaker Parameters,”Available at http://sound.westhost.com/tsp.htm, 2007.[13] J. Pakarinen, M. Tikander, and M. Karjalainen, “Wave digitalmodeling of the output chain of a vacuum-tube amplifier,” inProc. of the Int. Conf. on Digital Audio Effects (DAFx-09),Como, Italy, Sept. 1–4, 2009.DAFX-4DAFx-62