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Comparison of Germanium Bipolar Junction Transistor Models for Real-Time Circuit Simulation


Abstract and Figures

The Ebers-Moll model has been widely used to represent Bipolar Junction Transistors (BJTs) in Virtual Analogue (VA) circuits. An investigation into the validity of this model is presented in which the Ebers-Moll model is compared to BJT models of higher complexity , introducing the Gummel-Poon model to the VA field. A comparison is performed using two complementary approaches: on fit to measurements taken directly from BJTs, and on application to physical circuit models. Targeted parameter extraction strategies are proposed for each model. There are two case studies , both famous vintage guitar effects featuring germanium BJTs. Results demonstrate the effects of incorporating additional complexity into the component model, weighing the trade-off between differences in the output and computational cost.
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Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Ben Holmes
Sonic Arts Research Centre,
School of Electronics, Electrical Engineering
and Computer Science
Belfast, U.K
Martin Holters
Department for Signal Processing and
Helmut Schmidt University
Hamburg, Germany
Maarten van Walstijn
Sonic Arts Research Centre,
School of Electronics, Electrical Engineering
and Computer Science
Belfast, U.K
The Ebers-Moll model has been widely used to represent Bipolar
Junction Transistors (BJTs) in Virtual Analogue (VA) circuits. An
investigation into the validity of this model is presented in which
the Ebers-Moll model is compared to BJT models of higher com-
plexity, introducing the Gummel-Poon model to the VA field. A
comparison is performed using two complementary approaches:
on fit to measurements taken directly from BJTs, and on appli-
cation to physical circuit models. Targeted parameter extraction
strategies are proposed for each model. There are two case stud-
ies, both famous vintage guitar effects featuring germanium BJTs.
Results demonstrate the effects of incorporating additional com-
plexity into the component model, weighing the trade-off between
differences in the output and computational cost.
Recent developments in processing power have allowed for real-
time simulation of many audio circuits at a physical level, prompt-
ing the search for component models that achieve the highest ac-
curacy whilst maintaining real-time compatibility. In the field of
VA modelling, circuits featuring BJTs have been modelled suc-
cessfully with a variety of different component models. Simplified
models of the BJT are useful in applications featuring many BJTs.
A notable case is exemplified in the modelling of the Moog Ladder
filter in which current gain is presumed infinite, reducing a stage
of two BJTs to one nonlinear equation [1]. In circuits featuring
fewer BJTs, the large-signal Ebers-Moll model has been used [2].
This component model has been used in circuit models of com-
plete guitar pedals, including wah-wah [3] and phaser effects [4].
Despite having been replaced by their silicon counterpart in
most areas of electronics, vintage germanium BJTs (GBJTs) have
remained consistently popular in guitar pedals, particularly fuzz
effects. In previous work, two circuits featuring GBJTs have been
studied: the Fuzz-Face [5] and Rangemaster [6] guitar pedals, both
using the Ebers-Moll BJT model. However, each differs in how
the parameters are derived: the Fuzz Face using parameters ex-
tracted from a datasheet, and the Rangemaster using parameters
extracted through an optimisation procedure based on input/output
data. Comparisons between the output of the circuits and the mod-
els demonstrate a good fit in certain regions of operation, though
there are errors present in both models which remain unattributed.
The component with the most complex behaviour in each circuit
is the BJT which suggests it as a likely source of error.
A step further in complexity from the Ebers-Moll model is
a possible solution to this issue. Significant improvements have
been published, including: the Gummel-Poon model [7], the VBIC
model [8], and the MEXTRAM model [9]. These models have
not yet featured in the field of VA, leaving the question of what
difference they may make.
The aim of this paper is to provide an analysis of GBJTs within
audio circuits for the purpose of VA modelling. This analysis con-
sists of two primary sections: firstly the characterisation of models
based on measurements, followed by the analysis of each model
within the context of an audio circuit model. The analysed mod-
els include the Ebers-Moll model and models similar to Gummel-
Poon, considering both additional DC and AC effects. Procedures
for DC parameter extraction from measurements are discussed for
all models. We revisit the case studies already presented: the Dal-
las Rangemaster Treble Booster and the Dallas Arbiter Fuzz-Face.
In order to focus the analysis firmly on the differences between the
BJT models, comparisons are made only between circuit models,
as any separate circuit measurement would be subject to a range of
further system variances and errors.
The rest of the paper is structured as follows: Section 2 de-
scribes the compared BJT models, Section 3 discusses extraction
procedures for the DC parameters, Section 4 covers the case stud-
ies, the methodology, and results of the BJT model comparison,
and Section 5 concludes with suggestions for modellers working
on circuits featuring GBJTs.
This section describes the BJT models used in the analysis. Both
GBJTs that are studied are PNP, which is reflected in the descrip-
tion of the models. The difference with an NPN model is only in
notation, not behaviour.
In this work we define the term ‘external’ to refer to behaviour
modelled by additional components i.e. resistors and capacitors.
‘Internal’ will refer to the remaining terms, modelled as voltage
controlled current sources. This is illustrated by the differences
between Figure 1 (a) and (b), in which the BJT in (b) is mod-
elled by (a). External components values are modelled as being
independent of the BJT bias point i.e. constant. Combination of
the internal and external components will result in three models:
the Ebers-Moll model, a DC Gummel-Poon model, and an AC
Gummel-Poon (including capacitances).
Table 1 provides a reference for the name of each parameter in
each model. The effect of changes in each parameter value will be
discussed through the explanation of the extraction procedure in
Section 3. Only necessary discussion is included about each BJT
model; for a more comprehensive description see e.g. [10].
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
b b’
Figure 1: (a) Internal model schematic representation. (b) Com-
plete model using additional components.
2.1. Ebers-Moll
The Ebers-Moll model can be understood as modelling the two pn
junctions of the BJT as back to back diodes with coupling between
junctions. This can be expressed as
NfVt1), Ir=Is(e
NrVt1) (1)
Icc =IfIr, Ibe =1
If, Ibc =1
Ic=Icc Ibc, Ib=Ib e +Ibc (3)
where Icand Ibare the currents through the collector and base,
and Veb and Vcb are the voltages across the emitter- and collector-
base junctions. It is important to note that the remaining voltages
and currents can be found by Kirchoff’s circuit laws, i.e. Vec =
Veb Vcb and Ie=Ic+Ib.
Intermediate terms are used in the description of the model
to facilitate its extension and describe its behaviour in important
regions of operation. Ifand Irare forward and reverse currents
which can be considered independent when the controlling volt-
age of the other current is zero. The terms Icc ,Ibe and Ibc directly
describe the schematic representation of the model, illustrated in
Figure 1(a) as three current sources. Upon extension to the internal
Gummel-Poon this representation remains the same, only requir-
ing modification of the functions each current source represents.
2.2. Internal Gummel-Poon
To form the internal Gummel-Poon model several effects are added
which change the behaviour in both low and high current regions,
and also in response to changes in Vec. Terms defined in (1) hold
for the extended model whereas the intermediate current terms in
(2) are replaced. The internal Gummel-Poon is then expressed as
Ibe =1
NeVt1) (4)
Ibc =1
NcVt1) (5)
Icc =2
q1(1 + 1 + 4q2)(IfIr)(6)
Var Vcb
, q2=If
The new terms in Ibe and Ibc improve the modelling of the low
current behaviour often referred to as leakage. A dependence of
Icon Vec is introduced through q1where two parameters control
an increase to Icrelative to the positive and negative values of
Vec (consider the junction voltage that is dominant in each case).
Finally, a current gain roll-off is introduced to the model through
the inclusion of q2which reduces Icat high voltage values.
To reduce the internal Gummel-Poon to the Ebers-Mollmodel,
parameters must be set to specific values: Ise =Isc = 0 and
Vaf =Var =Ikf =Ikr =.
2.3. External components
Five additional components are included in this work: a resistance
at each terminal and a capacitance between both the base-emitter
junction and the base-collector junction. Figure 1(b) illustrates the
inclusion of these components to the internal BJT model.
More comprehensive models include two capacitances between
each junction [10], both dependent upon the voltage across the
junction; however, these are non-trivial to model using most VA
techniques. Because of this, and also to reduce the difficulty in
the measurement procedure, constant capacitance values were ex-
tracted from the datasheets of each GBJT.
DC parameter values for each model were extracted from mea-
surements of GBJTs. The extraction strategy consists of a direct
extraction followed by several targeted optimisations. This strat-
egy is based on existing work [12, 13], but both of these tools
are implemented in commercial systems which were unavailable.
Therefore a straightforward approach was developed to operate on
minimal measurements, with dedicated extraction procedures for
the Ebers-Moll and Gummel-Poon models.
3.1. Measurement Strategy
Three sets of measurements were performed on each BJT: forward
and reverse Gummel plots, which exposes the BJT behaviour in
forward and reverse bias conditions, and the common-emitter out-
put characteristic. Each measurement is designed to enable direct
extraction of certain parameters by exposing specific behaviour.
The circuits required for each measurement are illustrated in Fig-
ure 2. Table 2 contains the sourced currents and voltages for each
The common-emitter output characteristic is measured by spec-
ifying a base current, sweeping Vec, and measuring Ic. This is re-
peated for different values of Ibproviding several snapshots of the
relationship between Vec,Iband Ic.
The forward Gummel plot fixes Vec at a positive bias while Veb
is swept over a range and both Icand Ibare measured. This is sim-
ilar for the reverse Gummel plot, where Vec is biased negatively,
Vcb is swept, and Ieand Ibare measured. The applied method-
ology is described in [12], where Vec is biased between 2 V and
half of the maximum voltage supplied in the common-emitter out-
put characteristic. This creates a direct relationship between the
common-emitter characteristic at that value of Vec and the Gum-
mel plots, such that the voltages and currents should all be equal.
This approach therefore biases the BJT in the active regions which
provides confidence that the model will fit all measurements.
A Keithley 2602B Source Measure Unit was used to perform
the measurements, enabling simultaneous measurement and sourc-
ing of current and voltage. However, it should be noted that the
Gummel plots can be measured using only voltages by placing a
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Table 1: List of all parameters and extracted values from both the OC44 and AC128; constraints used in the intermediate optimisation
stages; initial values used for parameters that were not found through direct extraction.
Parameter Extracted Values Optim. Constraints Init. Values
OC44 AC128 Lower Lim. Upper Lim.
E.M. G.P. E.M. G.P.
IsSaturation current 2.029 µA 1.423 µA 23.75 µA 20.66 µA- - -
βfForward current gain 108.1 307.0 44.90 229.6 50 250 -
βrReverse current gain 8.602 20.27 4.568 14.66 3 20 -
NfForward ideality factor 1.062 1.022 1.168 1.133 - - -
NrReverse ideality factor 1.105 1.025 1.171 1.140 - - -
(Vt) Thermal voltage 25.5 mV 25.5 mV 25.5 mV 25.5 mV - - -
Vaf Forward Early voltage - 8.167 V -19.68 V - - -
Var Reverse Early voltage - 14.84 V -88.28 V - - -
Ikf Forward knee current (gain roll-off) - 43.82 mA -463.0 mA 10 µA 500 mA -
Ikr Reverse knee current (gain roll-off) - 611.7 mA -241.5 mA 10 µA 500 mA -
Ise BE junction leakage current - 30.54 nA -2.190 µA 0.1 fA 1 mA Is/2
Isc BC junction leakage current - 213.5 nA -7.546 µA 0.1 fA 1 mA Is/2
NeBE junction leakage emission coefficient - 1.316 - 1.796 0.5 4 2.3
NcBC junction leakage emission coefficient - 1.258 - 1.364 0.5 4 2.4
RbBase resistance - 32.83 Ω -1.885 Ω 1 Ω 250 Ω 25 Ω
ReEmitter resistance - 968.7 mΩ -306.4 mΩ 0.1 nΩ 2 Ω 10 mΩ
RcCollector resistance - 989.9µ-1.727 µ- - 10 mΩ
Ceb Emitter-base capacitance - 410pF - - - - -
Ccb Collector-base capacitance - 10 pF -100 pF - - -
Table 2: Ranges of the inputs to each measurement circuit. Specific
values of Ibare provided on each measurement plot.
Meaurement Input OC44 AC128
Forward Veb 0 - 0.7 V 0 - 0.8 V
Gummel Vec 2 V 2 V
Reverse Vcb 0 - 0.8 V 0 - 0.8 V
Gummel Vec 2 V 2 V
Common Ib3 - 50 µA26 - 1000 µA
Emitter Vec –5 - 5 V –5 - 5 V
shunt resistor over which to measure the voltage drop. This is im-
portant as it reduces the equipment required for measurements, and
as will be shown provides enough information to characterise the
Ebers-Moll model.
3.2. Direct Extraction
Direct extraction of parameters is used to provide initial estimates
upon which optimisation can then be performed. Estimates can be
made using simplifications as the optimisation performs the ma-
jority of the extraction. However, it is essential to start the optimi-
sation process in a position within the parameter space close to the
optimum to avoid local minima which may halt the optimisation
without providing the best model fit.
3.2.1. Ebers-Moll parameters
The Ebers-Moll parameters are extracted from the measured Gum-
mel plots. Figure 3 illustrates the effects of the forward parameters
and the saturation current Is. This behaviour is equivalent in the
reverse plot with the reverse parameters meaning that both regions
can be described by analysing only one, in this work the forward
region. To simplify the extraction procedure, the opposite current
term, in this case, Iris neglected, which is only valid if Vcb = 0.
As the measurement strategy actually enforces Vcb <0there will
be an error introduced into the direct extraction, but the error is
small, typically IrIs. Further, this error is removed during the
optimisation stages where there is no model simplification.
The thermal voltage Vtcan be found directly through measur-
ing the temperature of the room in which the measurement is taken.
This relies on the assumption that the measurements are taken in
such a way that avoids the BJT being heated by the current passing
through it, and that the BJT is allowed to settle at room tempera-
ture prior to measurement. Using the temperature in kelvin TK,
qwhere kis Boltzmann’s constant and qis the charge on
an electron.
Following this, the ideality factor Nfcan be found through
finding the gradient of the log of Ic. While the model shown in
Figure 3 is ideal, in measurements of BJTs the gradient of Icwill
not be constant so it is important to find a suitable point at which
to perform the extraction. One suitable method is to find the first
minimum of the absolute value of the second derivative of Ic. Ne-
glecting constant terms from Ifin (1) allows the formation of the
Rearranging this equation provides a value for Nf. A value for Is
can then be found at the same value of Veb, by solving the simpli-
fied expression of Icfor Is, i.e.
NfVt, Is= explog(Ic)Veb
Examining this equation for when Veb = 0 it is clear that Isis the
y-intercept of the Gummel-plot, as illustrated in Figure 3.
The extraction of βfrelies on the relationship between Icand
Ib, which from (1-3) can be expressed as Ic=βfIb. This re-
lationship is illustrated in Figure 4. As Veb approaches zero, Ib
decreases such that βfincreases very rapidly. This does not pro-
vide practical values of βfso values of βfbeneath the first turning
point with respect to Veb can be excluded. The maximum of the
βfis then used as the directly extracted parameter value.
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Figure 2: Measurement circuits for parameter extraction. Common-Emitter (left), Forward Gummel (middle), Reverse Gummel (right).
0 0.2 0.4 0.6 0.8 1
Figure 3: Example forward Gummel plot of the Ebers-Moll model
illustrating the effects of the forward parameters Nf,βfand also
Is. Currents Icand Ibare plotted logarithmically in the current
range against linear Veb.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
High current region
extracted βf
Figure 4: Example plot showing the relationship between Ic/Ib
and Veb. A nominal value of βfis extracted from the maximum
value, shown by the dashed line. High and low current effects
cause reduction of βf.
3.2.2. Gummel-Poon parameters
Values for Ik f and Ikr can be extracted from the current gain plots,
the forward of which is illustrated in Figure 4. The extracted values
for Ikf and Ikr are given by the value of Icand Ieat which the
current gain falls to half of its maximum value (respectively). If
necessary the curve can be extrapolated to find a value.
It was not found necessary to implement a direct extraction
technique for each parameter as initial values could be found through
manually tuning the leakage parameters and terminal resistances.
Due to the similarity of the GBJTs being modelled, several param-
eters could be initialised at the same value for each GBJT and pro-
vide a good starting point for the optimisation stage of the extrac-
tion. The values for each parameter initialised using this method
are shown in Table 1.
3.3. Optimisation Stages
After completing the direct extraction, each model’s fit to mea-
surements was optimised. Each optimisation was performed on a
specific input range, selected to emphasise the relevant behaviour
for each set of parameters being optimised. The ranges used can
be found in Table 3. Different strategies were required for charac-
terising the Ebers-Moll model and DC Gummel-Poon.
Two optimisation algorithms were used from MATLAB’s op-
timisation toolbox, fminsearch which uses the Nelder-Mead
simplex method [14], and fmincon, which uses the interior-point
method [15]. The Nelder-Mead simplex method is useful in this
scenario due to it’s ability to handle discontinuous surfaces. This
enabled the use of objective functions that would return an infi-
nite value if the parameters supplied were negative, preventing
non-physical parameter sets. Experimentally it was found that
this combination provided better convergence properties than us-
ing the interior-point method with a similar boundary. However,
the interior-point method was useful in scenarios in which more
complex boundaries were required to ensure the parameters re-
trieved would be a suitable starting point for the next simulation.
The final stage of characterising each model was performed with
the Nelder-Mead simplex method.
The same objective function was used for each optimisation
with the only change being the data compared. The objective value
is normalised with respect to both the number of data points and
the values of the data points. An example of this can be expressed
R(θ, y) = 1
where Ris the objective value, yis the measured data, ˆy(θ)is the
modelled data for a given parameter set θ, and Nis the number of
data points.
3.3.1. Ebers-Moll
Following the direct extraction, one optimisation stage was used in
the extraction of the Ebers-Moll parameters. This was performed
on the Gummel plots, using a low voltage range (see Table 3) to
match the first ‘ideal’ region in which the gradient of the collector
current is constant.
3.3.2. Gummel-Poon
The optimisation procedure for the DC Gummel-Poon model is il-
lustrated in Figure 5. After the direct extraction stage, three stages
of optimisations are used. The intermediate optimisation stages
use constraints implemented using the interior-point method. Con-
straints for the intermediate optimisations can be seen in Table 1.
The first optimisation stage works on the current gain of the
BJTs, given by e.g. Ic/Ibfor the forward case. This significantly
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Direct Extraction
Optimise on Gain
Optimise on
Optimise on
Gummel and
Measurements Parameters
No optimisation
Interior-point method
Nelder-Mead simplex method
Forward and
reverse Gummel,
Forward and
reverse Gummel
Forward and
reverse Gummel
Forward and
reverse Gummel,
Is, Nf, Nr, βf,
βr, Ikf, Ikr, Vaf,
βf, βr, Ikf, Ikr,
Rb, Re, Ise, Isc,
Ne, Nc
Ikf, Ikr, Rb, Re
All parameters
Figure 5: Optimisation strategy used to extract the parameters of
the Gummel-Poon model. The arrow indicates the flow of stages of
extraction. The resultant parameters of each stage are passed on
to the next.
reduces the effects of Is,Vaf ,Var,Rc,Nf,Nr, reducing the pa-
rameter set and thus the number of dimensions in the optimisation.
A second stage was used to further tune a subset of these param-
eters to the Gummel plots. Finally, all parameters were optimised
against all of the data sets. A weighting was applied to the objec-
tive function in the final optimisation, making the objective value
of the common-emitter characteristic 10×higher than that of the
Gummel plots.
3.4. Results
Multiple GBJTs were used in the comparison: 4 AC128 and 3
OC44 BJTs. Figures 6 and 7 show the results of the best fit of
the DC Gummel-Poon to the measurements as determined by the
objective function of the final optimisation. As these models had
the best fit to the data, they were used in the comparison in the case
studies. See Table 1 for the parameters of each model.
Thermal effects were noticed in the measurements despite con-
siderations to reduce the effects. Post processing was used to re-
duce the observable effect of these; however there remains a possi-
bility that some thermal error remains in the measurements which
would affect the extracted parameters.
In the high-current region of the Gummel-plots the model de-
viates from the measurements. During the measurement stage, a
current limit was enforced to prevent damage to components or
the measurement system, set slightly above the maximum current
specified by the GBJT datasheets. This limited the amount of high-
current data that could be gathered. Further, the common-emitter
plots were taken at low currents to ensure they were close to that
of the ‘ideal’ region of operation, which reduces the amount of
data about the high-current region. If the measurements were to
Table 3: Voltage ranges upon which each optimisation for both
models were performed. Gummel plots were used in both the
penultimate and ultimate stages for the Gummel-Poon model, and
are labelled 1 and 2 to differentiate.
Model Measurement Input Lower
Veb, Vcb 10 mV 200 mV
Gummel- Current gain Veb, Vcb 110 mV 700 mV
Poon Gummel
plots 1
Veb, Vcb 100 mV 700 mV
plots 2
Veb, Vcb 50 mV 600 mV
Vec 5 V 5 V
be repeated it might be sensible to increase the base currents in the
common-emitter characteristic, and consider increasing the current
The schematics of the Dallas Rangemaster and Dallas Arbiter Fuzz-
Face can be found in Figures 8 and 9 respectively. These case
studies were selected because each biases the BJT in different re-
gions, exhibiting different behaviour. Three tests were performed
on each case study: informal listening tests, waveform compar-
isons, and a comparison of the computational requirements. Each
circuit was modelled using the Nodal DK-method (for reference
see e.g. [3]) although it should be noted that the use of a different
simulation technique would yield similar results for both audio and
waveform comparisons. Computation time would however require
evaluation for different simulation techniques. For each test all po-
tentiometers of both case studies were set to maximum. Other po-
tentiometer settings were tested but are omitted as they illustrate
no substantial difference from those presented.
4.1. Informal listening tests
Several guitar signals were processed by both case studies at 8×
oversampling as a means of comparing each model. Listeners
agreed that differences could be heard between each model, with
the Ebers-Moll model having the most high frequency content due
to distortion and the AC Gummel-Poon having the least. Audio
examples can be found on the first author’s website1.
4.2. Waveform comparison
An objective comparison of each BJT model is achieved here using
time-domain waveforms. Sinusoids at different frequencies and
amplitudes were processed by both case studies and each model.
To remove transient behaviour from the results, the final period
of each of these signals are shown in Figure 10 and 11. Plots at
1200 Hz show the largest difference for the AC effects, illustrat-
ing the low-pass type behaviour of the capacitances. Differences
due to the increased DC complexity are most prominent at lower
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Figure 6: Forward and reverse Gummel plots and the common-emitter characteristic of the OC44 BJT.
0 1 2 3 4 5
Ib = 75, 501, 750, 1000 μA
-5 -4 -3 -2 -1 0
0.1 0.2 0.3 0.4 0.5
Current (A)
Current Gain
0.2 0.4 0.6
Current (A)
Current Gain
Meas. Ib
Meas. r
Meas. Ib
Meas. f
Vec (V)
Figure 7: Forward and reverse Gummel plots and the common-emitter characteristic of the AC128 BJT.
9 V
1 kΩ
4.7 nF
68 kΩ
470 kΩ
10 kΩ
3.9 kΩ
47 µF
10 nF
1 MΩ
Figure 8: Schematic of the Rangemaster circuit.
9 V
1 kΩ
6.8 nF
36 kΩ
62 kΩ
11 kΩ
680 kΩ
22 µF
1 kΩ
500 kΩ
Figure 9: Schematic of the Fuzz Face circuit.
Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
Table 4: Simulation time required to process one second of signal,
average iterations per sample, and sub-iterations per sample of
circuit models processing a guitar chord using different BJT mod-
els. The Rangemaster was tested over a peak voltage range of
0.12 V, the Fuzz-Face over a range of 10 100 mV.
Model Rangemaster Fuzz-Face
DC E.M. 95.9 3.53/0.20 341.6 3.62/0.04
AC E.M. 75.8 3.52/0.13 376.5 3.60/0.01
DC G.P. 371.8 3.47/0.07 819.1 3.04/0.03
AC G.P. 357.0 3.45/0.05 769.9 2.99/0.01
4.3. Computational efficiency
To understand the cost of increasing the complexity of the BJT
model, computational requirements of each model were compared.
The nonlinear equation of the circuit models was solved using
damped Newton’s method as described in [16], which uses an in-
ner iterative loop to aid in convergence. This provides three met-
rics: time needed for one second of simulation, average iterations,
and average sub-iterations.
A fourth model was included for this test: the Ebers-Moll
model with Ceb and Ccb (AC Ebers-Moll) to provide an improved
assessment of the cost of the capacitances. A guitar signal was
processed by both case studies and each BJT model, with the peak
amplitude of the signal set to 20 different levels. Computation time
was then measured by MATLAB’s tic/toc stopwatch functions.
The results are shown in Table 4. It is clear from the results that
increasing the DC complexity causes a significant increase in com-
putation time, whereas including additional capacitances carries
little cost. As iterations and sub-iterations decrease with increas-
ing model complexity the increase in computation must be due to
the increased complexity of evaluating the model equations. De-
crease in computation cost when including the capacitances can be
attributed to the reduction in high frequencies reducing the stress
placed on the iterative solver, outweighing the increase in the com-
plexity from including additional components.
A comparison of BJT models has been presented with a focus on
GBJTs. Each model was characterised by extracting parameters
from measured data using a multi-step optimisation strategy. The
resultant models were compared through the use of two case study
circuits covering both moderately and highly distorted circuit out-
puts. The circuit models were compared using three metrics: au-
dible and waveform differences, and computational efficiency.
Results show that increase in model complexity does make a
change to the behaviour of GBJTs in audio circuit models. This
work has primarily focused on improving DC characterisation;
however, the results show that AC effects are at least equally im-
portant. The improved DC characterisation has a significant in-
crease in computational cost whereas the cost of the AC effects
are minimal. These results indicate that any first extension to the
Ebers-Moll model should be AC effects, and further extensions
should then concern DC effects.
The core motivating factor for implementing and characteris-
ing more sophisticated BJT models was to reduce the error present
in VA circuits featuring GBJTs. A fitting next step is to now use
these models in conjunction with the design of models based on
specific circuits. An implementation of the Gummel-Poon model
has been included in ACME.jl2emulation tool for modellers inter-
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Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017
1.998 1.9985 1.999 1.9995 2
Vout (V)
(a) A = 0.1 V, F = 500 Hz
DC Gummel-Poon
AC Gummel-Poon
1.9992 1.9994 1.9996 1.9998 2
Vout (V)
(b) A = 0.1 V, F = 1200 Hz
1.995 1.996 1.997 1.998 1.999 2
Vout (V)
(c) A = 0.5 V, F = 200 H z
1.998 1.9985 1.999 1.9995 2
Vout (V)
(d) A = 0.5 V, F = 500 Hz
1.9992 1.9994 1.9996 1.9998 2
Vout (V)
(e) A = 0.5 V, F = 1200 H z
1.995 1.996 1.997 1.998 1.999 2
Time (s)
Vout (V)
(f) A = 1 V, F = 200 H z
1.998 1.9985 1.999 1.9995 2
Time (s)
Vout (V)
(g) A = 1 V, F = 500 H z
1.9992 1.9994 1.9996 1.9998 2
Time (s)
Vout (V)
(h) A = 1 V, F = 1200 Hz
Figure 10: Single cycles of the Rangemaster output’s response to a sinusoidal input signal.
1.998 1.9985 1.999 1.9995 2
Vout (V)
(a) A = 1 m V, F = 500 Hz
DC Gummel-Poon
AC Gummel-Poon
1.9992 1.9994 1.9996 1.9998 2
Vout (V)
(b) A = 1 m V, F = 1200 Hz
1.995 1.996 1.997 1.998 1.999 2
Vout (V)
(c) A = 10 mV, F = 200 Hz
1.998 1.9985 1.999 1.9995 2
Vout (V)
(d) A = 10 m V, F = 500 Hz
1.9992 1.9994 1.9996 1.9998 2
Vout (V)
(e) A = 10 mV, F = 1200 Hz
1.995 1.996 1.997 1.998 1.999 2
Time (s)
Vout (V)
(f) A = 100 mV, F = 200 Hz
1.998 1.9985 1.999 1.9995 2
Time (s)
Vout (V)
(g) A = 100 mV, F = 500 Hz
1.9992 1.9994 1.9996 1.9998 2
Time (s)
Vout (V)
(h) A = 100 m V, F = 1200 Hz
Figure 11: Single cycles of the Fuzz Face output’s response to a sinusoidal input signal. The fuzz control of the circuit was set to maximum.
... These transistors are less reliable than their silicon counterparts but are preferred by guitarists and pedal builders as they produce a smoother, warmer distortion. The Rangemaster (or similar) has been widely studied in existing VA literature [19], [34]- [37], but to the author's knowledge, never in the context of antialiasing. The transistor is a 2-port nonlinearity. ...
... The nonlinear behaviour of BJT transistors can be described via the Ebers-Moll model, which relates the base-emitter voltage, v be , and base-collector voltage v bc to the current at the base, i b , and the current at the collector, i c [34], [37]: ...
... where I s is the saturation current, β f is the forward current gain, β r is the reverse current gain and V t is the thermal voltage. The values used for these parameters are the same as those in [34], in which they were derived from direct measurements. ...
Full-text available
Aliasing is an inherent problem in virtual analogue modelling when simulating nonlinear systems such as guitar amplifiers and distortion effects units. Such systems introduce harmonics into the signal, which in the discrete-time domain can exceed the Nyquist frequency, resulting in unpleasant aliasing distortion. Recent research has shown that aliasing can be significantly reduced by using the antiderivatives of the nonlinear function, and that this method can be applied to systems with state as well as memoryless nonlinearities. In this work, the application of antiderivative antialiasing in the state-space modelling of several nonlinear circuits will be outlined in detail. Existing literature has focused on one-port nonlinearities, so in this work a method for two-port nonlinearities is proposed and demonstrated by example. Furthermore, a second order antialiasing method for state-space models is presented. The antialiasing methods were found to significantly improve the signal to noise ratio and reduce aliasing at low oversampling rates. In the case of scalar nonlinearities, the methods introduced no notable extra computational cost, but for two-port nonlinearities the processing time increased with the order of antialiasing. Finally, the suitability of antiderivative antialiasing in a real-time context was demonstrated through the development of a virtual analogue guitar effects plug-in.
... Several Ph.D. theses have been written about this topic [102][103][104]. Further investigations, case studies and refinements of the method have been made by Holters et al. [42][43][44][45][46]55,65,69,71,76,85,105,106]. All of these works have in common that a state-space system is constructed from the circuit diagram using some form of circuit analysis. ...
... Additionally Holters proposed a method to automatically decompose the large statespace matrices into smaller ones to be able to solve the system quicker [105]. Several other publications address the computational load of the whitebox approaches and propose improvements to ensure real-time capability like [42,49,80,85,106,113,117]. A complete guitar amplifier circuit could be modeled with these approaches but without extensive optimization and simplification the digital model would not be real-time capable. ...
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Digital systems gain more and more popularity in todays music industry. Musicians and producers are using digital systems because of their advantages over analog electronics. They require less physical space, are cheaper to produce and are not prone to aging circuit components or temperature variations. Furthermore, they always produce the same output signal for a defined input sequence. However, musicians like vintage equipment. Old guitar amplifiers or legendary recording equipment are sold at very high prices. Therefore, it is desirable to create digital models of analog music electronics which can be used in modern digital environments. This work presents an approach for recreating nonlinear audio circuits using system identification techniques. Measurements of the input- and output-signals from the analog reference devices are used to adjust a digital model treating the reference device as a ‘black-box’. With this technique the schematic of the reference device does not need to be known and no circuit elements have to be measured to recreate the analog device. An appropriate block-based model is chosen, depending on the type of reference system. Then the parameters of the digital model are adjusted with an optimization method according to the measured input- and output-signals. The performance of the optimized digital model is evaluated with objective scores and listening tests. Two types of nonlinear reference systems are examined in this work. The first type of reference systems are dynamic range compressors like the ‘MXR Dynacomp’, the ‘Aguilar TLC’, or the ‘UREI 1176LN’. A block-based model describing a generic dynamic range compression system is chosen and an automated routine is developed to adjust it. The adapted digital models are evaluated with objective scores and a listening test is performed for the UREI 1176LN studio compressor. The second type of nonlinear systems are distortion systems like e.g. amplifiers for electric guitars. This work presents novel modeling approaches for different kinds of distortion systems from basic distortion circuits which can be found in distortion pedals for guitars to (vintage) guitar amplifiers like the ‘Marshall JCM900’, or the ‘Fender Bassman’. The linear blocks of the digital model are measured and used in the model while the nonlinear blocks are adapted with parameter optimization methods like the Levenberg–Marquardt method. The quality of the adjusted models is evaluated with objective scores and listening tests. The adjusted digital models give convincing results and can be implemented as real-time digital versions of their analog counterparts. This enables the musician to safe a snapshot of a certain sound and recall it anytime with a digital system like a VST plug-in or as a program on a dedicated hardware.
... The physical parameter values for the Mullard OC44 presented inTable 3.3 were extracted by Holmes at al. using measurements and optimization methods[29]. By using other parameter values in the Ebers-Moll equations, the Rangemaster can be simulated with a variety of different transistors, each of which provides a unique sonic character. ...
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This paper presents in detail the state-space approach to virtual analog modeling. A variety of different numerical methods are derived and implemented so that their performance may be compared. Four different guitar effects circuits are analyzed and simulated, and the detailed MATLAB code for each is provided.
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This work addresses the real-time simulation of nonlinear audio circuits. In this thesis, we use the port-Hamiltonian (pH) formalism to guarantee power balance and passivity. Moreover, we adopt a continuous-time functional framework to represent "virtual analog" signals and propose to approximate solutions by projection over time frames. As a main result, we establish a sufficient condition on projectors to obtain time-continuous power-balanced trajectories. Our goal is twofold: first, to manage frequency-bandwidth expansion due to nonlinearities, we consider numerical engines processing signals that are not bandlimited but, instead, have a "finite rate of innovation"; second, to get back to the bandlimited domain, we design "virtual analog-to-digital converters". Several numerical methods are built to be power-balanced, high-order accurate, with a controllable regularity order. Their properties are studied: existence and uniqueness, accuracy order and dispersion, but also, frequency resolution beyond the Nyquist frequency, aliasing rejection, reproducing and Peano kernels. This approach reveals bridges between numerical analysis, signal processing and generalised sampling theory, by relating accuracy, polynomial reproduction, bandwidth, Legendre filterbanks, etc. A systematic framework to transform schematics into equations and simulations is detailed. It is applied to representative audio circuits (for the UVI company), featuring both ordinary and differential-algebraic equations. Special work is devoted to pH modelling of operational amplifiers. Finally, we revisit pH modelling within the framework of Geometric Algebra, opening perspectives for structure encoding.
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In this work we explore optimising parameters of a physical circuit model relative to input/output measurements, using the Dallas Rangemaster Treble Booster as a case study. A hybrid meta-heuristic/gradient descent algorithm is implemented, where the initial parameter sets for the optimisation are informed by nominal values from schematics and datasheets. Sensitivity analysis is used to screen parameters, which informs a study of the optimisation algorithm against model complexity by fixing parameters. The results of the optimisation show a significant increase in the accuracy of model behaviour, but also highlight several key issues regarding the recovery of parameters.
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Iterative solvers are required for the discrete-time simulation of nonlinear behaviour in analogue distortion circuits. Unfortunately, these methods are often computationally too expensive for real-time simulation. Two methods are presented which attempt to reduce the expense of iterative solvers. This is achieved by applying information that is derived from the specific form of the nonlin-earity. The approach is first explained through the modelling of an asymmetrical diode clipper, and further exemplified by application to the Dallas Rangemaster Treble Booster guitar pedal, which provides an initial perspective of the performance on systems with multiple nonlinearities.
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In this study, a famous boxed effect pedal, also called stompbox, for electrical guitars is analyzed and simulated. The nodal DK method is used to create a non-linear state-space system with Matlab as a physical model for the MXR Phase 90 guitar ef-fect pedal. A crucial component of the effect are Junction Field Effect Transistors (JFETs) which are used as variable resistors to dynamically vary the phase-shift characteristic of an allpass-filter cascade. So far, virtual analog modeling in the context of audio has mainly been applied to diode-clippers and vacuum tube cir-cuits. This work shows an efficient way of describing the non-linear behavior of JFETs, which are wide-spread in audio devices. To demonstrate the applicability of the proposed physical model, a real-time VST audio plug-in was implemented.
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The nodal DK method is a systematic way to derive a non-linear state-space system as a physical model for an electrical circuit. Unfortunately, calculating the system coefficients requires inversion of a relatively large matrix. This becomes a problem when the system changes over time, requiring continuous recomputation of the coefficients. In this paper, we present an extension of the DK method to more efficiently handle variable circuit elements. The method is exemplified with the Dunlop Crybaby wah-wah effect pedal, as the continuous change of the potentiometer position is an extremely important aspect of the wah-wah effect.
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An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented.
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The Nelder{Mead simplex algorithm, rst published in 1965, is an enormously pop- ular direct search method,for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder{Mead algorithm. This paper presents convergence properties of the Nelder{Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder{Mead algo- rithm converges to a nonminimizer. It is not yet known,whether the Nelder{Mead method,can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions. Key words. direct search methods, Nelder{Mead simplex methods, nonderivative optimization AMS subject classications. 49D30, 65K05 PII. S1052623496303470
Most electric guitar players make use of different amplifiers in combination with auxiliary effect units. These electronic or electro-acoustic devices affect the instrument's tone in a complex manner and give the play a unique sound. The term "amp modeling" refers to signal processing algorithms that intend to imitate the behavior of these devices as true to original as possible - a topic that can be found both in commercial products and academic research. In this paper the simulation of the "Fuzz-Face" is presented, a popular nonlinear analog circuit from the 1960s that distorts the guitar signal with germanium transisors. The presented modeling is based on a state-space description with minimal system order. After performing the system analysis in continuous time domain, a trapezoidal rule discretization is executed. The simulation results are compared to a reference and show good match. The processing in real-time, the computational complexity, and the quality of the sound reproduction are discussed.
We present in this paper a compact model of bipolar transistors, suitable for network analysis computer programs. Through the use of a new charge control relation linking junction voltages, collector current, and base charge, the model includes high injection effects. The performance substantially exceeds that of existing models of comparable complexity. For low bias and with some additional idealization, the model reduces to the conventional Ebers-Moll model.
To specify the second-level nonlinear Ebers-Moll model for the bipolar transistor, 12 parameters must be measured, in addition to the five parameters needed for the first-level Ebers-Moll model. Besides a curve tracer and a thermometer, which is the minimal test equipment for measuring the first-level parameters, the second-level parameters require at least a capacitance bridge, a pulse generator, a high-speed oscilloscope, and a small-signal measurement system such as an S-parameter setup. A power supply is also required.
From the Publisher:This new book, written by Andre Vladimirescu, who was instrumental in the development of SPICE at the University of California Berkeley, introduces computer simulation of electrical and electronics circuits based on the SPICE standard. Relying on the functionality first supported in SPICE2 that is now supported in all SPICE programs, this text is addressed to all users of electrical simulation. The approach to learning circuit simulation is to interpret simulation results in relation to electrical engineering fundamentals; the book asks the student to solve most circuit examples by hand before verifying the results with SPICE. Addressed to both the SPICE novice and the experienced user, the first six chapters provide the relevant information on SPICE functionality for the analysis of linear as well as nonlinear circuits. Each of these chapters starts out with a linear example accessible to any new user of SPICE and proceeds with nonlinear transistor circuits. The latter part of the book goes into more detail on such issues as functional and hierarchical models, distortion analysis, basic algorithms in SPICE and related options parameters, and, how to direct SPICE to find a solution when it does not converge to a solution. The approach emphasizes that SPICE is not a substitute for knowledge of circuit operation but a complement. The SPICE Book is different from previously published books in the approach of solving circuit problems with a computer. The solution to most circuit examples is sketched out by hand first and followed by a SPICE verification. For more complex circuits it is not feasible to find the solution by hand but the approach stresses the need for the SPICE user to understand the results. Readers gain a better comprehension of SPICE thanks to the importance placed on the relation between EE fundamentals and computer simulation. The tutorial approach advances from the hand solution of a circuit to SPICE verification and simulation results interpret