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Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017

COMPARISON OF GERMANIUM BIPOLAR JUNCTION TRANSISTOR MODELS FOR

REAL-TIME CIRCUIT SIMULATION

Ben Holmes

Sonic Arts Research Centre,

School of Electronics, Electrical Engineering

and Computer Science

Belfast, U.K

bholmes02@qub.ac.uk

Martin Holters

Department for Signal Processing and

Communications,

Helmut Schmidt University

Hamburg, Germany

martin.holters@hsu-hh.de

Maarten van Walstijn

Sonic Arts Research Centre,

School of Electronics, Electrical Engineering

and Computer Science

Belfast, U.K

m.vanwalstijn@qub.ac.uk

ABSTRACT

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.

1. INTRODUCTION

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Despite having been replaced by their silicon counterpart in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.

2. BJT MODELS

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].

DAFX-152

Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017

c

Ibc

b

e

Ibe

Icc

Rb

b b’

c’

Rc

c

e’

Re

e

Ccb

Ceb

(b)(a)

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

If=Is(e

Veb

NfVt−1), Ir=Is(e

Vcb

NrVt−1) (1)

Icc =If−Ir, Ibe =1

βf

If, Ibc =1

βr

Ir(2)

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

[11]

Ibe =1

βf

If+Ise(e

Veb

NeVt−1) (4)

Ibc =1

βr

Ir+Isc(e

Vcb

NcVt−1) (5)

Icc =2

q1(1 + √1 + 4q2)(If−Ir)(6)

where

q1=1

1−Veb

Var −Vcb

Vaf

, q2=If

Ikf

+Ir

Ikr

.(7)

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.

3. PARAMETER EXTRACTION

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

measurement.

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

DAFX-153

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 - 989.9µΩ-1.727 µΩ- - 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 Ir≤Is. 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,

Vt=TKk

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

expression

dlog(Ic)

dVeb

=1

NfVt

.(8)

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.

Ic=IsexpVeb

NfVt, Is= explog(Ic)−Veb

NfVt.(9)

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.

DAFX-154

Ib

+

−Vec

+

−Veb

+

−Vec

+

−Vcb

+

−

Vec

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

Veb

10-10

10-5

100

10

Is

5

Current

Ic

Ib

NfVt

1

βf

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

Veb

50

100

150

Ic/Ib

High current region

Low

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

by

R(θ, y) = 1

N

N

X

n=1y[n]−ˆy(θ)[n]

y[n]2

(10)

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

DAFX-155

Direct Extraction

Optimise on Gain

Optimise on

Gummel

Optimise on

Gummel and

common-emitter

characteristic

Measurements Parameters

No optimisation

Interior-point method

Nelder-Mead simplex method

Forward and

reverse Gummel,

common-emitter

characteristic

Forward and

reverse Gummel

plots

Forward and

reverse Gummel

plots

Forward and

reverse Gummel,

common-emitter

characteristic

Is, Nf, Nr, βf,

βr, Ikf, Ikr, Vaf,

Var

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

limit

Upper

limit

Ebers-

Moll

Gummel

plots

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

Gummel

plots 2

Veb, Vcb 50 mV 600 mV

Common-

emitter

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

limit.

4. CASE STUDIES

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

amplitudes.

1http://bholmesqub.github.io/DAFx17/

DAFX-156

0 2 4

0

5

10

Ic

Sim.

Meas.

-5 -4 -3 -2 -1 0

-0.5

0

0.2 0.4 0.6

10-6

10-4

10-2

Current (A)

0

50

100

150

200

Current Gain

0.2 0.4 0.6

10-8

10-6

10-4

10-2

Current (A)

0

5

10

15

Current Gain

Ib

Meas.

Meas. Ib

f

Meas. f

Mod.

Mod.

Mod.

Ic

Ic

Ib = 8, 25, 38, 50 μA

(mA)

Ic(mA)

Vec (V)

Vec (V)

Vcb (V)

Veb (V)

Figure 6: Forward and reverse Gummel plots and the common-emitter characteristic of the OC44 BJT.

0 1 2 3 4 5

Vec

0

50

100

Ic

Ib = 75, 501, 750, 1000 μA

Mod.

Meas.

-5 -4 -3 -2 -1 0

-5

0

0.1 0.2 0.3 0.4 0.5

Veb

10-6

10-4

10-2

Current (A)

0

20

40

60

80

100

120

Current Gain

0.2 0.4 0.6

Vcb

10-6

10-4

10-2

Current (A)

0

1

2

3

4

5

6

Current Gain

Ib

Meas.

Meas. Ib

r

Meas. r

Mod.

Mod.

Mod.

Ib

Meas.

Meas. Ib

f

Meas. f

Mod.

Mod.

Mod.

Ie

Ie

Ic

Ic

(mA)

(V)

Ic(mA)

Vec (V)

(V)(V)

Figure 7: Forward and reverse Gummel plots and the common-emitter characteristic of the AC128 BJT.

+

−

Vcc

9 V

+

−

Vin

Rin

1 kΩ

C1

4.7 nF

R2

68 kΩ

R1

470 kΩ

Vol

10 kΩ

R3

3.9 kΩ

C2

47 µF

C3

10 nF

R4

1 MΩ

Figure 8: Schematic of the Rangemaster circuit.

+

−

Vcc

9 V

+

−

Vin

Rin

1 kΩ

C5

6.8 nF

C3

2.2µF

R4

36 kΩ

R3

62 kΩ

R1

11 kΩ

R2

680 kΩ

C2

0.1µF

C1

22 µF

Fuzz

1 kΩ

Vol

500 kΩ

Figure 9: Schematic of the Fuzz Face circuit.

DAFX-157

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

0.1−2 V, the Fuzz-Face over a range of 10 −100 mV.

Model Rangemaster Fuzz-Face

Sim.

time/s

(ms)

Mean

Iter./Sub-

iter.

Sim.

time/s

(ms)

Mean

Iter./Sub-

iter.

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.

5. CONCLUSION

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-

ested in further investigation.

6. REFERENCES

[1] A. Huovilainen, “Non-linear digital implementation of the Moog

ladder ﬁlter,” in Proc. of the International Conference on Digital

Audio Effects (DAFx-04), Naples, Italy, 2004, pp. 61–64.

[2] J. J. Ebers and J. L. Moll, “Large-signal behavior of junction tran-

sistors,” Proceedings of the IRE, vol. 42, no. 12, pp. 1761–1772,

1954.

[3] M. Holters and U. Zölzer, “Physical Modelling of a Wah-Wah Pedal

as a Case Study for Application of the Nodal DK Method to Circuits

with Variable Parts,” in Proc. of the 14th Internation Conference on

Digital Audio Effects, Paris, France, Sept. 2011, pp. 31–35.

[4] F. Eichas, M. Fink, M. Holters, and U. Zölzer, “Physical Modeling

of the MXR Phase 90 Guitar Effect Pedal,” in Proc. of the 17 th

Int. Conference on Digital Audio Effects, Erlangen, Germany, Sept.

2014, pp. 153–156.

[5] K. Dempwolf and U. Zölzer, “Discrete State-Space Model of the

Fuzz-Face,” in Proceedings of Forum Acusticum, Aalborg, Den-

mark, June 2011, European Acoustics Association.

[6] B. Holmes and M. van Walstijn, “Physical model parameter opti-

misation for calibrated emulation of the Dallas Rangemaster Treble

Booster guitar pedal,” in Proc. of the 19th Internation Conference on

Digital Audio Effects, Brno, Czech Republic, Sept. 2016, pp. 47–54.

[7] H. K. Gummel and H. C. Poon, “An integral charge control model

of bipolar transistors,” Bell System Technical Journal, vol. 49, no. 5,

pp. 827–852, 1970.

[8] C. C. McAndrew, J. A. Seitchik, D. F. Bowers, M. Dunn, M. Foisy,

I. Getreu, M. McSwain, S. Moinian, J. Parker, D. J. Roulston, and

others, “VBIC95, the vertical bipolar inter-company model,” IEEE

Journal of Solid-State Circuits, vol. 31, no. 10, pp. 1476–1483, 1996.

[9] R. Van der Toorn, J. C. J. Paasschens, and W. J. Kloosterman, “The

Mextram bipolar transistor model,” Delft University of Technology,

Technical report, 2008.

[10] I. Getreu, Modeling the Bipolar Transistor, Tektronix, 1976.

[11] A. Vladimirescu, The SPICE book, J. Wiley, New York, 1994.

[12] F. Sischka, “Gummel-Poon Bipolar Model: Model description, pa-

rameter extraction.,” Agilent Technologies, 2001.

[13] J. A. Seitchik, C. F. Machala, and P. Yang, “The determination

of SPICE Gummel-Poon parameters by a merged optimization-

extraction technique,” in Proc. of the 1989 Bipolar Circuits and

Technology Meeting. 1989, pp. 275–278, IEEE.

[14] J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Conver-

gence properties of the Nelder–Mead simplex method in low dimen-

sions,” SIAM Journal on optimization, vol. 9, no. 1, pp. 112–147,

1998.

[15] R. H. Byrd, J. C. Gilbert, and J. Nocedal, “A Trust Region Method

Based on Interior Point Techniques for Nonlinear Programming,”

Mathematical Programming, vol. 89, no. 1, pp. 149–185, 2000.

[16] B. Holmes and M. van Walstijn, “Improving the robustness of the it-

erative solver in state-space modelling of guitar distortion circuitry,”

in Proc. of the 18th International Conference on Digital Audio Ef-

fects, Trondheim, Norway, Dec. 2015, pp. 49–56.

2https://github.com/HSU-ANT/ACME.jl

DAFX-158

1.998 1.9985 1.999 1.9995 2

-1

0

1

Vout (V)

(a) A = 0.1 V, F = 500 Hz

Input

Ebers-Moll

DC Gummel-Poon

AC Gummel-Poon

1.9992 1.9994 1.9996 1.9998 2

-2

0

2

Vout (V)

(b) A = 0.1 V, F = 1200 Hz

1.995 1.996 1.997 1.998 1.999 2

-2

0

2

Vout (V)

(c) A = 0.5 V, F = 200 H z

1.998 1.9985 1.999 1.9995 2

-2

0

2

4

6

Vout (V)

(d) A = 0.5 V, F = 500 Hz

1.9992 1.9994 1.9996 1.9998 2

-2

0

2

4

6

Vout (V)

(e) A = 0.5 V, F = 1200 H z

1.995 1.996 1.997 1.998 1.999 2

Time (s)

-2

0

2

4

Vout (V)

(f) A = 1 V, F = 200 H z

1.998 1.9985 1.999 1.9995 2

Time (s)

-2

0

2

4

6

Vout (V)

(g) A = 1 V, F = 500 H z

1.9992 1.9994 1.9996 1.9998 2

Time (s)

-2

0

2

4

6

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

-0.05

0

0.05

Vout (V)

(a) A = 1 m V, F = 500 Hz

Input

Ebers-Moll

DC Gummel-Poon

AC Gummel-Poon

1.9992 1.9994 1.9996 1.9998 2

-0.05

0

0.05

Vout (V)

(b) A = 1 m V, F = 1200 Hz

1.995 1.996 1.997 1.998 1.999 2

-0.2

0

0.2

0.4

Vout (V)

(c) A = 10 mV, F = 200 Hz

1.998 1.9985 1.999 1.9995 2

-0.2

0

0.2

0.4

Vout (V)

(d) A = 10 m V, F = 500 Hz

1.9992 1.9994 1.9996 1.9998 2

-0.2

0

0.2

0.4

Vout (V)

(e) A = 10 mV, F = 1200 Hz

1.995 1.996 1.997 1.998 1.999 2

Time (s)

-0.2

0

0.2

Vout (V)

(f) A = 100 mV, F = 200 Hz

1.998 1.9985 1.999 1.9995 2

Time (s)

-0.2

0

0.2

Vout (V)

(g) A = 100 mV, F = 500 Hz

1.9992 1.9994 1.9996 1.9998 2

Time (s)

-0.2

0

0.2

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.

DAFX-159