“On the engineers’ new toolbox”
or
Analog Circuit Design, using Symbolic Analysis, Computer Algebra, and Elementary
Network Transformations
(2011 Re-Release)

arXiv:1101.0270v1 [cs.SC] 31 Dec 2010

Eberhard H.-A. Gerbracht

Abstract—In this paper, by way of three examples – a fourth
order low pass active RC filter, a rudimentary BJT amplifier,
and an LC ladder – we show, how the algebraic capabilities of
modern computer algebra systems can, or in the last example,
might be brought to use in the task of designing analog circuits.
ACM Classification: I.1 Symbolic and algebraic manipulation; J.2
Physical sciences and engineering; G.2.2 Graph theory
Mathematics Subject Classification (2000): Primary 94C05; Secondary 94C15, 68W30, 13P10, 05C85
Keywords: analog circuits, filter design, polynomial equations, computer algebra, delta-wye transformation.

I. I NTRODUCTION
For those about to embark on the study of this paper,
let us start with a warning: This article is not a report on
ground-breaking results. Neither will we present spanking
new circuits, ripe for patent. Rather in spirit of SM2 ACD’s
explicit policy to be “a forum for the discussion of new
ideas and methodologies”, we would like to advocate the
(re)introduction1 of computer algebra into the design process
of analog circuits. In order to do this, we present some
examples, where computer algebra (together with numerics)
was used in the design, especially the sizing of the given
circuits.
These examples do not represent a systematic approach, but
they should be considered as tentative steps towards getting
to know the capabilities (and some of the shortcomings) of
computer algebra systems (CAS), and making these tools bring
forth fruit in the daily work of electrical engineers.
II. T HE GENERAL ANSATZ
One of the principal problems in analog circuit design2 is
to determine element values in a given circuit in such a
way that the resulting network function is in accordance
with some previously given performance specifications. In
This article first appeared in: SM2 ACD’08 Proceedings of the Xth International Workshop on Symbolic and Numerical Methods, Modeling and
Applications to Circuit Design, Erfurt, Germany, October 07 to 08, 2008;
pp. 127-134. Due to the low distribution of these proceedings, the author has
decided to make the article available to a larger audience through the arXiv.
The author’s current (as of December 31st, 2010) address is Bismarckstraße
20, D-38518 Gifhorn, Germany. Current e-mail: e.gerbracht@web.de
1 Reintroduction, because some symbolic analysis tools were and still are
closely associated to existing computer algebra systems, although nowadays
they do not make much use of the advanced algebraic capabilities (it is
debatable if they ever did), but rely more on their numeric proficiencies.
2 See e.g. [1].

more mathematical terms: suppose the network function is
demanded to be a rational function of form
a0 + a1 s + · · · + am sm
,
(1)
fdes (s) =
b 0 + b 1 s + · · · + b n sn
with the numbers m, n, a1 , . . . , am , b1 , . . . , bn determined by
the initial specifications.
Then we need to find an analog circuit Γ built from
elements, which themselves are described by parameters
p1 , . . . , pk , such that its network function
Pm
i
µ=0 Aµ (p1 , . . . , pk )s
,
(2)
HΓ (s) = Pn
i
ν=0 Bν (p1 , . . . , pk )s
where the terms Aµ and Bν are polynomials in p1 , . . . , pk ,
satisfies
HΓ (s) = fdes (s).
(3)

In other words: at the end of the day, many design problems
result in a system of nonlinear equations
Aµ (p1 , . . . , pk ) − aµ = 0,
Bν (p1 , . . . , pk ) − bν = 0,

(4)

1 ≤ µ ≤ m, and 1 ≤ ν ≤ n for variables p1 , . . . , pk .
If indeed the terms Aµ , Bν are polynomials3 with coefficients
in some number field, the set of solutions, otherwise also
known as the zero set of (4), forms what is called an affine
algebraic variety, an object which belongs to the mathematical
subdiscipline of Algebraic Geometry. Thus when trying to
solve these equations, we could profit both from the centuries
of experience, mathematicians have gained in this particular
field, and from the more recent results and algorithmic instruments developed – the concept of Gröbner bases, as well as the
Buchberger algorithm and its “relatives” which are realized in
nearly all of the current computer algebra systems. The result
would be exact (“symbolic”) descriptions of the sought after
solution sets, from which we could deduce numeric results
afterwards.
Now, when we take a look at the advances in symbolic
methods applied to circuit design, we can make a peculiar observation: nowadays it has become nearly routine to calculate
any given network function as a multivariable rational function
of the complex frequency s and the symbolic circuit elements
(even though there is still the problem of the combinatorial
3 Thus for example we do not talk about the problem of biasing a transistor,
since then we would have to solve equations containing the exponential
function (as in the Ebers-Moll model of the BJT), or equations which contain
roots, and moreover are piecewisely defined (as for FETs).

2

explosion for large circuits). However, when it comes to
finding solutions for the nonlinear equations arising from the
task of sizing a particular circuit, even if it is relatively small,
researchers, as well as designers, fall back upon numerical
means and various approximative methods with their various
drawbacks.
III. A N E XAMPLE - S IZING AN ACTIVE 4 TH ORDER LOW
PASS RC FILTER
In section 6.3.3. of [2] the following filter circuit was
presented:
C
R

4

4

R

R

3

C2
2

K

R1

U in
C

Fig. 1.

C
3

1

U

a

The aim there was to construct a fourth order Butterworth filter
with this topology.
Using one of the available symbolic analysis programs4, we
see that the circuits symbolic transfer function is given by
K
Ua
=
U in
D(s)

(5)

with D(s) = 1+a1 s+a2 s2 +a3 s3 +a4 s4 being a polynomial,
and
a1 = ((1−K)(C4 +C2 )+C3 +C1 )R4 +((1−K)C2 +C3 +C1 )R3 +
((1−K)C2 +C1 )R2 +C1 R1 ,

a2

=

C2 C1 R1 R2 +(C3 +C2 )C1 R1 R3 +(C4 +C3 +C2 )C1 R1 R4 +
C3 ((1−K)C2 +C1 )R2 R3 +((C4 +C3 )((1−K)C2 +C1 )R2 R4 +
C4 (C3 +(1−K)C2 +C1 )R3 R4 ,

a3

=

C3 C2 C1 R1 R2 (R3 +R4 )+C4 C2 C1 R1 (R2 +R3 )R4 +

a4

=

C 4 C 3 C 2 C 1 R1 R2 R3 R4 .

for the special case of the fourth order Butterworth filter. Thus
we got rid of the precalculated approximate values for the
ai , by using instead the defining algebraic equations given by
matching the coefficients from
D(s) · D(−s) =

Circuit topology for fourth order low pass RC filter

H(s) :=

numbers to arbitrarily chosen parameters, and the choice of a
starting vector for the Newton-Raphson method might hinder
from finding a solution, or even worse, might make a solution
impossible5
Furthermore, questions like “Do alternatives exist? How
many are there?”, cannot be answered either, and changing
any of the numerical parameters forces us to redo most of the
calculations.
When we tried to solve this design task, first of all we
started with a slightly different approach, using the Feldtkeller
equation
D(s) · D(−s) = 1 + s8
(7)

(6)

C4 C3 C1 (R1 +R2 )R3 R4 +(1−K)C4 C3 C2 R2 R3 R4 ,

In order to specify element values in such a way that H(s) becomes the transfer function of a fourth order Butterworth filter,
we can proceed from these data by matching the coefficients
with those numerical values, which can be found in any table
for filter design, and then try to solve the resulting equations by
numerical means. Since we only have four equations for nine
unknowns, there is the need to reduce the existing degrees of
freedom by more or less arbitrarily attaching numerical values
to some of the element values. In fact, that is what was done
in [2], where K was set equal to 2 and all the Ci were set
to 1 (all element values in this example are considered to be
normalized); afterwards some variant of the Newton-Raphson
method was used, to deduce one single solution.
The disadvantages of such an approach are obvious: Using
rounded values (from precalculated tables) introduces errors
right from the start. Both, the assignment of (quite arbitrary)
4 For the analysis of this and all other circuits within this paper, we use
SapWin 3.0, because of its affordability and its easy availability through the
world wide web [3].

1 + (2a2 − a21 )s2 + (a22 − 2a1 a3 + 2a4 )s4
+ (2a2 a4 − a23 )s6 + a24 s8

with 1 + s8 . Consequently we enlarged the set of equations
(6) by adding
2a2 − a21 = 0, a22 − 2a1 a3 + 2a4 = 0, 2a2 a4 − a23 = 0, a4 = 1
(8)
to it, and regarding a1 , . . . , a4 as further unknowns.
Even though in this paper we propose the usage of the
algebraic capabilities of a CAS, at this point we used Mathematica (Version 6.0.1.0) as a numerical engine – not in
order to approximate one solution, but to find approximates
for all solutions. Moreover as was done in [2], we also fix
K = 2, and Ci = 1 for 1 ≤ i ≤ 4. Since obviously
the enlargement of the set of equations results in a bigger
set of solutions, we were left with the final task of “postprocessing” this zero set, by singling out those particular
solutions for which a1 , . . . , a4 determine a polynomial D(s),
all of whose zeroes have negative real part (= filter stability),
and such that all the Ri are non-negative (= realizability
by classic elements). Indeed, we observe that from all the
possible solutions calculated, only nine satisfy the restrictions
concerning the ai , and again only one of these,
K

=

2, C1 = C2 = C3 = C4 = 1,

R1
R2

≈
≈

0.133933818297194652631087580090,
3.893036697318392402871746149006,

R3
R4

≈
≈

2.479192111455558403082198766696,
0.773590398536329977043175927841.

results in the Ri all being positive.
So, where does computer algebra enter this picture?
First of all Mathematica’s NSolve-procedure, which we
used in the above task, at its core heavily depends on algebraic
algorithms: the fact that it is able to calculate all solutions
stems from the algebraic variety, which was defined by (6) and
(8), being zero dimensional. That is, it consists of finitely many
5 E.g. this is the case, when we use the preassigned values from [2], and try
to find element values, such that the above circuit attains the characteristics
of a fourth order Chebyshev filter.

3

points. Again we can be sure of this fact, only because there
is an algorithm – the Buchberger algorithm6 – which is used
by Mathematica as a subprocedure of NSolve and which for
any system of algebraic equations produces particularly wellbehaved equivalent systems of equations – so-called Gröbner
bases – which allow much more insight into the properties of
the zero set, and may be used for a subsequent elimination of
variables7 .
Thus
e.g.,
when
we
apply
Mathematica’s
GroebnerBasis-command to (6) and (8), with the
values of K, C1 , . . . , C4 fixed as above, and the variables
ordered lexicographically by a1 < . . . a4 < R1 < · · · < R4
(which amounts to saying that first a1 is to be eliminated,
then a2 , etc.), then the resulting Gröbner basis contains the
polynomial
P = 1 − 160 R44 + 13872 R48 − 788512 R412 + 31505120 R416

−920274816 R420 + 20065991808 R424

32
−328437088768 R28
4 + 4000414289152 R4
40
−35535204282368 R36
4 + 223781766674432 R4
−956822102532096 R444 + 2535921430958080 R48
4
52
−3050522934050816 R4 − 2341746368053248 R456
+10182414501412864 R460 − 1806331484831744 R64
4
68
72
−13996296348106752 R4 + 2888816545234944 R4

in R4 of degree 72, which implies that there are only
finitely many possible values for R4 . Furthermore this Gröbner
basis contains polynomials p1 (R1 , R4 ), p2 (R2 , R4 ), and
p3 (R3 , R4 ), each of which being linear in the respective Ri ,
and otherwise consisting only of a high degree polynomial in
R4 . Since these polynomials as part of a Gröbner basis have to
be zero, too, consequently each zero of P determines precisely
one solution vector (R1 , . . . , R4 ).
Those solutions, which result in the additional specifications for the ai to be fulfilled, originate from a factor of
the polynomial P of degree 18, which can be calculated
using Mathematica’s Factor-procedure, when we allow the
coefficients
to be sums of integers and integer multiples of
√
2. This factor polynomial is
√
√
√
−10 + 7 2 + (−12 + 8 2)R42 + (280 − 196 2)R44
√
√
+(520 − 352 2)R46 + (−2824 + 1900 2)R48
√
√
+(−10816 + 6656 2)R410 + (−13904 + 7368 2)R412
√
√
+(−8288 + 5952 2)R414 + (1280 + 6592 2)R416
+10368R418.
Substituting T = R42 , we see that its structure is effectively
determined by a polynomial of degree 9, which results in the
number of admissable solutions being nine.
6 The Buchberger algorithm can be seen as a generalization of both the
Euclidean algorithm (to the multidimensional case) and the Gauß algorithm
(to polynomial equations of degree > 1).
7 In most cases, especially if the zero set is zero dimensional, there exist
Gröbner bases which are in triangular form, a property which generalizes
the analogous property of sets of linear equations, which can be brought to
triangular form using the Gauß algorithm. For more details see [4], [5].

To finish this section, let us remark that by modifying the
Feldtkeller equation to
D(s) · D(−s) = 1 + T4 (s) · T4 (−s),
where T4 (s) denotes the fourth order Chebyshev polynomial,
and slightly changing the initial values for the Ci , after some
trial-and-error we were able to calculate the following set of
parameter values, with which we can make the initially given
circuit into a Chebyshev filter
K
R1
R2
R3
R4

=
≈

2, C1 = 1, C2 = 2, C3 = 2, C4 = 1,
0.263638090854794185461593787592,

≈

3.249013399873649633437777451232.

≈
≈

0.624164765447879000316525786179,
2.645185226758545312744750592839,

The question remains for each filter characteristic: which
set of prescribed values for K, C1 , . . . , C4 imply solutions for
R1 , . . . , R4 , which make sense in an electrical engineering
point of view, i.e., for which positive K, C1 , . . . , C4 do we
get positive R1 , . . . , R4 ?

IV. A NOTHER E XAMPLE : S IZING

A BJT A MPLIFIER WITH
RESPECT TO BREAK FREQUENCIES

From [6] - a students’ textbook in analog electronics - we
take our next example: a rudimentary BJT amplifier circuit
with only a few additional classic elements for biasing and
sizing purposes (see figure (IV)). The biasing resistances are

Fig. 2.

Rudimentary topology for a BJT amplifier

given by RB = 310kΩ, and RC = 2kΩ. The biased transistor
is modelled by concrete h-parameters
H=


h11
h21




672Ω
0
h12
,=
96
35 · 10µS
h22

(9)

and its capacitances CBC = 11pF, and CBE = 320pF.
(s)
With denoting its transfer function by H(s) = N
D(s) , a full
symbolic analysis with SapWin 3.0. gives
N (s) = Ce h21 RB RC · s − CBC Ce h11 RB RC · s2 ,

4

and D(s) = a0 + a1 s + a2 s2 + a3 s3 , with
a0

=

−h11 −RB +h12 h21 RC −h11 h22 RC −h22 RB RC ,

a1

=

−CBC h11 RB −CBE h11 RB −Ce h11 RB −Ca h11 RC
−CBC h11 RC −Ca RB RC −CBC RB RC +CBC h12 RB RC
−CBC h21 RB RC +CBC h12 h21 RB RC +CBE h12 h21 RB RC
+Ce h12 h21 RB RC −CBC h11 h22 RB RC −CBE h11 h22 RB RC
−Ce h11 h22 RB RC −Ce h11 Rs −Ce RB Rs +Ce h12 h21 RC Rs
−Ce h11 h22 RC Rs −Ce h22 RB RC Rs ,

=

a2

−Ca CBC h11 RB RC −Ca CBE h11 RB RC −CBC CBE h11 RB RC
−Ca Ce h11 RB RC −CBC Ce h11 RB RC −CBC Ce h11 RB Rs
−CBE Ce h11 RB Rs −Ca Ce h11 RC Rs −CBC Ce h11 RC Rs

Fig. 3. Bode Plot of initially given BJT amplifier, according to (10)

−Ca Ce RB RC Rs −CBC Ce RB RC Rs +CBC Ce h12 RB RC Rs
−CBC Ce h21 RB RC Rs +CBC Ce h12 h21 RB RC Rs

as H(s), and possesses two (new) poles p1 and p2 . Introducing
a new parameter k (since due to the simplicity of our design
e
topology, we cannot specify the maximal attenuation of H,
as we will see below), we thus are asked to determine the
parameters k, Rs , Ca , and Ce in such a way that

+CBE Ce h12 h21 RB RC Rs −CBC Ce h11 h22 RB RC Rs
−CBE Ce h11 h22 RB RC Rs ,

=

a3

−Ca CBC Ce h11 RB RC Rs −Ca CBE Ce h11 RB RC Rs
−CBC CBE Ce h11 RB RC Rs .

With regard to the concretely given transistor parameters at the
bias point, the nominator and denominator of H(s) become
N ′ (s) = (290625000000Ces(−1000000000000 + 77s)),

=

(11)

This is equivalent to

N ′ (s)·(s−p1 )·(s−p2 )−D′ (s)·k·s·(77s−1000000000000) = 0.
(12)
Factoring out common terms on both sides of the −sign, we are left with a third order polynomial in s
with coefficients being integer polynomials in the variables
{k, Rs , Ca , Ce , p1 , p2 }. To solve (12) we need to set these coefficients to zero, which consequently results in the following
system of four algebraic equations9 (the i-th equation resulting
from the coefficient of si ):

and
D′ (s)

k · s · (77s − 1000000000000)
N ′ (s)
e
= H(s)
=
.
′
D (s)
(s − p1 ) · (s − p2 )

1623139843750000000
+3590505265625 s
+3033906250000000000000 Cas
+1088390625000000000000 Ces
+1623139843750000000 CeRs s
+7161 s2
+673378125000000 Cas2 + 22378125000000 Ces2

0=

+2034375000000000000000000CaCe s2
+3590505265625CeRs s2
2

+3033906250000000000000CaCe Rs s
+7161Ce Rs s3 + 673378125000000CaCe Rs s3 ,
respectively. Thus, e.g. setting Rs = 0Ω, Ca = 300pF, and
Cb = 0.1µF, we get
H(s) =

465000s(−1000000000000+77s)

,
(10)

25970237500000000000+1813435834250000s+1015651791s2

the poles being (approximately8)
p1 ≈ −14437.758,

and p2 ≈ −1, 771051.964.

This results in the Bode plot shown in figure 3.
We will now use this example as a problem in circuit design.
Our task shall be: with the bias point remaining unchanged,
find parameters Rs , Ca , and Ce in such a way that a new
e
transfer function H(s)
is achieved, which has the same zeroes
8 Clearly, due to the nominator being a quadratic function, we would also
be able to calculate exact values.

1623139843750000000 k − 290625000000 Cep1 p2 ,

0 = 3590505265625 k + 3033906250000000000000 Cak
+ 1088390625000000000000 Cek + 290625000000 Cep1
+ 290625000000 Cep2 + 1623139843750000000 CekRs ,
0 = −290625000000 Ce + 7161 k
(13)
+ 673378125000000 Cak + 22378125000000 Cek
+ 2034375000000000000000000 CaCe k
+ 3590505265625 CekRs
+ 3033906250000000000000 CaCe kRs ,
0 = 7161 Ce kRs + 673378125000000 CaCe kRs .
Alhough this system is nonlinear, using its procedure Solve,
Mathematica is able to calculate four symbolic solutions for
k, Rs , Ca , and Ce as functions of p1 and p2 , voicing the explicit caveat: “Equations may not give solutions
for all "solve" variables.”. Nevertheless, indeed
there are only four different solutions.
9 This is why we had to introduce the additional parameter k, since we
consider p1 and p2 to be fixed.

5

The first is irrelevant for the concrete problem at hand from
an electrical engineering point of view, since it is given by
Ce = 0 and k = 0,

(14)

and thus results in a trivial transfer function.
The second one, though it leads to nontrivial terms for Ce , k
and Rs , furthermore implies
11
Ca = −
,
(15)
1034375000000
which again has to be considered absurd from an engineering
point of view10 .
Finally, there are two solutions left which lead to two
different valid sets of design parameters, both of them having
in common that Rs = 0.
Here we will not present the explicit formulas for Ca , Ce , k,
but we will give a numerical example, and we will further
show, how the formulas for Ca (and beyond this our claim,
that there are only four solutions) can be derived with the help
of Gröbner bases.
First, for the numerical example let p1 = −10, and p2 =
−1000. Then the two remaining approximate solutions of the
above set of equations are
(Ca , Ce , k) ≈ (53.4989µF, 1.49102µF, 2.66969 · 10−9 ) (16)
and
(Ca , Ce , k) ≈ (53.5000µF, 149.12881µF, 2.67017 · 10−7 ).
(17)
Indeed, the corresponding Bode plots for the respective
circuits can be seen in figure 4.

(13) is equal to the zero set of its Gröbner basis, we thus see
that there truly are only four possibilities for the values of Ce .
Factoring this particular polynomial (again with the help of
Mathematica) we get the following factors:
Ce ,
and
1182300335490970802500000000000000
− 2591834572818828634150000000 p1
− 2591834572818828634150000000 p2

+ 5681810493667293811609 p1 p2
+ 1737403713997011459000000000000 p1 p2 Ce ,

and finally
100852820307500000000000000
− 22384370986950000000 p1
− 22384370986950000000 p2

+ 49070938130697 p1 p2
− 67626498450000000000000000000 p1 Ce

Finally, applying Mathematica’s GroebnerBasis-command to the right-hand sides of equations (13), with the order
of variables chosen to be k < Rs < Ca < Ce , results in a
Gröbner basis consisting of 11 polynomials with exactly one of
them containing only the variables Ce , p1 , p2 . This particular
polynomial is of degree four in the variable Ce . Keeping in
mind that the zero set of the original polynomials constituting
10 Unless we allow for NICs (negative impedance converters) to be added
to the initial circuit topology.

(19)

− 67626498450000000000000000000 p2 Ce
+ 163214120034000000000000 p1 p2 Ce
+ 45346707000000000000000000000000 p1 p2 Ce2 .

Further analysis shows that Ce = 0 results in k = 0, and the
solution of the linear (with respect to Ce ) equation implies the
negative value of Ca according to (15). Thus the two solutions,
whose existence we claimed above, have their origin in the
quadratic equation (19).
V. W HAT I F EQUATIONS BECOME TOO
O UTLOOK

Fig. 4. Bode plot of BJT amplifier circuit with original parameters (dotted),
parameters (16) (dashed), and (17) (thick)

(18)

COMPLICATED ?

AN

Symbolic methods always suffer from one major deficiency
– the “combinatorial explosion” of terms – which most of the
time results in both, a massive, nearly insatiable demand for
space, and an extraordinary high amount of computing time.
This observation also applies to all Gröbner basis methods: It
is known that by necessity the problem of finding a Gröbner
basis in the general case needs an exponential amount of
space11 . On the other hand, in recent years there have been
major improvements with regard to the run-time behaviour of
the original Buchberger algorithm12, which have resulted in
faster and thus more powerful algorithms to compute Gröbner
bases.
As we could see in the above examples, even moderately
sized small circuits already give rise to mid-sized equations13,
which again – by way of the methods from computer algebra –
result in even bigger symbolic expressions for the sought-after
11 For more details and/or a mathematically more precise formulation of
this statement, see [7].
12 See e.g. the results by J.-C. Faugére in [8] and [9].
13 Indeed we must not regard the results, which have been presented up until
now, as large – the maximal computing time used in the above examples in
the worst case, which was the calculation of all the solutions of the Chebyshev
RC filter, was ten minutes.

6

design parameters. Thus the “new toolbox” computer algebra,
the usage of which I try to recommend with this article, seems
to be doomed from the start.
Still, bad behaviour in general does not mean that the instruments from computer algebra always behave badly – there
might be classes of problems for which results are computed
fast, or can be written down very economically. As electrical
engineers know, there are close (but in the author’s opinion by
far not completely understood) connections between a circuit’s
topology and the algebraic structure of its network function,
which may be taken advantage of with regard to design tasks.
Take on the other hand researchers in symbolic methods,
who are accustomed with the phenomenon of swelling expressions arising in the analysis of circuits. As a workaround
a number of them have proposed so-called “sequences of
expressions”14. It is interesting to note that even electrical
engineers without experience in methods of symbolic analysis
may already be well versed in this “sequential approach”, if
only they knew classic filter theory, in particular the Cauer
canonical forms of LC-circuits.
In form of an example, let us show, what we mean. Suppose
we are given an LC-ladder network, consisting of three
inductances and three capacitances, as shown in figure 5.

where Z2 (s) is the impedance of the circuit, that is left, when
L1 and C1 have been removed. Continuing like this, taking
away one more circuit element in each step, this process can
be repeated, until we get to one last impedance Z5 (s) (which
would just consist of C3 ).
Reinserting the various Zi we get
Z(s) = L1 s + Z1 (s)
= L1 s +
= ...,

1
C1 s + Z21(s)

and thus are able to calculate Z(s).
When conversely one wants to realize an LC transimpedance function as an LC ladder, classic theory15 tells us to
expand the given rational function into a continued fraction (an
algorithm which in case of rational functions always comes to
an end after a finite number of steps) and read off the element
values directly from this expression16, which according to
theorems in classic filter theory all are positive.
If we want to bring together the capabilities of a CAS for
solving algebraic equations and the sequence of expression
approach in our example, we first observe that we can easily
calculate the form of the intermediate impedances Zi , i.e. we
have
1
1
∗s2 + 1
= , Z4 (s) = L3 s + Z5 (s) =
,
C3 s
∗s
∗s
1
∗s3 + ∗s
1
= C2 s +
=
,...,
Z3 (s)
Z4 (s)
∗s2 + 1
a4 s4 + a2 s2 + 1
Z1 (s) =
, and finally
a5 s5 + a3 s3 + a1 s
Z(s) = L1 s + Z1 (s)
(a5 · L1 )s6 + (a3 · L1 + a4 )s4 + (a1 · L1 + a2 )s2 + 1
=
a5 s5 + a3 s3 + a1 s
Z5 (s) =

Fig. 5.

LC ladder in Cauer canonical form

As is well known its impedance Z(s) can be described by
a continued fraction
1

Z(s) = L1 s +

.

1

C1 s +

(20)
Zdes (s) =

1

L2 s +
C2 s +

1
1
L3 s +
C3 s

That, in fact, is just short hand for a sequence of expressions.
This can be seen by deducing (20) with a process of alternatively combining impedances in series and in parallel. Thus
we have
Z(s) = L1 s + Z1 (s),
where Z1 (s) denotes the impedance of the ladder circuit, with
L1 having been removed. Further:
1
=
Z1 (s)

1
1
C1 s

+

with ∗ and the ai denoting polynomials in the circuit parameters C1 , C2 , C3 , L2 , L3 .
A sensible design task would now be to specify an
impedance function

1
1
= C1 s +
,
Z2 (s)
Z2 (s)

(21)

[10].

(22)

with real numbers k, A1 , . . . , A6 , and to demand Z(s) =
Zdes (s). Analogously to the above example of the BJT amplifier, by matching coefficients, this results in six nonlinear
equations, which have to be solved for the variables a1 , . . . , a5
and L1 . Here we only present the (easy) solution (determined
by Mathematica again) for the first step, which is
A3 · A6
A1 · A6
, a4 = A4 −
,
a2 = A2 −
A5
A5
A5
A1
A3
a5 =
, a1 =
, a3 =
,
k
k
k
A6 · k
.
L1 =
A5
15 See

e.g. [11].
nowadays a computer algebra system might be of help for
calculating this expansion without much effort, see [12].
16 Again,

14 See

k (A6 s6 + A4 s4 + A2 s2 + 1)
,
A5 s5 + A3 s3 + A1 s

7

To calculate further element values, this procedure needs to
be repeated for the other Zi . Thus one has to solve next the
equations resulting from Z11(s) = C1 s + Z21(s) , where Z1 (s)
should still be described in terms of the coefficients ai , and
Z2 (s) is given by the ansatz
Z2 (s) =

b 4 s4 + b 2 s2 + 1
b 3 s3 + b 1 s

(23)

with new unknowns b1 , . . . , b4 . This has to be continued until
we have equations for each set of coefficients of Zi in terms
of the coefficients appearing in Zi−1 , plus additional equations for the element values. Starting from those equations,
successive reinserting leads to the desired expressions for the
element parameters.
One has to concede that – compared to the traditional
continued fraction expansion – this procedure looks very
cumbersome. But in fact, if one looks a little bit closer, it
is equivalent to repeatedly doing polynomial division as part
of a continued fraction expansion. Furthermore our approach
might be generalized to other impedance functions which
need not come from ladders, but still ultimately result from a
sequence of series, respectively parallel reductions applied to
the initially given circuit, i.e. the process of replacing two
impedances in series, or in parallel by a single equivalent
impedance17. Beyond this there is a further generalization to
handle Delta-Wye and Wye-Delta transformations.
On the other hand by results from Epifanov and Truemper, it
is known that every planar two terminal graph can be reduced
to a single edge by these four kinds of transformations (plus
removing loops and pendant edges). Moreover there is an
algorithm, developed by Feo and Provan18, which calculates
the sequence of necessary graph transformations. Since to
each of these transformations there corresponds an equation,
which gives the value of the resulting element(s) in terms
of the values of previously existing elements, there always
is a sequence of equations which leads from the initially given
element values to the driving point impedance of a planar two
terminal circuit19 .
Circuit design for (midsized to large) planar two terminal
circuits, which consist of impedances only, thus becomes the
task of “reversing” these equations, in an analogous manner
to what we did with LC ladders.
The author of this paper firmly believes, that computer algebra systems will be a valuable tool in tackling this particular
problem. The challenge remains to find classes of circuits for
which the sequence of reversed equations can be effectively
calculated, or to prove – what might be also possible, and
would be an interesting result in itself – that, besides the
classic, continued- or partial-fraction-topologies attached to
the names of Cauer and Foster, no such further classes exist.
17 However we need to be careful with the claim for generality, since if we
combine e.g. two resistances R1 , R2 in series to a single resistance R, from
the resulting equation R = R1 + R2 , when only given the value of R, we
cannot deduce uniquely the values of the Ri .
18 As a reference for these results in graph theory, see the original paper
[13].
19 This algorithm has been successfully implemented in a student’s minor
thesis [14] at the TU Braunschweig in 2003.

R EFERENCES
[1] Georges Gielen and Willy Sansen, Symbolic Analysis for Automated
Design of Analog Integrated Circuits, Kluwer Academic, Boston, 1991.
[2] Tran chi Hieu, Anwendung symbolischer Methoden beim Entwurf
linearer Analogschaltungen, Ph.D. thesis, Technische Universität Braunschweig, 2002.
[3] University of Florence. Department of Electronics and Telecommunications, “Sapwin 3.0 - Symbolic Analysis Program for Window,”
http://cirlab.det.unifi.it/Sapwin/, viewed September
1, 2008.
[4] David Cox, John Little, and Donal O’Shea, Ideals, Varieties, and
Algorithms, Springer, New York, 2 1997.
[5] David Cox, John Little, and Donal O’Shea, Using Algebraic Geometry,
Springer, New York, 2 2004.
[6] Uwe Naundorf, Analoge Elektronik, Hüthig, Heidelberg, 1 2001.
[7] Joachim von zur Gathen and Jürgen Gerhard, Modern Computer
Algebra, Cambridge University Press, Cambridge, 1 1999.
[8] Jean-Charles Faugére, “A new efficient algorithm for computing Gröbner
bases (F4 ),” J. Pure Appl. Algebra, vol. 139, pp. 61–88, 1999.
[9] Jean-Charles Faugère, “A new efficient algorithm for computing Gröbner
bases without reduction to zero (F5 ),” Mora, Teo (ed.), ISSAC 2002.
Proceedings of the 2002 international symposium on symbolic and
algebraic computation, Lille, France, July 07–10, 2002. Pp. 75-83, ACM
Press, New York, NY., 2002.
[10] Benedykt S. Rodanski and Marwan Hassoun, “Symbolic Analysis,” in
The Circuits and Filters Handbook, Wai-Kai Chen, Ed., chapter 24, pp.
753–779. CRC Press, Boca Raton, Florida, 2nd edition, 2002.
[11] Louis Weinberg, Network Analysis and Synthesis, McGraw-Hill, New
York, 1962.
[12] Frank Garvan, The Maple Book, Chapman & Hall/ CRC, Boca Raton,
FL, 2001.
[13] Thomas A. Feo and J. Scott Provan, “Delta-Wye Transformations and
the Efficient Reduction of Two-Terminal Planar Graphs,” Operations
Research, vol. 41, pp. 572–582, 1993.
[14] Sven Domann, “Implementierung des Feo-Provan-Algorithmus zur
Stern-Dreieck-Reduktion planarer Graphen,” Studienarbeit, Technische
Universität Braunschweig, 2003.

N OTE ADDED

TO THE

E LECTRONIC V ERSION

In this electronic document keywords, ACM and MSC
classifications have been added. (March 14th (Pi Day), 2009)

Eberhard H.-A. Gerbracht received a Dipl.-Math.
degree in mathematics, a Dipl.-Inform. degree in
computer science, and a Ph.D. (Dr. rer.nat.) degree in mathematics from the Technical University
Braunschweig, Germany, in 1990, 1993, and 1998,
respectively.

From 1992 to 1997, he worked as Research
Fellow and Teaching Assistant at the Institute
for Geometry at the TU Braunschweig. From
1997 to 2003 he was an Assistant Professor in
the Department of Electrical Engineering and
Information Technology at the TU Braunschweig. After a two-year
stint as a mathematics and computer science teacher at a grammar
school in Braunschweig and a vocational school in Gifhorn, Germany,
he is currently working as tutor, and independent researcher in various
areas of mathematics. His research interests include combinatorial
algebra, C*-algebras, the history of mathematics in the 19th and early
20th century and applications of computer algebra and dynamical
geometry to graph theory, calculus, and electrical engineering.
Dr. Gerbracht is a member of the German Mathematical Society
(DMV), the Society of Computer Science Teachers in Lower Saxony
within the Gesellschaft für Informatik (GI-NILL), and founding
member of the society ”Web Portal: History in Braunschweig www.gibs.info”.

