# Multiobjective Optimization Tool for a Free  Structure Analog Circuits Design Using Genetic  Algorithms and Incorporating Parasitics

Authors: Yaser M. A. Khalifa; Badar Khan; Faisal Taha
Journal: Journal of Artificial Evolution and Applications
 (2008)

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## Abstract

This paper presents a novel approach for a free structure analog circuit design using genetic algorithms (GAs). A major problem in a free structure circuit is its sensitivity calculations as a polynomial approximation for the design is not available. A further problem is the effect of parasitic elements on the resulting circuit_s performance. In a single design stage, circuits that are produced satisfy a specific frequency response specifications using circuit structures that are unrestricted and with component values that are chosen from a set of preferred values including their parasitic effects. The sensitivity to component variations for the resulting designs is performed using a novel technique and is incorporated in the fitness evaluation function. The extra degrees of freedom resulting form unbounded circuit structures create a huge search space. The application chosen is an RLC ladder filters circuit design.


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## Body

## 1. Introduction

Computer-aided design tools for analog circuits
consist almost invariably of circuit analysis or simulation packages; no
general-purpose analog circuit synthesis tools are available. Conventional
numerical optimization techniques can be used to obtain combinations of
component values that satisfy a given design specifications. In addition,
conventional optimization techniques operate on circuits after having their
desired responses approximated by a suitable polynomial, using ideal components
with unrestricted values, and only operating on predefined circuit structures.
The restrictions introduced by predefining the circuit structure waste extra
degrees of freedom in the design process and can conceal many novel design
structures that can satisfy the target response and at the same time enhance
the design process.In previous papers [1–5], evolutionary
computation algorithms are applied to fixed structure circuit designs to
satisfy certain specifications. As mentioned earlier, the predetermination of
the circuit structure required in conventional design techniques is a limiting
factor when considering a powerful search technique such as genetic algorithm. A few attempts
have been made to include structure synthesis in genetically-based optimization
techniques such as GA [6, 7] and genetic programming (GP) [8]. However, these
techniques suffered huge delays and memory usage and needed to be used in
conjunction with circuit analysis packages such as PSPICE [9, 10].There is still a need, however,
for more efficient algorithms that are faster, self-sufficient and can function
independently. An efficient measure of performance based on the sensitivity for
component variations for the resulting circuits and the incorporation of the
parasitic effects associated with components is still needed. This is
particularly important in the development of new generations of analog
programmable analog arrays (APAAs) programming tools. In this paper, an
efficient GA technique for free structure analog circuit design is described.
The technique incorporates a sensitivity measure to compare between potential
solutions and incorporates parasitic effects.


## 2. Genetic Algorithms

The field of genetic algorithms [11–13] was founded
by Holland in the early 1970s. In [12], Holland emphasizes the ability of simple
representations to encode complicated structures and the power of simple
transformations to improve that structure. These representations are combined
in what is called in biology a chromosome (also referred to as individuals). A
number of these chromosomes will constitute a population. Syntactic operations are then used to alter and improve
these coded solutions.The
standard genetic algorithms invented by Holland
are the basic form from which other improved algorithms have been subsequently
developed. As mentioned earlier, the problem to be optimized has to be encoded
into a chromosome-like structure in which each gene or set of genes represents
a significant parameter towards the desired solution. A population would consist
of a prespecified number of these chromosomes.Starting
from an initial randomly-generated
population, a selection mechanism is used to identify and select members that
exhibit better characteristics. Those selected chromosomes are then acted upon
by genetic operators, such as Mutation and Crossover, which are explained in
later sections, to reproduce and generate the subsequentgenerations. The procedure continues until either a time limit is
reached or is achieved.


### 2.1. Crossover Operator

TheCrossover operator is considered to be
the most important operator in genetic algorithms. It is also a special
characteristic of genetic algorithms over many other search techniques that are
inspired by the process of biological optimisation.In
nature,Crossover occurs when two
parents exchange parts of their corresponding chromosomes. In genetic
algorithms, the same idea forms the basis of the Crossover operator, where useful parts of different parent
chromosomes should be combined in order to produce an offspring that benefits
from the advantageous bit combinations of both parents. Two-point Crossover is an
operator that randomly selects two crossover points within a chromosome then
interchanges the two parent chromosomes between these points to produce two new
offspring.Consider the following two parents selected for Crossover. The “|” symbols
indicate the randomly chosen crossover points:Parent 1:110|010|10.Parent 2:001|001|11.After interchanging the parent chromosomes between the crossover points, the
following offspring are produced:Offspring 1:110|001|10.Offspring 2:001|010|11.


### 2.2. Mutation

Mutation is a genetic operator that is used to change individual bits in the chromosome
at a specified probability. The motivation for using this operator is that it helps
to introduce genetic diversity to new generations. While it is desirable in
nature to keep the level of mutation low so that an individual produced by the
mutation does not differ too much from its ancestor, this is not always the
case in applications of genetic algorithms. Chosen mutation rates typically can
range from 0.001 to 0.05. Rates beyond these limits are not recommended in most
genetic algorithms applications.


## 2.1. Crossover Operator

TheCrossover operator is considered to be
the most important operator in genetic algorithms. It is also a special
characteristic of genetic algorithms over many other search techniques that are
inspired by the process of biological optimisation.In
nature,Crossover occurs when two
parents exchange parts of their corresponding chromosomes. In genetic
algorithms, the same idea forms the basis of the Crossover operator, where useful parts of different parent
chromosomes should be combined in order to produce an offspring that benefits
from the advantageous bit combinations of both parents. Two-point Crossover is an
operator that randomly selects two crossover points within a chromosome then
interchanges the two parent chromosomes between these points to produce two new
offspring.Consider the following two parents selected for Crossover. The “|” symbols
indicate the randomly chosen crossover points:Parent 1:110|010|10.Parent 2:001|001|11.After interchanging the parent chromosomes between the crossover points, the
following offspring are produced:Offspring 1:110|001|10.Offspring 2:001|010|11.


## 2.2. Mutation

Mutation is a genetic operator that is used to change individual bits in the chromosome
at a specified probability. The motivation for using this operator is that it helps
to introduce genetic diversity to new generations. While it is desirable in
nature to keep the level of mutation low so that an individual produced by the
mutation does not differ too much from its ancestor, this is not always the
case in applications of genetic algorithms. Chosen mutation rates typically can
range from 0.001 to 0.05. Rates beyond these limits are not recommended in most
genetic algorithms applications.


## 3. Implementation

### 3.1. Chromosome Adaptation and Structure

In [1–3], the
predetermination of the circuit structures has helped in reducing the
chromosome length. This was achieved by incorporating the order of the
chromosome genes into the node connections information. However, in the case of
free structure circuit synthesis this technique cannot be used.In the case of free structure design, the solution
space is much larger and the optimization process is composite. It involves the
optimization of two different but highly correlated problems; the structure as
well as the sizing of a circuit. Hence, all specifications of the electrical
circuit including the structure and the sizing of all the components should be
included in the chromosome representation.The specifications of the
electrical circuit include the number of components in the circuit, the type of
each component, and a list of connections between the components. Circuit
components can include a variety of different types of components, including
resistors, capacitors, and inductors.Given the above information, the chromosome for each
circuit comprises a numberM of groups of equal bit lengths, and an extra group
of bits representing the number of nodes in the circuit; see Figure 1.Figure 1
Chromosome structure.Each gene in the chromosome specifies a component and
contains four fields (i) type of component (L,C, etc.), (ii) a pointer to a
menu containing component value, (iii) and (iv) circuit nodes to which the
component is connected; see Figure 1. Two-point crossover rate of 100% and a
mutation rate of 0.5% were used for the GA operators.It is generally desirable for a
low-cost design to use the least number of components. For this reason a Null
component, representing an open circuit, is introduced into the component type
menu.


### 3.2. The Design Program

On
entry the user is required to supply the following input data.Type of filter (low pass,
high pass, band pass)(i)
ω1, αmin⁡.(ii)
ω2, αmax⁡.(iii)
N, the
maximum number of nodes. Optional and can be used to limit the search to a
certain number of designs.(iv)
M, the
maximum number of components (optional). The execution of the program starts by calculating N,
maximum number of nodes. Then, genes representing elements are then decoded one
at a time, splitting them into four fields as explained in Section 3.1.Upon determining type, value and
connection nodes, the component value is inserted in the corresponding nodal
admittance matrix (NAM)
location [14]. When all elements are decoded, contents of the resulting entries
in NAM are passed to the calculation function, in which lower-upper (LU) decomposition
and solutions to the matrix equations are solved. Figure 2 shows a pseudocode
for the program.Figure 2
Pseudocode for a
single generation evaluation.


## 3.1. Chromosome Adaptation and Structure

In [1–3], the
predetermination of the circuit structures has helped in reducing the
chromosome length. This was achieved by incorporating the order of the
chromosome genes into the node connections information. However, in the case of
free structure circuit synthesis this technique cannot be used.In the case of free structure design, the solution
space is much larger and the optimization process is composite. It involves the
optimization of two different but highly correlated problems; the structure as
well as the sizing of a circuit. Hence, all specifications of the electrical
circuit including the structure and the sizing of all the components should be
included in the chromosome representation.The specifications of the
electrical circuit include the number of components in the circuit, the type of
each component, and a list of connections between the components. Circuit
components can include a variety of different types of components, including
resistors, capacitors, and inductors.Given the above information, the chromosome for each
circuit comprises a numberM of groups of equal bit lengths, and an extra group
of bits representing the number of nodes in the circuit; see Figure 1.Figure 1
Chromosome structure.Each gene in the chromosome specifies a component and
contains four fields (i) type of component (L,C, etc.), (ii) a pointer to a
menu containing component value, (iii) and (iv) circuit nodes to which the
component is connected; see Figure 1. Two-point crossover rate of 100% and a
mutation rate of 0.5% were used for the GA operators.It is generally desirable for a
low-cost design to use the least number of components. For this reason a Null
component, representing an open circuit, is introduced into the component type
menu.


## 3.2. The Design Program

On
entry the user is required to supply the following input data.Type of filter (low pass,
high pass, band pass)(i)
ω1, αmin⁡.(ii)
ω2, αmax⁡.(iii)
N, the
maximum number of nodes. Optional and can be used to limit the search to a
certain number of designs.(iv)
M, the
maximum number of components (optional). The execution of the program starts by calculating N,
maximum number of nodes. Then, genes representing elements are then decoded one
at a time, splitting them into four fields as explained in Section 3.1.Upon determining type, value and
connection nodes, the component value is inserted in the corresponding nodal
admittance matrix (NAM)
location [14]. When all elements are decoded, contents of the resulting entries
in NAM are passed to the calculation function, in which lower-upper (LU) decomposition
and solutions to the matrix equations are solved. Figure 2 shows a pseudocode
for the program.Figure 2
Pseudocode for a
single generation evaluation.


## 4. Least-Squared Error Evaluation

The error measured is defined as the difference
between the specified response constraint,HUpp(ω) and HLow(ω),
denoted by the shading in Figure 3, where HUpp(ω) is the upper boundary of the frequency
response, and HLow(ω) is the lower boundary, and the actual
response, Ha(ω),
from the design represented by GA chromosomes. The actual frequency response, Ha(ω),
will hence be prescribed by its unique combination of component values.
Therefore, (1)H(ω,R1,R2,…,Rm,C1,C2,…,Cn,L1,L2,…,Lv). For simplicity, all types of components will be
declared as k. Thus H becomes (2)H(ω,k1,k2,…,kq), where q=m+n+v.Figure 3
Design template.Hence, the error function will depend on the region it
is applied for. For the passband the error function would be(3)err(ω,k1,k2,…,kq)=HLow(ω)−Ha(ω,k1,k2,…,kq), while in the stopband the error function would be (4)err(ω,k1,k2,…,kq)=HUpp(ω)−Ha(ω,k1,k2,…,kq). The least-squares error criterion E is used as a
performance index because of mathematical simplicity and ease of programming.
The error is defined as (5)E=∫luerr(ω,k1,k2,…,kq). Substituting
from (4) in (5), LSE would be (6)LSE=∫ωlωu{H(ω)−Ha(ω,k1,k2,…,kq)}2, where H(ω) is the boundary response HUpp(ω) or HLow(ω).
Since the calculations must be performed numerically, the integration in (6)
will be replaced by summation. Therefore, (7)LSE=∑i=1m[H(ω)−Ha(ω,k1,k2,…,kq)]2, where m is
equally spaced frequency points assumed over the range of integration. This is
chosen because of ease in programming.


### 4.1. Specified Weighted Errors

Equation (7) gives equal weights for all
frequency points within the range. This does not necessarily reflect the
prescribed performance, for example, in the case of numerically small errors in
the passband. This numerically small error in the passband will be more likely
to affect the accuracy of a solution than a relatively larger error encountered
in the stopband.  Hence, a weight
function can be inserted in (7) to enable us to emphasize or de-emphasize the
error in the various regions of the frequency spectrum. Therefore, (7) can be
modified to be (8)LSE=∑i=1m{w(ω)⋅[H(ω)−Ha(ω,k1,k2,…,kq)]}2. The weight function w(ω) has been arbitrarily chosen
to be two in the passband region and unity in the stopband region. Any other
design weight functions can be easily applied instead.Although the passband will be characterized by a
specific passband ripple, the upper boundary of the response represents the
maximum power transfer level, the claimed results of LC filter theory.
Therefore, testing only the suitability of the lower limit of the response
would be sufficient, as the upper limit will never be exceeded in the passband.In the stopband region, only the upper limit of the
specified attenuation level needs to be considered, since any lower value of
attenuation meets the specifications. The transition region is considered as a
“do not care” region. Thus, it is sufficient to test just the passband and the
stopband for confirmations. The different frequency band boundaries are shown
in Figure3. The fitness is taken as the reciprocal of the error function.


### 4.2. Performance Improvements by Controlling Connection Fields

The number of nodes comprising the circuit is
determined at the beginning of the decoding process to control the connection
node numbers of the circuit components.The fields determining the
connection nodes are decoded in a way to produce node numbers withinN. This is
achieved by linear-scaling down the whole range of values of allowed node
numbers to only that of N. This helps to avoid circuit structures that would
have components connected to nodes that do not exist in the defined structure
and which are hence unacceptable and would be a total waste of time to
evaluate.


## 4.1. Specified Weighted Errors

Equation (7) gives equal weights for all
frequency points within the range. This does not necessarily reflect the
prescribed performance, for example, in the case of numerically small errors in
the passband. This numerically small error in the passband will be more likely
to affect the accuracy of a solution than a relatively larger error encountered
in the stopband.  Hence, a weight
function can be inserted in (7) to enable us to emphasize or de-emphasize the
error in the various regions of the frequency spectrum. Therefore, (7) can be
modified to be (8)LSE=∑i=1m{w(ω)⋅[H(ω)−Ha(ω,k1,k2,…,kq)]}2. The weight function w(ω) has been arbitrarily chosen
to be two in the passband region and unity in the stopband region. Any other
design weight functions can be easily applied instead.Although the passband will be characterized by a
specific passband ripple, the upper boundary of the response represents the
maximum power transfer level, the claimed results of LC filter theory.
Therefore, testing only the suitability of the lower limit of the response
would be sufficient, as the upper limit will never be exceeded in the passband.In the stopband region, only the upper limit of the
specified attenuation level needs to be considered, since any lower value of
attenuation meets the specifications. The transition region is considered as a
“do not care” region. Thus, it is sufficient to test just the passband and the
stopband for confirmations. The different frequency band boundaries are shown
in Figure3. The fitness is taken as the reciprocal of the error function.


## 4.2. Performance Improvements by Controlling Connection Fields

The number of nodes comprising the circuit is
determined at the beginning of the decoding process to control the connection
node numbers of the circuit components.The fields determining the
connection nodes are decoded in a way to produce node numbers withinN. This is
achieved by linear-scaling down the whole range of values of allowed node
numbers to only that of N. This helps to avoid circuit structures that would
have components connected to nodes that do not exist in the defined structure
and which are hence unacceptable and would be a total waste of time to
evaluate.


## 5. Sensitivity Comparison

In [5], the author has developed
a rapid feasible region estimation technique using GA.  The technique is modified to provide a
measure of comparison among the resulting circuit designs using the free
structure GA technique described in the previous sections.A hierarchical approach for
circuit design using GA is implemented. Once convergence occurs and a family of
solutions emerges,
usually 4–6 in average, the
structures of these circuits are fixed. For this stage, a uniform closely
spaced distribution of component values is used to help determining estimates
for near and far limits of component values within which the required circuit
specifications are met. At the end of this stage, the least minimumxmin⁡i and largest maximum xmax⁡i boundaries for each of the parameters are taken to be the estimate of
the feasible region for that specific parameter. A measure for the transfer
function sensitivity to variations in component xi is (9)Δxi=xmax⁡i−xmin⁡ixi; and for m, the total
number of components in the design is (10)SP=∑i=1mΔxi. The new fitness function will
result from multiplying SP by its current value.


## 6. Extension to Include Parasitic Effects

Parasitics
can include wire and wounding resistances and inductors in lumped circuit
components as well interlayer capacitances and node to ground voltages. It can
also include resistances associated with polysilicon and doped semiconductor
regions when used as conductors, and capacitances associated with any
crossover, from any conductor to substrate, and with any depletion region in a
reverse-biased pn junction, in integrated circuits. Careful layout design can
eliminate some of these parasitics, but avoiding them all together is not
feasible. Other factors which can also affect the amount of parasitics are the
area of the circuit and the packaging density of the chip [15].Parasitics can cause significant
deviations in circuit performance and should be accounted for in early stages
of the design, if their effects are to be minimized. In this chapter, GAs, as
described earlier, are modified to include some additional information the
designer might want to consider for each circuit component. Parasitic effects
considered in this paper include inductor resistance as shown in Figure4, and
series and parallel resistances, and series inductance associated with a capacitor
as shown in Figure 5.Figure 4Figure 5To
implement the parasitic effects into the circuit design optimisation problem
using the GA, an additional parameter is added into the component menu and this
represents the parasitic effect of the associated component and the quality
factorQ; see Figure 6.ArbitraryQ-factors considered in the GA implementation.
(a)(b)(c)(d)This
method will avoid adding to the complexity of the structure and will make it
possible to add or remove any effect from the component menu without having to
alter the chromosome structure.In
the implementation considered in this work, arbitrary loss factors for
components are assumed. The technique is based on the idea to perturb the
element values of the lossy filter with the aim of making the magnitude
function of the lossy filter proportional to that of the ideal filter at some
discrete frequencies in the passband and the stopband. There is no restriction
on theQ's of the elements, provided that they are high enough, and the
elements need not have identical Q's.However,
the methods outlined need preknowledge of the values of theQ's for each
element, which might not be feasible in practical situation, where designers
quite commonly depend on statistical distribution data for their Q values. It
is still feasible to specify classes of Q's, for example, low, medium, and
high; but these will reflect a band of randomly distributed values for each class rather than
specific values.It
is very likely that the designer will decide on what class ofQ's is needed for
a circuit design rather than specific Q values. For this reason, the method
used in this work assumes a random distribution of Q values within each class
of Q's; see Figure 6.


### 6.1. Incorporation of Parasitic Effects in Circuit Designs

In
implementing the parasitic effects into the circuit design optimisation problem
using the GA, two methods can be applied. The first is to include whatever
effects required to be considered in the chromosome structure itself; see Figure7.Figure 7
Additional effects included
in the chromosome structure.This
has the following limitations, firstly it will add to the complexity of thegene structure and the GA operation.
This will eventually put limits on the number of additional effects a designer
might be able to consider. In addition to that, inserting or removing an effect
will require a change in the chromosome structure. Last but not least, effects
considered for each type of elements are different and hence different gene
lengths will be required to represent each type of elements.The
second method to represent any parasitics into the GA implementation is to
associate these effects within the menu listings of elements. Additional
effects can then be introduced and any further calculations of these effects
can then be made, for example, loss factors; see Figure8. This method will
avoid adding to the complexity of the chromosome structure and will make it
possible to add or remove any effect from the component menu without having to
alter the chromosome structure.Figure 8
Additional effects included
in the component list menu.


## 6.1. Incorporation of Parasitic Effects in Circuit Designs

In
implementing the parasitic effects into the circuit design optimisation problem
using the GA, two methods can be applied. The first is to include whatever
effects required to be considered in the chromosome structure itself; see Figure7.Figure 7
Additional effects included
in the chromosome structure.This
has the following limitations, firstly it will add to the complexity of thegene structure and the GA operation.
This will eventually put limits on the number of additional effects a designer
might be able to consider. In addition to that, inserting or removing an effect
will require a change in the chromosome structure. Last but not least, effects
considered for each type of elements are different and hence different gene
lengths will be required to represent each type of elements.The
second method to represent any parasitics into the GA implementation is to
associate these effects within the menu listings of elements. Additional
effects can then be introduced and any further calculations of these effects
can then be made, for example, loss factors; see Figure8. This method will
avoid adding to the complexity of the chromosome structure and will make it
possible to add or remove any effect from the component menu without having to
alter the chromosome structure.Figure 8
Additional effects included
in the component list menu.


## 7. Results

The GA used has a crossover rate of 100% and a
mutation rate of 5% throughout all runs. The GA was allowed to run for one
thousand generation in each run. However, convergence was reached after an
average of one hundred and fifty generations. Each generation had fifty
chromosomes or individuals. Each generation took 10 seconds on average in
processing. In these implementations, the desired design template is the
starting point of the design process, whereas in conventional designs
polynomial approximation for the desired response is used [15]. This has the limitation of
constraining the feasible solution to sets of design parameters that would
satisfy the polynomial approximation. By
designing directly from the template specifications, a designer would avoid
extra degree of approximation a polynomial fitting would introduce. At the same
time, it provides an extra degree of freedom which would enlarge the solution
space considerably to include any design that would reside
within the design template without necessarily being represented by a polynomial. Also the GA search starts from
an arbitrary point in the solution space and needs not to have any prior
knowledge of the feasible region of the required design.


### 7.1. Low-Pass Design

Different low-pass design
specifications were considered. One had a passband ripple of 1 dB, minimum
attenuation of −100 dB at a transition ration of 10. An average group of five
solutions emerged; the frequency response of one of these solutions is plotted
in Figure9.  The circuit constructed by
the GA is shown in Figure 10. It is
noticed that well-known π and T ladder structures have also been
generated by the GA. A different design specification of 0.5 dB passband ripple
and −60 dB stopband attenuation was tested. Figure 11 shows the resulting circuit design.Frequency response examples for low-pass design template
with high quality factor (a) passband and (b) stopband.
(a)(b)Figure 10
Genetically designed circuit with a passband ripple of 1 dB and stopband edge
at −100 dB.Figure 11
Genetically designed filter circuit with 0.5 dB passband
ripple and −60 dB stopband attenuation.In a different run, two circuits that were designed
to meet achieve a maximum attenuation of 1 dB in the passband and a minimum
attenuation of −100 dB in the stopband; see Figure12, Table 1. A comparison between the
two circuits was conducted based on the tolerance rage to changes in component
values. Figure 9 shows the two circuits used to test the GA program.Table 1
(a) Table shows components, nodes, ranges, and
actual values for each component in circuit a. (b) Table shows components, nodes, ranges, and actual values for
each component in circuit b.
(a)
Circuit 1RangeComponentActual valueNodes 1Nodes 2LowHighC11.80331.282.28C22.20222.22.72C32.20112.22.72L12.20121.682.65L41.00230.982.72L31.00210.51.09C52.70222.73.22R11.00330.481.4L22.203212.52C61.00330.481.48(b)
Circuit 2RangeComponentActual valueNodes 1Nodes 2LowHighL41.00322.22.27R10.10330.10.62C21.80221.82.32C11.20330.771.57C32.20111.682.44L22.20230.572.36L14.70124.75.22R20.15330.150.67L31.00120.621.23(a)
and (b) are filter circuit with 1dB passband ripple and attenuation of −100 dB
at the stopband edge used to test GA program.
(a)(b)Even though
the two circuits have a very close components values, the range generated by
the GA program was different. The differences in ranges for each component in
the circuits are shown in Table2. We also note that C2 and C5 in circuit (a) are
in parallel, which is equivalent to C2 in circuit (b).Table 2
Differences between circuits’ components.
Circuit bCircuit aComponentRangeDifferenceComponentRangeDifferenceL42.202.270.07L42.202.720.52R1∥R20.060.320.26R10.481.400.92C21.82.320.52C2∥C51.201.470.27C10.771.570.80C1∥C60.350.890.54C31.682.440.76C32.202.720.52L20.571.360.79L21.001.520.52L14.705.220.52L11.682.650.97L30.621.230.61L30.501.090.59Average difference0.54Average difference0.61As
shown in Table2, the two circuits vary with the difference at some components,
but in general, circuit (a) has a higher tolerance range for its component
values than circuit (b). This gives circuit (a) fitness leads over circuit (b).


### 7.2. High-Pass Design

The
high-pass filter design is done directly without having to use a low-pass
prototype and then performs frequency transformations. A design template of 1 dB passband ripple and attenuation of −60 dB at the stopband edge with a
transition ratio of 10 is considered. A number of four solutions on average
emerged two of these circuits are shown in Figure13. The frequency responses
of the resulting circuits are shown in Figure 14.(a) and (b) are genetically derived
high-pass filter circuit with 1dB passband ripple and attenuation of −60 dB at
the stopband edge.
(a)(b)Genetically derived high-pass filter circuit.
(a)(b)


## 7.1. Low-Pass Design

Different low-pass design
specifications were considered. One had a passband ripple of 1 dB, minimum
attenuation of −100 dB at a transition ration of 10. An average group of five
solutions emerged; the frequency response of one of these solutions is plotted
in Figure9.  The circuit constructed by
the GA is shown in Figure 10. It is
noticed that well-known π and T ladder structures have also been
generated by the GA. A different design specification of 0.5 dB passband ripple
and −60 dB stopband attenuation was tested. Figure 11 shows the resulting circuit design.Frequency response examples for low-pass design template
with high quality factor (a) passband and (b) stopband.
(a)(b)Figure 10
Genetically designed circuit with a passband ripple of 1 dB and stopband edge
at −100 dB.Figure 11
Genetically designed filter circuit with 0.5 dB passband
ripple and −60 dB stopband attenuation.In a different run, two circuits that were designed
to meet achieve a maximum attenuation of 1 dB in the passband and a minimum
attenuation of −100 dB in the stopband; see Figure12, Table 1. A comparison between the
two circuits was conducted based on the tolerance rage to changes in component
values. Figure 9 shows the two circuits used to test the GA program.Table 1
(a) Table shows components, nodes, ranges, and
actual values for each component in circuit a. (b) Table shows components, nodes, ranges, and actual values for
each component in circuit b.
(a)
Circuit 1RangeComponentActual valueNodes 1Nodes 2LowHighC11.80331.282.28C22.20222.22.72C32.20112.22.72L12.20121.682.65L41.00230.982.72L31.00210.51.09C52.70222.73.22R11.00330.481.4L22.203212.52C61.00330.481.48(b)
Circuit 2RangeComponentActual valueNodes 1Nodes 2LowHighL41.00322.22.27R10.10330.10.62C21.80221.82.32C11.20330.771.57C32.20111.682.44L22.20230.572.36L14.70124.75.22R20.15330.150.67L31.00120.621.23(a)
and (b) are filter circuit with 1dB passband ripple and attenuation of −100 dB
at the stopband edge used to test GA program.
(a)(b)Even though
the two circuits have a very close components values, the range generated by
the GA program was different. The differences in ranges for each component in
the circuits are shown in Table2. We also note that C2 and C5 in circuit (a) are
in parallel, which is equivalent to C2 in circuit (b).Table 2
Differences between circuits’ components.
Circuit bCircuit aComponentRangeDifferenceComponentRangeDifferenceL42.202.270.07L42.202.720.52R1∥R20.060.320.26R10.481.400.92C21.82.320.52C2∥C51.201.470.27C10.771.570.80C1∥C60.350.890.54C31.682.440.76C32.202.720.52L20.571.360.79L21.001.520.52L14.705.220.52L11.682.650.97L30.621.230.61L30.501.090.59Average difference0.54Average difference0.61As
shown in Table2, the two circuits vary with the difference at some components,
but in general, circuit (a) has a higher tolerance range for its component
values than circuit (b). This gives circuit (a) fitness leads over circuit (b).


## 7.2. High-Pass Design

The
high-pass filter design is done directly without having to use a low-pass
prototype and then performs frequency transformations. A design template of 1 dB passband ripple and attenuation of −60 dB at the stopband edge with a
transition ratio of 10 is considered. A number of four solutions on average
emerged two of these circuits are shown in Figure13. The frequency responses
of the resulting circuits are shown in Figure 14.(a) and (b) are genetically derived
high-pass filter circuit with 1dB passband ripple and attenuation of −60 dB at
the stopband edge.
(a)(b)Genetically derived high-pass filter circuit.
(a)(b)


## 8. Conclusion

This paper has presented a novel
circuit design tool by using the GA strong search abilities in both the
structure and sizing domains
of a circuit design problem. A successful design of the chromosome structure
has been obtained to include the full circuit specifications. A novel technique
to compare circuit sensitivities to changes in component values among different
circuit structures was developed.It is noticed that well-knownπ and T ladder structures have also been
generated by the GA. That is due to the low sensitivity such structures have,
to small changes in component values. This feature was detected by the GA as a
result of the sensitivity comparison technique developed in this paper.So far, the technique described
in this paper is limited either passive circuits, or active circuits that are
based on passive models. A further extension to include active circuits and
three terminal components is underway and will appear shortly in another paper.



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*Source: 761380-2008-09-08.xml*