New Op-Amp Circuits Realizations Using Genetic

Algorithm Nariman A. Khalil1, Rania F. Ahmed1, Rania. A.

Abulsoud1

1Electrical Department,

Faculty of Engineering,

Fayoum University, Fayoum, Egypt

Ahmed M. Soliman2 2Electronics and Electrical Communication Engineering

Department, Cairo University, Giza, Egypt

Abstract—Genetic algorithm (GA) applications in analog

design circuits play an important role with promising results. This

algorithm is utilized to generate equivalent circuits for the well-

known Tow-Thomas (TT) filter which is the most commonly used

active circuit. This paper introduces a proposed methodology

based on the genetic algorithm and the symbolic representation to

generate equivalent op-amp configurations for well-known filters.

This methodology is applied to the TT filter to generate 168

different configurations. The generated realizations are simulated

using PSPICE simulation and the simulation results showed that

some generated circuits have better performance compared with

the original configuration.

Keywords— Nullor; Symbolic analysis; Genetic Algorithm; TT

filter.

I. INTRODUCTION

The symbolic analysis is a powerful tool to analyze electronic circuits, where all part of the circuit elements is considered as symbols. The nullor is quite useful for the analysis, synthesis, and design procedures, as it facilitates modeling the behavior of any active device disregarding the particular realization of the active blocks [1-5]. For instance, synthesis methodologies based on symbolic representation and genetic algorithms (GAs) have been introduced in [6, 7].

The Genetic Algorithm (GA) is such an optimization technique which operates on the principle of "survival of the fittest". GA has the capability to generate new design solutions from a population of existing solutions, and discarding the solutions which have an inferior performance or fitness [6]. A GA starts from high-level descriptions to automatically synthesize analog circuits. However, the automatic synthesis of analog circuits from high-level specifications is recognized yet as a challenging problem [6-9].It is worth noting that the genetic algorithm with nullator-based descriptions was applied to generate Voltage Followers (VFs) circuits and in [8]. This method described how an automatic system can deal with huge search spaces to design practical VFs by performing evolutionary operations from nullator-based descriptions. Also, generation of Voltage Mirrors circuits (VMs) based on the genetic algorithm was presented in [9]. Moreover, [6] introduced a new genetic algorithm to synthesis the negative type CCII- blocks by superimposing VFs and current followers (CF). The GA is utilized in this work to propose a genetic based methodology to generate equivalent op-amp configurations for a specific op-amp filter. The proposed methodology is applied to well-known filter TT filter

The Tow-Thomas (TT) second order filter using operational amplifiers (op-amps) has been reviewed in [10, 11]. TT bi-quad of Fig. 1 is an active-RC topology used to realize both low pass and band pass bi-quadratic filtering. This topology has been widely used because it is simple, versatile, ad require few components.

The paper is organized as follows; the proposed methodology is described in section II. In section III, the methodology is applied to TT filter to get the equivalent circuits. Some of generated circuits are simulated in section IV. Finally, section V concludes the work.

Fig. 1. Tow-Thomas (TT) second order filter using three Op Amps [10, 11]

II. METHODOLOGY

The flowchart of the proposed algorithm is shown in Fig. 2. Firstly, draw the pathological equivalent circuit of the target filter by replacing the op-amps with their pathological representation as shown in Fig. 3. Then, number all the nodes of the equivalent circuit. Second, write the gene code of each op-amp in the form 𝐺𝑒𝑛|𝑜𝑝𝑛

= Y.X.𝑂𝑛.Z.0.𝑃𝑛 where Y and X are the

interconnecting nodes of the nullator, O represents a nullator, Z and 0 are the interconnecting nodes of the norator and P represents a norator. The genes then are arranged together to form a chromosome. Then, write the chromosome for the circuit in the form: Chromosome=𝐺𝑒𝑛|𝑜𝑝1

.𝐺𝑒𝑛|𝑜𝑝2.𝐺𝑒𝑛|𝑜𝑝3

.

……….𝐺𝑒𝑛|𝑜𝑝𝑛. Third, using these gens, we can define the

number of nullators “m” that contain a common node. Fourth, make the crossover between norators to get all possible equivalent circuit. Note: all passive elements change their position as the norator change its position. The number of generated circuits is equal to n! as in [12] where n is the number of genes. Fifth, the generated circuits are then tested for the optimum ω0 and Q to get the best realizations and reject the bad ones. Fifth, utilizing the pass circuits and based on nullator tree, apply genetic operations (mutation and crossover) to the interconnecting nodes of the nullators to generate new circuits. Then, the first four steps are applied to the generated circuits from step five. So, one can also get more new realizations. The

978-1-5090-5721-4/16/$31.00 ©2016 IEEE ICM 2016321

number of resulted circuits depends on number nullators in the tree. The generated circuit is then tested for the optimum ω0 and Q

Fig. 2. Flow chart for the proposed algorithm

Fig. 3. The nullor representation of op-amp [12]

III. DIFFERENT REALIZATIONS FOR THE TT CIRCUIT

The TT filter is shown in Fig. 1. The transfer functions of the two output responses (band pass and low pass responses) are given by:

VBP

Vin= −

sG4

C1

s2 +G1

C1s +

G2G3

C2C1

V LP

Vin=

G2G4

C2C1

s2 +G1

C1s +

G2G3

C2C1

The center frequency and the quality factor are as follows:

ω0 = √G2G3

C1C2 And Q =

1

G1√

G2G3C1

C2

The proposed methodology is applied to the TT circuit configuration as follows:

Step 1: Draw the pathological equivalent circuit as shown in Fig. 4.

Step 2: Number the nullators norators nodes then, write the chromosome for the circuit in the form like this

𝐺𝑒𝑛|𝑜𝑝1= 2.0. 𝑂1. 3.0. P1

𝐺𝑒𝑛|𝑜𝑝2= 4.0. O2. 5.0. P2

𝐺𝑒𝑛|𝑜𝑝3= 6.0. O3. 7.0. P3

Chromosome1|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.0. 𝑶𝟐. 5.0. 𝑷𝟐. 6.0. 𝑶𝟑. 7.0. 𝑷𝟑

Step 3: notice from Gen1, Gen2 and Gen3 that that O1, O2 and O3 nullator tree with having the common node (0) as shown in Fig. 5.

Step 4: crossover between norators to get all possible solutions. Using

Chromosome1|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.0. 𝑶𝟐. 5.0. 𝑷𝟐. 6.0. 𝑶𝟑. 7.0. 𝑷𝟑

Chromosome2|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.0. 𝑶𝟐. 5.0. 𝑷𝟑. 6.0. 𝑶𝟑. 7.0. 𝑷𝟐

Chromosome3|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟐. 4.0. 𝑶𝟐. 5.0. 𝑷𝟏. 6.0. 𝑶𝟑. 7.0. 𝑷𝟑

Chromosome4|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟐. 4.0. 𝑶𝟐. 5.0. 𝑷𝟑. 6.0. 𝑶𝟑. 7.0. 𝑷𝟏

Chromosome5|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟑. 4.0. 𝑶𝟐. 5.0. 𝑷𝟐. 6.0. 𝑶𝟑. 7.0. 𝑷𝟏

Chromosome6|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟑. 4.0. 𝑶𝟐. 5.0. 𝑷𝟏. 6.0. 𝑶𝟑. 7.0. 𝑷𝟐

All passive components that connecting to norator changes its position as the norator change its position. Using the six chromosomes, there are six possible realizations for the TT circuits. The first realization is the original realization shown in Fig. 1 and the other five realizations are shown in Fig. 6.

Step 5: using the first chromosome,

Chromosome1|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.0. 𝑶𝟐. 5.0. 𝑷𝟐. 6.0. 𝑶𝟑. 7.0. 𝑷𝟑

First, we make crossover to any node number in the nullators’ tree. The resulted chromosome as follow:

Chromosome7|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.2. 𝑶𝟐. 5.0. 𝑷𝟐. 6.2. 𝑶𝟑. 7.0. 𝑷𝟑

The final realization is shown in Fig. 7(a). then repeat step 3 only on the norators that pairing with nullators in the tree like this

Chromosome8|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟏. 4.2. 𝑶𝟐. 5.0. 𝑷𝟑. 6.2. 𝑶𝟑. 7.0. 𝑷𝟐

Chromosome9|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟐. 4.2. 𝑶𝟐. 5.0. 𝑷𝟏. 6.2. 𝑶𝟑. 7.0. 𝑷𝟑

Chromosome10|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟐. 4.2. 𝑶𝟐. 5.0. 𝑷𝟑. 6.2. 𝑶𝟑. 7.0. 𝑷𝟏

Chromosome11|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟑. 4.2. 𝑶𝟐. 5.0. 𝑷𝟐. 6.2. 𝑶𝟑. 7.0. 𝑷𝟏

Chromosome12|𝑇𝑇 = 2.0. 𝑶𝟏. 3.0. 𝑷𝟑. 4.2. 𝑶𝟐. 5.0. 𝑷𝟏. 6.2. 𝑶𝟑. 7.0. 𝑷𝟐

The final realization is shown in Fig. 7(b) to Fig. 7(f) respectively. After every generated gene repeat step 3, so the total number of generated circuits is

= 27*3! + 3! = 168

We can apply genetic operations (mutation and crossover) to get all possible circuit. All possible genes are summarized in table I.

Fig. 4. TT circuit using nullators and norators

Fig. 5. Nullator tree

322

(a)

(b)

(c)

(d)

(e)

Fig. 6. The op-amp realizations for the TT equivalent circuits (a) The op-amp

realization of chromosome 2, (b) the op-amp realization of chromosome 3, (c)

The op-amp realization of chromosome 4, (d) the op-amp realization of chromosome 5, (e) the op-amp realization of chromosome 6.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 7. The op-amp realizations for the TT equivalent circuits (a) The op-amp

realization of chromosome 7, (b) The op-amp realization of chromosome 8, (c)

The op-amp realization of chromosome 9, (d) The op-amp realization of

chromosome 10, (e) The op-amp realization of chromosome 11, (f) The op-amp realization of chromosome 12

TABLE I. ALL POSSIBLE GENES FOR TT AFTER STEP 4

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

323

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟒. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟎. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟒. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟎. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟔. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟐. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

𝟐. 𝟔. 𝑶𝟏. 𝟑. 𝟎. 𝑷𝟏. 𝟒. 𝟐. 𝑶𝟐. 𝟓. 𝟎. 𝑷𝟐. 𝟔. 𝟎. 𝑶𝟑. 𝟕. 𝟎. 𝑷𝟑

IV. SIMULATION RESULTS

In this section, spice simulations are carried out using the op

amp 741 and using supply voltages of ±15V.

A. TT FILTER

The TT circuit of Fig. 1 is designed for Q= 5 and f0 = 10 kHz taking C1=C2= 500 pF, R4 =R3= R2 = 31.9kΩ, R = 10kΩ and R1= 150kΩ. The input signal is a sinusoidal input voltage source of 1V magnitude. Fig. 8 represents the magnitude response results of the TT equivalent circuits.

Fig. 8. Response of BP TT

TABLE II. SUMMARIZED RESULTS OF SIMULATIONS FOR EQUIVALENT

CIRCUITS OF TT

The equivalent

configuration Center frequency (kHz)

Quality factor

Chromosome 1 9.97255 5.10871

Chromosome 2 10.15359 5.08550

Chromosome 3 10.17965 4.92152

Chromosome 4 10.06507 5.02838

Chromosome 5 10.17484 4.32630

Chromosome 6 10.41956 4.65435

Chromosome 7 10.04397 4.87726

Chromosome 8 10.13557 4.84579

Chromosome 9 10.20106 5.15268

Chromosome 10 10.13905 4.83872

Chromosome 11 10.19252 4.48573

Chromosome 12 10.40283 4.44862

Apart of the resulted TT equivalent circuits are simulated and the results are summarized in Table II. Comparing among these results, one can find that the lowest ∆Q and ∆ω0belongs to the configuration of Chromosome 4 which has the lowest error as shown in Fig.6(c) and better than the original configuration (chromosome 1). The highest error in the quality factor is found in the realization of Chromosome 6 and Chromosome 12 which is equal to 0. 4 as shown in Fig.6 (e) and Fig.7 (f) respectively. Where the highest ∆ω0 =0. 32630 kHz is found in the realization of Chromosome 6 shown in Fig.6 (e).

V. CONCLUSIONS

New application for the genetic-based methods to generate equivalent circuits was presented. The proposed methodology generated a family of TT equivalent circuits. First, a new generation methodology; based on the nullor representation and the GA, was introduced. This methodology was applied on the TT circuits, to generate 168 TT equivalent circuits. The generated circuits have been confirmed using P-Spice simulations and the simulation results showed that some generated circuits have better performance compared with the original configuration. Moreover, it was shown that the greatest benefit of the proposed genetic methodology is to generate the large number of equivalent realizations to the required circuit (e.g. filter) programmatically. It is worth noting that the proposed methodology represents a systematic generation method and can be applied to any other op amp circuit.

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