World Journal of Wireless Devices and EngineeringVol.
1, No.1 (2017), pp. 39-46http://dx.doi.org/10.21742/wjwde.2017.1.
1.07Print ISSN: 2207-5968, eISSN: 2207-5976 WJWDECopyright ? 2017GV School PublicationA Review on Passive Network Synthesis using Cauer FormAbhilasha Sharma, and Teena Soni1Assistant Professor, Madhav Institute of Technology and Science, GwaliorAbstractNetwork synthesis involves the methods used to determine an electric circuit that satisfy certain specifications. Different methods may also be used to synthesize circuits, all of which may be optimal. Hence the solution to a network synthesis problem is never unique.
In this paper, significant methods in network synthesis theory is presented. Among all these method we will focus on the Cauer Realization, is a technique which is used to synthesize the passive network.Keywords: Positive Real function, Cauer form 2, Partial fraction, Factor Division.1.
IntroductionThe circuit design has been always a challenge with respect to the current and voltage relationship and the energy issues. In network synthesis there are far too many design approaches to enumerate here, and plenty of scope for anyone to invent new ones. It is a less clearly defined problem and then, there are many ways to approach a solution. There is need of judgement as well as calculation. It is (arguably at least) more difficult, more open-ended, more interesting than analysis. Yet analysis remain vital – to test the design ideas before putting them into practice.Because so much of synthesis is concerned with filter design, it is easy to suppose that all design is restricted to the frequency domain. Not so.
Often there are clear requirements in both domains, and usually they conflict. For example, a filter?s performance may be specified as a particular pass-band in the frequency domain, and at the same time it may be required to have a restricted transient response. This could occur in a radar system where bursts of oscillation are used, and the response to one burst must die away before the next pulse is received. Usually the requirements are in conflict and a synthesis procedure consists in finding a compromise.2. Network analysis vs synthesisA typical “analysis problem” is represented as:In analysis there are standard tools for a systematic approach:• Circuit theoryArticle history:Received (November 05, 2016), Review Result (January 12, 2017), Accepted (February 22, 2017)A Review on Passive Network Synthesis using Cauer Form40 Abhilasha Sharma and Teena Soni• Fourier/Laplace transforms• ConvolutionWhereas, in synthesis the situation is far less clear.
In this case we haveIn synthesis there are too many design approaches:? Foster form? Cauer form? Brune?s method realization? Bott Duffin method etc.3. Elements for the synthesisThese are:• Passive circuits,• Active Circuits and• Digital Filters.3.1. Passive Circuits• Components: These are restricted to the use of normal L,C and R electrical components (lumped or distributed) for the use in filters, equalizers, etc.
• Causality: The response of these networks is determined by the signal information reaching them up to the output instant, i.e., they are limited by causality: the effect cannot precede the cause.• Limited memory: Another restriction is that of severely limited memory: The energy dissipation time constants place restrictions on how much data from the past can influence the output at t = 0, i.e. the present.• Stability: Another restriction (which is also sometimes an advantage of passive circuits) is stability; i.e.
they do not have energy sources and consequently there cannot be growing oscillations. This restriction means that poles and zeros of immittances (impedances or admittances) must be in the left-hand half-plane.On the positive side, passive circuits are simple, reliable, stable and they can handle high powers compared with circuits with Op-amps.On the negative side, inductors, which are often necessary, are not compatible with integrated circuit technology, where circuits are made more or less in a plane. Inductors areWorld Journal of Wireless Devices and EngineeringVol.
1, No.1 (2017)pp.39-46Copyright ? 2017GV School Publication 41bulky and 3-dimensional. Another fundamental limit arises because, if you scale an inductor down in size, the Q-factor falls; a 10:1 linear scaling gives a 100:1 reduction in the Q-factor.3.2.
Activecircuits• Components: Active circuits include an energy source, such as an amplifier, as well as resistors and capacitors. Inductors can and usually are avoided.Active circuits are not limited by stability, – e.g. they can have an exponentially growing response (within certain limits). This means that poles and zeros of immittance functions can be located anywhere in the complex s-plane.
They can be made without the need of using inductors. This is particularly useful since in this way they are compatible with I.C.
technology. Capacitors can be scaled down without affecting the Q-factor.4. Why use analog passive, LC ladder networks?Many applications today use digital processing in lieu of analog processing and the GHz spectrum is finding increasing use in applications such as wireless communications. However, operation at high frequencies requires analog filtering and processing circuits simply because using digital techniques is neither realistic nor economical.
Another advantage that analog devices have over their digital counterparts is their ability to operate with wide instantaneous bandwidths and moderately high dynamic ranges at microwave frequencies 1. Analog circuits with passive elements are generally preferred unlike active components, as they do not require an excitation source. Passive LC networks are also more advantageous as compared to active networks since they have a high tolerance to component variances and are simple. Also analog passive circuits can be used as prototypes for designing active networks, interface circuits, transmission lines and other complex networks with discrete components or on chips.Most importantly, passive LC circuits generally operate in the range of 102 to 109 Hz 1. As we will be dealing with high frequency applications (of the order of GHz) in this project, we felt that it was best to use analog passive LC circuits.
Figure 1. Examples of lattice (top) and ladder (below) networksTwo of the most commonly synthesized network structures are lattice networks and ladder networks as shown in Figure 1.1. Lattice structures are relatively simple but balanced circuits.
This means that they do not have a common ground between input and output. Also becauseA Review on Passive Network Synthesis using Cauer Form42 Abhilasha Sharma and Teena Soniof tolerance requirements they are usable only when the specified transfer function has a zero on or near the „jw’ axis 2. Although this problem can be solved by applying balanced to unbalanced conversion methods using transformers (e.g. the Weinberg synthesis procedure), all of these techniques only lead to relatively complicated parallel networks 2.
On the other hand, ladders are popular structures for circuits because the shunt or series LC arms are directly related to the transmission zeros by LCtransmission. This makes circuit tuning not only a simpler process but also making the loss poles relatively insensitive to element variations as compared to balanced networks 2. Hence ladders are preferred over lattices.5. Positive real functionIn order to synthesize a driving point function into a passive network using resistors, inductors and capacitors, it must be positive real; a fact that was first demonstrated by Otto Brune 3. This means that the following properties must be satisfied,• It must be a rational function of the complex frequency, s.• The poles must lie on the left hand side of the jw axis or on the imaginary axis (stable function).
• The poles on the jw axis must be simple (multiplicity of 1). The denominator polynomial must be Hurwitz.• Complex poles and zeros must occur in conjugate pairs.Most of the transfer function synthesis methods in literature can be considered to be realizations of driving point impedances 8. Hence the same conditions of positive realness are applicable for transfer functions also. For a driving point function, the zeros should have negative or zero real parts. However in the case of synthesis of transfer functions, there is no restriction on the location of zeros. But to synthesize lossless circuits (those with only inductors and capacitors), the zeros must lie exclusively on the imaginary (jw) axis 11.
6. Different methods to realize the PR functionDue to the exhaustive literature present on the synthesis of two-terminal networks, we will not discuss every one of these techniques in detail. Instead we will focus only on a few important methods.
The synthesis of transfer impedance can be considered to be the realization of the associated driving point impedance at the zeros of transmission 2. Also, since networks serve the purpose of point-to-point (or multipoint) transmission of information, two-port networks (or in general, n-port networks) are more practical. The design of these higher port networks has its roots in the synthesis of one-port networks. Notable among the driving point impedance synthesis methods are those by Brune, Bott Duffin, Darlington and Cauer.
The first method for the synthesis of passive networks was proposed by Brune. The main idea behind this method is the removal of the zeros located at the origin, infinity and on the jw axis. The Bott Duffin method is quite similar to Brune?s method, but is more complex as it does not use transformers.
Kuh and Miyata proposed transfer function simplification by splitting the input function into a sum of functions easier to realize. Darlington?s method synthesizes resistively terminated reactive networks (containing inductors, capacitors and transformers). This method uses surplus factors and requires ideal components.Cauer?s method also requires the use of transformers in case a negative inductance is encountered but is by far the easiest and simplest method to implement. This method is basedWorld Journal of Wireless Devices and EngineeringVol.
1, No.1 (2017)pp.39-46Copyright ? 2017GV School Publication 43on continued fraction expansion or in Foster?s representation; the function may be split into partial fractions to realize a ladder structure.In addition, there are many other techniques for network synthesis based on one or more of the earlier methods. They differ in the procedure followed to attain the impedance (or admittance) functions to be realized. Some of these techniques include Guillemin?s transfer admittance synthesis (which realizes the impedance function as the summation of a series of functions each having a single numerator terms), Lucal?s method (decomposition of driving point functions to realize the conditions for RC synthesis) and so on. A more detailed description on classical and modern synthesis methods is provided in literature 4.
7. Cauer synthesis techniqueThe Cauer synthesis methods (Cauer I and Cauer II) are based on the continued fraction expansion method and involve the removal of alternate series and shunt elements as needed. The zeros at infinity are realized using the Cauer I synthesis method.The important point to note in the Cauer I synthesis method is that the starting driving point function should always be such that the degree of the numerator is greater than the degree of the denominator 5. Let Z1(s) be the original driving point function.
The numerator is divided by the denominator to yieldZ1(s) = sL1 +Z2(s).Z2(s) is the remainder and the order of this function is one less than Z1(s). The next step is to produce an admittance branch. This is done byY2(s) = sC2 + Y3(s).Here Y2(s) = 1/ Z2(s) and Y3(s) is the remainder driving point function.
The synthesis is carried out by removing series inductors and shunt capacitors each element accounting for one zero at infinity 6.On the other hand, the zeros at the origin are realized using the Cauer II method which is also a continued fraction expansion but in this case the polynomials are arranged in ascending order of their powers. In the case of the Cauer II synthesis technique the starting driving point function is always chosen such that the denominator is an odd polynomial 5. The procedure to determine the series capacitors and shunt inductors is the same as that described for the Cauer I synthesis method. In this case, the synthesis is carried out by removing series capacitors and shunt inductors where each element accounts for one zero at the origin 6.The Cauer I and Cauer II techniques, however, can not be used if there are finite zeros on the imaginary axis (transmission zeros). The continued fraction expansion is valid only for transfer functions with zeros at the origin or at infinity 6. The transmission zeros coincide with the poles of the driving point impedance, they can be realized as series impedance branches or shunt admittance branches 7.
8. Zero shifting procedureHowever, when the transmission zeros do not occur at the poles of the driving point impedance function or any of its remainders, the zero shifting technique must be used. The pole locations are created by introducing redundant elements 7.A Review on Passive Network Synthesis using Cauer Form44 Abhilasha Sharma and Teena SoniFigure 2. Shunt resonant and anti-resonant sectionsIn other words, the basic idea is to manipulate the driving point impedance function (z22) by pulling out a series inductance or shunt capacitance so that it exhibits a transmission zero at the frequency of the pole of the driving point function. This zero can then be removed as a shunt resonant or a series anti-resonant section while synthesizing the driving point function as shown in Figure 2.However, in the event that transfer impedance function H12(s) is not minimum phase, it can be realized as a cascade of a minimum phase network and an all pass network.
9. ConclusionFor network synthesis problems Cauer synthesis method is preferred because it is a simple technique based on continued fraction expansion, which results in an unbalanced ladder network. Another desirable feature about Cauer synthesized networks is that it is possible to synthesize a circuit such that no transformers are required.Future work can address the issue of how the variation of individual component values affects the sensitivity of the synthesized networks. The sensitivity could be examined in more detail.
It would be interesting to see how the change in component values affects poles, zeros and stability of the original transfer function. So in this context we will move towards the MATLAB tool.References1 M. Piedade, and M.
Silva , “An improved method for the evaluation of filter sensitivity performance”, IEEE Transactions on Circuits and Systems., Vol. 33, No. 3, pp. 332-335, March, (1986).2 J.
E. Storer, “Passive network synthesis”, McGraw-Hill, New York, (1957).3 L.Weinberg, “Network analysis and synthesis”, McGraw-Hill, New York, (1962).4 W.C.
Yengst, “Principles of modern network synthesis”, Macmillan, New York,(1964).5 H. Lam, “Analog and Digital filters”, Prentice Hall, (1979).6 G.
Daryanani, “Principles of active network synthesis and design”, Wiley, New York, (1976). 12.7 H. Lam, “Analog and Digital filters”, Prentice Hall, (1979).8 E.S. Kuh and D.
O. Pederson, “Principles of Circuit Synthesis”, McGraw-Hill, New York, (1959).9 S.S. Khilari, “Transfer Function and Impulse Response Synthesis using Classical Techniques” University of Massachusetts Amherst, February, (2014).World Journal of Wireless Devices and EngineeringVol.1, No.
1 (2017)pp.39-46Copyright ? 2017GV School Publication 4510 G. Bogdan, V. Radu, F.
Octavian, B. Cristian, M. Constantin, and A. Petre, “Design of multi-band microwave filter with asymmetric polygonal microstrip loops”, 2016 8th International Conference on Electronics, Computers and Artificial Intelligence (ECAI), (2016).A Review on Passive Network Synthesis using Cauer Form46 Abhilasha Sharma and Teena Soni