Foster's reactance theorem is an important theorem in the fields of electrical network analysis and synthesis. The theorem states that the reactance of a passive, lossless two-terminal one-port network always strictly monotonically increases with frequency. It is easily seen that the reactances of inductors and capacitors individually increase with frequency and from that basis a proof for passive lossless networks generally can be constructed. The theorem can be extended to admittances and the encompassing concept of immittances. A consequence of Foster's theorem is that zeros and poles of the reactance must alternate with frequency.

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Full Name Comment goes here. Are you sure you want to Yes No. Utkarsh Raghav. Show More. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. We will now look at this from the other side — can we construct physical objects that correspond to a given abstraction? We will first look at the relatively simple problem of constructing passive circuits that correspond to a given network function. By this time, you are of course aware that the function itself should have certain characteristics if it is to correspond to a realisable network.

In the special but all-important case of the design of filters to meet specific requirements, we have to go one step backwards to obtain a system function that ensures such compliance, before we can obtain a realisation. We will then look at a few standard methods for the realisation of functional blocks such as filters using active circuits. The treatment up to this has been limited to linear circuits.

We have seen that the state space approach allows us to model non-linear systems as well as linear systems. As far as circuits are concerned, we still need mechanisms for the modelling of non-linear elements, even the simplest of them; the diode.

We will now step two steps back and investigate some elementary methods for the modelling of non-linear elements. We also need at least some idea about the sensitivity of network characteristics to changes in component values.

We will look at how this helps us in the design of robust systems. We have also studied how this can be done. We are now going to explore how a network can be synthesised, given its network function.

We will first look at one port networks, defined by either impedance or admittance functions and later go on to investigate two port networks. Let us recall some of the common properties of passive RLC functions. They are real, rational functions. The resulting matrices are symmetric.

Let us now look at each of these types of networks. The LC network functions have these additional properties: They are simple, that is there are no higher order poles or zeros. Poles and zeros alternate. The origin and infinity are always critical frequencies, that is, there will be either a pole or a zero at both the origin and at infinity. The multiplicative constant is positive. The properties of RC network functions are: The poles and zeros of an RC driving point function lie on the non- positive real axis.

They are simple. The slopes of impedance functions are negative, those of admittance functions are positive. Foster and Cauer first proposed the realisation of these networks in various canonical forms, in the s.

We will now examine each of these forms. There are four canonical forms, relating to the realisation of LC, RC and RL networks as follows: 1 st Foster form 2 nd Foster form 1 st Cauer form 2 nd Cauer form Partial fraction expansion of impedance function b Partial fraction expansion of admittance function b Continued fraction expansion about the point at infinity b Continued fraction expansion about the origin b 3.

Chapter 3 — Synthesis of analogue circuits Considering the pole-zero properties of LC network functions, we can assume that the impedance function will have a zero or pole at the origin and pairs of complex conjugate poles and zeros.

The admittance function will have a pole or zero at the origin. These functions may be expanded as partial fractions to yield realisations of Foster-form networks.

They may also be subjected to continued fraction expansion to yield Cauer-form networks. We will consider realisations of each of these forms. Chapter 3 — Synthesis of analogue circuits We will now consider each of the four examples used to illustrate the Foster forms.

This too results in a realisation where the series arms are impedances while the shunt arms are admittances. We will consider the same examples as before. However, in this realisation the series arms contain capacitors while the shunt arms have inductors. With these properties, the form of the impedance function may be represented by Chapter 3 — Synthesis of analogue circuits There are four possible combinations that we need to study.

We used the following general form of the impedance of an RC function Chapter 3 — Synthesis of analogue circuits We now consider a function of the same form as the admittance function of an RL network The remainder function, that is, Z1 s is still positive real, for its real part is still positive and all its poles are on the non-positive left half of the s-plane. Z1 s is a minimum reactance function. We now repeat Step 1, and then Step 2, repeatedly until the remainder is both minimum reactive and minimum susceptive.

This may be illustrated by a flow chart as follows: Start Remove reactance to yield minimum reactive remainder Remove susceptance to yield minimum susceptive remainder Realise the remainder Stop Yes No.

Is remainder both minimum reactive and minimum susceptive? Chapter 3 — Synthesis of analogue circuits Note: Steps 1 and 2 may be interchanged. We should choose the sequence that gives rise to a simpler realisation. Chapter 3 — Synthesis of analogue circuits Brune synthesis Brune synthesis starts with the Foster preamble, to remove reactive and susceptive components to yield a minimum reactive and minimum susceptive function.

This would be of the form: 0,,, Z4 s is of the same form as Z s , except that it is of order n-2 instead of n. We have completed one cycle of the Brune cycle. There are two possible cases to be considered in going through this cycle. One is the possibility that L1 is negative, leaving Z2 s positive real, and the other is when L1 is positive, leaving Z2 s non-positive real.

In either case, we will end up with the structure shown at the end of the Brune cycle, with either L1 or L3 negative. Stop Yes No. Chapter 3 — Synthesis of analogue circuits 3. As there are no resistance elements, there will be no filter loss in such a filter. Let us see what happens when we connect such a lossless LC filter between a source and a load. R1 E1 R2E2 P The figure shows the voltage and power distribution before and after the insertion of a lossless LC filter.

This is also known as the transmission ratio. This general form is used in the specification of the Butterworth filter to denote the attenuation of power over the frequency range of interest. In general, we consider the y-axis as representing power or the square of the voltage, and the x- axis as representing the normalised frequency. Example: Assume that we want to design a filter with a pass band of 0 — 5 MHz. Chapter 3 — Synthesis of analogue circuits 99 5.

Instead of 3dB attenuation corresponding to half-power , let us assume that we wanted the attenuation in the pass band to be limited to 0. Further example: Low pass filter, with the following specifications: Pass band 0 — 20 kHz.

In this case, we will have to resort to numerical methods to factorise the expression into two parts. As before, we will end up with a filter with three components. However, for filters with tighter specifications, the Chebyschev approximation yields lower degree implementations than the Butterworth approximation.

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## Network synthesis

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HP 6N136 PDF

## Foster's reactance theorem

Network synthesis is a design technique for linear electrical circuits. Synthesis starts from a prescribed impedance function of frequency or frequency response and then determines the possible networks that will produce the required response. The technique is to be compared to network analysis in which the response or other behaviour of a given circuit is calculated. Network synthesis was a great leap forward in circuit design.

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