Dual impedance

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Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance is its reciprocal, or algebraic inverse . For this reason the dual impedance is also called the inverse impedance. Another way of stating this is that the dual of is the admittance .

Electrical impedance intensive physical property

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

Contents

The dual of a network is the network whose impedances are the duals of the original impedances. In the case of a black-box network with multiple ports, the impedance looking into each port must be the dual of the impedance of the corresponding port of the dual network.

Port (circuit theory) pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy

In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit, that meets the port condition; the currents flowing into the two nodes must be equal and opposite.

This is consistent with the general notion duality of electric circuits, where the voltage and current are interchanged, etc., since yields [1]

In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and current in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the duality of electricity and magnetism.

Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.

Scaled and normalised duals

In physical units, the dual is taken with respect to some nominal or characteristic impedance. To do this, Z and Z' are scaled to the nominal impedance Z0 so that

Characteristic impedance ratio of the amplitudes of voltage and current of a single wave propagating along the line

The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

Z0 is usually taken to be a purely real number R0, so Z' is changed by a real factor of R02. In other words, the dual circuit is qualitatively the same circuit but all the component values are scaled by R02. [2] The scaling factor R02 has the dimensions of Ω2, so the constant 1 in the unitless expression would actually be assigned the dimensions Ω2 in a dimensional analysis.

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.

Duals of basic circuit elements

[3]
Element Z Dual Z'
Resistor R Dual Z 1.PNG
Resistor R
Conductor G = R Dual Z 2.PNG
Conductor G = R
Conductor G Dual Z 2.PNG
Conductor G
Resistor R = G Dual Z 1.PNG
Resistor R = G
Inductor L Dual Z 3.PNG
Inductor L
Capacitor C = L Dual Z 4.PNG
Capacitor C = L
Capacitor C Dual Z 4.PNG
Capacitor C
Inductor L = C Dual Z 3.PNG
Inductor L = C
Series impedances Z = Z1 + Z2 Dual Z 5.PNG
Series impedances Z = Z1 + Z2
Parallel admittances Y = Z1 + Z2 Dual Z 6.PNG
Parallel admittances Y = Z1 + Z2
Parallel impedances 1/Z = 1/Z1 + 1/Z2 Dual Z 6.PNG
Parallel impedances 1/Z = 1/Z1 + 1/Z2
Series admittances 1/Y = 1/Z1 + 1/Z2 Dual Z 5.PNG
Series admittances 1/Y = 1/Z1 + 1/Z2
Voltage generator V Dual Z 7.PNG
Voltage generator V
Current generator I = V Dual Z 8.PNG
Current generator I = V
Current generator I Dual Z 8.PNG
Current generator I
Voltage generator V = I Dual Z 7.PNG
Voltage generator V = I

Graphical method

There is a graphical method of obtaining the dual of a network which is often easier to use than the mathematical expression for the impedance. Starting with a circuit diagram of the network in question, Z, the following steps are drawn on the diagram to produce Z' superimposed on top of Z. Typically, Z' will be drawn in a different colour to help distinguish it from the original, or, if using computer-aided design, Z' can be drawn on a different layer.

Computer-aided design constructing a product by means of computer

Computer-aided design (CAD) is the use of computers to aid in the creation, modification, analysis, or optimization of a design. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, and to create a database for manufacturing. CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term CADD is also used.

  1. A generator is connected to each port of the original network. The purpose of this step is to prevent the ports from being "lost" in the inversion process. This happens because a port left open circuit will transform into a short circuit and disappear.
  2. A dot is drawn at the centre of each mesh of the network Z. These dots will become the circuit nodes of Z'.
  3. A conductor is drawn which entirely encloses the network Z. This conductor also becomes a node of Z'.
  4. For each circuit element of Z, its dual is drawn between the nodes in the centre of the meshes either side of Z. Where Z is on the edge of the network, one of these nodes will be the enclosing conductor from the previous step. [4]

This completes the drawing of Z'. This method also serves to demonstrate that the dual of a mesh transforms into a node and the dual of a node transforms into a mesh. Two examples are given below.

Example: star network

A star network of inductors, such as might be found on a three-phase transformer Graphic method 1.svg
A star network of inductors, such as might be found on a three-phase transformer
Attaching generators to the three ports Graphic method 2.svg
Attaching generators to the three ports
Nodes of the dual network Graphic method 3.svg
Nodes of the dual network
Components of the dual network Graphic method 4.svg
Components of the dual network
The dual network with the original removed and slightly redrawn to make the topology clearer Graphic method 5.svg
The dual network with the original removed and slightly redrawn to make the topology clearer
The dual network with the notional generators removed Graphic method 6.svg
The dual network with the notional generators removed

It is now clear that the dual of a star network of inductors is a delta network of capacitors. This dual circuit is not the same thing as a star-delta (Y-Δ) transformation. A Y-Δ transform results in an equivalent circuit, not a dual circuit.

Example: Cauer network

Filters designed using Cauer's topology of the first form are low-pass filters consisting of a ladder network of series inductors and shunt capacitors.

A low-pass filter implemented in Cauer topology Graphic method 7.svg
A low-pass filter implemented in Cauer topology
Attaching generators to the input and output ports Graphic method 8.svg
Attaching generators to the input and output ports
Nodes of the dual network Graphic method 9.svg
Nodes of the dual network
Components of the dual network Graphic method 10.svg
Components of the dual network
The dual network with the original removed and slightly redrawn to make the topology clearer Graphic method 11.svg
The dual network with the original removed and slightly redrawn to make the topology clearer

It can now be seen that the dual of a Cauer low-pass filter is still a Cauer low-pass filter. It does not transform into a high-pass filter as might have been expected. Note, however, that the first element is now a shunt component instead of a series component.

See also

Related Research Articles

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Butterworth filter

The Butterworth filter is a type of signal processing filter designed to have a frequency response as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".

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Electronic filter electronic circuit that removes unwanted components from the signal, or enhances wanted ones, or both

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Stub (electronics) short electrical transmission line

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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 proof of the theorem was presented by Ronald Martin Foster in 1924, although the principle had been published earlier by Foster's colleagues at American Telephone & Telegraph.

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Zobel network type of filter section based on the image-impedance design principle

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Lattice phase equaliser

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m-derived filters or m-type filters are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

Prototype filter

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

mm'-type filters, also called double-m-derived filters, are a type of electronic filter designed using the image method. They were patented by Otto Zobel in 1932. Like the m-type filter from which it is derived, the mm'-type filter type was intended to provide an improved impedance match into the filter termination impedances and originally arose in connection with telephone frequency division multiplexing. The filter has a similar transfer function to the m-type, having the same advantage of rapid cut-off, but the input impedance remains much more nearly constant if suitable parameters are chosen. In fact, the cut-off performance is better for the mm'-type if like-for-like impedance matching are compared rather than like-for-like transfer function. It also has the same drawback of a rising response in the stopband as the m-type. However, its main disadvantage is its much increased complexity which is the chief reason its use never became widespread. It was only ever intended to be used as the end sections of composite filters, the rest of the filter being made up of other sections such as k-type and m-type sections.

General m<sub>n</sub>-type image filter

These filters are electrical wave filters designed using the image method. They are an invention of Otto Zobel at AT&T Corp.. They are a generalisation of the m-type filter in that a transform is applied that modifies the transfer function while keeping the image impedance unchanged. For filters that have only one stopband there is no distinction with the m-type filter. However, for a filter that has multiple stopbands, there is the possibility that the form of the transfer function in each stopband can be different. For instance, it may be required to filter one band with the sharpest possible cut-off, but in another to minimise phase distortion while still achieving some attenuation. If the form is identical at each transition from passband to stopband the filter will be the same as an m-type filter. If they are different, then the general case described here pertains.

The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram. It is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.

An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

Bartlett's bisection theorem is an electrical theorem in network analysis attributed to Albert Charles Bartlett. The theorem shows that any symmetrical two-port network can be transformed into a lattice network. The theorem often appears in filter theory where the lattice network is sometimes known as a filter X-section following the common filter theory practice of naming sections after alphabetic letters to which they bear a resemblance.

RLC circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

Frequency selective surface

A frequency-selective surface (FSS) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

A symmetrical lattice is a two-port electrical wave filter in which diagonally-crossed shunt elements are present – a configuration which sets it apart from ladder networks. The component arrangement of the lattice is shown in the diagram below. The filter properties of this circuit were first developed using image impedance concepts, but later the more general techniques of network analysis were applied to it.

References

  1. Ghosh, pp. 50–51
  2. Redifon, p.44
  3. Guillemin, pp. 535–539
  4. Guillemin, pp. 49–52
    Suresh, pp. 516–517

Bibliography