Image impedance

Last updated April 08, 2024

Image impedance is a concept used in electronic network design and analysis and most especially in filter design. The term image impedance applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, Z_i 1, seen looking into port 1 when port 2 is terminated with the image impedance, Z_i 2, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical (or anti-symmetrical) with respect to the ports.

Derivation

As an example, the derivation of the image impedances of a simple 'L' network is given below. The 'L' network consists of a series impedance, $Z$ , and a shunt admittance, $Y$ .

The difficulty here is that in order to find $Z$ _i 1 it is first necessary to terminate port 2 with $Z$ _i 2. However, $Z$ _i 2 is also an unknown at this stage. The problem is solved by terminating port 2 with an identical network: port 2 of the second network is connected to port 2 of the first network and port 1 of the second network is terminated with $Z$ _i 1. The second network is terminating the first network in $Z$ _i 2 as required. Mathematically, this is equivalent to eliminating one variable from a set of simultaneous equations. The network can now be solved for $Z$ _i 1. Writing out the expression for input impedance gives:

Z_{i1}=Z+{\frac {1}{2Y+{\frac {1}{Z+Z_{i1}}}}}

and solving for $Z_{i1}\ ,$

Z_{i1}^{2}=Z^{2}+{\frac {Z}{Y}}

$Z$ _i 2 is found by a similar process, but it is simpler to work in terms of the reciprocal, that is image admittance $Y$ _i 2,

Y_{i2}^{2}=Y^{2}+{\frac {Y}{Z}}~.

Also, it can be seen from these expressions that the two image impedances are related to each other by:

{\frac {Z_{i1}}{Y_{i2}}}={\frac {Z}{Y}}~.

Measurement

Directly measuring image impedance by adjusting terminations is inconveniently iterative and requires precision adjustable components to effect the termination. An alternative technique to determine the image impedance of port 1 is to measure the short-circuit impedance Z_SC (that is, the input impedance of port 1 when port 2 is short-circuited) and the open-circuit impedance Z_OC (the input impedance of port 1 when port 2 is open-circuit). The image impedance is then given by,

Z_{i1}={\sqrt {Z_{\mathrm {SC} }Z_{\mathrm {OC} }}}

This method requires no prior knowledge of the topology of the network being measured.

Usage in filter design

When used in filter design, the 'L' network analysed above is usually referred to as a half section. Two half sections in cascade will make either a T section or a Π section depending on which port of the L section comes first. This leads to the terminology of Z_i T to mean the Z_i 1 in the above analysis and Z_i Π to mean Z_i 2.

Relation to characteristic impedance

Image impedance is a similar concept to the characteristic impedance used in the analysis of transmission lines. In fact, in the limiting case of a chain of cascaded networks where the size of each single network is approaching an infinitesimally small element, the mathematical limit of the image impedance expression is the characteristic impedance of the chain.^[1]^[2]^[3] That is,

Z_{i}^{2}\rightarrow {\frac {Z}{Y}}

The connection between the two can further be seen by noting an alternative, but equivalent, definition of image impedance. In this definition, the image impedance of a network is the input impedance of an infinitely long chain of cascaded identical networks (with the ports arranged so that like impedance faces like). This is directly analogous to the definition of characteristic impedance as the input impedance of an infinitely long line.

Conversely, it is possible to analyse a transmission line with lumped components, such as one utilising loading coils, in terms of an image impedance filter.

Transfer function

The transfer function of the half section, like the image impedance, is calculated for a network terminated in its image impedances (or equivalently, for a single section in an infinitely long chain of identical sections) and is given by,

A(i\omega )={\sqrt {\frac {Z_{I2}}{Z_{I1}}}}e^{-\gamma }

where $γ$ is called the transmission function, propagation function or transmission parameter and is given by,

\gamma =\sinh ^{-1}{\sqrt {ZY}}

The ${\sqrt {\frac {Z_{I2}}{Z_{I1}}}}$ term represents the voltage ratio that would be observed if the maximum available power was transferred from the source to the load. It would be possible to absorb this term into the definition of $γ$ , and in some treatments this approach is taken. In the case of a network with symmetrical image impedances, such as a chain of an even number of identical L sections, the expression reduces to,

A(i\omega )=e^{-\gamma }\,\!

In general, $γ$ is a complex number such that,

\gamma =\alpha +i\beta \,\!

The real part of $γ$ , represents an attenuation parameter, $α$ in nepers and the imaginary part represents a phase change parameter, $β$ in radians. The transmission parameters for a chain of n half sections, provided that like impedance always faces like, is given by;

\gamma _{n}=n\gamma \,\!

As with the image impedance, the transmission parameters approach those of a transmission line as the filter section become infinitesimally small so that,

\gamma \rightarrow {\sqrt {ZY}}

with $α$ , $β$ , $γ$ , $Z$ , and $Y$ all now being measured per metre instead of per half section.

Relationship to two-port network parameters

ABCD parameters

For a reciprocal network ( $AD - BC =1$ ), the image impedances can be expressed^[4] in terms of ABCD parameters as,

Z_{I1}={\sqrt {\frac {AB}{CD}}}

Z_{I2}={\sqrt {\frac {DB}{CA}}}

.

The image propagation term, $γ$ may be expressed as,

\gamma =\cosh ^{-1}{\sqrt {AD}}

.

Note that the image propagation term for a transmission line segment is equivalent to the Propagation constant of the transmission line times the length.

Image filter sections

Unbalanced
L Half section	T Section	Π Section

Ladder network

Balanced
C Half-section	H Section		Box Section

Ladder network

X Section (mid-T-Derived)		X Section (mid-Π-Derived)

N.B.	Textbooks and design drawings usually show the unbalanced implementations, but in telecoms it is often required to convert the design to the balanced implementation when used with balanced lines.	edit

Related Research Articles

<span class="mw-page-title-main">Characteristic impedance</span> Property of an electrical circuit

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.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

In radio engineering and telecommunications, standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances. However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

In electrical engineering and electronics, a network is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.

In electronics, a two-port network is an electrical network or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port.

An attenuator is an electronic device that reduces the power of a signal without appreciably distorting its waveform.

Impedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Y-parameters, H-parameters, T-parameters or ABCD-parameters.

The Π pad is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the Greek capital letter pi (Π).

Constant k filters, also k-type filters, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

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 passband to a pole of attenuation just inside the stopband. 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 stopband 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.

Filters designed using the image impedance methodology suffer from a peculiar flaw in the theory. The predicted characteristics of the filter are calculated assuming that the filter is terminated with its own image impedances at each end. This will not usually be the case; the filter will be terminated with fixed resistances. This causes the filter response to deviate from the theoretical. This article explains how the effects of image filter end terminations can be taken into account.

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.

A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a transmission line or waveguide used in electrical engineering of length one-quarter wavelength (λ), terminated with some known impedance. It presents at its input the dual of the impedance with which it is terminated.

An antimetric electrical network is an electrical network that exhibits anti-symmetrical electrical properties. The term is often encountered in filter theory, but it applies to general electrical network analysis. Antimetric is the diametrical opposite of symmetric; it does not merely mean "asymmetric". It is possible for networks to be symmetric or antimetric in their electrical properties without being physically or topologically symmetric or antimetric.

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

The T pad is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the letter "T".

Iterative impedance is the input impedance of an infinite chain of identical networks. It is related to the image impedance used in filter design, but has a simpler, more straightforward definition.

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.

Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.

References

↑ Lee, Thomas H. (2004). "2.5. Driving-point impedance of Iterated Structure". Planar microwave engineering: a practical guide to theory, measurement, and circuits. Cambridge University Press. p. 44.
↑ Niknejad, Ali M. (2007). "Section 9.2. An Infinite Ladder Network.". Electromagnetics for high-speed analog and digital communication circuits.
↑ Feynman, Richard; Leighton, Robert B.; Sands, Matthew. "Section 22-7. Filter". The Feynman Lectures on Physics. Vol. 2. If we imagine the line as broken up into small lengths Δℓ, each length will look like one section of the L-C ladder with a series inductance ΔL and a shunt capacitance ΔC. We can then use our results for the ladder filter. If we take the limit as Δℓ goes to zero, we have a good description of the transmission line. Notice that as Δℓ is made smaller and smaller, both ΔL and ΔC decrease, but in the same proportion, so that the ratio ΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) as ΔL and ΔC go to zero, we find that the characteristic impedance z0 is a pure resistance whose magnitude is √(ΔL/ΔC). We can also write the ratio ΔL/ΔC as L0/C0, where L0 and C0 are the inductance and capacitance of a unit length of the line; then we have ${\sqrt {\frac {L_{0}}{C_{0}}}}$ .
↑ Pedro L. D. Peres, Carlos R. de Souza, Ivanil S. Bonatti, "ABCD matrix: a unique tool for linear two-wire transmission line modelling", The International Journal of Electrical Engineering & Education, vol. 40, iss. 3, pp. 220–229, 2003, archived 4 March 2016.

Matthaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[lee2004-1] Lee, Thomas H. (2004). "2.5. Driving-point impedance of Iterated Structure". Planar microwave engineering: a practical guide to theory, measurement, and circuits. Cambridge University Press. p. 44.

[niknejad2007-2] Niknejad, Ali M. (2007). "Section 9.2. An Infinite Ladder Network.". Electromagnetics for high-speed analog and digital communication circuits.

[3] Feynman, Richard; Leighton, Robert B.; Sands, Matthew. "Section 22-7. Filter". The Feynman Lectures on Physics. Vol. 2. If we imagine the line as broken up into small lengths Δℓ, each length will look like one section of the L-C ladder with a series inductance ΔL and a shunt capacitance ΔC. We can then use our results for the ladder filter. If we take the limit as Δℓ goes to zero, we have a good description of the transmission line. Notice that as Δℓ is made smaller and smaller, both ΔL and ΔC decrease, but in the same proportion, so that the ratio ΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) as ΔL and ΔC go to zero, we find that the characteristic impedance z0 is a pure resistance whose magnitude is √(ΔL/ΔC). We can also write the ratio ΔL/ΔC as L0/C0, where L0 and C0 are the inductance and capacitance of a unit length of the line; then we have ${\sqrt {\frac {L_{0}}{C_{0}}}}$ .

[4] Pedro L. D. Peres, Carlos R. de Souza, Ivanil S. Bonatti, "ABCD matrix: a unique tool for linear two-wire transmission line modelling", The International Journal of Electrical Engineering & Education, vol. 40, iss. 3, pp. 220–229, 2003, archived 4 March 2016.

[1]

[2]

[3]

[4]