Commensurate Line Circuit
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Commensurate line circuits are electrical circuits composed of
transmission 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 transmis ...
s that are all the same length; commonly one-eighth of a
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
.
Lumped element The lumped-element model (also called lumped-parameter model, or lumped-component model) is a simplified representation of a physical system or circuit that assumes all components are concentrated at a single point and their behavior can be de ...
circuits can be directly converted to
distributed-element circuit Distributed-element circuits are electrical circuits composed of lengths of transmission lines or other distributed components. These circuits perform the same functions as conventional circuits composed of passive components, such as capacitors, ...
s of this form by the use of Richards' transformation. This transformation has a particularly simple result;
inductor An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s are replaced with transmission lines terminated in short-circuits and
capacitor In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
s are replaced with lines terminated in open-circuits. Commensurate line theory is particularly useful for designing
distributed-element filter A distributed-element filter is an electronic filter in which capacitance, inductance, and resistance (the elements of the circuit) are not localised in discrete capacitors, inductors, and resistors as they are in conventional filters. Its pu ...
s for use at
microwave Microwave is a form of electromagnetic radiation with wavelengths shorter than other radio waves but longer than infrared waves. Its wavelength ranges from about one meter to one millimeter, corresponding to frequency, frequencies between 300&n ...
frequencies. It is usually necessary to carry out a further transformation of the circuit using Kuroda's identities. There are several reasons for applying one of the Kuroda transformations; the principal reason is usually to eliminate series connected components. In some technologies, including the widely used
microstrip Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as ''substrate''. Microstrip lines are used to convey microwave-freq ...
, series connections are difficult or impossible to implement. The frequency response of commensurate line circuits, like all distributed-element circuits, will periodically repeat, limiting the frequency range over which they are effective. Circuits designed by the methods of Richards and Kuroda are not the most compact. Refinements to the methods of coupling elements together can produce more compact designs. Nevertheless, the commensurate line theory remains the basis for many of these more advanced filter designs.


Commensurate lines

Commensurate lines are
transmission 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 transmis ...
s that are all the same electrical length, but not necessarily the same
characteristic impedance 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 wave travelling in one direction along the line in the absence of reflections in th ...
(''Z''0). A commensurate line circuit is an electrical circuit composed only of commensurate lines terminated with
resistor A resistor is a passive two-terminal electronic component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active e ...
s or short- and open-circuits. In 1948, Paul I. Richards published a theory of commensurate line circuits by which a passive
lumped element The lumped-element model (also called lumped-parameter model, or lumped-component model) is a simplified representation of a physical system or circuit that assumes all components are concentrated at a single point and their behavior can be de ...
circuit could be transformed into a
distributed element In electrical engineering, the distributed-element model or transmission-line model of electrical Electrical network, circuits assumes that the attributes of the circuit (Electrical resistance, resistance, capacitance, and inductance) are distri ...
circuit with precisely the same characteristics over a certain frequency range.Levy & Cohn, p. 1056 Lengths of lines in
distributed-element circuit Distributed-element circuits are electrical circuits composed of lengths of transmission lines or other distributed components. These circuits perform the same functions as conventional circuits composed of passive components, such as capacitors, ...
s, for generality, are usually expressed in terms of the circuit's nominal operational wavelength, λ. Lines of the prescribed length in a commensurate line circuit are called ''unit elements'' (UEs). A particularly simple relationship pertains if the UEs are λ/8. Each element in the lumped circuit is transformed into a corresponding UE. However, ''Z''0 of the lines must be set according to the component value in the analogous lumped circuit and this may result in values of ''Z''0 that are not practical to implement. This is particularly a problem with printed technologies, such as
microstrip Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as ''substrate''. Microstrip lines are used to convey microwave-freq ...
, when implementing high characteristic impedances. High impedance requires narrow lines and there is a minimum size that can be printed. Very wide lines, on the other hand, allow the possibility of undesirable transverse resonant modes to form. A different length of UE, with a different ''Z''0, may be chosen to overcome these problems. Electrical length can also be expressed as the
phase change Phase change may refer to: * Phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term ...
between the start and the end of the line.
Phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...
is measured in angle units. \theta, the mathematical symbol for an angle variable, is used as the symbol for electrical length when expressed as an angle. In this convention λ represents 360°, or 2π
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s. The advantage of using commensurate lines is that the commensurate line theory allows circuits to be synthesised from a prescribed frequency function. While any circuit using arbitrary transmission line lengths can be analysed to determine its frequency function, that circuit cannot necessarily be easily synthesised starting from the frequency function. The fundamental problem is that using more than one length generally requires more than one frequency variable. Using commensurate lines requires only one frequency variable. A well developed theory exists for synthesising lumped-element circuits from a given frequency function. Any circuit so synthesised can be converted to a commensurate line circuit using Richards' transformation and a new frequency variable.


Richards' transformation

Richards' transformation transforms the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
variable, ω, according to, : \omega \to \tan (k \omega) or, more usefully for further analysis, in terms of the
complex frequency In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the complex-valued freq ...
variable, ''s'', : s \to \tanh (k s) :where ''k'' is an arbitrary constant related to the UE length, θ, and some designer chosen reference frequency, ωc, by : k \omega_c = \theta. :''k'' has units of time and is, in fact, the
phase delay In signal processing, group delay and phase delay are functions that describe in different ways the delay times experienced by a signal’s various sinusoidal frequency components as they pass through a linear time-invariant (LTI) system (such as ...
inserted by a UE. Comparing this transform with expressions for the driving point impedance of stubs terminated, respectively, with a short circuit and an open circuit, : \begin Z_\mathrm &= j Z_0 \tan(k \omega) \\ Z_\mathrm &= -j Z_0 \cot (k \omega) \\ \end it can be seen that (for θ < π/2) a short circuit stub has the impedance of a lumped
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the ...
and an open circuit stub has the impedance of a lumped
capacitance Capacitance is the ability of an object to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are two closely related ...
. Richards' transformation substitutes
inductor An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s with short circuited UEs and
capacitor In electrical engineering, a capacitor is a device that stores electrical energy by accumulating electric charges on two closely spaced surfaces that are insulated from each other. The capacitor was originally known as the condenser, a term st ...
s with open circuited UEs. When the length is λ/8 (or θ=π/4), this simplifies to, : \begin Z_\mathrm &= j Z_0 \\ Z_\mathrm &= -j Z_0 \\ \end This is frequently written as, : \begin Z_\mathrm &= j L \\ Z_\mathrm &= \\ \end ''L'' and ''C'' are conventionally the symbols for inductance and capacitance, but here they represent respectively the characteristic impedance of an inductive stub and the characteristic
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the multiplicative inverse, reciprocal of Electrical impedance, impedance, analogous to how Electrical resistanc ...
of a capacitive stub. This convention is used by numerous authors, and later in this article.


Omega-domain

Richards' transformation can be viewed as transforming from a
s-domain In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the complex-valued fre ...
representation to a new domain called the Ω-domain where, : \Omega = \tan ( k \omega ) If Ω is normalised so that Ω=1 when ω=ωc, then it is required that, : k \omega_c = \theta = and the length in distance units becomes, : \ell = Any circuit composed of discrete, linear, lumped components will have a
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
''H''(''s'') that is a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
in ''s''. A circuit composed of transmission line UEs derived from the lumped circuit by Richards' transformation will have a transfer function ''H''(''j''Ω) that is a rational function of precisely the same form as ''H''(''s''). That is, the shape of the frequency response of the lumped circuit against the ''s'' frequency variable will be precisely the same as the shape of the frequency response of the transmission line circuit against the ''j''Ω frequency variable and the circuit will be functionally the same. However, infinity in the Ω domain is transformed to ω=π/4''k'' in the ''s'' domain. The entire frequency response is squeezed down to this finite interval. Above this frequency, the same response is repeated in the same intervals, alternately in reverse. This is a consequence of the periodic nature of the
tangent function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
. This multiple passband result is a general feature of all distributed-element circuits, not just those arrived at through Richards' transformation.


Cascade element

A UE connected in cascade is a
two-port network In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two ''pairs'' of Terminal (electronics), terminals to connect to external circuits. Two terminals consti ...
that has no exactly corresponding circuit in lumped elements. It is functionally a fixed delay. There are lumped-element circuits that can approximate a fixed delay such as the
Bessel filter In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group delay (i.e., maximally linear phase response), which preserves the wave shape of filtered signals in ...
, but they only work within a prescribed
passband A passband is the range of frequency, frequencies or wavelengths that can pass through a Filter (signal processing), filter. For example, a radio receiver contains a bandpass filter to select the frequency of the desired radio signal out of all t ...
, even with ideal components. Alternatively, lumped-element
all-pass filter An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to it ...
s can be constructed that pass all frequencies (with ideal components), but they have constant delay only within a narrow band of frequencies. Examples are the lattice phase equaliser and bridged T delay equaliser. There is consequently no lumped circuit that Richard's transformation can transform into a cascade-connected line, and there is no reverse transformation for this element. Commensurate line theory thus introduces a new element of ''delay'', or ''length''. Two or more UEs connected in cascade with the same ''Z''0 are equivalent to a single, longer, transmission line. Thus, lines of length ''n''θ for integer ''n'' are allowable in commensurate circuits. Some circuits can be implemented ''entirely'' as a cascade of UEs:
impedance matching In electrical engineering, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or ...
networks, for instance, can be done this way, as can most filters.


Kuroda's identities

Kuroda's identities are a set of four equivalent circuits that overcome certain difficulties with applying Richards' transformations directly. The four basic transformations are shown in the figure. Here the symbols for capacitors and inductors are used to represent open-circuit and short-circuit stubs. Likewise, the symbols ''C'' and ''L'' here represent respectively the
susceptance In electrical engineering, susceptance () is the imaginary part of admittance (), where the real part is conductance (). The reciprocal of admittance is impedance (), where the imaginary part is reactance () and the real part is resistance ( ...
of an open circuit stub and the reactance of a short circuit stub, which, for θ=λ/8, are respectively equal to the characteristic
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the multiplicative inverse, reciprocal of Electrical impedance, impedance, analogous to how Electrical resistanc ...
and characteristic impedance of the stub line. The boxes with thick lines represent cascade connected commensurate lengths of line with the marked characteristic impedance. The first difficulty solved is that all the UEs are required to be connected together at the same point. This arises because the lumped-element model assumes that all the elements take up zero space (or no significant space) and that there is no delay in signals between the elements. Applying Richards' transformation to convert the lumped circuit into a distributed circuit allows the element to now occupy a finite space (its length) but does not remove the requirement for zero distance between the interconnections. By repeatedly applying the first two Kuroda identities, UE lengths of the lines feeding into the
ports Ports collections (or ports trees, or just ports) are the sets of makefiles and Patch (Unix), patches provided by the BSD-based operating systems, FreeBSD, NetBSD, and OpenBSD, as a simple method of installing software or creating binary packages. T ...
of the circuit can be moved between the circuit components to physically separate them. A second difficulty that Kuroda's identities can overcome is that series connected lines are not always practical. While series connection of lines can easily be done in, for instance, coaxial technology, it is not possible in the widely used microstrip technology and other planar technologies. Filter circuits frequently use a
ladder topology Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their t ...
with alternating series and shunt elements. Such circuits can be converted to all shunt components in the same step used to space the components with the first two identities. The third and fourth identities allow characteristic impedances to be scaled down or up respectively. These can be useful for transforming impedances that are impractical to implement. However, they have the disadvantage of requiring the addition of an ideal transformer with a turns ratio equal to the scaling factor.


History

In the decade after Richards' publication, advances in the theory of distributed circuits took place mostly in Japan. K. Kuroda published these identities in 1955 in his Ph.D. thesis. However, they did not appear in English until 1958 in a paper by Ozaki and Ishii on
stripline In electronics, stripline is a transverse electromagnetic (TEM) transmission line medium invented by Robert M. Barrett of the Air Force Cambridge Research Centre in the 1950s. Stripline is the earliest form of planar transmission line. De ...
filters.Levy & Cohn, p. 1057


Further refinements

One of the major applications of commensurate line theory is to design
distributed-element filter A distributed-element filter is an electronic filter in which capacitance, inductance, and resistance (the elements of the circuit) are not localised in discrete capacitors, inductors, and resistors as they are in conventional filters. Its pu ...
s. Such filters constructed directly by Richards' and Kuroda's method are not very compact. This can be an important design consideration, especially in mobile devices. The stubs stick out to the side of the main line and the space between them is not doing anything useful. Ideally, the stubs should project on alternate sides to prevent them coupling with each other, taking up further space, although this is not always done for space considerations. More than that, the cascade connected elements that couple together the stubs contribute nothing to the frequency function, they are only there to transform the stubs into the required impedance. Putting it another way, the
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
of the frequency function is determined solely by the number of stubs, not by the total number of UEs (generally speaking, the higher the order, the better the filter). More complex synthesis techniques can produce filters in which all elements are contributing. The cascade connected λ/8 sections of the Kuroda circuits are an example of impedance transformers, the archetypical example of such circuits is the λ/4 impedance transformer. Although this is double the length of the λ/8 line it has the useful property that it can be transformed from a
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
to a
high-pass filter A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency ...
by replacing the open circuit stubs with short circuit stubs. The two filters are exactly matched with the same cut-off frequency and mirror-symmetrical responses. It is therefore ideal for use in
diplexer A diplexer is a passive device that implements frequency-domain multiplexing. Two ports (e.g., L and H) are multiplexed onto a third port (e.g., S). The signals on ports L and H occupy disjoint frequency bands. Consequently, the signals on L an ...
s. The λ/4 transformer has this property of being invariant under a low-pass to high-pass transformation because it is not just an impedance transformer, but a special case of transformer, an impedance inverter. That is, it transforms any impedance network at one port, to the inverse impedance, or
dual impedance Dual Electrical impedance, impedance and dual network are terms used in Network analysis (electronics), electronic network analysis. The dual of an impedance Z is its reciprocal, or algebraic inverse Z'=\frac. For this reason, the dual impedance i ...
, at the other port. However, a single length of transmission line can only be precisely λ/4 long at its resonant frequency and there is consequently a limit to the
bandwidth Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
over which it will work. There are more complex kinds of inverter circuit that more accurately invert impedances. There are two classes of inverter, the ''J''-inverter, which transforms a shunt admittance into a series impedance, and the ''K''-inverter which does the reverse transformation. The coefficients ''J'' and ''K'' are respectively the scaling admittance and impedance of the converter. Stubs may be lengthened in order to change from an open circuit to a short circuit stub and vice versa. Low-pass filters usually consist of series inductors and shunt capacitors. Applying Kuroda's identities will convert these to all shunt capacitors, which are open circuit stubs. Open circuit stubs are preferred in printed technologies because they are easier to implement, and this is the technology likely to be found in consumer products. However, this is not the case in other technologies such as coaxial line, or twin-lead where the short circuit may actually be helpful for mechanical support of the structure. Short circuits also have a small advantage in that they are generally have a more precise position than open circuits. If the circuit is to be further transformed into the
waveguide A waveguide is a structure that guides waves by restricting the transmission of energy to one direction. Common types of waveguides include acoustic waveguides which direct sound, optical waveguides which direct light, and radio-frequency w ...
medium then open circuits are out of the question because there would be radiation out of the aperture so formed. For a high-pass filter the inverse applies, applying Kuroda will naturally result in short circuit stubs and it may be desirable for a printed design to convert to open circuits. As an example, a λ/8 open circuit stub can be replaced with a 3λ/8 short circuit stub of the same characteristic impedance without changing the circuit functionally. Coupling elements together with impedance transformer lines is not the most compact design. Other methods of coupling have been developed, especially for
band-pass filter A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range. It is the inverse of a '' band-stop filter''. Description In electronics and s ...
s that are far more compact. These include parallel lines filters, interdigital filters, hairpin filters, and the semi-lumped design combline filters.


References


Bibliography

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"Microwave filter design using radial line stubs"
''1988 IEEE Region 5 Conference: Spanning the Peaks of Electrotechnology'', p. 68-72, IEEE, March 1988. * Helszajn, Joseph, ''Synthesis of Lumped Element, Distributed and Planar Filters'', McGraw-Hill, 1990 . * Hunter, Ian C., ''Theory and Design of Microwave Filters'', IET, 2001 . * Kumar, Narendra; Grebennikov, Andrei; ''Distributed Power Amplifiers for RF and Microwave Communications'', Artech House, 2015 . * Lee, Thomas H., ''Planar Microwave Engineering'', vol. 1, Cambridge University Press, 2004 . * Levy, Ralph; Cohn, Seymour B.
"A History of Microwave Filter Research, Design, and Development"
''IEEE Transactions on Microwave Theory and Techniques'', vol. 32, iss. 9, pp. 1055–1067, September 1984. * Maloratsky, Leo, ''Passive RF & Microwave Integrated Circuits'', Elsevier, 2003 . * Matthaei, George L.; Young, Leo; Jones, E. M. T. ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'' McGraw-Hill 1964 . * Ozaki, H.; Ishii, J.
"Synthesis of a class of strip-line filters"
''IRE Transactions on Circuit Theory'', vol. 5, iss. 2, pp. 104–109. June 1958. * Richards, Paul I.
"Resistor-transmission-line circuits"
''Proceedings of the IRE'', vol. 36, iss. 2, pp. 217–220, 1948. * Sisodia, M. L., ''Microwaves: Introduction To Circuits, Devices And Antennas'', New Age International, 2007 . * Wen, Geyi, ''Foundations for Radio Frequency Engineering'', World Scientific, 2015 . * Wiek, Martin, ''Fiber Optics Standard Dictionary'', Springer, 1997 {{ISBN, 0412122413. Filter theory Distributed element circuits Microwave technology Circuit theorems Analog circuits Linear filters Electronic design