Distributed-element circuit
Distributed-element circuits are
Conventional circuits consist of individual components manufactured separately then connected together with a conducting medium. Distributed-element circuits are built by forming the medium itself into specific patterns. A major advantage of distributed-element circuits is that they can be produced cheaply as a
A phenomenon commonly used in distributed-element circuits is that a length of transmission line can be made to behave as a resonator. Distributed-element components which do this include stubs, coupled lines, and cascaded lines. Circuits built from these components include filters, power dividers, directional couplers, and circulators.
Distributed-element circuits were studied during the 1920s and 1930s but did not become important until World War II, when they were used in radar. After the war their use was limited to military, space, and broadcasting infrastructure, but improvements in materials science in the field soon led to broader applications. They can now be found in domestic products such as satellite dishes and mobile phones.
Circuit modelling
Distributed-element circuits are designed with the
There is no clear-cut demarcation in the frequency at which these models should be used. Although the changeover is usually somewhere in the 100-to-500 MHz range, the technological scale is also significant; miniaturised circuits can use the lumped model at a higher frequency.
Construction with transmission lines
The overwhelming majority of distributed-element circuits are composed of lengths of transmission line, a particularly simple form to model. The cross-sectional dimensions of the line are unvarying along its length, and are small compared to the signal wavelength; thus, only distribution along the length of the line need be considered. Such an element of a distributed circuit is entirely characterised by its length and characteristic impedance. A further simplification occurs in commensurate line circuits, where all the elements are the same length. With commensurate circuits, a lumped circuit design prototype consisting of capacitors and inductors can be directly converted into a distributed circuit with a one-to-one correspondence between the elements of each circuit.[3]
Commensurate line circuits are important because a design theory for producing them exists; no general theory exists for circuits consisting of arbitrary lengths of transmission line (or any arbitrary shapes). Although an arbitrary shape can be analysed with Maxwell's equations to determine its behaviour, finding useful structures is a matter of trial and error or guesswork.[4]
An important difference between distributed-element circuits and lumped-element circuits is that the frequency response of a distributed circuit periodically repeats as shown in the
Advantages and disadvantages
Distributed-element circuits are cheap and easy to manufacture in some formats, but take up more space than lumped-element circuits. This is problematic in mobile devices (especially hand-held ones), where space is at a premium. If the operating frequencies are not too high, the designer may miniaturise components rather than switching to distributed elements. However, parasitic elements and resistive losses in lumped components are greater with increasing frequency as a proportion of the nominal value of the lumped-element impedance. In some cases, designers may choose a distributed-element design (even if lumped components are available at that frequency) to benefit from improved quality. Distributed-element designs tend to have greater power-handling capability; with a lumped component, all the energy passed by a circuit is concentrated in a small volume.[6]
Media
Paired conductors
Several types of transmission line exist, and any of them can be used to construct distributed-element circuits. The oldest (and still most widely used) is a pair of conductors; its most common form is
Coaxial
Planar
The majority of modern distributed-element circuits use planar transmission lines, especially those in mass-produced consumer items. There are several forms of planar line, but the kind known as
Waveguide
Many distributed-element designs can be directly implemented in waveguide. However, there is an additional complication with waveguides in that multiple
Mechanical
In a few specialist applications, such as the mechanical filters in high-end radio transmitters (marine, military, amateur radio), electronic circuits can be implemented as mechanical components; this is done largely because of the high quality of the mechanical resonators. They are used in the radio frequency band (below microwave frequencies), where waveguides might otherwise be used. Mechanical circuits can also be implemented, in whole or in part, as distributed-element circuits. The frequency at which the transition to distributed-element design becomes feasible (or necessary) is much lower with mechanical circuits. This is because the speed at which signals travel through mechanical media is much lower than the speed of electrical signals.[11]
Circuit components
There are several structures that are repeatedly used in distributed-element circuits. Some of the common ones are described below.
Stub
A stub is a short length of line that branches to the side of a main line. The end of the stub is often left open- or short-circuited, but may also be terminated with a lumped component. A stub can be used on its own (for instance, for impedance matching), or several of them can be used together in a more complex circuit such as a filter. A stub can be designed as the equivalent of a lumped capacitor, inductor, or resonator.[12]
Departures from constructing with uniform transmission lines in distributed-element circuits are rare. One such departure that is widely used is the radial stub, which is shaped like a sector of a circle. They are often used in pairs, one on either side of the main transmission line. Such pairs are called butterfly or bowtie stubs.[13]
Coupled lines
Coupled lines are two transmission lines between which there is some electromagnetic coupling. The coupling can be direct or indirect. In indirect coupling, the two lines are run closely together for a distance with no screening between them. The strength of the coupling depends on the distance between the lines and the cross-section presented to the other line. In direct coupling, branch lines directly connect the two main lines together at intervals.[14]
Coupled lines are a common method of constructing power dividers and directional couplers. Another property of coupled lines is that they act as a pair of coupled resonators. This property is used in many distributed-element filters.[15]
Cascaded lines
Cascaded lines are lengths of transmission line where the output of one line is connected to the input of the next. Multiple cascaded lines of different characteristic impedances can be used to construct a filter or a wide-band impedance matching network. This is called a stepped impedance structure.
Cavity resonator
A
Dielectric resonator
A dielectric resonator is a piece of dielectric material exposed to electromagnetic waves. It is most often in the form of a cylinder or thick disc. Although cavity resonators can be filled with dielectric, the essential difference is that in cavity resonators the electromagnetic field is entirely contained within the cavity walls. A dielectric resonator has some field in the surrounding space. This can lead to undesirable coupling with other components. The major advantage of dielectric resonators is that they are considerably smaller than the equivalent air-filled cavity.[19]
Helical resonator
A helical resonator is a
Fractals
The use of
Fractals that have been used as a circuit component include the
Taper
A taper is a transmission line with a gradual change in cross-section. It can be considered the limiting case of the stepped impedance structure with an infinite number of steps.
Tapers can be used to match a transmission line to an antenna. In some designs, such as the horn antenna and Vivaldi antenna, the taper is itself the antenna. Horn antennae, like other tapers, are often linear, but the best match is obtained with an exponential curve. The Vivaldi antenna is a flat (slot) version of the exponential taper.[32]
Distributed resistance
Resistive elements are generally not useful in a distributed-element circuit. However, distributed resistors may be used in
Circuit blocks
Filters and impedance matching
Filters are a large percentage of circuits constructed with distributed elements. A wide range of structures are used for constructing them, including stubs, coupled lines and cascaded lines. Variations include interdigital filters, combline filters and hairpin filters. More-recent developments include fractal filters.[35] Many filters are constructed in conjunction with dielectric resonators.[36]
As with lumped-element filters, the more elements used, the closer the filter comes to an
Impedance matching for narrow-band applications is frequently achieved with a single matching stub. However, for wide-band applications the impedance-matching network assumes a filter-like design. The designer prescribes a required frequency response, and designs a filter with that response. The only difference from a standard filter design is that the filter's source and load impedances differ.[39]
Power dividers, combiners and directional couplers
A directional coupler is a four-port device which couples power flowing in one direction from one path to another. Two of the ports are the input and output ports of the main line. A portion of the power entering the input port is coupled to a third port, known as the coupled port. None of the power entering the input port is coupled to the fourth port, usually known as the isolated port. For power flowing in the reverse direction and entering the output port, a reciprocal situation occurs; some power is coupled to the isolated port, but none is coupled to the coupled port.[41]
A power divider is often constructed as a directional coupler, with the isolated port permanently terminated in a matched load (making it effectively a three-port device). There is no essential difference between the two devices. The term directional coupler is usually used when the coupling factor (the proportion of power reaching the coupled port) is low, and power divider when the coupling factor is high. A power combiner is simply a power splitter used in reverse. In distributed-element implementations using coupled lines, indirectly coupled lines are more suitable for low-coupling directional couplers; directly coupled branch line couplers are more suitable for high-coupling power dividers.[42]
Distributed-element designs rely on an element length of one-quarter wavelength (or some other length); this will hold true at only one frequency. Simple designs, therefore, have a limited bandwidth over which they will work successfully. Like impedance matching networks, a wide-band design requires multiple sections and the design begins to resemble a filter.[43]
Hybrids
A directional coupler which splits power equally between the output and coupled ports (a 3 dB coupler) is called a hybrid.[44] Although "hybrid" originally referred to a hybrid transformer (a lumped device used in telephones), it now has a broader meaning. A widely used distributed-element hybrid which does not use coupled lines is the hybrid ring or rat-race coupler. Each of its four ports is connected to a ring of transmission line at a different point. Waves travel in opposite directions around the ring, setting up standing waves. At some points on the ring, destructive interference results in a null; no power will leave a port set at that point. At other points, constructive interference maximises the power transferred.[45]
Another use for a hybrid coupler is to produce the sum and difference of two signals. In the illustration, two input signals are fed into the ports marked 1 and 2. The sum of the two signals appears at the port marked Σ, and the difference at the port marked Δ.
Circulators
A circulator is usually a three- or four-port device in which power entering one port is transferred to the next port in rotation, as if round a circle. Power can flow in only one direction around the circle (clockwise or anticlockwise), and no power is transferred to any of the other ports. Most distributed-element circulators are based on ferrite materials.[48] Uses of circulators include as an isolator to protect a transmitter (or other equipment) from damage due to reflections from the antenna, and as a duplexer connecting the antenna, transmitter and receiver of a radio system.[49]
An unusual application of a circulator is in a
Passive circuits, both lumped and distributed, are nearly always
for an ideal three-port circulator, showing that circulators are non-reciprocal by definition. It follows that it is impossible to build a circulator from standard passive components (lumped or distributed). The presence of a ferrite, or some other non-reciprocal material or system, is essential for the device to work.[51]
Active components
Distributed elements are usually passive, but most applications will require active components in some role. A microwave
History
Distributed-element modelling was first used in electrical network analysis by Oliver Heaviside[54] in 1881. Heaviside used it to find a correct description of the behaviour of signals on the transatlantic telegraph cable. Transmission of early transatlantic telegraph had been difficult and slow due to dispersion, an effect which was not well understood at the time. Heaviside's analysis, now known as the telegrapher's equations, identified the problem and suggested[55] methods for overcoming it. It remains the standard analysis of transmission lines.[56]
Warren P. Mason was the first to investigate the possibility of distributed-element circuits, and filed a patent[57] in 1927 for a coaxial filter designed by this method. Mason and Sykes published the definitive paper on the method in 1937. Mason was also the first to suggest a distributed-element acoustic filter in his 1927 doctoral thesis, and a distributed-element mechanical filter in a patent[58] filed in 1941. Mason's work was concerned with the coaxial form and other conducting wires, although much of it could also be adapted for waveguide. The acoustic work had come first, and Mason's colleagues in the Bell Labs radio department asked him to assist with coaxial and waveguide filters.[59]
Before World War II, there was little demand for distributed-element circuits; the frequencies used for radio transmissions were lower than the point at which distributed elements became advantageous. Lower frequencies had a greater range, a primary consideration for broadcast purposes. These frequencies require long antennae for efficient operation, and this led to work on higher-frequency systems. A key breakthrough was the 1940 introduction of the cavity magnetron which operated in the microwave band and resulted in radar equipment small enough to install in aircraft.[60] A surge in distributed-element filter development followed, filters being an essential component of radars. The signal loss in coaxial components led to the first widespread use of waveguide, extending the filter technology from the coaxial domain into the waveguide domain.[61]
The wartime work was mostly unpublished until after the war for security reasons, which made it difficult to ascertain who was responsible for each development. An important centre for this research was the MIT Radiation Laboratory (Rad Lab), but work was also done elsewhere in the US and Britain. The Rad Lab work was published[62] by Fano and Lawson.[63] Another wartime development was the hybrid ring. This work was carried out at Bell Labs, and was published[64] after the war by W. A. Tyrrell. Tyrrell describes hybrid rings implemented in waveguide, and analyses them in terms of the well-known waveguide magic tee. Other researchers[65] soon published coaxial versions of this device.[66]
George Matthaei led a research group at
Planar formats began to be used with the invention of stripline by Robert M. Barrett. Although stripline was another wartime invention, its details were not published[71] until 1951. Microstrip, invented in 1952,[72] became a commercial rival of stripline; however, planar formats did not start to become widely used in microwave applications until better dielectric materials became available for the substrates in the 1960s.[73] Another structure which had to wait for better materials was the dielectric resonator. Its advantages (compact size and high quality) were first pointed out[74] by R. D. Richtmeyer in 1939, but materials with good temperature stability were not developed until the 1970s. Dielectric resonator filters are now common in waveguide and transmission line filters.[75]
Important theoretical developments included
References
- ^ Vendelin et al., pp. 35–37
- ^
- Nguyen, p. 28
- Vendelin et al., pp. 35–36
- ^ Hunter, pp. 137–138
- ^ Hunter, p. 137
- ^ Hunter, pp. 139–140
- ^
- Doumanis et al., pp. 45–46
- Nguyen, pp. 27–28
- ^
- Hura & Singhal, pp. 178–179
- Magnusson et al., p. 240
- Gupta, p. 5.5
- Craig, pp. 291–292
- Henderson & Camargo, pp. 24–25
- Chen et al., p. 73
- ^
- Natarajan, pp. 11–12
- ^ Ghione & Pirola, pp. 18–19
- ^ Ghione & Pirola, p. 18
- ^
- Taylor & Huang pp. 353–358
- Johnson (1983), p. 102
- Mason (1961)
- Johnson et al. (1971), pp. 155, 169
- ^
- Edwards & Steer, pp. 78, 345–347
- Banerjee, p. 74
- ^ Edwards & Steer, pp. 347–348
- ^
- Magnusson et al., p. 199
- Garg et al., p. 433
- Chang & Hsieh, pp. 227–229
- Bhat & Koul, pp. 602–609
- ^ Bhat & Koul, pp. 10, 602, 622
- ^ Lee, p. 787
- ^ Helszajn, p. 189
- ^ Hunter, pp. 209–210
- ^ Penn & Alford, pp. 524–530
- ^
- Whitaker, p. 227
- Doumanis et al., pp. 12–14
- ^ Janković et al., p. 197
- ^ Ramadan et al., p. 237
- ^ Janković et al., p. 191
- ^ Janković et al., pp. 191–192
- ^ Janković et al., p. 196
- ^ Janković et al., p. 196
- ^ Janković et al., p. 196
- ^ Zhurbenko, p. 310
- ^ Garg et al., pp. 180–181
- ^
- Garg et al., pp. 404–406, 540
- Edwards & Steer, p. 493
- ^
- Zhurbenko, p. 311
- Misra, p. 276
- Lee, p. 100
- ^
- Bakshi & Bakshi
- pp. 3-68–3-70
- Milligan, p. 513
- ^
- Maloratsky (2012), p. 69
- Hilty, p. 425
- Bahl (2014), p. 214
- ^ Hilty, pp. 426–427
- ^ Cohen, p. 220
- ^
- Hong & Lancaster, pp. 109, 235
- Makimoto & Yamashita, p. 2
- ^ Harrell, p. 150
- ^ Awang, p. 296
- ^ Bahl (2009), p. 149
- ^ Maloratsky (2004), p. 160
- ^ Sisodia & Raghuvansh, p. 70
- ^ Ishii, p. 226
- ^ Bhat & Khoul, pp. 622–627
- ^ Maloratsky (2004), p. 117
- ^ Chang & Hsieh, pp. 197–198
- ^ Ghione & Pirola, pp. 172–173
- ^
- Chang & Hsieh, p. 227
- Maloratsky (2004), p. 117
- ^
- Sharma, pp. 175–176
- Linkhart, p. 29
- ^
- Meikle, p. 91
- Lacomme et al., pp. 6–7
- ^ Roer, pp. 255–256
- ^ Maloratsky (2004), pp. 285–286
- ^ Bhat & Khoul, pp. 9–10, 15
- ^ Kumar & Grebennikov, pp. 153–154
- ^ Heaviside (1925)
- ^ Heaviside (1887), p. 81
- ^ Brittain, p. 39
- ^ Mason (1930)
- ^ Mason (1961)
- ^
- Johnson et al. (1971), p. 155
- Fagen & Millman, p. 108
- Levy & Cohn, p. 1055
- Polkinghorn (1973)
- ^ Borden, p. 3
- ^ Levy & Cohn, p. 1055
- ^ Fano & Lawson (1948)
- ^ Levy & Cohn, p. 1055
- ^ Tyrrell (1947)
- ^
- Sheingold & Morita (1953)
- Albanese & Peyser (1958)
- ^ Ahn, p. 3
- ^ Matthaei (1962)
- ^ Matthaei (1963)
- ^ Matthaei et al. (1964)
- ^ Levy and Cohn, pp. 1057–1059
- ^ Barrett & Barnes (1951)
- ^ Grieg and Englemann (1952)
- ^ Bhat & Koul, p. 3
- ^ Richtmeyer (1939)
- ^ Makimoto & Yamashita, pp. 1–2
- ^ Richards (1948)
- ^
- First English publication:
- Ozaki & Ishii (1958)
- ^ Levy & Cohn, pp. 1056–1057
- ^ Cohen, pp. 210–211
Bibliography
- Ahn, Hee-Ran, Asymmetric Passive Components in Microwave Integrated Circuits, John Wiley & Sons, 2006 ISBN 0470036958.
- Albanese, V J; Peyser, W P, "An analysis of a broad-band coaxial hybrid ring", IRE Transactions on Microwave Theory and Techniques, vol. 6, iss. 4, pp. 369–373, October 1958.
- Awang, Zaiki, Microwave Systems Design, Springer Science & Business Media, 2013 ISBN 981445124X.
- Bahl, Inder J, Fundamentals of RF and Microwave Transistor Amplifiers, John Wiley & Sons, 2009 ISBN 0470462310.
- Bahl, Inder J, Control Components Using Si, GaAs, and GaN Technologies, Artech House, 2014 ISBN 1608077128.
- Bakshi, U A; Bakshi, A V, Antenna And Wave Propagation, Technical Publications, 2009 ISBN 8184317220.
- Banerjee, Amal, Automated Electronic Filter Design, Springer, 2016 ISBN 3319434705.
- Barrett, R M, "Etched sheets serve as microwave components", Electronics, vol. 25, pp. 114–118, June 1952.
- Barrett, R M; Barnes, M H, "Microwave printed circuits", Radio TV News, vol. 46, 16 September 1951.
- Bhat, Bharathi; Koul, Shiban K, Stripline-like Transmission Lines for Microwave Integrated Circuits, New Age International, 1989 ISBN 8122400523.
- Borden, Brett, Radar Imaging of Airborne Targets, CRC Press, 1999 ISBN 1420069004.
- Brittain, James E, "The introduction of the loading coil: George A. Campbell and Michael I. Pupin", Technology and Culture, vol. 11, no. 1, pp. 36–57, January 1970.
- Chang, Kai; Hsieh, Lung-Hwa, Microwave Ring Circuits and Related Structures, John Wiley & Sons, 2004 ISBN 047144474X.
- Chen, L F; Ong, C K; Neo, C P; Varadan, V V; Varadan, Vijay K, Microwave Electronics: Measurement and Materials Characterization, John Wiley & Sons, 2004 ISBN 0470020458.
- Cohen, Nathan, "Fractal antenna and fractal resonator primer", ch. 8 in, Frame, Michael, Benoit Mandelbrot: A Life In Many Dimensions, World Scientific, 2015 ISBN 9814366064.
- Craig, Edwin C, Electronics via Waveform Analysis, Springer, 2012 ISBN 1461243386.
- Doumanis, Efstratios; Goussetis, George; Kosmopoulos, Savvas, Filter Design for Satellite Communications: Helical Resonator Technology, Artech House, 2015 ISBN 160807756X.
- DuHamell, R; Isbell, D, "Broadband logarithmically periodic antenna structures", 1958 IRE International Convention Record, New York, 1957, pp. 119–128.
- Edwards, Terry C; Steer, Michael B, Foundations of Microstrip Circuit Design, John Wiley & Sons, 2016 ISBN 1118936191.
- Fagen, M D; Millman, S, A History of Engineering and Science in the Bell System: Volume 5: Communications Sciences (1925–1980), AT&T Bell Laboratories, 1984 ISBN 0932764061.
- Fano, R M; Lawson, A W, "Design of microwave filters", ch. 10 in, Ragan, G L (ed), Microwave Transmission Circuits, McGraw-Hill, 1948 OCLC 2205252.
- Garg, Ramesh; Bahl, Inder; Bozzi, Maurizio, Microstrip Lines and Slotlines, Artech House, 2013 ISBN 1608075354.
- Ghione, Giovanni; Pirola, Marco, Microwave Electronics, Cambridge University Press, 2017 ISBN 1107170273.
- Grieg, D D; Englemann, H F, "Microstrip—a new transmission technique for the kilomegacycle range", Proceedings of the IRE, vol. 40, iss. 12, pp. 1644–1650, December 1952.
- Gupta, S K, Electro Magnetic Field Theory, Krishna Prakashan Media, 2010 ISBN 8187224754.
- Harrel, Bobby, The Cable Television Technical Handbook, Artech House, 1985 ISBN 0890061572.
- Heaviside, Oliver, Electrical Papers, vol. 1, pp. 139–140, Copley Publishers, 1925 OCLC 3388033.
- Heaviside, Oliver, "Electromagnetic induction and its propagation", The Electrician, pp. 79–81, 3 June 1887 OCLC 6884353.
- Helszajn, J, Ridge Waveguides and Passive Microwave Components, IET, 2000 ISBN 0852967942.
- Henderson, Bert; Camargo, Edmar, Microwave Mixer Technology and Applications, Artech House, 2013 ISBN 1608074897.
- Hilty, Kurt, "Attenuation measurement", pp. 422–439 in, Dyer, Stephen A (ed), Wiley Survey of Instrumentation and Measurement, John Wiley & Sons, 2004 ISBN 0471221651.
- Hong, Jia-Shen G; Lancaster, M J, Microstrip Filters for RF/Microwave Applications, John Wiley & Sons, 2004 ISBN 0471464201.
- Hunter, Ian, Theory and Design of Microwave Filters, IET, 2001 ISBN 0852967772.
- Hura, Gurdeep S; Singhal, Mukesh, Data and Computer Communications: Networking and Internetworking, CRC Press, 2001 ISBN 1420041312.
- Ishii, T Koryu, Handbook of Microwave Technology: Components and devices, Academic Press, 1995 ISBN 0123746965.
- Janković, Nikolina; Zemlyakov, Kiril; Geschke, Riana Helena; Vendik, Irina; Crnojević-Bengin, Vesna, "Fractal-based multi-band microstrip filters", ch. 6 in, Crnojević-Bengin, Vesna (ed), Advances in Multi-Band Microstrip Filters, Cambridge University Press, 2015 ISBN 1107081971.
- Johnson, Robert A, Mechanical Filters in Electronics, John Wiley & Sons Australia, 1983 ISBN 0471089192.
- Johnson, Robert A; Börner, Manfred; Konno, Masashi, "Mechanical filters—a review of progress", IEEE Transactions on Sonics and Ultrasonics, vol. 18, iss. 3, pp. 155–170, July 1971.
- Kumar, Narendra; Grebennikov, Andrei, Distributed Power Amplifiers for RF and Microwave Communications, Artech House, 2015 ISBN 1608078329.
- Lacomme, Philippe; Marchais, Jean-Claude; Hardange, Jean-Philippe; Normant, Eric, Air and Spaceborne Radar Systems, William Andrew, 2001 ISBN 0815516134.
- Lee, Thomas H, Planar Microwave Engineering, Cambridge University Press, 2004 ISBN 0521835267.
- Levy, R; Cohn, S B, "A History of microwave filter research, design, and development", IEEE Transactions: Microwave Theory and Techniques, pp. 1055–1067, vol. 32, iss. 9, 1984.
- Linkhart, Douglas K, Microwave Circulator Design, Artech House, 2014 ISBN 1608075834.
- Magnusson, Philip C; Weisshaar, Andreas; Tripathi, Vijai K; Alexander, Gerald C, Transmission Lines and Wave Propagation, CRC Press, 2000 ISBN 0849302692.
- Makimoto, M; Yamashita, S, Microwave Resonators and Filters for Wireless Communication, Springer, 2013 ISBN 3662043254.
- Maloratsky, Leo G, Passive RF and Microwave Integrated Circuits, Elsevier, 2004 ISBN 0080492053.
- Maloratsky, Leo G, Integrated Microwave Front-ends with Avionics Applications, Artech House, 2012 ISBN 1608072061.
- Mason, Warren P, "Wave filter", U.S. patent 2,345,491, filed 25 June 1927, issued 11 November 1930.
- Mason, Warren P, "Wave transmission network", U.S. patent 2,345,491, filed 25 November 1941, issued 28 March 1944.
- Mason, Warren P, "Electromechanical wave filter", U.S. patent 2,981,905, filed 20 August 1958, issued 25 April 1961.
- Mason, W P; Sykes, R A, "The use of coaxial and balanced transmission lines in filters and wide band transformers for high radio frequencies", Bell System Technical Journal, vol. 16, pp. 275–302, 1937.
- Matthaei, G L, "Interdigital band-pass filters", IRE Transactions on Microwave Theory and Techniques, vol. 10, iss. 6, pp. 479–491, November 1962.
- Matthaei, G L, "Comb-line band-pass filters of narrow or moderate bandwidth", Microwave Journal, vol. 6, pp. 82–91, August 1963 ISSN 0026-2897.
- Matthaei, George L; Young, Leo; Jones, E M T, Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964 OCLC 830829462.
- Meikle, Hamish, Modern Radar Systems, Artech House, 2008 ISBN 1596932430.
- Milligan, Thomas A, Modern Antenna Design, John Wiley & Sons, 2005 ISBN 0471720607.
- Misra, Devendra K, Radio-Frequency and Microwave Communication Circuits, John Wiley & Sons, 2004 ISBN 0471478733.
- Natarajan, Dhanasekharan, A Practical Design of Lumped, Semi-lumped & Microwave Cavity Filters, Springer Science & Business Media, 2012 ISBN 364232861X.
- Nguyen, Cam, Radio-Frequency Integrated-Circuit Engineering, John Wiley & Sons, 2015 ISBN 0471398209.
- 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.
- Penn, Stuart; Alford, Neil, "Ceramic dielectrics for microwave applications", ch. 10 in, Nalwa, Hari Singh (ed), Handbook of Low and High Dielectric Constant Materials and Their Applications, Academic Press, 1999 ISBN 0080533531.
- Polkinghorn, Frank A, "Oral-History: Warren P. Mason", interview no. 005 for the IEEE History Centre, 3 March 1973, Engineering and Technology History Wiki, retrieved 15 April 2018.
- Ramadan, Ali; Al-Husseini, Mohammed; Kabalan Karim Y; El-Hajj, Ali, "Fractal-shaped reconfigurable antennas", ch. 10 in, Nasimuddin, Nasimuddin, Microstrip Antennas, BoD – Books on Demand, 2011 ISBN 9533072474.
- Richards, Paul I, "Resistor-transmission-line circuits", Proceedings of the IRE, vol. 36, iss. 2, pp. 217–220, 1948.
- Richtmeyer, R D, "Dielectric resonators", Journal of Applied Physics, vol. 10, iss. 6, pp. 391–397, June 1939.
- Roer, T G, Microwave Electronic Devices, Springer, 2012 ISBN 1461525004.
- Sharma, K K, Fundamental of Microwave and Radar Engineering, S. Chand Publishing, 2011 ISBN 8121935377.
- Sheingold, L S; Morita, T, "A coaxial magic-T", Transactions of the IRE Professional Group on Microwave Theory and Techniques, vol. 1, iss. 2, pp. 17–23, November 1953.
- Sisodia, M L; Raghuvanshi, G S, Basic Microwave Techniques and Laboratory Manual, New Age International, 1987 ISBN 0852268580.
- Taylor, John; Huang, Qiuting, CRC Handbook of Electrical Filters, CRC Press, 1997 ISBN 0849389518.
- Tyrrell, W A, "Hybrid circuits for microwaves", Proceedings of the IRE, vol. 35, iss. 11, pp. 1294–1306, November 1947.
- Vendelin, George D; Pavio, Anthony M; Rohde, Ulrich L, Microwave Circuit Design Using Linear and Nonlinear Techniques, John Wiley & Sons, 2005 ISBN 0471715824.
- Whitaker, Jerry C, The Resource Handbook of Electronics, CRC Press, 2000 ISBN 1420036866.
- Zhurbenko, Vitaliy, Passive Microwave Components and Antennas, BoD – Books on Demand, 2010 ISBN 9533070838.