# 1.4: Distributed Circuits

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- 41002

RF and microwave engineering has as its basis transmission line effects also known as distributed effects. As is well known, the voltage and current at one point, \(\mathsf{A}\), cannot instantaneously affect the voltage and current at another spatially separated point \(\mathsf{B}\). However if points \(\mathsf{A}\) and \(\mathsf{B}\) in a circuit are sufficiently close then a circuit can nearly always be treated as a lumped element circuit. What constitutes sufficiently close is related to the distance between \(\mathsf{A}\) and \(\mathsf{B}\), \(d\), compared to a wavelength, \(\lambda\). If \(d <\lambda /100\) then nearly always the circuit can be considered as being lumped. If \(d ≥\lambda /10\) then distributed effects, limitations imposed by the finite time-of-flight, must be considered and the circuit is always regarded as being distributed. When \(d\) is between \(\lambda /100\) and \(\lambda /10\) then it is not clear whether the circuit must be regarded as distributed or can be regarded as lumped. It is of course much simpler to analyze and design with lumped-element circuits.

Distributed and transmission-line effects are synonymous. While a distributed circuit may be difficult to analyze, distributed effects can be exploited to realize a very large array of elements that usually have no equivalent at lower frequencies. For example, the fields from two adjacent transmission lines can overlap so that part of the signal (energy) from one of the lines appears on the other. This could be a problem in some situations but can be used to couple some of the power from one line onto another. Many novel circuit elements are based on this coupling effect.

Distributed effects and transmission line effects result from a signal, voltage or current, at one point in space not being able to instantaneously change the voltage and current at another point in space. To be more physical, voltage and current can be replaced by electric and magnetic field. The finite speed of light is encapsulated in Maxwell’s equations which were developed between 1861 and 1862 [8, 9]. For microwave engineers Maxwell’s equations are the most convenient and complete description of reality. They are the classical limit of a more fundamental theory called quantum electrodynamics. Maxwell’s equations are important to the physical understanding of transmission line effects and distributed effects, but in reality are very difficult to manipulate and use in calculations. What are much more useful are the transmission line equations, or telegrapher’s equations, developed by Heaviside in 1887 [10].

The telegrapher’s equations use voltage and current which are related to the integral of electric field and integral of magnetic field respectively. The teleprapher’s equations relate the time derivative of voltage (or current) to the spatial variation of current (or voltage). This is analogous to how Maxwell’s equations relate the electric and magnetic fields. Heaviside also introduced the concept of phasors with the result that the four dimensions of Maxwell’s equations, time and three spatial dimensions, are reduced to just one spatial dimension, along the length of a transmission line. Microwave designers, as with all circuit designers, is that they design based on lumped-element circuits that are naturally described by voltage and current. What is particularly important about the telegrapher’s equations is that they enable direct relationship of transmission line circuits to lumped-element circuits. Never-the-less the telegraphist’s equations are not always sufficient to understand distributed circuits so (sometimes reluctantly) microwave engineers must have working familiarity with Maxwell’s equations to understand every situation that can be encountered. Putting this another way, microwave engineers want to express circuit quantities as voltage and currents rather than electric and magnetic fields but are always aware that microwave circuits are intricately connected to electric and magnetic fields and energy is stored in electric and magnetic fields. The telegraphist’s equations provide a link between the voltage/current and EM field worlds and introduce concepts of traveling wave voltages and currents which further aid the connection.