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18 DESIGN007 MAGAZINE I DECEMBER 2019 a single point. This is quite different from the properties of an ideal transmission line, which also consists of these three elements, but they are distributed continuously through the di- electric materials along its length. The distrib- uted model is used when the wavelength be- comes comparable to the physical dimensions of the circuit, making the lumped model inac- curate. This typically occurs at high frequen- cies, where the wavelength is very short. How- ever, it can also occur on very long, low-fre- quency transmission lines, such as high-volt- age power lines. The three primary elements now include distributed capacitance, induc- tance, and conductance (G). The lumped element model completely fails at one-quarter wavelength (a 90° phase change), with not only the value but the very nature of the component itself being unpre- dictable. Due to this wavelength dependency, the distributed system model is used mostly at higher frequencies. It is important to realize that the terms lumped and distributed are not properties of the system itself. These properties are related to the size of the circuit, compared to the wave- length of the voltages and currents passing through it. So, a resistor is, or isn't, a lumped element (even though it is usually meant to be one), depending on the frequency of the ap- plied signals. Lumped systems are described by ordi- nary differential equations because, due to the small size of the system (compared to the wavelength), the spatial derivatives can be neglected and we only need to consider time derivatives. On the other hand, for distribut- ed systems, we need to take electromagnetic wave propagation into account to get spatial as well as time derivatives, which leads to par- tial differential equations in the frequency domain. A transmission line can be represented by an infinite number of segments, incorporating series resistive and inductive elements with shunt capacitive and conductive elements, as in Figure 2. And because of the restricted ve- locity of propagation in the medium, the sig- nal does not know what the termination is at the end of the line. It can only see the imped- ance of the line, which, by design, should be matched to the driver. What forms the electromagnetic field in the transmission line? This is a question that even Google can't an- swer (until now). Here's how I see it: An elec- tric field is produced when voltage is applied across an IC output driver. When a signal var- ies this voltage, there is a surge of current that produces a magnetic field. This electromagnet- ic energy then transmits the signal, at about half the speed of light—limited by the dielec- tric medium—down the length of the transmis- sion line following the trace. The energy ra- diates into the surrounding dielectric material and couples into nearby structures, creating a distributed system of parasitic elements. The electromagnetic fields are not restricted to the multilayer substrate, and, if adequate care is not taken, may emit radiation causing electro- magnetic interference. Figure 2: Transmission line represented by a series of R-L-C-G elements.