The simplistic approach to analyzing electronic circuits is to use the lumped element model. This methodology assumes that the attributes of the circuit—resistance, capacitance, and inductance—are concentrated into idealized electrical components connected by a network of perfectly conducting wires. However, in reality, that is not the case.
As the frequency and rise time increase, these elements become distributed continuously through the substrate along the entire length of the trace. The copper trace and the adjacent dielectric materials become a transmission line, the skin effect forces current into the outer regions of the conductor, and frequency-dependent losses impact on the quality of the signal. The PCB trace now behaves as a distributed system with parasitic inductance and capacitance characterized by delay and scattered reflections. The behavior we are now concerned about occurs in the frequency domain. In this month’s column, I will discuss the difference between the lumped element model and the distributed system. (Fig. 1)
In my previous column, “The Frequency Domain,” we saw that impedance is defined in both the time and frequency domains. In the time domain, the impedance of a resistor (R) can be represented by a relationship between voltage and current (Ohm’s Law). Similarly, an ideal capacitor (C) has a relationship between the stored charge and the voltage across its plates. And the behavior of an ideal inductor (L) is defined by how fast the current traveling through it changes in the time domain.
We group these three elements (RLC) in a category called lumped circuit elements, in the sense that their properties can be lumped into 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 dielectric materials along its length. The distributed model is used when the wavelength be-comes comparable to the physical dimensions of the circuit, making the lumped model inaccurate. This typically occurs at high frequencies, where the wavelength is very short. How-ever, it can also occur on very long, low-frequency transmission lines, such as high-volt-age power lines. The three primary elements now include distributed capacitance, inductance, 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 unpredictable. 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 wavelength 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 applied signals.
Lumped systems are described by ordinary 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 distributed systems, we need to take electromagnetic wave propagation into account to get spatial as well as time derivatives, which leads to partial differential equations in the frequency domain.
To read this entire column, which appeared in the December 2019 issue of Design007 Magazine, click here, or download the PDF to your library for further reference.