4.3: Physical Circuits and The Lumped Circuit Model

Based on "From Conservation to Kirchhoff: Getting Started in Circuits With Conservation and Accounting," by Bruce A. Black, 14 February 1996, published in the Proceedings of the 1996 ASEE Frontiers In Engineering Education Conference.

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One of the most important and common applications of the conservation of charge equation is in the study of electrical circuits. It is a common experience in physics lab to connect electrical components we commonly call resistors, capacitors, and inductors along with various current and voltage sources, and to then study the behavior of this physical circuit. To predict the behavior of these physical circuits, we must have an accurate and mathematically tractable model. A distinct engineering science called circuit theory has been developed to model and analyze physical circuits.

Circuit theory is the vast collection of specialized techniques and results for analyzing and designing physical circuits that satisfy the lumped circuit model. The key assumptions underlying the lumped circuit model are listed below:

• Lumped circuits are constructed from discrete lumped circuit elements that are physically distinct and are connected only by wires.
• Electrical and magnetic energy is stored or converted to other forms of energy only within the circuit elements, i.e. we assume that no electric fields exist in the space outside the elements; that is, no electric fields exist either between the elements or between the elements and ground.
• No time-varying magnetic fields intersect any of the circuit loops.
• Lumped circuits are physically "small", i.e. the time scales of interest are much greater than the time for an electrical disturbance to propagate at the speed of light or for charges and currents to redistribute inside the devices.

Typical lumped circuit elements include resistors, inductors, capacitors, and voltage and current sources connected by wires. Remember that lumped circuit elements are models for the real physical components. Although physical circuits never exactly match the corresponding lumped circuit, the power of circuit theory is its ability to accurately predict the behavior of physical circuits for a wide range of conditions. In fact, their use is so prevalent and so accepted that we often forget the assumptions inherent in the model.

When the conservation of charge equation is applied to a lumped circuit element, the key lumped circuit assumption is that "no electric fields exist in the space outside the elements." If a system boundary is not pierced by electric field lines, then it is impossible to change the amount of net charge inside the system, and charge cannot accumulate. This is a consequence of Gauss's Law, which studied in physics, and is always true for a lumped circuit element. This result is commonly referred to as Kirchhoff's current law (KCL): \[\sum_{\text {in}} i_i = \sum_{\text {out}} i_e \quad \text { [Kirchhoff's Current Law] } \nonumber \] which follows directly from Eq. \(4.1.7\) when \(\dfrac{dq_{sys}}{dt} = 0\).

Note that we have not discussed Kirchhoff's voltage law (KVL): The summation of voltage drops around any circuit loop is zero. This is an extremely powerful tool for circuit analysis and is a direct consequence of the lumped circuit modeling assumptions. Although all of the circuits studied in this course must satisfy KVL because they are lumped circuits, we will not use this tool directly in solving for currents and voltages in our circuits. A more detailed discussion of this important result will be discussed in ES 203 - Electrical Systems.