8.1: Introduction

In this chapter we introduce the analysis of AC RLC circuits configured in parallel. AC parallel circuits echo the characteristic of their DC counterpart and many of the solution techniques from DC analysis will be applicable here. This includes the use of Ohm's law and Kirchhoff's current law, along with the current divider rule. Generally, as with the series circuits presented in the previous chapter, reactance values will need to be computed from capacitor and inductor values before the main analysis may begin. Here, as in the most of the remaining chapters, we shall be concerned with determining the circuit response based on a source with a single frequency of excitation, in other words, a simple sine wave.

In electrical engineering, RLC circuits are fundamental components consisting of resistors (R), inductors (L), and capacitors (C). When analyzing such circuits, we often encounter second-order differential equations due to the interplay of energy-storage elements (inductor and capacitor) and energy-dissipating elements (resistor).

The behavior of an RLC circuit can be described by Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL), leading to differential equations that govern the dynamics of the circuit. For example, the voltage across the capacitor and the current through the inductor can be described by second-order differential equations, which arise from the relationships between voltage, current, and the derivative of charge or magnetic flux.

These second-order differential equations typically take the form of linear constant coefficient differential equations, where the coefficients depend on the values of resistance, inductance, and capacitance in the circuit. Solving these differential equations allows us to understand the transient and steady-state behavior of the RLC circuit in response to different input signals or initial conditions, making it a crucial aspect of circuit analysis and design in electrical engineering.

Parallel circuits are in many ways the complement of series circuits. The most notable characteristic of a parallel circuit is that it has only two nodes and each component is connected from one node to the other. There are no other connections with which to create a voltage divider. Consequently, all components see the same voltage. Currents divide among the components in proportion to their conductance/susceptance (i.e., in inverse proportion to their resistance/reactance).

The key to this is to remember that all computations involve vector quantities. This can lead to some surprising results for the uninitiated. For example, due to the 180 degree phase differential between inductors and capacitors, it is possible for an individual branch current to be greater than the source current. This does not violate Kirchhoff's current law, as we shall see. Indeed, it is reminiscent of a similar situation in AC series circuits where an individual component voltage can be greater than the source voltage without violating Kirchhoff's voltage law.

To clarify our analyses, we shall make considerable use of both time domain plots of currents as well as phasor diagrams. We will cover the phasor in chapter 10. 

I strongly recommend reviewing the basic methods of solving second-order differential equations. Here is the link to Khan Academy.