Application of Complex Variable in the analysis of ac RLC Circuit 2022

Application of Complex Variable in the analysis of ac RLC circuit
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Introduction
An RLC circuit refers to an electrical circuit that is made of a resistor (R), a capacitor (C) and an
inductor (L) which can either be connected in parallel or series. They form current harmonic
oscillators which resonates in a similar way as simple LC circuit. Introduction of a resistor to an
LC circuit introduces damping and the peak resonant frequency. AC RLC circuits are those
which are driven by alternating sources (ac voltages or currents). A typical ac RLC circuit is
shown below. Historical figures such as Heinrich Hertz, a German Physicist and Olive Lodge
experimentally demonstrated resonance properties of RLC circuits driven by ac sources (Krantz,
2012. Because RLC circuits yielded differential equations as the scientists strove to solve for the
current flowing through the circuit, with the affordability of complex variable, scientists adopted
the use of this technique in exploring the characteristics of the RLC circuits,
Components of the RLC circuit
Resistor
Resistors offers electrical resistance with an SI unit of ohm (Ω). Resistors are passive
components meaning that they do not require any power supply to operate. A resistor is
characterized by two main values, its resistance and its power rating. The power rating is given
in watts and it represents the maximum amount of power the resistor can dissipate as heat
without breaking down (U.A. Bakshi, 2008). There many different types of resistors depending
on the construction and chemical composition. These include Carbon resistors, wire-wound
resistors, metal film resistors and thermistors (Mayergoyz & Lawson, 2012). For a current to
flow through a resistor, a voltage potential must be applied across the terminals of the resistor. A
resistor introduces damping into a circuit and reduces resonant frequency in the circuit. The
symbol of a resistor is shown below (Ablowitz & Fokas, n.d.).
Capacitor
In its most basic form, a capacitor is an electrical device made up of two plates in parallel and
separated by a medium called a dielectric. The dielectric can be air or an electrolyte. Unlike the
resistor, the capacitor is an energy storage element. Energy is stored in a capacitor in the form of
an electric field between the two plates (). The two main characteristics of a capacitor are its
capacitance which has the SI unit of Farad (F) and the voltage rating in volts (V) (Krantz, 2012).
The relationship between the three variables associated with a capacitor can be expressed as
follows:
C= Q
V , Q=CV

The voltage rating of a capacitor indicates the maximum voltage across the terminals above
which the capacitor is damaged. Application of potential across a capacitor causes electrical
charge to accumulate on the capacitor plates. There are many types of capacitors including:
electrolytic capacitors (aluminium and tantalum), ceramic capacitors. They can also be classified
into fixed and variable capacitors just like resistors (MAHADEVAN & CHITRA, 2018). The
symbol of a capacitor is shown below.
Inductor
Like the capacitor, an inductor is an energy storage element. However, unlike in a capacitor
where energy is stored in the electric field, in an inductor, energy is stored in a magnetic field
(Bird, 2017). The magnetic field is established by the electric current flowing through the
inductor coil (Krantz, Gavosto, & Peloso, 2018). An inductor is characterized by its inductance
which has the SI unit Henry (H). The inductance is defined as the tendency of the inductor to
oppose changes in the current flowing through it. A field is developed in the coil that establishes
an electromotive force that acts so as to reduce the current if it was rising or to increase it if it
was decreasing (). The inductance of a coil can be estimated using the following formula:
L= N2 μA
I
L is the coil inductance in henrys, μ is the permeability of the core, A is the Cross-sectional area
of the coil in m2 and I is the current through the coil in amperes.
The symbol of an inductor is shown below.
Problem statement from the ac RLC network solutions
Consider a series RLC circuit shown below.