Parallel RLC Circuit Laboratory Report Levy V. Medina II 3 BS Computer Engineering ECCE Department Ateneo de Manila University Loyala Heights, Quezon City [email protected] Julius Jay R. Sambo 4 BS Computer Engineering ECCE Department Ateneo de Manila University Loyala Heights, Quezon City [email protected] Abstract— The aim of this experiment is to familiarize the researchers with the different types of natural response associated with parallel RLC circuits. This activity also involves the concept of Second Order Differential Equations. Second Order Differential Equations are a type of differential equation with the second power as its highest exponent. [2] Second Order Differential Equations often express the voltage or current of a storage device in a circuit with 2 storage devices. Differential Equations can show the respective voltage or current of the component over time, often divided into the transient response and the forced response. The activity will be done by simulating the given circuit, taking note of the graph and comparing the important values in the graph with the theoretical computed equation with varying resistances. The researchers were able to perform the experiment successfully. Familiarization with second order differential equations was achieved and how it can be applied was learned.[1] Index Terms—Parallel RLC circuits; natural response; second order differential circuits I. INTRODUCTION An RLC circuit is an electrical circuit that utilizes the following components connected in either series or parallel: a resistor, an inductor, and a capacitor. Since there are two independent energy storage elements, these types of circuits can be described through the use of second order differential equations. For this particular lab experiment, we will be focusing on parallel RLC circuits and the natural response associated to them. Like a series RLC circuit, the natural response of the circuit can take one of the following three forms. (1) the overdamped response, whose roots are real and distinct, (2) the critically damped response, whose roots are equal , real and repeated, and (3) the underdamped response, which has complex roots. A parallel RLC circuit’s natural response will take one of the three forms mentioned based on the relative magnitudes of α and ω o or whatever constants are used. II. THEORETICAL INFORMATION Since a parallel RLC circuit provides a second ordcr differential equation, solving for the total response of either the inductor current or the capacitor voltage will provide a natural (or transient) response, and if applicable, a forced or steady-state response. Like in the previous experiments, the second order differential equation takes on the form: ! ! + ! + ! = (1) where x(t) is a voltage v(t) or a current i(t). To find the natural response, we set the forced response f(t) to zero, and then substitute the s-equation (Aes t ) in order to get the characteristic equation: ! + ! + ! = 0 2 . Using the quadratic equation, we find the roots s 1 and s 2: ! , ! = − ! ± ( ! ! − 4 ! ) 2 3 . By using the result of the equation inside the square root ( ! ! − 4 ! ), we can categorize the natural response according to their roots. Case 1 wherein ! ! − 4 ! > 0, will produce real roots and is called the overdamped response. Its solution will follow the form: ! = ! ! ! ! + ! ! ! ! 4 . Case 2 wherein ! ! − 4 ! = 0, will produce repeated roots and is called the critically damped response. Its solution will follow the form: ! = ( ! + ! ) !" 5 . Case 3 wherein ! ! − 4 ! < 0, will produce complex roots and is called the underdamped response. Its solution will follow the form: ! = !" [ ! cos βt + ! sin βt ] 6 .