Parallel RLC Circuit Laboratory Report

Levy V. Medina II

3 BS Computer Engineering ECCE Department

Ateneo de Manila University Loyala Heights, Quezon City

levymedina3@gmail.com

Julius Jay R. Sambo 4 BS Computer Engineering

ECCE Department Ateneo de Manila University Loyala Heights, Quezon City

juliusj.sambo@gmail.com

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 ωo 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:

𝑑!𝑥𝑑𝑡!

+ 𝑎!𝑑𝑥𝑑𝑡+ 𝑎!𝑥 𝑡 = 𝑓 𝑡          (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 (Aest) in order to get the characteristic equation:

𝑠! + 𝑎!𝑠 + 𝑎! = 0                           2 . Using the quadratic equation, we find the roots s1 and s2:

𝑠!, 𝑠! =−𝑎!  ± ( 𝑎! ! − 4𝑎!)

2           3 .

By using the result of the equation inside the square root ( 𝑎! ! − 4𝑎! ), we can categorize the natural response according to their roots. Case 1 wherein 𝑎! ! − 4𝑎! > 0, will produce real roots and is called the overdamped response. Its solution will follow the form:

𝑥! = 𝐴!𝑒!!! + 𝐴!𝑒!!!           4 . Case 2 wherein 𝑎! ! − 4𝑎! = 0, will produce repeated

roots and is called the critically damped response. Its solution will follow the form:

𝑥! = (𝐴!𝑡 + 𝐴!)𝑒!"           5 . Case 3 wherein 𝑎! ! − 4𝑎! < 0, will produce complex

roots and is called the underdamped response. Its solution will follow the form:

𝑥! = 𝑒!"[𝐴! cos βt + 𝐴! sin βt ]       6 .

III. METHODOLOGY

A. Materials

As in the previous laboratory experiments, only a computer simulated circuit was utilized. The program Multisim was used for this experiment. In this simulated circuit, the following components were used: a 1 Ohm resistor, a 2V voltage source, inductors of different values, a 0.25F capacitor, a varying current source and an oscilloscope.

B. Procedures

1. Using Multisim, construct the circuit shown in Figure 1.

Figure 1. Parallel RLC circuit

2. Observe the voltage across the resistor using the oscilloscope with the following component configurations: Is = -3A, Vs = 2V, C= 0.25F, and L = 1.33H.

3. Start the simulation and record the resulting graph.

4. Repeat steps 2 and 3 for the inductor and current pair values: -3A and 1mH, -1A and 1H.

5. Tabulate and record data for further analysis.

IV. RESULTS AND DISCUSSION

Table 1. Graphs formed by the different circuits

Is L Sketch of Transient V -3A

1.33H

-3A

1mH

-1A

1H

As stated in the methodology, we constructed the circuit in a circuit simulation program. Table 1 shows the graph of the different circuit configurations stated in the experiment. We choose transient as the analysis type since we are looking at the natural response of the circuit.

The various graphs of the circuit configurations show that all represent an underdamped response, which means it follows the third case, which produces complex roots. This is further verified by the computation for the theoretical natural response function of each circuit configuration. The results for which can be seen in Table 2. For the forced response function, which we based on the graphs formed by the circuit, we get zero as in all cases, the graph approaches zero. For the theoretical forced response value, there is no expression to the right of the equation, indicating that there is no steady-state response.

Table 2. Summary of Values

Is L Theoretical Natural Response Function

Forced Response Function

Theoretical Response

Value -3A 1.33H 𝑒!!![𝐴! cos 1.42𝑡

+ 𝐴! sin 1.42𝑡 ] 0 0

-3A 1mH 𝑒!!![𝐴! cos 28𝑡+ 𝐴! sin 28𝑡 ]

0 0

-1A 1H 𝑒!!![𝐴! cos 2𝑡+ 𝐴! sin 2𝑡 ]

0 0

Solving for the natural and forced response of the circuit: Through KCL, we get this general second order

differential equation which we can use to get the natural response for each circuit configuration:

𝐶 !!!!!!!

+ !!  !"!!"+  !!

!= 0

Using the characteristic equation, we get the following: 𝐶𝑠! + !

!  𝑠 +  !

!= 0

Case 1: Is = -3A, L = 1.33H Substituting the values to the equation we get:

0.25𝑠! +  𝑠 +  1

1.33= 0

Case 2: Is = -3A, L = 1mH Substituting the values to the equation we get:

0.25𝑠! +  𝑠 +  1

1×10!!= 0

Case 3: Is = -1A, L = 1H Substituting the values to the equation we get:

0.25𝑠! +  𝑠 +  1 = 0 With these equations, we got the values placed in Table

2.

V. CONCLUSION

Through the simulation of the circuit through Multisim, the researchers were able to observe the natural response in the parallel RLC circuit.

VI. REFERENCES

[1] Johnson, D., Johnson, J., & Hilburn, J. (n.d.). Electric Circuit Analysis (Second ed.) [2] Pauls Online Notes : Differential Equations - Second Order DE's. (n.d.). Retrieved September 29, 2015.

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