Synopsis
In this experiment, AC circuits, capacitors, and inductors are reviewed. Circuit analysis to cover sinusoidal signals is done. The experiment is drawn from lecture 5 and six learned on series and parallel configuration of RLC circuits, that is, Resistor –capacitor, RC; Resistor-inductor, RL; series Resistor-Inductor-Capacitor; RLC; and parallel RLC circuits.
Introduction
The three basic passive components, resistor, inductor, and capacitor have varied phase relationships when connected in a sinusoidal AC circuit. Analysis of a series RCL circuit is similar to dual series RL and RC circuit. Series RCL circuits are categorized as second order circuit because they have two energy storage components; inductor and capacitor.
Figure 1. Series RCL circuit
In an ohmic resistor, voltage waves are in phase with the current. In the inductor, voltage waveforms lead the current waveforms by 90 electrical degrees giving the expression of ELI. In the capacitor, the voltage waveform lag the current waveforms by 90 degrees giving the ICE expression.
A parallel RCL circuit is the inverse of series combination. Analysis of an RCL circuit is more complex than that of a series combination. In this configuration, the circuit elements are assumed pure. The supply voltage, Vs is common in the three components while the IS is divided into three parts (through the inductor IL, capacitor IC, and resistor IR). The current from the supply is the vector sum of the three branches (Zahn, 1979).
Figure 2. Parallel RCL circuit
Theory
In series RCL circuit, the instantaneous current though the loop is the same in each circuit element. The inductive and capacitive reactance are functions of the supply frequency. The sinusoidal response of series RCL circuit vary with frequency, f. individual voltage drops across the elements are out of phase with each other. In the capacitor, the instantaneous current leads the voltage while in the inductor, the voltage leads the current. Both the current and voltage in a resistor are in phase. The figure below illustrates the components and respective voltages and currents (Perret, 2013).
Figure 3.
The amplitude of Vs the three components in a series combination is made up of 3 component voltages; VR, VC and VC. Vector diagrams have current vector as their reference.
Figure 4. Graphical representation of individual voltage and current in an RCL circuit
In parallel combination of the passive components, the voltage across the elements is common, Vs. Vs is therefore used as the reference vector while the three current vectors are drawn relative to this. Resultant vector is obtained by adding three vectors, IL, IR, and IC. The resultant angle between Vs and Is is the phase angle of the circuit.
Figure 5. Phase angle of the circuit.
Objective
The aim of this lab experiment was to develop a basic understanding of the frequency response in a series RC and RL circuits.
Material/equipment
Signal generator
Multimeter
Oscilloscope
Procedure
a) Experiment 1
An Agilent function generator was selected to produce a sin wave of peak to peak voltage, Vpp = 10 volts at a frequency of 1 Hertz as the input voltage. The output of function generator was connected to a one-ohm resistor and a one microfarad capacitor connected as shown in figure 6 below. Channel 1 of an oscilloscope was selected and connected to the input voltage. Channel 2 was connected to point “an” as shown in the diagram. Necessary adjustments were done to view both the input and the output characteristics.
Figure 6
The observed measurements were recorded in Table 1.
While keeping Vpp at 10 volts, the oscilloscope was tuned and connected such that the output voltage and input voltage were clearly viewed on the oscilloscope. The circuit was completed using a one-kiloohm resistor and a one microfarad capacitor as shown in figure 7 below. Table 2 was filled with the calculated values
Figure 7
Experiment 2
The oscilloscope was connected with the appropriate settings while keeping Vpp at 10 volts. The circuit was completed with a one-kiloohm resistor and a 50 mH inductor as shown in figure 8 below. The measurements were used to fill in Table 3 below.
Figure 8
The experiment was repeated with a one-kiloohm resistor and 50 mH inductors as shown in figure 9 below with 10 volts of Vpp. The values obtained were used to fill table 4 below.
Results
Calculations
Experiment 1
Given that
If
Then
The form factor Vrms = 0.7071Vp and Vav = 0.637Vp
The current
When Vp is 10 volts, the current at channel 2 is 10/1000 = 0.01 Amperes
A resistor was used in this experiment thus there are no imaginary numbers
Experiment 2
Given Vmax is 160 v and 10 v
Vp = 10-v, Ip =
The inductor gives the that results into imaginary values of current and voltage.
Discussion
In this experiment, passive components are analyzed in both parallel and series connection. The outputs are viewed on an oscilloscope and calculations are done. The values show the imaginary when inductors and capacitors are combined and real values when the resistor is used alone. The frequency of the AC supply leads to the sinusoidal relationship of the capacitors and inductors.
Conclusion
The experiment was a success. The parallel and series connections of passive elements were analyzed separately and their values viewed on an oscilloscope. The data was analyzed mathematically to verify the connection variation.
References
Perret, R. (2013). Power Electronics Semiconductor Devices. New York, NY, John Wiley & Sons
Zahn, M. (1979). Electromagnetic field theory: a problem-solving approach. New York, Wiley.
