

RLC Series and Parallel Circuits Lab
Procedure:
Figure 1: Series  Parallel RLC Circuit
1. Calculate I_{1}, I_{2}, I_{3} and Vo for the circuit shown in Figure 1 for f = 1kHz and f = 10kHz.
2. Use PSpice to determine I_{1}, I_{2}, I_{3}, and Vo in Figure 1 at f = 1kHz and f = 10kHz.
3. Construct the circuit shown in Figure 1 and measure I_{1}, I_{2}, I_{3} and Vo. Note, currents are measured
indirectly by measuring the voltage across an appropriate resistor and applying Ohm’s Law.
4. By inspection, estimate Vo for:
a. f = 0Hz
b. f = ∞Hz
5. Use PSpice to plot Vo vs. frequency (amplitude and phase) for 100Hz ≤ f ≤ 100kHz (use .AC analysis).
6. Measure the magnitude of Vo at f = 100Hz and f = 100kHz. Plot actual data points on PSpice plot for f = 100kHz, 1kHz, 10kHz, and 100kHz.
Procedure & Data:
1. In the given experiment of a series/parallel RLC circuit (see Figure 1) the characteristics of the voltages and currents at
specific and over a range of frequencies was examined using three separate techniques. First hand calculations were performed to
mathematically describe the circuit’s behavior at a frequency of 1kHz and 10kHz with a 4volt peek input sine wave. These calculations
were performed in several steps starting with the following. In determining the circuits overall and piece wise components the
individual and total impedance was calculated for each component of the circuit (see Table 1) using the following equations.
Table 1: Calculated Impedance Values for RLC Series  Parallel Circuit
Table 2: Calculated Voltages & Currents for RLC Series  Parallel Circuit
2. Secondly the use of Pspice analysis was incorporated in the break down of the circuit’s characteristics. In using Pspice simulations to test and confirm
the calculated values it also provided several graphical plot which help in determining the over all function of the circuit (see Graph I & 2). Several
properties of Pspice were used including that of the transient analysis (.TRAN). By using the .TRAN analysis statement Pspice incorporates a very detailed
differential equation solver routine that may cause initial problems within a circuit using time dependent components such as capacitors and inductors. As
found the capacitors and inductor had an initial effect on the circuit at time equal to zero. Thus requiring several hand calculations to determine the effects
of each time dependent component when time is set equal to zero (see Table 3). To solve for these initial conditions the follow equations were used.
I_{IC} = Isin(ωt ± Θ) (E.6)
V_{IC} = Vsin(ωt ± Θ) (E.7)
By setting time equal to zero each equation breaks down to an initial current or voltage multiplied by the sine of the phase angle. With close inspection of
equations (E.6) & (E.7) at time equal to zero our variable omega cancels out, which may lead one to assume that frequency has no effect on initial conditions.
Although upon closer examination the values of the current and voltages effecting the time dependent components were originally calculated based upon the specific
frequency of either 1kHz or 10kHz.
Table 3: Calculated Initial Conditions on Time Dependent Variables
3. Third, the circuit (Figure 1) was constructed in a laboratory setting using components with a 5% error probability. The input sine wave was set using a function
generator at 1kHz, 10kHz and a 4volt peek waveform. The input signals frequency and voltage were then confirmed with the use of an oscilloscope. Measured data was
then taken with respect to the variables Vo, I_{1}, I_{2}, I_{3} for both frequencies. Measuring Vo was a direct measurement although for
the currents required an indirect approach. By measuring the voltage across R_{1}, R_{2}, R_{3} and applying ohms law (E.4) the currents
were simply calculated for each of the Variables (see Table 4).
Table 4: Measured Voltages & Currents for RLC Series  Parallel Circuit
4. Once measured data was taken, Pspice computer simulations run, and hand calculations all agreed with one another the characteristics of the circuit with respect to a
wide range of frequencies was examined. Upon inspection of Vo vs. frequency it was predicted that as Vo approached 0Hz, the voltage across Vo would approach zero. Also
as the frequency of Vo approached infinity, the voltage across Vo would approach 4volts.
5. To better understanding of the characteristics of the circuit (see Figure 1) Pspice simulations were again used. Different then in Part 2 where the transient analysis
was invoked, in Part 5 of the experiment an AC analysis (.AC) were performed. Unlike the .TRAN analysis the AC analysis does not involve differential equations in it's
calculations. Rather the AC analysis uses the same approach as the hand calculations in Part 1. This approach alleviates the need for initial conditions and allows greater
ease to examine Vo as a frequency vs. phase angle, voltage, and current (see Graphs 3, 4, 5).
On examination of graph 3, Vo vs. frequency over a period of 100Hz  100kHz it was confirmed that as Vo approaches 0Hz, Vo approaches zero. And as Vo approaches infinite
frequency, Vo stays constant with a positive magnitude of 2.775volts. In examination of Vo as frequency approaches infinity all frequency dependent components resembled one
of the following.
In a capacitor, as frequency increases the impedance of the capacitor gets very small and at very large frequencies the impedance of a capacitor approaches zero.
Graph 4 examines each variable Vo, I_{1}, I_{2}, I_{3} over a frequency range of 100Hz  100kHz. Taking into consideration the above characteristics
of inductors and capacitors one can visually confirm the properties of the components. Graph 5 examines the variables Vo, I_{1}, I_{2}, I_{3} over
the same frequency range of 100Hz  100kHz as a function of phase angle. By comparing the individual outputs, especially that of Vo, Graph 5 very quickly confirms the over
all circuits characteristics. The output phase angle Vo and both phase angles of the capacitors at high frequencies (above 100kHz) all have the same phase angle of zero. As
for the inductor at high frequencies (above 100kHz) has a phase angle of 90deg. Therefore, the over all characteristics of the circuit is capacitive.
6. Part 6 involved a final check of Graph 3 and actual measured values of Vo at frequencies of 100Hz and 100kHz. Measurements were taken using an oscilloscope at both
frequencies and plotted directly on Pspice Graph 3.
Conclusion & Discoveries:
Phase shift are independent of initial conditions. Therefore, actual capacitors and inductors initial conditions may have either an initial voltage or current associated
with them as well as a phase shift. It was discovered that even though correcting the voltage or current of a component, the phase shift error is unable to be corrected.
At extremely high frequencies a capacitors impedance approaches zero allowing current to flow unrestricted through the element. Capacitors at very high frequencies resemble
a short circuit.
At extremely high frequencies an inductors impedance approaches infinite resistance allowing no current to flow through the element. So inductors at very high frequencies
resemble an open circuit.
The Pspice transient analysis (.TRAN) initial conditions always need to be calculated when examining the circuit’s behavior at time equal to zero. Although by examining
the circuits behavior after some time t > 0, initial conditions become obsolete.
The relationship between frequency of capacitors and inductors becomes very important in examining the circuit in Figure 1. It was discovered that by incorrectly
calculating the current I1(I1 = E / R1 + Xc1) at frequency 1kHz the Pspice simulations and measured values all agreed within a 5% error. Although when the frequency
was increased to 10kHz, calculating I1 incorrectly resulted in large errors of the hand calculations when compared to Pspice simulation and measured data. By performing
three separate analysis of any circuit one can more accurately discover errors.
In closing, the overall effectiveness of a circuit depends greatly upon it frequency and the behavior of its frequency dependent components. So by examining any given
RLC circuit and knowing the input frequency and either input total voltage or current one can determine generally the characteristics of the circuit’s possible outputs.
Electrical Engineering lab key words: RLC series circuit, RLC parallel circuit, circuit behavior, resistor, inductor, capacitor,
seriesparallel, voltage, current, amplitude measurement, phase measurement, PSpice analysis, discrete frequency, computer simulations, voltage, current,
phase, high frequency, resistance, power, work and efficiency; Ohm’s and Kirchhoff’s laws; series and parallel circuit principles. RLC circuits, equivalent
circuits, seriesparallel DC resistive networks, sinusoidal AC voltage, phasors, average and effective values, impedance, AC series parallel
circuits, AC power, AC network analysis, AC network theorems, dependent sources, transformers. Network theorems applied to DC circuits:
source conversions, Thevenin, Norton, superposition; capacitance; magnetic circuits; inductance; transient analysis of RC and RL circuits;
sinusoidal waveforms; reactance and impedance; series, parallel, and seriesparallel AC circuits.
