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TA’s Name:
ABSTRACT
In an LRC circuit where a resistor, capacitor and inductor are connected in series is an example of a resonant circuit. Such a circuit can be tuned in order to achieve resonance. At resonance the phase angle of the input voltage in the circuit is zero. When an AC input voltage is applied to such a circuit the phase angle will be affected since the impedance of the capacitor and inductor are dependent on the frequency of the input voltage. For this experiment the phase difference between voltage and current for components in a series LRC resonant circuit. From the results of the experiment it is clear that for the resistor the voltage and current are in phase. For the capacitor, the current through it leads the input current while the inductor current lags the input current.
Objectives:
The main objective of the experiment is to determine the phase difference between voltage and current for components in a series LRC resonant circuit. The lab is aimed at demonstrating the concept of voltage and current phase difference in reactive components.
Procedure
1. The resistance r for the inductor provided for the experiment was measured and recorded
2. The circuit was connected as shown in the figure shown below. The values of R, C and L used were 20 Ω, 10 µF and 85 mH respectively
Figure 1: Figure showing the different components as connected in the circuit
3. A sine wave with a frequency of 100 Hz and an amplitude of 2 volts peak-to-peak was input into the circuit connected. Both waves were showing on the scope simultaneously and the resonant frequency f0 (where the current waveform was a maximum and the phase shift is zero) was determined and recorded.
4. The resulting waveforms were sketched and described as resonance was achieved
5. The frequency of the signal generator was set just below resonance at a point where the voltage across the resistor R was about half its value while at resonance. The voltage across the LRC circuit was observed and it was determined whether it was leading or lagging.
6. The voltage across each component was measured using a DVM as well as the input voltage
7. The waveform for each component in the LRC circuit was digitized. This was achieved by loading the setup file as directed. The Pasco probe B was connected to each component in turn and the waveform across each component determined. Probe A was checked to ensure that it triggered properly. The resulting graph was copied and passed to Graphical Analysis. The sign of the phase shift was noted in table 1.
Experimental Data [15 Points]
Resistance of Inductor:
The measured resistance of the inductor was 3.8 Ω
Resonance Frequency reading:
The resonance frequency was determined to be 247.2 Hz
Measurements of Voltages around circuit using DVM:
The voltages measured around the circuit using the DVM were as follows:
V vs. t includes VResistor, VCapacitor,, VInductor, VInput, and VTotal [10 Points]
The figure below shows the graph of V vs. t derived from the experiment conducted in the lab. The chart shows the waveforms for VResistor, VCapacitor,, VInductor, Input, and VTotal.
Results
Statement of Necessary Equations
In a series circuit the current flowing through all components is
i (t) = I0 sin ω t(1)
The voltage across each component in the circuit is given by:
vj (t) = VJ sin (ωt + Φj)(2)
Based on ohms law the voltage in terms of reactance can be expressed as:
Vj = IO Xj (3)
The phase is given by:
tan Φj = Xj / R(4)
For series LRC circuits the reactance can be determined as:
Z = √ (R2 + XL – XC) 2) (5)
The phase is given by:
tan-1 (XL – XC) / (R)(6)
Define the parameters
Explain derivation (where do equations come from?)
The equations are derived based on ohms law which states:
V = I R
Where,
V is the voltage across a component
I is the current flowing through a component
R is the resistance of the component
Theoretical Resonant Frequency Calculation
At resonance the value of XL = XC
Therefore,
ω0 = 1/ (√LC)
We know that ω = 2 π f
Therefore,
f0 = ω0 / 2 π
= (1/ (√LC)) / 2 π
= (1/ √ ((1 x 10-5)*(0.85 x 10-1))) / 2 π
= 172.60 Hz
Percent Error for Resonant Frequency
(247.2 – 172.60)/ 247.2
= 30%
Kirchhoff’s Law with RMS Voltage Values
The Kirchhoff’s Voltage Law states that the sum of all voltage drops across a closed circuit equal to zero.
∑ Vj = 0
Error Propagation
The main source of error between the
_____ Discussion and Analysis [20 Points]
_____ Calculation of all phase angles from the Graph [6 Points]
_____ Calculation of theoretical phase angle for Vinput and VResistor [4 Points]
Phase angle for Vinput
Phase angle = tan-1 (XL – XC) / (R)
XL = ωL and XC = 1/ωC
f = 100 Hz
ω = 2 * π * 100
= 628.4
XL = ωL
= 628.4 * 85 mH
XC = 1/ (628.4 * 10 µF)
Z = √ (R2 + XL – XC) 2) (5)
= 107.625 ohms
tan-1 (XL – XC) / (R)(6)
= -79.29 degrees
Phase angle for VResistor
For the voltage across the resistor, the phase angle = 0
Percent Error for Phase Angle [1 point]
(80 – 79.29) / 79.29
= 0.8954 %
Does the Resistor current Lead or Lag the Input current? [1 Point]
The resistor current is in phase with the input current-
_____ Does the Capacitor current Lead or Lag the Input current? [1 Point]
The capacitor current leads the input current.
_____ Does the Inductor current Lead or Lag the Input current? [1 Point]
The inductor current lags the input current.
_____ Do the waveforms obey Kirchhoff’s Laws? [4 Points]
Yes, the waveforms obey Kirchhoff’s laws.
_____ How can you tell when the phase changes sign? [2 Points]
One can tell when the phase changes sign when the crest and trough of the wave is in the opposite direction when compared to the original wave.
_____ Conclusion [10 Points]
_____ Was Objective Completed [3 Points]
_____ Application to Real World [2 Points]
_____ Overall report is well written and follows syllabus in format [5 points]
_____ Total [Out of 85 Points]
