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Stephen Natanael Basatua U2040050A Experiment 8: LRC Circuit Aim To study an LRC circuit and obtain the resonance frequency of the LRC circuit Theoretical Background A series LRC circuit applied with an AC voltage has a property called the impedance. It is a sum of the value of reactance and resistance of a circuit that measures the flow that has opposite direction to the current on the circuit. The impedance Z is expressed as

√

Z = R 2+ ( X L − X C ) Where

X L=ωL

and

X C=

2

(1)

1 ωC

The voltage across the circuit will be affected by Z, at resonance where

X L= X C ω=

we can write (2) as the angular resonance frequency of the circuit

Figure 1 Series LRC Circuit

To better understand this, we can illustrate impedance Z imaginary impedance diagram

Figure 2 Phasor Diagram

on a phasor diagram inside real-

1 √ LC

Stephen Natanael Basatua U2040050A Notice that the reactance contributes to the phase angle ϕ . An inductor will make the current lag by

π , while a capacitor will make the opposite. 2

Apparatus 1. 2. 3. 4. 5. 6.

850 Interface Electronics board LCR Meter Capacitors 10Ω Resistor Voltage sensor connector for USB 850

Procedures Part I: Determining the resonant frequency for specific values of L and C Measure the actual value of inductor in the electronic circuit board, resistors and capacitors used in the experiment using the LCR meter and estimate the resonant frequency from theory. Part II: Determining resonant frequency from scan over frequencies 1. Set up the series LRC circuit in Figure 1 using a 10 resistor, 100F capacitor and the inductor in the electronic box. 2. For AC signal, use output 1 of USB 850 interface and connect to circuit box using cords. The convention is to use red for positive /live and black for negative/ground. This will be our signal voltage Vs. 3. Connect the analog input A of USB 850 interface to monitor voltage drop across resistor in circuit. This will be the output voltage Vo 4. Launch the CAPSTONE software on computer after turning on the USB 850 interface. a. On clicking on “Hardware Setup”, you will see an icon for the 850 Interface with all channels, the software has successfully detected the interface. b. Click on channel A and select “Voltage Sensor” from drop-down menu. c. Click on Output 1 and select “output voltage-current sensor” from drop-down menu. d. Select scope display in the main window. e. Select sampling rate of 500 Hz. Choose “Common rate” from the drop down menu to ensure data is sampled from all sensors at 500Hz. 5. Click on “Signal generator” setup and select the settings below for ‘850 Output 1’: i) Output: Sine wave, ii) Amplitude: 3 V , iii) Frequency: 40 Hz 6. Increase frequency in steps of 10Hz or 20Hz to ~250-300 Hz and monitor the amplitude of the voltage across resistor for each frequency.

Stephen Natanael Basatua U2040050A 7.Repeat for the 330F and 47 F capacitors.

Part III: Phase portrait method to determine resonance 1. For same circuit with 100F capacitor, change scope display so that the X axis is not time, but the signal voltage Vs. The output voltage will be displayed along the Y-axis. 2. Set the signal voltage in the signal generator menu at resonant frequency observed in Part II. 3. Do a fine adjustment of frequency until a straight line plot is obtained. This indicates that the two voltages are in phase. 4. Repeat above step for other capacitor values. Part IV: Measurement of voltages across components at resonance Monitor the voltage across all components of the circuit i.e. L, C and R at the resonant frequency using the different capacitors.

Results Notice that the values of the capacitance differ a bit than 47µf, 100 µf, and 330 µf. We will use instead 49.74 µf, 97,43 µf, and 345,2 µf which are the actual values when measured in lab. Therefore we expect to get the resonance frequency of 0.6839∗103 rad / s , 0.4689∗103 rad / s , 0. 2581∗103 rad / s respectively.

Stephen Natanael Basatua U2040050A Graph 1 Output Voltage vs Frequency graph

We can see the voltage drop for each graph as we increase the frequency according to the procedures mentioned earlier. The drop for each graph (in this case we can only do so by calculations and graph interpretation) represents the voltage where resonant frequency occurs through equation. Theoretically we should be expecting to obtain resonance frequency at

40 . 17 s−1 , 75 . 6 1 s−1

, and 105 .8 s−1 with the same order as before (for 49.74 µf, 97.43 µf, and 345.2 µf). In practice we managed to obtain 38 . 85 s−1 , 73 .85 s−1 , 104 .3 s−1 The signal voltages for each capacitor with the same order as before are: 1.443V, 2.324V, and 3.279V At the resonance frequency, the voltages across inductor, capacitor respectively are 2.82V, 2,81V and we know that they will cancel each other out due to the 180degree phase difference. The voltage across the circuit now will be just from ohm’s law which is V=IR

Error Analysis From equation (2) we can estimate the uncertainty of the angular resonance frequency to be

δω=

ω 2

√(

)( ) 2

δL δC + L C

2

For each capacitors 49.74 µf, 97.43 µf, and 345.2 µf, the uncertainty will be 0.0509rad/s, 0.0284rad/s, and 0.0143 rad/s respectively

Conclusion

ω Best ( 49.74 μ F )=( 0.6839∗10 ± 0.0509)rad / s 3

ω Best ( 97.43 μ F ) =(0.4689∗10 ± 0.0284 )rad /s 3

ω Best ( 345.2 μ F ) =( 0.2581∗103 ± 0.0143 )rad / s

Reference

1. RLC Series Circuit. (n.d.). Hyperphysics. Retrieved March 25, 2021, from http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcser.html

2. Series RLC Circuit: Analysis & Example Problems. (n.d.). Electrical A2Z. Retrieved March 25, 2021, from https://electricala2z.com/electrical-circuits/series-rlc-circuitanalysis-example-problems/

Stephen Natanael Basatua U2040050A

3. Parallel Resonant Circuits. (n.d.). Hyperphysics. Retrieved March 25, 2021, from http://hyperphysics.phy-astr.gsu.edu/hbase/electric/parres.html#c1...