LRC Circuit Electromagaesm Labs 1.. Introduction IE Brief Description of the Cancegrs The purpose of this experiment is to study an inductorresistor-capacitor circuit [LRC circuit), most importantly we will study how a resonant LRC circuit is created. This is done by examining the current through the circuit as a function ofthe frequency of the applied voltage. When a vibrating mechanical system is set in motion, it vibrates at its natural frequency. However, a mechanical system can be forced to vibrate at a different frequency. The amplitude of vibration, and hence the energy transferred to the system, depends on the difference between the natural frequency and the frequency of forced vibration. The amplitude becomes very large when the difference between the natural and forced frequency becomes very small. This is known as resonance and the natural frequency of the system is sometimes called the resonant frequency. At resonance. relatively little energy is required to get a large amplitude. One example of resonance is when a singer's amplied voice is used to shatter a glass. Electrical resonance is analogous to mechanical resonance. The energy transferred to a system is maximized at resonance. The amplitude of the AC current [la] in a series LRC circuit is dependent on the amplitude of the applied voltage {Val and the V impedance {Z}: fl] = f' Since the impedance depends on frequency, the current varies with frequency 2' = .y (XL - XE): + R2, where X; 2 toll is defined as the 1 inductive reactance, Xe = c is dened as the capacitive reactance, R is the Lt} resistance. and m is the angular frequency [= Inf. where f is the frequency}. The current will be at its maximum when the circuit is driven at its resonant frequency: m3\" = E. This is obtained when X=Xr and thus the impedance [Z] is reduced to R and hence to its lowest possible value. Therefore, the current will be at its largest value, hence the voltage across the resistor will be at its maximum as well. II. In-Lab: Equipment, Experimental Procedure, Data Taking and Preliminary Data Analysis a) Equipment needed (for face to face): The equipment needed in this lab are: 1. Power amplifier 2. Voltage Sensor 3. Multi-meter 4. Patch Cords 5. A 430 F capacitor (connect 330 uF and a 100 uF in parallel) 6. 10 0 resistor: use the decade box 7. Inductor Coil: use the 800 turns coil (Self-inductance L=13.5mH) b) Experimental Procedure & Data Taking: Use the power amplifier to produce an alternating current through the LRC circuit. And then use the Voltage Sensor to measure the voltage drop VR (potential difference) across the resistor in the circuit. Both the voltage VR and the current depend on the impedance in the circuit, which varies with frequency. In this lab, you need to do the following: 1. Change the frequency of the output signal from the power amplifier. What you need to watch for is the voltage VR, if it is at its maximum, then you succeeded in finding the resonant frequency. 2. Calculate the theoretical resonant frequency for your circuit, using the values of the inductance and capacitance and compare it to your measured resonant frequency. Computer Setup: 1. Connect the Pasco interface to the computer using the USB cable 2. Connect the Power Amplifier to Pasco interface and make sure to use analog channel. 3. Open the Cap file: LabE09_LRC_def.cap 4. Enable analog out of the power amplifier by choosing "Set Up Sensors Pasco" from the Experiment menu. Click the CH4 menu in the resulting sensorsdialog. and select Analog Out. Select the \"sine" waveform. and set the frequency and voltage output to maximum. Equipment Setup: 1. Before making any connections measure the resistance of the resistor (decade box) and record the value here. R 2. 2. Connect the resistor the inductor the capacitor and the power amplifier 111 series. 3. Connect the differential voltage across the resistor, see picture below- . 13ers discarding 1. In the \"analog out\" of the power amplier. set the output voltage the maximum value (31!) and set the output frequency to 31'} Hz. NOTE: These settings are the default settings of the LoggerPro le. But. you can change the frequency through the small window located in the bottom left side of the display. 2. Start collecting data. 3. Determine the maximum voltage VR across the resistor: highlight the whole curve and click on the \"stat" icon. A pop-up window will appear. Write down the maximum value in the Table l. Frequency 85 4. Repeat this measurement 5 times and record the values in Table 1. 5. Keep on increasing the frequency by 10Hz up to 120Hz, and every time repeat the process to find the value of voltage and record it in Table 1. Table 1 Freq (Hz) Trial 1: VR Trial 2: VR Trial 3: VR Trial 4:VR Trial 5: VR| Mean Value 30 40 50 60 70 80 90 100 110 120 6. Look at the data Table 1, and estimate the approximate resonant frequency f.= HZ (where voltage across the resistor reaches a maximum). 7. Try to narrow the estimated value of the resonant frequency by narrowing the frequency interval: for example, add and then subtract SHz from the estimated value of the resonant frequency, and in each case measure the maximum voltage VR Record the values in Table 2.Table 2 Freq (Hz) Trial 1:VR |Trial 2: VR Trial 3: VR Trial 4: VR Trial 5: VR | Mean Value fe-5 f+5 8. Estimate the resonant frequency from Table 2 and record it here: exp res (Hz) 9. Set the frequency for the voltage output of the power amplifier to f = 30Hz, then using a multi-meter measure the following voltages: a. The voltage across the Power Amplifier and record the value in Table 3 b. The voltage across the Resistor and record the value in Table 3 c. The voltage across the Capacitor AND Inductor and record the value in Table 3 d. The voltage across the Capacitor and record the value in Table 3 e. The voltage across the Inductor and record the value in Table 3 10.Set the frequency of the power amplifier to the resonant frequency fres an the repeat the measurements in step 9. 1 1. Repeat the measurements in step 9, by choosing a different frequency for the voltage output of the power amplifier, USE f = 120Hz. Table 3 Item V(30Hz) V (fres ) V(120Hz) Power Amplifier Resistor: VR C-L: VC-L Capacitor: Vc Inductor: VL c) Preliminary Data Analysis: You need to answer these questions before you leave the lab:]. Append the data from Table 2 to the data from Table l and create a graph of Ist'yif'f'{volts}: versus the frequency fin]. 2. From the previous graph, estimate the experimental resonant frequency ip and record its value in table 4. 3. Calculate the resonant angular frequency wig? = 21rfsp and record its value in table 4. 4. Calculate the theoretical resonant angular frequency using the values of the . . 1 . . inductance and capacitance: m3\" = and record its value in table 4. E M f\"? (Resonant frequency (Hill) I mfg; [Resonant angular frequency (radis m3\" (Theoretical resonant angular frquLJency {radish 5. Using the values obtained in table 3, and for every frequency, calculate: a. The ratio of the voltage across the capacitor over the voltage across the resistor and fill out table 5. b. The ratio ofthe voltage across the inductor over the voltage across the rcsistor and fill out table 5. Ind tic tor 6. You also need to calculate the capacitance of the capacitor and the inductance of the inductor using the ratios from Table 5: [record these values in table 6} c. For the capacitance: It is clear that the ratio - Vc = AC where XC = VR XR w * C and XR = R, therefore -C = 1 , hence C = 1 ( *C*R VR d. For the inductance: It is clear that the ratio - = =L where X, = ( * L VR XR and XR = R, therefore -4 - (*L VR R , hence L= VL VR Table 6 Item C or L(30Hz) C or L (fres) C or L (120Hz) Capacitor Inductor III. Post-Lab: Final Data Analysis and Conclusion 1. How does your measured value for resonant angular frequency compare to the theoretical value for resonant angular frequency? Remember, the accuracy of the measurement is: (1 - res - th Exp -wo x100% 2. Is the plot of voltage versus frequency symmetrical about the resonant frequency? Explain. 3. What is your conclusion from this lab?Table 1. Maximum Voltage Measurements for Varying Frequencies Table 3. Measure Voltages Across Components at Varying Frequencies Table 5: Freq. (Hz) Trial 1 (VR) Trial 2 (VR) Trial 3 (VR) Trial 4 (VR) Trial 5 (VR) Mean Item V(30Hz) V(f-res-exp) V(120Hz) Item (Vc or L)/VR (30H (Vc or L)/VR (f-r. (Vc or L)/VR (120Hz) 30 1.873 1.885 1.885 1.876 1.885 1.8808 Power Amplifie 2.5 2.5 2.2 Capacitor 1.22 0.5 0.3 40 1955 .958 1.955 1.958 1.955 1.9562 Resistor: VR 0.9 1 1 Inductor 0.89 1 1.2 50 1.983 1.976 1.976 1.976 1.98 1.9782 C-L: VC-L 1.5 1.2 1.3 60 1.976 1.98 1.98 1.976 1.98 1.9784 Capacitor: VC 1.1 .5 0.3 70 1.976 1.976 98 1.98 1.98 1.9784 Inductor: VL 0.8 1.2 Table 6: 80 1.964 1.961 1.955 1.964 1.961 1.961 Item C or L(30Hz) Cor L (f-res) Cor L (120Hz) 90 1.952 1.958 1.955 1.955 1.955 1.955 Capacitor(IF) 421.42 515.0645408 429.2204506 100 1.943 1.946 1.949 1.943 1.946 1.9454 Inductor (mH) 48.57 27.32159856 16.39295914 110 1.928 1.93 1.931 1.934 1.934 1.9316 120 1.922 1.928 1.925 1.925 1.928 .9256 Table 2. Maximum Voltage Measurements for Narrowed Frequency Interval Freq. (Hz) Trial 1 (VR) Trial 2 (VR) Trial 3 (VR) Trial 4 (VR) Trial 5 (VR) Mean Table 4: fe-5 = 55 1.983 1.983 1.983 1.983 1.983 1.983 f exp(Resonant frequency (Hz)) 60 Accuracy=(1-((WT-we)/WT))*10 fe+5 = 65 1.983 86 1.976 1.98 1.976 1.979 exp(Resonant Angular frequency (Hz) 376.99 90.83 wo (Theoretical resonant angular frequency (rad/s)) 415.1 VR (f) 1.98 50 60 $ 70 1.96 80 40 90 100 1.94 110 Voltage across R (V) 120 1.92 1.9 1.88 30 1.86 20 40 60 80 100 120 140 Frequency (Hz)