# RESONANCE AND FILTERS

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1 RESONANCE AND FILTERS Experiment 1, Resonant Frequency and Circuit Impedance For more courses visit

2 OBJECTIVES 1. To verify experimentally our theoretical predictions concerning the resonant frequency of a series circuit 2. To show experimentally, that at series resonance, the current in a circuit is maximum and the impedance is at minimum

3 INTRODUCTION The impedance of any series RLC circuit depends on the signal frequency. The following formula illustrates the formula for the impedance at any given frequency.

4 At resonance X T is equal to zero. This is because X C and X L cancel each other out. X C will decrease with frequency X L will increase with Frequency There is one frequency where X L = X C It is at this frequency where X T = 0, which is called the resonant frequency (f 0 ) and the circuit condition is called resonance

5 At resonance (X T = 0), and the previous equation for total impedance can be rewritten as Z 0 = R. X T is sometimes written as X. Z 0 is the total impedance at resonance The subscript O is often used to designate the resonate circuit condition

6 Use the formula below to calculate Resonant Frequency

7 SEVERAL IMPORTANT FACTS When the signal frequency is below resonant frequency (f O ), the circuit is capacitive (that is X C is greater than X L ). When the signal frequency is above resonant frequency (f O ), the circuit is inductive (that is X L is greater than X C ). At resonance, the circuit is resistive (X C = X L ).

8 The greater the LC product, the lower the resonant frequency of the circuit A practical approach to finding the f O of a series RLC circuit is outlined in the following steps 1. Apply a signal voltage to the circuit 2. Vary the signal frequency

9 3. While the frequency is varied, measure the voltage drop across the resistance a. When the voltage drop reaches its maximum value, the circuit is at resonance. The signal frequency that produces the maximum or peak voltage drop is the resonant frequency

10 SEVERAL PRECAUTIONS MUST BE TAKEN The output voltage of the signal source must be kept fairly constant The voltmeter should have a high sensitivity rating, (that is a high ohms/volt rating) Take your time!

11 Make sure the resistance scale is zeroed, on the Analog Multimeter, before making resistance measurements Use the voltage doubler circuit when making voltage measurements The voltage doubler circuit may not be necessary when using a Digital Meter

12 PARTS REQUIRED 1 107mH ferrite core inductor µF Mylar capacitor (103) µF Mylar capacitor (333) Ω ½ W resistor (red-violet-red-gold) µF Electrolytic capacitor

13 PROCEDURE Caution: Take your time! When using the Analog Meter: Make sure the resistance scale is zeroed before making resistance measurements; and use the AC voltage doubler circuit when making voltage measurements. The voltage doubler is optional when using a Digital Meter

14 PROCEDURE 1. Measure and record the value of the 2700W resistor 2. Mark your frequency generator knob as shown in the series RLC response characteristics graph, which is shown on the next slide

15 SERIES RLC RESPONSE CHARACTERISTICS GRAPH

16 3. Construct the circuit using the following schematic diagram

17 VOLTAGE DOUBLER SCHEMATIC CKT

18 VOLTAGE DOUBLER PICTORIAL DIAGRAM

19 1422, EXP 1, SERIES RLC CKT PICTORIAL

20 a. Make sure to use the correct components! b. L = 107mH, C = 0.01µF and R = 2700Ω 4. Turn on the trainer, set the generator to the X10 range and turn the FREQ knob to its maximum counterclockwise (CCW) position 5. Set your meter on the 10V scale and connect it across the 2700Ω resistor

21 a. When using an Analog meter, connect a 1000µF capacitor in series with the sine-wave frequency generator output to block the d-c, as seen in the schematic and pictorial diagram. b. The majority of digital meters have a capacitor built in the meter to accomplish this Your results may vary compared to our results, depending on your trainers signal generator. Our results varied by meter used, trainer, and the rated value of the components, so do not be overly concerned if this occurs.

22 DATA TABLE FOR EXPERIMENT 1

23 6. Rotate the FREQ knob slowly until the voltage drop (E R ) across the resistor reaches a maximum value. Using a soft lead pencil, mark the position on the frequency knob on the frequency scale. a. This mark indicates the resonant frequency of the circuit b. Estimate the frequency and record it in the data table

24 c. Also record the values of E R and the applied voltage E at the resonant condition d. Then turn off the trainer 7. Use the f O equation and calculate the resonant frequency where L =.0107 H and C = 0.01 x 10-6 F and record the value in the data table

25 8. Compare the estimated resonant frequency to the calculated resonant frequency. Remember: The frequency calibration of the generator is not exact. If your answer is close, you have demonstrated the ability to predict the resonant frequency

26 9. Using the calibration marks of the generator as a guide, set the FREQ knob indicator to the first calibration mark. a. Note: the knob should be rotated to the fullest CCW position b. Record the voltage drop across the resistor (E R ) and applied voltage E in the data table c. Continue this procedure at each calibration mark and then shut off the trainer

27 10. Plot the readings, using the Response Characteristics graph of a Series RLC circuit a. Plot the voltage drop across the resistor (E R ) for each frequency mark obtained in step 9 b. Also plot the value of the voltage drop (E R ) obtained at the resonant frequency in step 6.

28 RESPONSE CHARACTERISTICS GRAPH

29 11. Now determine the impedance at each of the frequency calibration marks, which is a two step process. a. Calculate the current flowing in the circuit at each calibration mark by applying Ohm s Law, I = E/R. b. Then determine the impedance (Z = E/I) at each calibration mark. c. Record these values in the following table

30 DATA TABLE FOR STEP 11

31 12. Erase the pencil marks on your trainer 13. Repeat steps 2 through 5, but replace the capacitor with a 0.033µF capacitor 14. Calculate the resonant frequency and record the new resonant frequency in step Calculate the LC product; multiply the value of L by the value of C for steps 6 and 12 and record the data in table 1

32 16. Record the value of f O obtained in steps 6 and 13, as well as the LC product in the following table. Note the f O decreases as the LC product increases STEP LC Product Resonant Frequency 6 13

33 RESULTS The following data shows it is possible to predict the resonant frequency of a series RLC circuit, and then to confirm the theoretical results with experimental data In the 1 st circuit we used the following components (L = 0.107H and C = 1 x 10-8 F) We predicted f O = 4868 Hz which compared favorably to the experimental estimate of 4600 Hz

34 In the 2 nd circuit we used the following components (L = 0.107H and C = 3.3 x 10-8 F) We predicted f O = 2679 Hz which compared favorably to the experimental estimate of 2400 Hz

35 The data table for step 11 illustrates that the impedance of the circuit will have its minimum value at the resonant frequency The data table for steps 15 and 16 illustrates that the LC product increases as the resonant frequency decreases.

36 RESULTS - DATA TABLE FOR EXPERIMENT 1

37 DATA TABLE FOR STEP 11

38 LC PRODUCT AND f O TABLE RESULTS STEP LC Product Resonant Frequency x Hz x Hz

39 FINAL DISCUSSION The problem of determining resonant frequency can be determined in two ways 1. Use the resonant frequency formula

40 2. We also used the practical approach when the circuit was constructed. We made use of the principle that the voltage drop across the resistor will reach its maximum value only at resonance We were able to see both methods were affective as demonstrated in the results. An exact reading of the practical f O is not possible due to the imperfect calibration of the signal generator

41 RESPONSE RESULTS GRAPH OF SERIES RLC CKT

42 We were also able to see another characteristic of the Series RLC circuit in the Response Results Graph of Series RLC Circuit As the frequency was increased, we were able to see the voltage drop across the resistor increase reaching a maximum value at resonance

43 At the same time, we were able to view the impedance of the circuit decrease to a minimum value at resonance and is equal to the resistance of the circuit.

44 VOLTAGE DOUBLER CONVERSION GRAPH

45 QUESTIONS?

46 RESOURCES Rubenstein, C.F. (2001, January). Resonance and Filters. Lesson : Experiment 1, Resonant Frequency and Circuit Impedance. Cleveland: Cleveland Institute of Electronics.

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