1422-1 RESONANCE AND FILTERS

Experiment 1, Resonant Frequency and Circuit

Impedance

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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

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.

At resonance XT is equal to zero. This is

because XC and XL cancel each other out.

XC will decrease with frequency

XL will increase with Frequency

There is one frequency where XL = XC

It is at this frequency where XT = 0, which is

called the resonant frequency (f0) and the

circuit condition is called resonance

At resonance (XT = 0), and the previous

equation for total impedance can be

rewritten as Z0 = R. XT is sometimes

written as X.

Z0 is the total impedance at resonance

The subscript “O” is often used to designate

the resonate circuit condition

Use the formula below to calculate

Resonant Frequency

SEVERAL IMPORTANT FACTS

When the signal frequency is below

resonant frequency (fO), the circuit is

capacitive (that is XC is greater than XL).

When the signal frequency is above

resonant frequency (fO), the circuit is

inductive (that is XL is greater than XC).

At resonance, the circuit is resistive

(XC = XL).

The greater the LC product, the lower the

resonant frequency of the circuit

A practical approach to finding the fO of a

series RLC circuit is outlined in the

following steps

1. Apply a signal voltage to the circuit

2. Vary the signal frequency

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

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!

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

PARTS REQUIRED

1 107mH ferrite core inductor

1 0.01µF Mylar capacitor (103)

1 0.033µF Mylar capacitor (333)

1 2700Ω ½ W resistor (red-violet-red-gold)

1 1000µF Electrolytic capacitor

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

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

SERIES RLC RESPONSE CHARACTERISTICS GRAPH

3. Construct the circuit using the following

schematic diagram

VOLTAGE DOUBLER SCHEMATIC CKT

VOLTAGE DOUBLER PICTORIAL DIAGRAM

1422, EXP 1, SERIES RLC CKT PICTORIAL

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

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.

DATA TABLE FOR EXPERIMENT 1

6. Rotate the FREQ knob slowly until the

voltage drop (ER) 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

c. Also record the values of ER and the applied

voltage E at the resonant condition

d. Then turn off the trainer

7. Use the fO 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

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

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

(ER) and applied voltage E in the data table

c. Continue this procedure at each calibration

mark and then shut off the trainer

10. Plot the readings, using the Response

Characteristics graph of a Series RLC

circuit

a. Plot the voltage drop across the resistor (ER)

for each frequency mark obtained in step 9

b. Also plot the value of the voltage drop (ER)

obtained at the resonant frequency in step 6.

RESPONSE CHARACTERISTICS GRAPH

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

DATA TABLE FOR STEP 11

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

1

15. 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

16. Record the value of fO obtained in steps

6 and 13, as well as the LC product in

the following table. Note the fO

decreases as the LC product increases

STEP LC Product Resonant

Frequency

6

13

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 1st circuit we used the following

components (L = 0.107H and C = 1 x 10-8 F)

We predicted fO = 4868 Hz which compared

favorably to the experimental estimate of

4600 Hz

In the 2nd circuit we used the following

components (L = 0.107H and C = 3.3 x

10-8 F)

We predicted fO = 2679 Hz which compared

favorably to the experimental estimate of

2400 Hz

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.

RESULTS - DATA TABLE FOR EXPERIMENT 1

DATA TABLE FOR STEP 11

LC PRODUCT AND fO TABLE RESULTS

STEP

LC Product

Resonant

Frequency

6

1.07 x 10-9 4868 Hz

13

3.53 x 10-9 2679 Hz

FINAL DISCUSSION

The problem of determining resonant

frequency can be determined in two ways

1. Use the resonant frequency formula

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 fO is not possible due to the imperfect calibration of the signal generator

RESPONSE RESULTS GRAPH OF SERIES RLC CKT

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

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.

VOLTAGE DOUBLER CONVERSION GRAPH

QUESTIONS?

RESOURCES

Rubenstein, C.F. (2001, January).

Resonance and Filters. Lesson 1422-1:

Experiment 1, Resonant Frequency and

Circuit Impedance. Cleveland:

Cleveland Institute of Electronics.

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All Rights Reserved /July 2012