OBJECTIVES:

1. To understand what happens to a circuit when it reaches a resonant frequency.
2. To understand why the current peaks at the resonant frequency.
3. To graphically show the various voltages in the circuit at resonance and on both sides of resonance.

BACKGROUND & THEORY:

In the following R-L-C circuit of ideal components, the current (I_s) will reach a maximum when the reactances (X_L and X_C) are equal in magnitude, and the impedance (Z_eq) reaches a minimum at a specific frequency. This frequency is called the resonant frequency.

Note the following equations which express this phenomenon mathematically.

 

To solve for the resonant frequency (f_r), you merely set the two reactances equal as shown below.

Solving for -- omega and then f_r

EQUIPMENT AND PARTS:

Resistors: 68 W, 1.0 kW
Capacitor 2.5 µF
Inductor 47 mH
Breadboard
H-P LCR Meter
Oscilloscope
Signal Generator (Oscillator) PROCEDURE:
Digital Multi meter (DMM)

1. Measure the actual value of the R and the dc resistance of the inductor with the DMM. Measure the values of L and C using the LCR meter at 1 kHz and record the dissipation factor (D) as well as the inductance or capacitance.  The dissipation factor is equal to the cotangent of the phase angle of the impedance of the inductor or capacitor.

2. Construct the following circuit with the 1.0 kW resistor. Note that the inductor is not "ideal", and that it has a resistance value due to the length of the wire in its coil. You measured this resistance in step 1.

3.  Set the signal generator output level, V_S, to approximately 1.7 volts.  Then for each of the following frequencies:  100 Hz, 200 Hz, 500 Hz, 1 kHz, 2 kHz, and 5 kHz,  measure and record each of the five voltages (V_S, V_R, V_L, V_C, and V_LC) using the DMM.

4. a. Adjust the frequency until the phase angle is zero.  Use the XY display and watch for the elliptical pattern to become a straight line. This is the phase resonant frequency.  You should expand the scales to make this measurement more sensitive, since you don't need the ends of the ellipse to see when it flattens to a single line.  Expanding the X scale makes the distance between the sides of the ellipse larger and therefore a more sensitive measure of the resonant frequency.  You also need to expand the Y scale to make the sides of the ellipse steeper and the difference between their zero crossings easier to see.

b. Measure and record all five voltages at this resonant frequency.

c. With the voltmeter across the L-C combination to measure, V_LC, change the frequency slightly in each direction from the resonant frequency and observe how the voltage changes.  Is this voltage a minimum near this frequency? Record the frequency at which the value of, V_LC, is the smallest.  This is the voltage resonant frequency.  How does this frequency compare to the frequency at which the phase angle went to zero?

d. The V_LC voltage will increase if the frequency is changed either direction from this point.  Record the upper frequency above the resonant frequency at which this voltage has increased 10% from its minimum value.  Then record the lower frequency below the resonant frequency at which this voltage has increased 10% from its minimum value.

Note that, V_LC, is minimum and, V_R, and therefore, I_R,  are maximum at the resonant frequency.
 
  Repeat steps 3 and 4 with the 68 W resistor replacing the 1.0 k Ohm resistor while keeping V_S at about 1.3 volts

1. Divide each of the four voltages (V_R, V_L, V_C, and V_LC) by V_S for each frequency. This "normalizes" your data to give the value you would have gotten if V_S had been set at exactly 1 volt for all measurements. Then plot these normalized voltages versus frequency using a logarithmic scale for frequency. (In the newer versions of the Quatro Pro or the Excell spreadsheets this is done by clicking on the x axis of the graph to select it and then right clicking on it to get the properties menu.  Then select x-axis properties.  From the x-axis properties menu select scale.  On the scale menu you will see the log and linear scale selections.)   Plot this graph using a spreadsheet program.  Plot one graph for each of the two resistance values.  The graph should be similar to the example below.

2. Draw to scale(2 inches/volt) phasor diagrams for the circuit at each of these frequencies: 200 HZ, the resonant frequency, and 2000 HZ, showing that the sum of voltages V_R, V_C, and V_L is equal to V_S.  Use the resistor voltage as the reference for zero phase angle as shown in the Example below.  In this example the frequency is below the resonant frequency where the capacitor voltage is larger than the inductor voltage.  This causes the source voltage to have a negative phase angle relative to the resistor voltage.  For frequencies above the resonant frequency the source voltage will be at a positive phase angle relative to the resistor voltage and the inductor voltage will be larger than the capacitor voltage.  Use the law of cosines to calculate all the angles in each of the two triangles formed in these diagrams.  Then determine the phase angle between V_R and V_L from the angles in the two triangles for each of the three diagrams.

For frequencies below resonance where V_C is larger than V_L the diagram can be drawn like this
 

or like this

3. Using your measured voltage values and your knowledge of trigonometry (Cosine law), calculate the phase angle between V_R and V_S at each frequency for both resistance values.

4. Plot a graph of the phase angle of the source voltage relative to the resistor voltage for both resistor values  using a logarithmic scale for frequency.

5. Use your measured voltages and resistance value to calculate the impedance of the series L-C part of the circuit at each frequency for both resistance values. Note:  I  = V_R/R, and  Z_LC = V_LC/I.  Then plot Z_LC versus frequency.  Plot the Z_LC  impedance for both circuits on the same graph using a logarithmic scale for frequency.  Are they the same?  Why?
Example

6. Is it possible for any of the voltages to be greater than V_S? Why?

7. Calculate the phase angle between the positive direction of the V_R phasor (current phasor) and the positive direction of the V_L phasor at each frequency.  This will require you to solve for the angle between V_R and V_LC and the angle between V_LC and V_L using the Law of Cosines.  From these angles and your knowledge of geometry you can find the required angle.  Refer to your three phasor diagrams for guidance.  Plot this phase angle using a logarithmic scale for frequency.   Since V_L is the sum of a voltage across the ideal inductance of the inductor and the voltage across the resistance of the inductor, the part of the voltage that is in phase with the resistor voltage can be used to estimate the equivalent ac resistance of the inductor and the part that is 90 degrees out of phase can be used to estimate the equivalent ideal inductance of the real inductor. The diagram below illustrates the equivalent circuit modeled with IDEAL components.

Calculate these two values for each frequency. On another graph plot the resistance and inductive impedance using a logarithmic scale for frequency.

8.  Calculate the impedance of the Capacitor and the impedance of the inductor at each frequency and plot them using a logarithmic scale for frequency.

CONCLUSIONS:

Your conclusions should include, but not be limited to answers to these questions:

a. What are your observations of the behavior of this circuit as a function of frequency?

b. Can you suggest some possible uses of this type of circuit?