Microsoft Word - LAB-10 LRC-Resonance.docx
LRC Circuit-Resonance - Page 1 of 8
Written by Chuck Hunt
LRC Circuit Resonance
Introduction
The cu
ent through a series LRC circuit is examined as a function of applied frequency and the
effects of changing the values of the resistance, inductance, and capacitance are observed. The
phase difference between the applied voltage and the cu
ent is measured below resonance, at
esonance, and above resonance.
Theory
Figure 1: LRC circuit
An ideal inductor, a capacitor, and a resistor are connected in series with a sine wave power
supply. Since it is a series circuit, the cu
ent will be common to all the components and given by
I = Imax cos(ωt) (1)
The voltage V0 across the power supply has a phase shift (φ) with respect to the cu
ent.
V0 = E = Emax cos(ωt + φ) (2)
The voltage across the resistor is simply:
VR = IR = Imax R cos(ωt) (3)
The voltage across the capacitor is:
?! =
?
? =
∫ ? ??
? = ?"#$ *
1
??- sin?? = ?"#$ *
1
??- cos .?? −
?
22
(4)
The capacitor voltage lags behind the resistor voltage (and cu
ent) by ? 2⁄ or 90⁰.
The voltage across the ideal inductor is:
LRC Circuit-Resonance - Page 2 of 8
Written by Chuck Hunt
?% = ?
??
?? = −?"#$
(??) sin?? = ?"#$(??) cos .?? +
?
22
(5)
The inductor voltage leads the resistor voltage (and cu
ent) by ? 2⁄ or 90⁰. An actual inductor
also has some resistance, so the measured voltage across the inductor will have slightly smaller
phase shift.
The addition of voltages with different phases is analogous to addition of vectors with different
directions. The result is called a “phasor” diagram (but has nothing to do with Star Trek
weapons):
Figure 2: Phasor diagram illustrating phase-dependent addition of voltages. As time
increases, the whole voltage pattern rotates (increasing phase ??) and the observed
voltages are the horizontal components of each “vector” voltage.
The three components obey the AC analogs of Ohm's Law: VR = IR, VL max = Imax XL, VC max =
ImaxXC, where XL and XC are the AC analogs of resistance called the inductive reactance and the
capacitive reactance. The capacitive reactance and the inductive reactance are given by:
?! =
&
'!
(6)
XL = ωL (7)
The maximum cu
ent and total voltage are then related by
ℰ"#$ = ?"#$? = ?"#$>?( + (?% − ? XXXXXXXXXX)
where ? = >?( + (?% − ?))( is called the impedance and is the AC analog of resistance for the
entire circuit.
The phase shift (the angle between VR and V0 in the phasor diagram) is related to the other
variables by
LRC Circuit-Resonance - Page 3 of 8
Written by Chuck Hunt
tan ? = ('!('")
*
(9)
If the frequency is varied, the inductive reactance and capacitive reactance also vary. At
esonance, the cu
ent is maximum and thus the impedance is at its minimum. The minimum
impedance (Equation 7) occurs when XL = XC, yielding Z = R. Setting Equation 3 equal to
Equation 4 yields the resonant frequency:
?+,-? =
.
#$%0
(10)
?+,- =
.
√23
(11)
Equipment:
PC, SW850 interface
1 each voltage probe
1 each small circuit experiment board Model CI-6512
4 each electrical lead cables
LRC Circuit-Resonance - Page 4 of 8
Written by Chuck Hunt
Setup:
1. Connect the BNC-to-Banana cord to the #2 Signal Generator and connect the red cord to
one end of the 2.5 mH inductor on the circuit board. Connect the other end of the
inductor to the 560 pF capacitor in series and the 1.0 kΩ in series. Then connect the black
cord to the open end of the resistor.
2. Connect a Voltage Sensor to Channel A on the 850 interface and attach the leads across
the resistor, making sure the black cable from the voltage sensor is connected to the
grounded side of the resistor.
3. Connect a Voltage Sensor to Channel B on the 850 interface and attach the leads across
the leads of the Output #2 cable, making sure the black cable from the voltage sensor is
connected to the black side of the signal generator.
4. Open the Signal Generator 850 Output 2 and choose the Sine Wave at a frequency of
10,000 Hz and an amplitude of 7 V. Leave the output on AUTO.
Figure 1: Series LRC Circuit
XXXXXXXXXXFigure 2: LRC Circuit with Sensors
5. Create a table as shown below.
LRC Circuit-Resonance - Page 5 of 8
Written by Chuck Hunt
The first three columns and the last column are User-Entered data sets. The fourth
column is a calculation:
V Ratio = [Resistor VA (V)]/[Output VB (V)]
In the table shown, the first data set has been renamed “1k Ohm”.
6. Create a graph of V Ratio vs. Frequency in kHz.
7. Create an oscilloscope with both voltages on the same axis (this is done by choosing
similar measurement in the measurement selector on the axis). Select ms for the units of
time on the horizontal axis.
8. Set the common sample rate to 10 MHz.
Procedure: Plotting the Resonance Curve
In this part of the lab, you will vary the frequency of the applied voltage and record the response
(cu
ent) of the circuit. The response is measured by measuring the voltage across the resistor
since the cu
ent is in phase with this voltage and it is proportional to it.
One further complication is that you must divide the resistance voltage by the output voltage (of
the 850) to account for any changes in the output voltage. This works because if the output
voltage doubles, then the resistance voltage also doubles and the ratio VR/Vo remains constant.
This is faster than trying to adjust the output voltage each time to keep it constant.
1. Begin with the signal generator set on 10 kHz. Record this frequency in Table I. Click on
Monitor. If the trace is rolling left or right, click the trigger on the oscilloscope. Adjust
the horizontal scale on the scope so about three cycles show.
2. Stop monitoring and use the coordinates tool to measure the amplitude of each of the
voltages and type the results in Table I. The coordinate tool should show three significant
figures for voltage, the snap to pixel distance should be 1 (snap disabled) and the delta
tool should be on. If not, right click on the tool and change tool properties. Increase the
vertical scale to make the signal (especially VA) you are measuring as large as possible.
Always measure the positive peak. Set the coordinate tool so the horizontal line is tangent
to the peaks. Leave the vertical scale expanded until after step 3. That will make it easier
to measure the phase shift.
3. To find the phase shift between the two voltages, use the delta tool on the coordinates
tool to measure the difference in time between the first two points where the two signal
cross the x axis (V=0) with a negative slope. First position the delta tool on the first place
this happens and then spread the horizontal scale so that less than one cycle shows so that
you can see the x axis crossings more clearly. Record this phase shift in time in Table I in
column 5. Then click the Scale to Fit icon and decrease the horizontal scale so about
LRC Circuit-Resonance - Page 6 of 8
Written by Chuck Hunt
three cycles are visible. The cross-hairs should always be on VB and the delta tool on VA
so that the sign of the phase shift time given by the delta tool will be co
ect. When
cu
ent (in phase with VA = VR) is to the left of VB (= total voltage), then total voltage
lags cu
ent (negative phase shift) since total voltage shows up later in time.
4. Increase the frequency of the output by 10 kHz and repeat the measurements.
5. Continue to increase the frequency in steps of 10 kHz up to 250 kHz. Above 250 kHz
continue by 25 kHz steps up to 500 kHz (why is this OK? Hint: examine Figure 2).
6. To find the resonance peak more precisely, we need more data near the peak (see Figure
2). We want readings at 5 kHz intervals for the region within 20 kHz of the peak. For
example, if the peak is at 140 kHz, we would like to add points at: 125 kHz, 135 kHz,
145 kHz, & 155 kHz. To do this, on the Table I, click and drag to highlight the 130 kHz
ow.