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Microsoft Word - LAB-10 LRC-Resonance.docx LRC Circuit-Resonance - Page 1 of 8 Written by Chuck Hunt LRC Circuit Resonance Introduction The current 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 current is measured below resonance, at resonance, 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 current 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 current. 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 current) by ? 2⁄ or 90⁰. The voltage across the ideal inductor is: LRC Circuit-Resonance - Page 2 of 8 Written by Chuck Hunt ?% = ? ?? ?? = −?"#$ (??) sin?? = ?"#$(??) cos .?? + ? 22 (5) The inductor voltage leads the resistor voltage (and current) by ? 2⁄ or 90⁰. An actual inductor also has some resistance, so the measured voltage across the inductor will have slightly smaller phase shift. The addition of voltages with different phases is analogous to addition of vectors with different directions. The result is called a “phasor” diagram (but has nothing to do with Star Trek weapons): Figure 2: Phasor diagram illustrating phase-dependent addition of voltages. As time increases, the whole voltage pattern rotates (increasing phase ??) and the observed voltages are the horizontal components of each “vector” voltage. The three components obey the AC analogs of Ohm's Law: VR = IR, VL max = Imax XL, VC max = ImaxXC, where XL and XC are the AC analogs of resistance called the inductive reactance and the capacitive reactance. The capacitive reactance and the inductive reactance are given by: ?! = & '! (6) XL = ωL (7) The maximum current and total voltage are then related by ℰ"#$ = ?"#$? = ?"#$>?( + (?% − ?))( (8) 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 resonance, the current 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 Figure 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 (current) of the circuit. The response is measured by measuring the voltage across the resistor since the current 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 correct. When current (in phase with VA = VR) is to the left of VB (= total voltage), then total voltage lags current (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 row.