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General Physics Experiment 1 Resistor Inductor Capacitor Series Circuits Objectives: < < To observe the relationships between the voltage and current across resistors, inductors, and capacitors in series combinations as the frequency of the source is varied. To observe resonance in an R-L-C circuit. Equipment: < < < < < < Cables, 3, with din connectors Pasco Signal Generator PI-9587 Pasco Circuit Board with 100 and 330 microfarad Capacitors, 8 millihenry Inductor having about 5 S resistance, and a 10 S Resistor Multimeter Capacitance meter with 200 microfarad range Pasco Signal Interface II with Science Workshop and Graphical Analysis software Physical Principles: The voltage, V, across a resistor varies in phase with a current, I, through it and is related to it by V R ' VRo sin(Tt) (1) In this equation VR is the voltage across the resistor, VR0 is the maximum voltage or amplitude, and T = 2Bf with f the frequency of the current. The voltage across an ideal capacitor (one with negligible resistance) reaches a maximum one quarter cycle after the current does while the voltage across an ideal inductor (one with negligible resistance) reaches a maximum one quarter cycle before the current does. The voltage builds up on the capacitor as the current deposits charge on it, and when it is charged the current is zero. Since the current is increasing most rapidly when it is zero the voltage on the inductor is greatest then. Voltages across ideal capacitors and inductors vary with time according to the equations below. VC ' Q ' VCo sin(Tt & 90E) C (2) VL ' L )I ' VLo sin(Tt % 90E) )t (3) Experiment 1 Page 1 where C is the capacitance in farads and L is the inductance in henries. The amplitudes of the voltages across the three elements discussed are given by VRo ' RIo (4) 1 I TC o (5) VLo ' T L Io (6) V Co ' where T = 2B f and f is the frequency. The coefficients of Io in equations (5) and (6) are called the capacitive and inductive reactances, XC and XL. XC ' 1 TC XL ' T L and (7) For a series R-L-C circuit the current and voltage amplitudes are related by V S ' Io Z (8) where Z is called the impedance and is determined by Z ' R 2 %(X L & X C )2 (9) The voltage leads the current if XL is larger than XC by an angle given by N ' arctan XL & XC (10) R and lags the current by the angle N given by (9) (a negative angle) when XL is smaller than XC. When XC = XL the resulting current amplitude is maximum and the circuit is in resonance. The resonance condition is TL ' 1 TC (11) which gives T' 1 LC and f ' 1 2B L C (12) The above equations are correct if ideal capacitors and inductors are used, where their resistances are small compared to their inductances. For capacitors this is generally true but for inductors it is true only for higher frequencies. To correct for the effect of the inductor resistance, the value R in equations (9) and (10) must be replaced with R + RL. Experiment 1 Page 2 For an R-L circuit, as shown in figure 1, the following phasor diagram and equations can be written tanNL ' tanNS ' L T RL L T R%RL X (13) Z Z N NS (14) R L TL L R R L For an R-C circuit, as shown in figure 2, the following equation Figure 1 Phasor diagram for a R-L and diagram can be written. series circuit. cotNS ' RC T (15) X R R Procedure: N S 1 Z T C Series R-L Circuit Caution: Do not turn on the Pasco Signal Generator until the lab instructor checks your circuit. Figure 2 Phasor diagram for a series R-C circuit. With an ohmmeter measure and record the values of the resistance R (about 10 ohms) of the resistor and RL (about 5 ohms) of the inductor. Set up a series R-L circuit, with channel A of the Pasco interface box connected across the resistor. Be certain that the RED probe wire is on the positive side of the resistor. Connect channel B of the interface box across the inductor again with the RED probe wire on the positive side of the inductor. Connect channel C across the signal generator terminals with the RED probe wire on the positive Figure 3 Pasco Signal Generator. side of the inductor. Set the multimeter to 20 volt AC operation. Ask the lab instructor to check your circuit. Turn the voltage on the signal generator all the way down, and then turn it on. Now, raise the voltage on the signal generator until the voltage displayed on the multimeter is approximately 3.5 Volts. Experiment 1 Page 3 Never exceed 4 Volts on the multimeter. For frequencies of about 30 Hz, 60 Hz, 100 Hz, 150 Hz, and 200 Hz complete the table 1 by following the instructions listed below. Science Workshop Setup Double click on the Science Workshop icon. Click on the din plug icon (to the right of the graph icon), drag it onto analog channel A and double click on Voltage Sensor. In the same way drag the din plug icon onto analogue channels B and C, selecting Voltage Sensor in each case. Double Click on Sampling Options and click on Time under Stop Conditions. Enter a stop time of 0.1s and click on OK. Set the Periodic Samples rate to 10,000 and click on OK. Click on REC to collect a set of data. Click on the graph icon and drag it onto the Voltage Figure 4 AC circuit board for R - L connections tR tL tS Cursor icon Rescale icon Input selection icon Figure 5 Graphs of the voltages in an R - L circuit, channel A shows the resistor voltage, channel B shows the inductor voltage and channel C shows the signal generator voltage. Experiment 1 Page 4 icon below channel A. Click on the rescale icon and a graph of the resistor voltage versus time is shown. Click on the Input selection box at the lower left of the graph window (second row, second icon), click on Analog B and click on Voltage to display the inductor voltage graph. Repeat this to select and display the signal generator voltage from Analog C. Click on the cursor icon and place the vertical cursor line through the zero crossing point for the resistor voltage then read the value below the time axis. Repeat this for the zero crossing just before that of the resistor voltage for the inductor and signal generator voltages as shown in figure 5 and record these values in table 1 as tR , tL and tS . Compute the phase shifts in radians from NL = T(tR-tL) and NS = T(tR-tS). Plot a graph of tan NL on the y axis versus T on the x axis. From equation (13) the slope = L/RL. Determine the value of the inductance from this relation. Plot a graph of tan N on the y axis versus T on the x axis. From equation (14) the slope = L/(R+RL). Determine the value of the inductance from this relation. The two inductances calculated should be very close to each other. Calculate the % error between them. Table 1 Series R-L circuit f T tR tS tL NL NS tan NL tan NS Series R-C Circuit Turn the amplitude on the signal generator down and replace the inductor with a 330 µF capacitor. Then increase the gain until the voltage is about 3.5 volts. Repeat the measurements performed in the R-L experiment and complete table 2. Do this for frequencies of 20 Hz, 30 Hz, 50 Hz, 80 Hz, and 150 Hz. You need not include phase shifts for the capacitor in table 2 since they are all very nearly 90E. In your journal calculate at least one capacitor phase shift. Click on the cursor icon and place the horizontal cursor at the top of each voltage, then read the value at the left of the voltage axis. Experiment 1 Page 5 Plot a graph of VRo/VCo on the y axis vs T on the x axis and compute the value of the capacitance from t h e relation slope = RC. Dividing equation (4) by equations (5) gives Vro/VCo = RC T showing that the slope is RC Plot a graph of 1/tanN on the y axis vs T on the x axis. From equation (15) the slope = L/(R+RL). Determine the value of the inductance from this relation. These values should be close to each other. Find the % error between them. Figure 6 AC circuit board for R - L connections Table 2 Series R-C circuit f T tR tS NS VR0 VC0 VR0'VC0 tan NS Resonance in a Series R-L-C Circuit Turn the source voltage amplitude to zero. Add the inductor to the circuit so everything is connected in series. Increase the amplitude to about 3.5 volts. For a frequency of about one fifth of the resonant frequency describe and explain the phase and voltage relationships. Repeat this for a frequency of about 5 times the resonant frequency. Find resonance by adjusting the frequency until the voltage across the resistor is maximum. Compare this to the theoretical resonant frequency. Experiment 1 Page 6