eactance and Impedance Theory esistors, inductors, and capacitors all have the effect of modifying the size of the current in an AC circuit and the time at which the current reaches its maximum value (in relation to the voltage). However, each of these circuit elements affects the current in a different way. First, consider a resistive circuit ( only). Here, the current is given by where I e and V e are the effective (rms) values of current and voltage respectively. The current is also in phase with the voltage - it reaches its maximum at the same time as the voltage. Next, consider an inductive circuit. Here, the inductor L generates a self-induced back emf so as to oppose any change in the current through it. We call this opposition to the change in current the inductor s reactance X L, X L = ωl (1) where ω = 2πf. As with resistance, the unit of reactance is an ohm (Ω). With this quantity, the current is given by X L but the current is 90 behind the voltage (the voltage reaches its maximum one quarter cycle before the current). We say that the voltage leads the current. Finally, consider a capacitive circuit. Here, the capacitor C supplies a current to the circuit so as to oppose any change in the voltage across it. We call this opposition to the change in voltage the capacitors reactance X C. Here, X C = 1 ωc (2) X C but the current is 90 ahead of the voltage. We say that the current leads the voltage. In general, the total impedance Z in a circuit containing series resistance, inductance, and capacitance is given by 1
Z = 2 + (X L X C ) 2 and we have Z. The phase angle φ between the current and the voltage is obtained from Apparatus tan φ = X L X C. (3) Computer, Pasco 750 interface, Voltage sensor, esistor, Inductor, Capacitor, Patch cords, DMM s, Data Studio software. Procedure esistor 1. Connect a DMM (ammeter) and the resistor in series with the signal generator output of the 750 as shown below. Connect a second DMM (voltmeter) across the resistor. Scale the ammeter at 20mA AC; the voltmeter at 2V AC. Signal Generator V e I e 2. Open the activity eactance and Impedance. You should see a Signal Generator window similar to Figure 1. Make sure that you are producing a Sine Wave with Amplitude 1.000V and frequency 200Hz. When ready to collect data, click the Start button in Data Studio. 3. ecord the effective current in the circuit as well as the effective voltage across the resistor (these are the values shown on the DMM s). 4. Increase the frequency by 200Hz and repeat Step 3. Continue until you reach 1400Hz. 5. Click the Stop button and disassemble the circuit. Connect the resistor directly to the signal generator output. 6. Plug the DIN connector of the voltage sensor into analog Channel A of the 750. Connect the leads of the voltage sensor to the resistor (the black lead of the sensor should be connected to the side of the resistor which is connected to the ground output). 2
Figure 1: Signal Generator 7. eset the frequency to 200Hz and open the Oscilloscope window (it should be along the bottom of the Data Studio window). 8. Click the Start button. You should see the full voltage across and the current through the resistor. Make sure the scope settings are: Output Voltage 0.5V/div, Current 0.005A/div, and sweep 1ms/div. 9. Save this oscilloscope window output to a flash drive - for inclusion in the report. 10. Use the Smart Tool to obtain values for V max and I max. Note that you will need to open separate Smart Tools for voltage and current. Inductor This procedure is the largely the same as for the resistor, with the following exceptions: The original circuit will use an inductor rather than a resistor. escale the ammeter to 200mA. escale the oscilloscope current display to 0.1A/div. In addition to V max and I max, obtain the times at which these occur (these maxima should be adjacent). Capacitor This procedure is the largely the same as for the inductor, with the following exceptions: The original circuit will use a capacitor rather than an inductor. escale the ammeter to 2A AC. For the oscilloscope trace, reset the frequency to 50Hz. escale the current to 0.05A/div and the sweep to 5ms/div. Total Impedance 1. Connect the resistor, inductor, and capacitor in series with the signal generator as shown below. Connect the voltage sensor across all three elements. 3
Signal Generator L V Voltage Sensor C 2. eset the frequency to 2000Hz. 3. The oscilloscope settings are 0.5V/div, 0.005A/div, and 0.1ms/div. 4. Save the trace as well. 5. Obtain V max, I max, and the times at which these occur. Analysis esistor 1. Does vary with ω; i.e., is there a relationship between the two? Support your answer. 2. Include the oscilloscope trace in your report. With V max and I max, calculate V e and I e. These are related by V e = V max 2 I e = I max 2. Do these values seem reasonable based on the data collected in the procedure? Why or why not? Is there a phase difference between V and I? Is this what you expected? Inductor 1. Confirm Equation 1 graphically. Use the slope to determine the inductance L of the inductor used in the procedure. 2. Include the oscilloscope trace in the report. As above, calculate and discuss V e and I e. 3. Calculate and discuss the phase angle φ between V and I. Proceed as follows: (a) Calculate the time difference between V max and I max as t = t I max t V max. (b) Use This will be in radians; convert to degrees. φ = ω t (c) What should be the phase difference between I and V? How close were you? 4
Capacitor 1. Confirm Equation 2 graphically. use the slope to calculate the capacitance C of the capacitor used in the procedure. 2. Include the oscilloscope trace in the report. Calculate and discuss V e and I e. 3. Calculate and discuss φ. Total Impedance 1. Include the oscilloscope trace in the report. Calculate and discuss V e and I e. 2. Calculate and discuss φ. Use Equation 3 to determine another value of φ for comparison. 5
Pre-Lab: eactance and Impedance Name Section Answer the questions at the bottom of this sheet, below the line - continue on the back if you need more room. Any calculations should be shown in full. 1. What is ω in terms of f? 2. What is the relationship between reactance and frequency for an inductor? 3. What is the unit of inductance? 4. You measure 45.7mA through an inductor with a voltage of 2.12V across it. What reactance does it offer? 5. What is the phase difference between current and voltage for an inductor? Which leads, I or V? 6. What is the relationship between reactance and frequency for a capacitor? 7. What is the unit of capacitance? 8. What is the phase difference between current and voltage for a capacitor? Which leads? 9. For a capacitor at f = 100Hz, you measure t I max as 0.0301s and t V max as 0.0325s. What is the phase angle between I and V? 6