Inductors in AC Circuits


 Caren Robbins
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1 Inductors in AC Circuits Name Section Resistors, 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. A capacitor in a circuit has a reactance, as does an inductor. When these are combined with resistance in the circuit, there is impedance which determines the size and phase of the current. This impedance can be thought of roughly as the AC equivalent of resistance. However, each of these circuit elements affects the current in a different way. Theory First, consider an inductor L by itself in a circuit with an alternating source of potential V AC as shown below. The inductor generates a selfinduced 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. This reactance varies directly with both the frequency f of the applied potential and the inductance L of the inductor and is given by X L =2 fl (1) As for resistance, the unit of reactance is the ohm (Ω). With this quantity, the effective potential V e across the inductor and the effective current I e through it are related by V e =I e X L (2) However, the maximum current I max is 90º (one quarter cycle) behind the maximum voltage V max. We say that the voltage leads the current. The effective (rms) value of the voltage is related to the maximum value by V e = V max 2 (3) A similar relationship exists between I e and I max. In general, the total impedance Z in a circuit containing series resistance, inductance, and capacitance is given by The effective current in this type of circuit is and the phase angle ϕ between the current and the voltage is Z= R 2 X L X C 2 (4) V e =I e Z (5) Sp07 Page 1 of 6
2 tan = X L X C R (6) Apparatus Computer, Pasco 750 interface, Power amplifier, Inductor, Wires, DMMs, Data Studio software. Procedure Reactance 1. If it is not already, plug the DIN connector from the power amplifier into Channel A on the 750 interface. Plug the DIN connector from the voltage sensor into Channel B (this sensor will not be used in this procedure). Turn on the interface but do not turn on the power amplifier yet. 2. Connect a DMM (ammeter) and the inductor to the outputs of the power amplifier as shown below. Connect a second DMM (voltmeter) across the inductor. Turn on both DMMs. The ammeter should be set to read a full scale current of 200mA AC; the voltmeter a full scale voltage of 2V AC. 3. Open the activity entitled Inductive Reactance. You should see a Signal Generator window similar to the one here. Make sure that you are producing a Sine Wave with Amplitude 1.000V and at a frequency of 200Hz. If this is the case, then turn on the power amplifier. When ready to collect data, click the Start button in Data Studio. 4. Record the current in the circuit as well as the voltage across the inductor in Table 1. Use Equation2 to calculate the reactance at this frequency. 5. Increase the frequency by 100Hz and repeat Step 4. Continue until you reach 1500Hz. 6. Turn off the power amplifier and the DMMs. Graph the inductive reactance vs. the frequency. Plot the straight line of best fit over your data and determine the slope of the line. Sp07 Page 2 of 6
3 Frequency (Hz) Current (ma) Table 1 Inductor Data Voltage (V) Reactance (Ω) Phase 1. Remove the DMMs from the previous circuit so that the solenoid is connected directly to the outputs of the power amplifier. 2. Connect the leads of the voltage sensor to the solenoid. The red lead of the sensor should be connected to the side which is connected to the red output of the power amplifier (and black to black). 3. In the Signal Generator window, adjust the frequency to 100Hz (1.000V Sine Wave). 4. Open the Oscilloscope window (should be along the bottom of the Data Studio window). Turn on the power amplifier and click the Start button when ready to collect data. 5. You should see the voltage across and the current through the solenoid. Make sure the scope settings are: Output Voltage 0.5V/div, Current 0.2A/div, and sweep 5ms/div. 6. Draw what you see on the scope in Diagram 1 to scale. Use the voltage and current settings to estimate the currents and voltages in the diagram. Sp07 Page 3 of 6
4 Diagram 1 Voltage and Current Traces for the Inductor V max (V) V e (V) I max (A) I e (A) Analysis 1. Why did you graph reactance vs. frequency? What does this graph show you? How does it show you this? 2. What is the theoretical slope of the line on your graph according to Equation 1? Sp07 Page 4 of 6
5 3. Use the slope of your line to calculate the inductance L of the inductor. Show all work. 4. Discuss the traces you made in Diagram 1. Is this what was expected? Why or why not? Sp07 Page 5 of 6
6 PreLab: Inductors in AC Circuits Name Section 1. Using Equation 1, derive the units of inductive reactance. 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? Sp07 Page 6 of 6
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