Experiment 1 Inductors in AC Circuits Preparation Prepare for this week's experiment by looking up inductors, self inductance, enz's aw, inductive reactance, and R circuits Principles An inductor is made of wire formed into a coil The inductor has an inductance,, which is a property that depends on the geometry of the wire coils When a time varying current flows through the coils an emf is generated such that dφ ε = - B, dt where ΦB is the magnetic flux through the coil and is equal to Φ B = I An ideal inductor has no resistance, so the voltage across it is due only to the induced emf If the current through the inductor is varying sinusoidally so that I = Isin(ω t), where ω = π f, then the voltage across the inductor will be dφb d di = ε = = ( I ) = ( Isin(ω t) ) = Iω cos(ω t), dt dt dt π = Icos(ω t) = I sin ω t+ Comparing the equations above we see that the inductive reactance,, is a function of the frequency such that = ω = π f Consider an inductor in a single loop circuit with a resistor, R, and a sinusoidally varying voltage source of amplitude Z and frequency ω Now a real inductor will have some resistance, R, which must be included along with the resistor, R, in the circuit Then the total impedance, Z, for an R circuit will be Z = (R+R ) +
This means that the voltage across the inductor is always the sum of two voltages that are π/ out of phase as given by = I +R Rearranging this equation to solve for the inductive reactance gives = -R I Figure 1 An R circuit We can note that the current amplitude, I, is related to the amplitude of the voltage across the resistor by the expression R I = R Substituting this back into the expression for the inductive reactance gives R = -R R So we see that reactance can be found experimentally if the current and voltages across the inductor and resistor are known The phase angle between the current and the signal generator voltage is given by: tanφ = R+R The voltage across the entire circuit is given by Z ( ) = I Zsin ω t+φ The magnitude of the current will be Z I = Z Because the inductive reactance depends on frequency, this is a filter circuit In this case the inductive reactance increases with increasing frequency so the magnitude of the current and the resistor voltage both decrease as the frequency is increased
Equipment 1 protoboard 1 multimeter and leads 1 oscilloscope 1 signal generator with power cord three prong adapters 3 BNC connectors small wires 1 kω resistor 1 5 mh inductor 3 18" banana wires, one red, two black 4" banana wires, one red, one black spade lugs 1 resistor substitution box Procedure Measure and record voltages to at least two significant digits Measure and record the resistor, inductor resistance, and the inductance to three significant figures 1 Use the impedance bridge to measure the inductance of the inductor Use the DMM to measure the resistance of the resistor and the inductor If your inductor shows infinite resistance consult your instructor Use ground isolators (three to two prong adapters) on the oscilloscope and the signal generator Build your circuit as shown in Figure Figure Set the generator voltage to a convenient value like 4 pp Make sure that the signal generator voltage remains constant; it may drift Be sure that the calibration knobs remain off, otherwise the data will be worthless 3 Wire the resistor and inductor in series with the inductor closer to ground Record the frequency and the peak-to peak-voltages across the resistor and the inductor for 11 frequencies between 5 khz and 5 khz in increments of khz Use the same steps as the last experiment: 1 Set the frequency and record its value Check the calibration knobs on the 'scope 3 Check the signal generator voltage and adjust it if necessary 4 Measure and record R 5 Measure and record 6 Go to step 1
4 If the inductor voltage does not change as you change the frequency, call your lab instructor 5 Wire the inductor and the resistance substitution box in series as you did in the last experiment Be sure that the resistor box is not set to zero resistance Set the frequency to about 5 Hz Set the 'scope and adjust the frequency so that one complete wave covers 4 divisions on the screen This means that, on the screen, 4 divisions equals a phase shift of π radians Two divisions equal π radians, etc Figure 3 6 Calculate the value of R that will make φ=π/4 and set the sub box to that value Observe the phase shift on the screen The two traces should be separated by half a division Set the box to a smaller value and record the phase angle Set the box to a larger value and record the phase angle 7 Be sure your instructor approves your data before you put your equipment away Data Data for the first part of the experiment should consist of the measured inductance and resistances, the frequencies, and the voltages For the second part it should consist of the frequency, the substitution box settings, and the phase angles Analysis 1 Calculate the theoretical and experimental reactances for each frequency and find the percent error for each Calculate the current for each frequency Calculate the phase shift for each frequency For the part of the experiment where you kept frequency constant and changed the resistance calculate the theoretical and experimental phase shifts Calculate the current for each resistance 3 Use semilog paper to graph out / in for both the resistor and the inductor 4 Calculate the frequency for which your out / in for the resistor and the inductor should be equal Does this agree with your data?
Questions 1 Draw the schematics for this experiment and explain how you took the data Write at least a paragraph in which you derive the equation for the voltage across an inductor in an AC circuit Discuss the frequency dependence of Discuss what happens to the current in an R circuit as the frequency changes How does this compare to the way the current changes as a function of frequency in an RC circuit? 3 Write out the units for f and and show that the unit of is ohms 4 What happened to the current this time as the frequency increased when the resistance was constant? What happened to the phase angle? 5 Discuss the similarities and difference between RC and R filters If it applies to you, write "I have not cheated on this lab report" and sign your name Grading 4 pts Data and Analysis pts Each for questions 1, 3, and 4 5 pts Each for questions and 5