page 1

Physics 405 Lab NAME _____________________________

Reactance and Resonance

In this experiment you will investigate the reactance of an inductor, the reactance of a capacitor,and resonant circuits.

Equipment required: Osciloscope and two probes, function/signal generator, CADET elec-tronic breadboard, 150mH inductor, 1000Ω resistor, 0.033µF capacitor and wire for connections.

Review of impedance:In AC circuits, the ratio of the voltage to current is referred to as impedance, the analog ofresistance. If one speaks only about inductors or capacitors, then the term “reactance” is used.Impedance is represented by Z, while reactance is represented by XL for inductors, or XC for

capacitors. The frequency dependence of these functions is XL = 2 π f L and XC = , where

f is frequency of the oscillating power supply. At resonance: where L is in Henries

and C is in Farads.

1. Inductive Reactance

figure 1

First, measure the resistance of both the inductor and resistor and record on page 2. Also, measurethe L value of the inductor. Next, connect the circuit as shown in figure 1, with the inductor andresistor in series. The circuit consists of an inductor (L about 150 mH), with some internalresistance and the resistor. Note that the inductor is represented by both L and RL. Similarly thegenerator provides a source of voltage and an “internal resistance”, Rg. This Rg can not bedetermined by us, so we shall ignore it in this experiment. It will, however, reduce Vinput as youwork at higher frequencies. Adjust the generator to produce a sine wave. Connect a cable betweenthe generator output and a connector on the CADET. To measure Vinput and VR, set the oscilloscopeprobe input to AC. Remember that if there is a 10x attenuator in the probe the actual voltage is 10times larger than what is measured on the display. Make sure that both probe grounding clips areconnected to ground. The voltage from the generator produces current through the inductor and theresistor. Because the circuit is a series circuit, the current is the same in both elements. Bymeasuring the voltage developed across the resistor, we measure the current , that is, I=VR/R .

12πfC-------------

f1

2π LC-------------------=

Rg RL L

RV input

VR

functiongenerator

inductor

page 2

DATA TABLES

Wire the circuit as shown in figure 1. Read the frequency from the dial on the signal generator. Setthe generator to produce a sine wave. Measure Vinput and VR at each frequency using theoscilloscope. KEEP Vinput CONSTANT at each frequency by adjusting the generator amplitudecontrol (not the scope.) Although keeping Vinput is not essential it will allow you to easily seechanges that occur at high frequencies. Calculate the currents and gain later.

Measure with multimeter: RL= _____________, L = ______________, R = _____________

Frequency Vinput VR IR Ipredicted log freq Gain=20 log (VR/Vinput)

40 1.6

80 1.9

100 2.0

200 2.3 400 2.6

800 2.9

1000 3.0

2000 3.3

4000 3.6

8000 3.9

10,000 4.0

20,000 4.3

40,000 4.6

80,000 4.9

Measure these two columns Calculate these three columns

page 3

Analysis (do at home, during lab skip to part 2.)The current, I, in the circuit can be calculated using two methods: with Vinput and Z (the impedance)or with VR and R.

I = VR / R or I= Vinput / Z = , where RT = R + RL, and XL = 2πfL.

a) Calculate, IR = VR/R for all frequencies in the table.

b) Calculate, Ipredicted = for f = 100, 1000 and 10,000 Hz.

c) How does the current as measured behave as the frequency is increased? Look at your tableto see the effect. This effect is due to the inductance. Is this the result you would expect?

d) Calculate the gain in decibels using 20 log (VR / Vinput). For example, if VR / Vinput = 1/2 thengain = -6dB (or 1/4 the power). Fill in the dB column.

e) Plot gain vs. log frequency. The curve should be level at low frequencies and then decrease athigh frequencies. At which frequency has it dropped about 3dB down from the flat region? (Thelog-log representation is commonly used for representing the gain of a circuit network.)

f 3dB = _______________________. Compare this value to f = Hz.

2. Capacitive Reactance

figure 2.

a) Measure C of the capacitor using the DMM: C = ________________________________.Replace the inductor in figure 1 by the capacitor (see figure 2). This time scan through thefrequencies and note how the current (proportional to VR) changes with frequency, becoming amaximum when XC becomes small with respect to R. Here we will see that capacitors pass highfrequencies better than low frequencies, that is, low frequency signals have smaller voltage dropsacross C. At what frequency is VR about one half of Vinput?At what frequency is VR about 70% of Vinput (the 3dB point)? How does this compare to f = 1/(2πRC)? Remember to hold Vinput constant. (Continue on part b when you have seen the effect.)

Vinput

RT2 XL

2+

------------------------

Vinput

RT2 XL

2+

------------------------

R2πL----------

C

RV inputVR

page 4

figure 3.

b) Now switch the positions of the resistor and capacitor as shown in figure 3. This situation issimilar to that found in long wires, where C represents the capacitance found between the wires byvirtue of their position. How does the voltage drop in the capacitor changes as the frequencyincreases? Does that the voltage across C (which could represent a transmitted signal) decrease or increase asf increases? For this configuration, how does f = 1/2πRC, compare to the frequency where you find VR to beabout 70% of the value of Vinput?

3. Resonance

figure 4.

Construct the circuit shown in figure 4. The scope measures the peak-to-peak voltage amplitudeacross R, and hence the current in the circuit. Scan the frequency until you see the peak to peakvoltage become a maximum. Record this resonant frequency.

f = ______________________________ Hz.

Compare this with f = = _____________________ Hz.

Explain what effects can limit the precision and accuracy of your measurements.

1

2π LC-------------------

C L

RV inputVR

R

CV input

VC