Resonance and radio waves - Prelab questions

1. Draw a diagram of a standard capacitor. What properties determine the amount ofcharge a capacitor can store?

2. When should we use the RMS (root mean square) measurement of voltage, and howdoes its value differ from a direct DC voltage measurement?

3. Calculate the resonant frequency for the circuit shown in Figure 3, with the followingvalues.

L = 100 H C= 0.01 F and R= 1000 (neglect the inductor resistance at this stage).

4. Calculate the Q value (quality factor) for the above circuit, hence the bandwidth, .

5. Why do FM stations drop out at shorter distances from their source than AM?

6. Could we use this radio to tune in to digital, rather than analogue, radio stations?Explain.

1

Resonance and radio waves - Theory

The Rubbery Ruler

The Rubbery Ruler1 is a wide range, conformable, capacitive displacement transducer, firstdevised and created here in the School of Physics2. The Rubber Ruler consists of a bifilar3

helix of insulated conductive material (see Figure 1). This results in an extraordinarily longrange of elongation, which can be more than double its initial length. Changes in Rulerlength result in predictable changes in its capacitance. Since the change in capacitance ispredictable, and related to its to length, it can be used to monitor changes in size in hard-to-measure shapes (e.g.: chest, muscles, limb extension, fruit growth...)

Figure 1: The unstretched and elongated Rubbery Ruler, showing the increased wire separa-tion and double helix structure. The capacitance changes with extension.

Electrically the Rubbery Ruler can be described as a distributed capacitor. The two wiresof the double helix constitute the electrodes of a variable gap capacitor. As the Ruler isstretched, the two wires of the core separate in a uniform, reversible fashion controlled bythe elastomeric covering. Signal processing techniques can convert capacitance changes intoa frequency, current or voltage.

Inductors and Capacitors in circuits

Across capacitors and inductors, the voltage is out of phase with the current going throughit. This can mean that circuits employing capacitors can seem to have a lower total voltagethan their input voltage. If we change the frequency of the input voltage, we may find thatthe input and total voltage are closer, but still do not add up like in a circuits made onlyof resistors. We therefore need to be careful when measuring signals in circuits containingcombinations of resistors, capacitors and inductors.

Filtering Signals

When dealing with DC signals, a pair of resistors in series forms a voltage divider. Theoutput voltage then depends on the relative values of the two resistors as:

Vout =R2

R1 +R2Vin (1)

1I think its a trademark - thus the repeated double capital.2Theres a plaque commemorating it in the level 7 tea room!3A bifilar coil is an electromagnetic coil that contains two closely spaced, parallel windings.

2

VinR1

R2

Vout

Figure 2: A simple DC voltage divider.

This is an ideal circuit, in which no output current is drawn, and holds for both AC andDC circuits. However, when reactive components (inductors or capacitors) are used insteadof resistors, we need to start referring to the impedance (symbol Z) of the components,rather than the resistance. Impedance, unlike resistance, will vary with the applied voltagefrequency. When working with reactive circuits it is common to define the frequency interms of (omega), where

= 2f (2)

with f is the applied AC voltage frequency.

The impedance, ZC , of a capacitor can be defined as

ZC =1

jC(3)

where C is the capacitance, in farads (F), of the capacitor. We use the term j to representthe imaginary component of the current through the capacitor to distinguish it from typicalcurrent. We should note that the voltage across a capacitor lags behind the current throughthe capacitor by a phase of /2.

The impedance, ZL, of an inductor is given by

ZL = jL (4)

where L is the inductance, in henrys (H), of the inductor. We can also note that for aninductor the voltage leads the current by /2.

Imagine we now replace one of the resistors shown in Figure 2 with one of these reactivecomponents. As the impedance of these components changes with frequency, we can makea voltage divider where the output voltage depends on the input frequency. We have nowcreated a filter.

Resonance

Imagine a circuit as shown in Figure 3.

The total impedance of a series RLC circuit is a complex quantity, and can be expressed inthe form

Z = R + jL+1

jC(5)

3

CR

L

Figure 3: A simple R (resistor) L (inductor) C (capacitor) circuit.

Looking at the total impedance, and recalling that the inductor impedance, ZL = jL andthe capacitor impedance, ZC = 1jC vary in exactly opposite ways with frequency, there mustbe a frequency at which the resultant impedance is simply R - a minimum. This frequency iscalled the circuits resonant frequency and is given by

joL+1

joC= 0 (6)

which can be written aso =

1LC

(7)

At this frequency, the phase angle will be zero (the two imaginary components have cancelledout). A typical plot of the way the magnitude of the current varies with frequency for variousresonant circuits is shown in Figure 4. This plot is normalised to a maximum current of 100mA. Note that it reaches a peak for a given frequency and tails off either side.

Quality factor, Q

A useful measure of the properties of reactive circuits is the quality factor, Q. This is ameasure of how sharp a resonance peak is, or how quickly the current rises to its peak

mA100.0

50.0

0.0

Q=10

Q=5

Q=2Q=1

frequency Hz100 1000

Figure 4: A plot of frequency versus current, showing changes in the amplitude and thequality factor Q.

4

and falls again as you scan through frequency. Its value can be obtained from Figure 4 bymeasuring the width of the resonance peak at 1

2of its maximum, this value is commonly

known as the full width at half maximum, or FWHM, and can represent a measure of thesharpness of any peak. The quality factor is defined as

Q =o

(8)

it can also be represented as

Q =1

R

L

C(9)

As we have seen, the larger the Q value, the sharper the peak. From Equation 9, we can seethat a larger resistance, R, will cause a broader peak. Remember also that for any inductor,L, there will always be a real resistance associated, because inductors are essentially justwound wires, and all wires have some intrinsic resistance. We must add this this resistance,RL, in series with the inductor when calculating a value for Q.

In our case, a high Q factor relates to how well our circuit can isolate specific frequencies.We would like to isolate frequencies rather well, which is difficult using broad peaks withlow Q factors.

Using an RLC circuit to tune in a radio signal and building an AM radio

Can we use our knowledge and these components to build a radio?

In order to tune to a station we need to filter out all unwanted carrier frequencies eitherside of the desired station.

An AM (amplitude modulation) radio signal is a superposition of a high frequency wavecalled a carrier wave (this is the RF signal) and a signal in the audio frequency range(only about 100 Hz to 7.5 kHz for AM in Australia). The carrier frequency range in thestandard commercial AM radio band is roughly 550 kHz to 1600 kHz.

For high fidelity (Hi-Fi) sound it is necessary to have a bandwidth of 20 kHz. On the AMfrequency range however, the maximum modulating frequency is limited to 7.75 kHz, tomake room for more stations. FM radio, on the other hand, is transmitted at around 100MHz, with bandwidth up to 20 kHz, which is why FM radio sounds significantly clearerthan AM.

Essentially, our circuit will consist of an antenna connected to an LCR circuit which in-cludes our Rubbery Ruler. As we change the length of the Ruler, the resonant frequency ofthe circuit will change and well scan across the AM band. If we choose our componentsappropriately well have a nice bandwidth and will isolate one station at a time.

5

Resonance and radio waves - Procedure

NOTE: Do not stretch the ruler beyond the furthest point on the board. Handle theruler carefully. Once you are done with a measurement, relax the ruler to its naturalextension.

Capacitance with extension

1. Take measurements of the capacitance of the Ruler across its range directly using amultimeter.

Question 1 Produce a plot to show the relationship between ruler extension and capaci-tance. What is the relationship? Is the change in capacitance between points very large?

The Rubbery Ruler in a circuit

Rubbery ruler

V1 V21 k

Figure 5: Circuit for constructing your I vs V plot

1. Build the circuit depicted in figure 5, setting the signal generator to 20 kHz.

2. Vary the amplitude of the input voltage (use the amplitude control on the signal gener-ator) and construct a table of the V1 and V2 values you measure.

3. V1 is a measure of the voltage across the whole circuit, which we are assuming to beequal to the voltage across the Rubbery Ruler.

4. V2 is a measure of the voltage across the resistor. You need to convert your V2 read-ings to values of current in the circuit.

5. Obtain at least 5 readings of V1 and V2 at this fixed length. Convert your V2 readingsinto current, and plot the V vs I graph for this length (capacitance) in Excel.

Question 2 Is the V vs I graph Ohmic? Explain.

Question 3 Explain how to convert our measurement of voltage at V2 into a measurementof current in the circuit.

6. Repeat this process for two other extensions of the Rubbery Ruler.

6

7. Fit a linear trend line to your V vs I graph. For reactive circuits, V = IZ.

8. Use your trendline and the relationship below to determine a capacitance for eachrubbery ruler extension.

|Z| = vi

=1

C(10)

9. Note that these values only apply at this frequency of f = 20 kHz.

Question 4 Show how the output voltage V1 compares to the voltage purely across the ca-pacitor. Use your capacitance values, and compare the impedance of the Rubbery Ruler withthe resistance of the resistor, then use the equation for the voltage divider.

Measuring the ruler impedance with frequency

1. Choose a single extension, and leave the Rubbery Ruler fixed at that position.

2. Set the signal generator so it produces voltage at a frequency of 5 kHz.

3. Produce a V vs I graph as in the section above.

4. Similarly produce graphs for 10 kHz and 40 kHz input frequencies.

5. Calculate the impedance and capacitance of your capacitor for each input frequency.

Question 5 What comments can you make about the change in impedance and capacitancewith frequency? Is the relationship similar to the relationship between these values andextension?

Question 6 How do your impedance values compare with resistance values? Just for thesake of learning resistor colour charts, draw the resistor colour-chart equivalent of two ofyour impedance values.

Using a circuit with the Rubbery Ruler to tune to a radio signal

A very simple AM radio receiver circuit is shown in Figure 6. It contains a parallel LCcircuit, including the Rubbery Ruler, to tune to the resonant frequency of the desired carrierwave. (Most radios use a variable inductor rather than capacitor, but the Rubbery Ruler isfar more interesting for our purposes.) The resonant frequency of an RLC circuit is given byequation 7, while the quality factor is given by:

Q =oL

R(11)

with R the resistance of the circuit.

In order to extract the modulating signal from the carrier signal, the signal will be (rectified)by passing through a germanium diode. This passes only the positive half-wave through

7

the circuit. In this case the property needed to demodulate the incoming signal is called thesquare law, where the output voltage is proportional to the square of the input voltage.

After rectification the signal will be passed through a low pass filter to remove the carrier andany side carriers from the signal. This should just leave the amplitude of the desired audiosignal remaining.

Question 7 Why have we chosen this type of diode, specifically? Refer to the square lawand the properties that make germanium appropriate.

Question 8 You have a variety of inductors you can choose from. Use Equation 7, yourmeasured values of C and a frequency from the table of stations to determine which inductoryou will use.

Building the radio

Antenna

Ge diode

Tuning Detection Filtering

L 47 M 10 pF

Figure 6: Full radio circuit showing various stages of signal processing.

1. Choose a middle or average capacitance from your early measurements as a startingpoint. This will allow you to tune the circuit over a wider range than starting at anend point.

2. Build the circuit shown in figure 6 using the provided equipment (breadboard, wires,etc) and your choice of conductor.

3. The output of this circuit will still be too weak to drive a speaker, so we will use aseries of operational amplifiers (op-amps) to boost the output.

4. Identify the stations you can tune to. How can you improve the circuit and the soundquality?

Question 9 Draw exactly the wiring components on the breadboard. How are the rows andcolumns of the breadboard electrically connected?

Question 10 What were the difficulties in building and tuning the circuit? Do you think thewiring or the components themselves are more important in listening to a station?

8

Local Radio FrequenciesRadio National 621 kHz3AW 693 kHzABC Melbourne 774 kHzSport 927 927 kHzNewsradio 1026 kHz3RPH 1179 kHzSBS Radio - Special Broadcasting Service 1224 kHz3CW- (Chinese Languages Mandarin and Cantonese) 1341 kHz3KND 1503 kHzRete Italia - Italian Language Radio 1593 kHz3ME - Arabic Language Radio 1638 kHz

9