Experiment 3 - The RLC Tuned Circuit.

Experiment 3 — Resonant Tuned Circuit.

This experiment shows you some of the properties of circuits which contain resistors, capacitors, and inductors.

Resonance is an important physical phenomenon. It can occur in all sorts of systems, from a swinging mechanical pendulum to an optical cavity. In each case it requires a situation where energy is periodically transferred back and forth between two possible ‘reservoirs’. In the case of a pendulum, energy is transferred from gravitational potential (i.e. the height of the pendulum mass) to kinetic and back again. In an optical cavity the transfer is between the electric and magnetic fields inside the cavity.

When processing electronic signals in analogue form, we often need to use a filter to select (or reject) a specific band of frequencies. One of the easiest ways of making a filter for this sort of task is to combine a resistance, capacitance, and inductance. Diagram 4 shows a typical arrangement. You should assemble this circuit for measurement in this experiment. Use a 1,000 pF capacitor for ‘C’, a 2·2 mF inductor for ‘L’, and a 10

resistor for ‘R’. Use the signal generator to provide the input signal,

As with the earlier experiments, your circuit should be laid out in a similar way to the circuit shown in the photographs which you can see by clicking on the image of a camera.

You should use the generator output which is typically labelled 50

or 600

In this case, the lower wire of the circuit has an earth symbol attached to remind you that this is meant to be the earth/zero-volts line. In practice you will connect the line to earth via the outer leads of the co-axial cables used for the signal generator and ‘scope. Use both ‘scope leads and channels to observe both

and

at the same time.

Although this circuit doesn't look anything like an optical cavity, it is working in a similar way. The capacitor can store energy in the form of an electric field in between its plates. The inductor can store energy in the form of a magnetic field around its coil. If you put some energy into the circuit it will tend to be moved back and forth between these two components at a frequency which depends upon their values.

Start by applying a large square-wave input signal with a frequency of a few hundred hertz. You should see the output voltage ring after the abrupt input voltage changes which occur at each square wave edge. This ringing is a damped resonance which occurs whenever you abruptly try to alter the state of a resonant system. The time taken to settle down depends upon the amount of damping which, here, depends upon the resistance, R, in the circuit. Note that the frequency of the ringing doesn't depend upon the input square wave frequency. It is characteristic only of the resonant frequency of the circuit.

Sketch the output waveform and use the 'scope to estimate the ringing frequency,

, by timing each cycle.

(Caution: the time-base readings will be only be correct if the ‘scope display is correctly calibrated. Check to see if there is a time knob or switch setting marked something ‘calibrate’ and ensure it is set to the calibrated position before making any timing measurements.)

Now switch over to using sine waves. You should find that the ratio of

depends upon the sinewave frequency

Find the frequency,

, where

is a maximum.
Note this frequency.

How does

compare with

The circuit can be thought of as being in two parts:

Part 1: a resonant circuit made with the L, C, and 22 resistor
Part 2: an input series 100k resistor.

The properties of the resonant circuit can be examined using this arrangement because the impedance of a resonant circuit is frequency dependent. In effect, you have made a potential divider using a resistor (the 100k

) whose resistance doesn't depend upon the signal frequency, and a resonant circuit whose impedance does depend upon the frequency. As a result

varies with the input frequency in a way which reveals the frequency dependent behaviour of the resonant circuit.

Take the data to form a table of

and

for a range of frequencies, f, from about

Plot a graph of normalised values of

versus the frequency.

(Here, normalised means divide all the

values by the maximum value which occurs at

. This means that when the graph is plotted its peak value will appear to be unity.)

Note the frequencies,

where

falls to

of its peak value.

Note the difference,

between these two frequencies. This value of

indicates the ‘range’ of frequencies the circuit will pass though if used as a bandpass filter.

In theory, the resonant frequency of a weakly damped resonant circuit should be given by

Calculate the theoretical value of

for your circuit using this expression. Compare the result with your measured

and

values and say how much they differ in percentage terms.

You can also use your data to measure two more properties of your circuit.

i) The circuit “Q".

The Q (or quality factor) of a resonant system is a measure of how ‘sharp’ a resonance is. It is an important property of a system because it depends upon how quickly the system loses stored energy. The circuit you have been experimenting upon can be used as band pass filter which will let through signal frequencies

, but reject frequencies which are very different to

. The band width of the filter — i.e. the width of the frequency range passed by the filter — depends upon its Q.

In principle, the quality factor of your resonant circuit can be calculated in two ways

where R is the dissipation resistance of the resonant circuit, and

is the measured frequency width of the resonant peak (at the points

below its peak).

Take the measured

and

values from your graph and use expression 3 to calculate a value of Q.

Compare this with the value you get if you use expression 4 and the values of the components you are using. You may well find that these results for Q aren't the same!

Part of the reason for this difference is the fact that the inductor also has a resistance, which you haven't taken into account. The other resistances (the 100k

, and the input resistance of the 'scope) also have some effect even through they look as if they're ‘outside’ the resonant circuit. However, the main problem is one called the ‘skin effect’. This makes a.c. signals tend to prefer to flow in the outer ‘skin’ of a conductor. The higher the frequency, the thinner the skin the current is confined to. In effect, for an a.c. signal you could remove the metal inside the wire just leaving a hollow tube of metal. As a result the wire behaves as if it is becoming thinner (and hence more resistive) as you increase the frequency. This means that the behaviour of an inductor – which contains a long wire thin wire wound into a coil – can be very different to a plain inductance.

Many textbooks will leave you with the impression that you can calculate Q just from knowing L and R. The above comparison should serve as a warning that the actual value of the dissipation resistance of a circuit isn't always obvious. This is because the resistance actually experienced by the a.c. signals may not have the value you expect. In practice, it is better to discover the Q by measuring

and

and then calculating

.

ii) The peak impedance.

At any particular frequency, f, the resonant circuit will have an impedance which we can call

. This reaches its maximum value,

, at the resonant frequency. As your circuit is a sort of potential divider you can expect that

where

is the input 100k

series resistor.

Use your (un-normalised) measurements to calculate a value for

at the resonant frequency,

Note for those who know something about a.c. circuit theory. When a circuit contains inductors or capacitors its impedance,

, is generally complex. This means that the alternating currents and voltages in it don't always share the same phase. When using the 'scope to measure

and

you may have noticed their relative phases — as well as sizes — changing when you altered the signal frequency. This means that, strictly speaking, in the above equation

, and

should all be considered as complex numbers. At resonance, however, the impedance of a circuit always becomes ‘real’ — i.e. purely resistive — so we don't need to worry about this complication.

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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.