5 RLC and Bandpass circuits

This lab is based on an experiment in Curtis Meyer’s Basic Electronics: Carnegie Mellon Lab Manual.

The goals of this lab are to

• gain familiarity with the frequency response of a simple RLC circuit. In particular, to inves-tigate resonance.

• build and investigate a band-pass filter.

Length: 2 lab sessions

5.1 Background

5.1.1 RLC Filters

RLC circuits play a large role in the modern world. An important use is in receivers, where theyselect out a particular frequency, the resonant frequency !0. This frequency is characterized by theinductance and capacitance of the circuit, !0 = 1/

pLC.

If we consider a series RLC circuit, there are three possible output voltages. These are just the threevoltages across the three components of the circuit.

Figure: The series RLC circuit.

We can use an impedance analysis to determine the gain of the circuit. Here the total impedance is

Ztot = R+ j!L+1

j!C

= R

1� j

!RC

!

✓1� !

2

!LC

◆�

The gain depends on which component is used for the output.

GX(!) =ZX

Ztot

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If we look at the gain to be across the resistor,

GR(!) =R

Ztot

=1

1� j(!RC/!)(1� !

2/!

2LC).

The magnitude is given by

|GR(!)| =!/!RC

[(!/!RC)2 + (1� !

2/!

2LC)

2]1/2

=1

[1 + (!L/R)2(1� !

20/!

2)2]1/2

The phase is given by

tan[�R(!)] =!RC

!

✓1� !

2

!

2LC

◆

=!L� 1

!C

R

5.1.2 Bandpass Filters

As discussed in Sections 3.7 and 3.8 of the textbook, there are many occasions when we need tocouple one functional block of circuitry to another. In fact, its hard to think of a situation where thisis not necessary! In the case of a high-pass filter connected to a low-pass filter, we create a band-passfilter. Such a circuit will attenuate signal both above and below some characteristic frequency.

The new feature is that we need to worry about the input impedance of the second filter relative tothe output impedance of the first filter. In order for the overall gain of the combined circuit to bethe product of the two individual gains, we must have that

|(Zin)2| � |(Zout)1|.

See your textbook for a more detailed discussion of this circuit.

5.2 Procedure

Reminder: At the beginning of each section below, enter into your lab notebook a summary ofwhat you are setting out to do. Great lengths of verbiage are not necessary, but some orientingexplanation is. This should be standard practice in any lab notebook!

Reminder 2: All measured values require uncertainties. Provide a brief explanation of how youdetermined these uncertainties.

This lab is an excellent place to compare theory to your measurements and such comparisons areexpected. Your measurements should be compared to quantitative calculations of the expectedbehavior of the circuits and the results plotted on top of your data.

5.3 Quick check

1. Function generator: Is it set to “High-Z”? Is the voltage o↵set set to 0 V ?

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2. Oscilloscope probe: Is it set on the right magnitude (1x, 10x, 100x, . . . )?

3. Oscilloscope channels: Use DC coupling on your oscilloscope.

4. Circuit board and probes: Are all of your grounds connected to the same point?

5.4 RLC filter

Set up the series RLC circuit shown below.

Vin Vout

L C

R

Figure: Series RLC circuit.

Use an inductor of a few mH, a capacitance on the order of 0.01µF, and a resistor of 100⌦. Use aa sine wave of reasonable amplitude (say, 5 Volts) as the input voltage. Take the output across theresistor.

Question: Calculate the theoretical resonance frequency of your circuit, !0. Give this both inrad/sec and cycles/sec.

Question: Consider the equation for the gain. What are its values for (a) ! ⌧ !0, (b) ! = !0 and(c) ! � !0.?

Measure vout and vin over a frequency range that extends at least two decades below and abovethe calculated resonance frequency. Choose frequencies so that so that your measurements will beroughly equally-spaced on a logarithmic frequency axis. Choose more points near !0 to accuratelymap the behavior.

Set up the scope to display both signals. Use the measure function to measure the amplitudes (youstill need to pay attention to uncertainties here). Use the cursors to measure time shift and calculatephase shift.

As you take the data, plot the gain, |G(f)| = |vout|/|vin|, on a Bode plot and the phase shift betweenvout and vin on semi-log scales.

Analysis Does your experimental data agree with what is expected theoretically? In addition, providethe following:

Create a Bode plot of your data. Write down an expression for the theoretically expected gain ofthis circuit. Include this theoretical function to your plot.

Create a phase response plot of your data. Write down an expression for the theoretically expectedphase shift of this circuit. Include this theoretical function to your plot.

What are the magnitude and phase of the gain at the resonance frequency?

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5.5 Bandpass filter

Build a bandpass filter by taking the output of a high pass RC filter and putting it into a low passRC filter with the same characteristic frequency.

Figure: A bandpass filter. The Thevenin equivalent circuit for the source is on the left. The nextstage is a high pass filter, with output va. The output va is connected to the low pass filter, withan output of vo. This output is, in turn, connected to a complex load ZLD.

For the low pass stage, use approximately a 20k⌦ resistor and 0.008µF capacitor. Choose the otherresistor and capacitor values such that

RhpChp = RlpClp

Rhp � rs ! Rhp > 20rs

Rlp � Rhp ! Rlp > 20Rhp.

See section 3.8 in your textbook for how these constraints are determined.

Measure the frequency response of your bandpass filter. Use a a sine wave of reasonable amplitude(say, 5 Volts) as the input voltage.

Analysis What is the frequency response of your bandpass filter? What are the input and outputimpedances of each stage of the filter?

Create a Bode plot of your data. Create a phase response plot of your data.

Calculate and compare the input impedance of the first stage to the output impedance of the functiongenerator (typically 50⌦). Which is larger?

Calculate and compare the output impedance of the first stage to the input impedance of secondstage . Which is larger?

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