Experiment 5
RC filters and transfer functions
Another common kind of time-dependent signals is the continuous AC current of a certain
frequency. RC circuits respond to signals of different frequencies differently, and a convenient
way to describe this is in terms of the so-called frequency response, or transfer function. As a
circuit usually also affects the phase of the output relative to that of the input signals, a more
detailed Bode plot is useful, involving both the amplitude and phase components of the [complex]
transfer function.
5.1
Using a scope to measure frequency response
We have used the oscilloscope to capture time-dependent transient signals; for continuous sunusoidal AC
signals all we need to do is to measure the peak-to-peak amplitude of the incoming and outgoing signals.
The ratio G = Vout /Vin is called gain; in general, this is a complex number as both input and output
signals have both amplitude and phase.
Assemble the following circuit. Vary the frequency
setting of the function generator, and use the scope to
measure the amplitude of both input and output signals.
You may need to adjust the time base, and the amplitude
gain for each channel independently, to see both signals
clearly.
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Vary the frequency from 100 Hz to 10 MHz while
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Figure 5.1: Gain of an RC circuit
keeping the input amplitude constant; monitor
what happens to the amplitude of the output signal. Notice for what frequencies the signal passes
through the circuit, and for what frequencies the
output is greatly attenuated compared to the input.
? Would you call this a low-pass or a high-pass filter?
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Vary the frequency again, this time paying attention to the phase of the two signals. Notice how the
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phase of the output signal shifts relative to that of the input signal whenever there is a variation of
amplitude gain |G| with frequency.
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Use Circuit −→ Schematic Options to display the node numbers on the schematics. Then use
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Analysis −→ AC Frequency to simulate response curves for the circuit. Select both nodes that
correspond to the input and output of the RC circuit and you will see two response curves, in
different colours, one for each node.
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EXPERIMENT 5. RC FILTERS AND TRANSFER FUNCTIONS
5.2
The Bode plotter
Replace the scope with a special instrument called the
Bode plotter, as shown in the diagram. The Bode plotter
calculates directly the ratio of the two response curves
you had simulated in the previous step. It is usually
convenient to express the attenuation factor Vout /Vin in
logarithmic units. The decibel (dB) is defined as:
dB = 20 ∗ log10
Figure 5.2: Bode plotter
Vout
.
Vin
Likewise, frequency ranges are usually more significant
as logarithmic ratios. A range f0 to 2f0 is defined as an
octave; a range f0 to 10f0 constitutes a decade. Filter
response is typically expressed in dB/octave. A first order filter has a rolloff of 6 dB/octave or 20 dB/decade; a
second order filter has 12 dB/octave and so on.
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Use the cursor to examine the measured Bode plot. Determine “the 3 dB point”, i.e. the frequency
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for which the output signal falls exactly 3 dB below the input signal. Determine where the phase
of the output signal is shifted exactly 45◦ relative to the input. Compare and comment on the two
‘measurements”. You may need to change the number of points the Bode plotter is using to scan
through the frequency range; see Analysis −→ Analysis Options . Save the results in a file.
? Knowing that a capacitor behaves as an open circuit for DC, and conducts well for high frequencies,
what do you expect to happen when R and C are interchanged in our circuit?
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Interchange R and C. Repeat the Bode analysis, and save the results again.
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? Examine the data files. What is the point where the two amplitude curves intersect? What are their
slopes?
5.3
Multi-stage filters
A combination of two different RC filters in series has the
transfer function that is the product of the two individual
ones. On the logarithmic, or dB scale, the gains of the
two stages simply add.
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Assemble the circuit, and examine its gain using
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Figure 5.3: Two-stage filter
the Bode plotter. Determine the point of maximum
gain, and the width of the bandpass region, defined
as the frequency range over which the gain drops
3 dB from its maximum value.
? On the dB scale, the response curve does not look
very impressive. Calculate the intensity attenuation factor (on the linear scale) that corresponds
to -3 dB and -6 dB.
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Switch the vertical axis of the Bode plotter to linear scale and adjust the vertical plot limits to see
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the linear gain curve clearly. Verify your calculations from the previous step.