Frequency response of RC circuits, part 1
A common kind of time-dependent signal is the continuous AC current of a certain frequency,
represented by a sine wave. RC circuits respond differently to signals of different
frequencies f and a convenient way to describe this behaviour is in terms of the circuit
frequency response, or transfer function.
As a filter circuit usually affects both the amplitude and the phase of an input signal,
the transfer function describes both the gain G(f)=Vout/Vin
and the phase φ(f) relationship of the output signal Vout
relative to that of the input signal Vin.
To generate plots of G(f) and φ(f), begin by determining the theoretical centre
frequency f0 of the circuit, then select a series of frequency values that
span from f~ f0/20 to f~ 20f0, concentrating on the region near
f0. A dozen or so well chosen values are typically sufficient.
The waveform generator, FG
The waveform, or function, generator provides a precision alternating current (AC) signal that can
be used to analyse electrical
circuits. You will be using a sine wave during this analysis as it consists of a single
discrete frequency. Other waveforms such as the square wave, traingle and ramp,
have different harmonic contents consisting of an infinite series of
sine frequencies with varying amplitudes.
The sine wave amplitude and frequency are adjustable. An offset that shifts the AC
signal by a constant DC voltage can also be added to the signal; for the filter
analysis verify that the FG offset is set to zero.
Comparative analysis of two signals
To determine the gain and phase relationship of Vout to Vin,
setup CH1 as described above, then adjust the CH2 vertical position so that the signal
grounds overlap at the bottom of the screen.
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For Vin, measure the CH1 peak or use the value set by the FG, they should agree.
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For Vout, set the CH2 gain to maximize the vertical range of the half wave, then use the
cursor to measure the peak.
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For Δt, measure the time difference between the two signals where
they cross the ground axis; the displayed voltage values should be zero.
The slope of the AC sine wave is greatest where the signal crosses the x-axis, making
the time determination more accurate. Following the same reasonimg, measuring time delays
using the peaks of the sine waves is not the ideal choice.
Plotting the gain as a function of frequency
To generate the G(f) plot, measure for each of the chosen frequencies
the peak-to-peak amplitude Vin(f) of the incoming and Vout(f) of the outgoing
signal. Calculate their ratio G(f)=Vout(f)/Vin(f), called the gain,
and plot log(G) as a function of log(f). Gains are conventionally plotted on a log scale
in decibels (dB), where dB = 20 log(Vout/Vin). Recall also that an
octave is a logarithmic unit that refers to a frequency ratio of 2:1; a decade
refers to a frequency ratio of 10:1.
Plotting the phase as a function of frequency
To generate the φ(f) plot, measure for each of the chosen frequencies
the amount of time Δt that the output signal is delayed from the input signal, then
calculate and plot the corresponding phase shift φ(f), in degrees, as a function of log(f).
With the period T=1/f, φ=360*Δt/T.
The phase shift refers to the shift of the output signal relative to the input signal.
If the output leads, or precedes the input in time, then
Δt and the phase shift φ will be positive. If the output lags, or follows the input
in time, then Δt and the phase shift φ will be negative. Be careful to give φ(f)
the correct sign.
The Lissajous curve
Typically, the scope displays CH1 and CH2 as y-axis waveforms as a function of time
along the x-axis. Setting the scope to XY mode will display the CH1 signal as a voltage
Vx along the x-axis and the CH2 signal as a voltage Vy
along the y-axis. The resulting Lissajous figure provides an alternate way
to measure the signal gains and the phase shift between the two signals.
Prelab preparation
Review the slides, then use the appropriate equations and the given nominal component values
to determine the theoretically predicted centre frequency f0 and phase shift
φ(f0) for the low-pass and high-pass filters.
For the band-pass filter, determine f0 and the decay time constant τ.
You will use these nominal results during your lab session to properly analyse the filters.
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