A common kind of time-dependent signal is a varying AC voltage of a certain frequency, represented by a sine wave. RC circuits respond differently to signals of different frequencies f.

The RC circuit as a voltage divider

This dependence arises because the Ohmic resistance R does not vary with the frequency of an applied voltage but the resistance of the capacitor C, or capacitive reactance X_C, does vary as X_C(f)=1/(2*π*f*C). Since the R and C components in series make up a voltage divider, as the frequency f of a sine wave of constant amplitude at the input of the filter is varied, a sine wave of the same frequency f but different amplitude will appear at the output node.

The transfer function

A convenient way to describe this frequency 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)=V_out/V_in and the phase φ(f) relationship of the output signal V_out relative to that of the input signal V_in.

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, as shown in the sample plots.

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 V_out to V_in, setup CH1 as described above, then adjust the CH2 vertical position so that the signal grounds overlap near the the bottom of the screen. Verify that Voltage setting of both CH1 and CH2 is set to 1X.

• For V_in, measure the CH1 peak or use the value set by the FG, they should agree.
• For V_out, set the CH2 gain to maximize the vertical range of the half wave, then use the cursor to measure the peak.
• For Δt, measure the time difference between the two signals where they cross the ground axis with the same slope; 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 V_in(f) of the incoming and V_out(f) of the outgoing signal. Calculate their ratio G(f)=V_out(f)/V_in(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(V_out/V_in). 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.

A first-order low-pass RC filter (one R,C) will approach a 6dB/octave, 20dB/decade rolloff in gain as f increases past 2f₀. The plot of this region should be approximately linear on a log/log scale, as shown.

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 V_x along the x-axis and the CH2 signal as a voltage V_y 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 for the low-pass filter, then determine the expected gain G(f) and phase shift φ(f₀) at f₀ by setting f=f₀. You will use these nominal results during your lab session to properly analyse the filter.