Electronics and Signal Processing

Lab 3 Summative

Signals Lab: RLC Circuits

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This is the summative lab: it is assessed by a lab report (see the instructions at the end). Before the lab, read the Background section below. The error propagation rules you will need are in the Lab Guide, and you will use your results from Lab 1 and Lab 2.

L3.1 Goals

L3.2 Background

RC circuits as high-pass filters

rc highpass

As we saw in class (Introduction to Signal Processing), a series RC circuit can function as a high-pass filter when the output is taken across the resistor. At low frequencies the capacitor acts effectively as an open circuit, and at high frequencies as a short circuit, allowing high-frequency signals to pass through to the resistor.

Recall from class that the response function \(h(\omega)\), which measures the amplitude attenuation of a pure sine wave with angular frequency \(\omega\), is \[\begin{equation} \tag{L3.1} \keyeq{\displaystyle h(\omega) = \frac{V_\text{out}}{V_\text{in}} = \frac{RC\omega}{\sqrt{1 + (RC\omega)^2}}} \end{equation}\] where \(V_\text{in}\) and \(V_\text{out}\) are the voltage amplitudes of the incoming and filtered sine waves. The cutoff angular frequency \(\omega_c\), defined by \(h(\omega_c) = 1/\sqrt 2\), is \[\begin{equation} \tag{L3.2} \omega_c = \frac{1}{RC} , \qquad\text{or, as a frequency,}\qquad f_c = \frac{1}{2\pi RC} . \end{equation}\] For frequencies much larger than \(f_c\), \(h \approx 1\) and the signal passes through unattenuated. For frequencies much smaller than \(f_c\), the signal drops linearly with frequency: \(h(\omega) \approx RC\omega = \omega/\omega_c = f/f_c\). Figure L3.1 shows this behaviour for values of \(R\) and \(C\) similar (but not identical) to those we will encounter in this lab.

rc highpass response

Figure L3.1. Frequency response of the RC high-pass filter, \(h(\omega)\) as a function of frequency, with the low-frequency asymptote \(h = RC\omega\) and the high-frequency asymptote \(h = 1\) for reference.

RLC circuits as band-pass filters

rlc bandpass

A series RLC circuit can function as a band-pass filter when the output is taken across the resistor. The inductor and capacitor form a resonant circuit that lets signals near the resonance frequency pass, while attenuating both low- and high-frequency signals.

Recall that we derived in class (the resonance curve of the driven RLC circuit) that the frequency response across the resistor of a series RLC circuit is \[\begin{equation} \tag{L3.3} \keyeq{\displaystyle h(\omega) = \frac{1}{\sqrt{\left(\dfrac{\omega^2 - \omega_0^2}{\omega\,\Delta\omega}\right)^2 + 1}}} \end{equation}\] where \(\omega_0\) is the resonant frequency and \(\Delta\omega\) is the full width at half maximum, or simply the bandwidth: \[\begin{equation} \tag{L3.4} \omega_0 = \frac{1}{\sqrt{LC}} , \qquad \Delta\omega = \frac{R}{L} . \end{equation}\] The two frequencies at which \(h = 1/\sqrt 2\) are separated by exactly \(\Delta\omega\); when \(\omega_0 \gg \Delta\omega\) they lie at \(\omega_0 \pm \frac12\Delta\omega\). The quality factor \[\begin{equation} \tag{L3.5} Q = \frac{\omega_0}{\Delta\omega} \end{equation}\] measures the sharpness of the peak: values much larger than 1 mean sharper peaks. When \(R\), \(L\) and \(C\) are tuned so that \(Q\) is large, the RLC circuit only lets a very narrow band of frequencies pass, and the circuit is said to “resonate” at those frequencies. Figure L3.2 shows this behaviour for parameters similar (but not identical) to those studied in this experiment. Note that the quality factor in this experiment is relatively small, \(Q \approx 1.9\), so the passband is relatively wide.

rlc bandpass response

Figure L3.2. Frequency response of the RLC band-pass filter, \(h(\omega)\) as a function of frequency, with the resonance frequency \(f_0\) and the bandwidth \(\Delta f = \Delta\omega/2\pi\) marked.

Ringdown in an RLC circuit

In class (RL, RLC and AC Circuits), we studied what happens in an RLC circuit if the applied voltage is suddenly dropped to zero: the circuit becomes mathematically equivalent to a damped harmonic oscillator, (5.3). The voltage across the resistor is \[\begin{equation} \tag{L3.6} \keyeq{V_\text{out}(t) = V_0\,e^{-t/\tau}\sin(\omega' t - \phi)} \end{equation}\] where the decay time \(\tau\) and the ringdown frequency \(\omega'\) are \[\begin{equation} \tag{L3.7} \tau = \frac{2L}{R} , \qquad \omega' = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2} = \sqrt{\omega_0^2 - \frac{1}{\tau^2}} . \end{equation}\] This formula holds provided \(\omega_0\tau = 2Q > 1\), which will be satisfied here. The parameters of (L3.6) are \(V_0\), which is determined by the initial voltage, and \(\phi\), which is determined by the initial rate of change, if any, of the voltage. We will set things up so that \(\phi = 0\).

In this experiment, we will use the wave generator to produce a sudden drop in the voltage using a step function, and use the oscilloscope probe to measure the voltage response across the resistor and verify (L3.6). We can measure the period \(T = 2\pi/\omega'\) of the oscillations by noting that the sine goes through zero at \(t = 0, \tfrac{T}{2}, T, \dots\). Since \(e^{-t/\tau} > 0\), \(V_\text{out}(t)\) in (L3.6) can only go through zero when the oscillating part does. So you can measure the period by noting where \(V_\text{out}\) goes through zero.

To find \(\tau\), we use the fact that the sine is periodic: \(\sin(\omega' t) = \sin\big(\omega'(t + T)\big)\). If we choose some time \(t_1\) (say, when the function is at a peak), the oscillating part has the same value at \(t = t_1 + T\); only the exponential \(e^{-t/\tau}\) has changed. Calling \(V_1 = V_\text{out}(t_1)\) and \(V_2 = V_\text{out}(t_1 + T)\), we see that \[\frac{V_1}{V_2} = e^{T/\tau} .\] Taking logarithms of both sides and rearranging gives \[\begin{equation} \tag{L3.8} \keyeq{\displaystyle \tau = \frac{T}{\ln(V_1/V_2)}} \end{equation}\] So by measuring \(V_1\) and \(V_2\), and using the measured value of \(T\), we can calculate \(\tau\). Figure L3.3 illustrates the procedure for values similar (but not identical) to those we will encounter in this experiment.

rlc ringdown response

Figure L3.3. RLC ringdown after a step-function voltage drop. A peak \((t_1, V_1)\), the point \((t_1 + T, V_2)\) one period later, and the period \(T\) are marked. The red dashed lines show the exponential envelope \(V_0\,e^{-t/\tau}\).

Internal resistance

Just as in Lab 2, the wave generator has an internal resistance of about \(600\,\mathrm{\Omega }\), which is significant compared to the resistance \(R_R\) of the resistor and \(R_L\) of the inductor. The value of \(R\) in the RLC circuit is the sum of all three: \[\begin{equation} \tag{L3.9} R = R_R + R_L + R_\text{int} . \end{equation}\]

L3.3 Materials

L3.4 Instructions

RLC response curve

rlc circuit photo

Figure L3.4. The series RLC band-pass circuit on a breadboard.

RLC ringdown

RC high-pass response curve

rc circuit photo

Figure L3.5. The series RC high-pass circuit on a breadboard (the same layout as in Lab 1).

L3.5 Analysis and discussion

Show your derivations for all calculations. When computing an error for a derived quantity, use the rules in Error propagation in the Lab Guide.

L3.6 Instructions for the lab report

Submit a lab report that includes: