Download Experiment 4 – Series RLC Resonant Circuit Physics 242

Experiment 4 – Series RLC Resonant Circuit
Physics 242 – Electronics
Introduction
RLC resonant circuits are useful as tuned filters where the resonant peak can serve as a narrow
pass band of the filter. We will investigate the properties of a resonant bandpass filter in this
experiment.
Procedure and Questions
1. Set up the circuit shown above left, with L = 2.6 H (approx.), C = 0.5 nF (approx.), and R = 2
k. As usual, measure the components individually using the LCR meter; record both the
inductance and series resistance values for the inductor. Apply an input sine wave of 1 V peakto-peak amplitude to the input (be careful not to exceed 1 V p-p), and measure the output voltage
amplitude (peak-to-peak) as a function of frequency. Choose frequencies spaced closely enough
to map out the shape of the peak near resonance, and also measure at more widely spaced
frequencies across the full range of frequencies that you can measure.
𝑉
𝑉
Plot |𝑉𝑜 | vs. f (Hz) using your data. Also plot the theoretical curve of |𝑉𝑜 | calculated from circuit
𝑖
𝑖
theory (using your measured component values). Note the real inductor can be represented as a
pure inductance in series with its intrinsic coil resistance 𝑅𝐿 as shown above right; because it is
not negligible, you will need to include the inductor's resistance 𝑅𝐿 in your calculations. To
display both the theory curve and your data points, it might be easiest to use Mathematica
(KGraph will also work); see the appended example Mma code.
Does your observed value of the resonant frequency 𝑓𝑟𝑒𝑠 agree closely with the predicted value
1
𝑓𝑟𝑒𝑠 = 2 𝜋 𝐶? What is the percent difference?
√𝐿
If you see a noticeable discrepancy in the fit, particularly in the resonant frequency, see if
adjusting the capacitance value will fix the problem. What addition to the capacitance is needed
to give an optimal fit of the data? Can you speculate about where this additional capacitance
could be coming from?
2. Measure the phase shift ∆𝜙 = 𝜙𝑜𝑢𝑡𝑝𝑢𝑡 − 𝜙𝑖𝑛𝑝𝑢𝑡 at a frequency 100 Hz below the resonant
frequency. To make the measurement, you will need to display the voltage 𝑉𝑖 and the voltage 𝑉𝑜
at the same time on your oscilloscope, with leads connected to channel 1 and channel 2 inputs on
the 'scope. Measure the time difference Δ𝑡 (horizontal scale) between the sine waves and the
Δ𝑡
time period t of one of the waves; the phase difference ∆𝜙 is given by ∆𝜙 = 2𝜋 𝑡 . Be sure to
note whether the output wave leads or lags behind the input wave.
Compare your measured phase difference to the theoretical value, calculated from circuit theory
using your measured component values. Note: Calculate the phase difference between 𝑓𝑟𝑒𝑠 and
1
𝑓𝑟𝑒𝑠 − 100 𝐻𝑧, where 𝑓𝑟𝑒𝑠 = 2 𝜋 𝐶 . This way the phase difference will not depend on whether
√𝐿
your experimental value of 𝑓𝑟𝑒𝑠 agrees exactly with the predicted value, which it might not. Find
the percent difference between experimental and theoretical phase shifts.
3. Measure the resonant frequency 𝜔𝑟𝑒𝑠 and the bandwidth B using the different resistance
values R from the following list (one at a time): 2 k, 5 k, 10 k, 20 k. To measure the
resonant frequency, find the frequency at which the amplitude 𝑉𝑜 is maximum. To find the
bandwidth B, measure the frequency difference between the half-power points, the frequencies
𝑉
where the amplitude is reduced by √2, i.e. 𝑉𝑜 = 𝑖 .
√2
Plot theoretical Q vs. experimental Q for your measurements. Experimental Q is given by
𝜔
𝑄𝑒𝑥𝑝 = 𝐵𝑟𝑒𝑠. Theoretical Q is calculated from circuit theory and component values; we showed
1
𝐿
in class that it is given by 𝑄𝑡ℎ𝑒𝑜𝑟𝑦 = 𝑅 √𝐶 where R represents the total resistance (the sum of the
discrete resistor and the inductor's intrinsic resistance). If theory and experiment are in
agreement, the slope of the plot should be unity—does your plot indicate that theory and
experiment are in agreement?

(* Example code to display data points and theory curve *)
(* First the data set is defined. *)
datalist =
{{2000,.016},
{3000,.036},
{3500,.058},
{4000,.142},
{4200,.24},
{4400,.67},
{4600,.31},
{4800,.158},
{5000,.104},
{5200,.081},
{5400,.069},
{6000,.039},
{6500,.036},
{7500,.022}};
plot1=ListPlot[datalist,PlotMarkers{Automatic,10}];
(* Next the theory curve is defined. *)
r = 2000.0;vi = 1.0; rL = 1000.0;L = 2.6; c = 500.0 10-12;
r
rL 2
2
fL
2
1
fc
2
g[f_]:=(vi r)/
;
plot2=Plot[g[f],{f,2000,7500},PlotRange{All,All}];
(* Finally, the data points and theory curve are displayed together. *)
Show[plot1,plot2]
0.6
0.5
0.4
0.3
0.2
0.1
3000
4000
5000
6000
7000