ECE 2111 Signals and Systems
Spring 2009, UMD
Experiment 6: Fourier Series and Superposition
Objective:
In this experiment, the characteristics of the Fourier series and the superposition principle will be
investigated. The sinusoidal components of periodic signals will be measured with a spectrum
analyzer and compared with theoretical predictions which will be derived using Fourier series
analysis.
The validity of the superposition principle will be demonstrated by comparing the measured
response of a simple RC circuit, to a square wave input, with the calculated response of the RC
circuit, to the same square wave input.
The Superposition Property:
Superposition is valid only in linear circuits, i.e. in circuits whose behavior is described by a linear
differential equation or an algebraic equation. Conversely, if superposition is experimentally proven
for a given circuit, then it is possible to state that the circuit is linear.
Suppose that the response of a linear circuit to each of the individual input signals x1, x2, x3, … xm
is
y1, y2, y3, … ym. According to the superposition property, if the input is now a1 x1 + a2 x2 + a3
x3 + … + am xm , the response is given by: a1 y1 + a2 y2 + a3 y3 + … + am ym , where each ai is a
constant.
The superposition property and Fourier series are important tools for calculating the response of
linear circuits to a periodic input signal. A periodic wave is a summation of complex exponential
functions, which are harmonics, each with a different frequency, amplitude, and phase. The transfer
function of a linear circuit, which has an amplitude and phase response, is a function of frequency,
such that:
If a single harmonic is input into the linear circuit, the output will be the same harmonic times the
transfer function at the frequency of the harmonic, that is:
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The amplitude of the output harmonic is:
while the phase is:
If the input signal is made up of several harmonics, the output signal will be the summation of each
of the input signal harmonics multiplied by the transfer function H(f) at the frequency of the
individual harmonics:
PRELAB
Complete the following problems, showing all your work:
1). Given the following RC circuit,
Figure 1. RC circuit for testing superposition
Compute and Plot: The amplitude response and the phase response of the circuit, i.e.
2). Given x(t) =1 Vp-p, no DC, 100 kHz, square wave; compute the exponential Fourier series. In
particular evaluate the coefficients for the first ten harmonics, i.e., find Xn for n = 1, 2, …, 10
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3). Given: The RC circuit in fig.1, and the input, x(t) = 1Vpp, no DC, 100kHz, square wave,
Sketch: The output of the RC circuit in the time domain.
4). Given: The RC circuit in fig. 1, and the input, x(t) = 1Vpp, no DC, 100kHz, square wave,
Calculate: The coefficients for the first ten output harmonics, i.e. find Yn for n = 1, 2... 10.
Note that if the input is defined as:
then the output is given by:
thus,
5). Given x(t) =1 Vp-p, no DC, 100 kHz, Triangular wave; compute the exponential Fourier series.
In particular evaluate the coefficients for the first ten harmonics, i.e., find Xn for n = 1, 2, …, 10
6). Given: The RC circuit in fig.1, and the input, x(t) = 1 Vpp, no DC, 100 kHz, triangular wave,
Sketch: The output of the RC circuit in the time domain.
7). Given: The RC circuit in fig. 1, and the input, x(t) = 1 Vpp, no DC, 100 kHz, triangular wave,
Calculate: The coefficients for the first ten output harmonics, i.e. find Yn for n = 1, 2, . . , 10.
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PRACTICAL PART
Set up all measurements in chart form. Draw realistic sketches on graph paper. Label horizontal and
vertical axes with its corresponding units (e.g. volts, kHz, rad, sec).
1). Using a signal generator set the signal: x(t) =1 Vpp, 100 kHz, no DC, square wave.
Measure the amplitude of all the harmonics of x(t), up to 1 MHz using the spectrum analyzer.
Harmonic
Fundamental
Frequency (kHz)
Amplitude (dBm)
Amplitude (volts)
2). Prepare the RC circuit in fig.1, and the input, x(t) = 1 Vpp, no DC, 100 kHz, square wave.
Measure the amplitude of all output harmonics up to 1MHz, using the spectrum analyzer.
Harmonic
Fundamental
Frequency (kHz)
Amplitude (dBm)
Amplitude (volts)
3). Using a signal generator set the signal x(t) = 1Vpp, 100KHz, no DC, triangular wave.
Measure the amplitude of all the harmonics of x(t), up to 1MHz using the spectrum analyzer.
Harmonics
Fundamental
Frequency (kHz)
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Amplitude (dBm)
Amplitude (volts)
4). Prepare the RC circuit in fig.1 and the input x(t) = 1Vpp, no DC, 100 kHz, triangular wave.
Measure the amplitude of all output harmonics up to 1 MHz, using the spectrum analyzer.
Harmonic
Fundamental
Frequency (KHz)
Amplitude (dBm)
Amplitude (volts)
REPORTING RESULTS
Before plotting any measured amplitude spectrum, convert all dBm values into Volts values. All
graphs and plots must be on graph paper or done by graphing program with voltage and frequency
scales displayed.
1). Plot: The calculated and measured amplitude of the harmonics obtained in the theoretical and
practical parts for the input signal x(t) = square signal. This is the amplitude line spectrum. Draw
both results in the same graph.
Compare: The differences between the calculated and the measured amplitude line spectrum.
Discuss the differences.
2). Plot: The calculated and measured output amplitude of the harmonics, Yn for n = 1, 2, . . , 10
(when the input signal is x(t) = square signal), obtained in the theoretical and practical parts. This
is the amplitude line spectrum of the output signal. Draw both results in the same graph.
Compare: The differences between the calculated and the measured amplitude line spectrum.
Discuss the differences.
3). Plot: The calculated and measured amplitude of the harmonics obtained in the theoretical and
practical parts for the input signal x(t) = triangular signal. This is the amplitude line spectrum.
Draw both results in the same graph.
Compare: The differences between the calculated and the measured amplitude line spectrum.
Discuss the differences.
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4). Plot: The calculated and measured amplitude of the output harmonics, Yn for n = 1, 2, . . , 10
(when the input signal is x(t) = triangular wave), obtained in the theoretical and practical parts.
This is the amplitude line spectrum of the output signal. Draw both results in the same graph.
Compare: The differences between the calculated and the measured amplitude line spectrum.
Discuss the differences.
Post Experiment (Report) Requirements:
1- Every student must have his own individual lab report.
2- The report should include the following:
a) An Introduction that summarizes the significance of what was done.
b) Background on the topic that was experimented if needed.
c) Results with detailed explanations are needed.
d) Answer the questions if there are any.
3- All reports should be word processed and should also have the assigned cover page.
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