ECE 2111 Signals and Systems
Spring 2009, UMD
Experiment 7: Linear Time Invariant Systems
PRELAB
I. General Case
Consider the RC circuit shown in Fig.1.
1). Find the differential equation that describes this system, i.e. the relationship between the input
x(t) and the output y(t).
2). Taking the Fourier Transform of the differential equation, compute the transfer function of the
system, this is: H(w) = Y(w) / X(w). Use the following properties of the Fourier Transform:
x(t)
y(t)
dx(t)/dt
X(w)
Y(w)
jw X(w)
3). Plot the frequency response of the above system using the transfer function found in (2):
Consider R = 2.2KΩ and C = 0.001µF
a) the amplitude response: abs(H(w))
b) the phase response: angle(H(w))
4). For an input signal: 1 Vpp, no DC, 100 kHz, sine wave, sketch the output of the circuit in the
time domain.
5). For an input signal: 1 Vpp, no DC, 100 kHz, square wave, sketch the output of the circuit in the
time domain.
6). For an input signal: 1 Vpp, no DC, 100 kHz, triangular wave, sketch the output of the circuit in
the time domain.
Note: For the questions 4 to 6, note that the given RC circuit is a low pass filter (an integrator.)
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II. Circuit for the Experiment
Given the following RC circuit:
Fig. 2 the RC circuit to be used in the laboratory
PRACTICAL PART
1). Prepare the RC circuit of Fig. 2. Measure the amplitude response, and phase response of the RC
circuit, i.e.
To obtain the amplitude and phase response of the circuit, connect as an input signal a sine wave,
1Vpp, no DC, and measure the amplitude (volts), and phase (degrees) of the output signal using the
oscilloscope.
To visualize the effect of the circuit on a sinusoidal input do the following:
- Connect the signal x(t) (sine wave, 1Vpp, no DC), to the input of the circuit, also connect at this
node one of the channels of the oscilloscope.
- Connect the output of the circuit y(t) to another channel of the oscilloscope. Set the amplitude
scales of both channels to the same value.
Start turning the knob of the function generator in order to sweep the frequency of the input signal.
Start from around 10 kHz and sweep it to reach about 1 MHz. Observe what is happening to the
output signal. Record the magnitude and the phase of the output signal, and compare it with the
input signal.
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Note: fc is the cutoff frequency of the low pass filter circuit given in Fig.2, fc = 1/(2πRC).
To obtain quantitative results of what you are visualizing fill out the following table:
Input signal x(t)
x(t): sine, 1Vpp, no DC
Frequency
(KHz)
0.1 fc =
0.5 fc =
1.0 fc =
1.5 fc =
2.0 fc =
2.5 fc =
3.0 fc =
5.0 fc=
10 fc =
20 fc=
Amplitude
(volts)
Output signal y(t)
Frequency
(KHz)
Amplitude
(volts)
Phase: [0 - π] or ( 0 - 180o )
Measure the phase shift between
the input and output signals
Using the values from the above table, sketch the amplitude and the phase of H(f)
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2). Prepare the RC circuit in Fig.2, and the input, 1Vp-p, no DC, 100 kHz, Sine wave.
Measure and Sketch: The output of the RC circuit in the time domain.
3). Prepare the RC circuit in Fig.2, and the input, 1Vpp, no DC, 5 kHz, Square wave.
Measure and Sketch: The output of the RC circuit in the time domain.
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4). Prepare the RC circuit in Fig.2, and the input, 1Vpp, no DC, 100 kHz, Square wave.
Measure and Sketch: The output of the RC circuit in the time domain.
5). Prepare the RC circuit in Fig.2, and the input, 1Vp-p, no DC, 5 kHz, Triangular wave.
Measure and Sketch: The output of the RC circuit in the time domain.
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6). Prepare the RC circuit in Fig.2, and the input, 1Vp-p, no DC, 100 kHz, Triangular wave.
Measure and Sketch: The output of the RC circuit in the time domain.
Post Experiment (Report) Requirements:
1- Every student must have his own individual lab report.
2- The report should include the following:
a) Results with detailed explanations are needed.
b) Answer the questions if there are any.
c) Conclusion - what did you learn in this experiment? Please write only a few lines.
3- All reports should be word processed and should also have the assigned cover page.
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