Network Theory
LAB 4: Bilinear Transfer Function
Objective
• To study the magnitude and phase responses of basic bilinear transfer functions.
Components & Equipment
• Signal Generator
• Oscilloscope
• Digital Multimeter
• Resistors: 10 kΩ
• Inductor: 100 mH
• Capacitor: 0.01 μF
• Breadboard
Deliverables
A complete lab report including the following:
• Pre-Lab analysis of the experiments
• Findings/Results and discussions
• Relevant plots
• Date Due: 14th February, 2025
1. Theory
A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate
others.
Categories:
• Passive filters include only passive components—resistors, capacitors, and inductors.
• Active filters consist of active elements (such as transistors and op amps) in addition to
passive elements R, L, and C.
Filters are circuits which perform signal processing functions, specifically to remove unwanted
frequency components from the signal, to enhance wanted ones, or both.
Electronic filter include high-pass, low-pass, band-pass, and band-stop (band-rejection; notch)
The filter is described mathematically by the transfer function which can be displayed in the form of
an amplitude and a phase response
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Network Theory
Cut-off Frequency
The frequencies, fc, are known as the half-power frequencies and also as the lower and upper break
or cutoff frequencies. Additionally, they are known as the 3dB frequencies and are further identified
as the frequencies where the output power is down to 0.5 of its mid-band value and the output voltage
is down to 0.707 of its mid-band value. This can also be seen graphically as the point where the log-
log representation is 3dB lower than the pass band.
Figure 3.1
• Passband - The passband (blue zone) allows spectral components of a signal to pass through.
Modification of the signal in this frequency range should be avoided.
• Stopband - In the stopband (red zone), the filter attenuates the corresponding frequency
components of the signal.
• Transition - The transition (yellow zone) separates the passband and the stopband. It should
normally be as small as possible. The design of the transition phase is a defining criterion for
the selection of the filter type and its parameterization.
• Passband ripple - The ripple in the passband describes the waviness of the amplitude response
in the passband.
Cutoff frequency ωc, 1
𝜔=
𝑅𝐶
Low Pass Filters Bandpass Filter
1 𝑅
𝐻(𝜔) =
1 + 𝑗𝜔𝑅𝐶 𝐻(𝑠) = 𝐿𝑠
𝑅 1
𝑠 2 + 𝐿 𝑠 + 𝐿𝐶
High pass filters Bandstop Filter
𝑗𝜔𝑅𝐶 1
𝐻(𝜔) = 𝑠 2 + 𝐿𝐶
1 + 𝑗𝜔𝑅𝐶 𝐻(𝑠) =
𝑅 1
𝑠 2 + 𝐿 𝑠 + 𝐿𝐶
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Network Theory
Figure 3.2
Figure 3.3
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Network Theory
Experiments
PreLab
1. Derive the voltage transfer functions for the networks in the Figures 3.4 and 3.5
2. Calculate the theoretical cutoff frequency
3. Using MATLAB, obtain a plot of the voltage transfer function.
4. Assume
a. R= 1 kΩ and C= 0.01 μF.
b. R= 1 kΩ and C=1 μF.
Procedure
1. Obtain resistors and capacitors and measure and record the values
2. Setup the circuit as shown in figure 3.4. Use functional generator for the supply voltage
3. Connect both FGen and Oscilloscope’s channel 1 to terminal 1 of the circuit. The oscilloscope
connection will measure vin
4. Connect channel 2 of the oscilloscope to measure the filter’s output voltage vo
5. Adjust the signal generator to 5 V peak-to-peak.
6. Oscilloscope operation
a. On the oscilloscope, turn on the measurement functions for CH1 and CH2.
b. Vary the frequency from 50 Hz to 20 kHz in steps indicated in table 3.1 and record
the values.
c. You may press stop to freeze the display when taking cursor measurements.
d. For the values of vin and vo measurements, be sure the time scale is sufficient to show
at least a few cycles, or the instrument may not properly calculate the peak-to-peak
and RMS values.
e. Use the oscilloscope’s cursor to measure the phase shift Δt and then calculate the
phase angle between vin and vo.
7. Fill in table 3 (both measured and calculated from the measured values)
8. Sketch the bode plot of the filter’s output voltage (both magnitude and phase) using the data
in table 3.1 as follows. Show all the calculations.
9. Discuss the results
10. Repeat the above steps for the circuit in figure 3.5
11. Modify the circuit as shown in figure 3.6 and repeat all the steps above
12. Compare the results and discuss the operation of these three circuits (Figures 3.4, 3.5, & 3.6)
13. Explain any differences between the experimental results and the calculated results
Note: You may substitute the values of the components (resistors, capacitors, and inductors) in the
figure below
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Network Theory
Figure 3.4: Circuit A Figure 3.5: Circuit B
Figure 3.6: Circuit C
Table 3.1: Measured values
Frequency vin vo Tn(jω) Am(ω) Δt Φ()
(KHz) (v) (v) =Vin/Vo =20logTm(jω) (μs) (degrees)
dB
0.05
0.1
0.5
1.0
2.0
3.0
4.0
6.0
12.0
20.0
Cutoff
