Alpha Laboratories ECSE-2010 Spring 2020 Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020 Rensselaer Polytechnic Institute Troy, New York, USA 1 LABORATORY 6: Transfer Functions, Filters, and Bode Plots Material covered: • First Order Filters • Multi-stage Circuits • Bode Plots • Design Problem Part A: First Order Filters Overview Notes: Decibels: When considering the magnitude of the transfer function, a log-log plot is useful for gaining physical insight to the circuit. Typically, the vertical axis has units of Decibels (dB). The magnitude of the transfer function can be converted to dB by using the relationship () () () ( ) s H s V s V in out log 20 log 20 = [dB]. Using this formula, when we plot the transfer function of a first order circuit, we can make some observations. Cutoff frequency: The cutoff frequency of a first order circuit is defined as the frequency at which the amplitude of the output voltage is 2 1 the amplitude of the input voltage. In other words the ratio of output to input is () () 2 1 = s V s V in out . Substituting that value into the above dB expression, at the cutoff frequency the expression evaluates to -3dB. The cutoff frequency is often referred to as the 3dB point. In first order circuits, the cutoff frequency occurs when the magnitudes of the real part and the imaginary part of transfer function denominator are the same. Passband: The frequency range where the output signal has approximately a constant amplitude. Stopband: The frequency range where the output signals are attenuated relative to passband . In the stopband of a first order circuit, a 20dB difference occurs every
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Alpha Laboratories
ECSE-2010 Spring 2020
Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020
Rensselaer Polytechnic Institute Troy, New York, USA
1
LABORATORY 6: Transfer Functions, Filters, and Bode Plots
Material covered:
• First Order Filters
• Multi-stage Circuits
• Bode Plots
• Design Problem
Part A: First Order Filters
Overview Notes:
Decibels: When considering the magnitude of the transfer function, a log-log plot
is useful for gaining physical insight to the circuit. Typically, the vertical axis has
units of Decibels (dB). The magnitude of the transfer function can be converted to
dB by using the relationship ( )
( )( )( )sH
sV
sV
in
outlog20log20 =
[dB]. Using this formula,
when we plot the transfer function of a first order circuit, we can make some
observations.
Cutoff frequency: The cutoff frequency of a first order circuit is defined as the
frequency at which the amplitude of the output voltage is 2
1the amplitude of the
input voltage. In other words the ratio of output to input is ( )
( ) 2
1=
sV
sV
in
out .
Substituting that value into the above dB expression, at the cutoff frequency the
expression evaluates to -3dB. The cutoff frequency is often referred to as the 3dB
point. In first order circuits, the cutoff frequency occurs when the magnitudes of
the real part and the imaginary part of transfer function denominator are the same.
Passband: The frequency range where the output signal has approximately a
constant amplitude.
Stopband: The frequency range where the output signals are attenuated relative to
passband . In the stopband of a first order circuit, a 20dB difference occurs every
Alpha Laboratories
ECSE-2010 Spring 2020
Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020
Rensselaer Polytechnic Institute Troy, New York, USA
2
time the frequency changes one order of magnitude. For example, in a low pass
filter with a cutoff frequency of 1kHz, the dB value of |H(jω)| at 50kHz would
have a value 20dB lower than the the dB value of |H(jω)| at 5kHz. This attribute is
called a reduction of 20dB/decade. (We will discuss these concepts in class.)
We will be performing frequency sweeps in the upcoming labs. We will need to
configure the voltage component to an AC voltage source by right clicking and
putting 1V in the AC analysis section as shown in the figure below.
You must choose the AC Analysis tab in the simulation profile.
Alpha Laboratories
ECSE-2010 Spring 2020
Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020
Rensselaer Polytechnic Institute Troy, New York, USA
3
To perform frequency sweeps;
1) Edit the simulation command and select the AC Analysis Tab
2) Change the “Type of sweep:” to Decade
3) To generate smoother, more accurate plots, choose 100 points per decade.
4) You will then need to select the frequency limits. These limits will likely
depend on your circuit. For the first order circuits, we want the lower limit to
be well below the cutoff frequency and the upper limit to be well above the
cutoff frequency. For the above circuit, we will use a range of 1Hz-1E6Hz
5) LTSpice will output a graph with both amplitude and phase plotted on the
right and left hand axes respectively.
a. The amplitude is expressed in dB.
b. Phase is expressed in degrees.
Alpha Laboratories
ECSE-2010 Spring 2020
Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020
Rensselaer Polytechnic Institute Troy, New York, USA
4
A.1. RL Filter Circuit Calculations
Determine the transfer function for the voltage across the resistor,
HVR(s) = ______________________
Determine the transfer function for the voltage across the inductor,
HVL(s) = ______________________
What is the cutoff frequency for this circuit? _________________
As the frequency approaches zero (DC), the voltage across the inductor
approaches _________
As the frequency approaches infinity, the voltage across the inductor
approaches _________
Alpha Laboratories
ECSE-2010 Spring 2020
Written by J. Braunstein Revised by S. Sawyer Spring 2020: 1/28/2020
Rensselaer Polytechnic Institute Troy, New York, USA
5
When measuring the voltage across the inductor, is this circuit a low pass filter or a
high pass filter?
A.1. RL Filter Circuit Simulation and Experiment
Build the circuit. Set your source amplitude to 2Vpp (1V amplitude) with a 0V
offset and adjust the frequency as indicated in the table. Measure the output
voltage amplitude and phase across the inductor for various frequencies and fill in
the following table with your calculations, LTSpice, Analog Discovery Board