Instructor's
Notes:
The resonant filter is a circuit element containing a
resistor with resistance R, a capacitor with capacitance C, an
inductor with inductance L and a voltage source. Its response can be modeled by a
second-order, constant-coefficient, non-homogeneous second order differential
equation
where q(t) is the charge on the capacitor at
time t and Vin(t) is the (known) voltage from the input
source. The equation can be reforumlated in terms of current, i(t),
using the relation that i'(t) = q(t). Differentiating, we obtain an
equivalent equation in terms of the current in the circuit,
.
Let Vout(t) represent the voltage
drop across the resistor. The relationship between the voltage and current
is given by Vout(t) = R i(t). Thus, the second order
differential equation equation, represented in terms of Vout is
.
A program has been developed which takes as parameters the circuit elements, the source,
and the initial conditions. This program provides an interactive utility
for a numerical simulation of the effect on the output by varying the parameters
and the input signal.
The program displays three components; the control
panel, a graph of the input signal, and a graph of the output signal,
represented by y(t). These are
displayed in Figure 1.
Values for L, C, and R
are controlled by sliders. Initial data are controlled by user input and
there are options for setting the viewing scale. Three input signals of
the form 
are allowed with controls for amplitude A, frequency
,
and phase shift 
.
The initial value problem is solved using a choice of the fixed step size Fourth
Order Runge-Kutta algorithm or the variable step size Runge-Kutta-Fehlberg
algorithm and the output function is graphed. Each numerical routine
provides user control over step size. A slider controls the end time and
the preferred frequency for the filter is calculated and displayed.
The accompanying help file, accessible by clicking the help
button on the control panel, provides detailed instructions. (To see the help
file click here.)
DEMO 1: A basic demo involves
simply illustrating the solution to an initial value problem that models the
resonant filter with one input signal. Holding all parameters fixed except
one, it is easy to investigate the effect on the output signal from changing the
value of one of the physical parameters. Such an experiment is shown in
Figure 2.
 |
 |
| Figure 2. Effect
on output from varying C. |
Aside from showing the numerically computed solution to the second order differential equation
that describes the resonant filter, there are three other useful demonstrations that this software can
facilitate.
DEMO 2: Solutions of a second order differential equation with constant coefficients and a sinusoidal forcing
function decompose into two components--a transient solution and a steady state solution. Simply sliding the bars for the initial data shows quite clearly the presence of a transient solution that depends on the initial data, and a steady state solution that is unaffected by changes in the initial data.
Figure 3 shows the output signal for an RLC filter with R = 0.52, L
= 0.215, C = 0.215, y(0) = 1,
y'(0) = 0, and input signal 
obtained by varying the initial conditions. Observe the initial transient
solution and the steady state that does not change.
|
y(0) = 1, y'(0) = 0
|
|
y(0) = .5, y'(0) = 0
|
|
y(0) = -1, y'(0) = 2
|
|
Figure 3. Effect on
output from varying initial conditions. |
DEMO 3: This demo is useful for
illustrating the phenomenon of resonance in an electrical device. Although
resonance is undesirable in many mechanical systems, such as a spring mass
system, many electrical devices such as radios would not function properly
without the phenomenon of resonance.
Choose an input signal with a certain
frequency. Fix the resistance and inductance. Slide the capacitance
slider and observe the changes in the output curve as the preferred frequency
changes. Notice that the amplitude of the steady state signal is largest when
the preferred frequency is close to the frequency of the input signal. We
can think of the capacitor as a tuner.
We can use the software to simulate a
primitive radio. Assume that volume is roughly proportional to amplitude
and suppose we wish to pick up a radio station at a particular frequency (the
input frequency). By adjusting the capacitor so that the amplitude is the
highest (this will occur when the preferred frequency is the same as the input
frequency), we have simultaneously tuned in the station broadcasting at our
preferred frequency and tuned out stations broadcasting at other frequencies.
DEMO 4: Choose
three different input signals with different frequencies. Set the resistance to
a low value, and slide either the inductance or capacitance values so that the
preferred frequency varies across the frequencies of your input signals. You
will be able to visually pick out the frequencies that correspond to the input
frequencies. It is another nice demonstration of a primitive method for tuning a
radio.
In a course on scientific simulation, we had students construct a program like this. They were given tables of data formed by the sum of periodic signals and used their program to determine the frequencies of the original signals.
Material Included:
A zipped file with all requisite material is available; see below. This file contains the program, which should run on any recent
PC, the help file which describes the program interface as well as the mathematical details of the model and the resonance phenomenon.
Also included is a complete commented source code for those who wish to modify the program. There is no installation process; you need only unzip the package in a convenient directory.
To download the zipped file click
here.
Once the files have been extracted, the
utility is the executable file entitled LRC Circuit.exe. To run the
program, double click the program icon.
This software can be used either as a classroom
demonstration or in a lab situation. It is recommended that if you use this
with students in a lab that you prepare a handout of experiments for students
to perform and report on.
Other Resource:
http://www.math.ubc.ca/~feldman/demos/demo5.html
is a java applet that illustrates resonance in an RLC circuit.
References:
Davis, Paul, Differential Equations for
Mathematics, Science, and Engineering, Prentice Hall, 1992.
Edwards, C. Henry and David E. Penney, Differential
Equations: Computing and Modeling, Second Edition, Prentice
Hall, 2000.
Boyce, William E. and Richard C. Diprima, Elementary
Differential Equations, 6th Edition, John Wiley & Sons, 1996.
Zill, Dennis G., A
First Course in Differential Equations with Modeling Applications, 7th
Edition, Brooks-Cole, 2000.
