Solving differential equations using operational amplifiers
Introduction:
An operational amplifier ("op amp") is a differential-input, high gain voltage amplifier,
usually packaged in the form of a small integrated circuit. The term "operational" dates back
to the early days of analog computers when these devices were employed in circuits that
performed mathematical operations such as addition, subtraction, integration, and the solution
of differential equations. Today's op amps are used in a much wider variety of circuits and
operate at considerably lower voltages and powers; however, the name remains. The modern
operational amplifier is a very useful and versatile building block for thousands of circuits in
applications as diverse as audio, video, communications, process control and instrumentation.
Analog computation:
There are two main types of computer. What you probably think of when you say
"'computer" is a digital computer. Numbers are represented by sets of 1's and 0's where 1 and
0 are represented by two different voltages, (typically +5V, and 0V). Operations are simple
logical operations (i.e. AND, OR, etc.) or arithmetic operations (i.e. addition or subtraction).
Calculus-type operations are very complicated to do.
The other type is the analogue computer, where numbers are represented by continuously
varying quantities. Typically, voltage is the quantity used. You are already familiar with
OPAMP circuits for different mathematical operations using various voltages. These circuits
can be combined to solve various differential equations. Some exemplary circuits are given in
this experiment. The heart of the technique is the op amp integrator circuit shown in Figure 1.
The output of the circuit is given as follows:
1
1
Vout = −
Vin dt = −
Vin
∫
RC
jωRC
Fig. 1
The following experiments are exercises in problem-solving. It is instructive to think of each
experiment as consisting of four steps. The first step is the building of a 'model' of the
problem. The model is a mathematical description of the problem derived from known
physical laws and/or experimental data after making reasonable simplifications. It will
include a set of algebraic and/or differential equations, initial conditions, values of
coefficients and range of variables of interest. The second step is to rearrange the equations to
get a set of 'computer equations' suitable for interpretation. The third step is to interpret the
equations term-by-term and from there construct the computer. The fourth step is running the
computer and recording the output. The output, a plot of voltage against time, may be viewed
on a SCOPE or may be plotted on an x-y recorder, with the x-input connected to a 'Time
Generator".
Experiment 1: A Time Generator (The Integrator Revisited)
Recall that the integrator integrates with respect to real time. Some processes of interest may
vary with real time too quickly or too slowly to be computed conveniently. Therefore, the
passage of real time itself can be simulated by a voltage varying linearly (with time) and the
output of the computer can be expressed with respect to this voltage. The voltage simulating
time can also aid in displaying the output by providing the time base of an oscilloscope or the
x-drive of an xy plotter.
Fig. 2
For the circuit in Fig. 2, with Vi = -1 Volt, D.C. and S a switch which is closed at time, t = 0,
and Vi0 = 0, the output voltage T seconds after the closure is
Also, the voltage has increased at the rate of 1 volt/sec. Clearly, the voltage is now simulating
time (in second) at a ratio of 1:1. The circuit of Fig. 2 can also be thought of as a clock with
displays the passage of time following the closure of S. The 1:1 ratio can be changed by
changing the forward gain of the integrator.
Procedure: Set up the circuit of Fig. 2 using an Op-Amp. Use a multimeter or a SCOPE to
display the output. To obtain Vi use a D.C. power supply. Momentarily short circuit the
capacitor, C. This procedure sets the initial condition of 0 volts. Close the switch and record
the output Vo(t) on the meter or oscilloscope.
Explain your observations.
To return to zero, the starting condition short circuit C momentarily. The output should return
to zero. Once at zero, you are ready for another run.
With the construction of this real time computer, you now have a device which will prove
useful in the experiments to follow.
ࢊ࢟
Experiment 2 – Solving ࢊ࢚ ൅ ࢑࢟ ൌ ࢉ
This is an equation which may describe damped oscillation or charge on a capacitor or a
simple radioactive decay.
Let’s take the case where y(t) is displacement in meter and assume arbitrary values of
k and c such that the equation has the following form
ࢊ࢟
൅ ૞࢟ ൌ ૛
ࢊ࢚
As an analog, we can let the output of the integrator be proportional to displacement, i.e.
Vo(t) = K y(t)
Note that in order to maintain linear operation and obtain an accurate solution, we must
always be careful to scale output voltages so as not to saturate the op amp. Assuming that
VSAT ≈ 12 Volts for the op amp and that the maximum of y(t) ≤ 12m, we can design an op
amp circuit to simulate the system by scaling the output voltage of the integrator to be 1
Volt/m, i.e., Vo(t) = y(t) or K = 1 Volt/m. Figure 3 shows one of many possible op amp
circuits that can be used to simulate the given differential equation.
Fig. 3
Procedure:
Set up the circuit in Fig. 3. Note that three op amps are used: one as a summing amplifier, one
as an integrator, and one as an inverting amplifier. Also, switches and DC voltages, shown as
batteries, are included in the circuit in order to (1) the initial condition of y(0) = -1m (Vo2(0)
= -1V) to the feedback capacitor during "reset", i.e., t < 0, and (2) the step input of 2m (V1 =
2V) to the input of the integrator during "operation", i.e., t ≥ 0. Connect the y voltage output
to the y input of a SCOPE, and connect the x input to the time generator circuit of
Experiment 1.
The analog computer simulation shown in Figure 3 was obtained as follows. First, the output
of the integrator was chosen to represent the displacement, i.e., Vo2 = y(t). Choosing Ri = 1
MΩ and C = 1 µF so that RiC = 1 second,
Vo1 = - dVo2/dt = - dy/dt = -2 + 5 y = - 2 + 5 Vo2
Resetting the time generator circuit as in Experiment 1, observe the plots of y vs t obtained
(as a solution to this equation) by switching on the circuit. Find solutions for various values
of k and various initial values of y.
Experiment 3 – Solve
ࢊ૛ ࢟
ࢊ࢚૛
൅ ࢑૚
ࢊ࢟
ࢊ࢚
൅ ࢑૛ ࢟ ൅ ࢉ ൌ ૙
Try to set up a circuit to solve the above equation.
Hint: You will need two integrators, a summer and some inverting circuits to realise the
solution. Choose some arbitrary values of constants and keep in mind that the opamp will
invert the sign at its output.
Suggested References:
1. Supplementary manual for operational amplifier (4th sem Advanced electronics lab
manual)
2. Jacob Millman, Microelectronics
3. J. Millman and C.C. Halkias, Integrated Electronics