A reversible, first-order chemical reaction can be represented as follows
If reactant and product are conserved
then, after substitution, we obtain
4/ General differential equation
For many (lumped-parameter or compartmental) systems, the input x(t) and the output y(t) are related by an nth order differential equation of the form
We seek a solution of this system for an input that is of the form
where X and s are in general complex quantities and u(t) is the unit step function. We will also assume that the system is at rest for t<0, so that y(t) = 0 for t<0.
We will seek the solution for t>0 by finding the homogeneous (unforced solution) and then a particular solution (forced solution).
VI. HOMOGENEOUS SOLUTION
1/ Exponential solution
Let the homogeneous solution be
, then the homogeneous equation is
To solve this equation we assume a solution of the form
Since
we have
2/ Characteristic polynomial
The equation can be factored to yield
We are not interested in the trivial solution
Therefore, we can divide by
to obtain the characteristic Polynomial
This polynomial of order N has N roots which can be exposed by writing the polynomial in factored form
3/ Natural frequencies
The roots of the characteristic polynomial
are called natural frequencies. These are frequencies for which there is an output in the absence of an input. For example, imagine striking a tuning fork. After the tuning fork has been struck there is no further input, but the tuning fork keeps vibrating — at its natural frequency.
If the natural frequencies are distinct, i.e., if
for i ≠ k, the most general homogeneous solution has the form
Example — natural frequencies of a network
The differential equation relating v(t) to i(t) is
The characteristic polynomial is
The roots of this quadratic characteristic polynomial, the natural frequencies, are
The locus of natural frequencies is plotted for L=C = 1 with R increasing in the directions of the arrows. Note the natural frequencies are in the left-half plane for all values of R>0. As
, the natural frequencies approach the imaginary axis. What is the physical significance of this behavior?
VII. PARTICULAR SOLUTION
Now we need to find a particular solution to the differential equation
where for t>0
and s does not equal one of the natural frequencies. We assume that
and we solve for Y . Substitution for both x(t) and y(t) in the differential equation yields, after factoring,
1/ System function
a/ System function — derivation
After dividing both sides of the equation by
we can solve for Y which has the form