A standard form of a first order differential equation is:
(1) $$\tau \frac{dy}{dt} + y = k * x(t)$$
The laplace transform of this:
(2) $$G(s) = \frac{Y(s)}{X(s)} = \frac{k}{\tau s+1}$$
but sometimes it is given as
(3) $$H(s) = \frac{1}{\tau s +1} = \frac{a}{s+a}$$
A standard form of a second order differential equation is:
(4) $$\tau ^{2} \frac{d^{2}y}{dt^{2}}+2 \tau \zeta \frac{dy}{dt} + y = k * x(t)$$
The laplace transform of this:
(5) $$G(s) = \frac{Y(s)}{X(s)} = \frac{k}{\tau^2s^2 + 2\tau\zeta s+1}$$
but sometimes this is given as
(6) $$H(s) = \frac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n^2}$$
Here are my questions:
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What is the physical meaning of "first" and "second order"? (apart from the fact that the highest power of the differential in the first is 1 and in the second is 2). How do I know if a system is first or second order?
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Where do equations (1) and (4) come from? Why were these decided to be the "standard form"? What is so special about this form and how were these equations derived?
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When given a first order system, why is sometimes equation (2) given, and sometimes equation (3) as the transfer function for this system? Likewise, when given a second order system why is equation (6) usually given, when the laplace transform is actually equation (5)?
Best Answer
A 1st order system has one energy storage element and requires just one initial condition to specify the unique solution to the governing differential equation. RC and RL circuits are 1st order systems since each has one energy storage element, a capacitor and inductor respectively.
A 2nd order system has two energy storage elements and requires two initial conditions to specify the unique solution. An RLC circuit is a 2nd order system since it contains a capacitor and an inductor
Consider the homogeneous case for the 1st order equation:
$$\tau \frac{dy}{dt} + y = 0$$
As is well known, the solution is of the form
$$y_c(t) = y_c(0) \cdot e^{-\frac{t}{\tau}}$$
which gives physical significance to the parameter \$\tau\$ - it is the time constant associated with the system. The larger the time constant \$\tau\$, the longer transients take to decay.
For the 2nd order system, the homogeneous equation is
$$\tau^2\frac{d^2y}{dt^2} + 2\tau \zeta \frac{dy}{dt} + y = 0$$
Assuming the solutions are of the form \$e^{st}\$, the associated characteristic equation is thus
$$\tau^2s^2 + 2\tau\zeta s + 1 = 0 $$
which has two solutions
$$s = \frac{-\zeta \pm\sqrt{\zeta^2 -1}}{\tau}$$
which gives physical meaning to the damping constant \$\zeta\$ associated with the system.
The transient solutions are, when \$\zeta > 1\$ (overdamped), of the form
$$y_c(t) = Ae^{\frac{-\zeta +\sqrt{\zeta^2 -1}}{\tau}t} + Be^{\frac{-\zeta -\sqrt{\zeta^2 -1}}{\tau}t} $$
when \$\zeta = 1\$ (critically damped), the solutions are of the form
$$y_c(t) = \left(A + Bt\right)e^{-\frac{\zeta}{\tau}t} $$
and when \$\zeta < 1\$ (underdamped), the solutions are of the form
$$y_c(t) = e^{-\frac{\zeta}{\tau}t}\left(A\cos \left(t\sqrt{1 - \zeta^2}\right) + B\sin \left(t\sqrt{1 - \zeta^2}\right) \right)$$
Different disciplines have different conventions and standard forms. Equation (2) looks to me like control theory standard while equation (3) looks like signal processing standard.
Standard forms evolve to fit the needs of a discipline. Further, if a particularly influential person or group develops and uses a particular convention, that convention often becomes the standard. It might be educational to peruse older textbooks and journals to get a sense of how notation and standard forms evolve.