What is the significance of the standard form of 1st and 2nd order transfer functions?

Question

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:

• 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?

• 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?

• 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)?

Answer 1

1. The order of a transfer function is determined by the highest order of the denominator. This order gives the number of poles and - thus - determines the roll-off characteristics of the transfer function (magnitude) as well as the amount of phase shift for rising frequencies.

2. The standard form is very important because it allows to find characteristic parameters by visual inspection and/or simple calculations: Order of the filter, formula for pole frequency, formula for pole-Q. These parameters are the design entries for designing a filter and can be found for all classical responses in filter tables.

3. Both forms of the equations can be used, of course. However, the last form is more convenient because you immediately can identify the pole frequency wn. This characteristic frequency is related to the cut-off frequency wc (which normally is given) by a fixed factor which depends on the desired characteristic (Example 1: Butterworth second order, wc=wn; example 2: Chebyshev second order, 0.5dB ripple, wp=1.2313*wc).

EDIT: I forgot to mention that the pole quality factor (pole Q or Qp) is related to the damping factor ζ (as given in your formulas) by the relation: ζ=1/(2*Qp).

Summary: The last form (Equ. 6) contains the most important lowpass quantities as parameters (Ao, wp, Qp), which also can be measured (Bode diagram for magnitude and phase). In practice, this form is compared with the form which is derived directly from the circuit - and, thus it is possible to see how all the parameters depend on the particular parts in the circuit.

Answer 2

What is the physical meaning of "first" and "second order"? ... How do I know if a system is first or second order?

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

Where do equations (1) and (4) come from?

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

$$y_c(t) = e^{-\frac{\zeta}{\tau}t}\left(A\cos \left(t\sqrt{1 - \zeta^2}\right) + B\sin \left(t\sqrt{1 - \zeta^2}\right) \right)$$

When given a first order system, why is sometimes equation (2) given, and sometimes equation (3) as the transfer function for this system?

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.