Some helpful expressions for the transient analysis and design of second order systems.

By: Signal Processing Group Inc. technical staff.            October 2010. Chandler, Arizona.

1.0

1.0 The shape of a second order transient response is as shown below. ( It is for a stable second system)

Figure 1.0

The transient response of a second order system

The quantities of interest and importance are:

The overshoot

The settling time

The rise time

The parameters governing the transient response are:

4.0       The natural undamped frequency ω_n

5.0       The damping ratio ζ

6.0 The time t.

ω_n  and ζ are derived from the characteristic equation of a second order system in the Laplace variable “s”.

s² + 2 ω_n ζs + ω_n²  = 0                                                                                      Eqn.1.0

The quantity  a = ω_n ζ is called the damping constant or simply the damping of the system and controls  the rate of rise and fall of the transient response. The inverse of a, 1/a is proportional to the time constant of the system.

When ζ = 0, the response is simply a sinusoidal signal. In this case ω_n  is the frequency of the sinusoid.

When ζ = 1, the system is critically damped and a = ω_n,

In other cases 

                                                            ω = ω_n √(1- ζ²)                                    Eqn 2.0

The following are the various cases of damping with respect to the damping ratio.

0 .0 < ζ < 1.0                                       Underdamped response.

ζ =1.0                                                  Critically damped response

ζ >1.0                                                  Overdamped response

ζ = 0.0                                                 Undamped response ( sinusoidal output)

ζ < 0.0                                                 Negatively damped.

In practice only stable systems are of interest ( not considering oscillators as a functional system).

In figure 2.0 some responses which are functions of the damping ratio are shown.           The independent variable is ω_n,t.           

Note the dependence of the response on the damping ratio ζ.

                                                                        Figure 2.0

                                                Transient response as a function of ζ

So what ζ should one use ? The answer is that in the absence of any other consideration use

ζ between 0.5 and 0.8.

2.0

Overshoots occur at times given by:

                                                t = nπ / [ω_n √(1- ζ²)]                                                     Eqn 3.0

From this identity, the first maximum occurs at:

                                                t = π / [ω_n √(1- ζ²)]                                                      Eqn 4.0

The odd values of n define the overshoots. Even values of n define         the undershoots.

Even though the over and undershoots occur at periodic intervals, the response is not periodic.

The general equation for under and overshoots is:

            y(t) = 1.0 +{ e^{- nπ ζ /√(1- ζ* ζ)} / √(1- ζ²)}sin(nπ – tan^-1 [√(1- ζ²)/- ζ]                  Eqn 5.0

            n =1,2,3, ….

The expressions for other quantities such as delay time, rise time, etc are not so exact. The following expressions can be used however.

Delay time:

                        t_d = [1.0 + 0.6 ζ + 0.15 ζ² ]/ ω_n                                                            Eqn 6.0           

Rise time: ( 10% to 90%):

                        t_r = [1.0 + 1.1 ζ + 1.4 ζ² ]/ ω_n                                                                                   Eqn 7.0

Settling time: ( Note 1.0):

                        t_s = {(- 1/ ζ) ln[0.05[√(1- ζ²)]}/ ω_n                                                                         Eqn 8.0

For small values of ζ, equation 8.0 can be simplified to:

                        t_s = 3/ ζ ω_n                                                                                                                          Eqn 9.0

Here 0.0 < ζ < 1.0

Note 1.0

Settling time is notoriously difficult to calculate analytically. The expressions above are approximations that can be used to get approximate results. Numerical simulations can be used to confirm or deny the results. The issue is, that for most second order ( or higher order) systems accurate simulations take a long time. Approximate expressions can be a help in evaluating parameters quickly before launching long simulation runs.

The idea is that when designing or analyzing second order systems, one should obtain the two parameters , ζ  ω_n  through analytical or empirical means and then use them to analyze the system.

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