O VERDAMPED R ESPONSES

An overdamped response is a second-order response with lag whose damping coefficient ( ζ ) is greater than 1. By algebraically manipulating the transfer function of a second-order system with lag (see Section 14-5), the transfer function of an overdamped response can be expressed as:

 A 1   A 2  Hp () s =  τ 1 s + 1   τ 2 s + 1 

Industrial Text & Video Company 1-800-752-8398

www.industrialtext.com

S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

Critically damped ( ζ = 1) Overdamped ( ζ > 1)

Underdamped ( ζ < 1)

Figure 14-49. Overdamped, critically damped, and underdamped responses.

In this equation, which is a function of two first-order systems (i.e., two time lags), the terms A 1 and A 2 represent the gains. By substituting the term K OD for the total overdamped system gain A 1 A 2 , this function can be simplified to:

For an overdamped system ( ζ > 1), solve for (a) K sys , (b) ω n , (c) τ sys , and (d) ζ using the transfer functions for a second-order response and an overdamped response.

()= n

n ( s ζω + ω (Second-order transfer function) n )

Hp () s =

 τ 1 s + 1   τ 2 s + 1  (Overdamped response)

S OLU T I ON

Multiplying the terms in the overdamped transfer function yields the equation:

Industrial Text & Video Company 1-800-752-8398

www.industrialtext.com

S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

12 s + [ ( τ 1 + τ 2 ) s + 1 ]

Dividing by the term τ 1 τ 2 generates the following equation, which has a denominator in the form of a second-order lag transfer function:

AA 1 2

Hp =

() s

s 2 ( ττ + + ) 1 ττ 2 12 s + 1

Therefore, this equation is equal to the second-order lag equation:

() s = 2 ( 2

s + 2 ζω n s + ω n )

because the denominator of the polynomial can be separated into two real factors, where τ 1 and τ 2 are the two time constants. Thus, the relationship of the terms in these two equations is:

6 K 7 sys 8

(a) Knowing that the term K OD is equal to the overdamped system gain A 1 A 2 , the term K sys for an overdamped system is:

(b) 2 For an overdamped system, ω

n is equal to:

Solving for ω n generates:

Industrial Text & Video Company 1-800-752-8398

www.industrialtext.com

S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

(c) Earlier, we explained that τ sys is equal to 1 over the frequency ω n . Using the information from part (b), τ sys for an overdamped system is:

(d) The damping coefficient terms for a second-order system with lag and an overdamped system relate as follows:

2 ζω ) n = (

Solving for ζ yields:

ζω n =

( )( ) ττ 12 2 ω n

An overdamped second-order transfer function in real time (the time domain) is described by the inverse Laplace transform (see Table 14-2):

This indicates two exponential decaying responses—one at a rate of τ 1 and the other at the rate of τ 2 . Figure 14-50 illustrates the form of these two exponential responses, along with the response of H (t) , which is a function of

a combination of these two responses. Note that, as indicated in the time

domain transfer function term ( e τ 1 τ − 2 e ), the curve of H (t) is equal to the curve

of the τ 1 response minus the curve of the τ 2 response.

Industrial Text & Video Company 1-800-752-8398

www.industrialtext.com

S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

e –t τ

1 Gain K

OD = A H

A –t

1 2 2.75 τ 1 – τ ( e – e 2 )

t Figure 14-50. Real-time transfer function of an overdamped second-order process.

Figure 14-51 illustrates a second-order system response to a step input with amplitude B. As shown in this figure, the output of the time domain transfer function in response to a step input is similar in form to the second-order response curve shown in Figure 14-50. The overdamped response may also follow the shape of a first-order system response curve if one of the time

( ( τ 1 s + 1)( τ 2 s + 1) (

Figure 14-51. Second-order response to a step input.

Industrial Text & Video Company 1-800-752-8398

www.industrialtext.com

S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

constants is significantly longer than the other (i.e., τ 1 >> τ 2 ). Figure 14-52 illustrates a case like this, where one of the exponential components ( τ 2 ) dies out much more rapidly than the other ( τ 1 ). Thus, the response to the unit step is heavily damped, causing a sluggish response similar to a first-order one. The system response is heavily damped because the value of ζ in this system becomes large (see the value of ζ in Example 14-4). Two first-order systems with different time lags, which are connected in series (or cascaded), will produce this type of overdamped second-order response (see Figure 14-53). In a cascaded system, the output of one part of the system depends on the input to another part of the system.

e –t τ

1 – e τ 2 ) Process Transfer Function [ Hp (t) ]

Second-Order τ System

1 >> τ 2 Out ( s ) H ( s )

τ 1 >> τ 2 t

Response to a unit step

Figure 14-52.

A heavily damped response.

O VERDAMPED R ESPONSES