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A second-order lag response exhibits oscillations that occur while the output signal is settling into its final steady-state value. This type of response is caused by a step change in the input or a disturbance in the process.

A second-order transfer function with lag is characterized by a second-order differential equation that is represented in Laplace form as:

Out 2 A ω

Hp n ()

In ( s + 2 ζω n s + ω n )

where: A = the gain

ω n = the resonant, or natural, frequency of oscillation in radians/second

ζ = the damping coefficient Figure 14-45 illustrates this second-order, oscillating response to a step

input. The frequency term ω n is the factor that determines how quickly the response oscillates above and below the desired outcome. The damping coefficient ζ is the factor that suppresses the oscillation over time, so that the response finally levels off at the desired outcome value. The complete

n represents the system gain (K sys ), which specifies the total amplitude of the response signal given its frequency.

numerator term A 2 ω

Damping – e ζω nt

Figure 14-45. Second-order response to a step input.

The amplitude of the oscillation of a second-order response dies off exponen- −ζω n tially due to the damping of the factor e t , which is part of the inverse

Laplace transform representation (time domain) of the system. If the damping −ζω n coefficient ( t ζ ) is equal to 0, then the term e will be 1 and the response will

oscillate indefinitely in a sinusoidal manner at a frequency of ω n, instead of leveling out. Thus, the damping coefficient determines the shape of the response (see Figure 14-46).

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S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

ω n t (in radians)

Figure 14-46. Damping coefficient effect on the oscillation of a second-order response.

Unlike a first-order system, a second-order system has two lag times ( τ 1 and τ 2 ), which are related to the frequency of oscillation ( ω n ). These two lag times combine to create a system second-order time constant τ sys , which is equal to:

sys =

As used in Laplace and time domain second-order response equations, the term ω n represents frequency. This frequency is expressed in radians per

second. However, this frequency can also be expressed in degrees. A second-order response is a sinusoidal response, meaning that it fluctuates above and below the final outcome (set point) value once every 2 π periods (see Figure 14-47). Therefore, the response period is characterized by the

equation ω n (see Figure 14-48). In degrees, this same period is expressed as

f n , where f n is the frequency in hertz. Therefore:

Figure 14-47. Sinusoidal response of a second-order system around the set point.

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S ECTION PLC Process Process Responses C HAPTER 4 Applications

and Transfer Functions 14

2 π or 1

Figure 14-48. Response period of a sinusoidal curve.

Solving for ω n yields: ω n = 2 π f n

So, the radian/sec frequency term ω n is equivalent to the degree frequency

term 2 π f n .

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