**Nikola Šerman,**prof. emeritus

*Linear Theory*nserman@fsb.hr

*Transfer Function Concept Digest*

**10. ELEMENTARY BLOCKS**

#### 10.3 First Order Lag

When the pole of the transfer function (10.2) is a real negative number *p = – ω_{0}* and constant

*K*equals

*ω*, the transfer function becomes:

_{0}In control engineering, this elementary block is named *first-order lag* or PT1. Its only parameter is referred to as *time constant *denoted by * T , *where . It is presented in fig. 10.9 in that form.

**Fig 10.9** Block presentation of the first-order lag

First-order lag as an autonomous component is most often made by closing a negative unity feed-back around an integrator, as is shown in fig. 10.10.

**Fig 10.10** First-order lag internal structure

Relationship between the output *y(t)* and input *x(t)* in the time domain is defined by the equation:

First-order lag step response is shown in fig. 10.11.

**Fig 10.11** First order lag step response to unity step at the input (time constant *T = 1s*).

The step response provides an information on a shape of the monotonic summands in the complex system’s transients – chapter 5. The shape is defined by time constant *T *influence on the respective exponential function *e ^{σ0t}* with

*σ*, (for all the converging parts of the system transients). A time period between point A and point B is always equal

_{0}< 0*T*. Point A is any point where the tangent touches the curve. Point B is the point where the same tangent intersects the horizontal asymptote. Point A can be located anywhere at the curve itself.

Frequency response of a first-order lag in Nyquist diagram is presented in fig. 10.12. It is always a halve-circle with the unity diameter regardless of the time constant *T * value.

**Fig 10.12** First-order lag frequency response in Nyquist diagram.

A first-order lag frequency response in Bode diagram is shown in fig. 10.13.

In a low frequency range, where , a first-order lag resembles the properties of static gain (*α _{dB}* = 0,

*φ*= 0°).

In a high-frequency range, where , the characteristics coincide with their asymptotes that are resembling the properties of an integrator. The slope of the amplitude-frequency characteristic asymptote equals -20dB/decade, and the phase shift amounts -90° in the whole high-frequency range. Any variation in *T* would move the characteristics in the horizontal direction. Their shape always remains the same.

**Fig 10.13** Frequency response of a first-order lag in Bode diagram for *T* = 1. Red straight lines denote high-frequency asymptotes.