# PT1 element

As **PT _{1} -element** a refers LZI - transmission member in the control engineering , which is a proportional having transmission characteristics with 1st order delay. A common example in electrical engineering is the low pass (1st order) .

## Transfer function

The transfer function of the PT1 element results from its differential equation :

The corresponding complex Laplace transfer function in the image area has the form:

Here, K, K> 0, the transmission constant or the gain factor and T, T> 0, the time constant . For T <0 the system would have an exponentially increasing behavior.

## Bode diagram

With the PT _{1} link is the frequency response . Therefore, the following applies to the amplitude and phase response in the Bode diagram :

### Amplitude response

Identifies the buckling or Eckkreisfrequenz , then the amplitude response can be roughly divided into two areas divided:

or logarithmized, in decibels :

For angular frequencies below the corner angular frequency, the amount characteristic of the PT _{1} element is parallel to the 0 dB line at a distance of K _{dB} and for high angular frequencies it falls at 20 dB / decade. The two asymptotes intersect at the buckling angular frequency ω = ω _{0} . The actual value of the amplitude response deviates there by −3 dB from the asymptotic approximation. With ω = 0.5 ω _{0} or ω = 2 ω _{0} , the deviation is only −1 dB.
_{}_{}

The corner angular frequency is calculated from the pole of the transfer function, i.e. the zero of the denominator 1 + Ts. The pole is and is called the eigenvalue , the magnitude of which describes the corner angular frequency ω _{0} .
_{}

### Phase response

The phase shift of the PT _{1} element is 0 ° for small angular frequencies, −90 ° for large angular frequencies and _{0} −45 ° for the bending angular frequency ω .

For the asymptotic approximation, a straight line is drawn that begins at 0 ° a decade before the arcuate frequency and ends at −90 ° a decade after the arcuate frequency.

## Step response

The step response of the PT _{1} element is described by

and has the course of an exponential function. The course approaches the final value K. After the time t = T the value is 0.63 K and after t = 3 T it is already 0.95 K, but theoretically there is always a minimal deviation from the final value. The tangent at the origin intersects the value of the gain factor K after the time T. The amount of the time constant T determines the speed of the link.

## Locus

The locus ( ) of the PT _{1} element runs from point K on the positive real axis through the fourth quadrant for point 0.
_{}

Complex conjugate expanding yields

so that the real and imaginary part can be explicitly represented:

This is how the amount and phase are calculated

The extreme values result as follows:

## Discrete-time PT1 element

The behavior of a PT1 element can be calculated in a time-discrete manner using the simplest integration method ( Euler explicitly ) . It is the basis for emulating this transmission link in digital signal processing .

From the above differential equation follows with the than the increment sampling the difference equation :

From this you get

With the initial value we get :

This leads to the corresponding z-transfer function of the PT1 system.