# PT1 element

PT 1 link in the structure diagram

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 :

${\ displaystyle T \ cdot {\ dot {y}} (t) + y (t) = K \ cdot u (t)}$

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

${\ displaystyle G (s) = {\ frac {K} {1 + T \ cdot s}}}$

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

Bode diagram of a PT 1 link (K = 2, T = 1)

With the PT 1 link is the frequency response . Therefore, the following applies to the amplitude and phase response in the Bode diagram : ${\ displaystyle G (j \ omega) = {\ frac {K} {1 + j \ omega T}}}$

${\ displaystyle | G (j \ omega) | = {\ frac {K} {\ sqrt {1+ \ omega ^ {2} T ^ {2}}}}}$
${\ displaystyle \ varphi (\ omega) = - \ arctan (\ omega T)}$

### Amplitude response

Identifies the buckling or Eckkreisfrequenz , then the amplitude response can be roughly divided into two areas divided: ${\ displaystyle \ omega _ {0} = {\ frac {1} {T}}}$

${\ displaystyle G (j \ omega) = {\ begin {cases} K, & {\ mbox {if}} \ omega \ ll \ omega _ {0} \\ {\ frac {K} {\ frac {\ omega } {\ omega _ {0}}}}, & {\ mbox {if}} \ omega \ gg \ omega _ {0} \ end {cases}}}$

or logarithmized, in decibels :

${\ displaystyle | G | _ {\ mathrm {dB}} = {\ begin {cases} 20 \ log K, & {\ mbox {if}} \ omega \ ll \ omega _ {0} \\ 20 \ log K -20 \ log {\ frac {\ omega} {\ omega _ {0}}}, & {\ mbox {if}} \ omega \ gg \ omega _ {0} \ end {cases}}}$

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 . ${\ displaystyle - {\ frac {1} {T}}}$

### 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

Step response of a PT 1 element (K = 2, T = 1)

The step response of the PT 1 element is described by

${\ displaystyle a (t) = K (1- \ mathrm {e} ^ {- {\ frac {t} {T}}})}$

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

Locus of a PT 1 link (T = 1, K = 2)

The locus ( ) of the PT 1 element runs from point K on the positive real axis through the fourth quadrant for point 0. ${\ displaystyle 0 \ leq \ omega \ leq \ infty}$${\ displaystyle \ omega \ to \ infty}$

${\ displaystyle F (\ mathrm {j} \ omega) = {\ frac {K} {1+ \ mathrm {j} \ omega T_ {1}}}}$

Complex conjugate expanding yields

${\ displaystyle F (\ mathrm {j} \ omega) = {\ frac {K} {1+ \ mathrm {j} \ omega T_ {1}}} \ cdot {\ frac {1- \ mathrm {j} \ omega T_ {1}} {1- \ mathrm {j} \ omega T_ {1}}} = {\ frac {K- \ mathrm {j} \ omega T_ {1} K} {1+ \ omega ^ {2 } T_ {1} ^ {2}}}}$

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

${\ displaystyle \ mathrm {Re} \ left \ {F (\ mathrm {j} \ omega) \ right \} = {\ frac {K} {1+ \ omega ^ {2} T_ {1} ^ {2} }}}$
${\ displaystyle \ mathrm {Im} \ left \ {F (\ mathrm {j} \ omega) \ right \} = {\ frac {- \ omega T_ {1} K} {1+ \ omega ^ {2} T_ {1} ^ {2}}}}$

This is how the amount and phase are calculated

${\ displaystyle \ left | F \ left (\ mathrm {j} \ omega \ right) \ right | = {\ frac {K} {\ sqrt {1+ \ omega ^ {2} T_ {1} ^ {2} }}}}$
${\ displaystyle \ varphi (\ omega) = \ varphi \ left (F (\ mathrm {j} \ omega) \ right) = \ arctan (- \ omega T_ {1}) = - \ arctan (\ omega T_ {1 })}$

The extreme values ​​result as follows:

${\ displaystyle \ mathrm {Re} \ left \ {F (\ mathrm {j} \ omega \ to 0) \ right \} = K}$
${\ displaystyle \ mathrm {Im} \ left \ {F (\ mathrm {j} \ omega \ to 0) \ right \} = 0}$
${\ displaystyle \ mathrm {Re} \ left \ {F (\ mathrm {j} \ omega \ to \ infty) \ right \} = 0}$
${\ displaystyle \ mathrm {Im} \ left \ {F (\ mathrm {j} \ omega \ to \ infty) \ right \} = 0}$
${\ displaystyle | F (\ mathrm {j} \ omega \ to 0) | = K}$
${\ displaystyle \ varphi (\ mathrm {j} \ omega \ to 0) = 0 ^ {\ circ}}$
${\ displaystyle | F (\ mathrm {j} \ omega \ to \ infty) | = 0}$
${\ displaystyle \ varphi (\ mathrm {j} \ omega \ to \ infty) = - 90 ^ {\ circ}}$

## 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 : ${\ displaystyle \ Delta t}$

${\ displaystyle T \ cdot {\ frac {y_ {n} -y_ {n-1}} {\ Delta t}} + y_ {n} = Ku_ {n}}$

From this you get

${\ displaystyle {\ frac {T + \ Delta t} {\ Delta t}} \ cdot y_ {n} = {\ frac {T} {\ Delta t}} \ cdot y_ {n-1} + Ku_ {n} }$

With the initial value we get : ${\ displaystyle y_ {0}}$${\ displaystyle y_ {n}}$

${\ displaystyle y_ {n} = y_ {n-1} + {\ frac {\ Delta t} {T + \ Delta t}} \ cdot (Ku_ {n} -y_ {n-1}), \ quad n = (1,2, \ dotsc, n _ {\ text {max}})}$

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