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: ${\displaystyle \omega _{0}={\frac {1}{T}}}$

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

or logarithmized, in decibels :

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

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.

See also

• Regulator
• P element
• I-link
• D link
• PT2 element
• PID controller
• Dead time element