# Time constant

The time constant (Greek (tau) or ) is a characteristic quantity of a linear dynamic system , which is described by an ordinary differential equation or by an associated transfer function. It has the dimension of a time; its unit of measurement is usually the second. ${\ displaystyle \ tau}$${\ displaystyle T}$ ${\ displaystyle G (s)}$

Functional diagram of a PT 1 element
after an input jump .

A dynamic system is a functional unit for processing and transmitting signals; the system input variable is defined as the cause and the system output variable as the temporal effect. Typical input signals for testing the system behavior are the pulse function , step function and increase function . ${\ displaystyle u (t)}$${\ displaystyle y (t)}$

In electrical engineering, the time behavior of a first-order delay element (e.g. an RC element - low pass ) with a step response with an exponential asymptotic course is generally known. The time constant determines the course over time. After a period of approx. 3 time constants has elapsed, the output signal has reached approx. 95% of the size of the input signal when the system gain is up. ${\ displaystyle T = R \ cdot C}$${\ displaystyle K = 1}$

In principle, the time course of the output signal of a transmission system of any order depends on the type of transmission system and the input signal and does not only refer to time delay elements ( elements). ${\ displaystyle PT_ {1}}$

The term time constant results from the description of a linear dynamic system by means of an ordinary differential equation with constant coefficients . To make it easier to calculate the time-dependent system behavior, the differential equation describing the system is subjected to the Laplace transformation and from this the signal ratio is formed as a transfer function . ${\ displaystyle G (s)}$

The transfer function in the time constant representation arises as follows:

• Laplace transform of the ordinary higher order differential equation,
• Formation of the transfer function .${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {\ text {numerator polynomial (s)}} {\ text {denominator polynomial (s)}}}}$
The poles and zeros of the transfer function are the most important parameters of the system behavior.${\ displaystyle s_ {pi}}$${\ displaystyle s_ {ni}}$
• Factoring the polynomials into the pole zero representation: ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = k \ cdot {\ frac {(s-s_ {n1}) (s-s_ {n2}) \ dotsm (s-s_ {nm})} {(s-s_ {p1}) (s-s_ {p2}) \ dotsm (s-s_ {pn})}}}$
• Conversion of the pole and zero point representation through numerical values ​​of the poles and zero points into the time constant representation,
• The values ​​of the poles and zeros of a linear factor can take three forms: zero, negative real, negative conjugate complex.
This means that in the numerator and denominator of the transfer function, different basic forms of linear factors and 2nd order factors with different system behavior can arise.${\ displaystyle 3 \ cdot 2 = 6}$
${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = k \ cdot {\ frac {T_ {v} \ cdot s \ cdot (T \ cdot s + 1) \ cdot (T ^ {2} \ cdot s ^ {2} + 2DT \ cdot s + 1)} {T_ {n} \ cdot s \ cdot (T \ cdot s + 1) \ cdot (T ^ {2} \ cdot s ^ {2} + 2DT \ cdot s + 1)}}}$.

The time constant corresponds to the coefficient before the complex Laplace variable . They are generally calculated from the reciprocal of a negative real pole or a zero of the denominator polynomial or numerator polynomial of the transfer function as: ${\ displaystyle T}$ ${\ displaystyle s}$${\ displaystyle s_ {p}}$${\ displaystyle s_ {n}}$

${\ displaystyle T_ {i} = {\ frac {1} {| s_ {pi} |}}}$or .${\ displaystyle T_ {i} = {\ frac {1} {| s_ {ni} |}}}$

## Determination of the time constants from the polynomials of a linear dynamic transmission system of higher order

System descriptions through transfer functions can arise from: ${\ displaystyle G (s)}$

For the sake of simpler calculation and easier understanding, the ordinary differential equation describing the system is subjected to a Laplace transformation and can therefore be calculated algebraically . According to the Laplace differentiation theorem, a first order derivative of the differential equation is replaced by the Laplace variable as a complex frequency . Higher derivatives nth order are in accordance with the atomic number by replacing. ${\ displaystyle s}$${\ displaystyle n}$${\ displaystyle s ^ {n}}$

Example of an ordinary higher order differential equation of a transmission system with constant coefficients and : ${\ displaystyle a_ {i}}$${\ displaystyle b_ {i}}$

${\ displaystyle a_ {n} y ^ {(n)} + \ ldots + a_ {2} {\ ddot {y}} + a_ {1} {\ dot {y}} + a_ {0} y = b_ { m} u ^ {(m)} + \ ldots + b_ {2} {\ ddot {u}} + b_ {1} {\ dot {u}} + b_ {0} u}$

This general form of the differential equation is subjected to a Laplace transform:

${\ displaystyle {a_ {n} \ cdot s ^ {(n)} \ cdot Y (s) + \ ldots + a_ {2} \ cdot s ^ {2} \ cdot Y (s) + a_ {1} \ cdot s \ cdot Y (s) + a_ {0} \ cdot Y (s) = b_ {m} \ cdot s ^ {(m)} \ cdot U (s) + \ ldots + b_ {2} \ cdot s ^ {2} \ cdot U (s) + b_ {1} \ cdot U (s) + b_ {0} \ cdot U (s)}}$.

The transfer function is obtained by applying the Laplace differentiation theorem to the system-describing ordinary differential equation . By determining the pole and zero positions of the numerator and denominator polynomial, the factorial representation (linear factors) of the transfer function is created. ${\ displaystyle G (s)}$

The transfer function G (s) is formed from the ratio of the output variable to the input variable. There must be no initial values ​​of the internal energy storage ( state space representation ) of the system.

${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {b_ {m} \ cdot s ^ {m} + \ dotsb + b_ {2} \ cdot s ^ {2} + b_ {1} \ cdot s + b_ {0}} {a_ {n} \ cdot s ^ {n} + \ dotsb + a_ {2} \ cdot s ^ {2} + a_ {1 } \ cdot s + a_ {0}}} = {\ frac {\ text {numerator polynomial (s)}} {\ text {denominator polynomial (s)}}}}$

The Laplace variable is an independent variable in the complex frequency domain (image area, s-area) with a real part and an imaginary part. It allows any algebraic operations in the s-domain, but is only a symbol for a completed Laplace transformation and does not contain a numerical value. Exponents of s correspond to the degree of derivative of the differentials . ${\ displaystyle s = \ delta + j \ omega}$${\ displaystyle \ delta}$${\ displaystyle j \ omega}$

To determine the elementary individual systems of a transfer function G (s) of higher order, the polynomials of the numerator and denominator are factored by determining the zeros. ${\ displaystyle G_ {1} (s); G_ {2} (s); \ dots}$

If numerical values ​​of the coefficients are available, the poles and zeros can be calculated using various methods. The so-called pq formula for systems of the 2nd order is suitable for this . Ready-made programs available on the Internet for systems up to 4th order can be used with the call: "Determine zeros (solutions) of polynomials". ${\ displaystyle x ^ {2} + px + q = 0}$

The poles (zeros of the denominator) and zeros (zeros of the numerator) of the transfer function are the most important parameters of the system behavior. They are either zero (missing end term of the differential equation), real [ ]; and [ ] or conjugate complex [ ] and [ ]. ${\ displaystyle s_ {pi}}$${\ displaystyle s_ {ni}}$${\ displaystyle a_ {0}, \ b_ {0}}$${\ displaystyle s_ {n} = - \ delta}$${\ displaystyle s_ {p} = - \ delta}$${\ displaystyle s_ {n} = - \ delta \ pm j \ omega}$${\ displaystyle s_ {p} = - \ delta \ pm j \ omega}$

To determine the time constants, the polynomials of the transfer function are broken down into linear factors and 2nd order factors by determining the zeros. If numerical values ​​are given for the coefficients of the polynomials, the polynomials can be factored by determining the roots.

The decomposition of the higher order numerator and denominator polynomials by the poles and zeros results in multiple linear factors and multiple 2nd order factors. As a prerequisite for this, the polynomials must not have any gaps in the order of the sum elements according to the ordinal number.

If these factors are defined as independent individual transfer functions, the following elementary transfer functions arise depending on the type of poles and zeros:

• The final terms of the differential equation are zero: The resulting linear factor is a variable: both in the numerator and in the denominator.${\ displaystyle a_ {0}; b_ {0}}$${\ displaystyle K \ cdot s}$
• The poles or the zeros are negative real. From [ ] or [ ] the linear factor arises in time constant representation [ ] both in the numerator and in the denominator.${\ displaystyle s-s_ {n}}$${\ displaystyle s-s_ {p}}$${\ displaystyle T \ cdot s + 1}$
• The poles or the zeros are negatively conjugate complex. From [ ] or [ ] the factor arises in time constant representation [ ] 2nd order both in the numerator and in the denominator.${\ displaystyle s-s_ {n}}$${\ displaystyle s-s_ {p}}$${\ displaystyle T ^ {2} \ times s ^ {2} + 2DT \ times s + 1}$

Example of a transfer function with the polynomial representation, the pole-zero representation and the time constant representation:

{\ displaystyle {\ begin {aligned} G (s) & = {\ frac {Y (s)} {U (s)}} = {\ frac {b_ {m} s ^ {m} + \ ldots + b_ {2} s ^ {2} + b_ {1} s + b_ {0}} {a_ {n} s ^ {n} + \ ldots + a_ {2} s ^ {2} + a_ {1} s + a_ {0}}}: = k \ cdot {\ frac {(s-s_ {n1}) (s-s_ {n2}) \ dotsm (s-s_ {nm})} {(s-s_ {p1} ) (s-s_ {p2}) \ dotsm (s-s_ {pn})}}: = \\ &: = k \ cdot {\ frac {T_ {v} \ cdot s \ cdot (T \ cdot s + 1) \ cdot (T ^ {2} \ cdot s ^ {2} + 2DT \ cdot s + 1)} {T_ {n} \ cdot s \ cdot (T \ cdot s + 1) \ cdot (T ^ { 2} \ cdot s ^ {2} + 2DT \ cdot s + 1)}} \ end {aligned}}}.

### Time constants of the elementary individual systems in the numerator and denominator of the transfer function

The decomposition of the denominator polynomial results in time-delaying individual systems (linear factors) and delaying factors of the 2nd order. The decomposition of the numerator polynomial results in differentiating individual systems (linear factors) and differentiating factors of the 2nd order. In combination with the time-delaying systems of the denominator, the latter have no influence on the time behavior, but only on the signal amplitudes . ${\ displaystyle y (t)}$

• Time constants of the linear factors as variables with poles and zeros equal to zero:${\ displaystyle T \ cdot s}$
These linear factors arise from a Laplace transformation of a system-describing ordinary differential equation, the end terms of which or are missing.${\ displaystyle a_ {0}}$${\ displaystyle b_ {0}}$
The product term becomes the numerator and denominator, respectively . The time constants for the I element and for the D element shown in the table below in the next section are taken from the definition of the controller . In reality, they correspond to proportionality factors or with a rating of 1, if no other numerical values ​​have been given.${\ displaystyle (s-0)}$${\ displaystyle s}$${\ displaystyle {\ tfrac {1} {T_ {n}}}}$${\ displaystyle \ left ({\ tfrac {1} {T_ {n} \ cdot s}} \ right)}$${\ displaystyle T_ {v}}$${\ displaystyle (T_ {v} \ cdot s)}$${\ displaystyle K_ {I}}$${\ displaystyle K_ {D}}$
• Time constant of the linear factors with poles and zeros equal  :${\ displaystyle - \ delta}$
The definition of the time constant of an element or element is calculated as follows from the poles and zeros for numerical values ​​with negative real parts of .${\ displaystyle T}$${\ displaystyle PT_ {1}}$${\ displaystyle PD_ {1}}$${\ displaystyle s_ {pi}, s_ {ni}}$
Example for the definition of a time constant from the linear factor of the counter:
${\ displaystyle \ underbrace {(s-s_ {n})} _ {\ text {linear factor}} \: = \ underbrace {(s + | s_ {n} |)} _ {{\ text {term with}} s_ {n} = {\ text {negat.}}} = \ \ underbrace {| s_ {n} | \ cdot \ left ({\ frac {1} {| s_ {n} |}} \ cdot s + 1 \ right)} _ {{\ text {Term through}} | s_ {n} | {\ text {divid.}}} \ quad: = \ underbrace {{\ frac {1} {T}} \ cdot (T \ cdot s + 1)} _ {\ text {Time constant representation}}}$.
The time constant is generally calculated from the reciprocal value (reciprocal value) of a negative real zero or the denominator polynomial or numerator polynomial of the transfer function as: ${\ displaystyle s_ {p}}$${\ displaystyle s_ {n}}$
${\ displaystyle T_ {i} = {\ frac {1} {| s_ {pi} |}}}$or .${\ displaystyle T_ {i} = {\ frac {1} {| s_ {ni} |}}}$
• Time constants of the second order factor with complex conjugate poles and zeros:
Second order factors arise from the pole-zero representation with negative conjugate complex poles and zeros. If [ ] or [ ] is substituted for the complex conjugate zero from the zero position , the time constant representation is created by squaring to avoid imaginary quantities:${\ displaystyle s-s_ {n}}$${\ displaystyle s_ {n}}$${\ displaystyle s_ {n} = - \ delta \ pm j \ omega}$${\ displaystyle s_ {p} = - \ delta \ pm j \ omega}$
2nd order factor in pole zero representation after squaring:
${\ displaystyle [s ^ {2} -2 \ cdot \ delta \ cdot s + \ delta ^ {2} + \ omega ^ {2}]}$
The normal form of the time constant representation of the second order factor results with:${\ displaystyle T = {\ frac {1} {\ omega}}}$
${\ displaystyle [T ^ {2} \ cdot s ^ {2} + 2DT \ cdot s + 1] \ qquad D = {\ text {degree of damping}} \ 0
Conclusion:
• This second order factor applies to both the numerator and denominator polynomials and cannot be broken down into smaller mathematical expressions.
• The time course of a normalized step response of a 2nd order transmission system with complex conjugate poles ( -member) depends on the time constant and on the damping .${\ displaystyle PT2_ {kk}}$${\ displaystyle T}$${\ displaystyle D}$

### Establishment and behavior of the elementary individual systems

By assigning these factors in the numerator and denominator of the transfer functions, the following 6 different stable elementary systems can arise individually or multiple times: ${\ displaystyle G_ {i} (s)}$

${\ displaystyle G_ {1} (s) = (T \ cdot s) ^ {\ pm 1}; \ qquad G_ {2} (s) = (T \ cdot s + 1) ^ {\ pm 1}; \ qquad G_ {3} (s) = \ left (T ^ {2} \ cdot s ^ {2} +2 \ cdot D \ cdot T \ cdot s + 1 \ right) ^ {\ pm 1} \ quad}$
designation I-link D link ${\ displaystyle PT_ {1}}$-Element ${\ displaystyle PD_ {1}}$-Element ${\ displaystyle PT_ {2kk}}$-Link (vibrating link) ${\ displaystyle PD_ {2kk}}$-Element
Pole zeros ${\ displaystyle s_ {p} = 0}$ ${\ displaystyle s_ {n} = 0}$ ${\ displaystyle s_ {p} = - \ delta}$ ${\ displaystyle s_ {n} = - \ delta}$ ${\ displaystyle s_ {p1 / 2} = - \ delta \ pm j \ omega}$ ${\ displaystyle s_ {n1 / 2} = - \ delta \ pm j \ omega}$
Transfer function ${\ displaystyle {\ frac {Y} {U}} (s) = {\ frac {1} {T_ {n} \ cdot s}}}$ ${\ displaystyle {\ frac {Y} {U}} (s) = T_ {V} \ cdot s}$ ${\ displaystyle {\ frac {Y} {U}} (s) = {\ frac {K} {T \ cdot s + 1}}}$ ${\ displaystyle {\ frac {Y} {U}} (s) = T \ cdot s + 1}$ ${\ displaystyle {\ frac {Y} {U}} (s) = {\ frac {K} {T ^ {2} \ times s ^ {2} + 2TD \ times s + 1}}}$ ${\ displaystyle {\ frac {Y} {U}} (s) = K ({T ^ {2} \ cdot s ^ {2} + 2TD \ cdot s + 1})}$

In the time constant representation, the time constant corresponds to the coefficient in front of the complex Laplace variable . ${\ displaystyle T}$${\ displaystyle s}$

The calculation of the temporal behavior of an overall transmission system always requires that the number of factors in the denominator must always be equal to or greater than the number of factors in the numerator . ${\ displaystyle n \ geq m}$

Differentiating elements can fully compensate the time behavior of decelerating elements with the same time constants. The same goes for links and links, of course. ${\ displaystyle PD_ {1}}$${\ displaystyle PT_ {1}}$${\ displaystyle PD_ {2}}$${\ displaystyle PT_ {2}}$

Step response of various elementary transmission systems.

Test signals to check the system behavior:

Common test signals for transmission systems are: step function, return step, pulse function, rise function and sine function . These signals are also Laplace-transformed from the time domain into the image domain. See definition of test signals in the next section.

Time behavior of differentiating transmission elements:

The time behavior of the step response or the impulse response of a differentiating system of the numerator polynomial cannot be represented graphically alone because the change in the output signal takes place in the time domain . The time behavior of a differentiating system can only be represented graphically with an input signal as a rise function. Differentiating systems without so-called time-delaying parasitic elements cannot technically be produced as hardware. The required parasitic time constant of the time-delaying element must be much smaller than the time constant of the element or element. ${\ displaystyle 0 \ to 0 _ {+}}$${\ displaystyle PT_ {1}}$${\ displaystyle PT_ {1}}$${\ displaystyle D}$${\ displaystyle PD_ {1}}$

Time behavior of transmission elements with complex conjugate poles ( -member):${\ displaystyle PT_ {2kk}}$

These 2nd order transfer elements contain double poles in the s range . Depending on the amount of damping , when the system is excited by any input signal, a damped, oscillating output variable arises. The step response is often represented as a characteristic behavior in which the output variable exponentially asymptotically reaches a final value with a vibration superposition. ${\ displaystyle D}$

At , these systems can be broken down into two parts. ${\ displaystyle D \ geq 1}$${\ displaystyle PT_ {1}}$

The time constants associated with the system are in square form. ${\ displaystyle T}$

Transmission systems with linear factors or 2nd order factors with positive real part of the poles:

Positive real parts of the poles and zeros result in negative time constants.

Transfer elements with positive poles form unstable non-linear transfer functions, which can be achieved with z. B. can be called "unstable T1 members" or "unstable T2 members". Time constants can also be assigned to them. The output signal of these systems increases exponentially progressively after any positive input signal up to a limit and only returns when the input signal becomes negative ( feedback effect ). (For more details see controlled system # Characterization of the controlled systems ) ${\ displaystyle u (t)> 0}$

Calculation of the time behavior of transfer functions

Inverse Laplace transformation: The system output behavior of any transmission system in the time domain depends on the transfer function and the type of input signal . Using the inverse Laplace transformation, the time behavior can be found using Laplace transformation tables and the search term:${\ displaystyle y (t)}$${\ displaystyle G (s)}$${\ displaystyle U (s)}$
 ${\ displaystyle y (t) = {\ mathcal {L}} ^ {- 1} \ underbrace {\ left \ {G (s) \ cdot U (s) \ right \}} _ {\ text {search term}} }$
If the input signal is a standardized step function , then is .${\ displaystyle u (t) = 1}$${\ displaystyle U (s) = {\ frac {1} {s}}}$
Numerical calculation: With the help of numerical mathematics by calculating difference equations , the output signals can be calculated as numbered sequential equations of a dynamic system for given input signals . The time constants in the difference equations determine the behavior of the individual systems.${\ displaystyle u_ {ki}}$${\ displaystyle y_ {ki}}$${\ displaystyle T}$
Difference equations, approximating a continuous function, gradually calculate a sequence of values with the following elements for a small interval, the sequence of values at that point , whereby the calculated values ​​are numbered .${\ displaystyle y = f (x)}$${\ displaystyle y _ {(ki)}}$${\ displaystyle k = [0,1,2,3, \ dots]}$${\ displaystyle h = \ Delta x}$${\ displaystyle y _ {(k)} = [y _ {(0)}, y _ {(1)}, y _ {(2)}, y _ {(3)} \ dots]}$${\ displaystyle x _ {(k)} = [x _ {(0)}, x _ {(1)}, x _ {(2)}, x _ {(3)} \ dots]}$${\ displaystyle k}$${\ displaystyle y _ {(ki)}}$

### Test signals

Step responses from 4 decoupled PT1 elements connected in series, each with the same time constants.

The non-periodic (deterministic) test signals are of central importance in systems theory. With their help, it is possible to test a transmission system, check its stability or determine properties.

To calculate the time response of a transmission system, the transformed test signals in the image area can be multiplied by the system's transfer function instead . For the reverse transformation from into the time domain, the desired equation of the system response can be found with the help of the Laplace transformation tables. ${\ displaystyle U (s)}$${\ displaystyle G (s)}$${\ displaystyle Y (s)}$${\ displaystyle y (t)}$

Impulse responses from 4 decoupled PT1 elements connected in series, each with the same time constants.

The test signals have in common that they begin at the point in time and have an amplitude = 0 at. To distinguish the function of the signals, they are indicated with the characters δ ( pulse ), Ϭ ( jump ), a (rise) and s ( sine ). ${\ displaystyle t = 0}$${\ displaystyle t <0}$

The test signals are represented as an input variable and as a Laplace-transformed variable as follows. ${\ displaystyle u (t)}$${\ displaystyle U (s)}$

Term test signal
u (t)

Input signal image area
System response
y (t)

Impulse function δ or impact function, delta impulse
${\ displaystyle U _ {\ delta} (s) = 1 \,}$ Impulse response or
weight function
Step function σ
${\ displaystyle U _ {\ sigma} (s) = {\ frac {1} {s}}}$
Step response or
transition function
Increase function or
ramp
${\ displaystyle U_ {a} (s) = {\ frac {1} {s ^ {2}}}}$ Rise
response or ramp response
Sine function s
(periodic signal)
${\ displaystyle U_ {s} (s) = {\ frac {\ omega} {s ^ {2} + \ omega ^ {2}}}}$
Frequency response

## Basics of determining the time constants from the ordinary differential equation of the first order

A linear ordinary differential equation of 1st order with constant coefficients and reads: ${\ displaystyle a_ {i}}$${\ displaystyle b_ {i}}$

 ${\ displaystyle a_ {1} \ cdot {\ dot {y}} (t) + a_ {0} \ cdot y (t) = b_ {0} \ cdot u (t)}$.

The time constant can already be calculated from this form of the differential equation. ${\ displaystyle T = a_ {1} / a_ {0}}$

In general, the highest derivative of a differential equation is released for the determination of the zeros, in which all terms of the equation are divided by the associated coefficient, in this case . The new mathematically identical differential equation thus reads: ${\ displaystyle a_ {1}}$

${\ displaystyle {\ dot {y}} (t) + {\ frac {a_ {0}} {a_ {1}}} \ cdot y (t) = {\ frac {b_ {0}} {a_ {1 }}} \ cdot u (t)}$.

The transfer function G (s) of this differential equation reads for initial conditions equal to zero after applying the Laplace differentiation theorem:

${\ displaystyle s \ cdot Y (s) + {\ frac {a_ {0}} {a_ {1}}} \ cdot Y (s) = {\ frac {b_ {0}} {a_ {1}}} \ cdot U (s)}$.

The transfer function in the time constant representation results from the ratio of the output variable to the input variable : ${\ displaystyle Y (s)}$${\ displaystyle U (s)}$

${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {\ frac {b_ {0}} {a_ {1}}} {s + {\ frac { a_ {0}} {a_ {1}}}}} = {\ frac {b_ {0}} {a_ {1} \ cdot s + a_ {0}}} = {\ frac {\ frac {b_ {0 }} {a_ {0}}} {{\ frac {a_ {1}} {a_ {0}}} \ cdot s + 1}}}$.

With this form of the transfer function , the time constant can be read off directly as a coefficient in front of the Laplace variable with the ratio of the coefficients . ${\ displaystyle G (s)}$${\ displaystyle T}$${\ displaystyle s}$${\ displaystyle T = a_ {1} / a_ {0}}$

If you insert for and into the equation of the transfer function, you get the normal form of the transfer function of a delay element ( element) in the time constant representation: ${\ displaystyle {\ frac {a_ {1}} {a_ {0}}} = T}$${\ displaystyle {\ frac {b_ {0}} {a_ {0}}} = K}$${\ displaystyle PT_ {1}}$

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

### Creation of an ordinary first order differential equation from a hardware low pass

A delay element ( PT1 element ) described by an ordinary differential equation of the first order occurs most frequently in nature and in technology. It arises z. B. when heat flows into a medium or an electrical voltage is applied to an RC element . It is always of interest how the output variable of the system behaves as a function of time for a given input variable. The system behavior for a given input variable as a step function is particularly clear.

The most well-known dynamic system in electrical engineering, which is described by an ordinary differential equation of the first order, is the RC element as a resistor-capacitor circuit with the time constant . ${\ displaystyle T = R \ cdot C}$

Simple RC low-pass with
U e : input voltage
U a : output voltage

The general mathematical description of the RC element results from the application of Kirchhoff's laws .

For the hardware model as a low pass, the mesh equation of the voltages applies:

${\ displaystyle -U_ {e} + U_ {R} + U_ {C} = 0}$.

It is the input variable, the desired output size. If the equation for the charging current is inserted into the above equation for the voltage drop at R , the differential equation of the RC element results as a low pass: ${\ displaystyle U_ {e}}$${\ displaystyle U_ {C} = U_ {a}}$${\ displaystyle U_ {R}}$${\ displaystyle i = C \ cdot {\ frac {dU_ {C}} {dt}}}$

 ${\ displaystyle R \ cdot C \ cdot {\ frac {dU_ {C}} {dt}} + U_ {C} = U_ {e}}$

If the usual signal designations of systems theory are applied, the new signal designations are the ordinary differential equation: and . ${\ displaystyle U_ {e}: = u (t)}$${\ displaystyle U_ {a}: = y (t)}$

There is no numerator and denominator polynomial for a first order differential equation and the associated transfer function . It is already a linear factor in the denominator of the transfer function. Therefore the zero has no meaning. ${\ displaystyle G (s)}$

In the usual representation of the differential equation, the highest derivative of coefficients is left free by dividing all terms of the equation by the associated coefficient (here ). The new mathematically identical differential equation thus reads: ${\ displaystyle R \ cdot C}$

${\ displaystyle {\ dot {y}} (t) + {\ frac {1} {R \ cdot C}} \ cdot y (t) = {\ frac {1} {R \ cdot C}} \ cdot u (t)}$

The transfer function G (s) of this differential equation reads for initial conditions equal to zero according to the differentiation theorem:

${\ displaystyle s \ cdot Y (s) + {\ frac {1} {R \ cdot C}} \ cdot Y (s) = {\ frac {1} {R \ cdot C}} \ cdot U (s) }$.

Summarized as the ratio of the output variables to the input variable, the transfer function results in time constant representation:

 ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {\ frac {1} {R \ cdot C}} {s + {\ frac {1} { R \ cdot C}}}} = {\ frac {1} {R \ cdot C \ cdot s + 1}} = {\ frac {1} {T \ cdot s + 1}}}$.

The coefficient in front of the Laplace variable corresponds to the time constant . ${\ displaystyle s}$${\ displaystyle T = R \ cdot C}$

### Creation of the transfer function for a low-pass filter ( element) through the ratio of complex resistances${\ displaystyle G (p)}$${\ displaystyle PT_ {1}}$

In contrast to the transfer function , the frequency response can be measured with a linear transfer system. The frequency response is a special case of the transfer function. The transfer function can be converted into the frequency response with identical coefficients (time constants) at any time. The genesis of the frequency response and the transfer function are different, the spellings can remain identical. ${\ displaystyle G (s)}$ ${\ displaystyle G (p)}$${\ displaystyle p = j \ omega}$

${\ displaystyle G (p) = {\ frac {Y (p)} {U (p)}}}$

In the RC circuit shown, the ratio of the output voltage to the input voltage can also be defined as the ratio of the output impedance to the input impedance. If one sets the capacitance with the complex resistance, the following results for the complex resistance ratio as a transfer function : ${\ displaystyle p = j \ cdot \ omega}$${\ displaystyle {\ mathcal {Z}} _ {C} = {\ frac {1} {p \ cdot C}}}$${\ displaystyle G (p)}$

${\ displaystyle G (p) = {\ frac {U_ {a} (p)} {U_ {e} (p)}} = {\ frac {\ frac {1} {p \ cdot C}} {R + { \ frac {1} {p \ cdot C}}}} = {\ frac {1} {R \ cdot C \ cdot p + 1}} = {\ frac {1} {T \ cdot p + 1}}}$

The result corresponds to the term derived from the differential equation . ${\ displaystyle PT_ {1}}$

### Creation of the transfer function for a high pass through the ratio of the complex resistances${\ displaystyle G (p)}$

If the capacitance C is replaced by an inductance L in the RC circuit, a first-order high-pass filter is created when considering the input and output voltages of the system. With high frequency input signals, the inductance has a high complex resistance. As the frequency falls, the inductive resistance drops.

In the LC circuit shown, the ratio of the output voltage to the input voltage can also be defined as the ratio of the output impedance to the input impedance. If one sets the inductance with the complex resistance, the following results for the complex resistance ratio as a transfer function : ${\ displaystyle U_ {a} (p)}$${\ displaystyle U_ {e} (p)}$${\ displaystyle p = j \ cdot \ omega}$${\ displaystyle {\ mathcal {Z}} _ {L} = L \ cdot p}$${\ displaystyle G (p)}$

${\ displaystyle G (p) = {\ frac {U_ {a} (p)} {U_ {e} (p)}} = {\ frac {p \ cdot L} {p \ cdot L + R}} = {\ frac {{\ frac {L} {R}} \ cdot p} {{\ frac {L} {R}} \ cdot p + 1}}}$

The transfer function of the RL element is as follows : ${\ displaystyle T = {\ frac {L} {R}}}$

 ${\ displaystyle G (p) = {\ frac {U_ {a} (p)} {U_ {e} (p)}} = {\ frac {T \ cdot p} {T \ cdot p + 1}}}$

The result corresponds to a series connection of a -link with a D-link. For a normalized input step , the output signal jumps at the time to , and then falls for exponentially asymptotically to the value . ${\ displaystyle PT_ {1}}$${\ displaystyle u (t) = 1}$${\ displaystyle t = 0}$${\ displaystyle u (t) = 1}$${\ displaystyle t> 0}$${\ displaystyle y (t) = 0}$

The time behavior associated with this transfer function is for an input jump: ${\ displaystyle y (t)}$

 ${\ displaystyle y (t) = e ^ {- t / T}}$

The standardized equation applies to the input jump ; . ${\ displaystyle u _ {\ sigma} (t) = 1}$${\ displaystyle e = {\ text {Euler's number}} \ approx 2 {,} 71828}$

## Calculation of the time behavior of an element after an input jump${\ displaystyle PT_ {1}}$

Time behavior of the step response of an element with the time constant T = 1, K = 1.${\ displaystyle PT_ {1}}$

In the time domain, the output variable of the transfer function of the element is often represented as a step response. The normalized leap is Laplace transforms: . ${\ displaystyle y (t)}$${\ displaystyle RC}$${\ displaystyle PT_ {1}}$${\ displaystyle U (s)}$${\ displaystyle U (s): = {\ hat {U}} _ {\ sigma} (s) = {\ frac {1} {s}}}$

The transfer function for and the step response is: ${\ displaystyle Y (s)}$${\ displaystyle U (s): = {\ hat {U}} _ {\ sigma} (s)}$

${\ displaystyle {Y (s)} = {\ hat {U}} _ {\ sigma} (s) \ cdot {\ frac {1} {T \ cdot s + 1}} = {\ frac {1} { s \ cdot (T \ cdot s + 1)}}}$.

A possibly existing gain factor cannot be transformed. It also does not appear in the corresponding Laplace transformation tables of the inverse transformation and can be adopted unchanged in the time domain. ${\ displaystyle K}$

The solution in the time domain of the step response results from the correspondence tables of Laplace transformation tables for the expression: ${\ displaystyle y (t)}$

${\ displaystyle {Y (s)} = {\ frac {1} {s \ cdot (T \ cdot s + 1)}}}$:

gives the time behavior of the delay element with the added gain factor . ${\ displaystyle K}$

 ${\ displaystyle y (t) = K \ cdot (1-e ^ {- t / T}}$)

The standardized equation applies to the input jump to . . ${\ displaystyle y (t) = 0}$${\ displaystyle y (t) = 1}$${\ displaystyle e = {\ text {Euler's number}} \ approx 2 {,} 71828}$

Time behavior of the return${\ displaystyle {\ hat {u}} _ {\ sigma \ downarrow} (t)}$ from the initial value of the -link to . ${\ displaystyle y _ {(t = 0)} = 1}$${\ displaystyle PT_ {1}}$${\ displaystyle y (t) = 0}$

 ${\ displaystyle y (t) = e ^ {- t / T}}$
The standardized equation applies to the return from to .${\ displaystyle y (t) = 1}$${\ displaystyle y (t) = 0}$

Output values ​​of a -link of the jump and the jump back for one to five times time constants:${\ displaystyle PT_ {1}}$${\ displaystyle y (t) = 1-e ^ {- t / T}}$${\ displaystyle y (t) = e ^ {- t / T}}$

Time constant T Step response jump
: in [%]${\ displaystyle y (t)}$
Step response
return: in [%] ${\ displaystyle y (t)}$
T easy 63.2 36.8
${\ displaystyle 2 \ cdot T}$ 86.5 13.5
${\ displaystyle 3 \ cdot T}$ 95.0 5.0
${\ displaystyle 4 \ cdot T}$ 98.2 1.8
${\ displaystyle 5 \ cdot T}$ 99.3 0.7

This standardized equation applies to the input jump . is the gain factor . ${\ displaystyle {\ hat {u}} _ {\ sigma} (t) = 1: = 100 \, \%}$${\ displaystyle K}$${\ displaystyle e = {\ text {Euler's number}} \ approx 2 {,} 71828}$

## Calculation example for determining the time constants of an ordinary 2nd order differential equation

 Given: differential equation of a 2nd order timing element without differentials of the input variable . ${\ displaystyle u (t)}$ ${\ displaystyle a_ {2} \ cdot {\ ddot {y}} (t) + a_ {1} \ cdot {\ dot {y}} (t) + a_ {0} \ cdot y (t) = b_ { 0} \ cdot u (t)}$ Wanted: transfer function, poles, time constants. Application of the Laplace transformation of the differential equation according to the differentiation theorem: ${\ displaystyle a_ {2} \ cdot s ^ {2} \ cdot Y (s) + a_ {1} \ cdot s \ cdot Y (s) + a_ {0} (t) \ cdot Y (s) = b_ {0} \ cdot U (s)}$. Formation of the transfer function and exemption of the highest transformed derivative${\ displaystyle G (s) = Y (s) / U (s)}$ : ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {b_ {0}} {a_ {2} \ cdot s ^ {2} + a_ {1 } \ cdot s + a_ {0}}} = {\ frac {\ frac {b_ {0}} {a_ {2}}} {s ^ {2} + {\ frac {a_ {1}} {a_ { 2}}} \ cdot s + {\ frac {a_ {0}} {a_ {2}}}}} = {\ frac {\ text {numerator polynomial (s)}} {\ text {denominator polynomial (s)}}} }$. Given numerical values :${\ displaystyle a_ {i}}$ and for . ${\ displaystyle \ a_ {2} = 2, \ a_ {1} = 3, \ a_ {0} = 1}$${\ displaystyle b_ {0} = K = 1}$ The transfer function and exemption of the highest exponent (equation divided by ) is thus : ${\ displaystyle a_ {2} = 2}$ ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {1} {2 \ cdot s ^ {2} +3 \ cdot s + 1}} = {\ frac {0 {,} 5} {s ^ {2} +1 {,} 5 \ cdot s + 0 {,} 5}}}$. There are programs on the Internet (Google) that can calculate the zeros of polynomials up to the 4th order. The so-called pq formula can be used to solve the zeros (poles) of a 2nd order polynomial: Polynomial: ${\ displaystyle s ^ {2} +1 {,} 5 \ cdot s + 0 {,} 5 = 0 \ qquad {\ text {with}} p = 1 {,} 5; \ q = 0 {,} 5 }$ ${\ displaystyle s_ {p1; 2} = - {\ frac {p} {2}} \ pm {\ sqrt {{\ frac {p ^ {2}} {4}} - q}} = - {\ frac {1 {,} 5} {2}} \ pm {\ sqrt {{\ frac {1 {,} 5 ^ {2}} {4}} - 0 {,} 5}} = - 0 {,} 75 \ pm 0 {,} 25 \ qquad s_ {p1} = - 1; s_ {p2} = - 0 {,} 5}$. This enables a factorization of the polynomial and the transfer function in time constant representation. ${\ displaystyle G (s) = {\ frac {Y (s)} {U (s)}} = {\ frac {0 {,} 5} {s ^ {2} +1 {,} 5 \ cdot s +0 {,} 5}} = {\ frac {0 {,} 5} {(s-s_ {p1)} (s-s_ {p2})}} = {\ frac {0 {,} 5} { (s + 1) (s + 0 {,} 5)}} = {\ frac {1} {(s + 1) (2 \ cdot s + 1)}}}$. These equations are algebraically identical. Result: The two- link transmission system contains the time constants . ${\ displaystyle PT_ {1}}$${\ displaystyle T_ {1} = 1 {\ text {and}} T_ {2} = 2}$ A hardware simulation of this system with two RC elements in series requires a load-free decoupling.

Annotation:

The calculation of the time behavior of a higher order transfer function of a complex dynamic system for a given input variable consists of:

• Apply Laplace transformation tables for the corresponding time function with the normalized transfer function,
• or to convert a factorial form of the transfer function into a partial fraction representation , the additive terms of which can easily be transferred into the time domain,
• or via numerical calculation with difference equations , which are derived from the linear factors of the transfer function, in order to calculate the time behavior of for a certain input variable .${\ displaystyle y (t)}$${\ displaystyle u (t)}$
• In the case of transfer functions of higher order with a mixture of negative real zeros and negative conjugate complex zeros, the calculation of the time behavior from the equations of the Laplace transformation tables with the complex trigonometric functions and exponential functions can be quite complicated. The numerical calculation with difference equations or with the acquisition of commercial simulation programs is considerably easier.

## Standardized time constants and transition frequencies of filters

Time constant τ
in µs
Crossover frequency f c
in Hz
Equalization standard
7958 20th RIAA
3183 50 RIAA, NAB
1592 100 -
318 500 RIAA
200 796 -
140 1137 -
120 1326 MC
100 1592 -
90 1768 MC
75 2122 RIAA, FM USA
50 3183 NAB, PCM, FM Europe
35 4547 DIN
25th 6366 -
17.5 9095 AES
15th 10610 PCM

## Individual evidence

1. Author: Jan Lunze / Control Engineering 1; Springer Vieweg, Berlin, 8th edition 2014, ISBN 978-3-642-53943-5 ; Main chapter: transfer function, sub-chapter: time constants of the transfer function.
2. Author: Jan Lunze / Control Engineering 1; Springer Vieweg, Berlin, 8th edition 2014, ISBN 978-3-642-53943-5 ; Chapter: Description of the analysis of linear systems in the frequency domain.

## literature

• Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink. 11th edition. Verlag Europa-Lehrmittel, 2019, ISBN 978-3-8085-5869-0 .
• Jan Lunze: Control engineering 1. 6. Edition. Springer Verlag, Berlin 2007, ISBN 978-3-540-70790-5 . Control engineering 2nd 4th edition. Springer Verlag, Berlin 2006, ISBN 978-3-540-32335-8 .
• Michael Laible: Mechanical quantities, measured electrically. Basics and examples for technical implementation. 7th edition. Expert Verlag, Renningen 1980, ISBN 3-8167-2892-8 .
• Wolfgang Schneider: Practical control engineering. A text and exercise book for non-electrical engineers. 3. Edition. Vieweg + Teubner Verlag, Wiesbaden 2008, ISBN 978-3-528-24662-4 .
• Walter Kaspers, Hans-Jürgen Küfner: Measure Control Control. 3. Edition. Friedrich Vieweg & Sohn Verlag, Wiesbaden 1984, ISBN 3-528-24062-8 .
• David Halliday, Robert Resnick, Jearl Walker: Physics . Bachelor edition, 2nd edition. John Wiley & Sons Verlag, Weinheim 2013, ISBN 978-3-527-41181-8 .