How can a constant describe a complex system? Despite the diversity and complexity of many systems - whether in nature, technology or business - there is one particular constant that can do just that: the time constant. It allows dynamic behaviour to be predicted precisely.
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How can a single constant predict the behaviour of a complex dynamic system? This question may sound surprising at first, especially when we think of the diversity and complexity that many systems exhibit - whether in nature, technology or the economy. And yet there is a special kind of constant that can do just that: the time constant.
The time constant is a crucial parameter that determines how quickly or slowly a system reacts to change. It provides us with a direct insight into the behaviour of systems over time and is therefore a valuable tool for understanding, predicting or controlling dynamics.
This article examines what the time constant is, how it works in different contexts and why it is of central importance in science and technology.
In contrast to static systems, state variables in dynamic systems change over time. This change follows certain rules or equations that describe how the state variables (e.g. position, speed, temperature) change from one point in time to the next. Differential equations are the favoured mathematical tool here, as they directly describe the change in state variables in relation to time.
Systems are divided into linear and non-linear systems, depending on how state and input variables are related to each other. In linear systems, the superposition principle is applicable so that individual effects add up without influencing each other. In non-linear systems, this principle does not apply, which can result in complex behaviour.
Mathematically, a dynamic system can be described by one or more differential equations, which look like this:
$$\frac{dx(t)}{dt} = f(x(t), u(t))$$
with:
One possible solution to this differential equation is:
$$x(t) = x_0 \cdot e^{-\frac{t}{\tau}}$$
with:
Mathematically, the time constant \(\tau\) describes the time it takes for a system to reach about 63% (1-e-1 ≈ 0.6321) of the distance from the initial value to the new equilibrium value after a change has begun. This value results from the nature of exponential growth or decay. In the case of exponential decay, this means that after a time \(\tau\) about 37 % (e-1 ≈ 0.368) of the original value is still present.
In physical terms, the time constant can be seen as a measure of the "inertia" of a system. A system with a small time constant reacts very quickly to changes, while a system with a large time constant reacts slowly. It therefore provides information about how quickly a state can change, be it electrical current, mechanical movement or thermal energy.
An RC element is a basic circuit in electronics, consisting of a resistor \(R\) and a capacitor \(C\). This simple circuit is used in practice for frequency filtering, time delay and signal processing and is ideal as a real-life example of the abstract concept of the time constant.
An electrical circuit with PT1 element behaviour is shown below as an example. The system is made up of the voltage \(U\), which represents the differential quantity, the resistance \(R\) and the capacitiy \(C\). The flux quantity is the electric current \(I\). In the example shown below, the time constant results from \(\tau\) = \(RC\) to 1 second. In general, the time constant describes the duration of the discharging and charging process of the capacitor.
An electrical RC element can serve as a model for many other physical systems in which energy is stored or processed. Many dynamic processes can be better understood using this analogy.
An example also follows for a thermal system: The differential state here is the temperature. The capacity of the capacitor corresponds to the heat capacity, which describes the ability of a material to store heat. In this system, the resistance \(R\) is a thermal resistance, i.e. the thermal resistance that materials offer to the heat flow. As explained above, the time constant describes the rate of temperature change of an object. It is significantly influenced by the thermal resistance and the heat capacity of the object. Figure 2 illustrates this relationship with Modelica Multiphysics.
Modelica is an object-oriented, declarative programming language for modelling and simulating complex physical systems. Modelica can be used to create models that combine different physical domains (e.g. mechanics, electrics, hydraulics). The language makes it possible to efficiently simulate realistic, mathematically described systems, which makes it particularly useful for the development and optimisation of technical systems. The following examples have also been modelled with Modelica Multiphysics for illustrative purposes.
Another application example is a mechanical system consisting of a spring and a damper. The differential state here is the deflection of the spring and the damper. The capacitor corresponds to the spring with the spring constant \(c\), which stores energy. The resistance is realised by the damper with the damper constant \(d\) by dissipating energy through friction or other forces. The time constant of the RC element corresponds to the time required to dampen a movement or stabilise an oscillation.
The RC element analogy can also be found in a hydraulic system. The differential quantity is the filling level of the tank. The capacity of the capacitor corresponds to the capacity of the system to store fluid. The resistance corresponds to the flow resistance caused by a valve with pressure loss. The time constant describes how quickly the filling level of the tank changes.
To apply the analogy of RC elements to other systems can help to understand the behaviour of complex systems by drawing on the known behaviour of electrical circuits. Complex dynamic systems often consist of several state variables and therefore have several time constants. These systems can be viewed as networks of RC elements, with each time constant describing the response of a particular component.
To illustrate this, let us consider the cooling of a hot cylinder in air. The thermal resistance is neglected. Assume that the temperature distribution is homogeneous and the heat is dissipated through the surface of the cylinder:
We would now like to describe the system mathematically in order to determine the temperature of the cylinder as a function of time.
The change in thermal energy in the cylinder is caused by heat loss via the surface of the cylinder and can be described mathematically as follows:
$$\frac{dQ}{dt} = -\alpha A \big(T(t) - T_\infty\big)$$
with
$$Q = mc \big(T(t) - T_\infty\big)$$
results:
$$m c \frac{dT(t)}{dt} = -\alpha A \big(T(t) - T_\infty\big)$$
with
As already explained in the previous section, the shape of the RC element can be recognised here. By further summarising the variables, the thermal capacitance \(C\) is obtained. The thermal resistance \(R\) of this system is characterised by 1/\(\alpha A\), so that the equation can be transformed into:
$$C \frac{dT(t)}{dt} = -\frac{1}{R} \big(T(t) - T_\infty\big)$$
By further transformation, we obtain the familiar RC shape:
$$\frac{dT(t)}{dt} = -\frac{1}{RC} \big(T(t) - T_\infty\big)$$
Solution of the differential equation and introduction of the time constant
Once we have set up the differential equation, it must now be solved. By specifying the initial condition for the initial cylinder temperature, the differential equation can be solved by separation of the variables to:
$$T(t) = T_\infty + \left(T_0 - T_\infty\right)e^{-\frac{t}{R C}}$$
This equation describes the temporal change of the cylinder temperature \(T(t)\), which approaches the ambient temperature over time. The rate of cooling is determined by the thermal resistance multiplied by the thermal capacity \(RC\). We now introduce the time constant \(\tau\) = \(RC\).
In this context, large time constants mean that it takes longer for the temperature to cool down to the ambient temperature. IT therefore follows tat the greater the mass \(m\) of the bidy or the specific heat capacity \(c\) oft he body, the greater the time constant and the longer the temperature equalisation takes.
With small time constants, this process happens faster. This is therefore favoured by larger heat transfer coefficients \(\alpha\) or a larger surface area \(A\). This means that the time constant is a measure of the dynamics of a system.
The temperature curve is shown for a starting temperature of 150°C for different time constants in Figure 6. Here it can be seen that the yellow curve with the smallest time constant approaches the ambient temperature of 25°C the fastest. In contrast, the grey curve takes significantly longer to cool down to the ambient temperature.
More complex systems that have several differential states lead to systems of differential equations. These systems can be visualised in the state space representation:
$$\frac{dx(t)}{dt} = A x(t) + B u(t)$$
with:
The eigenvalues of the matrix \(A\) play a central role in the analysis of the dynamic behaviour of the system. The eigenvalues \(\lambda_i\) of the matrix \(A\) are the solutions of the characteristic equation:
$$det(A - \lambda I) = 0$$
where \(I\) is the unit matrix.
The eigenvalues influence the temporal behaviour of the system, as the state variables of the system are solved in the form of exponential functions of the eigenvalues.
Introduction of the time consant
The real part of the eigenvalues is closely linked to the time constant. It describes how quickly the system reacts to the input or a fault.
The time constant \(\tau_i\), which is assigned to the eigenvalue \(\lambda_i\), is calculated as
$$\tau_i = -\frac{1}{\operatorname{Re}(\lambda_i)}$$
The eigenvalues of the system matrix \(A\) determine the behaviour of the system. Negative eigenvalues lead to stability, positive eigenvalues indicate instability. Oscillations occur with complex eigenvalues, whereby the real part influences the damping or growth of these oscillations. Systems with several time constants exhibit complex dynamic behaviour, which is described by the superposition of the effects of the individual time constants. Overall, the analysis of the eigenvalues of the system matrix is a central aspect in understanding the dynamics of a system, including its stability and time behaviour.
As mentioned above, there are also non-linear differential equation systems. These systems are significantly more complex than linear systems, as the system behaviour depends heavily on the initial conditions and the structure of the non-linear model. The exponential type of approximation, which is typical for linear systems, often does not apply. Therefore, it is difficult to define a unique time constant as in linear systems.
The concept of the time constant can only be used on almost linear ranges. By a linear approximation around an equilibrium point, the system behaviour can be analysed and the local dynamics can be described. In highly non-linear systems, where the behaviour is chaotic or highly complex, there is no simple time constant that describes the entire system.
Example of a non-linear system
When a hot body is placed in an environment with a lower temperature, it cools down not only by conduction and convection, but also by thermal radiation. However, thermal radiation follows a non-linear relationship with temperature, which makes this system non-linear.
The Stefan-Boltzmann law describes the radiant power \(P\) that a body emits due to its temperature \(T\):
$$P = \sigma \cdot \epsilon \cdot A \cdot T^4$$
with:
The energy loss due to radiation therefore depends non-linearly on the temperature of the body, as it is proportional to the fourth power of the temperature.
To summarise, the definition of a time constant for non-linear systems is often not possible or can only be applied locally if the system is approximately linear in certain areas. In strongly nonlinear or chaotic systems, the concept of the time constant becomes less important and other concepts such as characteristic times, phase space analyses or Lyapunov exponents are used instead.
We have now learnt that a complex dynamic system usually has several time constants. These together influence the inherent dynamics of the system. What does this mean for a numerical simulation?
If there are time constants in a differential equation system that differ by several orders of magnitude, the system can be stiff. What exactly does this mean?
There is no standardised and clear definition of "stiffness". A dynamic system is stiff if it has very different time scales, i.e. very different time constants. This leads to numerical challenges. This means that special methods are required in numerical mathematics to ensure that the calculation remains stable and efficient. To ensure this, the step size usually has to be adapted to the smallest time constant. This makes the simulation extremely computationally intensive and inefficient. A simple example of this is, for example, several reactions that take place at different speeds at the same time.
In summary, it can be said that an understanding of the time constants in a dynamic system is essential in order to understand the dynamic behaviour. Using the RC analogy, time constants for systems in electrical engineering, thermodynamics, control engineering, etc. can be used to interpret the system behaviour.
Furthermore, an understanding of time constants is essential in the numerical simulation of dynamic systems, as these can lead to the simulation being cancelled, among other things.
We will explain how to debug such a system in our next article.
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