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As we will see in many of the examples to come, there are many different types of relationships between the through and across variables (Ohm’s law being just one of many).

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Modelica By Example
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The original © copyright in Modelica By Example tutorial text, code, and diagrams reproduced for educational purposes remains with Dr. Michael M. Tiller
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The second class of variables we will discuss are “through” variables (also called flow variables). Flow variables normally represent the flow of some conserved quantity like mass, momentum, energy, charge, etc.
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As we will see in many of the examples to come, there are many different types of relationships between the through and across variables (Ohm’s law being just one of many).
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https://mbe.modelica.university/components/connectors/#acausal-connections
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In this section, we’ll discuss relatively simple engineering domains. These are ones where a connector deals with only one through and one across variable. Conceptually, this means that only one conserved quantity is involved with that connector.
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In order to understand one specific class of connector semantics, it is first necessary to understand a bit more about acausal formulations of physical systems. An acausal approach to physical modeling identifies two distinct classes of variables.
The first class of variables we will discuss are “across” variables (also called potential or effort variables). Differences in the values of across variables across a component are what trigger components to react.
Typical examples of across variables, that we will be discussing shortly, are temperature, voltage and pressure. Differences in these quantities typically lead to dynamic behavior in the system.
The second class of variables we will discuss are “through” variables (also called flow variables). Flow variables normally represent the flow of some conserved quantity like mass, momentum, energy, charge, etc.
These flows are usually the result of some difference in the across variables across a component model. For example, current flowing through a resistor is in response to a voltage difference across the two sides of the resistor.
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In this section, we’ll discuss relatively simple engineering domains. These are ones where a connector deals with only one through and one across variable. Conceptually, this means that only one conserved quantity is involved with that connector.
The following table covers four different engineering domains. In each domain, we see the choice of through and across variables that we will be using along with the SI units for those quantities.
You may have seen a similar table before with slightly different choices. For example, you will sometimes see velocity (in ) chosen as the across variable for translational motion. The choices above are guided by two constraints.
The first constraint is that the through variable should be the time derivative of some conserved quantity.
The reason for this constraint is that the through variable will be used to formulate generalized conservation equations in our system. As such, it is essential that the through variables be conserved quantities.
The second constraint is that the across variable should be the lowest order derivative to appear in any of our constitutive or empirical equations in the domain.
So, for example, we chose position for translational motion because position is used in describing the behavior of a spring (i.e., Hooke’s law).
If we had chosen velocity (the derivative of position with respect to time), then we would have been in the awkward situation of trying to describe the behavior of a spring in terms of velocities, not positions.
An essential point here is that differentiation is lossy. If we know position, we can easily express velocity. But if we only know velocity, we cannot compute position without knowing an additional integration constant.
This is why we want to work with across variables that have not been overly differentiated.
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