**The through and across variables**

**Through variable:**A variable that measures a quantity that is transmitted through an element, e.g., current, force, fluid volumetric flow rate, heat flow rate.

**Across variable:**A variable that is measured as the difference between the states of the two terminals of an element, e.g., voltage, displacement, pressure, and temperature differences.

**Relationship between through and across variables:**A through variable associated with an element is proportional to the corresponding across variable or its integral/derivative:

where is the proportionality associated with the specific element in the system.

Specifically, typical elements in electrical and mechanical systems include:

- Spring, damper and mass in a mechanical system:

where is an inertial force (aka. fictitious, pseudo, or d'Alembert force). - Inductor, resistor, and capacitor in an electrical system:

Note that the proportionalities for the resistor and inductor are respectively and , which are defined differently from the others.

**Constitutive Laws**

Mechanical Systems | Electrical Systems | |

At a node | Newton's 2nd Law (conservation of momentum): (or ) | Kirchhoff's Current Law (conservation of charge): |

Around a loop | Spatial consistency (conservation of space): | Kirchhoff's Voltage Law (conservation of energy): |

Mechanical Systems | Electrical Systems | |

Inductive storage | spring: | inductor: , or |

Energy dissipater | damper: | resistor: , or |

Capacitive storage | mass: | capacitor: , or |

Generic | Mechanical | Electrical |

Conserved quantity | momentum | charge |

Through variable | force (momentum transfer rate): | current (charge transfer rate): |

Newton's 2nd law: | KCL: | |

Across variable | relative displacement between end points | voltage difference across element |

spatial consistency: | KVL: | |

Potential energy stored in spring (): | |

Kinetic energy stored in mass ( ): | |

Magnetic energy stored in inductor (): | |

Electrical energy stored in capacitor (): |

**Combination and Division Rules:**

**Series combination**When two elements are combined in series, the through variable is shared but the across variables are added:

From this we get the*division rule*for the across variable in a series combination:

For example, the displacements across two springs in series are inversely proportional to their individual and , but the voltages across two resistors in series are proportional to their individual and .Adding the above two across variables we get:

We therefore get the*combination rule*:

For example:

- Mechanical elements:

Masses can only be combined in parallel! - Electrical elements:

Sometimes the conductance can be used instead of resistance , then we have

- Mechanical elements:
**Parallel combination**When two elements are combined in parallel, the across variable is shared but the through variables are added:

From this we get the*division rule*for the through variable in a parallel combination:

For example, the forces through two springs in parallel are proportional to their individual and , but the current through two resistors in parallel are inversely proportional to their individual and .Adding the above two through variables we get:

We therefore get the*combination rule*:

For example:

- Mechanical elements:

- Electrical elements:

If conductance is used, then

- Mechanical elements:

Masses are always combined in parallel, because they share the same across variable displacement, but not necessarily same through variable force (unless they have the same mass). This is because the relative displacement between the two ends of a mass is always zero (rigid body), therefore the displacement of a mass can only be measured absolutely with respect to the reference, i.e., the ``ground''. In comparison, the voltage across a capacitor does not have to be zero. For this reason, if a mechanical system is to be modeled by an electrical system, the mass has to be modeled by a capacitor with one end grounded, so that its voltage is always measured absolutely with respect to zero.

**Example 1: RLC parallel circuit (KCL)**

The three elements are in parallel as they share the same across variable, the voltage . According to KCL, we have:

Taking derivative on both sides, we get:

**Example 2: Spring-damper-mass system**

The three elements are in parallel as they share the same across variable, the displacement . According to Newton's second law, we have:

As is the velocity, the above can also be written as: