Bond Graph Models for Electrical Systems

Mehrzad Tabatabaian

7 Bond Graph Models for Electrical Systems

An electrical system may consist of components such as resistors, inductors, capacitors, transformers, and batteries/source. The generalized BG elements and relations apply to the analysis of dynamics of electrical systems and components in a similar way that the mechanical components were treated; i.e., they are analogous (see Table 3-1). In other words, electric charge $Q$ is equivalent to the generalized displacement $q$ , electrical current to $i=\dfrac{dQ}{dt}=\.{Q}$ to the flow $f$ , and voltage $V$ to the effort $e$ . The inductor (with inductance $L$ ) is analogous to point mass and is represented by $I$ -element; the capacitor (with capacitance $c$ ) is analogous to a mechanical spring and is represented by $C$ -element; and resistor (with resistance $R$ ) is analogous to mechanical damper and is represented by $R$ -element. The generalized momentum $p_E$ or flux linkage is the integral of $V$ with respect to time. Therefore, we can write

$\begin{equation*} \begin{dcases} e\equiv V, voltage\\ f \equiv i=\.{Q}, current\\ e \cdot{f} \equiv{V} \cdot{i}, power\\ q \equiv{Q}=\int\textit{i dt}, charge\\ p \equiv{P_E}=\int{V}{dt}, \textit{electrical momentum} \end{dcases} \end{equation*}$

Using the constitutive relations, for an $I$ -element we have $e=I\.{f}$ or $V=L\dfrac{di}{dt}$ , and for a C-element we have $e=q/c$ or $V=Q/c$ . Similarly, for an R-element we have $e=Rf$ or $V=Ri$ . The energy associated with storage elements can be written using Equations (3.7) and (3.8), or for elements I and C, as $\dfrac{1}{2L} p^2_E=\dfrac{L}{2} i^2$ and $\dfrac{1}{2c} Q^2=\dfrac{c}{2} V^2$ , respectively. An advantage of the bond graph method is its analogous applicability to different domains using the common constitutive relations, as described above for electrical systems.
The analogy between mechanical and electrical systems can be summarized as follows:

For a system with series-connected components, we have equal effort for mechanical and equal flow for electrical systems. For example, when a spring and a damper are connected in series, they experience the same force, and when a capacitor and a resistor are connected in series, they experience the same current.
For a system with parallel-connected components, we have equal flow for mechanical and equal effort for electrical systems. For example, when a spring and a damper are connected in parallel, they experience the same velocity (or rate of displacement), and when a capacitor and a resistor are connected in parallel, they experience the same voltage.

In other words, the relations of efforts and flows are swapped according to the type of the physical system between mechanical and electrical systems.

$\begin{multiline} mechanical|_{parallel} \: \equiv \: electrical|_{series}\\ \: \\ mechanical|_{series}\:\equiv \: electrical|_{parallel} \end{multiline}$

Table 7-1 shows typical components for resistor/ $R$ , capacitor/ $C$ , Inductor/ $I$ , Transformer/ $TF$ , Electric motor/ $GY$ .

Table 7‑1 Typical electrical components and their corresponding BG elements
$R$ -Element (resistor)	$C$ -Element (capacitor)	$I$ -Element (inductor)	$TF$ -Element (transformer)	$GY$ -Element (electric motor)

As mentioned in chapter 4, the general guidelines for drawing BG model can be applied to electrical systems along with causality assignment rules. For electrical systems, we follow these guidelines, along with Kirchhoff’s circuit laws [24] and the PSC for building their BG models, as described in the following steps:

Assign voltage polarity ( $+V, -V$ ) for each element in the electrical circuit.
Assign current direction based on PSC for each element (see Figure 7‑1 and Figure 7‑2).
Assign 0-junction for each distinct voltage node in the circuit, according to Kirchhoff’s voltage law (KVL)—the algebraic sum of all voltage drops around a closed circuit is equal to zero.
Assign 1-junction for each element in the circuit, according to Kirchhoff’s current law (KCL)—the algebraic sum of all electrical currents entering and leaving a node is equal to zero). This is for taking care of relative voltage or drops related to each element located between two 0-junctions, since 1-junction is an effort summator.
Select a node in the circuit as a reference, i.e., the grounding, with zero voltage.
Assign $C$ -element for capacitors, $R$ -element for resistors, $I$ -element for inductors, $S_e$ for voltage, and $S_f$ for current sources.
Assign $TF$ -element for electrical transformers and $GY$ -element for electric motors.
Connect the elements with power bonds, assign causalities, and simplify by neglecting the bonds and the 0-junction which are connected to the ground source.

The above steps are based on KVL, and the process starts with assigning 0-junctions for each distinct voltage node. It is also possible to start with KCL and assign 1-junctions for the current in each closed-circuit loop and use 0-junctions in between for distribution of the current to corresponding circuit loops. The latter will result in a more simplified BG model and is recommended for complex circuits that involve several electric loops. In practice, we sometimes use a combination of these two approaches for building BG model for electrical systems.

In the following sections, we demonstrate the application of the procedure discussed above, with some worked-out examples.