{"id":68,"date":"2021-02-04T14:26:56","date_gmt":"2021-02-04T19:26:56","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/?post_type=chapter&p=68"},"modified":"2022-03-07T16:03:00","modified_gmt":"2022-03-07T21:03:00","slug":"bond-graph-models-for-electrical-systems","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/engineeringsystems\/chapter\/bond-graph-models-for-electrical-systems\/","title":{"raw":"Bond Graph Models for Electrical Systems","rendered":"Bond Graph Models for Electrical Systems"},"content":{"raw":"
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7.1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Overview<\/h1>\r\nAn 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\r\n\r\n\\begin{equation*}\r\n\r\n\\begin{dcases}\r\n\r\ne\\equiv V, voltage\\\\\r\n\r\nf \\equiv i=\\.{Q}, current\\\\\r\n\r\ne \\cdot{f} \\equiv{V} \\cdot{i}, power\\\\\r\n\r\nq \\equiv{Q}=\\int\\textit{i dt}, charge\\\\\r\n\r\np \\equiv{P_E}=\\int{V}{dt}, \\textit{electrical momentum}\r\n\r\n\\end{dcases}\r\n\r\n\\end{equation*}\r\n\r\nUsing the constitutive relations, for an $I$-element we have $e=I\\.{f}$ or $V=L\\dfrac{di}{dt}$, and for a C-<\/em>element we have $e=q\/c$ or $V=Q\/c$. Similarly, for an R<\/em>-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<\/em> and C<\/em>, 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.\r\nThe analogy between mechanical and electrical systems can be summarized as follows:\r\n