Example Network with a Semiconductor Switch

The derivation of ARRs from a bond graph of a hybrid system model that holds for all system modes is illustrated by means of a simple network with one switch and elements with a linear constitutive equation displayed in Fig. 4.1. 

Notice that the fixed conductance causality of the switch model and the inverted causality at the flow sensor Df / cause a causal conflict at junction O2 that has been resolved by attaching an auxiliary C-element C Ca in integral causality. However, that does not mean that the ARRs derived from this bond graph depend on the initial condition of this C-element. In the process of ARR formulation, its capacitance parameter Ca is set to zero. Instead of an auxiliary C-element a resistor R Ra with Ra 00 could also be used. Alternatively, the C element C Ci could be assigned integral causality, or the causality at the flow sensor Df / is not inverted. Clearly, the causal conflict vanishes if an additional effort sensor is used that replaces the auxiliary C-element C Ca- 

Furthermore, there is a causal path between resistor R R2 and the ON resistor of the switch Ron, which means that there is an algebraic loop for the current /sw As both resistors are linear, this algebraic loop can be solved symbolically. 

In comparison to the BG in Fig.4.2, the BG in Fig.4.3 reproduced from [9] does not reflect the series connection of a resistor and a semiconductor switch. The latter one is not explicitly visible in the BG. Additional information must be known that junction I3 is a controlled junction. Moreover, detectors are not in inverted causality although sensors deliver a measured and thus known variable into a DBG. Also note that an auxiliary resistor R Rs2 is needed to determine the effort at junction O4. 

The sum of power variables at the junctions in the bond graph of Fig. 4.2 read 

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