MULTIPORT SYSTEMS AND BOND GRAPHS
2.2 PORTS, BONDS, AND POWER
The devices sketched in Figure 2.1 can all be treated as multiport elements with ports that can be connected to other multiports to form systems. Further, when two multiports are connected, power can flow through the connected ports and the power can be expressed as the product of an effort and a flow quantity, as given in Tables 2.2–2.5. We now develop a universal way to represent multiports and systems of interconnected multiports based on the variable classifications in the tables.
Consider the separately excited dc motor shown in Figure 2.3. Physically, such motors have three obvious ports. The two electrical ports are represented by armature and field terminal pairs, and the shaft is a rotary mechanical port as sketched in Figure 2.3a. Figure 2.3b is a conventional schematic diagram in which the mechanical shaft is represented by a dashed line, the field coils are represented by a symbol similar to the circuit symbol for an inductance, and the armature is represented by a highly schematic sketch of a commutator and brushes. Note that the schematic diagram does not indicate what the detailed internal model of this subsystem or component will be. To write down equations describing the motor, an analyst must decide how detailed a model is necessary.
Figure 2.3c represents a further step in simplifying the representation of this engineering multiport. The namedc motoris used to stand for the device, and the ports are simply indicated by single lines emanating from the word representing the device. In a system in which several subsystems are connected, these port
FIGURE 2.3. Separately excited dc motor: (a) sketch of motor; (b) conventional schematic diagram; (c) multiport representation; (d) multiport representation with sign convention for power.
lines will be calledbonds. As a convenience, the effort and flow variables may be written next to the lines representing the ports. The notation can reflect the typical variables used in the particular energy domains involved. Whenever the port lines are either horizontal or vertical, it is useful to use the following convention:
• Efforts are placed eitherabove or to theleft of the port lines.
• Flows are placed eitherbelow or to theright of port lines.
• When diagonal lines are used, some judgment is required for placement of the effort and flow variables.
Note that Figures 2.3a, b, and c all contain the same information, namely, that the dc motor is a 3-port with power variables τ, ω, ef, if,ea, and ia. In Figure 2.3d, a sign convention has been added: The half-arrow on a port line indicates the direction of power flow at any instant of time when the product of the effort and flow variables happens to be positive.
For example, if ω is positive in the direction shown in Figure 2.3a and if τ is interpreted to be the torque on the motor shaft resulting from a connection to some other multiport and is positive in the direction shown in Figure 2.3a,
then when τ and ω are both positive (or, for that matter, both negative), the product τ ω is positive and represents power flowing from the motor to some other multiport coupled to the motor shaft. Thus, the half-arrow in Figure 2.3d points away from the dc motor. Similarly, when ef,if,ea, and ia are positive, power flows to the motor from whatever other multiports are connected to the field and armature terminals. Hence, the half-arrows associated with the field and armature ports point toward the motor.
Anytime one desires to be specific about the characteristics of a multiport—for instance, in equation form or in the form of tabulated data—then a sign conven- tion is necessary. The establishment of sign conventions is fairly straightforward for electric circuits or for the circuit-like parts of representations of multiports such as those of Figures 2.3a andb.
In mechanics, however, anyone who has struggled with the definition of forces and moments on interconnected rigid bodies using “free-body diagrams” knows that the establishment of sign conventions in mechanical systems is not trivial.
The problem is that the action and reaction forces show up as oppositely directed in most representations. Thus, in Figure 2.3a, one must decide whetherτ repre- sents the torqueon the motor shaft orfrom the motor onto some other multiport.
On diagrams such as Figure 2.3b, the mechanical signs are often not indicated at all, and it is up to the analyst to insert plus or minus signs in the equations without much help from the schematic diagram of the system.
When two multiports are coupled together so that the effort and flow variables become identical, the two multiports are said to have a commonbond, in analogy to the bonds between component parts of molecules. Figure 2.4 shows part of a system consisting of three multiports bonded together. The motor and pump have a common angular speed, ω, and torque at the coupling, τ. The battery and the motor have a common voltage and current defined at the terminals at which the battery leads connect to the motor armature. To represent this type of subsystem interconnection in the manner of Figure 2.3c or d is very straightforward; the
FIGURE 2.4. Partially assembled system.
FIGURE 2.5. Word bond graph for system of Figure 2.4.
joined ports are represented by a single line or bond between the multiports. This has been done in Figure 2.5. The line between the pump and motor in Figure 2.5 implies that a port of the motor and a port of the pump have been connected, and hence a single torque and a single angular speed pertain to both the pump and the motor. The half-arrow on the bond means that the torque and the speed are defined in such a way that when their product, τ ω, is positive, power is flowing from the motor to the pump. Thus, lines associated with isolated multiports indicate ports or potential bonds. For interconnected multiports, a single line represents the conjunction of two ports, that is, a bond.