SYSTEM MODELS
4.1 ELECTRICAL SYSTEMS
4.1.2 Electrical Networks
0
0
0 ea
ed
eb0 ec
e(t)
R1 R3
R2 R4 RL
+ +
+ +
− −
−
−
− a
b c
d ie
iR1
iR2
iR3
iR4 iL
+
0
0
0
0 ea
ed eb
ec R 1
Se
1
1 1
1
1
R
R R
R R1
R3
R4
R2 RL
e(t)
:
:
: :
:
:
0
0 ea
eb ec
1 R
Se
1
1
R
R R
R
R1 R3
R4 R2
RL e(t)
0 :
:
:
: : :
(a) (b)
(c) (d)
FIGURE 4.3. A Wheatstone bridge circuit. Example 3.
bonds at the ground voltage, ed =0. Steps 4 and 5 are done in Figure 4.3d. The bonds with no power are erased, and the bond graph simplifications have been done, where all 2-port 1-junctions with a “through” power convention have been reduced to single bonds.
The resulting final bond graph reveals a beautiful structural symmetry quite similar to that of the bond diagram for the hydrocarbon molecule called the
“benzene ring.” It was this similarity to chemical bond diagrams that prompted the “bond graph” name of our modeling approach. The idea is that physical systems consist of components or elements bonded together by power interactions much as atoms are bonded together in chemistry.
opposite to the current. Electrical gyrators are exhibited in Hall effect transducers (see Reference [1]), where voltage across a semiconductor material is related to a current through the material perpendicular to the voltage drop direction. The basic rules for bond graph construction remain unchanged.
Figure 4.4a shows the electrical symbol for a transformer, whereN indicates the turns ratio across the device. Positive voltage drops and current directions are chosen such that positive power isinto the device on the left side andout of the device on the right side.
Since ideal transformers and gryrators are power-conserving elements, it is logical to always define positive power such that it flows in on one side and out on the other. In Figure 4.4b, 1-junctions are used to establish the positive voltage drops across the input and output sides of the transformer. Notice that on the input side the voltage isea−eband on the output side the voltage on the transformer output bond isec−ed. These voltages are as defined in the schematic of Figure 4.4a. (It may require some thought to see that the sign-convention half- arrows on the two 1-junctions correctly imply the defined voltage differences at the two transformer bonds.)
Although the 2-port transformers and gyrators seem to involve four voltages, one at each terminal, they really relate only the voltage differences at the two ports. This means that we may have to choosetwo reference voltages on the two sides of the transformer. It is not necessary to have any direct metallic connection between the windings of a transformer so such anisolating transformer can be used to decouple the voltages in one part of a network from another part. If the two ground nodes are selected at the two sides of the transformer as in Figure 4.4b, then the transformer appears as in Figure 4.4c.
+ +
− −
a
b
c
d
ia ic
0
0 0
0
1 TF 1
ea
eb
ec
ed N
TF ea N ec
ia ic ea Nec Nia = ic
=
:
:
(a) (b)
(c)
FIGURE 4.4. The ideal electrical transformer and its bond graph representation.
In some networks, each side of a transformer is connected to a common ground. If this is the case,eb=ed=0. Then their associated bonds can be erased, some reductions can be performed, and the typical appearance of a transformer emerges as shown in Figure 4.4c.
An electrical network with an isolating transformer is shown in Figure 4.5a.
Positive voltage drops and current directions are shown in this figure. Notice
R3 R2
L1 C1
C2
a b d
c
f
ie iR1
iC1
iC2 iR2
iL1
e R1
e(t) +
− +
+
+ + +
+
+ + + +
+
−
−
−
−
−
− −
−
−
−
I(t) g
h j R4
C3 iR3
− iC3 iI
iR4 iL iR
Transformer of turns N (a)
0 0 0 0 0
0
0 0 0
ea eb
ec
ed ef eg
ee ej eh
(b)
0 0 0 0 0
0
0
0 0
ea eb
ec
ed ef eg
ee
ej eh
1 R:R1
Se TF Sf
1 1
R:R2
1
R3:R 1
C1:C L1:I 1
1 e(t)
1 1
C2
C
C3:C 1
1
R4:R
1 N I(t)
:
::
:
(c)
FIGURE 4.5. An electrical network with an isolating transformer.
0 0 0
0 eb
ec
ed eg
1 R:R1
C:C1
Se TF 1 Sf
1 R:R2
R3:R 1 1
L1:I
e(t) I(t)
1 C2
C
C:C3 R R4 N
:
:
: : :
(d)
FIGURE 4.5. (Continued)
that positive power flows in on the left side of the transformer and out on the right side and that two ground nodes are specified for the two parts of the network coupled only by the transformer. Following the bond graph construction procedure of the previous section, Figure 4.5b uses 0-junctions to expose the node voltages labeled in part a, and Figure 4.5c uses 1-junctions to establish the proper voltage drops across all the elements. Also in partc are dotted lines enclosing all the bonds with zero power. These are erased in Figure 4.5d, and bond graph simplifications have been performed to yield the final result.
A final example of the using the construction procedure is the network shown in Figure 4.6a. Most of the network is composed of circuit elements that must be looking pretty routine by now, but the new component is thevoltage-modulated current source, or “controlled current source,” labeled I(t) in the figure. The meaning of this schematic symbol is that the voltage,e, from the left side of the circuit modulates, with virtually no power, the current source that is an input to the right side of the circuit.
Active bonds were introduced in Section 2.4 and here an active bond is used to model a device such as a transistor in which a voltage on one terminal has a major effect on the current flowing out of another terminal. Almost no current is associated with the controlling voltage. Of course, this model is valid only for a limited range of voltages and currents, but the point is that, in the model, the voltage signal has zero current associated with it. Thus, the active bond with a full arrow resembles asignal in a block diagram without the back effect normal power bonds have.
Positive voltage and current directions are shown in the schematic diagram of Figure 4.6a. The labeled node voltages are shown using 0-junctions in Figure 4.6b, and 1-junctions are used in Figure 4.6c to establish voltage drops across the elements. Of particular interest is the establishment of the modulating voltage,e, and the use of the modulated flow source,Sf, for the current source generatingI(t).
Note that the 2-port 0-junction on which the voltageeappears has both an active bond and a normal bond. However, since the active bond implies zero current, the
e(t) R3 R2 L1
L2 C1 C2 I(t)
a
b
c d
f +
+ +
+ +
+
+ +
−
− −
− − − − −
− +
ie iR1
iC1 iC2
iR2 iR3 iL1
iL2 e
R1
(a)
0 0 0 0 0
0 0 0 0
ea
ea
eb
eb
ec
ec
ed
ed
ee
ee ef ef
ef ef
ef ef ef ef
(b)
0 0 0 0
0 0 0 0
0 1
C I
Se Sf
1 1
1 1
1 1
1
1
C 1 R
C1 C2 e I(t) L2
e(t)
e(t)
R3 I:L1
: : : : :
(c)
0 0 0
1
I R:R1
Se 1 Sf 1
I:L1 R:R2
C1:C C2:C I(t)
L2 R3:R :
: :
(d)
R:R1 R:R2
FIGURE 4.6. An electrical network with a controlled source.
normal bond also has zero current. (On any 2-port 0-junction the sum of the currents adds to zero—so if one current is zero, so is the other.) This normal bond is attached to a 1-junction, which also then has zero current on all of its bonds.
Altogether, the bond graph elements imply that the active bond is an effort signal (since it stems from a 0-junction) and that the effort ise=ec−ef (because of the signs on the 1-junction bonds) and that the currents on all the local bonds are zero (because of the active bond). The schematic diagram is intended to imply
all of this, but the bond graph makes these modeling assumption more explicit.
The rest of the network is processed in Figure 4.6candd in the same fashion as in the previous examples.
The zero ground voltage applies to the bonds enclosed by the dotted line, and the zero power bonds have been erased and simplifications performed to yield the final result of Figure 4.6d.
The network construction method presented here will allow you to model the most complicated electrical networks once you have acquired facility with the basic steps of node representation, element insertion, power definition, ground definition, and graph simplification. With practice you will begin to observe cer- tain recurrent patterns, and your ability to do more by inspection will increase.
Many interesting discoveries await you as you explore systems containing struc- tures likeladders,pi’s, andtees.
Speaking of ladders, let us make use of our ability to model the topology of circuits by bond graph junction structures in order to examine the bond graph form of a ladder network. An example of a resistive ladder circuit is given in Figure 4.7a. We are not really concerned with the nature of the 1-ports (source and resistances), but rather with their interconnection patterns (or circuit topology).
FIGURE 4.7. A ladder network example.
The bond graph of part b may be found by inspection, using arguments like
“. . .,R2in parallel,R3in series,R4in parallel,. . .” Or the formal construction procedure may be used. To display the system structure without regard to what is (or might be) connected to it, we use a bond graph like that of Figure 4.7c, which is a direct representation of the ladder structure. So ladders are nothing more than chains of alternating 3-port 0- and 1-junctions.
Next we shall use such a graph to explore the idea of dual topologies in circuits. A pi network is shown in Figure 4.8a in resistive form. The pi gets its name because of the Greek letter pi-like appearance of the schematic. The structure is represented as parallel–series–parallel in part b by the bond graph.
Resistances 1, 2, and 3 would go on the corresponding bonds to complete the model. In Figure 4.8c a tee network is shown, again in resistive form. The tee gets its name from the T-like appearance of the schematic. The bond graph in part d is a series–parallel–series type. We observe that partd can be obtained from part b by switching the roles of 0 and 1, and partb can be obtained from part d similarly. The formal idea behind the switching is that if the roles of voltage and current are interchanged, a dual network results. In terms of bond graph structure, this implies a switching of 0 and 1 elements. By this technique the topological dual of a complex circuit may be obtained from its bond graph in a very simple fashion.
The pi and tee structures are simplified versions of the ladder structure of Figure 4.7. We will see later in the text that electrical transmission lines and long hydraulic lines have models that can be represented by ladder structures that represent a large number of short segments cascaded in the manner of finite element representations. The pi and tee structures become truncations of the ladder structure that allow for reasonable inclusion of some of the transmission line dynamics without having the overall system become too large.
FIGURE 4.8. Truncations of the ladder structure—the pi and tee structures.