Electrical Circuits

SYSTEM MODELS

4.1 ELECTRICAL SYSTEMS

4.1.1 Electrical Circuits

In Chapter 3, the electrical resistance, inductance, and capacitance were shown to be bond graph —R, —I, and —C elements. It remains to determine how to use the junction elements (0- and 1-junctions) to construct an overall bond graph model of an electrical circuit. Sometimes, for simple circuits, it is easy to recognize that some elements have the same current (flow) and others have the same voltage (effort). For these circuits, bond graph construction can be accomplished by inspection. For example, Figure 4.1ashows a simple circuit with positive voltage drops and current directions defined and node voltages labeled.

a a b

c c c

+ + +

+

−

−

− −

C

L

R₁ R₂

i_C

i_L iR₁ iR₂

(a)

(b)

0 1

I : L R : R₁

R : R₂ C

C i_C

iR₁ i_L

iR₂

:

FIGURE 4.1. A simple electric circuit. Example 1.

For convenience, some node voltages are repeated to emphasize their association with specific elements. The circuit is grounded at the bottom, meaning that the voltage labeled cis considered to be zero,vc=0.

To arrive at the bond graph in part b, we argue as follows: The elements C and R₁ have the same voltage (v_a−v_c=v_a−0=v_a) and thus are attached to the same 0-junction (common effort junction); the elementsL and R₂ have the same current (iL=iR2), so they are attached to the same 1-junction (common flow junction). The bond joining the 0- and 1-junctions enforces the fact that the current through the inductor is the sum of the currents through the capacitor and the resistor, R1 (actually, with the sign convention shown in Figure 4.1a, iL= −iC−iR1). Notice that all the 1-portR,C, andI elements have the power half-arrows defined such that whenever the voltage drop across the element is in the direction defined as positive in Figure 4.1a, and the current is simultaneously in the defined positive direction, the power is flowing into the element. We always define positive power directions forR,C, andI such that this is true by pointing the half-arrows toward the elements.

Most of the time, electric circuits are too complex to model by inspection.

There may be some parts that are obviously in series or parallel (common current or common voltage), but constructing the overall bond graph model is much easier if a procedure can be followed that ensures success regardless of the complexity of the circuit. Here we present a foolproof circuit construction procedure and develop the procedure along with an example shown in Figure 4.2. This is a voltage-excited circuit, grounded at the bottom, and open circuited at the right side, exposing an output voltage,e_out.

Circuit Construction Procedure

1. Assign a power convention to the circuit schematic diagram.

This step must always be done regardless of the modeling procedure being used. On the circuit, this is done by showing the positive voltage drop and current directions. For the —I, —R, and —C elements, the positive voltage drop (+to−) is shown in the same direction as the positive current.

R₂ L₁

a b c

i_e i_C1 i_C2

iR₂ i_L1

e

e(t) +

+

− +

− −

C₁ +

− R₁ +

−

+ −

C₂

d +

+

−

−

C i_C3 L₂

e_out i_L2

(a)

0^a 0^b 0^c

0^e 0^e 0^e 0^e

0^d

(b)

(c)

0 0 0

0 0 0 0

0 eb

e_a ec

e_d

e_e 1

I R

C S_e:

e(t)

1 R₁

1 C₁

1

1

C₂ C L₁

1 R

R₂

1 1

C C₃ L₂I

1 S_f

e_out 0

e_e e_e

e_e

:

: : :

:

:

:

:

FIGURE 4.2. A complicated electric circuit. Example 2.

0 0 0

0

e_a e_b e_c

e_d 1

I R

S_e C e(t)

1 R₁

1 C₁

1

1 C₂ C

L₁

1 R

R₂

1 1

C C₃ I L2

1 S_f

e_out 0:

::

:

: :

:

:

:

(d)

0 0

0

e_b e_c

e_d R I

C S_e

e(t)

1 R₁

C₁

1

C₂ C L₁

1 R

R₂

C₃ C I L2

S_f e_out 0 0

: : :

: :

:

:

:

: (e)

FIGURE 4.2. (Continued)

This ensures that when the power is positive, power is flowingtoward the passive 1-ports.

For the source elements (Sf for current source and Se for voltage source), it is not automatic which directions should be chosen for positive voltage drop and positive current (and eventually which end of the bond should have the half-arrow). If positive current is defined such that the cur- rent moves “uphill” against the positive voltage as is done in Figure 4.2a, then if the current is positive, power will come from the source into the rest of the circuit. If either the positive voltage direction or current direction is chosen in the opposite direction, then positive power will flow toward the source and negative power will flow away from the source. There is abso- lutely nothing wrong with this, and, in fact, real time-varying sources may sometimes absorb power from the attached system and sometimes supply power to the attached system.

2. Label each node voltage on the circuit schematic diagram and use a 0-junction to represent each node voltage as shown in Figure 4.2b.

Node voltages are all the independent voltages that exist at points in the circuit, including the ground node voltage that later will be set to zero. The

node voltages in Figure 4.2b appear above and below or to the left and right of each circuit element. In the figure, the node voltages are labeled using letters. For convenience, the ground voltage,e, is repeated several times. Remember, every bond that touches a particular 0-junction has the voltage associated with that junction.

3. Establish the positive voltage drops across the circuit elements using 1-junctions.

Remember that 1-junctions add efforts (voltages) algebraically accord- ing to the power convention represented by the half-arrows. By properly directing the half-arrows on 1-junctions, the voltage drop can be established across each bond graph element. See the discussion about signs on 0- and 1- junctions in Section 3.3 and particularly the example leading to Eq. (3.21).

(This step will require practice to accomplish rapidly with confidence but this is a worthwhile endeavor.)

In every case the sign half-arrows flowthroughthe 1-junctions, indicat- ing that the circuit elements react to node voltagedifferences, never node voltage sums. Which way the arrows flow through the 1-junctions depends on the sign of the voltage difference established in step 1.

Figure 4.2c shows this construction. For example, for the —R element representing the resistor,R1, the effort on the bond attached to theR isea

−e_b, which corresponds to the positive voltage drop defined in the circuit schematic.

Notice that positive power flows out of the voltage source element and the voltage on the source bond isea−ee, as was defined in the schematic.

Also, the output voltage,eout, is exposed using a flow source,Sf, of zero current. (The idea of an open circuit is that no current flows even if there is a voltage across the terminals.) The voltage drop across this flow source is eout =e_c−e_e. The reader should check the other elements and ensure that all have their defined positive voltages.

4. Chose a ground or reference node and remove all bonds that have zero power.

Before the bond graph can be used for equation derivation or simulation, thereference or ground node must be established. (The reference node is the point at which we would put the black probe of a voltmeter if we were measuring voltages in the circuit. In other words it is the voltage we assume to be zero when we measure other voltages.)

The reference in Figure 4.2 has been chosen to be ee, which we now set to zero. Since every bond that touches a 0-junction has the identical voltage, all the bonds inside the dotted line in Figure 4.2chave zero voltage and therefore each of those bonds carries no power. We can either append an effort source of zero voltage to one of the 0-junctions representingee, or we can simply erase all the bonds that carry no power, with the result shown in Figure 4.2d.

5. Simplify the bond graph by using the bond graph identities defined in Chapter 3.

This is not an absolutely necessary step but it is generally useful. By removing any 2-port 0- and 1-junctions that happen to have a through power convention, a much neater picture emerges, as shown in Figure 4.2e. (These identities were discussed in Section 3.3.)

Also, the loop structure fromeb toec has been reduced using another type of bond graph identity. Noticing that L1 and C2 actually have the same voltage drop, we establish the voltage drop eb − ec once using a 1-junction, and then establish a 0-junction having this voltage difference.

Finally, the —I and —C elements associated withL1andC2are attached to this 0-junction, yielding the bond graph of Figure 4.2e.

For equation derivation or automated simulation as described in Chapter 5, it is not necessary to reduce the bond graph to its simplest form, but it does make the final bond graph easier to use and understand.

Note that the flow source introduced to expose the output voltage,e_out, may also be erased since there is no power associated with the source bond.

It is left in Figure 4.2eas a convenience to remind us that we are interested in that particular output voltage. Later we will see that any effort or flow on any bond can be simply established as an output, and we will not have to construct an artificial means to expose desired outputs.

The trick to creating arbitrary circuit connections exclusively out of series and parallel connections (or 0- and 1-junctions in bond graph terms) lies in first representingevery node voltage, including what will be the ground or reference voltage, by establishing a set of 0-junctions. Only after representing voltage differences using 1-junctions for each circuit element in step 3, do we set the ground voltage to zero.

This means that different bond graphs will result for a given circuit if we chose different ground nodes. This corresponds to the fact that if different points in a real circuit are used as a reference point for voltage measurements, the mea- sured voltages will have different values. The different bond graphs will correctly predict the voltages with respect to the different choices for the ground node.

As a final example of circuit modeling using the procedure just presented, consider the Wheatstone bridge shown in Figure 4.3a. This circuit is typically used with strain gages as the resistive elements R1 throughR4, and the voltage across the load resistance,RL, is the output that is indicative of any change in the bridge resistances. We are simply going to model this circuit as an exercise in using the bond graph circuit construction procedure. In Figure 4.3a, the positive voltage drop and current directions are shown along with labels for the node voltages. This is step 1 of the procedure.

Step 2 of the procedure is done in Figure 4.3b, where the node voltages are represented by 0-junctions. Step 3 is shown in Figure 4.3c, where 1-junctions are used to appropriately add voltages such that the proper positive voltage drops are across each element. Also shown inc is the dotted line surrounding all the

0

0

0 e_a

e_d

e_b0 e_c

e(t)

R₁ R₃

R₂ R₄ R_L

+ +

+ +

− −

−

−

− a

b c

d i_e

i_R1

i_R2

i_R3

i_R4 i_L

+

0

0

0

0 e_a

e_d e_b

e_c R 1

S_e

1

1 1

1

1

R

R R

R R₁

R₃

R₄

R₂ R_L

e(t)

:

:

: :

:

:

0

0 e_a

e_b e_c

1 R

S_e

1

1

R

R R

R

R₁ R₃

R₄ R₂

R_L 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.