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Introduction
Power variables
Standard elements
Power directionsBond numbers
Causality
System equations
Activation
Example models
Art of creating modelsFields
Mixed-causa lled fields
Differential causality
Algebraic loops
Causal loops
DualityMulti and Vector bond graphs
Suggested readings
Generation of system equations
In this section the method of generation of system equations is discussed. From
an augmented bond graph, using a step by step procedure, system equationsmay be generated. We take the model of a simple single degree of freedom
mass-spring-damper system as the starting point.
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The differential equations describing the dynamics of the system are written interms of the states of the system. All storage e lements (I and C) correspond to
stored state variables (P for momentum and Q for displacement, respectively)
and equations are written for their time derivatives (i.e. effort and flow). These
equations are derived in four steps as described below.
Observe what the elements (sources, I's, C's, and R's) are giving to the
system and write down their equations looking at the causalities and using
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variables for strong bonds .
Write down equations for the junctions and the two-port elements for the
variables for the strong bonds.
Replace the variables, which are expressed in terms of states in other
equations. Continue sorting and replacement till the right side of the entire
set of equations are expressed in terms of states and system parameters
only.
If some equations are s till not completely reduced, there is the existence of
some kind of a loop (algebraic loop, causal loop or differential causality, as
shall be discussed later). Try solving those as a set o f linear equations e ither
through substitution or matrix inversion.
And finally, erase all trivial equations other than those for derivatives of
state variables and write them in terms of state variables.
Thus using the following steps, the equations for the above system would be
Step 1 :
e1=SE1
f3=P3/M3
e2=K2*Q2
e4=R4*f4=R4*f3 (by junction rule, f4=f3 and strong bond number is 3)
Step 2 :e1-e2-e3-e4=0 or e3=e1-e2-e4
Step 3 :
e1=SE1
f3=P3/M3
e2=K2*Q2
e4=R4*P3/M3
e3=SE1-K2*Q2-R4*P3/M3
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Step 4 :
DQ2=f2=f3
DP3=e3
where prefix 'D' stand for time derivative d/dt leads to
DQ2=P3/M3
DP3=SE1-K2*Q2-R4*P3/M3
The difference between equations derived from bond graphs and otherwise are
that there will be `N' sets of first order differential equations, where `N' is the
number o f states . The term, number o f states means the number of lumped
parameter storage elements I and C with integral causality present in a system.
The equation of motion for the system discussed above by traditional method is
:
m d2x/dt2 + r * dx/dt + k*x = F(t).
The two state equations derived from a bond graph model (dropping suffixes)
are
dP/dt = -r/m * P - k*Q + SE,
dQ/dt = 1/m*P,
where, P is momentum of m*dx/dt, Q is displacement or x and SE is F(t)).From the second equation, P=m*dQ/dt, which when replaced in the first leads to
m*d2Q/dt2 = -r*dQ/dt - k*Q + F(t).
This equation corresponds to that derived through traditional method after
rearrangement.
However, manual derivation of equations for larger systems is not all that
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simple. For instance, derivation of system differential equations for the animated
bond graph model discussed in the causality section after proper causalling
would lead to formation of the so called algebraic loops. Similarly, complexities
and errors of various types, like Causal loops, Power loops, and Differential
Causalities may exist in the model of a system. The method for derivation of
their equations is described in corresponding sections.
Example -2
Here we take another example of a system with two-ports, whose model is
shown below.
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Then we follow the normal equation derivation steps.
Step 1 :
e1 = SE1
f2 = P2/M2
e3 = R3*f3=R3*f2=R3*P2/M2
f6 = P6/M6
e11 = K11*Q11
e12 = R12*f12 = R12*f9
f13 = P13/M13
Step 2 :
f1 = f2 = f4 = f3 = P3/M3
e2 = e1 - e3 - e4 = SE1 - R3*P 2/M2 - e4
e4 = MU*f5 = MU*f6 = MU*P6/M6
e5 = MU*f4 = MU*f2 = MU*P2/M2
f5 = f7 = f6 = P6/M6
e6 = e5 - e7f8 = r*f7 = r*P6/M6
e7 = r*e8 = r*e9
e8 = e10 = e9
f9 = f8 - f10 = r*P6/M6 - f13 = r*P6/M6 - P13 /M13
f10 = f14 = f13 = P13/M13
e13 = e10 + e14 = e9 + SE14
f11 = f12 = f9
e9 = e11 + e12 = K11*Q11 + R12*f9
Step 3 :
e1 = SE1
f2 = P2/M2
e3 = R3*f3=R3*f2=R3*P2/M2
f6 = P6/M6
e11 = K11*Q11
e12 = R12*f9 = R12*(r*P6/M6 - P13/M13)
f13 = P13/M13
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f1 = f2 = f4 = f3 = P3/M3
e2 = SE1 - R3*P 2/M2 - e4 = SE1 - R3*P 2/M2 - MU*P6/M6
e4 = MU*f5 = MU*f6 = MU*P6/M6
e5 = MU*f4 = MU*f2 = MU*P2/M2
f5 = f7 = f6 = P6/M6
e6 = e5 - e7 = MU*P2/M2 - r*(e11 + e12)
= MU*P2/M2 - r*(K11*Q11 + R12*(r*P6/M6 - P13/M13))
f8 = r*f7 = r*P6/M6
e7 = r*e8 = r*e9 = r*(e11+e12) = r*(K11*Q11 + R12*(r*P6/M6 - P13/M13))
e8 = e10 = e9 = e11+e12 = K11*Q11 + R12*(r*P6/M6 - P13/M13)
f9 = f8 - f10 = r*P6/M6 - f13 = r*P6/M6 - P13 /M13
f10 = f14 = f13 = P13/M13
e13 = e10 + e14 = e9 + SE14 = K11*Q11 + R12* (r*P6/M6 - P13/M13) + SE14
f11 = f12 = f9 = r*P6/M6 - P13/M13
e9 = e11 + e12 = K11*Q11 + R12*f9 = K11*Q11 + R12*(r*P6/M6 - P13/M13)
Step 4 :
DP2 = SE1 - R3*P 2/M2 - MU*P6 /M6DP6 = MU*P2/M2 - r*(K11*Q11 + R12*(r*P6/M6 - P13 /M13))
DP13 = K11*Q11 + R12*(r*P6/M6 - P13/M13) + SE14
DQ11 = r*P6/M6 - P13/M13
In matrix form, these equations may be w ritten as d{Y}/dt = [A]{Y} + [B]{u},
where, {Y} is a vector of states (P2,P6,P13 and Q11), [A] is square matrix, {u}
is the array of sources (SE1 and SE14} and [B] is a matrix of dimension N x M; N
being the number of states and M being the number of sources. The matrices [A]
and [B] are;
A =
-R3/M2 -MU/M6 0.0 0.0
MU/M2 -R12*r2/M6 R12*r2/M13 -r*K11
0.0 R12*r/M6 -R12/M13 K11
0.0 r/M6 -1/M13 0 ,
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B =
1 0
0 0
0 1
0 0 .
Activation
Some bonds in a bond graph may be only information carriers. These bonds are
not power bonds. Such bonds, where one of the factors of the power is masked
are called Activated bonds. As an example, let the system shown in the figure
below be considered.
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Here the velocity pick-up only carries the information of velocity to the amplifier
through which an electro-magnetic exciter applies force proportional to velocity
on the mass and the exciter does not carry the information of velocity and
impose it back on the mass as an reactive force. So on the bond representing
the velocity pick-up the information of force must be masked and on the bondrepresenting the exciter the information of the flow must be masked. A full arrow
somewhere on the bonds shows that some information is masked and what
information is masked may be written near that full arrow. According to this
convention the bond graph of the system is shown to the right of the system
drawn above.
The bond number 4 in the figure is the pick-up bond where information of effort
is masked and in bond number 5, which is the exciter bond, the flow is activated.
The concept of activation is very significant to depict feedback control systems.
The term activation initially seems a misnomer. However, Paynter's idea was
based on the fact that though the information of a factor of power is masked on
one end, an activated bond on the other end can impart infinite power which is
derived from a tank circuit used for both the measurement or actuation device
(for instance, the pick-up, the amplifier and the exciter, all have external power
sources).
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In the system shown above, a D.C. motor with externally energized field is
driving an elastic shaft with two disks and with a fluctuating resistive load. The
speed, picked up by a tachometer is fed-back to control the speed of the motor
by adjusting the armature current (increasing voltage alternatively). The bond
graph for this system is drawn as shown below.
Observers
Additional states can be added for measurement of any factor of power on a
bond graph model using the Observer storage elements. An effort activated C-
element would observe the time integral of flow (and consequently flow),
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whereas a flow activated I-element would observe the generalized momentum
(and consequently effot). Activated elements are percieved conceptual
instrumentations on a model. They dont interfere in the dynamics of the system
(i.e. their corresponding states never appear on the right-hand side o f any state
equation. A system with N states, M sources and L observers would have the
state equations of the form
d/dt{Y} = [A]{Y} + [B]{u} ,
d/dy{Z} = [C]{Y} + [D] {u} ,
whre {Y} is a vector of true states, {Z} is the vector of observed states, {u} is
a vector of sources, [A] is MxM matrix, [B] is MxN matrix, [C] is ZxM matrix and
[D] is ZxN matrix.
Here is an example of a single-degree-of-freedom system with instrumentations,
whose model with observer elements is shown below.
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The conceptual instrumentations shown here, assumes a spring of zero stiffness
(thus measuring displacement and not generating any reactive effort) and an
inductive coil within which the segments of the damper move to induce emf (a
measurement of force) without any reaction on the damper segment. The
equations for this model can be de rived as shown be low.
Step 1 :
f1 = SF1
e3 = K3*Q3
e5 = R5*f5
f6 = 0
e9 = SE9
e8 = 0
e10 = 0
f12 = P12/M12
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e11 = MU*f10 = MU*f12 = MU*P12/M12
Step 2 :
e12 = e7 + e9 - e8 + e11 - e10 = e2 + SE9 + MU*P12/M12
f2 = f1 - f7 = SF1 - P12/M12
f5 = f4 - f6 = f4 = f2 = SF1 - P12/M12
e2 = e3 + e4 = e3 + e5 = K3*Q3 + R5*f5 = K3*Q3 + SF1 - P12/M12
Step 3 :e12 = K3*Q3 + R5*(SF1-P12/M12) + SE9 + MU*P12/M12
e5 = R5*(SF1 - P12/M12)
Step 4 :
The State equations are
DP12 = e12 = K3*Q3 + R5*(SF1-P12/M12) + SE9 + MU*P12/M12
DQ3 = f3 = f2 = SF1 - P12/M12
and the observer equations are
DP6 = e6 = e5 = R5*(SF1 - P12/M12)
DQ8 = f8 = f12 = P12/M12
Bond graph modelingA simple mechanical system show n in the figure below is considered here for
explaining the modeling procedure.
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The output from the pump would be flow and pressure, while the input would be
torque and angular velocity applied to the pump shaft. The input power is
provided by an electric motor. The power flow diagram is shown in the figure
below.
Two transformations are identified. First from electric power to mechanical power
through the electric motor; followed by the mechanical pow er to the fluid power
through the pump. In the next step to construct the model, the components are
to be looked into more details. In this simple analysis of the pump it has its own
moving parts (inertia); oil flow has got its own compress ibility (capacitance)
because of which it can store energy; pump has leakage due to its geometric
tolerance and friction between the moving parts (resistance) through which it
can dissipate energy. The bond graph of the system with power directions and
causality is show n below.
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A two degrees of freedom mechanical system and its corresponding Bond graph
model shown be low can be explained in the following manner.
The bottom mass velocity is same as the rate of stretching of the spring and
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The bottom mass velocity is same as the rate of stretching of the spring and
damper connected to the ground. So, these are modeled as being connected to
the 1-junction. The middle spring and damper experience components of force,
say F1 and F2, respectively such that F=F1+F2, i.e., the force felt by the bottom
mass, the spring and damper in combination and the top mass is equal.
Alternatively, the distortion of the spring and the damper is dependent on
relative motion of the masses. Hence, one can easily smell the presence of a
force pass or flow summation junction, in other words the 0-junction here.
Further, this 0 junction has three branches corresponding to three componentsof distribution of effort.
Another 1-junction appears in the right, that signifies the equal relative velocity
(i.e., rate of compression here considering the power directions) felt by the
middle spring and the damper. The 1-junction at the top depicts the coherent
motion of the forcing device w ith the top mass.
For the above system, alternative methods can be applied to arrive at a bond
graph model, that may apparently look different. However, the final equationsderived from different models lead to the same set. It may be noted that bond
graphs for mechanical systems may be drawn using the method of flow balance
(kinematics) or force transmission, or a combination of the both. Since the basic
external elements used are the same and only the energy conserving junction
structure may vary, considering one factor of power automatically takes care of
the effect of the other factor. This is perhaps the greatest contribution of the
bondgraph theory.
Art of Creating Models
Representing systems in form of models is an art. Neat and good-looking models
are arrived at following certain established practices and reduction methods.
However, ultimately the model author dese rves the credit for putting pieces
together to create good visual effect Some reduction schemes which may be
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together to create good visual effect. Some reduction schemes, which may be
used to compress or uncompress bond graphs, are presented here. This work
out in the reverse in helping to understand the bond graphs drawn by others. In
this section the philosophy of model building and reduction is presented. Most of
the matter presented here are extracts from "Lecture notes on sytem modeling"
by Prof. A. Mukherjee and Prof. R.Karmakar of the Indian Institute of Technology,
Kharagpur.
Model reduction steps
A junction with only two bonds attached to it may be replaced by a single
bond provided this does not lead to attaching any two external elements (I,
C, R, SE, SF, GY or TF) to each other without a junction. These reductions
are summarized in the table below. In this table J stands for any junction (1
or 0) and E for an external element. Reader may verify these reductions by
simply writing the junction laws for such junction.
Two neighboring identical junctions may be merged, the internal bond
between two such junction is dropped in this process (see section E and F in
table).
Any 1-junction to which a source of flow is attached which has constant zero
value and no external element is attached may be removed from the bond
graph with all bonds attached to it (see section G in table).
Any 0-junction to which a source of e ffort with constant zero value is
attached and no external element is attached may be removed from thebond graph with all bonds attached to it (see section H in table).
Junction Structure Reduced Form
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A
B
C
D
E
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F
G
H
The reductions (3) and (4) must be done after the bond graph is already
reduced due to (1) and (2). In a sense there is a hierarchy in the reduction
process. Some more elementary equivalent forms shown in the table below.
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process. Some more elementary equivalent forms shown in the table below.
These equivalent forms may be established by w riting junction laws and
accounting for the fact that a source of flow (SF) can meet any demand of effort
by the system and a source of effort (SE) can meet any demand of flow. Thus
the following rules hold.
A zero source of flow may be removed from a 0 junction.
A zero source of effort may be removed from an 1 junction.
1-junction is a distributor of SF.
0 junction is a distributor of SE.
Junction Structure Reduced Form
A
B
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C
D
Two more reductions w idely used in bond graph modeling are shown below.
Junction Structure Reduced Form
E
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F
Bondgraphs for mechanical systems
To create a bond graph for a mechanical system it is advisable that a good
schematic sketch of the system or a word bond graph be made. A good word
bond graph is a great aid when one deals with systems in multi energy domain.
The following three methods are most often used effectively to create bond
graphs for mechanical systems :
Method of Flow Map
Method o f Effort Map
Method o f Mixed Map
- Method of flow map
The method of flow map is based on the fact that the single port mechanical C or
Relement may be attached to a 1-junction where the relative velocity of ends of
these elements are available. Single port I elements may be attached to 1-
junctions, where absolute velocities of generalized inertias are available.
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Following steps may be followed to create system bond graph by Method of
Flow Map.
Create 1-junctions depicting the components of velocities of inertial points.
Create 1-junctions depicting the motions of end points ofC and Relements.
Relate and connect these junctions by transformer (TF)elements if a lever
relations exist be tween them. Gyroscopic relations and feedbacks may be
incorporated using gyrator (GY) elements.
Create the relative velocities between the end points ofC and Relements.
0-junction may be used for adding or subtracting the velocities.
Attach C and Relements to 1-junctions where relative velocities of their end
points are created.
Reduce the bond graph us ing reduction process discussed earlier.
Often it may be of great advantage if superscripts or subscripts are shown at 1
junctions depicting the points of the system and components of motion.
However, these superscripts or subscripts are purely for book-keeping and are
ignored during subsequent processing.
Example 1 : Let us consider a system shown in the figure below. Two 1-
junctions are created depicting velocities at mass-point and ground excitation.
Difference of these two velocities are taken using a 0-junction and relativevelocity is established at the 1-junction shown as 1mv. The velocity depicted at
1mv is d(xm)/dt - d(xv)/dt. Next, one attaches the external e lements. An
alternative bond graph may be created by making two 1mv junctions and then
attaching C to one and Rto the other. This bond graph may then be reduced to
the earlier one.
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(a) (b)
(c) (d)
Creating bond graph of a spring-mass-damper system.
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(e) (f)
Alternative bond graph for the spring-mass-damper system.
In case V(t)=0, i.e., the end of the spring is attached to the inertial frame of
observation, the final bond graph forms given above (in figs (d) and (f)) may be
reduced to a very simple form. This reduction is shown below.
(a) (b)
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(c) (d) (e)
Bond graph of a spring-mass-damper system anchored to the ground.
(a) (b) (c)
Alternative bond graph for the ground anchored spring-mass-damper system.
Example 2 : An idealized car model with rigid body and flexible suspensions with
excitations from the road is shown in the figure below. It's bond graph is
created by Method of Flow Map as shown below the schematic diagram of the
system.
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Two dimensional model of a car
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(a )
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(b)
(a) bond graph model of the car, (b) reduced bond graph
- Method of effort map
When a generalized force applied on an inertial point is such that the positive
values of the force accelerates the inertial point in a positive sense, then the
power directions on the bonds attached to the 1-junction will be as shown in
the figure below. To its right, the case of a force is shown which when positive
produces acceleration in the negative coordinate direction of motion.
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The method of effort map may be completed by the following steps .
If necessary create a 0-junction for sources of effort. Such junctions may be
treated as d istributors of efforts.
Decide whether the tensile force in C is to be taken as positive or it is the
compressive force which is to be taken positive. In linear springs such a
choice may be arbitrary.
For R elements, decide what kind o f relative velocity (compress ive or
stretching) is to be taken as positive. In linear systems this choice may be
arbitrary.
Addition of forces may be done using 1-junction.
Linear forces may be converted to couples using transformer (TF) elements.
Similarly, angular velocities may be converted to gyroscopic forces using
gyrator (GY) elements.
Reduce the bond graph using the rules discussed earlier.
Example 3 : The system of shown below in figure (a) is modeled in steps as
shown in figures (b) and (c).
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(a) (b)
(c)
In this bond graph tensile force in the spring and tractile force on the damper is
taken as positive. C and Relements are directly put on 0-junctions depicting the
forces in them. An alternative bond graph is drawn for this system in the figure
below. In this bond graph the forces due to the spring and damper are added
and then distributed using a 0-junction. The graph is then reduced to a form
shown to the right.
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Example 4 : The car model discussed in the method of flow map is drawn using
Method of e ffort map in the figure below. Here the compressive forces in
suspension are taken to be positive. An alternative bond graph is shown in the
next figure and is then reduced to a s impler form.
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(a) Model for vertical dynamics of a car
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(b) Alternative model for vertical dynamics of a car
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(c) Reduced model
- Method of mixed map
The 1-junctions on which the forces accelerating generalized inertial points are
added correspond to velocities of these inertial point in Method of Effort Map.
Now if a part of a system is modeled by Method of Effort Map, then the 1-
junctions depicting these velocities may be used to create the bond graph for
the rest of the system following Method of Flow Map.
The following example illustrates this process.
Example 5 :For the system shown below, the bond graph of the car is completed
by Method of Effort Map. The idealized passenger placed on this car represented
as mass spring system as shown in the figure is then modeled by Method of
Flow Map.
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(a) Schematic diagram of car with a passenger
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(b) Bond graph of car with a passenger
Bond graphs of electrical circuits
Following three methods are found suitable while drawing bond graphs for
electrical circuits. They can also be extended to other circuit-morphic systems in
the field of hydraulics and electronics.
Method of Gradual Uncover,
Point Potential Method,
Mixed Network Method.
- Method of gradual uncover
h h d ( ll ) l l f
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In this method one may cover (conceptually) comparatively complex parts of
circuit and then treat them as macro impedances. Make the bond graph putting
such impedances at proper places. Then uncover them and put the details on
the bond graph. However one may have sub-covers covering complex parts of
macro impedances which may then be gradually uncovered. The following
example illustrates this process.
Example 6 : An electrical circuit is shown in the figure below and to its right a
covering scheme is shown. Then in 5 steps (c to g in figure) the bond graph is
created by gradual uncovering.
(a) (b)
(c) (d)
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(e )
(f)
(g)
- Point Potential Method
Method of Gradual Uncover fails in many cases. In bridge circuits, for example, a
satisfactory scheme of creating covers becomes impossible. The alternative Point
Potential method is guaranteed to work in every case. Following are the steps
of Point Potential Method.
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Each point of circuit where ends of various elements or sub-circuits are
tagged may be made into a 0-junction.
Any element or impedance through which current flows may be attached to a
1-junction between the 0-junctions.
A suitable tag point may be grounded. This is achieved by attaching a zerosource of effort to a 0-junction. Unless grounded, the circuit remains floating.
Each individual circuit must be separately grounded, for example circuits on
primary and secondary sides of a electrical transformer are treated as two
separate circuits and must be grounded on both sides.
Reduce the bond graph.
The following example illustrates this approach.
Example 7 : Let a bridge circuit shown in the figure be low be considered. Point
Potential Method produces the bond graph shown to the right. The reduced
bond graph and rearranged graph are shown next.
(a) (b)
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(c)
(d)
- Mixed Network method
The method of point potential and gradual uncover may be often mixed to a
great advantage. The complex impedances may be covered in the first go. Once
the overall structure is produced by Point Potential Method, the uncovering may
be taken up. The ultimate bond graph may then be reduced. The following
example illustrates this method.
Example 8 : Bond graph for the bridge circuit shown earlier is developed this
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p g p g p
way. The figure given below shows the scheme of covering. The bond graph is
then developed in stages (b) to (d) shown in the figure.
(a) (b)
(c) (d)
Hydraulic circuits are drawn more or less in the same manner as electrical
circuits. Before concluding, here are two important w arnings to the reader.
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Odd power loops should be avoided: If there is a loop of junctions then two
important points should be observed.
There should be even number of bonds forming the loop.
Bonds power directed clockwise or counter clockwise must be even innumber.
Causa l loops should be avoided. In a junction loop, not a ll junctions should
have their causalities determined by internal bonds (i.e. bonds which are
part of the loop).
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