Drawing Bond Graphs for Hydraulic and Electronic Components and Systems
4.3 Electronic Systems
4.3.1 Operational Amplifiers
Operational amplifi ers are sometimes called the “work horse” of electronic applications. Operational amplifi ers (op-amps) form the core of many sig-nal conditioning applications, such as amplifi ers, fi lters, integrators, dif-ferentiators, and so forth. Using the same basic op-amp and with proper combination of resistors, capacitors, and feedback loops, a large number of applications of op-amps can be implemented.
The op-amp is an integrated circuit (IC) that consists of many electronic components. Overall behavior of op-amps is characterized by a very high input resistance and a very low output resistance. In the open circuit mode (although op-amps are never used in this mode), they can provide voltage amplifi cation in the order of millions. By using proper external circuit elements (resistors and/or capacitors), the user can control this amplifi ca-tion factor to almost any desired value.
The operational amplifi er is an IC that is rectangular with eight pins used for specifi c connections. Figure 4.14 shows a schematic of it. A tri-angular symbol is used to represent the op-amp in an electronic circuit.
Figure 4.15 shows a schematic of what an op-amp actually looks like.
The specifi c purpose of each pin is shown in the fi gure as well. As was mentioned before, the internal circuit of an op-amp is rather complex and there is no need to go into the circuit details. Figure 4.16 shows a schematic that will capture its basic overall behavior. There are two input voltage sources shown in the fi gure as V + and V − . The resistance on the input side is connected to these two source pins, such that the voltage across the input resistance is (V +−V − ). This is represented as V in in the fi gure. On the output side of the op-amp is another voltage source con-nected to the output resistance. The voltage source on the output side is given as kV in , where “k” is the open circuit amplifi cation factor of the
Positive power supply, Vs+
Inverting input
Noninverting input
Negative power supply, Vs−
Output
FIGURE 4.14
Schematic of an op-amp as used in circuits.
Offset null No connection
Vs+ 1 a1
a2
a3
a4 b4
b3 b2 b1 5
6
7
8 2
3
4
Output
Of fset null Inverting input
Noninverting input
Vs−
FIGURE 4.15
Schematic of an op-amp as it looks.
FIGURE 4.16
Schematic showing how an op-amp is modeled.
V+
Vin
V− Rin
kVin
Vout Rout
op-amp. The output voltage (with respect to the ground) is measured at the output pin.
The equivalent bond graph model for the basic op-amp is shown in Figure 4.17. The left-hand side of the fi gure until the MSe represents the input to the op-amp. The right-hand side represents the output side.
The two Se elements represent the two voltage sources. The bond directions are chosen so that the voltage across the resistance on the input side is a difference of the two voltage values. The effort sensor shown in the model as the e inside a circle uses the effort signal in the bond as a signal source to the MSe element. MSe is a modulated source of effort.
Within Mse, the input effort signal is multiplied with the open circuit gain value to compute the output voltage, which is applied across the output resistance. A zero fl ow source is used at the output pin so that the effort value at that pin can be easily obtained in the model. The MSe model is expressed through the following statements:
Parameters
real open_circuit_amp; // open_circuit_amp is the open circuit amplification factor, k.
Variables real flow;
Equations
p.e = effort * open_circuit_amp; // effort information is brought in from the effort sensor in the model
flow = p.f;
where open _ circuit _ amp is the op-amp magnifi cation factor in the open circuit mode (designated as k in our earlier discussion). Simulation
FIGURE 4.17
Bond graph model of an op-amp.
MSe
results will show that the output voltage at Sf is the magnifi cation factor times the input voltage difference.
In actual applications, op-amps are almost never used in open circuit mode. They are used with resistors (and/or capacitors) usually connected in a feedback mode. The relative values of these external resistors/capacitors control the amplifi cation factors in actual applications. One possible confi gu-ration is shown in Figure 4.18. This is called an inverting amplifi er. In this case, it can be shown that the ratio of the output voltage to the input voltage at the negative terminal is equal to the negative ratio of R (feedback) and R (source) . The bond graph representation of this model is also shown in Figure 4.18.
Feedback resistance and the supply resistances are shown in the fi gure.
The parameters used for simulation are shown in Figure 4.19.
From the parameters it can be seen that the output voltage will be −5 (i.e., −10/2) times the input voltage. Plots of the varying input and output
V (input) R (source)
R (feedback)
V (output) + +
−
−
MSe
Input voltage 1 1 0 Supply resistanceR
Feedback resistanceR
1
0 1 Junction1
e 1
Output resistance lowR
Sf
0 Junction 20
Input resistance highR Se
Input voltage 2 Wave generator 1
Magnified MSe voltage source
0 Junction 5
1 Junction 2 0 flow source
FIGURE 4.18
Inverting amplifi er circuit and its bond graph model.
voltages are shown in Figure 4.20. The plot clearly shows that the input voltage is amplifi ed by a factor of −5. A particular advantage of this type of circuit is that the actual value of the op-amp application factor does not affect the fi nal circuit amplifi cation factor, which can be controlled through the ratio of the supply and the feedback resistances.
FIGURE 4.19
Parameter values used for simulating the inverting amplifi er.
Model
0 2 4 6 8 10
Time (s)
−10
−5 0 5 10
Output voltage Supply voltage (V)
FIGURE 4.20
Simulation results from the inverting amplifi er model.
4.3.2 Diodes
Diodes are another vital component in many electronic circuits. Diodes are semiconductor devices typically used to control the fl ow of current much the same way that one way valves are used to control fl uid fl ow in a hydraulic system. Under normal operating conditions, diodes can allow large currents when the potential across them is one way (forward bias) and little or no current when the potential difference is in the opposite direction (reverse bias). Diodes are constructed using a junction of p-type and n-type semiconductors in a manner shown in Figure 4.21. The fi gure also shows the symbol of a diode that is used in any circuit diagram, with the tip or vertex of the triangular symbol pointing toward the forward bias direction. For a full explanation of p-n junction behavior and how it applies to diode characteristics, one may refer to more fundamental texts on electrical circuits and/or electronics such as Rizzoni’s Principles and Applications of Electrical Engineering. For these junctions, the current fl ow-ing through the diode is somewhat like the one shown in Figure 4.22.
A mathematical representation of this is
I= I0(e
qV
kT − 1) (4.22)
p + −
+ −
n p n
FIGURE 4.21
PN junctions with forward and reverse bias and the symbol of a diode.
I
V
~0.6 V Breakdown voltage
Small reverse bias current ~ 10E-9 A
FIGURE 4.22
Current profi le in a diode.
where q is the charge of a single electron, 1.6E-19C, k is the Boltzmann’s constant, 1.381E −23 J/K, V is the applied voltage and T the absolute temper-ature (= 298K for normal ambient conditions), and I 0 is the revere bias leak-age current of the order of 1E-9 – 1E-15 A. The current grows exponentially after a forward bias voltage of about 0.6V is exceeded under forward bias condition. The current is small or negligible (1E-9 – 1E-15 A) in reverse bias until a breakdown voltage is reached when the current is infi nite. In many modeling applications, diodes are treated as ideal devices with infi nite cur-rent under positive bias and 0 curcur-rent under negative bias (i.e., short circuit under positive bias and open circuit for negative bias) conditions. This is one possible way of modeling the device in the bond graph method as well.
A more realistic way, perhaps, is to model the device as an actual diode using the diode current equation to model it as a resistive element with the cur-rent equation as the constitutive model. This approach could pose a problem with convergence because the current value shoots up to a very high value and may cause diffi culties with convergence when this model has to interact with other standard circuit elements. So the model recommended here is based on the concept of the diode’s behavior as an open circuit/short circuit resistive device. A standard resistance model is modifi ed as:
Variables
real r1; // r1 controls the resistance magnitude Equations
if (p.e>0.6) then // i.e. the forward bias voltage is greater than 0.6 V
r1 = 0.000000001; // the r1 value is such that the diode works as a short circuit
else
r1 = 10000000000; // the r1 value is such that the diode works as a open circuit
end;
p.f = p.e/r1; // the current is determined from the voltage difference and the diode resistance
One of the most common applications of a diode is in a rectifi er circuit, where an AC source may be rectifi ed into a DC voltage across a load resis-tor. The simple rectifi er circuit is shown in Figure 4.23. The bond graph representation of this circuit is shown in Figure 4.23 as well, where the resistance marked Diode is the diode model. Figure 4.24 shows simula-tion results from this model where the output across the load resistance is a value that is greater than or equal to 0 when a sinusoidal input is applied to the circuit. For negative applied voltage the output across the load resistance is zero since there is no current through the diode. For positive applied voltage the output across the load resistance matches the general profi le of the input voltage very closely.
In this chapter we have discussed how the various concepts of bond graph modeling may be applied to model the dynamic behavior of systems/components that are common in hydraulic applications as well as electronic applications.
R loadR
1 R
Diode MSe
MSe 1 AC source
Diode
AC R load
AC source
FIGURE 4.23
A simple rectifi er circuit and its corresponding bond graph.
Model
0 5 10 15 20
Time (s)
−5
−4
−3
−2
−1 0 1 2 3 4 5
Output voltage across R load Applied voltage
FIGURE 4.24
Simulation results from the rectifi er circuit.
Problems
4.1. Develop a bond graph model for the hydraulic circuit shown in Figure P4.1.
4.2. Develop a bond graph model of this or the hydraulic circuit shown in Figure P4.2.
25
d 10
FIGURE P4.1
Figure for Problem 4.1, hydraulic circuit.
60 m
100 m
200 m 30 m
FIGURE P4.2
Figure for Problem 4.2, hydraulic circuit.
4.3. Figure P4.3 is a circuit for a noninverting amplifi er. Develop a bond graph model for this integrator and then simulate the model.
Use a periodic signal to demonstrate that the amplifi cation factor is (1+R feedback /R source ).
4.4. Figure P4.4 is a circuit for an integrator. Develop a bond graph model for this integrator and then simulate the model. Use a square wave input to fi nd out what the output of this model is.
V (input) R (source)
R (feedback)
V (output) +
−
− +
R1
FIGURE P4.3
Figure for Problem 4.3, a noninverting amplifi er.
C (feedback)
−
−
+ V (input) +
R (source)
V (output)
FIGURE P4.4
Figure for Problem 4.4, an integrator.
4.5. Figure P4.5 shows the circuit for a low-pass fi lter. Using the bond graph model for this circuit, simulate its behavior. Typical values that may be used for the different components are: R (source) = 220 Ohms, R (feedback) = 68 kOhm, R (load) = 1 kOhm, C (feedback)
= 0.47 nF.
4.6. Figure P4.6 shows a circuit for a band pass fi lter. Using the bond graph model for this circuit, simulate its behavior. Typical values that may be used for the different components are: R (source) = 2.2 kOhms, R (feedback) = 100 kOhm, R (load) = 1 kOhm, C (feedback) = 1nF, C1 = 2.2 microF.
−
−
+ +
C (feedback) R (feedback)
V (input) R (source)
R (load) V (output)
FIGURE P4.5
Figure for Problem 4.5, a low-pass fi lter.
+ +
−
− C1
C (feedback) R (feedback)
V (input) R (source)
R (load) V (output)
FIGURE P4.6
Figure for Problem 4.6, a band-pass fi lter.
4.7. Figure P4.7 shows a circuit for a fi lter. Using the bond graph model for this circuit, simulate its behavior. From the results, can you decipher whether this is a low-pass or a high-pass fi lter?
Typical values that may be used for the different components are: R (source) = 9.1 kOhms, R (feedback) = 22 kOhm, R (load) = 2.2 kOhm, C (feedback) = 0.47microF.
5 10 15 20
15
−15 V
t(μs) +
+
−
− R (feedback)
V (input) C (source)
R (load) V (output)
FIGURE P4.7
Figure for Problem 4.7, circuit for a fi lter.
4.8. Figure P4.8 shows a circuit involving an op-amp. Using the bond graph model for this circuit, simulate its behavior. From the results, can you decipher whether this is a differentiator or an integrator?
Typical values that may be used for the different components are:
C1 = 1 microF, R (feedback) = 10 kOhm, R (load) = 1 kOhm. Using the profi le shown in the accompanying plot, determine what the output of this circuit is.
+ +
−
−
5 10 15 20
15
−15 V
t(μs) R (feedback)
V (input) C (source)
R (load)
V (output)
FIGURE P4.8
Figure for Problem 4.8, circuit involving an op-amp.
4.9. Figure P4.9 shows a rectifi er circuit where four diodes are in a circuit in the form of a wheatstone bridge. Develop a bond graph model for this circuit and for a sinusoidal voltage input determine the output across the load resistance shown in the fi gure.
4.10. Develop a bond graph model for the cramping circuit shown in the Figure P4.10. Use a sinusoidal input for the source and develop the output results from the simulation.
FIGURE P4.9
Figure for Problem 4.9, rectifi er circuit.
AC AC source
Diode 2
Diode 4 Diode 3
Diode 1
Load
FIGURE P4.10
Figure for Problem 4.10, cramping circuit.
Vout Diode
AC source
Vdc +−
R C
AC
4.11. In the diode circuit shown in Figure P4.11, the supply voltage is a saw-toothed wave as shown in the accompanying plot. Model this circuit using the bond graph method and simulate the model to determine the output of this circuit.
100 V 98 ohm
Vin
Diode
50V
Vout
− + 0.6 ohm
Time
FIGURE P4.11
Figure for Problem 4.11, circuit and waveform.
4.12. Develop a bond graph model for the hydraulic circuit shown in the Figure P4.12.
4.13. Develop a bond graph model for the hydraulic circuit shown in the Figure P4.13.
Tank 1
Tank 2
FIGURE P4.12
Figure for Problem 4.12, hydraulic circuit.
Tank 1
Tank 2
FIGURE P4.13
Figure for Problem 4.13, hydraulic circuit.
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