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Electronics
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K x M M B dxM M dt K X sin( t) M B X cos( t) M
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Figure 1238 Solution by repeated integration
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dxM dt
B M
K xM M
K M
Figure 1239 Analog computer simulation of suspension system
Let Table 121 represent the physical and simulation variables Considering time scaling rst, let t denote real time and , the computer time variable Then the time derivative of a physical variable can be expressed as dx dx d dx = = dt d dt d where is the scaling factor between real time and computer time: = t For higher-order derivatives, the following relationship will hold: d nx d nx = n n n dt d
Table 121 Actual and simulated variables in analog computers Physical system Physical variable, x Time variable, t Analog simulation Voltage, v Simulated time,
(1277)
(1278)
(1279)
12
Operational Ampli ers
While time scaling is likely to be prompted by a desire to speed up or slow down a computation, magnitude scaling is motivated by several different factors: 1 The relationship between physical variables and computer voltages (eg, calibration constants) 2 Overloading of the op-amp circuits (we shall see in Section 126 that one of the fundamental limitations of the operational ampli er is its voltage range) 3 Loss of accuracy if voltages are too small (errors are usually expressed as a percentage of the full-scale range) Thus, if the relationship between a physical variable and the computer voltage is v = x, where is a magnitude scaling factor, the derivative terms will be affected according to the relation dx 1 dv = dt dt (1280)
Note that different scaling factors may be introduced at each point in the analog computer simulation, and so there is no general rule with regard to magnitude scaling For example, if v = 0 x, it is entirely possible to have dx 1 dv = dt 1 dt
EXAMPLE 127 Analog Computer Simulation of Automotive Suspension
Problem
Implement the analog computer simulation of the automotive suspension system of Figure 1237
Solution
Known Quantities: Mass, spring rate, and damping parameters of automotive
suspension
Find: Component values of analog computer circuit of Figure 1239
B = 20 103 N/m-s F
Schematics, Diagrams, Circuits, and Given Data: M = 400 kg; K = 16 105 N/m; Assumptions: Assume ideal op-amps Express all resistors in M
and all capacitors in
Analysis: With reference to Figure 1239, we observe that the analog computer
simulation of the automotive suspension requires: a four-input summer, two integrators, two coef cient multipliers, and one sign inverter Expressing all resistors in M and all capacitors in F is useful because each integrator has a multiplier of 1/RC Using R = 1 M and C = 1 F results in 1/RC = 1 Figure 1240 depicts the integrator con guration The four-input summing ampli er uses 1-M resistors throughout, so that the gain for each input is also equal to 1 On the other hand, the two coef cient multipliers are required to have gains K/M = 4,000 and B/M = 50, respectively Thus we select 10-M and 25-k resistors for the rst coef cient multiplier, and 1-M and 20-k resistors for the second
Part II
Electronics
Finally, the inverter can be realized with two 1-M depicted in Figure 1240
resistors The various elements are
1 F 1 M 1 M vin + + (a) Integrator vout vb vc vd 1 M va 1 M 1 M +
1 M
+ vout
(b) Summer 10 M 1 M
25 k vin
+ + vout (c) Coefficient multiplier vin
20 k
+ + vout (d) Coefficient multiplier
Figure 1240 Analog computer simulation of suspension system
Comments: Note that it is not necessary to employ ve op-amps in this analog simulator The summing ampli er function and the rst integrator could, for example, be combined into a single op-amp, and one of the two coef cient multipliers could be eliminated This idea is explored further in the homework problems
EXAMPLE 128 Deriving a Differential Equation from an Analog Computer Circuit
Problem
Derive the differential equation corresponding to the analog computer simulator of Figure 1241
Solution
Known Quantities: Resistor and capacitor values Find: Differential equation in x(t)
1 M ; R4 = 25 k ; C1 = C2 = 1 F
Schematics, Diagrams, Circuits, and Given Data: R1 = 04 M ; R2 = R3 = R5 = Assumptions: Assume ideal op-amps Analysis: We start the analysis from the right-hand side of the circuit, to determine the
12
Operational Ampli ers
C1 R1
C2 R5 R3
f(t)
x(t)
Figure 1241 Analog computer simulation of unknown system
intermediate variable z as a function of x: R5 x = z = 400z R4 Next, we move to the left to determine the relationship between y and z: z= 1 R3 C2 1 R2 C1 y(t )dt or y= dz dt
Finally, we determine y as a function of x and f : y= or dy = x 25f dt Substituting the expressions into one another and eliminating the variables y and z, we obtain the differential equation in x: x = 400z dz dx = 400 = 400y dt dt d 2x dy = 400 (x 25f ) = 400 dt 2 dt and d 2x + 400x = 1, 000f dt 2
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