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(1268)
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and iF (t) = vout (t) RF (1269)
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Figure 1235 Op-amp differentiator
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so that the output of the differentiator circuit is proportional to the derivative of the input: vout (t) = RF CS dvS (t) dt (1270)
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Although mathematically attractive, the differentiation property of this opamp circuit is seldom used in practice, because differentiation tends to amplify any noise that may be present in a signal
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1211 Plot the frequency response of the ideal integrator in dB plot Determine the slope of the curve in dB/decade You may assume RS CF = 10 1212 Plot the frequency response of the ideal differentiator in a dB plot Determine the slope of the curve in dB/decade You may assume RF CS = 100 1213 Verify that if the triangular wave of Example 126 is the input to the ideal differentiator, the resulting output is the original square wave
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Prior to the advent of digital computers, the solution of differential equations and the simulation of complex dynamic systems were conducted exclusively by means of analog computers Analog computers still nd application in engineering
Part II
Electronics
practice in the simulation of dynamic systems The analog computer is a device that is based on three op-amp circuits introduced earlier in this chapter: the ampli er, the summer, and the integrator These three building blocks permit the construction of circuits that can be used to solve differential equations and to simulate dynamic systems Figure 1236 depicts the three symbols that are typically employed to represent the principal functions of an analog computer
CF dvout dt vout(0)
vin =
vin dv (t) vout = 1 out dt RSCF 0 dt The integrator
+ + vout
Initial condition
va vb vc
vout = (ava + bvb + cvc) va vb vc The summer
RS1 RS2 RS3
+ + vout
RS a vin vout = avin vin + + vout The coefficient multiplier
Figure 1236 Elements of the analog computer
The simplest way to discuss the operation of the analog computer is to present an example Consider the simple second-order mechanical system, shown in Figure 1237, that represents, albeit in a greatly simpli ed fashion, one corner of an automobile suspension system The mass, M, represents the mass of one quarter of the vehicle; the damper, B, represents the shock absorber; and the spring, K, represents the suspension spring (or strut) The differential equation of the system may be derived as follows: d 2 xM +B dt 2 dxR dxM dt dt
xM M xM xR
+ K(xM xR ) = 0
(1271)
Figure 1237 Model of automobile suspension
Rearranging terms, we obtain the following equation, in which the terms related to the road displacement and velocity xR and dxR /dt, respectively are the forcing
12
Operational Ampli ers
functions: M d 2 xM dxM dxR +B + KxM = B + KxR dt 2 dt dt (1272)
Assume that the car is traveling over a washboard surface on an unpaved road, such that the road pro le is approximately described by the expression xR (t) = X sin ( t) (1273)
It follows, then, that the vertical velocity input to the suspension is given by the expression dxR = X cos ( t) dt and we can write the equation for the suspension system in the form M d 2 xM dxM +B + KxM = B X cos ( t) + KX sin ( t) dt 2 dt (1275) (1274)
It would be desirable to solve the equation for the displacement, xM , which represents the motion of the vehicle mass in response to the road excitation The solution can be used as an aid in designing the suspension system that best absorbs the road vibration, providing a comfortable ride for the passengers Equation 1275 may be rearranged to obtain B dxM B K K d 2 xM = xM + X cos ( t) + X sin ( t) 2 dt M dt M M M (1276)
This equation is now in a form appropriate for solution by repeated integration, since we have isolated the highest derivative term; thus, it will be suf cient to integrate the right-hand side twice to obtain the solution for the displacement of the vehicle mass, xM Figure 1238 depicts the three basic operations that need to be performed to integrate the differential equation describing the motion of the mass, M Note that in each of the three blocks the summer and the two integrators the inversion due to the inverting ampli er con guration used for the integrator is already accounted for Finally, the basic summing and integrating blocks together with three coef cient multipliers (inverting ampli ers) are connected in the con guration that corresponds to the preceding differential equation (equation 1276) You can easily verify that the analog computer circuit of Figure 1239 does indeed solve the differential equation in xM by repeated integration Scaling in Analog Computers One of the important issues in analog computing is that of scaling Since the analog computer implements an electrical analog of a physical system, there is no guarantee that the voltages and currents in the analog computer circuits will be of the same order of magnitude as the physical variables (eg, velocity, displacement, temperature, or ow) they simulate Further, it is not necessary that the computer simulate the physical system with the same time scale; it may be desirable in practice to speed up or slow down the simulation Thus, the interest in time scaling and magnitude scaling
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