6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonAt one time, analog computers were the only machines that we had for performing simulations ofreal world systems. Although almost never used now, analog computation did stimulate a great dealof innovation in the design of low-offset, high DC gain amplification. Many of the techniques thatwere developed are still in use today.For our class exercise, we’re going to look at one building block that is very useful in analogcomputation: the three-mode integrator.CLASS EXERCISEDescribe the operation of this circuit in each of the following three modes: (A) switch (1) open andswitch (2) closed; (B) switch (1) closed and switch (2) open; (C) switch (1) and switch (2) open.21+−+−+−R2R2R1vAvBC6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonPage 2(Workspace)Analog computers were frequently used for simulating systems described by differential equations.Recall that an nthorder differential equation requiresninitial conditions to be specified. Mode (A) ofthe above circuit would be used for setting the initial condition on one of the integrators.Now, let’s see how we might go about simulating some common physical systems.Mass-Spring System with DampingWrite equation of motion: mx = F(t) − kx − bxmx +x = 0F(t )ForcingfunctionspringViscousdamping6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonPage 3Dividing through by the mass m, we get: x =1mF(t ) −bmx −KmxHow could we set up an electronic analog to this physical system? Consider:Note that the second-order equation required two integrators to realize. The initial conditions,namely x(0)and x(0), could be set with the aid of the 3-mode integrators. Using op-amps, we mightwind up with a realization like the following: xdt∫dt∫bmKm x xΣ1mF(t )+−−xf (t)m+−+− −x+−−+6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonPage 4It turns out that there is a general synthesis procedure for linear differential equations that take usfrom the equation to an equivalent block diagram. Suppose that we start with andnxdtn+ an−1dn−1xdtn−1+ … + a1dxdt+ a0x = f (t)Our first step is to solve for the highest order derivative: dnxdtn= −an−1andn−1xdtn−1− … −a1andxdt−a0anx +1anf (t)In the resulting block diagram, we make dnxdtn the output of a large summation junction, and followthat with a string of integrators to generate the lower order derivatives:As a final note, bear in mind that the independent variable of the system that we’re studying needn’talways be time. In some systems, the independent variable may be position, or temperature, or anynumber of things. In these cases, the proper interpretation of “time” in our analog simulation issimply applied to the simulation result.aoanan − 2anan −1anxdt∫dt∫dn −1xdtn −1dt∫dnxdtn−−−+1anf (t)6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonPage 5Our methods are not confined merely to linear differential equations. Consider modeling thecommon pendulum systemWriting out the torque about the point P: Iθ= − mgl sinθThe moment of inertia, I, for a mass at the end of a (massless) rod is ml2: ml2θ= − mgl sinθθ= −glsinθNormally at this point we invoke “small” displacements in and write the linearized equation θ= −glθThis system has a natural frequency of ω=gl. But what if we wanted to contend with the originalnonlinear system? Same as before: θ= −glsinθRigid, masslessrodmθˆθ+P6.301 Solid State CircuitsRecitation 11: Analog ComputationProf. Joel L. DawsonPage 6This would allow us to simulate the real system.Of course, we have the practical problem of realizing a sin(⋅)analog block. Barrie Gilbert of AnalogDevices, a legendary circuit designer, evidently came up with an ICthat performed exactly thisfunction!sin(⋅)glθdt∫ θdt∫