Analog computer programming is based essentially 

 on the electrical principle of the feedback loop. For 

 a simple illustration, let us return to the declining 

 exponential curve, as expressed in equation (1). 

 This expression may be further simplified by setting 

 F+il/ = Z, as in international notation and assuming 

 that our measurement of time begins at t, so that 

 /r = and {t — tr)=t. Expression (1) then becomes: 



Remembering that the analog computer requires 

 differential expressions, we differentiate the above 

 to find: 



clN, 

 dl 



-RZe-^' 



As in digital computation, we start with a "block 

 diagram" (ordinarily this step would be omitted in 

 sucli a simple circuit, but it is shown here to illustrate 

 the process). A few symbols are needed (arrows 

 indicate direction of information flow) : 



OPERATION PERFORMED 

 BY COMPUTER 



SYMBOL 



EQUATION 



Integrolion with respect to time 



Inversion (Ctionqe of sign) 

 Multiplication by o constant 



Summation 



"i" 



:|f^ 



y =y^«, dt ^ 



i c = y] 



y = Ax| 



y = X + X +x 



Using the first three of these symbols, we can now 

 construct a block diagram for our differential expres- 

 sion. We start by assuming that a voltage propor- 

 tional to dN ,/dt exists: 



dN, 



dt 



I.e. 



I 



'dt 



Nt + i.e. 



Considering only the rate at which ^V, changes, 

 and disregarding its absolute value, we can omit the 

 constant of integration (this constant will be added 

 in the circuit diagram) : 



dN, 

 "dT 



'dt 



-^ Nt = Re' 



zt 



But we know that 3^=— RZe~^', so we simply 



dt 

 assemble the derivative: 



dN| 



dt 



= -RZe 



Zt 



Nt = Re' 



■Zt 



The next step is to construct a circuit diagram 

 showing the actual computing elements and taking 

 account of "initial conditions" and any sign changes 

 that may occur. For this step we use additional 

 symbols : 



COMPUTING ELEMENT 



Integrating network witti 

 Summing Junction 



Potentiometer 



SYMBOL 



^ 



BLOCK DIAGRAM 

 EQUIVALENT 



Summing amplifier 



Still disregarding i. c, we have: 





dNi 

 dt 



= -RZe 



-zt 



-Nt=-Re' 



■zt 



■ZRe 



-zt 



<!> 



Finally we must supply the "initial condition" 

 voltage, i.e., which is equal to the constant of integra- 

 tion. We noted above, in the definition of the 



symbol 



that y= fo'.Tidt + 



i.e. and i.c.=y],=o. In our equation "y" is replaced 

 byA'^,. Since we set /r = 0, by definition N,],^ = R = 

 i.e. The completed diagram, assuming a computer 



ANALOG COMPUTER MODELS OF FISH POPULATIONS 



35 



