TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 315 



encoder. The integrator is required to generate a 15-volt ramp which is 

 linear and has a constant slope to within one part in 8,000. This ramp is 

 to have a slope of 5 millivolts per microsecond for an interval of 3,000 

 microseconds. 



The first step in the design is to determine the bandwidth over which 

 the negative feedback must be maintained in order to realize the desired 

 output voltage linearity. The relationship between the output and input 

 voltage of the integrator can be obtained from expression (11) by sub- 

 stituting (1/pc) for Zk and R for Zj (refer to Fig. 1). 



£l-C'outJ — 



pRC 



A/3 + Zr^'pC 



1 - AjS + 



-nN_ 

 R 



(21) 



where ce[£'ouT] and JSiii'iN] are the Laplace transforms of the output and 

 input voltages, respectively. In order to generate the voltage ramp, a 

 step voltage of amplitude E is applied to the input of the integrator. The 

 term Zy^ jR is negligible compared to unity at all frequencies. Therefore, 



£L-£'outJ — 



E \ A& 



+ 



EZ 



IN 



1 



'^-RC Ll - A&\ pR \\ - A^_ 

 It will be assumed that A/3 is given by the expression 



-K 



(22) 



A^ = 



V 



)0 + ^T 



(i + -M(i + ^ 



(23) 



Expression (23) implies that A/3 falls off at a rate of 6 db per octave at 

 low frequencies and 12 db per octave at high frequencies. The output 

 \ voltage of the integrator, as a function of time, is readily evaluated by 

 substituting (23) into (22) and taking the inverse Laplace transform of 

 the results. A good approximation for the output voltage is 



^OUT — 



E 



RC 



+ 



2K 



^-[(2w2+«l)(/2] ^;„ -x/W 



sm 



Vk> 



OJo 



■iC02M 



ER 



(24)^ 



IN 



R 



[1 _ e-(-i'W _!_ g-[(2<-2+.i)t/2i ^Qg ^Tkc.,!] 



The linear voltage ramp is expressed by the term — (Et/RC) . The 

 additional terms introduce nonlinearities. The voltage ramp has a slope 

 of 5 millivolts per microsecond for E = —21 volts, R = 42,000 ohms, 



* In evaluating jE'out it was assumed that Zm' was equal to a fixed resistance 

 Rin' , the low frequency input resistance to the first common emitter stage. A 

 complete analysis indicates that this assumption makes the design conservative. 



