Consider the following digital (switching) circuit. If a signal travels in the i

Question

Consider the following digital (switching) circuit. If a signal travels in the interconnect a. line with velocity u = 0.5c, where c is the speed of light, and the interconnect length is 100 [mum], find the maximum switching (clock) frequency that lumped circuit theory for the interconnect line is applicable. Formally, one dimensional wave phenomena are modeled with partial differential equations cast in the form of initial boundary value problems (IBVPs), that consist of a partial differential equation (PDE), sources/exceptions, and constraints of the unknown field/function at various boundary locations 1.e. boundary conditions (BCs), and time instances i.e. initial conditions (ICs). The mathematical solution of such IBVPs gives a unique response function i.e wave. For this problem, solve the following IBVP that models a vibrating string anchored at both ends Find f(x, t), that represent the location of the string, such that: PDE: partial differential^2 f(z, t)/partial differential x^2 - partial differential^2 f(z, t)/partial differential t^2 = 0, in 0 lessthanorequalto 1, and t > 0, BC: f(0, t) = 0 = f(1, t), for t > 0, IC: f(z, 0) = sin (pi x), partial differential f(z, t)/partial differential t = 0, in 0

Explanation / Answer

a. In this question, in the diagram, there are 2 CMOS circuits shown and output of first CMOS is serving as the input or GATE voltage of the 2nd CMOS.

Now, time taken for the output of CMOS1 to reach the gate terminal of CMOS 2 is t=100x10-6/(0.5X3X108)=6.67X10-13 sec and hence the corresponding frequency will be f=1/t=0.15X1013 Hz

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