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This expression is conveniently broken up into two parts In fact, right here we have the zero-state response and the zero-input response of the circuit Recall that the zero-state response is due to the input source (ie, the voltage source in this case) and not to any initial conditions (or the initial state ) of the circuit Since the Laplace transform of the voltage source (the input) is 4/s and the rst term on the right-hand side has an expression of this form, we recognize that the zero-state response for this circuit is given by 2 s(s + 1) Now recall that the zero-input response is the response of the circuit due to the initial conditions, which in this case is the initial current i(0) = 1 A This is the second term 1 s+1 This is the response of the circuit due to the initial state without the input voltage source Now we can compute the inverse Laplace transform to nd the current as a function of time We nd that it is i(t) = 2u(t) e t u(t) If you compute the Laplace transform of each term individually, you will nd that the zero-state response of the circuit is 2(1 e t ) and the zero-input response is e t EXAMPLE 13-11 Using Laplace transform methods, nd the current i(t) through the inductor shown in Fig 13-2 Then consider the natural response of the circuit and take R = 3, L = 1, and C = 1/2 Assume that the initial current is zero



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#include <stdlibh> long int labs(long int num);

The labs( ) function returns the absolute value of num Example This function converts the number entered at the keyboard into its absolute value:

long int get_labs() { char num[80]; gets(num); return labs(atol(num)); }

DECIBEL ABOVE 1 W The term dB is used for optical power referenced to 1 microwatt ( W) Thus, output power dB 1- W input

i( ) d + R i(t)

#include <stdlibh> long long int llabs(long long int num);

The llabs( ) function returns the absolute value of num It is similar to labs( ) except that it operates on values of type long long int Related Functions abs( ) and labs( ) ldiv

(1311)

#include <stdlibh> ldiv_t ldiv(long int numerator, long int denominator);





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The ldiv( ) function returns the quotient and remainder of the operation numerator/denominator in an ldiv_t structure The structure type ldiv_t has these two fields:

Many times when we work with fiber optics we will encounter the term power budget without an explanation of its meaning As we have just discussed decibels we will briefly discuss power budget, although we will return to this metric when we cover fiber in the next chapter TABLE 22 Relationship between Watts and Decibel-Milliwatts

long int quot; /* quotient */ long int rem; /* remainder */

Solving this equation for the current might not be pleasant, but the Laplace transform greatly simpli es the situation Now we take the Laplace transform of each piece On the left-hand side, we simply have the Laplace transform of the input voltage: v i (t) Vi (s) On the right-hand side, using Table 13-1, as a guide we nd L di 1 + dt C

Example This program displays the quotient and remainder of 10/3:

#include <stdlibh> #include <stdioh> int main(void) { ldiv_t n; n = ldiv(10L, 3L); printf("Quotient and remainder: %ld %ld\n", nquot, nrem); return 0; }

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Equating this result to Vi (s) and solving for I (s), we nd I (s) = sVi (s) + s L i(0) Vi (s) + L i(0) = 1 Ls 2 + Rs + 1/C Ls + R + Cs (1312)

Related Functions div( ) and lldiv( )

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