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10 14 10 13 X UNIT, XU HEF High Energy Freq 10 12 PICOMETER 10 11 10 10 ANGSTROM, a 10 9 NANO10 8

Related Functions asin( ), acos( ), atan2( ), atan( ), tan( ), sin( ), sinh( ), cos( ), and tanh( ) cosh

#include <mathh> float coshf(float arg); double cosh(double arg); long double coshl(long double arg);

coshf( ) and coshl( ) were added by C99 The cosh( ) family of functions returns the hyperbolic cosine of arg Example The following program prints the hyperbolic cosines of the values through 1, in increments of 1 one tenth:



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where n is the undamped natural frequency and is the damping ratio The dynamic behavior of the system is then described in terms of these two parameters and n To determine the behavior of a system, we look at the damping ratio There are three possibilities: If 0 < < 1, the roots are complex conjugates and the system is underdamped and oscillatory If = 1, the roots of the system are equal and the response is critically damped If > 1, the roots are negative, real, and unequal The system response is overdamped With this in mind, consider the RLC circuit second-order equation (628) that is speci c for a resistor, inductor, and capacitor in series: s2 +

#include <mathh> #include <stdioh> int main(void) { double val = -10; do {

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Page 394 printf("Hyperbolic cosine of %f is %f\n", val, cosh(val)); val += 01; } while(val<=10); return 0; }

R R LC And 2 n = = L 2L This provides us with the damped case relationships for an RLC second-order equation:

#include <mathh> float erff(float arg); double erf(double arg); long double erfl(long double arg);

erf( ), erff( ), and erfl( ) were added by C99 The erf( ) family of functions returns the error function of arg Related Function erfc( ) erfc

#include <mathh> float erfcf(float arg); double erfc(double arg); long double erfcl(long double arg);

LC If 0 < R 2L < 1, then the roots are complex conjugates and the system is underdamped and oscillatory

1014 1013

erfc( ), erfcf( ), and erfcl( ) were added by C99 The erfc( ) family of functions returns the complementary error function of arg Related Function erf( )

#include <mathh> float expf(float arg); double exp(double arg); long double expl(long double arg);

= 1, then the roots of the system are equal and the response is critically damped > 1, then the roots are negative, real, and unequal The system response is overdamped

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expf( ) and expl( ) were added by C99 The exp( ) family of functions returns the natural logarithm e raised to the arg power Example This fragment displays the value of e (rounded to 2718282):

printf("Value of e to the first: %f", exp(10));

108 107

As an example, we consider a series RLC circuit shown in Fig 6-15 If we apply KVL around the circuit, we obtain Ri + L di + vc = V dt

#include <mathh> float exp2f(float arg); double exp2(double arg); long double exp2l(long double arg);

exp2( ), exp2f( ), and exp2l( ) were added by C99 The exp2( ) family of functions returns 2 raised to the arg power Related Functions exp( ) and log( ) expm1

#include <mathh> float expm1f(float arg);

is satis ed Therefore, since the same current ows through the inductor, the voltage across the inductor is L di d 2vc = LC 2 dt dt

Page 396 double expm1(double arg); long double expm1l(long double arg);

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expm1( ), expm1f( ), and expm1l( ) were added by C99 The expm1( ) family of functions returns the natural logarithm e raised to the arg power, minus 1 That is, it returns earg 1 Related Functions exp( ) and log( ) fabs

Also, the same current ows through the resistor, meaning that the voltage across the resistor is Ri = RC dv c dt

#include <mathh> float fabsf(float num); double fabs(double num); long double fabsl(long double num);

fabsf( ) and fabsl( ) were added by C99 The fabs( ) family of functions returns the absolute value of num Example This program prints 10 10 on the screen:

#include <mathh> #include <stdioh> int main(void) { printf(''%11f %11f", fabs(10), fabs(-10)); return 0; }

Hence, in a series an RLC circuit with voltage source V is described by the differential equation LC d 2vc dv c + vc = V + RC dt 2 dt (626)

102 101 15

Related Function abs( )

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