# Cant understand from where the numbers come from

1. Jun 26, 2009

### electron2

suppose that the nonlinear resistor R has a characteristics specified by the equation:
$$v=20i+i^2+0.5i^3$$
express v as a sum of sinusoids for
$$i(t)=cos \omega _1t +2cos \omega _2t \\$$
the solution in the book is
http://i43.tinypic.com/eve807.gif

i get a differnt expression why??
$$v=20cos \omega _1t +40cos \omega _2t + cos^2 \omega _1t +4cos (\omega _1t) cos (\omega _2t)+4cos^2 \omega _2t+0.5cos^3 \omega _1t+2cos^2 \omega _1tcos \omega _2t+0.5cos^3 \omega _1t+2cos \omega _1tcos^2 \omega _2t+4cos^3 \omega _2t$$

2. Jun 26, 2009

### HallsofIvy

Staff Emeritus
You haven't completely reduced. For example, you have powers and products of cosines. Those can be reduced to first power by trig identitities: $cos^2(\omega_1t)= (1/2)(1+ cos(2\omega_1t))$ and $cos(\omega_1t)cos(\omega_2t)= (1/2)(cos((\omega_1+\omega_2)t)+ cos((\omega_1- \omega_2)t))$.
$cos^3(\omega_1t)$ can be written as $cos(\omega_1t)cos^2(\omega_1t)= cos(\omega_1t)(1/2)(1+ cos(2\omega_1t))$ which can be reduced, and so on.

3. Jun 26, 2009

### electron2

still i cant see i get numbers here
like they did
as far is i know only sin^2 a +cos^2 a =1

and my expression and not your has anything near that