From: retenshun@gmail.com   
      
   Crib notes or Cheat sheets.   
      
      
      
   "Trigonometry"   
      
   e^(iwt) = cos(wt) + isin(wt)   
      
   i^2 = -1 and +i = squareroot(-1).   
      
   -> 2(i)sin(wt) =    [e^(iwt) - e^(-iwt)].   
      
   -> 2cos(wt) = [e^(iwt) + e^(-iwt)].   
      
   -> 2sinh(wt) = [e^(-i(iwt)) - e^(i(iwt))] = [e^(wt) - e^(-wt)]= 2sin(-iwt)) =   
   -2sin(iwt).   
      
   -> 2cosh(wt) = (e^(-i(iwt)) + e^(i(iwt))) = (e^(wt) + e^(-wt)) = 2cos(iwt).   
      
   Further identities and such can be easier if   
      
   e^(iwt) = cos(wt) + isin(wt) is used instead of your noodle.   
      
   aside (that I found interesting):  e^(irwt) = cos(rwt) + isin(rwt) = (cos(wt)   
   + isin(wt))^r.   
      
      
   "polar notation".   
      
   "(angle)" being that "<" or "L" or  that character I'd rather not render here.   
      
   z = x + iy where x and y real. ( that is x the Real part and iy the Imaginary   
   part)   
      
   R (angle) theta = |square root(x^2 + y^2)| (angle) (tan^(-1)(y/x)).   
      
   If x = cos(wt) and y = sin(wt) that is z = e^(iwt) = x + iy = cos(wt) +   
   isin(wt)   
      
   Then   
      
   R (angle) theta = 1 (angle) (wt).   
      
   Also absolute value of z, either complex or real,   
      
   denoted as |z|>=0 and Real, is   
      
   |z| = | [(x + iy)(x -iy)]^(1/2) |  = |square root(x^2 + y^2)| = |square   
   root(z(z*))|.   
      
   z* = x - iy is the complex conjugate of z = x + iy always.   
      
   in elctrical engineering they often use j instead of i because i is used as a   
   variable for small signal current.   
      
   "A sinusoidal volatge"   
      
    souce can be represented, at first as, V(e^(iwt)).   
      
   Or better yet in polar notation as V(angle) wt.   
      
   And sometimes just plain V == V_rms (being the RMS voltage of a pure sine wave)   
      
   with a peak voltage V_0 = (2)^(1/2)V;  V_rms =V = V_0/ (|square root(2)|) )    
   with an associated w being w = 2(pi)f   
      
   where f is the frequency in Hertz (or cycles per second) of the source, V.   
      
   "Passive components"   
      
   "Impedence"   
      
   Z(w) = Resistor Impedence is R where R is the resistance ususally in Ohms.   
   note: 1/Ohm(s)  is a "mho", funny, eh?   
      
   Z(w) = Capacitor Impedence is 1/(jwC) where C is the capacitance usually in   
   Farads.   
      
   Z(w) = Inductance Impedence is jwL where L is the inductance usually in   
   Henries.   
      
    I (wi + phi) = V/Z(w).  In a loop with any one type of impedence.   
      
   phi = 90 if pure capacative load.   
      
   phi = -90 if pure inductive load.   
      
   phi = 0 for a pure resistive load.   
      
   (for straters, we can analyse these notions using a graph of x(real axis) vs.   
   y(imaginary) and then using a polar notaion we can see where a simple j has   
   the agle 90, -j = -90 and 1 has 0 degrees. anyway.)   
      
   Z_t  being the equavalent impedance (as in a simple loop) of any passive   
   circuit with one  voltage soure, again, V.   
      
   in series we add. Z_t = Z_1 + Z_2 + .... + Z_n.   
      
   in parallel we do this Z_t = 1 /  (1/Z_1 + 1/Z_2 + ... + 1/Z_m ).   
      
   and then we build upon these notions.   
      
   Good luck in your new year.   
      
   Simon Roberts (I had forgot these, please do not hesitate to correct errors.)   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)