From: retenshun@gmail.com   
      
   On Wednesday, August 30, 2017 at 4:37:57 PM UTC-4, Simon Roberts wrote:   
   > On Wednesday, August 30, 2017 at 3:38:00 PM UTC-4, Simon Roberts wrote:   
   > > best wishes in your first year: quick notes.   
   > >   
   > >   
   > >   
   > > "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 (I found this 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 y 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 electrical engineering they often use j instead of i because i is used   
   as a variable for small signal current.   
   > >   
   > > "A sinusoidal voltage"   
   > >   
   > >  The source can be represented as V_rms(e^(jwt)).   
   > >   
   > > Or usually in polar notation as V_rms(angle)beta where beta is easily set   
   to 0.   
   > >   
   > > And sometimes just plain V == V_rms (being the RMS voltage of a pure sine   
   wave) with a peak voltage V_peak = (2)^(1/2)V;  V_rms =V = V_pea   
   /|squareroot(2)|   with an associated w being w = 2(pi)f and constant.   
   > >   
   > > Where f is the frequency in Hertz (or cycles per second) of the source, V.   
   > >   
   > > "Passive components"   
   > >   
   > > "Impedance"   
   > >   
   > > Z = Resistor Impedance is R where R is the resistance usually in Ohms.   
   note: 1/Ohm(s)  is a "mho", funny eh?   
   > >   
   > > Z = Capacitor Impedance is 1/(jwC) where C is the capacitance usually in   
   Farads.   
   > >   
   > > Z = Inductance Impedance is jwL where L is the inductance usually in   
   Henries.   
   > >   
   > >  I(angle)(phi) = V_rms/Z.  In a loop with any one type of impedance.   
   phi is the phase compared with the phase of V being 0 degrees.   
      
   note V is strictly Real (like a constant). Or V(angle)(0). No phase or a phase   
   of 0.   
   > >   
   > > phi = 90 if pure capacative load.   
   > >   
   > > phi = -90 if pure inductive load.   
   > >   
   > > phi = 0 for a pure resistive load.   
      
      
   (for starters, we can analyse these notions using a graph of x(real axis) vs.   
   y(imaginary axis) and then using a polar notation we can see where a simple j   
   has the angle 90, -j = -90 and 1 has 0 degrees. anyway.)   
   > >   
   > > Z_t  being the equivalent impedance (as in a simple loop) of any passive   
   circuit with one voltage source, again, V.   
   > >   
   > > in series we add. Z_t = Z_1 + Z_2 + .... + Z_n. easiest if Z_t expressed as   
      
   Z_t = x_t + jy_t.   
      
   in parallel Z_t = 1 /  (1/Z_1 + 1/Z_2 + ... + 1/Z_m ).   
      
   that is 1/Z_t = (1/Z_1 + 1/Z_2 + ... + 1/Z_m ).   
      
   we can still use the notation Z_t = x_t + jx_t.   
   and then we build upon these notions as the voltage and current become more   
   dynamic (not a steady state sine wave).   
   > >   
   > > Good luck in your new year.   
   > >   
   > > Simon Roberts (I had forgot these, please, do not hesitate to correct)   
      
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