# Best wishes and Good luck in your first year.

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.)
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On Wednesday, August 30, 2017 at 4:37:57 PM UTC-4, Simon Roberts wrote:

e^(irwt) = cos(rwt) + isin(rwt) = (cos(wt) + isin(wt))^r.

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.

(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 = 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).

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to be honest my refresher but if it help i'm happy.
"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.
an aside I found interesting: e^(irwt) = cos(rwt) + isin(rwt) = (cos(wt) + isin(wt))^r. "polar notation". "(angle)" being that similar to "<" or "L" is a character I'd rather not render here. let s = x + iy where x and y real. R (angle) theta = |square root(x^2 + y^2)| (angle) (tan^(-1)(y/x)). If x = cos(wt) and y = sin(wt) that if s = e^(iwt) = x + iy = cos(wt) + isin(wt) then R (angle) theta = 1 (angle) (wt).
Also, the absolute value of s is denoted as |s|. |s| >=0 and is real.
|s| = | [(x + iy)(x -iy)]^(1/2) | = |sqt(x^2 + y^2)| = |sqr(s(s*))|. s* = x - iy is always the 'complex conjugate' of s = x + iy. In electrical engineering they often use 'j' for sqr(-1) instead of 'i' as 'i' is used as a variable for small signal current.
"A sinusoidal voltage" A sinusoidal voltage can be represented as V_p(e^(jwt)) where V_p is the real part and is actually the peak voltage of the waveform.
"Passive circuits with a steady state sinusoidal volatge"
Ususally Root Mean Square of a Voltage, V_rms = V is used.
Under these conditions V_rms(angle)wt is simplified to V == V_rms = V_p / sqr(2). please note w = 2(pi)f where f is the frequaency in units of cycles per second or just 1/seconds called Hertz
and w is "radians per second".
"Passive components" "Impedance, Z" Z = R is the the impedence of a resitor, R, usually in units of Ohms.
note: 1/Ohm(s) is a "mho", funny eh? Z = 1/(jwC) is the impedence of a capacitor, C, usually in units of Farads. Z = jwL is the impedance of an inductor, L, usually in units of Henries. I(angle)(phi) = V/Z. Where Z is in a simple a loop with V.
phi is the relative phase compared with V being 0 degrees.
phi = 90 degrees = pi/2 if the load, Z, is purely capacative. phi = -90 = -pi/2 if the load, Z , is purely inductive. phi = 0 if the load, Z, is purely resistive.
We can analyse these notions using a graph with an x-axis and a y-axis. For instance jwL falls of the y-axis (the imaginary axis) and therefore forms a 90 degree angle with the x- axis (the real axis) as this is the standard convention.
"total impedance, Z_t"
Z_t = Z_1 + Z_2 + .... + Z_n
if all Z_i on the right hand side of this equation are in series in the circuit.
Z_t = 1 / (1/Z_1 + 1/Z_2 + ... + 1/Z_m ) or 1/Z_t = (1/Z_1 + 1/Z_2 + ... + 1/Z_m )
if all Z_i on the right hand side of the equation(s) are in parallel in the circcuit.
And any passive circuit can be usually be reduced ultimatly to a load of impedance, Z, in a simple loop both parallel and in series with the voltage V, being a steady stae sinusoid.
Simon Roberts (I had forgot these, it is not all be clearly written, I know)
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On 01/09/2017 23:41, Simon Roberts wrote:>

If you could but be patient while I work out what is 2 ohms, then I'll tell you in half a "mho".
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