Crib notes or Cheat sheets.

"Trigonometry"

e^(iwt) = cos(wt) + isin(wt)

i^2 = -1 and +i = squareroot(-1).

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.)

"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.)