From: on Aug 30, 3:38 pm

All,

I'm an Advanced studying for my Extra, and so far I am getting 66% on
the practice exams without even studying after about 3 years of
homebrewing. I need a few extra points to make it over the top, and
since I am a builder (of sorts) I would like to do it using the
electrical principles part of the exam.

However, the questions pool provides the answers but not how to get
them. I'd rather be able to understand how to arrive at it without a
calculator.

You need some DEFINITIONS made clear first. See following...

Problem class 1: impedance and phase angle of RLC parallel circuit
where component values and frequency are known.

Problem class 2: impedance and phase angle of RLC series circuit where
component values and frequency are known.

Could I calculate these graphically using tip-to-tail summation of
impedance vectors?

...if you have some polar-coordinate graph paper, yes.

From what I understand, |Z| = 2 pi F L, |Z| = 2 pi / (F C), |Z| = R

WRONG. Impedance Z = Resistance R + j Reactance X for a series-
resonant circuit. Further, X_L = 2 pi F L and X_C = -1 / (2pi F C).

"|Z|" is MAGNITUDE of impedance; you can't do "phase calculations"
using just magnitudes of either Z or Y.

Note: Admittance Y is composed of conductance G = 1 / R (real
part) and susceptances B_L = -1 / (2 pi F L) and B_C = 2 pi F C
(imaginary part).


But how can I get the phase angle or the conjugate pair so that I can
do the vector addition?

A "conjugate match" (or "pair") results when the magnitude of
inductive reactance is exactly equal to the magnitude of capacitive
reactance. Their relative phase angles are 180 degrees and
opposed; they "cancel" each other. In a series-resonant circuit
that leaves you with ONLY the RESISTIVE part of the complex
number expression for impedance.

|Z| = SQRT ( R^2 + X^2 ), Impedance phase PHA = Tan (X / R)

You must assign a polarity to X in order to maintain relative
phase angles. Capacitive reactance is assigned a negative value
while inductive reactance is assigned a positive value.

For admittance Y (such as with a parallel-resonant circuit), the
magnitude (|Y|) is the same square-root of the sums of the
squares (and thus always positive), but the inductive susceptance
is negative and the capacitive susceptance is positive in value.

Doing "vector plotting" is generally NOT a good short-cut way
to get acquainted with either Y or Z...UNLESS you ALWAYS keep
in mind the relative phase of inductance and the relative
phase of capacitance...and forget the small frequency differences
where reactance/susceptance of L is bigger/smaller than reactance/
susceptance of C. AT RESONANCE the angles are EQUAL in magnitude
but opposed in phase; they equate to zero.

The rules of arithmetic of the "rectangular form" of complex
quantities is well-known and mentioned in all sorts of texts,
including the mathematics handbooks. Going through numbers with
rectangular form is no more harder/easier than trying to plot
vectors with polar form representation.

Assignment to the student: Learn the rectangular-form arithmentic
rules for complex quantities. ANY text that has anything to say
about complex quantities will have those rules.

A hint to keep from re-inventing the wheel. The HP 32S and 33
scientific pocket calculators have both forms' arithmentic rules
preprogrammed...AND they do both the real and imaginary part
calculations and answer displays like right now. The HP 33
costs about $50 new.