As the polar representation of a point is based
around the triangular form, we can use simple
geometry of the triangle and especially trigonometry and Pythagoras’s
Theorem on triangles
to find both the magnitude and the angle of the complex number. As we remember from
school, trigonometry deals with the relationship between the sides and
the angles of triangles
so we can describe the relationships between the sides as:
A
2
=X
2
+Y
2
A=√X
2
+Y
2
Also X=A cosө Y=A sinө
Using trigonometry again, the angle θ of A is given as follows.
Ө=tan
-1
y/x
Then in Polar form the length of A and its angle represents the complex number instead of a
point. Also in polar form, the conjugate of the complex number has
the same magnitude or
modulus it is the sign of the angle that changes, so for example the conjugate of 6
∠
30
o
would be 6
∠
– 30
o
.
Steady state Analysis of Series RLC circuits:
Thus far we have seen that the three basic passive components: resistance (R), inductance
(L), and capacitance (C) have very different phase relationships to
each other when connected
to a sinusoidal AC supply.
In a pure ohmic resistor the voltage waveforms are “in-phase” with the current. In a pure
inductance the voltage waveform “leads” the current by 90
o
, giving us the expression of: ELI.
In a pure capacitance the voltage waveform “lags” the current by 90
o
, giving us the
expression of: ICE.
This phase difference,Ф depends upon the reactive value of the components being used and
hopefully by now we know that reactance, ( X ) is zero if the
circuit element is resistive,
positive if the circuit element is inductive and negative if it is capacitive thus giving their
resulting impedances as:
