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1693 RLC Digibridge
Polar Form: Magnitude and Phase. A complex
number may be expressed in polar form as well as in
the Cartesian form used so far. For impedance and
admittance the relationships are:
where |Z| and |Y| are the magnitudes and
are the phase angle sin radians (see Figure 3-28).
The magnitude of a complex number is the square
root of the sum of the squares of the two parts so that
Note that |Z| = 1/|Y|
The phase angle of an impedance or an admittance
is the angle whose tangent is the ratio of the imagi-
nary part to the real part so that
and
36
The size of the phase angle of an admittance is
the same as that of the corresponding impedance,
but if we use the convention that a positive angle
is one in the counterclockwise direction then the
angle of an admittance is the negative of that of the
corresponding impedance, ,as one can see from the
equation
Euler's formula,
is useful when dealing with imaginary exponents
and can be used to show that
These phase relationships can be understood better
using the diagrams of Figure 3-28 which show both
inductive and capacitive impedance phasors (or planar
vectors) on both the Z and Y planes. Here the angle is
measured from the positive real axis (+r or +G) and
the angle δ (delta) is defined as the complement of Θ
and φ so one can see that:
So one can see that:
Operation
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