EVAL-AD5933EBZ Datasheet, PDF (28/32 Pages) Analog Devices – Evaluation Board for the 1 MSPS 12-Bit Impedance Converter Network Analyzer

EVAL-AD5933EB
In addition, care must be taken with the arctangent formula
when using the real and imaginary values to interpret the phase
at each measurement point. The arctangent function returns the
correct standard phase angle only if the sign of the real and
imaginary values are positive, that is, if the coordinates lie in the
first quadrant.
The standard angle is the angle taken counterclockwise from
the positive real x-axis. If the sign of the real component is
positive and the sign of the imaginary component is negative,
that is, the data lies in the second quadrant, then the arctangent
formula returns a negative angle, and it is necessary to add 180°
to calculate the correct standard angle.
Likewise, when the real and imaginary components are both
negative, that is, when the coordinates lie in the third quadrant,
the arctangent formula returns a positive angle, and it is
necessary to add 180° to the angle in order to determine the
correct standard phase.
Finally, when the real component is positive and the imaginary
component is negative, that is, the data lies in the fourth
quadrant, then the arctangent formula returns a negative angle,
and it is necessary to add 360° to the angle in order to calculate
the correct phase angle.
Therefore, the correct standard phase angle is dependant on the
sign of the real and imaginary components (see Table 8 for a
summary).
Preliminary Technical Data
Table 8. Phase Angle
Real
Imaginary
Positive Positive
Positive Negative
Negative Negative
Positive Negative
Quadrant
First
Second
Third
Fourth
Phase Angle (Degrees)
tan ^−1(I / R)× ¹⁸⁰
π
180
+
⎜⎝⎛
tan
−1
(I
/
R)
×
180
π
⎟⎠⎞
180
+
⎜⎝⎛
tan
−1
(I
/
R)
×
180
π
⎟⎠⎞
360
+
⎜⎝⎛
tan
−1
(I
/
R)
×
180
π
⎟⎠⎞
After the magnitude of the impedance (|Z|) and the impedance
phase angle (ZØ, in radians) are correctly calculated, it is
possible to determine the magnitude of the real (resistive) and
imaginary (reactive) components of the impedance (ZUNKNOWN).
This is accomplished by the vector projection of the impedance
magnitude onto the real and imaginary impedance axes using
the following formulas:
|ZREAL|= |Z| × cos(ZØ)
|ZIMAG|= |Z| × sin(ZØ)
where ZREAL is the real component, and ZIMAG is the imaginary
component.
Rev. PrC | Page 28 of 32