EVAL-AD5934EB
Preliminary Technical Data
Rev. PrC | Page 26 of 32
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.
and the repeated phase measurement with a 10 pF capacitive
impedance is shown in Figure 34.
The phase difference (that is, ZØ) between the phase response
of a capacitor and the system phase response using a resistor is
the impedance phase (ZØ) of the capacitor and is shown in
Figure 35.
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.
200
180
160
140
120
100
0
80
60
40
20
P
HAS
E
(
D
eg
rees)
0
15k
30k
45k
60k
75k
90k
105k
120k
FREQUENCY (Hz)
05
44
9-
03
4
220k
Ω
RESISTOR
10pF CAPACITOR
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 7 for a
summary).
Figure 34. System Phase Response vs. Capacitive Phase
Table 7. Phase Angle
–100
–90
–80
–70
–60
–50
0
–40
–30
–20
–10
P
H
AS
E
(
D
eg
rees
)
0
15k
30k
45k
60k
75k
90k
105k
120k
FREQUENCY (Hz)
05
44
9-
03
5
Real
Imaginary
Quadrant
Phase Angle (Degrees)
Positive Positive First
π
×
−
180
)
/
(
tan
1
R
I
Positive Negative Second
⎟
⎠
⎞
⎜
⎝
⎛
π
×
+
−
180
)
/
(
tan
180
1
R
I
Negative Negative Third
⎟
⎠
⎞
⎜
⎝
⎛
π
×
+
−
180
)
/
(
tan
180
1
R
I
Positive Negative Fourth
⎟
⎠
⎞
⎜
⎝
⎛
π
×
+
−
180
)
/
(
tan
360
1
R
I
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 (Z
UNKNOWN
).
This is accomplished by the vector projection of the impedance
magnitude onto the real and imaginary impedance axes using
the following formulas:
Figure 35. Phase Response of a Capacitor
It is important to note that the formula used to calculate the phase
and to plot Figure 34 is based on the arctangent function, which
returns a phase angle in radians. Therefore, it is necessary to
convert the calculated phase angle from radians to degrees.
|
Z
REAL
|= |
Z
| × cos(
Z
Ø)
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.
|
Z
IMAG
|= |
Z
| × sin(
Z
Ø)
where
Z
REAL
is the real component, and
Z
IMAG
is the imaginary
component.