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AD5934 Datasheet(PDF) 18 Page  Analog Devices 

AD5934 Datasheet(HTML) 18 Page  Analog Devices 
18 / 31 page AD5934 Data Sheet Rev. E  Page 18 of 31 Note that it is possible to calculate the gain factor and to calibrate the system phase using the same real and imaginary component values when a resistor is connected between the VOUT and VIN pins of the AD5934, for example, measuring the impedance phase (ZØ) of a capacitor. The excitation signal current leads the excitation signal voltage across a capacitor by −90 degrees. Therefore, an approximate −90 degrees phase difference between the system phase responses measured with a resistor and the system phase responses measured with a capacitive impedance exists. As previously outlined, if the user wants to determine the phase angle of the capacitive impedance (ZØ), the user first must determine the system phase response ( system ∇ ) and subtract this from the phase calculated with the capacitor connected between VOUT and VIN (Φunknown). Figure 22 shows the AD5934 system phase response calculated using a 220 kΩ calibration resistor (RFB = 220 kΩ, PGA = ×1) and the repeated phase measurement with a 10 pF capacitive impedance. One important point to note about the phase formula used to plot Figure 22 is that it uses the arctangent function that returns a phase angle in radians and, therefore, it is necessary to convert from radians to degrees. 0 20 40 60 80 100 120 140 160 180 200 60k 45k 15k 30k 0 75k 90k 105k 120k FREQUENCY (Hz) 220 kΩ RESISTOR 10pF CAPACITOR Figure 22. System Phase Response vs. Capacitive Phase 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 of the capacitor (ZØ) and is shown in Figure 23. In addition, when using the real and imaginary values to interpret the phase at each measurement point, care should be taken when using the arctangent formula. The arctangent function only returns the correct standard phase angle when the sign of the real and imaginary values are positive, that is, when the coordinates lie in the first quadrant. The standard angle is taken counterclockwise from the positive real xaxis. 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, the arctangent formula returns a negative angle, and it is necessary to add an additional 180° to calculate the correct standard angle. Likewise, when the real and imaginary components are both negative, that is, when data lies in the third quadrant, the arctangent formula returns a positive angle, and it is necessary to add an additional 180° to calculate the correct standard phase. When the real component is positive and the imaginary component is negative, that is, the data lies in the fourth quadrant, the arctangent formula returns a negative angle, and it is necessary to add an additional 360° to calculate the correct standard phase. 60k 45k 15k 30k 0 75k 90k 105k 120k FREQUENCY (Hz) –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 Figure 23. Phase Response of a Capacitor Therefore, the correct standard phase angle is dependent upon the sign of the real and imaginary components, which is summarized in Table 6. Table 6. Phase Angle Real Imaginary Quadrant Phase Angle 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 Negative Positive Fourth π ° × + ° − 180 ) / ( tan 360 1 R I Once 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) by the vector projection of the impedance magnitude onto the real and imaginary impedance axis using the following formulas: The real component is given by ZREAL = Z × cos(ZØ) The imaginary component is given by ZIMAG = Z × sin(ZØ) 
