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650 ⏐⏐⏐ SERIES AND PARALLEL ac CIRCUITS<br />

0°<br />

–30°<br />

–45°<br />

–60°<br />

–90°<br />

θT<br />

Applying the voltage divider rule to determine the voltage across the<br />

capacitor in phasor form yields<br />

VC �<br />

� �<br />

XC E ��90°<br />

����<br />

�R� 2 ��� X� 2 C� /�tan �1 Z<br />

�<br />

CE<br />

ZR � ZC (XC ��90°)(E �0°) XC E ��90°<br />

��� ��<br />

R � j XC R � j XC<br />

XC/R or VC �VC �vC � /�90° � tan �1 XC E<br />

��<br />

(X<br />

2 2 C/R)<br />

�R� ��� X� C�<br />

The magnitude of VC is therefore determined by<br />

X<br />

VC �<br />

CE<br />

��<br />

2 2 (15.13)<br />

�R� ��� X� C�<br />

and the phase angle v C by which V C leads E is given by<br />

vC ��90° � tan �1 � XC<br />

� ��tan<br />

R<br />

�1 R<br />

�� X<br />

(15.14)<br />

To determine the frequency response, X C must be calculated for each<br />

frequency of interest and inserted into Eqs. (15.13) and (15.14).<br />

To begin our analysis, it makes good sense to consider the case of<br />

f � 0 Hz (dc conditions).<br />

C<br />

a c<br />

1 5<br />

10<br />

15 20 f (kHz)<br />

θT<br />

( f )<br />

Circuit capacitive<br />

Circuit resistive<br />

FIG. 15.48<br />

The phase angle of the input impedance versus frequency for the circuit of<br />

Fig. 15.45.

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