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Chapter 12

Q. 12.16

The voltage across a capacitor having capacitance C=100 pF  is given by \nu(t)=V_0 \cos (2 \pi f t)  with \nu_0=5  V  and f=500 kHz . Find the current  i(t)  through the capacitor and draw a phasor diagram depicting the current and voltage, with the voltage as the phase reference.

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Verified Solution

The phasor \tilde{V} for the voltage and the admittance Y  of the capacitor are

\begin{aligned} &\tilde{V}=V_0 \angle 0=5 \angle 0 V, \\ &Y=j \omega C=j 314  \mu S =314  \mu S \angle(\pi / 2). \end{aligned}

From (12.24), the phasor for the current is

\begin{aligned} \tilde{I} &=Y \tilde{V}=(5  V \angle 0)[314  \mu S \angle(\pi / 2)] \\ &=1.57  mA \angle(\pi / 2). \end{aligned}

It follows that

i(t)=1.57  \cos (2 \pi  f  t+\pi / 2)  mA .

Figure  12.10  shows a phasor diagram for the current and voltage, where the voltage is taken as the reference. The current leads the voltage by \pi / 2 . Equivalently, the voltage lags the current by \pi / 2.

12-10