Name: AC Voltage Applied to a Resistor
Uploaded: 2019-07-19
Duration: 7 min 45 s
Description: A purely resistive circuit has very little inductance at its typical frequency. This video explains various aspects of it and also shows the plotting of curr...

Question

For a purely inductive ac circuit, show that the current lags the voltage by a phase angle of

\frac{π}{2}

radians. Hence, show that the power dissipated across the inductor is zero. Draw the phasor diagram.

Accepted Answer

Here, the ac voltage supply is $V = V_{\circ} \sin (ω t)$ that is applied across the inductor.

$V_{\circ}$ is peak voltage of the ac supply and $ω$ is the frequency of the ac voltage.

The current through the inductor varies and opposing induced emf is generated in the inductor.

So, the induced emf: $ε = - L \frac{d i}{d t}$ ............. $(1)$

By applying Kirchoff's law in the above circuit, we get: $V + ε = 0 \Rightarrow V_{\circ} \sin (ω t) - L \frac{d i}{d t} = 0$

$\Rightarrow d i = \frac{V_{\circ}}{L} \sin (ω t) d t$ ............. $(2)$

By integrating both side, we get the instantaneous value of current:

$\int d i = \int \frac{V_{\circ}}{L} \sin (ω t) d t \Rightarrow i = \frac{V_{\circ}}{L} \int \sin (ω t) d t = \frac{V_{\circ}}{ω L} (- \cos (ω t))$ ............. $(3)$

So, the peak current is $i_{\circ} = \frac{V_{\circ}}{ω L}$

Now, using equation $(3)$ we get: $i = \frac{V_{\circ}}{ω L} \sin (ω t - \frac{π}{2})$ ............. $(4)$

Hence, in inductor the current lags behind the voltage by a phase angle of $\frac{π}{2}$ .

The average power dissipated across the inductor will be: $P_{a v g} = \frac{V_{\circ} i_{\circ}}{2} \cos φ$ ............... $(5)$

Here, $φ$ is the phase difference between voltage and current at an instant that is $\frac{π}{2}$ , by substituting this in $(5)$ we get:

$P_{a v g} = \frac{V_{\circ} i_{\circ}}{2} \cos \frac{π}{2} = \frac{V_{\circ} i_{\circ}}{2} (0) = 0$

Hence, the power dissipated across the inductor is zero.

The phasor diagram for the given circuit:

For a purely inductive ac circuit, show that the current lags the voltage by a phase angle of $\frac{π}{2}$ radians. Hence, show that the power dissipated across the inductor is zero. Draw the phasor diagram.

Important Questions on Alternating Current

Learn and Prepare for any exam you want !

For a purely inductive ac circuit, show that the current lags the voltage by a phase angle of π2 radians. Hence, show that the power dissipated across the inductor is zero. Draw the phasor diagram.

Important Questions on Alternating Current

Learn and Prepare for any exam you want !

For a purely inductive ac circuit, show that the current lags the voltage by a phase angle of $\frac{π}{2}$ radians. Hence, show that the power dissipated across the inductor is zero. Draw the phasor diagram.